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Appendix a Basics of Fermi Liquid Theory

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Appendix a Basics of Fermi Liquid Theory Appendix A Basics of Fermi Liquid Theory A.1 Quasi-particles and Interactions The basics assumptions, outlined in Chap.2, above which the Fermi liquid theory is constructed, are equivalent to assume that the degrees of freedom of the vacuum of a non interacting Fermi gas are in a one-to-one correspondence with the states of the Fermi liquid, namely, one can recover a state of an interacting Fermi liquid by adiabatically turn on interactions in a non-interacting Fermi gas. As a consequence, the states of the interacting Fermi liquid are uniquely characterized by the distribution of particles N pσ ,(p is momentum and σ is spin), of the corresponding state of Free Fermi gas. N pσ is referred as the quasi-particle distribution function and quasi- particles, as all the fermionic particles, obey the exclusion principle. In general, the distribution function N pσ is a discontinuous function of p.How- ever, the microscopic properties of the Fermi liquid are successfully described by the mean quasi-particle distribution n pσ , that is a coarse-grained average of N pσ , and is a smooth function of p. In what follows we will assume that the energy of each state E{N pσ } is a func- tional of the distribution function. For each quasi-particle added to an unoccupied state {p,σ}, the total energy of the system will increase by an amount εpσ called the quasi-particle energy which is itself a functional of the distribution function. Without loss of generality, it is possible to assume that εpσ is a smooth function of p and consequently the energy E{n pσ } is a functional of the mean quasi-particle distribution function n pσ . Then, the quasi-particle energy is defined as the variation of the total energy per unit volume V with respect to n pσ : 1 δE = ε σ δn σ . (A.1) V p p pσ The mean quasi-particle distribution n pσ can be determined via thermodynamic arguments. In fact, a macroscopic thermal state at equilibrium and at temperature T has to satisfy the relation: © Springer International Publishing AG 2017 151 A. Amoretti, Condensed Matter Applications of AdS/CFT, Springer Theses, DOI 10.1007/978-3-319-61875-3 152 Appendix A: Basics of Fermi Liquid Theory δE = T δs + μδn (A.2) where δs is the variation of the entropy density, δn is the variation of the particle density, T is the temperature and μ is the chemical potential. Since quasi-particles are in one-to-one correspondence with the free Fermi gas, the entropy must have the same form, namely: kB s =− n σ ln n σ + (1 − n σ ) ln(1 − n σ ) , (A.3) V p p p p pσ where kB is the Boltzmann’s constant. Moreover, since the total number of particles is conserved, the density n is given by: 1 n = n σ . (A.4) V p pσ Now, substituting the variation of E (A.1), n and s with respect to n pσ , kB n p,σ 1 δs =− δn pσ ln ,δn = δn pσ , (A.5) V 1 − n pσ V pσ pσ in (A.2), and imposing that the relation have to be satisfied for every δn pσ , we obtain: = 1 . n pσ (ε −μ)/ (A.6) e pσ kB T + 1 Namely, the quasi-particles obey the usual Fermi-Dirac distribution.1 At T = 0(A.6) reduces to θ(εpσ − μ), which is the usual distribution function af a Fermi sea occupied up to the Fermi momentum pF . For small perturbation around the T = 0 equilibrium state, the quasi-particles distribution function varies only in the neighbourhood of the Fermi surface. Then, considering a state produced by adding a quasi-particle to the ground state, its energy measured relative to the ground state is given by: ε0 = ε { 0 }, pσ pσ n pσ (A.7) where the superscript 0 denote the ground state. The velocity of the quasi-particle at the Fermi surface (the Fermi velocity) is given by: 1 Note however that also εpσ is a functional of n pσ . Then the Eq.