Fermi liquid theory

Subir Sachdev

Department of Physics, Harvard University, Cambridge MA 02138, USA and

School of Natural Sciences, Institute for Advanced Studies, Princeton NJ 08540, USA

(Dated: September 20, 2021)

1

The conventional theory of metals starts from a theory of the free electron gas, and then

perturbatively accounts for the Coulomb interactions between the electrons. Already at leading

order, we find a rather strong effect of the Coulomb interactions: a logarithmic divergence in

the effective mass of the single-particle excitations near the Fermi surface. Further examination

of the perturbation theory shows that this divergence of an artifact of failing to account for the

screening of the long-range Coulomb interactions. Formally, screening can be accounted for by a

simple modification of the perturbative series: introduce a dielectric constant in the interaction

propagator, and sum only graphs which irreducible with respect to the interaction line. Once

screening is accounted for by this method, the effective mass of the single-particle excitations

becomes finite.

In this initial chapter we ask: is it possible to give a description of the interacting electron gas

which is valid to all orders in the Coulomb interactions? By “all orders in perturbation theory”

we are assuming the validity of perturbation theory, and cannot rule out non-perturbative effects

which could lead to the appearance of new phases of matter. But in this chapter, we present an all-

orders description of the electron gas. This starts by formalizing the definition of a “quasiparticle”

excitation, as a central ingredient in the theory of many-particle quantum systems.

I. FREE ELECTRON GAS

Let us start by recalling the basic properties of the free electron gas. We work in a second

quantized formalism with electron annihilation operators cpα where p is momentum and α =↑, ↓is the electron spin. The electron operator obeys the anti-commutation relation

[ckα, c†k′β]+ = δk,k′δαβ (1)

We assume the dispersion of a single electron is εp. The chemical potential is assumed to be

included in εp; so for the jellium model εp = ~2p2/(2m)− µ. Then the Hamiltonian is

H =∑p,α

εpc†pαcpα . (2)

The T = 0 ground state of this Hamiltonian is

|G〉 =∏

εp<0,α

c†pα|0〉 (3)

The equation εp = 0 defines the Fermi surface in momentum space, separating the occupied and

unoccupied states.

The elementary excitations of this state are of two types. Outside the Fermi surface we have

particle-like excitations

Particles: c†p,α|G〉, p outside Fermi surface, (4)

2

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energy

ParticlesHoles

FIG. 1. Fermionic excitation spectrum of a Fermi liquid as a function of momentum p along a fixed

direction from the origin.

while inside the Fermi surface we have hole-like excitations

Holes: cp,α|G〉, p inside Fermi surface. (5)

The energy of these excitations must be positive (by definition), and is easily seen to equal |εp|,as illustrated in Fig. 1.

From these elementary excitations, we can now build an exponentially large number of multi-

particle and multi-particle excitations. In the free electron theory, their energies are simple the

sum of the energies of the elementary excitations∑

p,α |εp|.

II. INTERACTING ELECTRON GAS

Our basic assumption is one of adiabatic continuity from the free electron gas. We imagine we

can tune the strength of the Coulomb interactions, and slowly turn them on from the free electron

theory. In this process, we assume that there is a correspondence between the ground states and

the elementary excitations of the free and interacting electron gas. So the state |G〉 in (3) evolves

smoothly to the unknown ground state of the interacting electron gas. And importantly, there

is also a correspondence in the excitations. In the ‘jellium’ model, with continuous translational

symmetry and a uniform background neutralizing charge, this correspondence is simply one-to-one:

a particle excitation with energy εp evolves into a ‘quasiparticle excitation’ with a modified value

of εp. And similarly, for a ‘quasihole’ with modified energy −εp. An important assumption will be

that εp remains a smooth function through the Fermi surface, and the energies of both particles

and holes is given by |εp|.In the presence of a lattice, the process of adiabatic evolution is more subtle, because we cannot

assume that εp is only a function of |p|. Consequently the shape of the Fermi surface can change

3

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"p = 0

FIG. 2. A point p0 on the Fermi surface, and its unit normal n.

in the adiabatic evolution, and a particle with momentum p can be inside the Fermi surface for

the free electron gas, and outside the Fermi surface for the interacting electron gas. We will see in

Section VII that even though the shape of the Fermi surface can evolve, the volume enclosed by the

