Fermi liquids and the renormalization group

Transcription

1 Fermi liquids and the renormalization group Carsten Honerkamp Institute for Theoretical Solid State Physics, RWTH Aachen, D Aachen, Germany JARA-FIT, Fundamentals of Future Information Technology Les Houches, April 2011 April 19, 2011

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3 Contents 1 Fermi liquids Fermi liquid: basics Thermodynamic properties in Fermi liquid theory Zero sound Quasiparticle scattering rate Low-energy coupling space Functional integral formalism for fermions Grassmann variables and coherent states for fermions Functional integral for the partition function Correlation functions and generating functionals Wick theorem Perturbation theory and quasiparticle properties General perturbation expansion Self-energy corrections up to second order Quasiparticle properties Interaction vertex One-particle irreducible vertex functions Effective interaction Perturbation series for the vertex General expressions for one-loop bubbles Small momentum and frequency transfer Summation of Cooper channel Summation of crossed particle-hole channel Competing instabilities and the need for full one-loop summations Wilson RG for fermions Reminder: Wilson RG for scalar fields Momentum-shell flow equations for the φ 4 -theory Flow equations RG flow in the φ 4 -model Fermionic version One-loop flow Simple expressions for one-loop bubbles Flow in the Cooper channel

4 4 CONTENTS g-ology model in one dimension /N-picture Connection to Fermi liquid theory Shortcomings of the simple flow scheme Field-theoretic renormalization group

5 Chapter 1 Fermi liquids 1.1 Fermi liquid: basics At low temperatures the charge transport and thermodynamical properties of conducting solids usually obeys laws that are strongly reminiscent of noninteracting fermions and that would follow from a Sommerfeld expansion of the Fermi distribution function at temperatures much lower than the Fermi energy. For example, the electronic contribution of the specific heat of a metal disappears linearly, C γt, and the uniform magnetic susceptibility χ and the compressibility κ roughly temperature independent (see Fig. 1.1 for experimental curves for specific heat and susceptibility for Sr 2 RuO 4 which is a wellcharaterized Fermi liquid or standard metal above its supercondcuting transition at T c 1K). The exact values of these observables (e.g. the prefactor γ are however different from the predictions for non-interacting electrons, even if band structure effects would taken into account (in fact one of the nicest example systems for interacting fermions in liquid Helium 3 where full translational invariance holds). This of course immediately leads to the suspicion that interaction effects are weak in some form. The Landau Fermi liquid concept that has been developed since the 1950s elevates this idea to a principle by proclaiming that the interacting state and the non-interacting state plus their respective excitations are adiabatically connected. This picture can be formulated by a few basic assumptions: The low-energy excitations of the interacting system can be described by quasiparticles with the same quantum numbers as non-interacting electrons, i.e. spin-1/2, momentum p and excitation energy ɛ p. The total (free) energy of the system can be expressed as a function of the occupation numbers n p,s of these quasiparticles. The lifetime τ p of these excitations becomes infinitely long as the excitations energy goes to zero. In fact, the scattering rate 1/τ p should vanish fast than ɛ p. Then we can think in terms of long-lived quasiparticles. The low-energy physics can be consistently explained using an energy 5

6 6 CHAPTER 1. FERMI LIQUIDS Figure 1.1: Low-temperature specfic heat and magnetic susceptibility of the layered metal Sr 2 RuO 4. Taken from Mackenzie and Maeno, Rev. Mod. Phys. 75 (2003). functional E[{δn p }] = p,s ɛ( p) δn p,s + 1 2V f p, p ss δn p,sδn p,s (1.1) p,s, p,s where the δn p,s measure the deviations of the occupation numbers from the ground state, i.e. the quasiparticle excitations, and the Landau function f p, p ss contains the interaction effects between the quasiparticles. At T = 0 there can be particle-like excitations with δn p,s > 0 and p > p F and hole-like exciations with δn p,s < 0 and p < p F. ɛ( p) is the quasiparticle dispersion. As the deviations δn p,s are only non-zero near the Fermi surface for low-energy excitations, the sums over the momenta are concentrated near the Fermi surface. It is important to note that interactions of the quasiparticle with the groundstate are considered as already included in the dispersion ɛ( p). This can be understood by starting with an energy functional of the full distribution function n p,s of the form E[{n p }] = p,s ɛ 0 ( p) n p,s + 1 2V p,s, p,s f ss p, p n p,sn p,s (1.2) which can be motivated microscopically. Here ɛ 0 ( p) is a bare dispersion that besides lattice effects only contains effects of higher-energetic modes, but no interaction effects of the modes near the Fermi level which are captured by the momentum summations in these formulae. Then we split up n p,s into a equilibrium part and deviations n p,s = n p,s + δn p,s, (1.3)

7 1.1. FERMI LIQUID: BASICS 7 Figure 1.2: Interacting distribution function n p,s. From Lecture Notes by Eduardo Fradkin (Urbana-Champaign). which leads to the energy functional for the deviations of the form 1.1, but with ɛ( p) = ɛ 0 ( p) + ss f p, p n p,s. (1.4) p, p s,s The second term can be understood as a selfenergy correction due to the interaction with the other particles in the ground state. The interaction f p, p ss in Eq. 1.1 only acts between two excitations. The energy of an additional quasiparticle is then ɛ( p) = ɛ( p) + ss f p, p δn p,s (1.5) p, p s,s The groundstate distribution function of he interacting system is in general different form the Fermi function with a step from 1 down to zero at T = 0. Here, suuming a generic Hamiltonian, one can see that interaction processes with small momentum transfer kick electrons out of the Fermi sea, forming particle-hole pairs. This will lead to an alteration of the step in the distribution function, as there is a finite probability for occupying states with p > p F. However, in order to comply with the idea of adiabatic continuity and also with the free-particle-like behavior of the quasiparticles, the step cannot be zero, it can only be reduced to a value smaller than one. Therefore one ends up with a distribution function as depicted in Fig This form can be understood using many-body perturbation theory, but originally Landau argued for it based on this phenomenological Fermi liquid theory. The Landau function can further be parameterized in terms of expansion parameters. Assuming spin rotational invariance of the interactions, the spin structure can be captured by f αβµν p, p = f s p, p δ αβδ µν + σ αβ σ µνf a p, p, (1.6)

8 8 CHAPTER 1. FERMI LIQUIDS or restricting the consideration to spin polarization along the z-axis by f s,s p, p = f p, p s + f p, p a, (1.7) f s, s p, p = f p, p s f p, p a. (1.8) As the interaction will in general depend only very mildly on the length of p and p we can ignore this dependence and put these momenta on the Fermi surface. Further, for rotational and translational invariance, the function should only depend on the angle θ between p and p at the Fermi level, and one can write f s/a (θ). We can then expand the angular dependence as f s/a (θ) = l=0,..., f s/a l P l (θ), (1.9) with Legendre polynomials P l. The Landau parameters f s/a l now allow one to express different observable quantities in a simple manner. Finally, these coefficients can be multiplied with ρ 0, the density of states at the Fermi level. This leads to the Fermi liquid parameters F s/a l = ρ 0 f s/a l. (1.10) 1.2 Thermodynamic properties in Fermi liquid theory The Fermi liquid parameters are basically the parameters that are responsible for the differences of the observables in the interacting system from that of the non-interacting one. For brevity we give two examples and list other results. The specific heat is obtained is obtained from the entropy via C v = T S T, (1.11) while the entropy can be computed from the distribution function via S = p,s [ n p,s ln n p,s + (1 n p,s ) ln(1 n p,s ) ]. (1.12) This is the formally the same equation as for non-interacting fermions. Due to the principle of an adiabatic connection between the free and the interacting excitations, their phase space should not change (while their dispersion appearing as argument of the distribution can change). With the entropy we can write down the free energy F = U T S µn and look for its minimum to determine n p,s. This gives after a short calculation interestingly 1 n p,s = exp( ɛ( p)/t ) + 1 = n F [ ɛ( p))], (1.13)

9 1.2. THERMODYNAMIC PROPERTIES IN FERMI LIQUID THEORY 9 with the quasiparticle addition energy defined in Eq In deriving this we have taken the functional derivative of the internal energy of Eq. 1.2, which gives δe[{n p }] = ɛ 0 ( p) + 1 f p, p ss δn p,s V n p,s p,s = ɛ( p) + 1 f p, p ss V δn p,s = ɛ( p) (1.14) p,s Note that Eq does not imply a sharp step of height 1 on the for the distribution function on the momentum axis, as the relation between momentum p and ɛ and is nontrivial due to the dependence of ɛ( p) and hence ɛ( p) on the equilibrium distribution function of the interacting system (i.e. the selfenergy correction to the bare dispersion that is included in ɛ( p)). Let us compute the change of the entropy for δn p,s. We get δs = 1 T ɛ( p)δn p,s. (1.15) p,s Now, let δn p,s be caused by temperature, δn p,s = dn F ( ɛ( p)) = n F ( ɛ( p)) ɛ( p) dt ɛ T δt + T p,s f ss p, p δn p,s. (1.16) The second term in the brackets is higher order an will be dropped. With the first part we get δs = 1 n F ( ɛ( p)) ɛ( p) 2 δt. (1.17) T ɛ T p,s We can now go over to energy integration using the density of states summed over spins, ρ = 2 p δ(e ɛ( p)), which we assume to be energy-independent equal to its value ρ 0 at the Fermi level. We can however not precisely determine the density of states, as this would require knowing the quasiparticle distribution and solving Eq This way we get (writing n F ( ɛ) = n F ( ɛ( p)) δs = s ρ 0 2 = ρ 0 de n F (E) E2 T 2 δt dx x 2 x ( 1 e x + 1 This gives the final result for the entropy change ɛ ) ) δt. (1.18) and the specific heat C V = T δs = π2 3 ρ 0δT (1.19) ( ) S T V = π2 3 ρ 0T. (1.20)

10 10 CHAPTER 1. FERMI LIQUIDS From this we see that the specific heat is basically given by the non-interacting result. The interaction effects are hidden in the value for the density of states. Now, let us consider the magnetic susceptibility. In a magnetic field, the groundstate quasiparticle energy ɛ p,s for spin s changes like (using γ = gµ B ) δɛ p,s = s 2 γh + 1 f s,s ( p, p )δn p V,s, (1.21) p,s where δn p,s = n p,s ɛ p,s δɛ p,s. (1.22) Now the term proportional to H will basically shift the Fermi surfaces of s = versus s =, and δn p,s is independent of the direction of p. This gives 1 f s,s ( p, p )δn p V,s = f s,s 0 δn s + f s, s 0 δn s = 2f0 a δn s. (1.23) p,s Putting this together and using that the p-sum over n p,s ɛ gives the density p,s of states at the Fermi level per spin, ρ 0 /2, gives δn s = 1 2 ρ 0 ( 12 ) γsh + 2f a0 δn s. (1.24) Therefore or [ γ ] (δn δn ) = ρ 0 2 H f 0 a (δn δn ), (1.25) (δn δn ) = ρ 0 γ 2 H 1 + F0 a (1.26) with F0 a = ρ 0f0 a. This leads to the magnetization M = γ 2 (δn δn ) and to the susceptibility χ = M H = ρ ( γ F0 a. (1.27) The denominator is the change with respect to the noninteracting result. Usually, F0 a is negative, and the magnetic susceptibility is enhanced. If F 0 a becomes 1, χ diverges. This corresponds to a ferromagnetic instability, and the ground state without field changes qualitatively by creating a spontaneous magnetization. This way we see how the Fermi liquid description creates its own stability criteria. A similar calculation lead to the expression for the compressibility κ = ρ 2 dρ/dµ, which is again changed by a denominator from the non-interacting value κ 0, κ = κ F0 s. (1.28) Again, F0 s < 1 causes an instability, which in this case can be interpreted as phase separation. The change of the effective mass due to the Fermi liquid interactions is given by m m = 1 + F 1 s 3. (1.29) ) 2

11 1.3. ZERO SOUND Zero sound So far we have seen that the observables of the Fermi liquid simply differ through renormalization factors from those of a free Fermi gas. Yet there is an important distinction from the free system when one considers the excitation spectrum. So far we have used the quasiparticle-excitations. These can be excited by providing sufficient extra energy and momentum to a filled state and thereby producing a quasiparticle outside the Fermi sphere, and leaving a hole-like quasiparticle inside the sphere. Hence, the quasiparticles are created in particle hole pairs. For small extra momentum q, the extra excitation energy ω is small as well, and for general q k F we will have 0 ω v F q for creating such a particle hole pair. The region in (q, ω)-space where particle-hole pairs can be created, the so-called particle-hole continuum, can be seen in Fig Note also, that for q > 2k F no PH pairs are possible near ω = 0, as then the final particle has to be further outside the Fermi sphere. In measurements of energy and possibly also momentum absorption in the Fermi liquid, the PH continuum can be observed. For example a phonon can spend its momentum and energy to create a PH pair, and can hence decay when its q and ω falls into the PH continuum. Besides a possible interaction-induced change of the Fermi velocity and of intensity changes the PH continuum is more or less the same as for the free system. The new feature in the Fermi liquid is the existence of a collective mode branch in the excitation spectrum, the so-called zero sound. This is a linear excitation mode with velocity larger than the Fermi velocity (for repulsive interactions). Therefore the zero sound mode is not damped by the PH continuum, at least in lowest order. It corresponds for an oscillating deformation of the Fermi sphere with an interaction-dependent angular profile. More precisely the solution is of the form δn p,s ( R, t) e i( q r ωt) cos θ s cos θ (1.30) with θ as angle between p and the propagation vector of the zero sound wave q. s > 1 is the ratio of the propagation speed of the wave and the Fermi velocity v F. As s > 1, the mode is above the PH continuum. Of course, ordinary sound waves (so-called first sound) do also deform the Fermi surface. Besides a different angluar profile of the oscillation (see Fig. 1.4), an important distinguishing feature between these two types of sound is however that the ordinary sound occurs in the hydrodynamic regime where the scattering rate τ 1 is much larger than the frequency ω, while the zero sound comes out best in the collision-less regime with τ 1 ω. The zero sound mode can be derived within Fermi liquid theory. We do not go through the derivation here, because the occurrence of this mode is very nicely and more easily seen in the random phase approximation of manybody perturbation theory. One important remark is that in charged Fermi liquids, i.e. for electrons in solids, the zero sound mode is pushed up to the plasma frequency, so that the gapless linear mode can only observed in fermionic quantum fluids such as Helium 3 or ultracold atom systems in traps.

12 12 CHAPTER 1. FERMI LIQUIDS Figure 1.3: Excitation spectrum of a Fermi liquid. Eduardo Fradkin (Urbana-Champaign). From Lecture Notes by Figure 1.4: Comparison of the FS deformations from zero and first sound. From Negele& Orland.

