Lecture notes on many-body theory

Transcription

1 Lecture notes on many-body theory Michele Fabrizio updated to 203, but still work in progress!! All comments are most welcome!

2 Contents Landau-Fermi-liquid theory: Phenomenology 5. The Landau energy functional and the concept of quasiparticles Quasiparticle thermodynamics Specific heat Compressibility and magnetic susceptibility Quasiparticle transport equation Quasiparticle current density Collective excitations in the collisionless regime Zero sound Collective excitations in the collision regime: first sound Linear response functions Formal solution Low frequency limit of the response functions High frequency limit of the response functions Charged Fermi-liquids: the Landau-Silin theory Formal solution of the transport equation Dirty quasiparticle gas Conductivity Diffusive behavior Second Quantization Fock states and space Fermionic operators Second quantization of multifermion-operators Fermi fields Bosonic operators Bose fields and multiparticle operators Canonical transformations More general canonical transformations

3 2.5 Examples and Exercises Application: fermionic lattice models and the emergence of magnetism Hubbard models The Mott insulator within the Hubbard model Spin wave theory in the Heisenberg model More rigorous derivation: the Holstein-Primakoff transformation Linear Response Theory Linear Response Functions Kramers-Kronig relations Symmetries Fluctuation-Dissipation Theorem Spectral Representation Power dissipation Absorption/Emission Processes Thermodynamic Susceptibilities Hartree-Fock Approximation Hartree-Fock Approximation for Fermions at Zero Temperature Alternative approach Hartree-Fock approximation for fermions at finite temperature Preliminaries Variational approach at T Mean-Field approximation for bosons and superfluidity Superfluid properties of the gauge symmetry breaking wavefunction Time-dependent Hartree-Fock approximation for fermions Bosonic representation of the low-energy excitations Application: antiferromagnetism in the half-filled Hubbard model Spin-wave spectrum by time dependent Hartree-Fock Linear response of an electron gas Response to an external charge Response to a transverse field Limiting values of the response functions Power dissipated by the electromagnetic field Reflectivity Random Phase Approximation for the electron gas Feynman diagram technique Preliminaries Imaginary-time ordered products

4 5..2 Matsubara frequencies Single-particle Green s functions Perturbation expansion in imaginary time Wick s theorem Perturbation theory for the single-particle Green s function and Feynman diagrams Diagram technique in momentum and frequency space The Dyson equation Other kinds of perturbations Scalar potential Coupling to bosonic modes Two-particle Green s functions and correlation functions Diagrammatic representation of the two-particle Green s function Correlation functions Coulomb interaction and proper and improper response functions Screened interaction and corresponding Dyson equation Irreducible vertices and the Bethe-Salpeter equations Bethe-Salpeter equation for the vertex functions The Ward identities Consistent approximation schemes Example: the Hartree-Fock approximation Some additional properties and useful results The occupation number and the Luttinger-Ward functional The thermodynamic potential The Luttinger theorem Landau-Fermi liquid theory: a microscopic justification Preliminaries Vertex and Ward identities Correlation functions Conserved quantities Coulomb interaction Kondo effect and the physics of the Anderson impurity model Brief introduction to scattering theory General analysis of the phase-shifts The Anderson Impurity Model Variation of the electron number Energy variation Mean-field analysis of the interaction From the Anderson to the Kondo model

5 7.3. The emergence of logarithmic singularities and the Kondo temperature Anderson s scaling theory Introduction to abelian Bosonization Interacting spinless fermions Spin wave theory Construction of the effective Hamiltonian Interactions Umklapp processes Bosonization of the Heisenberg model The Hubbard model

6 Chapter Landau-Fermi-liquid theory: Phenomenology One of the first thing that one learns in a Solid State Physics course is that the thermodynamic and transport properties of metals are well described in terms of non-interacting electrons; the Drude-Sommerfeld-Boltzmann theory of metals. Yet, this evidence is somehow surprising in view of the fact that actual electrons interact mutually via Coulomb repulsion, which is not weak at all. This puzzle was solved brilliantly in the end of the 50 s by Landau, as we are going to discuss in what follows. We will start by analyzing the case of a neutral Fermi system, like 3 He. Later we shall discuss charged Fermi systems, relevant to metals.. The Landau energy functional and the concept of quasiparticles Let us start from a non-interacting Fermi gas at very low temperature T. Right at T = 0 we know that the ground state is the Fermi sea that is obtained by filling with two opposite-spin electrons all momentum states with energy smaller than the chemical potential µ the Fermi energy. In other words, if n kσ is the occupation number of each momentum state n kσ being zero or one because of Pauli principle then the Fermi-sea occupation numbers are n 0 kσ = { if ɛ 0 k µ 0 if ɛ 0 k > µ. At finite temperature, the equilibrium values of the occupation numbers are given by the Fermi- Dirac distribution n 0 kσ = f ɛ 0 k µ = + e β ɛ0 µ k, 5

7 with β = /K B T the inverse temperature. Any excited state can be uniquely identified by the variation of the occupation numbers with respect to equilibrium, namely through and costs an energy δn kσ = n kσ n 0 kσ,. δe 0 [{δn pα }] = kσ ɛ 0 k µ δn kσ,.2 that is a simple functional of the δn kσ s. Now suppose that we switch on smoothly the interaction. Each non-interacting excited state will evolve smoothly into a fully-interacting one. The Landau hypothesis was that the non-interacting and the fully interacting states are adiabatically connected; in other words they are in one-to-one correspondence. This implies in particular that each fully-interacting excited state can be uniquely identified, like its non-interacting partner, by the deviation with respect to equilibrium of the occupation numbers δn kσ. Consequently, its excitation energy δe must be a functional of the δn kσ s, i.e. δe [{δn pα }]..3 Remark This hypothesis may look simple but it is actually deeply counter-intuitive. Suppose we have a non-interacting Hamiltonian H 0 and a fully-interacting one H = H 0 +H, and suppose to follow the spectrum, that we assume discrete, within a specific symmetry-subspace of the Hamiltonian Hλ = H 0 + λ H, with λ that increases smoothly from 0 to. The evolution of the spectrum must resemble that one drawn schematically in Fig.. with a series of avoided crossings that allow to follow adiabatically the n-th excited level at λ = 0 into the n-th excited level at λ =. Note that the crossing are avoided because the states have the same symmetry. If we were to consider levels of different symmetries nothing would prevent crossings. However, while the adiabatic evolution does occur and it is a well accepted phenomenon in models with discrete spectra, it is equally common wisdom that it can not happen when the spectrum is continuous, as in a bulk metal. Landau s revolutionary hypothesis that the adiabatic evolution also occurs in a bulk system, at least for the low energy excited states, and the fact that, by this simple assumption, one can justify and predict a lot of physical properties is one of the major achievements of the contemporary Condensed Matter Theory. Let us therefore assume that the excitation energy of the fully-interacting Fermi system is indeed a functional of the same δn kσ s as its non-interacting partner, and further suppose 6