Fermi surface is an adiabatic invariant. In the presence of a lattice, our basic assumption is that

there is a smooth function εp so that the Fermi surface is defined by εp = 0, and the excitation

energies of the quasiparticles and quasiholes is |εp|. Near, the Fermi surface, we assume a linear

dependence in momentum orthogonal to the Fermi surface: at a point p0 on the Fermi surface,

let the normal to the Fermi surface be the direction n (the value of pF can depend upon p0, see

Fig. 2), and so we can write for p close to p0

εp = vF (p− p0) · n, vF = |∇pεp| ≡ pF/m∗ , (6)

where pF = |p0|. This equation defines the Fermi momentum pF , the Fermi velocity vF , and the

effective mass m∗, all of which can depend upon the direction p0 in the presence of a lattice. Note

that ∇pεp = |∇pεp|n is a vector normal to the Fermi surface.

A further assumption in the theory of the interacting electron gas is that we can build up the

exponentially large number of other excitations also be composing the elementary excitations. (In

a finite system of size N , the number of elementary excitations is of order N , while the number

of composite excitations is exponentially large in N .) As we are interested in the thermodynamic

limit, we can characterize these excitations by the densities of quasiholes and quasiparticles. In

practice, it is quite tedious to keep track of two separate densities, along with a non-analytic

dependence of their excitation energy, |εp| on p. Both these problems can be overcome by a clever

mathematical trick; we emphasize that there is no physics assumption involved in this trick—it is

merely a bookkeeping device. We postulate that the interacting ground state has the same form

as the free electron ground state in (3). So the ground state has a density of quasiparticles n0(p)

4

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n0(p)

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n(p)

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n(p)

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0

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1

FIG. 3. Plot of the quasiparticle distribution functions n(p) and δn(p) of an excited state of the Fermi

liquid. Note that δn(p) has a discontinuity of unity at the Fermi surface.

given by

n0(p) = 1, p inside the Fermi surface

n0(p) = 0, p outside the Fermi surface (7)

as shown in Fig. 3. Then, an excited state is characterized by density of quasiparticles n(p), but

the excitation energy will depend only upon

δn(p) = n(p)− n0(p), (8)

where δn(p) has a discontinuity of unity at the Fermi surface. So for p outside the Fermi surface

δn(p) measures the density of quasiparticle excitations, while for p inside the Fermi surface −δn(p)

measures the density of quasihole excitations. (All of these densities can also depend upon the

spin of the quasiparticles or quasiholes, a complication we shall ignore in the following discussion.)

So the actual density of excitations with energy |εp| is |δn(p)|. For the total excitation energy,

which depends their product, we can drop the absolute value: this is one of the advantages of this

mathematical trick.

We assume we are at temperature T EF , so that the density of quasiparticles and quasiholes

is small. Our first thought would be that because of the low density, we can ignore the interactions

between the quasiparticles and quasiholes, and compute the total energy of the multiparticle/hole

excitations simply by adding their individual energies. An important observation by Landau was

that this is not correct. If we wish to work consistently to order (T/EF )2 in the total energy,

one (and only one) additional term is necessary; ignoring spin-dependence, we present the Landau

energy functional

E[δn(p)] =∑p

εpδn(p) +1

2V

∑p,k

Fp,k δn(p)δn(k) , (9)

5

where V is the volume of the system. At a temperature T EF , δn(p) is of order unity only in a

window of momenta with vF |p− pF | ∼ T where |εp| ∼ T . Then, as we perform the radial integral

in the first term in (3), we pick up a factor T from εp, and a second factor of T from the limits on

the integral: so the first term is of order T 2. Landau’s point is that the second term in (9) is also

of order T 2: there now are 2 integrals over radial momenta, and their product yields a factor of T 2.

This term describes the interaction between the quasiparticles and quasiholes, and is characterized

by the unknown Landau interaction function Fp,k. To order T 2, we can consistently assume that

all the quasiparticles and quasiholes are practically on the Fermi surface in the interaction term,

and so Fp,k depends only upon the directions of p and k.

Although the quasiparticles and quasiholes are assumed to interact Landau’s functional, the

interaction is conservative: i.e. it does not scatter quasiparticles between momenta, and change

the quasiparticle distribution function. The main effect of the interaction term is that the change

in the energy of the system upon adding a quasiparticle or quasihole depends upon the density

of excitations already present. We will consider scattering processes of quasiparticles later in

Section VI: these lead to a finite quasiparticle lifetime, but the correponding corrections to the

energy functional are higher order in T .