13 1.4. QUASIPARTICLE SCATTERING RATE Quasiparticle scattering rate The Fermi liquid concept is an important example of an effective theory, i.e. a simplified description of a complex many-particle system that is valid in a certain temperature range. Regarding the range of validity of the description, we will explore both ends. First, it is rather simple to understand why the description is only good at small energies or low temperatures. Let us consider the ground state of the Fermi liquid with a single excitation with energy ɛ( p) > 0 above the Fermi level. It can decay by scattering with another particle with energy ɛ( p ) to the final states k and k. The energies have to be conserved, ɛ( p) + ɛ( p ) = ɛ( k) + ɛ( k ), as demanded by the Golden Rule for the decay rate (dropping the spin arguments for avoiding complications), 1 τ k, k V int p, p 2 δ p p, k ( ɛ( p) + ɛ( p ) ɛ( k) ɛ( k ) ). (1.31) Now, as we are considering the ground state, the initial partner must have ɛ( p ) < 0. The only available, i.e. empty final states have ɛ( k) > 0 and ɛ( k ) > 0 (except for exceptions of measure zero). Therefore the sum ɛ( p) + ɛ( p ) = ɛ( k) + ɛ( k ) must be positive. This means that ɛ( p ) and ɛ( k) are between 0 and ɛ( p). The fourth energy is fixed anyway. The available phase space for the scattering can therefore be bounded from above by the product of two momentum-shells of thickness ɛ( p) below and above the Fermi surface, which goes like ɛ( p) 2. Note that not all regions of the phase space contribute, as in addition to energy conservation also the momentum conservation has to be fulfilled. We then have the following bound for the scattering rate 1 τ p Aɛ( p) 2. (1.32) The calculation at T > 0 gives an additional contribution BT 2. We will later revisit this result in second order perturbation theory. This argument shows that the quasiparticle lifetime at T = 0 rises quadratically with the excitation energy under quite general circumstances. Hence, only excitations near the Fermi edge will be stable quasiparticles, and only for these an effective description in terms of occupation numbers will make sense. The T 2 -dependence of the quasiparticle lifetime has direct consequences for the temperature dependence of the resistivity, which also goes like t 2 plus higher order corrections. Impurity scattering causes a nonzero resistivity ρ 0 at T = 0, so the main form of temperature dependence is ρ = ρ 0 + B T 2. In Fig. 1.5 we show experimental evidence for this behavior in Sr 2 RuO 4. It is actually not very easy to find experimental curves showing this expected behavior, there are many more plots for the rare occasions of disagreement with these standard expectations. 1.5 Low-energy coupling space The quasiparticle interaction in the Landau functional, f ss p, p can be understood from microscopic principles on various levels of sophistication. In order to get

14 14 CHAPTER 1. FERMI LIQUIDS Figure 1.5: Anisotropic resistivity in Sr 2 RuO 4, from Hussey et al. (Phys. Rev. B, 1998). The dotted line in the inset, showing the low-temperature T 2 dependence expected of a Fermi liquid, is for comparison with the data. a first idea let us start with a microscopic interaction conserving the spin, H int = 1 2 V ( p, p, p + q)c p+ q,s c p q,s c p,s c p,s. (1.33) p, p, q s,s It we now imagine that we are interested in low-energy and an low-temperature properties, only the states near the Fermi surface will play a role. We therefore only keep excitations in a thin shell of width Λ in energy around the Fermi surface. For simplicity let us first look at the situation in D = 2. There, we then have a thin ring of phase space for the excitations. We can now ask what constraints we get for the interaction processes, if we demand that both initial and final states should lie in this ring. In the limit of shell width going to zero we can identify three possible types of processes: Forward scattering: q 0, and the initial and final state with spin projection s are basically the same p. Exchange forward scattering p + q p, i.e. q = p p. Pair scattering: If the incoming wavevectors p and p add up to zero, all states in the ring are available for the final states. Fermi liquid theory builds on the first two types of processes, and indeed these processes turn out to be relevant for the low-scale behavior of the quasiparticle properties and observables under normal conditions. It ignores the the pair scattering, with consequences we will explore later on. In three dimensions and full rotational symmetry, there is a continuum of processes between forward and exchange forward scattering that also fully keeps

15 1.5. LOW-ENERGY COUPLING SPACE 15 the initial and final states in the low-energy shell. To see this, take the final states of the forward scattering (which are identical to the initial states), and rotate them around the axis parallel to their total momentum. A π-rotation takes the pair to the final states of the exchange forward process, but any angle in between 0 and 2π is a valid low-energy process. A deeper many-body analysis shows that these additional processes are however not relevant for the leading low-temperature behavior of the quasiparticle properties and the longwavelength response functions. This way we can restrict the the interaction Hamiltonian to the forward scattering form by just keeping the q = 0 and the q = p p terms, Hint forward = 1 V ( p, p, p)c 2 p,s c p,s c p,s c p,s p, p, s,s V ( p, p, p )c p,s c p,s c p,s c p,s. (1.34) With some algebra, in particular using the Fierz identity for the Pauli matrices σ σ αβ σ µν = 2δ αν δ βµ δ αβ δ µν, (1.35) and the definition of the many-electron spin operator (leaving out a usual 1/2 as prefactor) s p = σ αβ c p,α c p,α, (1.36) αβ p, p, s,s this can now be rewritten as with the identifications Hint forward = 1 f p, p s 2 n 1 p,sn + f p,s p, p a 2 s p s p (1.37) p, p p, p f s p, p = V ( p, p, p) 1 2 V ( p, p, p ) (1.38) f a p, p = 1 2 V ( p, p, p ) (1.39) for the effective interactions in charge (f s ) and spin (f a ) channel. Keeping only the σ z components of the spin part the interaction becomes Hint forward = 1 f p, p ss 2 n p,sn p,s (1.40) p, p At this stage, the n p,s s are in principle still operators, but obviously if we compute expectations values of H = H 0 + Hint forward, where H 0 = p,s ɛ0 ( p)n p,s, we can replace the operators by the distribution functions, i.e. classical expectation values. One might wonder why out of the sudden our theory becomes exactly solvable, and why there are no corrections left. This occurs due to our severe procedure of taking the limit q 0 exactly. This way, the interaction becomes infinitely ranged in real space, and these theories are exactly solved my mean-field ansatzes. In our case the distribution function is the mean-field

16 16 CHAPTER 1. FERMI LIQUIDS order parameter. In other words, the Landau theory is a classical limit of the many-particle theory, where operators can be replaced by real numbers in the form of distribution functions. In principle, the interactions between the quasiparticles in the low-energy shell near the Fermi surface are renormalized by multiple scattering processes involving higher-energy excitations as virtual intermediate states (this will be worked out more clearly). Such effective interactions are in general different from the interactions in the bare Hamiltonian. Just taking the bare values is the simplest approximation. The effective interactions also become frequency-dependent. Very near the Fermi surface, the frequency-to-zero ω 0 limit of the effective forward scattering interactions should be used to obtain the f-function.the more precise many-body theory (e.g. AGD, Negele- Orland)however shows that the limit q 0 and ω 0 do not commute, and that the limit with q 0 first should be taken for a correct description (otherwise some low-energy excitations would be counted twice). We will see later that a restriction to forward scattering processes that is understood in writing f ss p, p is only legitimate near the Fermi surface. These observations give an upper bound for the Fermi liquid description in energy and temperature. Regarding the lower bound, the Fermi liquid is often considered as a natural ground state of a many-fermion system that persists down to T 0. It is however known for almost 50 years that a many-fermion system with parity-symmetric dispersion ɛ( p) = ɛ( p) is quite generally unstable with respect to a Cooper instability at low scales. This omnipresence of so-called Kohn-Luttinger instabilities means that the generic ground state is Cooperpaired or superconducting, even without introducing phonons. Depending on the dispersion and on the interaction strength, the critical scales for the Kohn- Luttinger instabilities can however be rather low. Therefore the Fermi liquid state can be understood as an extended intermediate low-temperature regime that eventually becomes unstable with respect to pairing. In a real solid, impurities and additional interactions might also prevent the pairing from occurring. In the following we first introduce some field-theoretic formalism. This will allow us to identify the Fermi liquid as an asymptotic low-energy regime of a renormalization group procedure when certain dangerous channels are ignored. The functional renormalization group set-up then allows for a detailed search of instabilities of the Fermi liquid state.

17 Chapter 2 Functional integral formalism for fermions Here we want to describe how one can formulate the functional integral for fermionic quantum many-body problems. For such problems, one can also setup a field theory with generating functionals for correlation functions, just as the in case of of scalar or vector fields in classical statistical mechanics. However, for fermions in condensed matter there are two major changes. One is that we want to deal with a quantum field theory, and the fields will in addition to their spatial of momentum dependence also depend on an imaginary time τ [0, β = 1/T ] or Matsubara frequency ω n. We will see how this comes about naturally in the functional integral representation of the partition function. Second, for fermions one has to use anticommuting Grassmann-fields, in order to implement to fermionic commutation relations properly. 2.1 Grassmann variables and coherent states for fermions Let us start with a many-fermion system, where fermions are created and annihilated by operators a i and a i in quantum states i. The precise meaning of the index i can be specified later. For these operators, the commutation relations are { } a i, a j = δ ij and {a i, a j } = 0, { } a i, a j = 0. (2.1) The Hamilton operator is some function of these operators, H(a i, a i), and the partition function is Z = Tre βh(a i,a i). (2.2) While this trace can in principle be be computed within the operator formalism, dealing with higher powers of non-commuting operators in general cumbersome. In addition, many useful field-theoretical tools and approximations only become available if one derives path or functional integrals, where the operators are removed by the use of coherent states φ i which are eigenstates of the annihilation operators, a i φ i = φ i φ i. (2.3) 17

18 18CHAPTER 2. FUNCTIONAL INTEGRAL FORMALISM FOR FERMIONS In order to see the requirements for these eigenvalues, we can now form a twoparticle coherent state φ i φ j and operate on it with the anticommutator of zero value, {a i, a j } = 0, giving {a i, a j } φ i φ j = (φ i φ j + φ j φ i ) φ i φ j = 0. (2.4) Hence, for consistency, also the eigenvalues φ i, φ j should anticommute and their order how they are produced when acting with annihilation operators on a many-particle state has to be defined clearly. Such anticommuting numbers can be defined, and they go under the name Grassmann numbers. Here we collect some facts and rules for doing calculations with Grassmann numbers (without aiming at a thorough introduction). The anticommuting Grassmann numbers can be multiplied among themselves, with a minus sign upon interchange of the two individual Grassmann factors, and with complex numbers. The set of elements obtained by these operations forms a Grassmann algebra. If we have a finite number of states i = 1,... N, the full Grassmann algebra is spanned by a finite number of basis monomials, φ i1 φ i2... φ in with n N and i j [1,..., N] for j = 1,... n. Higher orders than N do not occur, as the Grassmann numbers are nilpotent, φ i φ i = 0, as follows from the anticommutation property. Power series then stop after some order, e.g., exp φ i = 1 + φ i, (2.5) as all higher orders vanish due to nilpotency. As the eigenvalues φ i replace fermionic annihilation operators, for consistency reasons, Grassmann numbers and annihilation operators anticommute as well, a i φ j = φ j a i. (2.6) We can also define conjugation of Grassmann variables by introducing φ i via φ i = φ i. (2.7) This leads to φ i a i = (a i φ i ) = (φ i φ i ) = φ i φ i, (2.8) i.e., φi is produced by acting with the corresponding creation operator to the left. The barred numbers anticommute with all other Grassmann numbers (with and without bars - note that there are no Kronecker δs as for operators) and operators. We also demand One also defines a derivative φ i ( φj φ k ) = (φ i a j ) = a j φ i. (2.9) φ i as anticommuting object, acting e.g. as ( ) φk φj = δik φj = φ φ j φ k. (2.10) i φ i Derivatives also anticommute among themselves.

19 2.1. GRASSMANN VARIABLES AND COHERENT STATES FOR FERMIONS19 Integration can also be defined by demanding dφ i 1 = 0 and dφ i φ j = δ ij, dφ i φj = 0. (2.11) This leads to the important integrals dφ i e φ i = 1, and dφ i d φ i e φ i φ i = 1 (2.12) A closer look shows that integration and differentiation have the same effect for Grassmann numbers. With this information at hand we are ready to formulate the fermionic coherent states. Let us write Then we have φ i = e φ ia i 0 = (1 φi a i ) 0 (2.13) a i φ i = a i (1 φ i a i ) 0 = φ ia i a i 0 = φ i(1 a i a i) 0 = φ i 0 The many- where the last equality is true because of the nilpotency of φ i. particle generalization for a set of φ i for i = 1,..., N is then ( ) φ = {φ i } = exp i φ i a i = φ i (1 φ i a i ) 0, 0. (2.14) Inequivalent φ φ differ by at least one of their components φ ( ) i, although differ is a little abstract as we are not dealing with usual numbers, and there is no metric defined. The conjugated state is then ( {φ i } = 0 exp ) ( ) a i φi = 0 exp φ i a i. (2.15) i i Let us again collect some facts about Grassmann coherent states. From the explicit form of the coherent states we see φ φ i = φ i φ. (2.16) The creation operators act like derivatives with respect to the corresponding Grassmann numbers, a i φ = φ i φ. (2.17) Conjugation yields now φ a i = φ i φ. (2.18)

20 20CHAPTER 2. FUNCTIONAL INTEGRAL FORMALISM FOR FERMIONS The coherent states for different {φ i } are not orthogonal, and the coherent states are not normalized to unity, φ φ = 0 i (1 φ i a i )(1 φ ia i ) 0 = i (1 φ i φ i) = e i φ i φ i. (2.19) The resolution of unity is then written as d φi dφ i e i φ i φ i φ i φ i. (2.20) While this shows at least completeness of the coherent states when all values for the Grassmann numbers are allowed, the non-orthogonality for different {φ i } means that these states are overcomplete. Another important formula is for Fock states i = a i 0 i φ φ j = φ j i φ. (2.21) The trace of an operator A can be expressed as Tr A = n n A n = = = d φ i dφ i e φ i i φ i n φ φ A n i d φ i dφ i e φ i i φ i φ A n n φ i n n d φ i dφ i e φ i i φ i φ A φ. (2.22) i Hence, the trace over the many-particle states can be rewritten as integral over the Grassmann fields. The matrix elements with coherent states ocurring in the trace can be evaluated simply, as normally any reasonable operator A can be expressed in terms of annihilation and creation operators, A = A(a i, a i), with normal ordering of the operators, i.e. the creation operators left of the annihilation operators. Then the creation operators can act leftwards on the coherent state φ, producing a factor φ i, while the annihilation operators act to the right, producing φ is. This leads to the formula φ A(a i, a j) φ = e i φ i φ i A( φi, φ i). (2.23) This formula allows one to remove all operators from matrix elements, one is left with the Grassmann numbers which are integrated over in the functional integral of the trace.