8 E 0 λ Figure.: Adiabatic evolution of a discrete spectrum within a specific symmetry-subspace. that the temperature is very small. In this case, any weak deviation from equilibrium must correspond to δn kσ, that justifies a Taylor expansion of.3, namely δe [{δn pα }] = kσ ɛ kσ µ δn kσ + 2 This is the famous Landau s energy functional. Note that kk σσ f kσ k σ δn kσ δn k σ + O δn 3..4 even though the two terms are apparently of different orders, in reality they are of the same order since the excitation energies ɛ kσ µ are of the same order as the deviations of the occupation numbers; ɛ kσ should not be confused with the non-interacting energies ɛ 0 kσ. Rigorously speaking, the δn kσ s that appear in.4 only serve as labels to identify excited states and they do not correspond to deviations of the occupation numbers of the real particles, as in the non-interacting case.2. Just for this reason, Landau coined for δn kσ the definition quasiparticle occupation number, as if the real-particle excitations were substituted in the presence of interaction by quasiparticle excitations. The Fermi gas of these quasiparticles is the so-called Landau-Fermi liquid. Remark This idea is actually ubiquitous in all Solid State Physics. Let us for instance recall briefly how phonons arise. One starts from a model of interacting ions and electrons. Because of their larger mass with respect to the electrons and of the Coulomb interaction among them and with the electrons, the ions localize, thus forming a lattice, and the low energy excitations become the small fluctuations around the equilibrium positions, whose quantization gives rise to phonons, which are bosons. In other words, one begins with real particles, the ions, yet their low-energy 7

9 excitations in the presence of interaction are new bosonic quasiparticles, the phonons, that turns out to be generally weakly interacting among each other and with the electrons. This allows to treat such an interaction within perturbation theory, unlike the original Coulomb interaction. In a more general context, since the only many-body systems that can be solved exactly in any dimensions are free bosons or fermions, it is a very natural, and fortunately successful, approach to attempt a description of the low-energy excitations of, even strongly, interacting models in terms of effective quasiparticles, bosons or fermions, whose interaction is weak enough to be treated perturbatively. Going back to the Landau energy functional.4, in the case in which spin isotropy is preserved the following equivalences hold: f k k = f k k, f k k = f k k. In this case, the choice of a spin quantization axis must not be influential, hence it is convenient to rewrite.4 in a form that is explicitly spin isotropic. For that purpose, let us introduce charge deviations from equilibrium and spin ones where the z-component is simply δρ k = δn k + δn k,.5 δσ k = δσ x k, δσy k, δσz k,.6 δσ z k = δn k δn k. With these definitions the second term of the energy functional.4 can be written as fkk S 2 δρ k δρ k + fkk A δσ k δσ k,.7 kk where f S = 2 f + f, f A = 2 f f,.8 as can be readily demonstrated by equating.4 and.7 assuming a deviation from equilibrium of the form δσ k = 0, 0, δσ z k. 8

10 .2 Quasiparticle thermodynamics Let us derive, starting from the Landau s energy functional, the thermodynamical properties of the quasiparticles. We note that, if we consider a deviation from equilibrium that consists, in the absence of interaction, to add δn particles in particular momentum-states, i.e. an excited state identified by δn kσ such that δn kσ = δ N,.9 kσ such a state must evolve in the Landau s hypothesis into a state with the same quasiparticle excitations δn kσ and furthermore, since the total number of particles is conserved along the adiabatic evolution, with the same deviation δn from equilibrium. In other words, although quasiparticles have not to be confused with real particles, still the sum of all quasiparticle deviations of the occupation numbers gives the variation of the total number of real particles, i.e. Eq..9. Remark It is important to note that the adiabatic evolution hypothesis guarantees that only true conserved quantities, like the total number of particles or the total spin, keep the same expressions in terms of quasiparticle occupation numbers as for the real particles. Conserved quantities refer, rigorously speaking, only to the fully interacting Hamiltonian, whereas the non-interacting one has generally much larger symmetry. In other words, the Landau s theory of Fermi liquid has only to do with conserved quantities but its use is not at all justified for non-conserved ones, even if they are conserved in the non-interacting limit. This fact is often underestimated and may lead easily to wrong conclusions. Since the labeling of the states is identical to non-interacting electrons, the phase-space volume counting is the same, hence also the formal expression of the entropy-change δs: [ ] δs = δ K B n kσ ln n kσ + n kσ ln n kσ..0 kσ Thermodynamic equilibrium implies an extremum of the free-energy F, namely that δf = δe T δs µ δn = 0.. Let us solve this equation by assuming that the deviation from equilibrium is induced by a deviation of the quasiparticle occupation numbers δn kσ, that amounts to impose δf = 0. δn kσ δn=0 9

11 By Eq..4 we find that δe δn kσ = ɛ k + k σ f kσ k σ δn k σ ɛ kσ,.2 is the quasiparticle energy in the presence of excited quasiparticles, including itself, while what it is needed is δe = ɛ k, δn kσ δn=0 that is obviously independent of δn. Since all terms in. have exactly the same expression as for non-interacting particles, one readily realizes that the quasiparticle equilibrium distribution is the Fermi-Dirac distribution n 0 kσ = n0 k = f ɛ k µ = + e β ɛ k µ,.3 the only difference with respect to non-interacting real particles being the renormalized band dispersion ɛ k. We must point out that f ɛ k µ corresponds to global equilibrium all quasiparticles are in equilibrium among themselves. Analogously, we could define a distribution function for the local equilibrium that minimizes the free-energy of a single excited quasiparticle in the presence of other excited quasiparticles. It is clear from Eq..2 that the local-equilibrium distribution function is the Fermi-Dirac distribution with argument the quasiparticle energy ɛ kσ, i.e. n 0 kσ = f ɛ kσ µ = + e β ɛ kσ µ, which is a functional of the occupation number deviations. In accordance, we can define a deviation with respect to local equilibrium as as opposed to that one with respect to global equilibrium δ n kσ = n kσ f ɛ kσ µ,.4 δn kσ = n kσ n 0 k = δ n kσ + n 0 kσ n0 k = δ n kσ + n0 k ɛ k ɛ kσ ɛ kσ = δ n kσ + f ɛ k µ ɛ k ɛ kσ ɛ kσ δ n kσ δ ɛ k µ k σ f kσ k σ δn k σ,.5 0