Landau’s central point is that the values of m∗ and Fp,k are sufficient to provide a description

of the low temperature properties of the interacting electron gas to order (T/EF )2, and all orders

in the strength of the underlying Coulomb interactions.

III. SPECIFIC HEAT

As a first application of Landau’s Fermi liquid theory, let us compute the specific heat. Assuming

a thermal distribution of excitations, we have, using the Fermi function f(ε) = 1/(eε/T + 1):

δn(p) = f(εp)− n0(p) (10)

Now using (7), the identity f(−ε) = 1− f(−ε), and approximating εp = vF (p− p0) · n at low T

close to the Fermi surface, we can easily show that δn(p) is an odd function of (p−p0) ·n at each

point on the Fermi surface p0. An immediate consequence is that the second term in (9) vanishes to

order T 2, because it has no further dependence upon momenta normal to the Fermi surface. More

physically stated, equal numbers quasiparticles and quasiholes are excited in thermal equilibrium,

and as the Landau interaction depends upon the total density of excitations, it does not contribute

to the total energy.

Consequently, the Landau interactions make no difference to the free energy, and so the specific

heat of the interacting electron gas is the same as that of the free electron gas, after replacing εp

by the true quasiparticle dispersion. So we have

CV = γT (11)

6

where the ‘γ-coefficient’ is given by

γ =π2k2BT

3g(0) (12)

with g(0) the density of quasiparticle states at the Fermi level. For the spinful jellium model in

d = 3

g(0) =m∗pF2π2~3

. (13)

IV. COMPRESSIBILITY

The compressibility measures the change in electron density in response to a change in the

chemical potential. As a change in the chemical potential has a different effect on quasiparticles

and quasiholes, the Landau interaction parameters now do have an important effect.

With the change µ → µ + δµ, we can argue from the Landau theory that the distribution of

quasiparticles is

δn(p) = f(ε∗p)− n0(p) (14)

where

ε∗p =δE

δn(p)

= εp − δµ+1

V

∑k

Fp,kδn(k) . (15)

We have now accounted for the fact that the energy of each quasiparticle depends upon the

density of other quasiparticles via the Landau interaction. We did not have to account for such a

dependence in the computation of the specific heat because the sum over k vanishes in that case.

Note that δn(p) appears on both the left and right hand sides of (14), and so has to be determined

self-consistently. This is rather similar to Hartree-Fock theory, where expectation values of fermion

bilinears appeared both in the Hartree-Fock Hamiltonian and in the self-consistency condition. The

difference, of course, is that the present considerations are exact at low T .

As T → 0, we expand the equations (14) and (15) to linear order in δµ. Using the low T identity

− ∂f

∂ε= δ(ε) (16)

we can conclude that the distribution function is of the form

δn(p) = Aδµ δ(εp) (17)

where A is a p-independent constant. Inserting (17) into (14) and (15), we obtain

A = 1− A

V

∑k

Fp,kδ(εk) . (18)

7

Combining all the results, we obtain the compressibility

dn

dµ=

1

δµ

1

V

∑p

δn(p)

=1

1 + F0

1

V

∑p

δ(εp)

=g(0)

1 + F0

, (19)

where

F0 =1

V

∑k

Fp,kδ(εk) . (20)

is the average of the Landau interaction parameter around the Fermi surface. For the case with

full rotational symmetry (as in jellium), we can decompose the Landau interactions into angular

momentum components F`, and F0 is then the s-wave component.

So there is a renormalization by a factor of 1/(1 + F0) of the compressibility from the Landau

interactions.

V. DYNAMIC RESPONSE FUNCTIONS

We can extend the ideas of Fermi liquid theory to include responses to time-dependent pertur-

bations. We place the Fermi liquid in an external potential V (r, t), and examine the response of

the quasiparticle distribution function. Then (15) is modified to

ε∗p = εp +1

V

∑k

Fp,kδn(k, r, t) + V (m, t) . (21)

where we have also allowed the quasiparticle distribution function to become time-dependent.

Provided the external potential is slowly varying in space, we can use semiclassical equations of

motion to describe the time and space evolution of the quasiparticles with average momentum p

and position r

d〈r〉dt

=∂ε∗p∂p

d〈p〉dt

= −∂ε∗p

∂r. (22)

From this we can write down the Boltzmann equation for the evolution of the quasiparticle distri-

bution function∂δn

∂t+d〈r〉dt

∂δn

∂r+d〈p〉dt

∂δn

∂p= Icol . (23)

The right hand side of (23) is the “collision” term, which scatters quasiparticles with around the

Fermi surface among themselves. The key assumption of Fermi liquid theory is that this scattering

rate is small, as we will see in Section VI. Neglecting Icol, we can solve (23) to obtain various

collective properties of Fermi liquids, including the existence of ‘zero sound’.