21 2.2. FUNCTIONAL INTEGRAL FOR THE PARTITION FUNCTION Functional integral for the partition function The grand-canonical partition function of our many-fermion problem shall be given by Z = Tre βh(a i ai) = d φ i dφ i e φ i i φ i φ e βh(a i ai) φ. (2.24) i For simplicity, we have absorbed the chemical potential into the Hamiltonian already. We can now split up the exponential function into e βh(a i ai) = ( ) 1 τ H(a i a i), (2.25) n=1,...,n with τ = β/n. The partition function is recovered in the( limit N. We ) can now insert N 1 resolutions of unity between the factors 1 τ H(a i a i). This gives N 1 integration measures (n) i d φ i dφ (n) i e (n) i φ i φ (n) i, with n = 1,... N 1. We can call the primary integration variable φ = φ 0 and φ = φ (N) (note that then the n = 0- and n = N-fields are not independent, and one has to integrate only over one of the two). This gives Z = lim N n=1,...n d i φ (n) i dφ (n) i e i Evaluating the matrix elements we get Z = lim d S (N) ( φ, φ) = τ N n=1,...n [ N n=1 i φ (n) i i φ (n) i ( φ (n) i (n) φ i φ (n) i φ (n) e δτh(a i ai) φ (n 1). dφ (n) i e S(N) ( φ,φ) ) φ (n 1) i + H τ ( φ(n) i (2.26) with (2.27), φ (n 1) i The first term in S (N) is a combination of the normalization factors in the resolutions of unity and the overlap in the matrix elements. It looks like a differential quotient. We can now let go N, or τ 0, and label the Grassmann numbers by a continuous index τ = n( τ), ranging from 0 to β. This way we get Grassmann fields φ i (τ) and φ(τ). For fermions, we have antiperiodic boundary conditions in the time τ, φ(β) = φ(0) and φ(β) = φ(0), (2.28) inherited from φ (0) = φ (N) and φ (0) = φ (N). Then the action becomes [ β S( φ, φ) = dτ φ i (τ) τ φ i (τ) + H ( φi (τ), φ i (τ) )] (2.29) 0 i In setting barred and unbarred fields in the Hamiltonian at the same time, we have ignored the time ordering that is in principle still present in taking ) ]

22 22CHAPTER 2. FUNCTIONAL INTEGRAL FORMALISM FOR FERMIONS (n) φ φ (n 1) the τ 0 limit of expressions like i i appearing in the Hamiltonian. This ordering can become essential to make sense out of seemingly divergent frequency sums, but for our purposes we can ignore this subtlety. The partition function can now be written as Z = D[ φ, φ] e β 0 dτ [ φ i i (τ) τ φ i (τ)+h( φ i (τ),φ i (τ))], (2.30) where the integration measure has to be understood as D[ φ, (n) φ] = lim d φ i dφ (n) i (2.31) N n=1,...n It is often more useful to Fourier transform along the τ-direction, by writing β β φ i (ω n ) = T 1/2 dτ e iωnτ φ i (τ), and φi (ω n ) = T 1/2 dτ e iωnτ φi (τ). The inversion is then 0 i 0 (2.32) φ i (τ) = T 1/2 ω n e iωnτ φ i (ω n ), and φi (τ) = T 1/2 ω n e iωnτ φi (ω n ). (2.33) The boundary conditions enforce that so-called fermionic Matsubara frequencies take the values ω n = (2n 1)πT, with n =,..., 2, 1, 0, 1, 2,.... (2.34) Now the partition function reads Z = D[ φ, φ] e ωn i [ iωn φ i (ω n)φ i (ω n)+h( φ i (ω n),φ i (ω n))]. (2.35) The integration measure remains invariant under this unitary transformation. For a moment let us go back to the action on the τ-axis. Note that we have transformed the partition function of the many-fermion system into a functional integral over Grassmann fields living on an imaginary time τ axis (a real-time path-integral would be obtained by setting τ it, with integral limits from initial time to final time). It differs from the action of a classical functional (or field) integral, as we derived previously for the Ising model as integral over all equal-time field configurations, Z cl. = D[φ( x)] e S[φ( x)] (2.36) as now the fields also vary along the τ-direction as well. This gives to us two interesting limits. First, imagine we go to zero temperature, β. Then the τ-integration becomes infinite in range. If i corresponds to some spatial (or momentum) degree of freedom x in D-dimensions, the field φ( x, τ) now lives in D + 1 dimensions. Hence we may expect that quantum systems at small or zero temperatures resemble those of classical systems in one dimension higher. Indeed this route has been explored for many examples, and many ground state properties of one-dimensional systems can be inferred from what is known from 2D finite-temperature classical statistical mechanics.

23 2.3. CORRELATION FUNCTIONS AND GENERATING FUNCTIONALS23 It may however also happen that the variation of the fields along the τ- direction does not give important contributions to the path integral, for example, when the characteristic frequency ω of the important degrees of freedom is much smaller than T, or ωβ 1. In this case the paths in the imaginary time do not wind significantly in the interval [0, β], and dropping this extra direction does not make much difference. Then one is left with fields that just depend on i or x, just as in the classical problem. Typically continuous phase transitions with symmetry breaking occur due to the softening of some mode whose characteristic frequency goes to zero. Then our argument tells us that close enough to the critical point, we always get into the regime ωβ 1, i.e. close enough to the transition the classical field theory description still holds even for a system of quantum origin, and the quantum character can only influence the properties of the system exactly at zero temperature, or for T > 0 somewhat away from the critical point. 2.3 Correlation functions and generating functionals Let us assume that the Hamiltonian has the free part that is diagonal in the quantum number i which could, e.g., correspond to a wavevector k in a translationally symmetric system, with H = H 0 + V = k ɛ( k) ψ( k, τ, s)ψ( k, τ, s) + V (2.37) appearing in the action of the system. V is the interaction, usually quartic in the fermion fields that are now called ψ( k, τ, s). s denotes the spin projection of the fermions, for electrons usually s =,. In the Matsubara representation, the quadratic part of the fermionic action has then the form S 0 = [ iω n + ɛ( k)] ψ( k, ωn, s)ψ( k, ω n, s) (2.38) k,ωn,s To make the notation more simple it is useful to use multi-indices K = ( k, iω n, s), and to write instead S 0 = Q(K) ψ(k)ψ(k). (2.39) K [ with the quadratic part Q(K) = iω n + ɛ( ] k). The partition function is simply Z = D[ψ, ψ]e S(ψ, ψ). (2.40) The single-particle Green s function G(K) is then given by G(K) = ψ(k) ψ(k) = D[ψ, ψ]ψ(k) ψ(k)e S(ψ, ψ) D[ψ, ψ]e S(ψ, ψ). (2.41)

24 24CHAPTER 2. FUNCTIONAL INTEGRAL FORMALISM FOR FERMIONS In the non-interacting case, we have Hence one can also write G 0 (K) = Q 1 (K) = 1 iω n ɛ( k). (2.42) e S((ψ, ψ) = e 1 ( ψ,g0 ψ) V ( ψ,ψ) (2.43) in the functional integral, which removes a sometimes unpleasant minus sign. Here we have used the short notation ( φ, η) = K φ(k)η(k). Higher-order correlation functions are defined as G (2n) (K 1,... K n, K 1,..., K n) = ( 1) n ψ(k 1 )... ψ(k n ) ψ(k 1)... ψ (K n ) D[ψ, ψ] = ( 1) n ψ(k1 )... ψ(k n ) ψ(k n)... ψ (K 1 ) e S(ψ, ψ) D[ψ, ψ]e S(ψ, ψ). It is now simple to see that these correlation functions can be obtained from a generating functional G(η, η) = Z 1 D[ψ, ψ] e S(ψ, ψ) e K [ η(k)ψ(k)+ η(k)ψ(k)], (2.44) by taking derivatives, D[ψ, ψ] G (2n) (K 1,... K n, K 1,..., K n) = ( 1) n ψ(k1 )... ψ(k n ) ψ(k n)... ψ (K 1 ) e S(ψ, ψ) D[ψ, ψ]e S(ψ, ψ) = ( 1) n δ 2n G(η, η) δ η(k 1 )... δ η(k n )δη(k n)... δη(k 1 ). (2.45) η= η=0 One can now use perturbation theory in the interactions V and finds that the functional G(η, η) generates non-connected diagrams as well connected diagrams. Taking the logarithm removes these disconnected parts, and the perturbation expansion has less terms. The generating functional for the connected correlation functions is usually (there is some freedom regarding the normalization of the functional integral) defined as { G c (η, η) = ln [ZG(η, η)] = ln D[ψ, ψ] e S(ψ, ψ) e } K [ η(k)ψ(k)+ η(k)ψ(k)]. (2.46) Expanding G c (η, η) in the source fields leads to a formal power series with the connected Green functions as coefficients, G c (η, η) = ln Z + ( η, G (2) c η) (2.47) 1 + (2!) 2 G (4) c (K 1, K 2 ; K 1, K 2) η(k 1 ) η(k 2 )η(k 2)η(K 1) K 1,K 2,K 1,K 2

25 2.4. WICK THEOREM Wick theorem One of the most important tricks of quantum field theory is Wick s theorem that allows one compute Gaussian averages over 2N fields as sum over all possible contractions of N pairs of fields, i.e. N free Green s functions. Written in general form, it states, using the free action S 0 = i,j ψ i Q ij ψ j ψ i1... ψ in ψjn... ψ j1 0 = D[ψ, ψ] ψi1... ψ in ψjn... ψ j1 e S 0(ψ, ψ) D[ψ, ψ]e S 0 (ψ, ψ) = P ( 1) P Q 1 (i1, jp 1)... Q 1 (in, jp N) (2.48). Here, the sum over P denotes the sum over all 2 N permutations P j of the N indices j1 to jn. The proof is quite simple using the generating functional of the theory (see Negele&Orland). In our case we get for N = 1 ψ i1 ψj1 0 = Q 1 (i1, j1) = G 0 (i1, j1), (2.49) in agreement with the previous definitions, and for N = 2 ψ i1 ψ i2 ψj2 ψj1 0 = Q 1 (i1, j1)q 1 (i2, j2) Q 1 (i1, j2)q 1 (i2, j1) = G 0 (i1, j1)g 0 (i2, j2) G 0 (i1, j2)g 0 (i2, j1). (2.50) 2.5 Perturbation theory and quasiparticle properties General perturbation expansion Here we consider perturbation theory in first and second order. This will allow us to see that there are well-defined quasiparticles near the Fermi level, and we will compute their effective mass. Let us first study the interacting Green s function, G(K) = ψ(k) ψ(k) = D[ψ, ψ]ψ(k) ψ(k)e S 0 (ψ, ψ) S I (ψ, ψ) D[ψ, ψ]e S 0 (ψ, ψ) S I (ψ, ψ). (2.51) In perturbation theory, one expands the exponent containing the interaction S I. The numerator then gives D[ψ, ψ] ψ(k) ψ(k)e S 0(ψ, ψ) S I (ψ, ψ) = D[ψ, ψ] ψ(k) ψ(k)e S 0(ψ, ψ) + D[ψ, ψ] [ S I (ψ, ψ) ] ψ(k) ψ(k)e S 0(ψ, ψ) + D[ψ, ψ] 1 [ SI (ψ, 2 ψ) ] 2 ψ(k) S ψ(k)e 0 (ψ, ψ) +...

26 26CHAPTER 2. FUNCTIONAL INTEGRAL FORMALISM FOR FERMIONS The denominator can be expanded as D[ψ, ψ] e S 0(ψ, ψ) S I (ψ, ψ) = D[ψ, ψ] e S 0(ψ, ψ) + D[ψ, ψ] [ S I (ψ, ψ) ] e S 0(ψ, ψ) + D[ψ, ψ] 1 [ SI (ψ, 2 ψ) ] 2 e S 0 (ψ, ψ) { = Z Z0 1 D[ψ, ψ] [ S I (ψ, ψ) ] e S 0(ψ, ψ) + +Z0 1 D[ψ, ψ] 1 [ SI (ψ, 2 ψ) ] } 2 e S 0 (ψ, ψ) +... = Z 0 {1 S I (ψ, ψ) [ S I (ψ, ψ) ] } We can now order everything and keep terms up to second order in S I. The first factor of Z 0 in the denominator can be used to form free averages 0 in the numerator. This gives (noting the minus sign in the definition of the free Green s function) (2.52) G(K) = G 0 (K) + S I (ψ, ψ) ψ(k) ψ(k) [ S I (ψ, ψ) ] 2 ψ(k) ψ(k) 0 +G 0 (K) S I (ψ, ψ) G 0(K) [ S I (ψ, ψ) ] G 0(K) S I (ψ, ψ) S I (ψ, ψ) ψ(k) ψ(k) 0 S I (ψ, ψ) (2.53) Here the second and third line come from expanding the denominator. We can now apply Wick s theorem tho the averages. For the first line, this will result in connected terms, where field from the S I are contracted with the two external fields ψ(k) and ψ(k), and in disconnected terms, where the latter two external fields are contracted to form G 0 (K). Then the first term in the second line is cancelled due to the disconnected contribution of the second term in the first line, while the third term in the first line removes the second term in the second line. The last term in the second line and the third line cancel as well. This cancellation of the disconnected diagrams can be shown to work at all orders and is content of the linked cluster theorem. Finally we are left with the connected parts of the first line, G(K) = G 0 (K) + S I (ψ, ψ) ψ(k) ψ(k) 0,c 1 2 [ S I (ψ, ψ) ] 2 ψ(k) ψ(k) 0,c (2.54) Self-energy corrections up to second order Now, the first correction gives, assuming K-diagonal Green s functions and summation over two spin directions for s = ±1/2 (the factor 2 in the denominator

27 2.5. PERTURBATION THEORY AND QUASIPARTICLE PROPERTIES27 goes away through the multiplicity of contractions) { S I (ψ, ψ) ψ(k, s) ψ(k, 1 [ s) 0,c = 2V (k, k, k) V (k, k, k ) ] } G(k, s ) V k G 0 (k, s) 2 = Σ HF (k) G 0 (k, s) 2. (2.55) Here we have given the label HF for Hartree-Fock approximation to the the first order selfenergy defined by the curly brackets. This selfenergy can also be represented diagrammatically. The first term is known as the Hartree term, the second as Fock term. Both represent one-loop diagrams with sum over the internal variable k (although the Foclk term doen not have a closed loop with spin sum in the strict sense of Feynman rules). The total sign is consistent with the Feynman rule that each order in perturbation theory gives a minus sign, and each fermion loop. So the Fock term without fermion loop has a minus sign, and the Hartree term of order one with one fermion loop has a plus and a factor of two from the spin sum. The second order term in Eq can also be incorporated into the selfenergy. Diagrammatically, it becomes clear that one gets two different contributions: duplications of the first order term, leading to G 0 (k, s)σ HF (k)g 0 (k, s)σ HF (k)g 0 (k, s) and genuine two-loop terms. This will be similar for high-order contributions. The iterations of the lower-order terms connected by G 0 are one-particle reducible. All diagrams that cannot be separated by cutting a single line are called one-particle irreducible (1PI). Dropping all 1PI terms with more than one loop, the perturbation series has the form G(k, s) = G 0 (k, s) + G 0 (k, s)σ HF (k) G 0 (k, s) + G 0 (k, s)σ HF (k) G 0 (k, s)σ HF (k) G 0 (k, s) +... = [ G 0 (k, s) Σ HF (k) G 0 (k, s) ] n = n=0 G 0 (k, s) 1 Σ HF (k) G 0 (k, s). (2.56) This leads to the Dyson equation G 1 (k, s) = G 1 0 (k, s) ΣHF (k) = iω ɛ( k) Σ HF ( k, iω). (2.57) If we includes higher order terms, the selfenergy appearing in the Dyson equation would be replaced by the one-particle irreducible self energy. We can assume that the interaction does not depend strongly on the Matsubara frequencies, as e.g. is the case for a screened Coulomb interaction. Then we can use the Matsubara sum T iω 1 iω ɛ( k) eiω0+ = n F [ɛ( k)]. (2.58)

28 28CHAPTER 2. FUNCTIONAL INTEGRAL FORMALISM FOR FERMIONS Then the Hartree-Fock selfenergy reads Σ HF ( k, iω) = 1 [ 2V ( V k, k, k) V ( k, k, ] k ) n F [ɛ( k )] = Σ HF ( k) (2.59) k Therefore, the first-order selfenergy is frequency independent and real, at least if the inetractions are real. This means that the first-order correction is merely a change in the dispersion relation, and does not lead to a modification of the spectral function or finite quasiparticle lifetimes. These effects enter at second order. Here one has various diagrams. The class of two-loop diagrams tha gives rise to finite lifetimes and frequency dependence is given by three diagrams. For constant interactions V (k 1, k 2, k 3 ) = g, their contribution is Σ (2) ( k, iω) = g2 V 2 = g2 V 2 G 0 ( k k, k iω,iω k, k, iω )G 0 ( k, iω )G 0 ( k + k k, iω + iω iω) n F (ɛ k )n F (ɛ k )[1 n F (ɛ k + k k )] + [1 n F (ɛ k )][1 n F (ɛ k )]n F (ɛ k + k k ) iω ɛ k ɛ k + ɛ k + k k There are two particles-one-hole and two-holes-one-particle contributions (first and second term in numerator) Quasiparticle properties For the quasiparticle- properties, the selfenergy must be evaluated at the corresponding pole of the Green s function, at ω = ɛ k. Taking the imaginary part after analytic continuation iω ω+iδ and using Im 1 x+iδ = iπδ(x) we get a quasiparticle lifetime of 1 = 2 ImΣ (2) ( τ k, iω = ɛ k + iδ) (2.61) k = 2πg2 } {n V 2 F (ɛ k )n F (ɛ k )[1 n F (ɛ k + k k )] + [1 n F (ɛ k )][1 n F (ɛ k )]n F (ɛ k + k k ) k, k δ ( ) ω ɛ k ɛ k + ɛ k + k k The detailed evaluation for the the three-dimensional case gives (Baym and Pethick) 1 ρ2 0 g2 [ τ k 8v F kf 2 /k ɛ 2 k + (πt ) 2]. (2.63) c This means that for T 0, the relative width of the quasiparticle with respect to the excitation energy becomes infinitely small. The ɛ 2 -dependence can be understood from a phase space argument. Let us consider the two-holes-oneparticle contribution with ɛ k, ɛ k > 0 and ɛ k + k < 0. Then in order to get k a contribution, in the δ-function we must have ω = ɛ k = ɛ k + ɛ k + ɛ k + k k = ɛ + ɛ + ɛ. (2.62) (2.60)

29 2.5. PERTURBATION THEORY AND QUASIPARTICLE PROPERTIES Im Σ / (i ω) = Im Σ(i ω), Re Σ(i ω) i ω / ε F Figure 2.1: Frequency dependence of the second-order self-energy for 2D dispersion ɛ = p 2 1 and g = 1.