12 where we have assumed to be close to equilibrium and at very low temperature. Now let us consider, as conventionally done, a space-isotropic system, where ɛ k = ɛ k and use the parametrization δn kσ = n0 k ɛ k Y lm Ω k δn lmσ δ ɛ k µ lm lm Y lm Ω k δn lmσ,.6 where Y lm Ω k are spherical harmonics identified by the Euler angles of the unit vector in the direction of k, being its modulus fixed by the δ-function to be right on the Fermi sphere. The δ-function in the right hand side of.6 derives from the fact that both sides of that equation must consistently be of first order in the deviation from equilibrium. In addition, we introduce the Legendre decomposition of f kk assuming that it depends only on the relative angle θ kk between the two momenta, their modula being on the Fermi sphere. Therefore f SA kk = l f SA l P l θ kk,.7 where P l θ kk are Legendre polynomials. In the non-abelian representation in which the energy functional is explicitly spin-isotropic, one readily finds through.5 the following relations between the charge/spin deviations at global and local equilibrium: δρ k = δ ρ k + 2 n0 k ɛ k δσ k = δ σ k + 2 n0 k ɛ k k fkk S δρ k,.8 k fkk A δσ k..9 One realizes that, because of ɛ k = ɛ k and of the identity dωk 4π P l θ kk Y l m Ω k = δ ll 2l + Y l m Ω k, it follows that k k fl S P l θ kk n0 k Y l ɛ m Ω k δρ l m k l fl S P l θ kk δ ɛ k µ Y l m Ω k δρ l m l m l Conventionally, the argument of the Legendre polynomials is indicated as cos θ [ : ]. Here, in order to simplify notations, we have decided to use as argument θ

13 = V 2 N ll m f S l δρ l m dωk 4π P l θ kk Y l m Ω k = V 2 N l m f S l δρ l m 2l + Y l m Ω k, and analogously for δσ k with f S f A, where N = N µ is the quasiparticle density of states N ɛ = V δ ɛ k ɛ, at the chemical potential. Therefore Eqs..8 and.9 have the simple solution δρ lm = δσ lm = where the Landau F -parameters are defined through F SA l It also common to define A-parameters through so that kσ δ ρ lm + F l S,.20 2l + δ σ lm + F l A,,.2 2l + = V N f SA l..22 A SA l = F SA l + F l A,.23 2l + Fl S δρ lm = A S l δ ρ lm,.24 Fl A δσ lm = A A l δ σ lm.,.25 We note that the above formulas are valid only for an isotropic system, like 3 He, or for a metal with an approximately spherical Fermi surface. In general, one must use, instead of spherical harmonics and Legendre polynomials, other basis functions appropriate to the symmetry of the lattice. However, in what follows we just take into account the isotropic case, unless otherwise stated. 2

14 .2. Specific heat Let us start the calculation of the quasiparticle thermodynamic properties at low temperature from the simplest one: the specific heat. It is easy to show through.0 we set K B = as well as = that c V = T V = T V = T S T N,V kσ T V n 0 k T ln n 0 kσ n 0 kσ S T µ,v dɛ N ɛ + µ ɛ 2 fɛ ɛ = T V + O T 2 = π2 3 kσ ɛ k µ 2 n 0 k ɛ k N T..26 This is the same expression as in the absence of interaction with the quasiparticle density of states N instead of the free-particle one N 0 N 0 = N 0 µ = kσ δ ɛ 0 k µ. Therefore, if c 0 V is the specific heat of the free particles, then c V c 0 V.2.2 Compressibility and magnetic susceptibility = N N Suppose that we perturb the system with a static homogeneous field that may modify the chemical potentials, µ σ = µ µ+δµ σ, for spin up and spin down real particles. At equilibrium, the quasiparticle occupation numbers will change accordingly as δn kσ = n0 k ɛ k δɛ kσ δµ σ,.28 where δɛ kσ = k σ f kσ k σ δn k σ, is the change in the quasiparticle energy due to the fact that the external field changes all quasiparticle occupation numbers. It is important to note that δµ σ that appears in.28 coincides with the field that acts on the real particles because the latter couples to conserved quantities, otherwise nothing could guarantee that the two are equal. Once again this demonstrates that this theory only addresses the response of conserved quantities. 3

15 By means of Eq..5 we find that δ n kσ = δn kσ n0 k ɛ k k f kσ k σ δn k σ = n0 δµ σ,.29 ɛ k σ k showing that the deviation from local equilibrium is the same as for free particles with dispersion ɛ k in the presence of the field. Compressibility The compressibility κ is defined by κ = V V P = n n 2 µ, where n is the total density. Since a variation in the total density of particles coincides with the variation in the quasiparticle one, we find through.20 that 2 δn = δn kσ = δρ k V V = kσ 2 4π N δρ 00 = 2 4π N δ ρ 00 + F0 S k = + F0 S The variation δ n with respect to a variation of chemical potential is the same as for free particles with density of states N, namely δ n δµ = N. Therefore we finally obtain that κ = n 2 N + F S 0 = + F0 S δ n. N N 0 κ0,.30 where κ 0 is the compressibility of the original free particles which have density of states N 0. Magnetic susceptibility A magnetic field B, e.g. in the z-direction, introduces a Zeeman term in the Hamiltonian δh = g µ B B 2 N N, 2 Note that Y 00Ω = 4π. 4

