Landau-Fermi liquid theory Shreyas Patankar Chennai Mathematical Institute Abstract We study the basic properties of Landau s theory of a system of interacting fermions (a Fermi liquid). The main feature of Fermi liquids is the notion of quasiparticles and the associated quasiparticle residues, giving rise to properties such as quadratic temperature dependance of transport properties and discontinuity in the ground state occupation number. 1 Interacting fermion system The basic properties of metals such as heat capacity, magnetic susceptibility etc. were explained by treating electrons in the metals as a gas of non-interacting fermions, by Pauli, Sommerfeld and others in 1928.[7] However, for several years it was a puzzle as to why a non-interacting theory, for a system of electrons which is clearly non-interacting, should be so successful. In 1956, Lev Landau developed a theory for interacting spin-1/2 fermions as an attempt to explain the stability of Fermi-gas theory against perturbations.[2] The Landa- Fermi liquid theory has been successful in explaining the behavior of several systems such as electrons in a metal, nuclear matter, liquid He-3 and so on. For such an intracting system, a generic Hamiltonian[3] could be written as, H = p,σ ɛ p c pσc pσ + p i,q,σ V (q)c p 1 qc p 2 +qc p1 c p2 (1) where the first term denotes the non-interacting kinetic energy and the second term denotes a two-point interaction. The key ideas behind Landau s theory are the principle of adiabaticity, and Pauli s exclusion principle. Landau argued that starting with the ground state of a non-interacting fermion system, if we slowly turn on the interaction, we will end up with the ground state of a Fermi liquid. Formally, we say that if the full Hamiltonian has the same symmetries of some unperturbed Hamiltonian, its ground states can be obtained by perturbatively transforming the simple ground state. 1 [6, 9] Similarly, the low-lying excitations of the non-interacting theory get mapped to the low-lying excitations of the Fermi liquid. Through this process, 1 The assumption rules out the possibility of a phase transition 1
certain dynamical variables such as spin and momentum of the excitations are conserved (the latter following from Pauli s exclusion principle). The low lying excitations of the Fermi liquid are called quasiparticles (or Landau quasiparticles). The quasiparticle excitations are obtained by adiabatic continuation of a non-interacting system, and hence, they need not be eigenstates of the new Hamiltonian. As such, they have a finite lifetime τ (arising only due to interactions with other quasi particles). We assume that these excitations are long-lived, that is, we consider only those excitations which satisfy τ ɛ where ɛ denotes the excitation energy. As the lifetime is solely dependent on two-particle scattering, let us try to estimate the scattering rate. 2 Suppose we have an incoming particle with energy above the Fermi surface, 3 ɛ 1 > ɛ F and suppose it collides with a particle with ɛ 2 < ɛ F, and results in two particles with energy ɛ 3, ɛ 4 > ɛ F. Then, we have the restriction that ɛ 2, ɛ 3 lie within ±(ɛ F ɛ 1 ), and ɛ 4 is fixed once the other three are. Thus, the volume of the available phase space is (ɛ 1 ɛ F ) 2. Fermi s golden rule states that the scattering rate is proportional to f V i 2 ρ, where V is the interaction potential, and ρ is the density of final states. Suppose we take the interaction to be Coulomb, then we can write V = V (q) 2 c k 1 +q c k 2 q c k 1 c k2, where V (q) is the k 1,k 2,q screened Coulomb potential. It can be shown[5] that V (q 0) = V 0 (a constant) and thence that scattering rate 1 τ ρ = (ɛ 1 ɛ F ) 2. At a small temperature above T > 0, the average value of excitations is kt. Thus, 1 τ T 2, and hence, all transport properties (such as resistivity) which depend on the scattering rate have quadratic dependence on temperature. This power law dependence is one of the key features of the Fermi liquid theory. 2 Two-point function As in any field theory, the two point-correlation function or the Green s function is an important parameter of Landau s theory. The two (space) point correlation function can be written as G αβ ( r 2, t 2 ; r 2, t 2 ). As spin is a conserved quantity, we can drop the spin indices. Let ψ( r, t) and ψ ( r, t) be fermion creation and destruction operators, and let m be quasiparticle excitation states 4 (corresponding to the full Hamiltonian). Then we have[1], G( r 2 r 1, t 2 t 1 ) = i 0 T ψ( r 2, t 2 )ψ ( r 1, t 1 ) 0 where T denotes the time-ordered product, defined (for any two Fermionic operators A, B) 2 Following passage is a summary of discussion presented in [5] 3 The Fermi energy is defined here to be the same as it would be without interactions. 4 That is, they are eigenstates of the full Hamiltonian 2