30 30CHAPTER 2. FUNCTIONAL INTEGRAL FORMALISM FOR FERMIONS This means that ɛ k must be positive and smaller than the smallest absolute value of the three internal energies, which put severe constraints in the internal lines. The integral can then be written as 1 τ(ɛ) = 4πg2 ɛ k ɛ k V 2 dɛ dɛ ρ(ɛ )ρ(ɛ )ρ(ɛ + ɛ ɛ) (2.64) 0 0 with the densities of states ρ(ɛ). If these do not vary strongly, we obtain from the two integrations 1 τ(ɛ) ɛ2 (2.65) These perturbative results, at least to second order, that the low-energy excitations are long-lived and behave almost as free particles. This results forms the basis to the Landau Fermi liquid. One should note that in the mathematical literature one can find proofs for the regularity of the self-energy and the convergence of the perturbation expansion to all orders in the interactions above a certain low temperature. In the Fermi liquid description, the effective mass of the quasiparticles is different from the non-interacting value. This effect can also be pulled out of the second order correction by the following argument. Let us first assume that the imaginary part of the selfenergy is not too large such that we can neglect it. Then the renormalized dispersion obtained from the poles of the Green s function G( 1 k, iω) = iω ɛ( k) Σ( k, iω) after analytic continuation iω ω+iη is given by (denoting the bare dispersion with ɛ 0 ( k)) ɛ r = ɛ 0 ( k) + Σ ( k, ɛ r ). (2.66) Here Σ ( k, ɛ r ) =ReΣ( k, ω = ɛ r ). Near the Fermi level, we can linearize the dispersion. Let k denote the deviation perpendicular to the Fermi surface. Then k = k dɛ r m r dk = k m + dσ dk + dσ dɛ r dɛ r dk. (2.67) This gives m r = m 1 dσ dɛ r 1 + m k dσ dk. (2.68) As the selfenergy is a power series in the interaction strength starting at the first order, it is clear that m r evolves smoothly from the bare m when the interaction strength is increased. This way one can in principle compute the quasiparticle mass from the microscopic Hamiltonian. Precise calculations of the self-energy show that it is possible to approximate its wavevector and frequency-dependence to some degree by linear functions. This leads to the form Σ( k, iω) = Σ( k F, 0) + iω iω Σ kf,0 + (k k F ) k Σ kf,0, (2.69)

31 2.6. INTERACTION VERTEX 31 where the k has to be understood as the component orthogonal to the Fermi surface and all coefficients might depend on the location on the Fermi surface if the system is not isotropic. Using this, we can now rewrite the Green s function as [ 1 1 iω Σ kf,0] G( k, iω) = = iω v F + k Σ kf,0 1 iω Σ kf,0 (k k F ) Z( k) iω v F,r (k k F ) with the quasiparticle weight (or renormalization factor) (2.70) Z( k) = [ 1 iω Σ kf,0] 1 (2.71) which is usually < 1, and the renormalized Fermi velocity v F,r = v F + k Σ kf,0 1 iω Σ kf,0 = v F 1 + ɛ Σ kf,0 1 iω Σ kf,0 = k F m r (2.72) For comparing (2.72) and (2.68) one has to know, that the linear frequencyslope Σ( k, iω) typically occurs in its imaginary part. Then, using the Cauchy- Riemann equations for the analytic selfenergy, one obtains that the iω-derivative of ImΣ( k, iω) equals the real-frequency derivative of ReΣ( k, ω), so that both definitions lead to basically the same result. As the Matsubara-frequency sum over G( k, iω) now gives Z( k)n F [ɛ r ( k)], the jump in the distribution function as function of k is diminished to Z. Together with an incoherent background contribution that rises with decreasing k, one obtains the Fermi-liquid distribution function as depicted in Fig Interaction vertex One-particle irreducible vertex functions One can derive RG flow equations for this fermionic functional G c (η, η), with the same structure as in the scalar case. For our purposes it will be later useful to again go over to to the one-particle irreducible vertex function γ (2m), which are generated by the Legendre transform Γ[ψ, ψ] = ( η, ψ) + ( ψ, η) + G c [η, η]. (2.73) The fields η, η are now functions of the fields ψ, ψ via ψ = G c / η and ψ = Gc / η. (2.74) Γ now generates one-particle irreducible vertex functions (see, e.g., Negele & Orland) Γ (2m) (K 1,..., K m ; K 1,..., K m) = m m ψ(k 1 )... ψ(k ψ] m ) ψ(k m)... ψ(k 1 )Γ[ψ,. (2.75) ψ= ψ=0

32 32CHAPTER 2. FUNCTIONAL INTEGRAL FORMALISM FOR FERMIONS In the non-interacting case one obtains Γ[ψ, ψ] = ln Z 0 ( ψ, G 1 0 ψ). The Legendre correspondence between the functionals G c and Γ yields relations between the connected Green functions G (2m) and the vertex functions Γ (2m). In particular, Γ (2) = G 1 = G 1 0 Σ, (2.76) where Σ is the one-particle-irreducible self-energy, and we retrieve the Dyson equation from the previous considerations. The connected two-particle Green function is related to the two-particle vertex by G (4) (K 1, K 2 ; K 1, K 2) = P 1,P 2,P 1,P 2 G(K 1, P 1 ) G(K 2, P 2 ) Γ (4) (P 1, P 2 ; P 1, P 2) G(P 1, K 1) G(P 2, K 2), (2.77) while the three-particle Green function G (6) = G 3 Γ (6) G 3 + G 3 Γ (4) GΓ (4) G 3 involves Γ (4) and Γ (6). More generally, the connected m-particle Green functions are obtained by adding all possible trees that can be formed with vertex functions of equal or lower order and G-lines (see Negele & Orland). Note that the expression (2.77) is a good example that the vertices generated by Γ are indeed one-particle irreducible (1PI). Here all 1PI-pieces contributing to G (4) (K 1, K 2 ; K 1, K 2 ) are kept in the full Green s functions on the legs. The effective action obeys the reciprocity relations Γ ψ = η, Γ = η. (2.78) ψ Effective interaction Another useful generating functional is the effective interaction { 1 V[χ, χ] = ln Z 0 } DψD ψ 1 ( ψ,g e 0 ψ) V [ψ+χ, ψ+ χ] e A simple substitution of variables ψ = ψ χ yields the relation. (2.79) or e V[χ, χ] = e ( χ,g 1 0 χ) e Gc[G 1 0 χ,g 1 0 χ] ln Z 0 (2.80) V[χ, χ] = G c [η, η] + ln Z 0 ( η, G 0 η), where χ = G 0 η, χ = G t 0 η. (2.81) Here G t 0 is the transposed bare propagator, that is, Gt 0 (K, K ) = G 0 (K, K). Hence, functional derivatives of V[χ, χ] generate connected Green functions with bare propagators amputated from external legs in the corresponding Feynman diagrams. The term ln Z 0 ( η, G 0 η) cancels the non-interacting part of G c [ η, η] such that V[χ, χ] = 0 for V [ψ, ψ] = 0. The effective interaction V can also be

33 2.6. INTERACTION VERTEX 33 expressed via functional derivatives, instead of a functional integral: e V[χ, χ] = 1 DψD Z ψ 1 ( ψ,g e 0 ψ) V [ψ+χ, ψ+ χ] e 0 = 1 e V [ η, η] DψD Z ψ 1 ( ψ,g e 0 ψ) ( η,ψ+χ)+(η, ψ+ χ) e 0 η= η=0 = e V [ η, η] e ( η,g 0η) e ( η,χ)+(η, χ) η= η=0 = e V [ η, η] e ( χ,g 0 χ) e ( η,χ)+(η, χ) η= η=0 = e G 0 e V [χ, χ], (2.82) with the functional Laplacian G0 = ( χ, G 0 χ ) = K,K Perturbation series for the vertex χ(k) G 0(K, K ) χ(k ). (2.83) Next we can study some aspects of the effective interactions, i.e. the 1PI vertex Γ (4), near the Fermi surface, that are contained in the connected part of the two-particle correlation function, as described previously The connected twoparticle Green function is related to the two-particle vertex by G (4) c (K 1, K 2 ; K 1, K 2) = P 1,P 2,P 1,P 2 G(K 1, P 1 ) G(K 2, P 2 ) Γ (4) (P 1, P 2 ; P 1, P 2) G(P 1, K 1) G(P 2, K 2), (2.84) Deriving the perturbation series for this, it becomes clear that Γ (4 ) should be understood as the effective interaction of quasiparticles in which the multiply scattering with and of higher-energy degrees of freedom has been included through virtual processes. To make the case more precise, let us choose a Hubbard-type interaction S I = U β 0 dτ k, k, q ψ ( k + q, τ) ψ ( k q, τ)ψ ( k, τ)ψ ( k, τ). (2.85) In Matsubara space this becomes S I = T U ψ (k + q) ψ (k q)ψ (k )ψ (k). (2.86) k,k,q Now, let us compute the connected two-particle Green s function in perturbation theory for s 1 = s 1 =, s 2 = s 2 =, i.e. the connected parts of the expectation value G (4) c (p,, p ; p + l, ; p l, ) = ψ (p)ψ (p ) ψ (p + l)ψ (p l) c = 1 n! ψ (p)ψ (p ) ψ (p + l)ψ (p l) ( S I ) n 0,c n

34 34CHAPTER 2. FUNCTIONAL INTEGRAL FORMALISM FOR FERMIONS Computing this for the first n by using the Wick theorem and ignoring the disconnected pieces we get G (4) c (p,, p ; p + l, ; p l, ) = G 0 (p)g 0 (p )δ l,0 UT G 0 (p)g 0 (p )G 0 (p + l)g 0 (p l) +O(U 2 ). (2.87) So, comparing with Eq we see that the first order vertex is just UT. The second order contributions are in general more involved and contains several terms that we will identify later as five possible one-loop diagrams. Here, for the restricted spin structure of the onsite interaction, some particle-hole-terms cancel and we obtain after using the wavevector and frequency conservation O(U 2 ) = G 0 (p)g 0 (p )G 0 (p + l)g 0 (p l) (2.88) [ T U 2 T G 0 (k)g 0 ( k + p + p ) + T ] G 0 (k)g 0 (k + p l + p). k k The first term in the brackets com from a so-called particle-particle diagram, where both lines in between the two interactions go in the same direction, the second term is a (crossed) particle-particle diagram, with opposite direction of the internal lines. For a general interaction with spin summation, there would be three additional particle-hole diagrams. In the next subsection we will compute these diagrams in more detail, and find out that the can give rise to divergent corrections at low T. In the next order, U 3, we either get products of one-loop diagrams, or two-loop diagrams with two internal summations. The two-loop diagrams are usually less important in dimensions more than one, as they do not lead to divergent corrections at low T. The products of one-loop diagrams can be summed up to some extent, as will be done in single-channel summations and using the RG machinery in the following. Another general observation that we can make is that through the perturbative corrections the vertex will acquire a wavevector and frequency dependence, i.e. become nonlocal in space and imaginary time, even if the bare interaction is just a constant onsite interaction U General expressions for one-loop bubbles Before continuing, let us compute the one-loop particle-particle and particlehole bubbles in more detail. For later use, and for getting clearer results, we will compute the diagrams with an infrared cutoof Λ imposed on the bare Green s function that cuts out the modes near the Fermi level for ɛ( k) Λ. The full results can be obtained letting Λ 0. The free propagator is then G 0,Λ (iω, k) = Θ( ɛ( k) Λ) iω ɛ( k). (2.89) The particle-hole diagram for incoming total frequency iν (bosonic) and total wavevector Q is given by L PP Λ (iν, Q) = k T iω n G 0,Λ (iω, k)g 0,Λ ( iω + iν, k + Q) (2.90)

35 2.6. INTERACTION VERTEX 35 First let us perform the Matsubara summation. For this purpose, we study the integral dz I R = R 2πi f(z)n F (z) = Res [f(z n )n F (z n )] (2.91) n where the right hand side is due to the Residue Theorem. The sum is over all poles encircled by the circle integral. As integration contour we choose a circle with radius R, which we let go to. The integrand is assumed to decay fast enough such that the integral I R vanishes when R. For the above particle-particle loop, we use z = iω and f(z) = Θ( ɛ( k) Λ) z ɛ( k) Θ( ɛ( k + Q) Λ) z + iν ɛ( k + Q) (2.92) for the two Green s functions. n F (z) is the Fermi function (without chemical potential) 1 n F (z) = e z/t (2.93) + 1 which has poles exactly at the Matsubara frequencies z n = i(2n 1)πT, with residues T. Using I R 0 we then find T iω n Θ( ɛ( k) Λ) iω n ɛ( k) = z k n F (z k ) Res Θ( ɛ( k + Q) Λ) iω n + iν ɛ( k + Q) [ Θ( ɛ( k) Λ) z ɛ( k) Θ( ɛ( k + Q) Λ) z + iν ɛ( k + Q) = Θ( ɛ( k) Λ)Θ( ɛ( k + Q) Λ) 1 n F [ɛ( k)] n F [ɛ( k + Q)] iν + ɛ( k) + ɛ( k + Q) (2.94). Here we have used n F (ɛ + iν) = n F (ɛ). Hence the particle-particle bubble is given by L PP Λ (iν, Q) = Θ( ɛ( k) Λ)Θ( ɛ( k+ Q) Λ) 1 n F [ɛ( k)] n F [ɛ( k + Q)] iν + ɛ( k) + ɛ( k +. Q) k Similarly we can derive for the particle-hole contributions ] (2.95) L PH Λ (iν, Q) = k Θ( ɛ( k) Λ)Θ( ɛ( k+ Q) Λ) T iω n G 0,Λ (iω, k)g 0,Λ (iω+iν, k+ Q) after performing the Matsubara summation (2.96) L PH Λ (iν, Q) = Θ( ɛ( k) Λ)Θ( ɛ( k + Q) Λ) n F [ɛ( k)] n F [ɛ( k + Q)] iν + ɛ( k) ɛ( k +. Q) k (2.97)