16 where N σ is the total number of spin-σ electrons, that acts as a difference of chemical potential for spin up and down electrons: δµ δµ = g µ B B. Since the total magnetization is also conserved, the quasiparticles acquire the same difference in chemical potentials. The magnetization per unit volume δm can therefore be calculated by the quasiparticles through δm = g µ B 2V = g µ B 4 4π N δ σz 00 + F A 0 k δn k δn k = g µ B 2V = δ m + F A 0. δσk z = g µ B k 4 4π N δσz 00 Once again, since the local equilibrium magnetization δ m is equivalent to the response of free particles with density of states N, instead of N 0 as for the original particles, we obtain that the magnetic susceptibility χ is χ = δm δb = N + F0 A N 0 χ0,.3 where χ 0 is the susceptibility the original free-particles..3 Quasiparticle transport equation Let us suppose to perturb the Landau-Fermi liquid by an external probe that varies in space on a wavelength /q and in time on a period /ω. If q k F, ω µ, where k F is the Fermi momentum ɛk F = µ, the probe will affect only quasiparticles extremely close to the Fermi-sea. In this limit, we can safely adopt a semi-classical approach and introduce a space- and time-dependent density of quasiparticle occupation numbers at momentum k k F and its space and time Fourier transform n kσ x, t, n kσ q, ω. Fixing both coordinate x and momentum k is allowed in spite of the Heisenberg principle x k, because k x q k F, so that the quantum indeterminacy of momentum k is much smaller than its typical value k F. 5

17 Since the quasiparticle occupation number is not a conserved quantity, its density should follow a Liouville equation n kσ x, t t + n kσx, t x x t + n kσx, t k k t = I kσ x, t,.32 where I kσ x, t is a collision integral and corresponds to the transition rate within a volume dx around x of all processes in which quasiparticles have a transition from any other state into k, σ minus the transition rate of the inverse processes. In the absence of the external probe, quasiparticles are at equilibrium and their density is uniform. Once the probe is applied, the density deviates from its uniform equilibrium value and acquires space and time dependence. In general the external force that acts on the real particles can not be identified with the force felt by the quasiparticles unless the external field couples to a conserved quantity. This is the case of a scalar potential coupled to the charge density, or a magnetic field that couples to the spin density. In those cases, the deviation from equilibrium of the energy becomes non-uniform hence can be generalized into δet = dx δex, t = dx ɛ k + V σ x, t δn kσ x, t kσ + dxdy δn kσ x, t f kσ k 2 σ x y δn k σ y, t,.33 kk σσ where V σ x, t represents the action of the external probe, V = V for a scalar potential and V = V for a magnetic field, and we assume that the quasiparticle interaction is instantaneous and only depends on the distance. The above expression provides also the proper definition of the quasiparticle excitation energy as ɛ kσ x, t = ɛ k + V σ x, t + dy f kσ k σ x y δn k σ y, t..34 k σ For neutral particles we can assume that f kσ k σ x y = f kσ k σ δx y, in which case ɛ kσ x, t = ɛ k + V σ x, t + f kσ k σ δn k σ x, t..35 k σ By means of the Hamiltonian equations for conjugate variables x t = H k, k t = H x, 6

18 Eq..32 can be finally written as n kσ x, t t + n kσx, t x ɛ kσ x, t k n kσx, t k ɛ kσ x, t x At linear order in the external probe V σ x, t, the occupation density and, consistently, where δɛ kσ x, t = n kσ x, t = n 0 kσ + δn kσx, t, ɛ kσ x, t = ɛ k + δɛ kσ x, t, δe δn kσ x, t ɛ k = V σ x, t + f kσ k σ δn k σ x, t. k σ Consequently, the linearized transport equation.36 reads where δn kσ x, t t = δn kσx, t t + δn kσx, t x + δn kσx, t x = δn kσx, t + δn kσx, t t [ x n0 k ɛ k V σ x, t ɛ k k x ɛ k k n0 k k ɛ k k n0 k k ɛ k k δɛ kσ x, t x [ V σ x, t x + ] δn k f σ x, t kσ k σ x k σ = δn kσx, t + v k δn kσx, t t [ x n0 k V σ x, t v k + ] δn k f σ x, t kσ k ɛ k x σ x k σ δn kσx, t t = I kσ x, t, + v k δn kσx, t x + n0 k ɛ k v k F kσ x, t = I kσ x, t,.36 + ] δn k f σ x, t kσ k σ x k σ.37 v k = ɛ k k is the quasiparticle group velocity. The Landau transport equation.37 resembles the conventional Boltzmann equation apart from the fact that the effective force F kσ x, t depends self-consistently on the occupation density. 7

19 .3. Quasiparticle current density Let us set V x, t = 0 in.37. We note that V δn kσ x, t = δρx, t, kσ where δρx, t is the deviation of the density with respect to the uniform equilibrium value, and moreover that I kσ x, t = 0, V kσ by particle conservation recall the meaning of the collision integral I kσ. Through Eq..37 and using the relation.29 between the deviations from global and local equilibrium, we find the following equation for the evolution of δρx, t in the absence of external probes: [ ] δρx, t + t x v k δn kσ x, t n0 k f kσ k V ɛ σ δn k σ x, t k kσ k σ = δρx, t t + x V v k δ n kσ x, t = kσ Since the integral over the whole space of δρx, t is the variation of the total number of real particles, which is conserved, then δρx, t should also satisfy a continuity equation δρx, t t + Jx, t x that allows to identify the current density Jx, t note that, at equilibrium, the current is zero hence Jx, t = δjx, t = 0, Jx, t = v k δ n kσ x, t V kσ [ ] = v k δn kσ x, t n0 k f kσ k V ɛ σ δn k σ x, t k kσ k σ [ = δn kσ x, t v k ] n 0 k f σ kσ k V σ v k ɛ kσ k σ k δn kσ x, t J kσ..39 V kσ This equation shows that the actual current matrix element J kσ is not the group velocity v k the moving quasiparticle induces a flow of other quasiparticles because of their mutual interaction. 8