8

VI. GREEN’S FUNCTIONS AND QUASIPARTICLE LIFETIME

For further discussion of the properties of the Fermi liquid, and the nature of its corrections

when we consider higher temperatures, it is useful to employ the language of Green’s functions. We

use the standard many-body Green’s function defined in Ref. [1]. The most convenient definition

starts from the Green’s functions defined in imaginary time τ (ignoring the electron spin α)

G(p, τ) = −⟨Tτcp(τ)c†p(0)

⟩(24)

where Tτ is the time-ordering symbol. We can then Fourier transform this to the Matsubara

frequencies ωn = (2n+1)πT/~, n integer, to obtain G(p, iωn). More generally, we can consider the

Green’s function in the complex z plane, G(p, z), obtained by analytic continuation of G(p, iωn).

This Green’s function obeys the dispersion relation

G(p, z) =

∫ ∞−∞

dΩρ(p,Ω)

z − Ω(25)

where ρ(Ω) = −(1/π)Im [G(p,Ω + i0+)] is the spectral density. We will also refer to the retarded

Green’s function GR(p, ω) = G(p, ω + i0+), and more generally GR(p, z) = G(p, z) for z in the

upper-half plane. Closely associated is the electron self-energy Σ(p, z), which is related to the

Green’s function by Dyson’s equation

G(p, z) =1

z − ε0p − Σ(p, z)(26)

where by ε0p we now denote the bare electron dispersion before the effects of electron-electron

interactions are accounted for.

The postulates of Fermi liquid theory described above have strong implications for the structure

of the Green’s function in the complex frequency plane. Specifically, the existence of long-lived

quasiparticles near the Fermi surface implies that the Green’s function has a pole very close to

the real frequency axis, at a frequency obeying Re(z) = εp for p close to the Fermi surface. The

existence of such a pole implies a free particle behavior of the Green’s function at long times,

representing the propagation of the quasiparticle. In this section, we wish to go beyond Fermi

liquid theory and include a finite quasiparticle lifetime by taking the pole just off the real axis.

Actually, there is an important subtlety in the statement “there is a pole in the Green’s function”

that we need to keep in mind. The spectral definition (25) implies that G(p, z) is an analytic

function for all z, with a branch cut on the real frequency axis, for an interacting system with

a reasonably smooth spectral density ρ(p,Ω). The pole is actually in a different Riemann sheet

from the definition (25), and is reached by analytically continuing across the branch cut. So the

retarded Green’s function GR(p, z) is analytic for all z in the upper-half plane, and the pole is

obtained when we analytically continue GR(p, z) to the lower-half plane (where it is not equal to

9

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p < pF<latexit sha1_base64="w0nzc8daIazL0WgiIy0s8kgXh+0=">AAAB7HicdVBNS8NAEJ3Ur1q/qh69LBbBU0lqtPUiRUE8VjBtoQ1ls922SzebsLsRSuhv8OJBEa/+IG/+G7dtBBV9MPB4b4aZeUHMmdK2/WHllpZXVtfy64WNza3tneLuXlNFiSTUIxGPZDvAinImqKeZ5rQdS4rDgNNWML6a+a17KhWLxJ2exNQP8VCwASNYG8mLL+Leda9Yssv2iV2tuciQOQw5d09du4qcTClBhkav+N7tRyQJqdCEY6U6jh1rP8VSM8LptNBNFI0xGeMh7RgqcEiVn86PnaIjo/TRIJKmhEZz9ftEikOlJmFgOkOsR+q3NxP/8jqJHtT8lIk40VSQxaJBwpGO0Oxz1GeSEs0nhmAimbkVkRGWmGiTT8GE8PUp+p80K2XnrFy5dUv1yyyOPBzAIRyDA1Woww00wAMCDB7gCZ4tYT1aL9brojVnZTP78APW2ye/OI6p</latexit>

p > pF

FIG. 4. The poles of the Green’s function GR(p, z) in the complex z plane. The poles are in the second

Riemann sheet, and the red line represents the branch cut implied by (25).

the G(p, z) defined by (25)). For p close to the Fermi surface in a Fermi liquid, this pole is at a

frequency z = εp − iγp where γp > 0 is related to the quasiparticle lifetime τp = 1/(2γp) because

it leads to exponential decay for the Green’s function in real time (the factor of 2 arises because

we measure the probability of observing a quasiparticle a time τp after creating it). Note that the

pole is in the lower-half plane of the analytically continued GR(p, z) for both signs of εp i.e. for

both quasiparticles and quasiholes: see Fig. 4.