36 36CHAPTER 2. FUNCTIONAL INTEGRAL FORMALISM FOR FERMIONS Let us now concentrate on vanishing external frequency, iν = 0, and for the particle-particle-diagram Q = 0. Then we get, using inversion symmetry ɛ( k) = ɛ( k), L PP Λ (0, 0) = Θ( ɛ( k) Λ)Θ( ɛ( k) Λ) 1 2n F [ɛ( k)] 2ɛ( k) k = Λ W dɛ ρ(ɛ) 1 2n F [ɛ] 2ɛ W + dɛ ρ(ɛ) 1 2n F [ɛ]. (2.98) Λ 2ɛ In the last step we have changed from the wavevector summation to an integral over the band energy and used that the cutoff functions cut out the region around the Fermi level with ɛ < Λ. If the temperature is much smaller than Λ, we can neglect the thermal smearing of the Fermi function and combine the two integrals to L PP Λ (0, 0) = W Λ dɛ ρ(ɛ) 1 ɛ ρ(0) ln W Λ. (2.99) ρ(0) is the density of states at the Fermi level, and we have assumed that the variation of ρ(ɛ) can be neglected in the respective energy window. For T Λ we can also obtain a simple estimate by approximating the Fermi function by 0 or 1 for ɛ > 0T or ɛ < T respectively, and by n F (ɛ) 1 2 (1 ɛ/t ) in between. The interval betwen T and T then gives a finite contribution for T 0, while the outer regions give together L PP Λ=0(0, 0) = W T dɛ ρ(ɛ) 1 ɛ ρ(0) ln W T. (2.100) Note that the sharp step of the Fermi function for T 0 is essential for this behavior. We learn that the particle-particle diagram is divergent at low T and small Λ. We face a logarithmic divergence which is quite common in many-body problems. This divergence at zero total incoming frequency and wavevector is common to all many-fermion system obeying ɛ( k) = ɛ( k) in any spatial dimension as long as the density of states at the Fermi level is nonzero. As the particle-particle channel is intimately connected with Cooper pairing, it means that the usual ground state of a metal with these properties is the superconducting state (Kohn-Luttinger effect). In practice however, the critical temperatures for the onset of this state might be extremely low. The analysis of the particle-particle channel at zero total incoming frequency and wavevector was the worst-case analysis, other contributions of this diagram type are usually smaller and regular in the low-scale or low-t limit. Now let us ask if we can get similarly divergent terms in the particle-hole channel. In order to get a analogous expression we now need to have ɛ( k + Q) = ɛ( k) for a dense set of k values, for k towards the Fermi level. This is

37 2.6. INTERACTION VERTEX 37 Figure 2.2: Dispersion of the one-dimensional model, with linearized dispersion and a left branch around k F and right branch branch around +k F. easily found in one dimension with a linear dispersion near the Fermi level and Q = ±2k F. To be more precise let us choose (see also Fig. 3.7) ɛ(k) = v F (k k F ), k > 0 and ɛ(k) = v F (k k F ), k < 0 (2.101) with Fermi velocity v F and Fermi wavevector k F. If we now shift a k < 0 near the left Fermi point k F by Q = 2k F we end up near the right Fermi point near +k F. Moreover, if ɛ(k) < 0 we find ɛ(k + 2k F ) = ɛ(k) > 0 and vice versa. This means that we get for the particle-hole diagram at zero transferred frequency and momentum transfer 2k F L PH Λ (0, 2k F ) = Θ( ɛ(k) Λ)Θ( ɛ(k) Λ) 1 + 2n F [ɛ(k)] 2ɛ(k) k { Λ = dɛ ρ(ɛ) 1 2n F [ɛ] W + dɛ ρ(ɛ) 1 2n } F [ɛ] W 2ɛ Λ 2ɛ left branch ρ l (0) ln W Λ. (2.102) Hence this particle-hole diagram has the opposite value as the particle-particle diagram but also diverges at low T and Λ. A slight subtlety is the restriction of the k-integration on the left branch around k F, as shifting the right branch by 2k F takes us usually far away from the Fermi level (except when 2k F = 2k F modulo a reciprocal lattice vector, this is the case at half-filling, when also Umklapp scattering has to be considered). Therefore we have introduced the notation for the density of states on one branch only, ρ l. For the particle-particle diagram there was no such restriction, and the density of states in front of the logarithm was that of both dispersion branches. The behavior of a log-divergent particle-hole bubble is now a specialty of one dimension, it higher dimensions a particular nested bandstructure or singularities in the density of states are needed to obtain a divergence of the particle-hole term. An example for this is the 2D Hubbard model on the square lattice at half

38 38CHAPTER 2. FUNCTIONAL INTEGRAL FORMALISM FOR FERMIONS band filling, where the dispersion is ɛ( k) = 2t(cos k x +cos k y ) = ɛ( k+ Q) with Q = (π, π). However, besides these special cases that have to be treated with more care, no divergencies occur in the particle-hole diagram. This is mainly because for al generic dispersion the energy denominator never approaches zero on a phasespace set of nonzero measure, i.e. if ɛ( k) goes to zero, ɛ( k+ Q) usually remains different from zero, and the k-integral just gives a finite number, no matter how sharp the Fermi function in the numerator gets Small momentum and frequency transfer Let us for later reference now look at the particle-particle diagram for Q 0 and ν 0. These two limits do not commute. If we set iν = 0 and then compute the limit Q 0 we get L PH Λ=0(0, Q 0) = k n F [ɛ( k)] n F [ɛ( k + Q)] k n F [ɛ( k)] ɛ ɛ( k) ɛ( k + Q) (2.103) k ρ 0 for T 0, (2.104) as the derivative of the Fermi function becomes a negative δ-function at the Fermi level in the T = 0 limit. This result would also be obtained from the double pole when one sets Q = 0 straight away. At finite T, the δ-function is temperature-smeared, and the density of states is averaged over a region of width T. Note that we we include an infrared cutoff Λ T, the loop contribution vanishes. The other limit is Q = 0, iν 0. Then one gets L PH Λ=0(iν, Q = 0) = k n F [ɛ( k)] n F [ɛ( k)] iν = 0. (2.105) Therefore also the limit iν 0 is different from Eq , also in the limit Λ Summation of Cooper channel It is not possible to give a closed expression for the full perturbation series to all orders in U. It is however not too difficult to see that the higher order terms, say to order U n will contain diagrams that are chains or ladders of n particleparticle diagrams. The internal summations within each segment of this chain are independent and the value of the diagram is basically the nth power of the

39 2.6. INTERACTION VERTEX 39 particle-particle bubble. More precisely we get Γ (4) PP ladder (p,, p ; p + l, ; p l, )/( T ) = U U 2 L PP Λ (p + p ) + U 3 [ L PP Λ (p + p ) ] = U [ UL PP Λ (p + p ) ] n n U = 1 + UL PP Λ (p + p ). (2.106) The result now strongly depends on the sign of U, in particular for p + p = 0 where the PP bubble diverges logarithmically. Here, L PP Λ (p + p = 0) W ρ 0 ln max(λ,t ). If U > 0 the vertex becomes very small near this particular wavevector and frequency combination in the limit T, Λ 0. Then the logdivergence of the bubble diagram is no longer significant, it basically switches off a part of the interactions. However, if U < 0, the ladder sum diverges, and the effective interaction including multiple scattering becomes infinitely strong. This leads to the formation of bound states of total wavevector zero, also known as Cooper pairs. This phenomenon is exactly the Cooper instability that expresses the instability of the Fermi sea with respect to attractive interactions Summation of crossed particle-hole channel Similar to the Cooper channel, also the second, PH-contribution in the perturbation series for the vertex is produced n arbitrary powers with alternating sign, and this particular part of the series can be summed to give (using q = p l p) Γ (4) PH ladder (p,, p ; p + l, ; p l, )/( T ) = U U 2 L PH Λ (l) + U 3 [ L PH Λ (q) ] = U [ UL PH Λ (q) ] n n U = 1 + UL PH. (2.107) Λ (q) In our language, L PH Λ (q) is negative. This means that the sign considerations from the Cooper case are exactly reversed. Now a positive U leads to an enhancement of the vertex, and if UL PH Λ (q) < 1 (2.108) we get an instability. As standard susceptibilities (i.e. expectation vales of two fermion bilinears) are contained in the two-particle Green s function, a divergence of the vertex Γ (4) corresponds to some divergent susceptibility. Here, the crossed particle-hole ladder for the given spin combination appears in the magnetic susceptibility, and a divergence a wavevector transfer q can be understood as spin-density-wave instability. The criterion in Eq is a generalized Stoner criterion for magnetic ordering. For generic dispersions, the particle-hole bubble remains finite, and a critical interaction strength U c 1/ L PH Λ (q) is needed for an instability. Yet, for particular band dispersions, also the particle-hole diagram can diverge logarithmically (or even stronger) at zero frequency transfer and certain wavevector

40 40CHAPTER 2. FUNCTIONAL INTEGRAL FORMALISM FOR FERMIONS transfers. We have seen an example in 1D. In 2D one gets a log 2 -divergence for the dispersion of the 2D square lattice with nearest neighbor hopping and chemical potential set to zero, where ɛ( k) = 2t(cos k x + cos k y ). Now, for Q = (π, π) we have ɛ( k) = ɛ( k + Q). The van Hove singularities density at the Fermi level at (π, 0) and (0, π) actually cause a log-square divergence of the PH bubble L PH ( Q, 0). The spin-density-wave state suggested by the instability in the crossed ladder summation is antiferromagnetic, with alternating magnetization between neighbored sites. For this dispersion, the particle-particle bubble at zero total momentum transfer diverges also like log-square Competing instabilities and the need for full one-loop summations We have just discussed two separate single-channel summations for the interaction vertex and argued that in some cases both ladders can lead to instabilities. Of course the full perturbation series contains many other diagrams, and in particular diagrams, where divergent PP and PH bubbles mix. Then the question is whether these contributions will significantly alter the expectations from the single-channel summations. Furthermore, one would like to know which instability is actually the strongest or leading and has the highest critical temperature. For this purpose one should sum all diagrams, or at least all one-loop diagrams and products of them together. The latter scheme goes under the name parquet summation and has been performed for special cases (like in D = 1). In general however, the RG framework is considered a more useful tool to accomplish this joint and unbiased summation of all possibly dangerous diagrams.

41 Chapter 3 Wilson RG for fermions 3.1 Reminder: Wilson RG for scalar fields Momentum-shell flow equations for the φ 4 -theory We first consider how the effective action of the φ 4 -theory develops when we integrate out high energy modes. The starting point is the partition function Z = Dφ e S Λ 0 (φ) (3.1) with the action S(φ) = V f Λ0 [ r 0 + c 0 ] k 2 φ( 2 k)φ( k) k + u Λ0 0 δ 4! k1, k 2, k 3, k1 + k k 2 + k 3 + k 4,0 φ( k 1 )φ( k 2 )φ( k 3 )φ( k 4 ). (3.2) 4 Here we have explicitly put short-length-scale cutoffs Λ 0 at the momentum integrals. This cutoffs is a consequence of the derivation of this effective action and its is roughly inversely proportional to the lattice scale a. The final results will hopefully not depend significantly on this vaguely determined parameter Flow equations The goal is now to integrate over high-energy modes with wavevectors k > Λ < Λ 0 in order to implement a scale change from the lattice scale a to some longer scale ba > a. This can be done by splitting the fields φ( k) into two parts, φ( k) = φ < ( k)θ(λ k ) + φ > ( k)θ( k Λ). (3.3) Here, φ < ( k) represents the low-energy or long-wavelength modes, and φ > ( k) are the high-energy and short-wavelength modes. Θ(x) is a step function with Θ(x) = 1 for x > 0 and Θ(x) = 0 otherwise. The quadratic part of the initial action, S 0,Λ0 is diagonal in k and hence nicely splits up into to separate parts, but in the quartic term S I short- and long-wavelength modes are coupled, S Λ0 (φ < + φ > ) = V f 0 + S 0,Λ0 (φ < ) + S 0,Λ0 (φ < ) + S I,Λ0 (φ < + φ > ). (3.4) 41

42 42 CHAPTER 3. WILSON RG FOR FERMIONS Note that the last terms contains different contributions, one with all four fields being φ < s, i.e. belonging to the long-wavelength modes, one with three φ < and one φ >, and do on. As the integral measure is just a product over all wavevectors, we can now split up the partition function (note that the inverse temperature β has been absorbed in f 0 and the other parameters in the action), Z = e V f 0 Dφ < e S 0,Λ 0 (φ < ) Dφ > e S 0,Λ 0 (φ > ) S I,Λ0 (φ < +φ > ) (3.5) The last integral is obviously only a functional of the long-wavelength modes φ <, as the short-wavelength modes are integrated over. The resulting expression Z Λ (φ < ) = Dφ > e S 0,Λ 0 (φ > ) S I,Λ0 (φ < +φ >) can be put back into the exponent by taking the logarithm of it, leading to Z = e V f 0 Dφ < e S 0,Λ(φ < )+log Z Λ (φ <) = Dφ < e S Λ(φ <). (3.6) Here the last equation defines the scale-dependent effective action, S Λ (φ < ) = V f 0 + S 0,Λ (φ < ) log Z Λ (φ < ). (3.7) We have also used the fact that the quadratic part of the long-wavelength modes only involves modes below Λ and have identified S 0,Λ0 (φ < ) = S 0,Λ (φ < ). Now one can expand the second term in effective action in powers of the fields. This gives additional contributions zeroth-order, second-order and fourth-order terms in the to the action of the long-wavelength modes, usually allowing one to define renormalized parameters of the effective action, i.e. f 0 f < r 0 r < c 0 c < u 0 u Λ. (3.8) Note however that it is a priori not clear that Z Λ can be adequately described by only these few parameters. In particular, new and higher order terms might be generated which need to be analyzed with care. The parameter changes described by Eq. 3.8 hold for a finite change of the upper bandwidth, Λ 0 Λ. By taking the derivative of the Λ-dependent parameters we can easily derive renormalization group differential equations or flow equations whose solution provides the trajectory of the effective action in the parameter space spanned by f <, r < etc.. The initial conditions for these RGDEs are given by f 0, r 0 etc. Now, let us compute the additional terms in the action in more detail, using a perturbation expansion of S I, by writing Z Λ (φ < ) = Dφ > e S 0,Λ 0 (φ > ) S I,Λ0 (φ < +φ > ) ( 1) ν = Dφ > e S 0,Λ 0 (φ > ) [ S I,Λ0 (φ < + φ > ) ] ν ν! ν=0 ( 1) ν = Z Λ0,0 [ S I,Λ0 (φ < + φ > ) ] ν Λ,0 (3.9) ν! ν=0