20 We observe that at very low temperature n 0 k σ ɛ k = f ɛ k µ ɛ k δ ɛ k µ, namely k lies on the Fermi surface. If the system is isotropic, then on the Fermi sphere v k = ɛ k k = v F k k = k F m with v F and m the Fermi velocity and effective mass, respectively, of the quasiparticles. Using an expansion like.6 in terms of spherical harmonics we find that, for instance the z-component of the current density, J z, is J z x, t = v k V z δ n kσ x, t where v 0 k kσ k k, = k z v F V k δ ɛ k µ Y lm Ω k δ n lmσ x, t kσ lm = v F 2 N dω δ ρ lm x, t 4π cos θ Y lm Ω lm = v F 2 2π N δ ρ 0x, t = v F 2 2π N [ = v F v 0 F = v F v 0 F + F S 3 + F S 3 V v 0 F 2 2π N δρ 0x, t kσ v 0 k + F S 3 ] is the group velocity of the free particles on the Fermi surface v 0 k = ɛ0 k k = v 0 F δρ 0 x, t z δn kσx, t,.40 k k = k F m k k..4 The x and y component can be found analogously and involve combinations of δn + x, t and δn x, t. Therefore in general it holds that Jx, t = v F v 0 + F S v 0 3 V k δn kσx, t F kσ = m m + F S 3 9 V kσ v 0 k δn kσx, t..42

21 As a concluding remark we notice that we could also follow the same analysis in connection with the spin-current density, the only difference being that F S would be substituted by F A. Translationally invariant systems In a system that is not only isotropic but also translationally invariant, the current matrix element is J k = k m, with m the mass of the real particles. Since the current is conserved, the total current per unit volume must be given in terms of quasiparticles by J z = V kσ k m δn kσ = k F 2m 2π N δρ Comparing.43 with.40 in the case in which the current density, hence the occupation density, are constant in time and space, i.e. J z x, t = J z and δρ 0 x, t = δρ 0, we conclude that, when translational invariance is unbroken, like in 3 He, the following relation holds m m = + F S 3,.44 namely the effective mass and the Landau parameter F S are not independent..4 Collective excitations in the collisionless regime The collision integral in the transport equation determines a typical collision time τ. If one is interested in phenomena that occur on a time scale smaller than τ, namely on frequencies ω /τ, then the collision integral can be safely neglected and the transport equation.37 in frequency and momentum space becomes ] [V σ q, ω + k σ f kσ δn k σ k σ q, ω ω q v k δn kσ q, ω + q v k n 0 k ɛ k = The solution of this equation in the absence of external field gives information about the collective excitations of the quasiparticle gas. These excitations can also be thought as collective vibrations of the Fermi sphere. From this point of view, let us suppose that, along the direction identified by the Euler angles θ, φ, the z-axis being directed along the momentum q, the Fermi momentum k F = k F sin θ cos φ, sin θ sin φ, cos θ for a spin σ quasiparticle changes into k F σ = k F + u σ θ, φ sin θ cos φ, sin θ sin φ, cos θ, 20

22 as if k F σ θ, φ = k F k F σ θ, φ = k F + δk F θ, σ = k F + u σ θ, φ. The occupation number for a spin σ quasiparticle with momentum k = k sin θ cos φ, sin θ sin φ, cos θ changes accordingly as We note that n kσ = n 0 kσ θ k k F σθ, φ n kσ = n 0 kσ n0 k k δk F = n 0 kσ + δn kσ. δn kσ = n0 k k δk F Inserting.46 into.45 with V = 0 we get = n0 k ɛ k ɛ k k δk F = n0 k ɛ k v F u σ θ, φ..46 ω v F q cos θ u σ θ, φ + v F q cos θ k σ f kσ k σ n 0 k σ ɛ k u σ θ, φ = For small vibrations of the Fermi sphere, we can assume that f kσ k σ only depends on the angle between k and k, f kσ k σ = f σσ θ kk, their modula being equal to the equilibrium value of the Fermi momentum k F. We note that, since n 0 k σ ɛ k δ ɛ k µ, it follows that 0 nk f σ kσ k σ u σ θ, φ = 2 V ɛ k σ k dωk 2 4π f σσ θ kk u σ Ω k = V N dωk 2 4π f σσ θ kk u σ Ω k = 2 4π k 2 d k 2π 3 δ ɛ k µ where, generalizing Eq..22, the F -parameters are defined through F σσ θ kk = V N f σσ θ kk. If furthermore we define, for spin-isotropic models, u S = u + u, u A = u u, dωk 4π F σσ θ kk u σ Ω k, 2

23 F = F = F S + F A, F = F = F S F A, we find, defining λ = ω/v F q, the following equation λ cos θ k u SA dωk θ k, φ k cos θ 4π F SA θ kk u SA θ k, φ k = We write, dropping the indices S and A, uθ, φ = lm F θ kk = l Y lm θ, φ u lm, P l θ kk F l, so that Eq..48 can be re-written as [ Y lm θ k, φ k u lm λ cos θ k cos θ ] k 2l + F l Since lm cos θ = 4π 3 Y 0θ, φ, = l is not a good quantum number of Eq..49, while m is. In particular m = 0 corresponds to longitudinal vibrations of the Fermi sphere, m = to dipolar vibrations and m = 2 to quadrupolar ones. We observe that a solution of.49 at ω = 0 would imply that a deformation of the Fermi sphere would be cost-free, namely that the Fermi sphere is unstable. At λ = 0, l is a good quantum number, so that a λ = 0 solution of.49 occurs whenever there is a value of l such that, see Eq..23, + F SA l 2l + = 0 = F SA l,.50 A SA l which is the condition for the instability of the Landau-Fermi liquid. We note that, since F SA l is generally non zero, the instability implies a singularity of the Landau A-parameters..4. Zero sound Let us assume, as commonly done, that only F SA 0 and F SA are not negligible, and study Eq..49 for longitudinal vibrations, m = 0. In this case, dropping for simplicity the labels S and A, we obtain λ cos θ uθ = lm λ cos θ Y lm θ k, φ k u lm = cos θ F 0 Y 00 u 00 + cos θ 3 F Y 0 u