Ultimately, this complexity can be succinctly captured by initially restricting attention to the G

Green’s function on the imaginary frequency axis. Then, the existence of the quasiparticle implies

that the Green’s function defined by (25) obeys

G(p, iω) =Zp

iω − εp + iγp sgn(ω)+Ginc(p, iωn) , (27)

where ε0p is the ‘bare’ electron dispersion,

εp = ε0p + Re [Σ(p, 0)] (28)

is the ‘renormalized’ quasiparticle dispersion, and

γp = −Im[Σ(p, εp + i0+)

]> 0 . (29)

Consistency of the above definitions requires that the inverse lifetime of the quasiparticle is much

smaller than its excitation energy, i.e.

γp |εp| , (30)

for p close the Fermi surface. The Fourier transform of G has a slowly-decaying contribution

which is just that of a free particle but with renormalized dispersion, and an amplitude suppressed

by Zp. Consequently, Zp is the quasiparticle residue, and it equals the square of the overlap

10

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0

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1<latexit sha1_base64="HtHKTRuix9Muq5hLZPzFD8VVD3Q=">AAAB9HicbVDLSgNBEOyNrxhfUY9eBoMQL2E3iHoMevEYwcRAsoTZSW8yZHZ2nZkNhCXf4cWDIl79GG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrVkA1Ci6xYbgR2EoU0igQ+BgMb6f+4wiV5rF8MOME/Yj2JQ85o8ZKvuwiKWedICTJ5LxbLLkVdwaySrwFKcEC9W7xq9OLWRqhNExQrduemxg/o8pwJnBS6KQaE8qGtI9tSyWNUPvZ7OgJObNKj4SxsiUNmam/JzIaaT2OAtsZUTPQy95U/M9rpya89jMuk9SgZPNFYSqIick0AdLjCpkRY0soU9zeStiAKsqMzalgQ/CWX14lzWrFu6xU7y9KtZtFHHk4gVMogwdXUIM7qEMDGDzBM7zCmzNyXpx352PemnMWM8fwB87nD5BekVI=</latexit>

ne(p)

<latexit sha1_base64="sqzBhzNX4g/yzrXSefYqb90/57M=">AAAB+HicbVBNS8NAEN3Ur1o/GvXoZbEInkpSRD0WvXisYD+wCWWz3bRLN5uwOxFqyC/x4kERr/4Ub/4bt20O2vpg4PHeDDPzgkRwDY7zbZXW1jc2t8rblZ3dvf2qfXDY0XGqKGvTWMSqFxDNBJesDRwE6yWKkSgQrBtMbmZ+95EpzWN5D9OE+REZSR5ySsBIA7v6MMi8MYHMC0Kc5PnArjl1Zw68StyC1FCB1sD+8oYxTSMmgQqidd91EvAzooBTwfKKl2qWEDohI9Y3VJKIaT+bH57jU6MMcRgrUxLwXP09kZFI62kUmM6IwFgvezPxP6+fQnjlZ1wmKTBJF4vCVGCI8SwFPOSKURBTQwhV3NyK6ZgoQsFkVTEhuMsvr5JOo+5e1Bt357XmdRFHGR2jE3SGXHSJmugWtVAbUZSiZ/SK3qwn68V6tz4WrSWrmDlCf2B9/gDiV5M+</latexit>

Zp

FIG. 5. The momentum distribution function of bare electrons in a Fermi liquid at T = 0. There is a

discontinuity of size Zp on the Fermi surface.

between the free and quasiparticle wavefunctions. The Ginc term is the ‘incoherent’ contribution,

associated with additional excitations created from the particle-hole continuum upon inserting a

single particle into the system: this contribution decays rapidly in time, and can be ignored relative

the quasiparticle contribution for the low energy physics.