43 3.1. REMINDER: WILSON RG FOR SCALAR FIELDS 43 Here the subscript at the expectation value means that only high-energy modes above Λ should be used. Z 0,Λ = Dφ > e S 0,Λ 0 (φ >) is the noninteracting partition function down to scale Λ. The result here has the same structure as the perturbation expansion written down in the last section, with the difference that now only the short-wavelength field φ < have a propagator and are integrated over, while the long-wavelength fields are fixed at this stage. This means that the φ < -fields appear as external fields in the diagrams. We can straightforwardly compute the first terms of Z Λ. This will give us again a structure of the type Z Λ = Z Λ,0 (1 + x) where x contains all orders of the perturbation series with ν 1. Then we can use the formula ln(1 + x) = x x 2 /2 + x 3 /3... (3.10) to put the result into the exponent. This logarithm series will again cancel disconnected diagrams. We will only keep one-loop diagrams, as they constitute the leading terms. To order ν = 0, we just get ln Z Λ,0, as the expectation value is properly normalized. Z Λ,0 is just a Gaussian integral for each wavevector k, and its logarithm is f 0,Λ = 1 Λ0 d D k 2 (2π) D log [ r 0 + c 0 k 2]. (3.11) Λ Here the superscript (ν=0) indicates that this contribution is of zeroth order in u 0. If we continue to integrate out modes and let Λ this way approach zero, this term would just collect the Gaussian fluctuations, as in our previous considerations of the specific corrections due to Gaussian fluctuations of the mean-field theory. The first order in u 0, i.e. ν = 1 gives us one interaction vertex with four fields in the expectation value. If all four fields are φ <, then there is nothing to average over, and we get Z Λ,0 times the direct interaction of the low-energy modes, S I,Λ0 (φ < ). So, the first term contributing to x is x 1 = Z Λ,0 S I,Λ0 (φ < ). (3.12) If there is only one high-energy field, the Gaussian average over the high-energy modes gives zero, as it also happens for three high-energy modes. The remaining diagram is one with two low-energy fields and two high-energy fields. Here there are various possibilities to pick ( 2) low-energy legs out of the four legs of the u 0-4 vertex, this gives a prefactor = 6. Hence this contribution is 2 x 2 = 6u 0 4! Λ0 Λ d D k 1 Λ0 (2π) D r 0 + c 0 k 2 = u 0 4 Λ d D k 1 (2π) D r 0 + c 0 k 2. (3.13) Here, the momenta of the low-energy legs need to be opposite in order to comply with momentum conversation. Note that this contribution x 2 does not depend on the momentum of the external low-energy legs. Another diagram would then have four high-energy modes, connected by two high-energy propagators. This two-loop diagram would give a correction

44 44 CHAPTER 3. WILSON RG FOR FERMIONS to the (φ < ) 0 -part, i.e. to the free energy, but in our approximation where only one-loop terms are considered, this contribution will be dropped. From the corrections of second order, ν = 2, we will only keep the one-loop term that connects two vertices with two high-energy legs each. This gives x 3 = 1 ( ) 2 Λ u0 4! Λ = u2 Λ Λ d D k (2π) D G( k)g( k + Q) d D k 1 1 (2π) D r 0 + c 0 k 2 r 0 + c 0 ( k + Q) 2 (3.14) Here the factor 1/2 comes from ν = 2, the 2 is due to the possibility of forming two contractions between the two high-energy legs of first and the the two of second vertex, and the 6 again reflects the number of possibilities to pick two low-energy legs out of four of a given vertex. The wavevector Q gives the total incoming momentum that flows into the one-loop diagram. We explicitly see that integrating out modes generates a momentum dependence of the effective interaction. There are a few more diagrams to second order. One is with two vertices that only contain low-energy legs. This is a disconnected contribution to x in second order that cancels against to powers of the direct low-energy vertex in the x 2 /2 in the logarithm expansion. Then there are terms with one lowenergy leg on one, say left, vertex, and three on the other, say right vertex. This means than two high-energy legs on the left vertex gets contracted with each other, and the remaining high-energy leg on the left vertex forms a propagator with the one high-energy leg on the right vertex. Such a contribution is left out at the current level, because the single propagator line connecting the two vertices i fixed by the external momenta. Now, in a further approximation that will be justified in the following, the external momenta are set to zero. This means that also the one internal line is at zero momentum, and gives hence no contribution as only modes above the cutoff are integrated over. Finally, there is another disconnected term with two high-energy legs on each vertex which gets contracted among each other without connecting to the second vertex. This also cancels against x 2 /2 terms. The contribution x 1 renormalizes f 0, while x 2 adds to r 0 in the quadratic part of the action, and x 3 renormalizes the φ 4 -interaction. c 0 is not renormalized, as the second order term in the low-energy fields is not momentumdependent. Taking into account the prefactors that occur in the usual parametrization of the action (1/2 for r 0 and 1/4! for u 0 ), we get the following renormalizations. f < = f r < = r 0 + u 0 2 Λ0 Λ Λ0 Λ d D k (2π) D log [ r 0 + c 0 k 2] (3.15) d D k 1 (2π) D r 0 + c 0 k 2 (3.16) c < = c 0 (3.17) u < = u 0 3u2 0 2 Λ0 Λ d D k (2π) D 1 [r 0 + c 0 k 2 ] 2 (3.18)

45 3.1. REMINDER: WILSON RG FOR SCALAR FIELDS 45 With this, we get the new action S Λ (φ) = V f < u< 4! Λ Λ k [ r < + c < k 2 ] φ < ( k)φ < ( k) k1,..., k 4 δ k1 + k 2 + k 3 + k 4,0 φ< ( k 1 )φ < ( k 2 )φ < ( k 3 )φ < ( k 4 ).(3.19) As we are looking for scale-invariant points that could describe continuous phase transitions, we now rescale the wavevectors in order to make the new action look like the original one. The scale change we have performed is the ratio b = Λ 0 /Λ > 1. Hence we write k k = Λ 0 Λ k = b k. (3.20) This restores the integral cutoff back to Λ 0 = bλ. The D powers of b we get from the integral measure and the c 0 k 2 can be absorbed into the fields by redefining φ ( k ) = ζ 1 b φ < ( k /b) with ζ b = b 1+D/2 Z b and Z b = c 0 c <. (3.21) In our current calculation the field renormalization Z b is still one, but this may differ for other problems or approximations. The rescaling is set up in order to keep the k 2 term invariant, but the r < term picks up an additional scaledependent factor from this change. We will see that this behavior is very natural for the Gaussian fixed point that correctly describes the phase transition for D > 4. In detail we have for the quadratic part 1 Λ 2 0 = 1 2 b D Λ = 1 2 Λ0 0 d D k [ (2π) D r < + c < ] k 2 φ < ( k)φ < ( k) 0 d D k [ (2π) D r < + c < b 2 ] k 2 b 2+D Z b φ ( k )φ ( k ) d D k [ (2π) D b 2 Z b r < + c 0 ] k 2 φ ( k )φ ( k ). (3.22) Now, we can write down the new renormalization descriptions by inserting r < and c <, and doing an analogous manipulation in the interaction term, [ r Λ = b 2 Z b r 0 + u Λ0 0 2 Λ u Λ = b 4 D Z 2 b u< = b 4 D Z 2 b d D ] k 1 (2π) D r 0 + c 0 k 2 [ u 0 3u Λ0 0 2 The energy density f 0 picks up a scale factor b D, [ f Λ = b D f Λ0 2 Λ Λ (3.23) d D ] k 1 (2π) D (r 0 + c 0 k 2 ) 2.(3.24) d D k (2π) D log ( r 0 + c 0 k 2) ]. (3.25)

46 46 CHAPTER 3. WILSON RG FOR FERMIONS In order to write down the RG flow equation for infinitesimal change of the scale, i.e. Λ = Λ 0 dλ or b = 1 db, we differentiate the integrals, d Λ0 dλ Λ d Λ0 dλ Λ d D k 1 (2π) D r 0 + c 0 k 2 = ΛD 1 K D r 0 + c 0 Λ 2 (3.26) d D k (2π) D 1 (r 0 + c 0 k 2 ) 2 = ΛD 1 K D (r 0 + c 0 Λ 2 ) 2 (3.27) K D is the surface of the D-dimensional unit sphere divided by (2π) D. Next we introduce the logarithmic flow parameter l by Λ = Λ 0 e l or b = e l. This leads to Λ d dλ = d (3.28) dl which brings some simplifications of the formulae by removing additional powers of Λ. This way we get d dl r l = (2 η l )r l + b 2 u l K D Λ D Z l 2 r l + c 0 Λ 2 (3.29) Here, η l = lz l Z l is the flowing anomalous dimension, which is zero in the current approximation, but in principle a finite η l at the fixed point will modify the decay of the correlations at T c. For the interaction we get d dl u l = (4 D 2η l )u l 3u2 l 2 K D Λ D (r l + c 0 Λ 2 ) 2 (3.30) Here we have replaced the quantities u 0 and r 0 with their flowing counterparts u l and r l, while c 0 is kept fixed naturally for the present case or by construction otherwise. Hence the equations can be used at every scale, also below the initial scale Λ 0. Note however that we always rescle such that the upper cutoff of the integral is Λ 0, which is Λ for an infinitesimal step. Hence we can set Λ = Λ 0 and b = 1 in the above equations, and do some more cosmetics by introducing dimensionless quantities r l = 1 c 0 Λ 2 0 r l and ũ l = K D u c 2 l. (3.31) 0 Λ4 D 0 The final equation we get are then (dropping the zero η l ) RG flow in the φ 4 -model d dl r l = 2 r l + 1 ũ l (3.32) r l d dl ũl = (4 D)ũ l 3 ũ 2 l 2 (1 + r l ) 2 (3.33) Let us now analyze what we learn from Eqs and Here, D = 4 plays a special role. For D > 4, the right hand side of the flow equation for u l is

47 3.1. REMINDER: WILSON RG FOR SCALAR FIELDS 47 negative for the physical values u l > 0, while the flow goes to larger values of l. This means that u l can only become smaller and flows towards zero. Now, if u l gets small, the seond term in the flow equation for r l will ultimately become negligible. Then the first term causes r l to grow, no matter what its sign is. This means that r l runs away from r l = 0, very much as we would expect for the correlation length for ξ < upon length rescaling. Indeed, if u l flows to zero, the system becomes more and more like a free Gaussian model. Therefore behavior for large l will be that of the mean-field theory, as we know already that Gaussian fluctuation effects are negligible for D > 4. Hence the critical point is at r = 0, and it is easy to check that r = u = 0 is a fixed point with the r-direction as unstable direction. This fixed point is called Gaussian fixed point and rules the critical behavior of the φ 4 -theory for D > 4. In Fig. 3.1 on the left side we show the flow in the two-dimensional parameter space. Beside sthe unstable fixed point with the relevant r-direction one can identify a line in the lower quadrant that separates flows to r l =, i.e. the high-temperature state, from flows to r l =, the low-temperature phase. Hence, all systems with initial parameters above this line have T > T c, while all systems with initial parameters below this line are below their T c. Obviously, and increase in the initial u 0 requires going to more negative r 0 in order to stay at T c. Of course we should keep in mind that our results are perturbative in u l, so the picture obtained here is only valid for small u l. Therefore the flow toward smaller u l is a great help, as it improves the validity of our theory, the better the larger l is. The opposite occurs for D < 4. Here, the first term on the right hand side of the flow equation for u l is positive, and at least for small u l the flow is away from zero. This is shown in the right figure of Fig This flow now means that the Gaussian fixed point does no longer describe the critical properties, as the systems flow away from it to nonzero u l on long scales. Instead one finds another fixed point with only one unstable direction which now can be used to explore the critical behavior. By setting the right hand sides of Eqs and 3.33 to zero we find this Wilson-Fisher fixed point at ũ 0 = 2 r r 0 = (4 D)ũ 3 (ũ ) 2 2 (1 + r ) 2 Now, let us write ɛ = 4 D. Then, to first order in ɛ we get ũ = 2 3 ɛ + O(ɛ2 ) (3.34) r = u 4 + O(ɛ2 ) = ɛ 6 + O(ɛ2 ). (3.35) From this we can infer two things First, as the fixed point values go back to the Gaussian fixed point r = u = 0 for ɛ = 4 D 0, the Wilson-Fisher fixed point emerges continuously out of the Gaussian one. In fact by looking at the picture, we understand that we can just move the WF fixed point on the

48 48 CHAPTER 3. WILSON RG FOR FERMIONS Figure 3.1: Flows of r l and u l in D > 4 (left plot) and D < 4 (right plot), taken from Kopietz s book. Only positive us are physical as otherwise the model is not bounded. Gaussian fixed point and recover the flow pattern for D > 4. This means that the neighborhood of the critical fixed point changes continuously, and so do the critical exponents. In turn this means that for larger ɛ = 4 D < 0 the fixed point is further away from the origin, and will occur at general, nonzero values of u. This makes the present RG approach that treated u l as perturbation less reliable. In this sense ɛ appears as small expansion parameter in this problem. One can try to improve the results systematically by going to higher order in ɛ (this would involve keeping two-loop terms), but the structure of the problem remains the same. Hence the whole RG calculation brings the new insight that for D < 4 there is a new non-gaussian fixed point with different critical exponents (no computed in detail here), but due to the problematic flow toward an interacting fixed point with nonzero u, the approach presented here remains a proof of principle, at least in the current approximation. There are different strategies to get reliable results for nonzero ɛ, i.e. in D = 3 and lower. We will later study the advantages of the functional RG treatment that works non-perturbatively in the interactions, and that allow one to study the case D < 4 without ɛ-expansion. An important remark concerns the validity of the effective model. Here, for one, we have neglected higher powers p in the fields and kept only terms up to p = 4. p = 6 will in principle be generated by the u 0 term, e.g. by a third-order contribution, and similarly all other higher powers will acquire nonzero values. Furthermore, the flow will in general generate a wavevector-dependence of these interactions. We can however use the rescaling in order to see that neglecting the p > 4-vertices and wavevector-dependences is justified at the Gaussian fixed point. Then by continuity we might also hope that the error is no too large for small ɛ, i.e. for D near but small than 4. Let us consider a general interaction of order p, denoted by u (p), depending on p 1 momenta k 1 to k p (with the last being fixed by momentum conservation) and expand it around the zero of momentum space. Now under our rescaling

49 3.1. REMINDER: WILSON RG FOR SCALAR FIELDS 49 Figure 3.2: Flows of r l and u l vs. l in D < 4, taken from Kopietz s book. The flow is first attracted by the fixed point, but then bends away when the system is not exactly critical. The first scale l c defines the scale where the systems starts to be controlled by the Wilson-Fisher fixed point, and where hence deviations from mean-field behavior are strong. The scale l is inversely proportional to the inverse (physical) correlation length ξ(t ) of the system, for T = T c, ξ. formulated in (3.20) and (3.21) we find that the k i = 0-component scales like ũ (p) (0, 0,... 0) = b p(1+d/2) D(p 1) Z p/2 b u (p) (0, 0,... 0) (3.36) The first powers multiplied by p come from the field rescaling, and the D(p 1) is from transforming the integration measure. As z b = 1, we get for p = 4 as exponent 4 D, for p = 6 this becomes 6 2G, and for p = 8 8 3D. This means that for D 4 all orders higher than p = 4 scale to zero automatically, and in general higher orders do scale more favorably than p = 4 in any dimension. Therefore, near D = 4, orders higher than p = 4 can be neglected. Note however that as soon the p = 4-term does no longer flow to zero, i.e. for D < 4, the tail of other perturbation series could change this naive tree-level scaling picture. Again, in this case only the continuity in ɛ gives us confidence in the correctness of the present calculation. A similar argument holds for the wavevector-dependence. Any power of k in the expansion of the interaction vertex brings in an additional scale factor b 1 when we express k by k = k /b. This means that any wavevector-dependence scales to zero already for p = 4.