24 We define, and, by noting that we get the formal solution In terms of this solution Y 00 = C = 4π F 0 u 00,.52 D = 3 3 4π F u 0,.53 4π 4π, cos θ = 3 Y 0, uθ = C cos θ + D cos2 θ..54 λ cos θ where 4π u 00 = 3 3 dω C cos θ + D cos 2 θ 4π λ cos θ = C I + D I 2,.55 dω 4π u 0 = 4π cos θ C cos θ + D cos2 θ λ cos θ = C I 2 + D I 3,.56 I n = Therefore the self-consistency condition that has to be satisfied is dz 2 z n λ z..57 C = F 0 C I + D I 2,.58 D = F C I 2 + D I 3,.59 which have a solution if By realizing that I 2 2 = F 0 I + F F I 3 I = + λ λ + 2 ln, λ I 2 = λ I, I 3 = 3 + λ2 I,.60 23

25 we finally get that.60 is satisfied if = F 0 + λ 2 A I,.6 where, by definition, We note that, for λ, A = F + F /3. I + λ 2 i π 2 λ, has a finite imaginary part. This implies that the frequency ω that solves.6 is necessarily complex, hence that it does not correspond to a stable collective mode but rather to a damped one. Therefore the only stable solutions of.6 must correspond to λ. For λ + δλ, with 0 < δλ, leading to I 2 ln e2 δλ 2, [ δλ 2 exp 2 + which is consistent with the assumption δλ only if F 0 + A 0 F 0 + A. ]..62 In other words, when the Landau parameters are positive and very small, the stable collective modes propagate like sound modes with a velocity [ ] v v F + 2 exp 2 +, F 0 + A slightly larger than the Fermi velocity. These modes correspond to a deformation of the Fermi sphere of the form uθ C cos θ + D cos2 θ cos θ + δλ, that are strongly peaked to θ = 0, namely in the direction of the propagating wave-vector q. In the opposite case of λ, I 3λ 2 + 5λ 4, 24

26 and the solution of.60 is readily found to be λ 2 = ω2 vf 2 q 2 = F0 3 + A + F, that requires the right hand side to be positive and much greater than one. Again this solution describes collective modes that propagate acoustically with a velocity this time much bigger than v F and that correspond to a deformation of the form that involves the whole Fermi sphere. uθ C cos θ + D cos2 θ, λ These collective modes that may emerge in the collisionless regime and propagate like sound modes were called by Landau zero sounds, as opposed to the first sound that exists in the collision regime. The existence of these new modes of a Landau-Fermi liquid depends on the quasiparticle interaction parameters F, that act as a restoring force absent for non-interacting electrons. Usually, for repulsive interactions, F0 S is positive and F 0 A negative, while A SA is negligible. 3 In this case only charge zero-sound, i.e. a Fermi-sphere deformation in the channel u S = u + u, is a stable collective mode, while a spin zero-sound in the channel is not. u A = u u,.5 Collective excitations in the collision regime: first sound For frequencies much smaller than the typical collision rate, the collision integral can not be neglected anymore and, as a result, any deformation of the Fermi sphere is destined to decay. The only exceptions are those excitations that correspond in the limit q 0 to conserved 3 For repulsive interaction, the effective mass m increases with respect to the non-interacting system the quasiparticles move more slowly than free ones. Furthermore, the magnetic susceptibility increases even more, since the slowed down quasiparticles with opposite spin repel each other, hence can react quite efficiently to a magnetic field. On the contrary, the compressibility κ decreases, as adding particles costs more energy. Since κ m = κ 0 m, + F0 S if follows that F S 0 > + m /m > 0 and F A 0 0. χ m = χ 0 m + F A 0, 25

27 quantities, in particular to charge and momentum densities, in which case it is guaranteed that kσ I kσ = kσ k I kσ = 0. Charge and momentum density excitations correspond to deformations of the Fermi sphere in the symmetric S-channel and with angular momenta l = 0 and l =, respectively, and m = 0. Through Eq..49, the excitations in this regime satisfy the equation 0 = cos θ λ Y 00 Ω u 00 + cos θ λ Y 0 Ω u 0 +F S 0 cos θ Y 00 Ω u 00 + F S 3 cos θ Y 0Ω u Note that this equation differs from Eq..5 for the zero sound because in that case F 0 and F are assumed to be non-zero, but the uθ solving the equation has all l-components, unlike in this case. To solve.64, we multiply both sides once by Y 00 Ω and integrate over the solid angle Ω, and do the same multiplying by Y 0 Ω, thus obtaining the following set of equations: λ u This set of equations has solution if λ 2 = 3 + F S 3 u 0 = 0, + F S 0 u00 λ u 0 = 0. ω2 v 2 F q2 = 3 + F S 0 + F S 3 which corresponds to acoustic waves propagating with velocity s, where s 2 = v2 F 3 + F S 0 + F S 3 We note that, for an isotropic system as that we are assuming, so that hence, through Eq..30, v F = k F, N = m k F m π 2, n = k3 F 3π 2, s 2 = nm κ v 2 F 3 = n m N + F S 3, One can readily realize that this is the ordinary sound velocity compatible with thermodynamics. When local thermodynamic equilibrium is enforced by collisions the hydrodynamic 26

28 regime the equations of motion for the charge density, ρx, t, through the continuity equation, and for the velocity field through the pressure P x, t, 4 ρx, t t vx, t m n t = n m m + F S 3 = P x, t, vx, t where n is the average density, can be solved using the local thermodynamic relation δp x, t = P δρx, t = δρx, t, ρ n κ where κ is the thermodynamic compressibility. The solution gives ordinary sound with a velocity that coincides with Linear response functions An external field induces deviations of the quasiparticle occupation numbers, and the linear relation between these deviations and the field responsible for them are the so-called linear response functions. Let us go back to the transport equation.37, and assume in what follows that the collision integral is negligible. After space and time Fourier transform, Eq..37 is 5 q v k ω δn kσ q, ω q v k n 0 k ɛ k f kσ k σ δn k σ q, ω k σ 4 We assume that the particle velocity field is related to the current as in Eq..42, where by definition n vx, t = V kσ v 0 k δn kσx, t. 5 In deriving the transport equation we assume implicitly the system to be initially, for instance at time t, at equilibrium. This implies that the external field that moves away from equilibrium must initially be absent, i.e. V x, t 0. A simple and convenient choice is to represent, for all times before the measure, performed e.g. at t = 0, an external field that oscillates with frequency ω as V x, t = V x, ω e i ω t+η t, with η an infinitesimal positive number. Consequently, the response of the system must follow the same time behavior, i.e. δn kσ x, t = δn kσ x, ω e i ω t+η t, so that the time dependence cancels out from.37. When taking the time derivative, δn kσ x, t t = i ω + iη δn kσ x, ω e i ω t+η t, which implies that the frequency ω has to be interpreted in whatever follows as ω + iη. 27