From (27), we can now compute the momentum distribution function ne(p) of the underlying

electrons;

ne(p) =⟨c†pcp

⟩, (31)

where we are dropping the spin index. For a free electron gas

ne(p) = θ(−ε0p), free electrons, T = 0, (32)

where θ(x) is the unit step function. So there is a discontinuity of size unity on the Fermi surface

in ne(p). For the interacting electron gas, it is important to distinguish ne(p) from the distribution

function of quasiparticles n(p) in (8). The quasiparticle momentum distribution function continues

to have a discontinuity of size unity on the Fermi surface εp = 0. For the electron momentum

distribution function at T = 0, we need to evaluate

ne(p) =

∫ ∞−∞

dω

2πG(p, iω)eiω0

+

(33)

Evaluating the integral in (33) using (27), we find a discontinuous contribution from the pole near

the Fermi surface. There is no reason to expect a discontinuity from Ginc, and so we obtain

ne(p) = Zp θ(−εp) + . . . , interacting electrons, T = 0, (34)

where . . . is the contribution from Ginc. We show a typical plot of ne(p) in Fig. 5. Because ne(p)

must be positive and bounded by unity, we have a constraint on the quasiparticle residue

0 < Zp ≤ 1 . (35)

11

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k

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k q

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p + q

FIG. 6. Decay of a quasiparticle with momentum p by scattering off a pre-existing quasiparticle with

momentum k to produce a quasiparticles of momena p + q and k − q.

Note that a small Zp is not an indication that the Fermi liquid theory is not robust: it merely

indicates a small overlap between the bare electron and the renormalized quasiparticle. Systems

with very small Zp can be very good Fermi liquids: we will study the heavy-fermion compounds

which are of this type. Rather it is a short quasiparticle lifetime, or large γp, and the failure of

(30), which is a diagnostic of the breakdown of Fermi liquid theory.

For an explicit evaluation of the inverse lifetime γp, we have to consider processes beyond those

present in Landau Fermi liquid theory. In particular, we have to evaluate the imaginary part of the

self energy in (29) for p near the Fermi surface. This requires a somewhat tedious to evaluation

of the relevant Feynman diagrams. For now, we will be satisfied here by ‘guessing’ the answer by

Fermi’s golden rule. Assuming only a contact interaction, U , between the quasiparticles, we can

write down the inverse lifetime

1

τp= 2γp = 2πU2 1

V 2

∑k,q

f(εk)[1− f(εp+q)][1− f(εk−q)]

× δ (εp + εk − εp+q − εk−q) . (36)

This obtained by employing Fermi’s golden role to the process sketched in Fig. 6, and including

probabilities that the initial states are occupied, and the final states are empty. The momentum

integrals in (36) are quite difficult to evaluate in general, but it is not difficult to see that the result

becomes very small for p near the Fermi surface and small T , because of the constraints imposed

by the Fermi functions and the energy conserving delta function. A simple overestimate can be

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made by simply ignoring the constraints from momentum conservation, in which case we obtain

γε ∼ U2[d(0)]3p−dF

∫ ∞−∞

dε1dε2dε3f(ε1)[1− f(ε2)][1− f(ε3)]

× δ(ε+ ε1 − ε2 − ε3)

= U2[d(0)]3p−dF ×π2T 2/4 for ε = 0

ε2/2 for T = 0 .(37)

More careful considerations of momentum conservations are needed to obtain the precise co-

efficients, but they show that the power-laws above in T and ε are correct. So at low temperatures,

γp ∼ T 2 is always much smaller than |εp| ∼ T , and this justifies Fermi liquid theory.

We can also use these results to give a formal definition of the Fermi surface using Green’s

functions. Notice that γε in (37) vanishes as ε → 0 at T = 0. This follows from the vanishing of

the phase space for the decay of an excitation with energy ε as ε → 0. This is actually a special

case of a more general phenomenon following from the stability of the ground state, and does not

even require excitations to be close to the Fermi surface. The more general statement is

Im[Σ(p,Ω + i0+)

]→ 0 as Ω→ 0 at T = 0 (38)

for any p, and its validity can be checked by examining the structure of the Feynman graph

expansion for Σ. We can now define the Fermi surface by the pole in the Green’s function which

is determined by

G−1(pF , i0+) = 0 at T = 0. (39)

By (38), the left hand side of (39) is real, and so the solution of (39) determines a surface of

co-dimension 1 in p space, which is the Fermi surface.

VII. THE LUTTINGER RELATION

We have already alluded to one of the most remarkable features of Fermi liquid theory: the

momentum space volume enclosed by the Fermi surface defined by (39) is independent of the

interactions, and depends only on the total electron density. Actually, this result is more general

than Fermi liquid theory, and holds also in non-Fermi liquids without quasiparticle excitations.