50 50 CHAPTER 3. WILSON RG FOR FERMIONS 3.2 Fermionic version For fermions, we can go through a very similar construction. We will mainly copy the step of integrating out high-energy modes. We will go go through the steps of wavevector-rescaling and subsequent field rescaling. The problem here is that in one should rescale momenta with respect to the Fermi surface, i.e. only the component perpendicular to the Fermi surface. In D > 1 this in incompatible with momentum conservation, and we prefer to keep this conservation law intact. While this removes the possibility to look for fixed points in the standard Wilsonian way, we will see that nevertheless we can gain a lot of information from the fermionic flows. Let us again consider an action of the form S = k ψ k,s ( ik 0 + ɛ( k))ψ k,s + T 2 k 1,k 2,k 3 s,s V (k Λ0 1, k 2, k 3 ) ψk1,s ψ k2,s ψ k4,sψ k3,s, (3.37) where the combined Matsubara- and wavevector-index k 4 = k 1 +k 2 k 3 is fixed by conservation laws. The bare propagator is diagonal in spin space, and the interaction term conserves the spin. Diagrammatically we can express V Λ0 by a rectangle with two long and two short sides. Each short side connects one incoming and one outgoing line with the same spin components, i.e. s stands on one short side, and s on the other. Now the fermionic fields are split with respect to the energy difference to the Fermi level ɛ( k) = 0 as where ψ k,s = ψ < k,s + ψ> k,s, (3.38) ψ < k,s = ψ k,sθ(λ ɛ( k) ) and ψ > k,s = φ k,sθ( ɛ( k) Λ) (3.39) are now called slow and fast modes. The action then splits up into S Λ0 (ψ < + ψ > ) = S 0,Λ0 (ψ < ) + S 0,Λ0 (ψ < ) + S I,Λ0 (ψ < + ψ > ), (3.40) i.e., provided that S 0 is just a sum over k, slow and fast modes only couple in the non-quadratic interaction part. Note that the last terms contains different contributions, one with all four fields being ψ < s, i.e. belonging to the longwavelength modes, one with three ψ < and one ψ >, and so on. As the integral measure is just a product over all wavevectors, we can now split up the partition function as Z = Dψ < e S 0,Λ 0 (ψ < ) Dψ > e S 0,Λ 0 (ψ > ) S I,Λ0 (ψ < +ψ >). (3.41) We can now, at least formally, perform the second functional integral over the

51 3.2. FERMIONIC VERSION 51 fast modes to obtain Z Λ (ψ < ) = Dψ > e S 0,Λ 0 (ψ > ) S I,Λ0 (ψ < +ψ > ) ( 1) ν = Dψ > e S 0,Λ 0 (ψ > ) [ S I,Λ0 (ψ < + ψ > ) ] ν ν! ν=0 ( 1) ν = Z Λ0,0 [ S I,Λ0 (ψ < + ψ > ) ] ν Λ,0. (3.42) ν! ν=0 In the last equation we have inserted the non-interacting partition function of the fast modes, Z Λ0,0 = Dψ > e S 0,Λ 0 (ψ >), (3.43) in numerator and denominator, to allow for the expectation values to appear. The change of the action of the low-energy modes that adds to S 0,Λ0 (ψ < ) will then be given by S Λ (ψ < ) = log Z Λ (ψ < ). (3.44) Eq is a straightforward recipe to obtain a perturbation series for the effective action parameters. Again, like in the scalar case, the logarithm leads to a cancellation of all disonnected and vacuum terms, and only connected diagrams remain. In the diagrams that arise then, only modes in the energy shell that is integrated out contribute on the internal lines, because only for those, the integral measure Dψ > e S 0,Λ 0 (ψ >) allows contractions. Z Λ (ψ < ) is again evaluated perturbatively by expanding the exponential of the interaction to some order n. In Fig. 3.3 we give all contributions up to second order in the interaction V Λ0 with two and four external slow fields ψ < (i.e. dropping the ψ < -independent terms). Except the constraint on the k-values of the internal lines, all the usual diagram rules for fermions (signs, spin sums) apply. We see that we get connected and disconnected diagrams. The latter cancel in taking the logarithm. This can be seen to any low fixed order from log(1 + x) = x 1 2 x x (3.45) Here we collect all terms of order n 1 into x. The cancellation is seen as follows. For example consider the n = 2 term with the squared vacuum diagram contributing to x. From n = 2 it carries a prefactor Now, we compare this with terms we get from subtracting x 2 /2 in the expansion of the logarithm. Here we get the vacuum diagram in one power already at n = 1, and if we square this term and divide by two, we exactly get 1/2 times the square of the vacuum diagram. There are no other contributions of this type to any order. Therefore this vacuum contribution cancels. The last term drawn in Fig. 3.3 (n = 2, four slow legs on one vertex) cancels also against the mixed term of the two last n = 1 contributions in x. Here we get two times the same term from the square, but also in the n = 2 contribution

52 52 CHAPTER 3. WILSON RG FOR FERMIONS Figure 3.3: Diagrams occurring for Z Λ (ψ < ) = 1 + x in the Wilson RG scheme described here. All terms of higher order n 1 are collected in x. one has two copies of them, because there are two possibilities to assign the slow legs to the tow interaction vertices. Therefore we can drop all disconnected diagrams. Among the connected terms we also have one-particle reducible terms. These also cancel for our momentum-shell RG, as due to momentum conservation the momentum of the internal fast line needs to be the same as that of the external slow line. This constraint cannot be fulfilled for all external legs at scales < Λ. Next we argue that one can restrict the analysis to one-loop order, at least in D > 1. The goal is to derive RG differential equations for the parameters of the effective action. This means that the thickness of the shell being integrated out in one step tends to zero, Λ = Λ 0 Λ dλ, and becomes a small parameter, that in the given context is also more useful than looking at the power of coupling constants. The main question is the dependence of the different diagrams on the shell-thickness dλ. A simple tadpole with the contraction of one internal line like for the first-order self-energy correction definitely scales dλ. The one-loop diagrams for the change of the interaction vertex superficially look like dλ 2, but we will see below, that for special wavevector combinations they can actually behave like dλ. So we will not drop them here, and discuss later some more restrictions. If we go to higher orders, we have products of one-loop diagrams and two-loop and higher-loop diagrams. These will also be of order dλ 2 and higher and will hence be negligible. This means that we can restrict the study to one-loop diagrams shown in Fig. 3.5.

53 3.2. FERMIONIC VERSION One-loop flow Here we want to study the one-loop corrections to the effective interactions, V Λ (k 1, k 2, k 3 ), in more detail. From the diagrams we observe that there are three or five different contributions, depending on how we represent the interaction (see Figs. 3.4 and 3.5). ). Note that in principle there is another tree+oneloop diagram in second order that formally contributes to the the flow of the interaction, coming from the contraction of an interaction with three slow fields and one fast leg with an interaction with three fast fields (two of them folded together) and one slow field. Momentum conservation in this case however means that the internal fast propagator connecting the two interactions needs to have the same energy than one slow external leg. Therefore, if this external leg is set to the Fermi level, the contribution of this diagram type is zero. Let us go back to the one-loop contributions with two internal lines. If we do not resolve the spin structure, and represent the interaction as circle with two incoming (k 1, k 2 ) and two outgoing (k 3, k 4 ) lines, there are three possibilities for grouping k 1 to k 4 in pairs. If k 1 and k 2 are on one, say left, side of the one-loop diagram, the two internal lines go both from the let to the right interaction. This will be called a particle-particle (PP) diagram, for reasons that will become clear soon. If k 1 and k 3 appear on the same side, the internal lines will be in opposite directions. This type of diagram is a particle-hole (PH) bubble. Another PH term arises if k 1 and k 4 are on the same side. Sometimes these two PH-diagrams are called zero-sound and zero-sound channel (e.g. see the review by Shankar), as they can summed up to give collective modes like the zero sound mode in Fermi liquid theory for small momentum transfer (see Fig. 3.4). Figure 3.4: Three one-loop diagrams for the interaction without resolving the spin structure. The internal lines need to be in the Λ-shell being integrated out. From Shankar s review, Rev. Mod. Phys. 66 (1994). If we resolve the spin structure of the interaction by using the rectangle with short and long sides, and study the case s s (which is sufficient to get the full interaction), we get again a PP diagram with k 1 and k 2 on one side, four diagrams in the direct PH channel with k 1 and k 3 on one side, and one crossed PH channel with k 1 and k 4 on one side (see Fig. 3.5). With this, we can let dλ go to zero and obtain the differential RG equations.

54 54 CHAPTER 3. WILSON RG FOR FERMIONS a) s s c) s' s' b) Figure 3.5: a) Interaction function V Λ (k 1, k 2, k 3 ) resolving the spin structure. b) One-loop diagrams for the selfenergy as in Eq c) One-loop diagrams for V Λ as in Eq The internal lines need to be in the Λ-shell being integrated out. For the self-energy, the equation in one-loop becomes d dλ Σ Λ(k) = dk [ 2V Λ (k, k, k) V (k, k, k ) ] S Λ (k ). (3.46) The equation for V Λ (k 1, k 2, k 3 ) with k 4 = k 1 + k 2 k 3 reads[?] d dλ V Λ(k 1, k 2, k 3 ) = T P P,Λ + T d P H,Λ + T cr P H,Λ (3.47) with the one-loop particle-particle contributions T P P,Λ and the two different particle-hole channels TP d H,Λ and T cr P H,Λ where T P P,Λ (k 1, k 2 ; k 3, k 4 ) = dk V Λ (k 1, k 2, k) L Λ (k, k + k 1 + k 2 ) V Λ (k, k + k 1 + k 2, k 3 ) (3.48) T d P H,Λ(k 1, k 2 ; k 3, k 4 ) = [ dk 2V Λ (k 1, k, k 3 ) L Λ (k, k + k 1 k 3 ) V Λ (k + k 1 k 3, k 2, k) +V Λ (k 1, k, k + k 1 k 3 ) L Λ (k, k + k 1 k 3 ) V Λ (k + k 1 k 3, k 2, k) ] +V Λ (k 1, k, k 3 ) L Λ (k, k + k 1 k 3 ) V Λ (k 2, k + k 1 k 3, k) (3.49) T cr P H,Λ(k 1, k 2 ; k 3, k 4 ) = dk V Λ (k 1, k + k 2 k 3, k) L Λ (k, k + k 2 k 3 ) V Λ (k, k 2, k 3 ) (3.50) In these equations, the product of the two internal lines in the one-loop diagrams is L Λ (k, k ) = S Λ (k)s Λ (k ) + S Λ (k)s Λ (k ) with the on-shell propagator { G0 (k) for ɛ(k) Λ dλ S Λ (k) =. (3.51) 0 else In general these one-loop bubbles depend on the values of the incoming and outgoing momenta. This means that even if the bare or initial interaction

55 3.2. FERMIONIC VERSION 55 V Λ0 is a wavevector- and frequency independent constant, the effective interaction will in general depend on wavevectors and frequencies. Therefore, we can no longer work with a single renormalized coupling constant, but we need to resolve the k-structure of the effective interaction. This automatically leads to functional differential equations, for flowing functions of several wavevectors and frequencies Simple expressions for one-loop bubbles Before continuing, let us quickly estimate the derivatives one-loop particleparticle and particle-hole bubbles in more detail. We already focus on the one-dimensional case with wavevector k in a certain range around the Fermi points of the dispersion. We will neglect self-energy terms for simplicity. The particle-hole diagram for incoming total frequency iν (bosonic) and total wavevector Q is given by L PP Λ (iν, Q) = T G Λ (iω, k)g Λ ( iω + iν, k + Q) (3.52) iω n k& k+ q dλ shell We do not write the vertices at the corners, as they will be treated as constant over the range of the internal loop integrations. The propagators are G Λ (iω, k) = 1 iω ɛ(k). (3.53) After performing the Matsubara summation the scale-derivative of the particleparticle bubble is given by L PP Λ (iν, Q) = k& k+ q dλ shell 1 n F [ɛ(k)] n F [ɛ( k + Q)] iν + ɛ(k) + ɛ( k + Q). (3.54) Similarly we can derive for the particle-hole contributions L PH Λ (iν, Q) = T G Λ (iω, k)g Λ (iω + iν, k + Q) (3.55) iω n k& k+ q dλ shell after performing the Matsubara summation L PH Λ (iν, Q) = k& k+ q dλ shell n F [ɛ(k)] n F [ɛ(k + Q)] iν + ɛ(k) ɛ(k + Q). (3.56) The restriction that both internal lines k and k + q have to lie in the shell to be integrated out is a severe one. Let us look at Fid. 3.6 for the example of the particle-hole bubble. We see that the space in k-space for both k and k + k + q being in the thin shells of width dλ scales like dλ 2, unless q = 0. A very similar same argument holds for the particle-particle bubble. This means that if we want to go over to a differential RG equation, only the PH bubbles at vanishing momentum transfer and the PP bubbles at total wavevector equal to zero give

56 56 CHAPTER 3. WILSON RG FOR FERMIONS Figure 3.6: Phase space in the particle-hole bubbles. The Fermi sphere of second loop particle is displaced by the momentum transfer Q. The overlap regions I, II, II and IV for both k and k + k + q being in the thin shells with width dλ scales like dλ 2. From Shankar, Rev. Mod. Phys. 66, (1994). non-vanishing contributions. This strongly reduces the right hand sides of the flow equations. Let therefore us now concentrate on vanishing external frequency, iν = 0, and for the particle-particle-diagram Q = 0. Then we get, using inversion symmetry ɛ(k) = ɛ( k), L PP Λ (0, 0) = = 1 2n F [ɛ(k)] 2ɛ(k) k dλ shell [ ρ(λ) 1 2n F [Λ] + ρ( Λ) 1 2n ] F [ Λ] dλ 2Λ 2Λ 1 2n F [Λ] ρ 0 dλ (3.57) Λ In the last step we have used ρ 0 as the density of states at the Fermi level, and we have assumed that the variation of ρ(ɛ) can be neglected in the respective energy window. The 1/Λ-dependence tells us that this contribution will become important for Λ 0. Now let us ask if we can get similarly divergent terms in the particle-hole channel. In order to get a analogous expression we now need to have ɛ(k +Q) = ɛ(k) for a dense set of k values, for k towards the Fermi level. This is easily found in one dimension with a linear dispersion near the Fermi level and Q = ±2k F. To be more precise let us choose (see also Fig. 3.7) ɛ(k) = v F (k k F ), k > 0 and ɛ(k) = v F (k k F ), k < 0 (3.58) with Fermi velocity v F and Fermi wavevector k F. If we now shift a k < 0 near the left Fermi point k F by Q = 2k F we end up near the right Fermi point

57 3.2. FERMIONIC VERSION 57 Figure 3.7: Dispersion of the one-dimensional model, with linearized dispersion and a left branch around k F and right branch branch around +k F. near +k F. Moreover, if ɛ(k) < 0 we find ɛ(k + 2k F ) = ɛ(k) > 0 and vice versa. This means that we get for the particle-hole diagram at zero transferred frequency and momentum transfer 2k F in this particular 1D situation L PH Λ (0, 2k F ) = ρl 0 dλ. (3.59) Λ Hence this particle-hole diagram has roughly the opposite value as the particleparticle diagram but also diverges at low T and Λ. A slight subtlety is the restriction of the k-integration on the left branch around k F, as shifting the right branch by 2k F takes us usually far away from the Fermi level (except when 2k F = 2k F modulo a reciprocal lattice vector, this is the case at halffilling, when also Umklapp scattering has to be considered). Therefore we have introduced the notation for the density of states on one branch only, ρ l. For the particle-particle diagram there was no such restriction, and the density of states in front of the logarithm was that of both dispersion branches. The behavior of a divergent particle-hole bubble is now a specialty of one dimension, it higher dimensions a particular nested bandstructure or singularities in the density of states are needed to obtain a divergence of the particle-hole term. However, as sen in the above phase space argument, we still need to consider the q 0 case, as this also has phase space dλ. The particle-hole bubble with q 0 is somewhat special. Here we have (for finite dλ) L PH Λ (iν, q 0)dΛ = 1 V k dλ-shell n F [ɛ( k + q)] n F [ɛ( k)] iν + ɛ( k + q) ɛ( k). (3.60) For iν 0 the limit q 0 gives zero, as the numerator cancels. For iν = 0 we get L PH Λ (iν = 0, q 0)dΛ = 1 n F [ɛ( k)] V ɛ(. (3.61) k) k dλ-shell