29 = q v k n 0 k ɛ k V σ q, ω..68 We stress once more that the external field that enters this equation can be identified with the actual field acting on the true particles only if it refers to conserved quantities, hence the charge, in which case V σ q, ω = V q, ω,.69 and the spin, Let us write V q, ω = V q, ω..70 δn kσ q, ω = q v k q v k ω through which Eq..68 transforms into.6. Formal solution X kσ q, ω k σ f kσ k σ q v k q v k ω n 0 k ɛ k X kσ q, ω,.7 n 0 k ɛ k X k σ = V σ q, ω..72 Let us write Eq..72 in matricial way like ] [Î ˆf ˆK X = V,.73 where the matrices have matrix elements I kσ k σ = δ kk δ σσ, f kσ k σ = f kσ k σ, K kσ k σ = q v k q v k ω n 0 k ɛ k δ kk δ σσ, and the vectors X and V X V kσ kσ = X kσ q, ω, = V σ q, ω. The formal solution of.73 is X = [Î ˆf ˆK] V

30 Let us suppose to introduce another matrix Â, with elements  = A kσ k σ kσ k σ q, ω,.75 that satisfies the set of equations ] [Î ˆf ˆK  = ˆf..76 One readily derives from this equation that  = ] [Î ˆf ˆK ˆf, hence that  ˆK = ] ] [ [Î ˆf ˆK ˆf ˆK = [Î ˆf ˆK ˆf ˆK Î] Î + = Î + [Î ˆf ˆK]. In other words so that ] [Î ˆf ˆK =  ˆK + Î,.77 X = [Î +  ˆK] V, that explicitly means X kσ q, ω = k σ [ δ kk δ σσ + A kσ k σ q, ω q v k q v k ω n 0 ] k V σ q, ω,.78 ɛ k hence that δn kσ q, ω = q v k q v k ω n 0 k ɛ k σ [ δ σσ + k A kσ k σ q, ω q v k q v k ω n 0 ] k V σ q, ω,.79 ɛ k which is the formal solution of.68. The functions A bkσ k σ q, ω are the so-called quasiparticle scattering amplitudes. We define as usual A kσ k σq, ω = A S kk q, ω + AA kk q, ω, A kσ k σ q, ω = AS kk q, ω AA kk q, ω, with σ σ. We also introduce a vertex λ σ that, in case of a scalar potential V σ q, ω = V q, ω, is defined as λ σ =, 29

31 while it is λ σ = δ σ δ σ, for a Zeeman-splitting magnetic field V σ q, ω = V q, ω δ σ δ σ = V q, ω λ σ. Upon multiplying both sides of.79 by the corresponding vertex and summing over k and σ and dividing by the volume, we find that the charge, δρq, ω, and spin, δσq, ω, density deviations at linear order in the corresponding fields are given by δρq, ω = χ S q, ω V q, ω δσq, ω = χ A q, ω V q, ω, where χ SA q, ω = 2 V k q v k q v k ω n 0 k ɛ k [ + 2 k A SA kk q, ω q v k q v k ω n 0 ] k..80 ɛ k Thus χ S q, ω is n 2 times the dynamical compressibility κq, ω, while χ A q, ω is the dynamical spin-susceptibility. Limiting cases We readily realize that, if q = 0 and ω 0, then the matrix ˆK = 0. In this case  = ˆf, namely A kσ k σ 0, ω = f kσ k σ..8 In the opposite limit, q 0 and ω = 0, Eq..76 reads explicitly A kσ k σ q, 0 pα f kσ pα n 0 p ɛ p A pα k σ q, 0 = f pα k σ, showing that A kσ k σ q, 0 is independent on q. By writing V N A kσ k σ q, 0 = l P l θ kk [ A S l ± A A l ], where the + sign refers to σ = σ and the one otherwise, one readily recognize that A S l A A l coincide with the Landau A-parameters of Eq..23. and 30

32 .6.2 Low frequency limit of the response functions For frequencies ω v F q, we can expand A kσ k σ q, ω as A kσ k σ q, ω = A kσ k σ q, 0 + δa kσ k σ q, ω,.82 where δa vanishes linearly at ω = 0. We note that q v k q v k ω = + ω, q v k so that the linear variation δa has to satisfy the equation δa kσ k σ q, ω pα f kσ pα n 0 p ɛ p δa pα k σ q, ω = pα f kσ pα n 0 p ɛ p ω A pα k q v σ q, p The integral over Ω p of the function q v p is singular, hence we need some regularization. Indeed, as discussed in the footnote [5], the frequency ω above is actually ω + iη, with η an infinitesimally small positive number. Therefore q v k q v k ω q v k q v k ω iη = + ω q v k iη + i π ω δ q v k + P ω q v k, where P indicates the principal value. Therefore, within the S and A channels, Eq..83 becomes the principal value averages to zero hence does not contribute δa SA kk q, ω 2 p f SA kp n 0 p δa SA ɛ pk p k q, ω = 2 π i ω p f SA kp At first order in ω, Eq..80 becomes we drop the indices S and A χq, ω 2 n 0 [ k + 2 ] A kk q, 0 n0 k V ɛ k ɛ k k Since +2 π i ω V +4 π i ω V +4 V kk k kk 0 δ q v k n0 k ɛ k 0 nk n 0 p ɛ p δ q v p A SA pk q, 0. [ + 2 k A kk q, 0 n0 k ɛ k n 0 k A kk q, 0 δ q v k ɛ k ɛ k ].84 nk n 0 k δa kk q, ω..85 ɛ k ɛ k dω 4π P lθ = δ l0, 3