Moreover there are deep connections of this ‘classic’ result to key ideas in the modern theories of

phases with fractionalization and anomalies.

We will present a proof of the Luttinger relation following the classic text book treatments, but

will use an approach which highlights its connections to the modern developments. Specifically,

there is a fundamental connection between the Luttinger relation and U(1) symmetries [2, 3]: any

many body quantum system has a Luttinger relation associated with each U(1) symmetry, and this

connects the density of the U(1) charge in the ground state to the volume enclosed by its Fermi

13

surfaces. This relation applies both to systems of fermions and bosons, or of mixtures of fermions

and bosons. However, the relation does not apply if the U(1) symmetry is ‘broken’ or ‘Higgsed’

by the condensation of a boson carrying the U(1) charge. As bosons are usually condensed at low

temperatures, the Luttinger relation is not often mentioned in the context of bosons. However,

there can be situations when bosons do not condense e.g. if the bosons bind with fermions to form

a fermionic molecule, and then the molecules form a Fermi surface: then we will have to apply the

Luttinger relation to the boson density [2].

We begin by noting a simple argument on why there could even be a relation between a short

time correlator (the density, given by an ‘ultraviolet’ (UV) equal-time correlator) and a long-time

correlator (the Fermi surface is the locus of zero energy excitations in a Fermi liquid, an ‘infrared’

(IR) property). In the fermion path integral, the free particle term in the Lagrangian is

L0c =

∑p

c†p

(∂

∂τ+ ε0p − µ

)cp, (40)

where we have now chosen to extract the chemical potential µ explicitly from the bare dispersion

ε0p. The expression in (40) is invariant under global U(1) symmetry

cp → cpeiθ , c†p → c†pe

−iθ (41)

as are the rest of the terms in the Lagrangian describing the interactions between the electrons.

However, let us now ‘gauge’ this global symmetry by allowing θ to have a linear dependence on

imaginary time τ :

cp → cpeµτ , c†p → c†pe

−µτ (42)

Note that in the Grassman path integral, cp and c†p are independent Grassman numbers and so

the two transformations in (42) are not inconsistent with each other. The interaction terms in the

Lagrangian are explicitly invariant under the time-dependent U(1) transformation in (42). The

free particle Lagrangian in (40) is not invariant under (42) because of the presence of the time

derivative term; however, application of (42) shows that µ cancels out of the transformed L0c , and

so has completely dropped out of the path integral. We seem to have reached the absurd conclusion

that the properties of the electron system are independent of µ: this is explicitly incorrect even

for free particles.

What is wrong with the above argument which ‘gauges away’ µ by the transformation in (42)?

The answer becomes clear from the expression for the total electron density

ρe =1

V

∑p

∫ ∞−∞

dω

2πG(p, iω)eiω0

+

. (43)

The transformation in (42) corresponds to a shift in frequency ω → ω + iµ of the contour of

integration, and this is not permitted because of singularities in G(p, iω). However, as show

14

below, it is possible to manipulate (43) into an ‘anomalous’ part which contains the full answer,

and a remainder which vanishes because manipulations similar to the failed frequency shift in (42).

The key step to extracting the anomalous part is to use the following simple identity which

follows directly from Dyson’s equation (26)

G(p, iω) = Ganom(p, iω) +GLW (p, iω)

Ganom(p, iω) ≡ i∂

∂ωln [G(p, iω)]

GLW (p, iω) ≡ −iG(p, iω)∂

∂ωΣ(p, iω) . (44)

The anomalous part is Ganom: it is a frequency derivative, and so its frequency integral in (43)

is not difficult to evaluate exactly after carefully using the eiω0+

convergence factor. Indeed, the

remaining contribution from GLW vanishes for free particles (which have vanishing Σ), and so the

contribution of Ganom to (43) must be the same as that for free particles, and measure the volume

enclosed by the Fermi surface; we will see explicitly below that this is indeed the case. Therefore,

establishing the Luttinger relation, i.e. the invariance of the volume enclosed by the Fermi surface,

reduces to establishing that the contribution of GLW to (43) vanishes.