58 58 CHAPTER 3. WILSON RG FOR FERMIONS Now, the ɛ-derivative is basically a temperature-smeared δ-function on the Fermi surface. If Λ T, we do not anything. Only if Λ gets of the order of T, there is a contribution to the flow. In any case the result for q 0 will not be of order 1/Λ, but of order Λ 0. The contribution is of order dλ, and will also remain in the differential RG equation Flow in the Cooper channel In simple cases it is straightforward to see that the integration of the Wilson RG equations in the present truncation contains the basic ladder and bubble summations known from perturbation theory. Here will treat as an example the particle-particle channel, where the RG recovers the Cooper instability. Superconductivity occurs in man-fermion systems because an attractive interaction, e.g. mediated by phonons, gets amplified by multiple scattering event in presence of a Fermi surface, most strongly for particle with opposite momenta. This leads to a divergence of the two-particle scattering at a critical temperature T c, the so-called Cooper instability, and and temperatures below the many-particle state becomes a superposition of Cooper pairs bound together by this attractive interaction, exhibiting superconductivity and the Meissner effect. Diagrammatically, the Cooper instability is obtained by summing up all particle-particle diagrams in a ladder summation, as seen in the previous chapter. One simple question is of course how the frg reproduces this instability. The connection to such ladder summations in the particle-particle channel can be seen if we drop all particle-hole terms on the right hand side of the flow equation for the interaction. Then the remaining equation has the structure PP V Λ (k 1, k 2, k 3 ) = V Λ (k 1, k 2, k) L Λ (k, k + k 1 + k 2 )V Λ (k, k + k 1 + k 2, k 3 ). Again, the sum over k is not written out. For simplicity, let us assume that the initial condition V Λ0 (k 1, k 2, k 3 ) does not depend on the wavevectors or frequencies, and let us focus on the channel k 1 + k 2 = 0, i.e. zero total wavevector and zero total momentum. Near the Fermi level and at T = 0, L PP (k, k) typically goes like 1/Λ, with T k Λ L PP Λ (k, k) ρ 0 /Λ, (3.62) as the particle-particle bubble at k 1 + k 2 = 0 diverges logarithmically in the infrared cutoff. These simplifications lead to the equation V Λ = ρ 0 Λ V 2 Λ V Λ = V Λ0 1 + ρ 0 V Λ0 log Λ 0 Λ (3.63) Let us first assume that V Λ0 is repulsive. We see that then the effective interaction V Λ gets reduced by the denominator > 1, i.e. repulsive interactions get screened in the particle-particle channel. This happens quite generally for all wavevector combinations, as L PP is always positive, but it is most pronounced for the case of zero incoming total wavevector we are concentrating on. In fact, due to the logarithm in the above formula, the initially repulsive interaction

59 3.2. FERMIONIC VERSION 59 disappears for Λ 0. Note that this latter drastic effect is only true in this simplified scheme where many terms have been dropped. If the initial vertex V Λ0 is attractive, this equation has a pole at Λ c = Λ 0 exp [ 1/ ρ 0 V Λ0 )]. (3.64) A comparison shows that with this scale the RG equation exactly reproduces the well-known Cooper instability scale. One has to replace the Debye frequency ω D with the initial scale Λ 0, and the exponent is exactly the dimensionless coupling constant. Like this example for the particle-particle Cooper channel, all other particleparticle and particle-hole instabilities obtained in ladder or bubble summations can be recovered in suitable simplifications of the full one-loop flow. Some more attention has to be paid for the case of particle-hole instabilities at small wavevector transfer, like ferromagnetic Stoner instabilities. As mentioned above, the deficit by construction in these cases can however be remedied by other flow parameters such as the temperature, interaction strength or a smooth frequency cutoff. More generally we can expand the pair scattering in the zero-total-momentum channel as (ignoring a possible frequency dependence) V Λ ( k, k k, k ) = l,l V l,l f l ( k)f l ( k ) (3.65) with a complete orthogonal basis set f l ( k) counted through with the index l. The V l are then given by the projections V l,l = k, k f l ( k)f l ( k )V Λ ( k, k k, k ) (3.66) If the basis functions to l and l have different transformation properties with respect to the underlying point group, the V ll are zero. A simple setting is obtained if we consider a rotationally invariant situation and focus onthe vicinity of the Fermi surface. Then the pair scattering should only depend on the angle θ k, k between the incoming and outgoing wavevectors, and we can use an expansion in terms of Legendre polynomials, V (θ) = P l (cos θ)v l with V l = 1 2 l=0 1 1 d cos θ V (θ). (3.67) This leads to the flow equation in the zero-total-momentum Cooper channel d dλ V l = where we have used the addition theorem P l (cos θ k, k ) = 4π 2l + 1 ρ 0 2π 2 (2l + 1) V 2 l, (3.68) l m= l Y lm (Ω k )Y lm (Ω k )

60 60 CHAPTER 3. WILSON RG FOR FERMIONS and the orthogonality of the spherical harmonics. Hence the different l decouple and run with individual flow equations. If only one of the V is negative, the flow will diverge as some exponentially small scale. We can ask where a negative V l could come from. A bare or screened Coulomb interaction does only have a positive V 0. Here one should however keep in mind that we have so far neglected the particle-hole channel, and indeed these usually create some wavevector structure in the effective pair scattering that will yield a negative V l for some l. This is the content of the so called Kohn-Luttinger-Theorem (1965), which states that the generic ground state of an interacting Fermi system is Cooper-paired. Kohn and Luttinger studied second-order particle hole corrections to the pair scattering and found that one necessarily gets a negative V l at some, possibly high l, leading to a possibly very low and hence likely unpbservable T c. We will however see that for lattice electrons the structures caused by the particle-hole channel do not need to by small, and that one can get substantial T c even for repulsive bare interactions. On the lattice, the matrix elements V ll within the same irreducible representation of the point group do not have to vanish, and the whole analysis gets more involved. One can however see in direct examples that this Kohn- Luttinger mechanism still works qualitatively unchanged, basically because it is very hard to change the positiveness of the right hand side of the flow equation, and all components get pressed down in the flow. The only difference is that form factor of the pairing can change during the flow g-ology model in one dimension We just saw that one spatial dimension, both Λ (0, 0) and L PH Λ (0, 2k F ) diverge at low Λ. Hence, in order to understand the RG flow of the interactions qualitatively, it is appropriate to focus on these two diagrams and to neglect all other terms. Now let us also simplify the coupling function by defining the following four components. g Λ 1 = V Λ(k F, k F, k F ) = V Λ ( k F, k F, k F ) g Λ 2 = V Λ(k F, k F, k F ) = V Λ ( k F, k F, k F ) g Λ 3 = V Λ(k F, k F, k F ) = V Λ ( k F, k F, k F ) g Λ 4 = V Λ(k F, k F, k F ) = V Λ ( k F, k F, k F ) L PP Then we can also approximate all coupling functions with ks in the neighborhood of ±k F by the gs for k at ±k F. This modeling with branch-dependent interactions is usually referred to as g-ology. For a band filling away from half band filling, 4k F is not a reciprocal lattice vector. Then the g 3 process with a total wavevector change of 4k F is not momentum-conserving and has to be dropped. Furthermore, it turns out that when we compute the particle-particle corrections to g4 Λ, the two dangerous diagrams L PP Λ (0, 0) and L PH Λ (0, 2k F ) do not enter. Hence g4 Λ will not flow too strongly at low scales, and we can ignore it for the time being. It remains to derive the flow equations for g1 Λ and

61 3.2. FERMIONIC VERSION 61 g2 Λ. They read (again ignoring all other one-loop terms besides L PP Λ (0, 0) and L PH Λ (0, 2k F )) [ ] ġ1 Λ = 2g1 Λ g2 Λ LPP,br. PH Λ (0, 0) + L Λ (0, 2k F ) 2g1 Λ 2 L PH Λ (0, 2k F ) (3.69) [ ] ġ2 Λ = 2g2 Λ 2 LPP,br. PH Λ (0, 0) + L Λ (0, 2k F ) + g1 Λ 2 L PP,br. Λ (0, 0) (3.70) L PP,br. Here, Λ (0, 0) denotes the particle-particle bubble with internal k-sum over one dispersion branch only (the other branch also occurs in the above equations, but produces a different combination of coupling constants). Now we can use L PP,br. Λ (0, 0) = L PH Λ (0, 2k F ) > 0 which holds for the linearized dispersion of the last subsection. The density of states per branch is ρ(0) br. = 1/(2πv F ). This gives ġ Λ 1 = ġ Λ 2 = 1 g1 Λ 2 πv F (3.71) 1 g1 Λ 2. 2πv F (3.72) For repulsive initial conditions, g1 Λ will now flow to zero for Λ 0, as one can see from the straightforward integration of the differential equation, g Λ 1 = g Λ g Λ 0 1 log(λ 0 /Λ)/(πv F ). (3.73) Then g2 Λ will basically stop to flow at low scales, and reach a fixed point value > 0. Actually it is easy to see that g 2 d ( g Λ dλ 2 g1 Λ /2 ) = 0 = g2 Λ g1 Λ /2 = g Λ 0 2 g Λ 0 1 /2, (3.74) and, as g2 Λ g 2 we get g 2 = gλ 2 gλ 1 /2. In this way we see that the balance between different one-loop diagrams leads to a finite flow of the interactions at low energy scales, very different from the higher-dimensional case where usually a Cooper instability occurs. Therefore, in one dimension the many-fermion system remains gapless, as no interactions diverge and no gap in the fermionic excitation spectrum is opened as a consequence. An analysis of the self-energy however shows that there are divergences of the frequency and momentum derivatives, and hence the Fermi liquid from higher dimensions is replaced by a Luttinger liquid without well-defined quasiparticles. Also, in the renormalization of the susceptibilities toward charge- and spin-density wave ordering, the log-divergent one-loop terms do not cancel, and hence these response functions diverge as power-laws at low temperatures or scales. In the case of repulsive interactions and half band filling, the Umklapp scattering g 3 enters and causes at flow to strong coupling. The resulting state has a gap in the single-particle excitation spectrum and can be called a Mott insulator, as no long-range ordering is accompanied by this. For attractive g 1 the flow leads again to strong coupling. In the one-dimensional case the divergence

62 62 CHAPTER 3. WILSON RG FOR FERMIONS Figure 3.8: Definition of the coupling constants g 1, g 2 and g 4. of zero-total-wavevector couplings is not directly followed by superconductivity, but the resulting Luther-Emery-liquid still has a spin gap similar to a singlet superconductor /N-picture We already argued that in order to have a particle-hole-loop contributing of order 1/Λ, one needs a special Fermi surface. This is because in the loop integrals it is in general difficult to have both internal lines at the same absolute value of the band energy, hence there will only be a small segment in wavevector space contributing to the summation. Actually, by dividing up the low-energy shell with band energies ɛ Λ around the Fermi surface one obtains a useful small parameter which allows one to distinguish the different diagrams in the low-scale limit. The division is shown in Fig We have divided up the ring of width 2 k with k = Λ/v F into N angular segments. We keep N = 2πk F / k = 2πv F k F /Λ. Let us denote the Fermi wavevectors in the centers of these segments by kf n. In the integration measure for the wavevector summations, we should replace the angular integral by a summation over the N sectors, times the sector width k 1/N Λ. The division into segments has the following effect. Let us take the wavevector difference of two such wavevectors q = kf n kf n, which can be considered as a usual wavevector transfer through a particle-hole bubble. Now in the internal summation, it will be added to the loop wavevector kf n in the red shells to be integrated out. It is now not difficult to see that, unless q Λ/v F, only O(1) sectors will give a non-vanishing contributions, for kf n in most of the O(N) sectors, kf n + q will lie outside the red integration regions. This means that the total contributions will be of order Λ/Λ = O(1), where the two positive power comes from the integration measure, and the negative power from the energy denominator. This again tells us that in the limit Λ small we can drop all particle-hole bubbles, except the one for q 0 even for finite dλ, without have to use the dλ 2 -argument. Two-loop terms and products of one-loop terms can also be found irrelevant by the same argument. The particle-hole bubble with q 0 is has been considered above. For

63 3.2. FERMIONIC VERSION 63 iν = 0 we get L PH Λ (iν = 0, q 0)dΛ = 1 V n F [ɛ( k)] ɛ(. (3.75) k) k dλ-shell Now, the ɛ-derivative is basically a temperature-smeared δ-function on the Fermi surface. If Λ T, we do not get anything. Only if Λ gets of the order of T, there is a contribution to the flow. In any case the result for q 0 will not be of order 1/Λ, but of order Λ 0, but the full angle O(N) around the Fermi surface will contribute. The contribution is of order dλ, and will also remain in the differential RG equation. Consider now the particle-particle channel with zero total momentum. Here all O(N) internal pairs of sectors with angle difference of π will contribute to the summation, and we will have a Λ in the denominator. This gives a contribution of order ΛNΛ O(1/Λ), which gets infinitely larger than the contribution from the particle-hole channel in the limit Λ 0. The argumentation here was performed for D = 2, but it also holds in other dimensions D > 2. The PH bubble always comes out one order higher in Λ than the PP bubble with zero total momentum. We can also analyze the one-loop self-energy diagrams of Hartree- and Focktype. They come out as order Λ 0, as after Matsubara summation one has n F ( k) in the wavevector sum, and all sectors contribute. As far as the one-loop bubbles in the Wilson scheme are concerned that means that we should drop almost all bubbles, expect the PP bubble with total momentum zero, and the PH bubbles for zero wavevector transfer. This means that the only interaction terms that flow are the Cooper processes with total momentum zero, and, very late in the flow for Λ T, the forward scattering processes (at least in the limit discussed). We would obtain the same conclusion by just keeping only the terms dλ in the differential RG equation. In fact one can also apply the N-sector-division scheme to the dλ-shells, with N 1/dΛ, in order to arrive at the same conclusions for any Λ. We note that this heavy reduction to just two channels appears to be a somewhat drastic conclusion that sheds some doubt of the generality of this scheme, and we will later argue that his way we miss some possibly important contributions Connection to Fermi liquid theory As already mentioned, the effective low-energy couplings can be divided into forward (+exchange forward) scatterings V F with zero momentum transfer and Cooper scatterings V C with zero total momentum. These are exactly the scattering processes that have a non-vanishing flow. In general, these couplings will mix on the right hand side of the flow equations. Now the sector scheme can be used to understand that the mixing disappears in the low-scale limit. For this purpose we define forward scattering in the sense that one outgoing sector and one incoming must agree. Cooper scattering then needs to have opposite sectors for the incoming lines, and also for the outgoing ones. If we insert a forward scattering process into the renormalization of

64 64 CHAPTER 3. WILSON RG FOR FERMIONS Figure 3.9: Division of the low-energy-shell into N 1/Λ segments. the Cooper scattering, the internal loop summation is killed by the constraint that one internal sector must equal an external sector for the forward scattering. Hence, even if we consider the PP diagram, we get a contribution O(1). This means that the impact of the forward scattering on the Cooper channel is sub-leading. Vice versa, the Cooper processes inserted into the PH channel are also subleading, as the internal sum is restricted by the requirement of zero total incoming momentum. In this sense, one gets, in the low-scale regime for small Λ, the schematic flow equations with the leading terms For the self-energy we get d dλ V c = 1 Λ O[V C] 2, (3.76) d dλ V F = O [ (VF 2 ) ]. (3.77) d dλ Σ = O [V F ]. (3.78) As we already discussed, the Cooper channel can give rise to instabilities at low scales. If we assume that we are above this scale, we can ignore the Cooper couplings. Then we are left with the forward interactions of Fermi liquid theory. One way to put the picture together is now to stop the flow at a nonzero Λ and use V F (without having picked up the low-scale flow at Λ T, corresponding to the limit iν 0 after q 0) and Σ as input for the Landau functional for