(A.6) is actually a quite complicated implicit functional equation for n pσ . Appendix A: Basics of Fermi Liquid Theory 153 0 ∂εσ v = p . (A.8) F ∂p p=pF The quasi-particle effective mass m∗ is consequently defined by the relation: p v = F . (A.9) F m∗ Note that m∗ is in general different from the bare mass of an electron m. In the neighbourhood of the Fermi surface the quasi-particle energy takes the form ε0 = μ + ( − ). pσ vF p pF (A.10) It is useful also to define the quasi-particles density at the Fermi surface, namely: 1 1 ∂ Ω m∗ pd−1 N(0) = δ(ε0 − μ) =− n0 = d F , (A.11) pσ pσ d−1 d V V ∂εpσ π pσ pσ where the last equivalence is obtained replacing the sum over p by an integral and taking εpσ as the variable of integration and Ωd is the solid angle in d spatial dimen- sions.2 By now we have not considered the possibility of interactions between quasi- particles. Switching on interactions, the interaction energy of two quasi-particle is defined as the amount f pσ,pσ /V that the energy of one (pσ ) changes due to the presence of the other (pσ ). Then, a variation of the distribution function produces a variation of εpσ given by 1 δε σ = f σ, σ δn σ . (A.12) p V p p p pσ In other words, f is the second order variation of the energy E (A.1) with respect to n pσ . Consequently, in presence of interactions the variation of the energy due to a variation δn pσ from its ground state can be written as: 1 0 1 δE = ε σ δn σ + f σ, σ δn σ δn σ , (A.13) V p p 2V 2 p p p p pσ pσ,pσ and the corresponding quasi-particle energy is: 0 1 ε σ = ε + f σ, σ δn σ . (A.14) p pσ V p p p pσ 2Remember that we are dealing with a d + 1-dimensional Fermi liquid. 154 Appendix A: Basics of Fermi Liquid Theory As it is evident, the presence of interactions between quasi-particles affects both the static properties of the Fermi liquid, like the effective mass m∗, both the transport properties, as we will discuss better in the following Sections. A.2 Thermodynamic Properties In this section we will review the low-temperature behaviour of the basic thermo- dynamical quantity of a Fermi liquid, namely the specific heat, the entropy and the chemical potential. Quite generally, the low temperature behaviour of the specific heat of a Fermi liquid is linear in temperature, with a coefficient given in terms of the effective mass of the quasi-particles at the Fermi surface. To see this, we calculate the variation of the quasi-particle entropy with respect to a variation of the temperature δT . Keeping into account (A.5) and (A.6), we obtain 1 δs = (ε σ − μ)δn σ . (A.15) TV p p pσ Moreover, from (A.5) we obtain also the following relation: ∂n pσ εpσ − μ δn pσ = − δT + δεpσ − δμ . (A.16) ∂εpσ T It is not difficult to prove that the leading low-T contribution to the previous formula is provided by the first term. Then, as long as we concern with the low-T behaviour, we can safely neglect δεpσ − δμ in (A.16). Finally, the variation of the entropy is: ∂ δ 1 2 n pσ T δs = (ε σ − μ) (A.17) p 2 V ∂εpσ T pσ Replacing the sum by an integral other the energies we have: Ω ∂ ε−μ 2 δ =− d dp d ε 1 δ s σ p ε ( π)d d ∂ε (ε−μ)/k T + T (A.18) d 2 e b 1 T =− 2 ( ) ∞ ∂ 1 2δ . kB N 0 −∞ dx ∂x ex +1 x T (A.19) Finally, we obtain the following expression for the entropy at low-T : π 2 s = N(0)k2 T . (A.20) 3 B Appendix A: Basics of Fermi Liquid Theory 155 Consequently, the specific heat at constant volume is: ∂s C = T = s V ∂ (A.21) T V Considering the free energy F = E − Ts, the first temperature variation (at constant volume) is −sδT , so that at low temperature π 2 F = E − N(0)k2 T 2 , (A.22) 0 3 B where E0 is the ground state energy density. Finally, we calculate the first correction to the chemical potential by using the thermodynamic relation μ =−(∂ F/∂n)T , obtaining: π 2 ∂ ∗ 2 μ( , ) = μ( , ) − 1 + n m T . n T n 0 kB ∗ (A.23) 4 3 m ∂n TF = 2 /( ∗ ) where TF pF 2m kB is the Fermi temperature. A.3 Quasi-particle Life-Time: The Fermi Liquid Stability Until now we have assumed that, even in the case where interactions between quasi- particles are switched on, there is a region of the phase space sufficiently near to the Fermi surface in which the Fermi liquid is stable, namely, the quasi-particles have a sufficiently long life-time in order for the particle description to be consistent. In this Section we will analyse this aspect in a more quantitative way, computing explicitly the quasi-particle life-time when interactions between quasi-particles are considered. To do this, we find quite instructive to take a different approach to what we have considered in the previous Sections, namely the renormalization group approach.
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