as Suppose t 2 > t 1. Then, T A(t 1 )B(t 2 ) = { A(t 1 )B(t 2 ) for t 1 > t 2 B(t 2 )A(t 1 ) for t 2 > t 1 G( r 2 r 1, t 2 t 1 ) = i 0 ψ( r 2, t 2 )ψ ( r 1, t 1 ) 0 = i 0 e iht 2 ψ( r 2 )e iht 2 e iht 1 ψ ( r 1 )e iht 1 0 = i 0 e iht 2 ψ( r 2 )e iht 2 m m e iht 1 ψ ( r 1 )e iht 1 0 m = i m e i(em E 0)(t 2 t 1 ) ψ 0m ( r 2 )ψ m0 ( r 1 ) In the absence of external fields, the spatial dependence can be factored out in terms of momentum eigenstates 5 as, ψ nm ( r) = ψ nm e i pnm r ψ nm( r) = ψ nme i pnm r Including time ordering, we get, i e i(em E 0)(t 2 t 1 ) e i pm ( r 2 r 1 ) ψ 0m 2 for t 2 > t 1 m G( r 2 r 1, t 2 t 1 ) = i e i(e m E 0)(t 2 t 1 ) e i p m ( r 2 r 1 (2) ) ψ 0m 2 for t 1 > t 2 m Note that (for a N-particle ground state) the states m are N +1 particle and m are N 1 particle states respectively. 6 Thus, we can write E m E 0 = ɛ m + µ and E m E 0 = ɛ m µ, where ɛ m, ɛ m are excitation energies and µ = E 0 (N + 1) E 0 (N) is the chemical potential. Writing r 2 r 1 = r and t 2 t 1 = t, we can write the Fourier transform[4], d 3 p dt G( p, ω) = (2π) 3 2π G( r, t)e iωt e i p r (3) We now use an identity from complex analysis. The integral of an oscillating function can be written as the limit: lim e ist δt 1 dt = i lim (4) δ 0 + δ 0 s + iδ 0 The position integrals in (3) will just give Delta functions in momentum. Thus we get, G( p, ω) = i m δ( p p m ) ω ɛ m µ + iδ ψ 0m 2 + i m δ( p + p m ) ω + ɛ m µ iδ ψ 0m 2 (5) 5 Adiabaticity implies that we can assign momentum eigenvalues to energy eigenstates, see [1] 6 The combination of ψψ ensures that we are dealing with at most single-particle (or hole) excitations. 3
2.1 Analytic structure For clarity, we use the notation A( p, E) = m ψ 0m 2 δ( p p m )δ(e ɛ m ) B( p, E) = m ψ 0m 2 δ( p + p m )δ(e ɛ m ) (Note that A, B are real and positive functions.) This gives, [ G( p, ω) = 0 A( p, E) ω E µ + iδ + B( p, E) ω + E µ iδ ] de (6) Using another result from complex analysis (due to the residue theorem), we have that, ds F (s) F (s) s + iδ = P ds iπf (0) s We can separate G( p, ω) into real and imaginary components, This gives, Re G( p, ω) = P Im G( p, ω) = Re G( p, ω) = 1 π P 0 [ A( p, E) B( p, E) + ω E µ ω + E µ { πa( p, ω µ) for ω > µ πb( p, µ ω) for ω < µ Im G( p, ω ) sgn(ω µ) ω ω ] de Observe that except for the sgn function, the integral has the form of the Kramer s-kronig relation. We know that a complex function which satisfies the Kramers-Kronig relation is analytic in the upper-half plane.[4] Thus, we can define[1] G in terms of two functions G R, G A 7 which are analytic in the upper and lower half planes respectively, as, Re G = Re G R = Re G A Im G R = Im G sgn(ω µ) Im G A = Im G sgn(ω µ) (7) G R, G A are called the retarded and advanced Green s functions respectively. 2.2 Poles and residues Observe from (5) that G(p, ω) as a function of ω has (simple) poles at ω = ɛ m + µ and ω = ɛ m + µ for all m, m, that is, every single-particle excitation (quasiparticle or hole) corresponds to pole of the Green s function. 7 As functions of ω, at constant p 4
Thus, for a given quasiparticle state m (or hole m ), we have a pole at the point ω = ɛ m + µ (or ω = ɛ m + µ) with residues Z m = ψ 0m 2 = m ψ 0 2 Z m = ψ 0m 2 = m ψ 0 2 This leads us to a physical interpretation for the quasiparticle residues, that they are the overlap of the quasiparticle and bare fermion states.[9] Further, from the above argument, we can see that the function G( p, ω) over the complex-ω plane has poles in the lower half plane for ω > µ and in the upper half plane for ω < µ, corresponding to quasiparticles and holes respectively. 3 Momentum distribution function For a given momentum state, we can write the number operator as ψ p ψ p (in the Schrödinger representation). That is, we can write, This can be re-written in the form, N( p) = S 0 ψ p ψ p 0 S (8) N( p) = lim 0 e iht 2 ψ p e iht 2 e iht 1 ψ t 2 t + p e iht 1 0 S 1 = lim t 2 t + 1 G( p, t 2 t 1 ) = 2i lim t 0 + Suppose we split the integral into two parts, G( p, ω)e iωt dω = µ G( p, ω)e iωt dω G( p, ω)e iωt dω + µ G( p, ω)e iωt dω Consider the case where p > p F, and hence, the function has a pole only on the ω > µ side. The first integral coincides with an integral over G A and the second integral coincides with an integral over G R. We use analyticity to shift the contours to more convenient forms. For the first integral, by Cauchy s theorem, the integral along first contour is the negative of the integral along the second contour in fig. 1 that is, we are integrating ω over the curve ω = α i where α varies over reals and is a large quantity. Thus, because of the presence of e iωt in the integral, we can make the horizontal part small by choosing a sufficiently large. Thus, we can replace the first integral with µ µ i G A ( p, ω)e iωt dω 5