33 dω 4π δ q v F cos θ = 2 q v F it follows from Eq..84 that dωk 4π dω k 4π δa kk q, ω + F 0 dωp 4π dω k 4π δa pk q, ω = π i ω V N 2 q v F F 0 A 0, namely dωk 4π dω k 4π δa kk q, ω = π i ω V N Therefore Eq..85 is found to be χq, ω N A 0 ω i π N A 0 2 q v F ω +i π N A 0 2 q v F ω i π N A q v F = N + F 0 i π = N + F 0 i π 2 q v F F 0 + F 0 A 0 = π i ω V N.6.3 High frequency limit of the response functions 2 q v F A 2 0. ω N A q v F ω N + F q v F Let us now consider the case ω v F q 0 when again the S and A indices are not explicitly shown A pk q, ω = f kk. In this case q v k q v k ω q v k q v k 2 ω ω 2, and, since the angular average of q v k vanishes, the response functions are χq 0, ω = 2 ω 2 V + 4 ω 2 V k kk n 0 k ɛ k q v k 2 q v k q v k n 0 k n 0 k f kk..87 ɛ k ɛ k 32

34 We note that 4 V kk 2 V k n 0 k ɛ k q v k 2 = N = 3 N v2 F q 2, 0 nk n 0 k f kk q v k q v k = N vf 2 q 2 ɛ k ɛ k l dω 4π v F q cos θ 2 F l dωk 4π dω k 4π cos θ k cos θ k P l θ kk = N v 2 F q 2, lm dωk F l 4π dω k 4π Therefore Eq..87 is 4π 3 Y 0 θ k = N v 2 F q 2 F 9. χq 0, ω = N v2 F q 2 3ω 2 4π 3 Y 0 θ k 4π 2l + Y lm θ k Y lm θ k + F In a system that is also translationally invariant, besides being isotropic, in the S channel the following relations hold N = m k F π 2 = mk F π 2 + F, 3 hence in agreement with the f-sum rule. v F = k F m = k F m n = k3 F 3π 2, + F 3 κq 0, ω = n m, q 2 ω 2,.89 33

35 .7 Charged Fermi-liquids: the Landau-Silin theory Now, instead of a neutral Fermi system, let us consider a charged one and discuss how all previous results change. The first thing to note is that the quasiparticle interaction f kσ k σ x y in Eq..33 will now contain, besides a short range contribution, also a long range Coulomb one, so that can be assumed to be of the form f kσ k σ x y = f kσ k σ δx y + e2 x y..90 Another novel feature that we have to take into account is the possible presence of a transverse electromagnetic field. If the system is translationally invariant or if inter-band matrix elements of the current density are negligible, the role of a transverse field is that the conjugate momentum is not anymore k but K = k e Ax, t, c where A is the transverse vector potential felt by the real electrons. Thus the quasiparticle occupation density in the semi-classical limit can be still parametrized by n Kσ x, t. Nevertheless, it is more convenient to define the occupation density in the equivalent way n Kσ x, t n kσ x, t = n K+ e c Ax,t σ x, t,.9 since, in the presence of A, the quasiparticle excitation energy, Eq..34, changes simply into From the above equation it follows that ɛ kσ x, t ɛ K+eA/c σ x, t = ɛ kσ x, t. ɛ K+eA/c σ x, t = ɛ kσx, t = v kσ x, t K k is the proper quasiparticle group velocity, and ɛ K+eA/c x, t x = e c v kσ ix, t A ix, t x i + V σx, t + dy f kσ k x σ x y n k σ y, t. y k σ Through the definition Eq..9, it follows that nkσ x, t nkσ x, t = + e t K x t k x c nkσ x, t nkσ x, t = + x K x k i nkσ x, t nkσ x, t = K k x. x n kσ x, t k Ax, t, t e n kσ x, t A i x, t c k i x, 34

36 where Putting everything together we find the following transport equation for charged electrons: I kσ x, t = n kσx, t t n kσ x, t k + e c +v kσ x, t n kσx, t x n kσx, t k ij Ax, t t + ij [ Vσ x, t + x k σ e n kσ x, t v kσ i x, t A ix, t c k j x j n kσ x, t k i A i x, t x j v kσ j x, t e c dy f kσ k σ x y n k σ y, t ] y We note that the two terms with the ij can be written as e n kσ x, t Ai x, t v kσ j x, t A jx, t c k i x j x i ij = e c v kσ x, t Hx, t n kσx, t, k Hx, t = Ax, t is the magnetic field. Since the external transverse electric field is Ex, t = c the final expression of the transport equation reads e n kσx, t k Ex, t e c Ax, t, t I kσ x, t = n kσx, t + v kσ x, t n kσx, t t x n kσx, t [ Vσ x, t + dy f kσ k k x σ x y n k σ y, t ] y k σ v kσ x, t Hx, t n kσx, t..92 k Let us now expand the transport equation.92 at linear order in the deviation from equilibrium n kσ x, t = n 0 kσ + δn kσx, t. First we assume an ac electromagnetic field acting as a perturbation, in which case e c v kσ x, t Hx, t n kσx, t e v k Hx, t n0 k k c k 35

37 = e c v k Hx, t n0 k v k = 0. ɛ k This shows that an ac magnetic field does not contribute to the linearized transport equation, that becomes I kσ x, t = δn kσx, t t where we recall that + n0 k ɛ k v k = δn kσx, t t + n0 k ɛ k v k + v k δn kσx, t x [ V σx, t e Ex, t x k σ + v k δ n kσx, t x [ V σx, t e Ex, t x δ n kσ x, t = δn kσ x, t n0 k ɛ k k σ dy f kσ k σ x y δn k σ y, t ] y ],.93 dy f kσ k σ x y δn k σ y, t, is the deviation from local equilibrium. In the presence of a dc magnetic field Hx, a subtle issue arises in connection with the Lorenz s force term v kσ x, t Hx, t n kσx, t. e c k Indeed, a dc field, unlike an ac one, can be assumed as integral part of the unperturbed Hamiltonian and, if large, can be taken as a zeroth order term, Hx = H 0 x. This requires to expand at linear order v kσ x, t and n kσ x, t/ k, v kσ x, t v kσ + k σ n kσ x, t k leading to the transport equation n0 k k + δn kσx, t, k I kσ x, t = δn kσx, t + v k δ n kσx, t t x e v k H 0 x, t c dy ] [f kσ k k σ x y δn k σ y, t, + n0 [ k v k V σx, t ] e Ex, t ɛ k x δ n kσx, t,.94 k which is the appropriate one to discuss properties like magnetoresistance, cyclotron resonances, etc. However, in what follows we will not consider such a physical situation of a large dc field, hence we will just focus on Eq