Figure 1: Contour for G A Figure 2: Contour for G R Following a similar procedure for the second integral, we can push the contour down everywhere except for the position of the pole, (see fig. 2) 8 That is, the second integral can be written as µ i µ G R ( p, ω)e iωt dω + 2πi Res [ G R (p, ω)e iωt] Observe the fact that G A = G R. Thus adding, we get, G( p, ω)e iωt dω = µ i µ Im G R ( p, ω)e iωt dω + 2πi Res [ G R (p, ω)e iωt] It can be shown [1] that the first term in this is negligible whenever δ ɛ(p) µ, that is, we are looking only at excitations close to the Fermi surface. Following similar calculations, we get for p < p F that, G( p, ω)e iωt dω = µ i µ Im G A ( p, ω)e iωt dω 2πi Res [ G R (p, ω)e iωt] (the contour rotates in the opposite sense about the pole to give a net minus sign) We can write the difference as, N(p > p F ) N(p < p F ) = Res [G(p, ω)] = Z p (9) Thus, we have that the jump in the momentum occupation number equals Z p, the quasiparticle residue at the Fermi surface. Observe that for Fermions, N(p) < 1, and we have already shown that The coefficients A, B in (6) are positive definite, and hence, Z > 0. Thus, we have 1 Z 0. Z = 1 corresponds to the case of non-interacting Fermi gas and Z = 0 corresponds to a non-fermi liquid (see fig. 3). 8 Note that the pole has this position only when ɛ(p) > µ, otherwise, the pole will be above the real line. 6
Figure 3: Momentum distribution function, for (a) non-interacting and (b) interacting Fermion system. 4 Breakdown of Fermi liquid theory Over the past thirty years, materials have been found which behave as Fermi liquids with quasiparticle mass renormalization of up to 10 3 (the so-called heavy Fermion systems)[7, 8] illustrating the robustness of Landau s theory. However, materials have also been found whose low-energy behavior do not fit the description of quasiparticle excitations. 9 Let us write a single-quasiparticle state as a perturbative expansion,[9] m = Zm 1/2 ψ 0 + N V (k 1 k 2 + k 3 p)c k 1 c k2 c k 3 δ(k 1 k 2 + k 3 ; p) 0 +... (10) k 1,k 2,k 3 where p corresponds to the momentum of the single-particle state m. The first term is the simple single particle creation whereas the second one arises due to the second term in (1). We say that Fermi liquid behavior breaks down for Z m 0. Eq. (10) shows us that this is equivalent to saying that the contribution to the excited state from many particle states (higher order terms in (10)) diverges. In media such as the high-t c cuprates, it is observed that the Fermi liquid behavior vanishes at a specific point when doping concentration is varied. The point at which this happens is a quantum critical point and is characterized by the presence of a sharp Fermi surface but no quasiparticle excitations.[8] 9 There are other instances where Fermi liquid theory does not apply, notably in one-dimensional interacting Fermion systems (Luttinger liquids) and systems with phase transitions, but these are expected from the underlying assumptions of the theory. 7
Acknowledgements I would like to thank Prof. K. Narayan, CMI for supervising my final year project. I would also like to thank Prof. G. Baskaran, IMSc, and Prof. Kedar Damle, TIFR, for helpful discussions. I also thank my friend and colleague Debangshu Mukherjee for comments and suggestions. References [1] A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, Methods of quantum field theory in statistical physics, Dover books on physics, Dover Publications, 1975. 2, 3, 4, 6 [2] Piers Coleman, Introduction to many body physics, Rutgers University (draft version), 2010. 1 [3] Richard P. Feynman and Jacob Shaham, Statistical mechanics: A set of lectures, W.A. Benjamin, Reading, Mass., 1972. 1 [4] E. M. Lifshitz and L. P. Pitaevskii, Statistical physics part 2, Elsevier, 1980. 3, 4 [5] P.W. Phillips, Advanced solid state physics, Frontiers in Physics Series, Westview Press, 2003. 2 [6] David Pines and Philip Noziéres, The theory of quantum liquids, W. A. Benjamin Inc., 1966. 1 [7] A. J. Schofield, Non-Fermi liquids, Contemporary Physics 40 (1999), 95 115. 1, 7 [8] T. Senthil, Critical fermi surfaces and non-fermi liquid metals, Phys. Rev. B 78 (2008), no. 3, 035103. 7 [9] C. M. Varma, Z. Nussinov, and Wim van Saarloos, Singular or non-fermi liquids, Physics Reports 361 (2002), no. 5-6, 267 417. 1, 5, 7 8