We consider the latter important step first. We would like to show that∑p

∫ ∞−∞

dω

2πGLW (p, iω) = 0 . (45)

We now show that (45) follows from the transformations of GLW under the gauge transformation

in (42) for an imaginary chemical potential

cp → cpe+iω0τ , c†p → c†pe

−iω0τ . (46)

The argument relies on the existence of a functional, ΦLW [G(p, iω)], of the Green’s function, called

the Luttinger-Wald functional, so that the self energy is its functional derivative

Σ(p, iω) =δΦLW

δG(p, iω). (47)

The existence of such a functional can be seen diagrammatically, in which the Luttinger-Ward

functional equals the interaction dependent terms for the free energy written in a ‘skeleton’ graph

expansion in terms of the fully renormalized Green’s function. Taking the functional derivative

with respect to G(p, ω) is equivalent to cutting a single G from all such graphs in all possible ways,

and these are just the graphs for the self energy. For a more formal argument, see Ref. [4]. An

important property of the Luttinger-Ward functional is its invariance under frequency shifts

Φ [G(p, iω + iω0)] = Φ [G(p, iω)] , (48)

15

for any fixed ω0. Here, we are regarding Φ as functional of two distinct functions f1,2(ω), with

f1(ω) ≡ G(p, iω + iω0) and f2(ω) = G(p, iω), and Φ evaluates to the same value for these two

functions. Now note that this frequency shift is nothing but the gauge transformation in (46);

therefore (48) follows from the fact that such frequency shifts are allowed in ΦLW . The singularity

on the real frequency axis is sufficiently weak so that the frequency shifts are legal in a Fermi

liquid; but we note that in the non-Fermi liquid SYK model, the Green’s functions are singular at

ω = 0, and the analogs of (45) and (48) do not apply. For the Fermi liquid, we can now expand

(48) to first order in ω0, using (47), and integrating by parts we establish (45).

Now that we have disposed of the offending term in (44), we can return to (43) and evaluate

ρe =i

V

∑p

∫ ∞−∞

dω

2π

∂

∂ωln [G(p, iω)] eiω0

+

. (49)

We will evaluate the ω integral by distorting the contour in the frequency plane. For this, we need

to carefully understand the analytic structure of the integrand. This is subtle, because there are

two types of branch cuts. One branch cut arises from the Green’s function: G(p, z) has a branch cut

along the real axis Im(z) = 0, with ImG(p, z) ≤ 0 for Im(z) = 0+, ImG(p, z) ≥ 0 for Im(z) = 0−

and ImG(p, z) = 0 for z = 0. The other branch cut is from the familiar ln(z) function: we take

this on the positive real axis, with a discontinuity of 2iπ. First, we account for the branch cut in

G(p, z), by distorting the contour of integration in (49) to pick up the discontinuity ImG(p, z)

ρe =−iV

∑p

∫ 0

−∞

dz

2π

∂

∂zln

[G(p, z + i0+)

G(p, z + i0−)

]. (50)

Note from (26) and (38) that on real frequency axis ImG(p, z + i0±) → 0 as z → 0 or −∞.

Consequently the only possible values of ln[G(p, z + i0+)/G(p, z + i0−)] are 0,±2πi as z → 0 or

−∞, from the branch cut of the logarithm. So we obtain from (50)

ρe =−i

2πV

∑p

ln

[G(p, i0+)

G(p, i0−)

]=

1

V

∑p

θ(−ε0p + µ− Σ(p, i0+)

)=

1

V

∑p

θ (−εp) . (51)

Because the branch cut of the logarithm to extends to z = +∞, only negative values of εp

contribute to the z integral extending from z = −∞ to z = 0. Eqn. (51) is the celebrated

Luttinger relation, equating the electron density to the volume enclosed by the Fermi surface of

the quasiparticles εp = 0. In the presence of a crystalline lattice, there can be additional bands

which are either fully filled or fully occupied: such bands will yield a contribution of unity or zero

respectively to (51).

16

To summarize, the Luttinger relation is intimately connected to the U(1) symmetry of electron

number conservation. Indeed, we can obtain a Luttinger relation for each U(1) symmetry of any

system consisting of fermions or bosons. The result follows from the invariance of the Luttinger-

Ward functional under the transformation in (46), in which we gauge the global symmetry to a

linear time dependence: in this respect, there is a resemblance to ’tHooft anomalies in quantum

field theories. If the U(1) symmetry is ‘broken’ by the condensation of a boson which carries U(1)

charge, then the Luttinger relation no longer applies.

A. Disordered systems

Our discussion of the Luttinger relation has so far assumed perfect crystalline symmetry, so

the quasiparticles energies εp are functions of the crystal momentum p. The Luttinger relation

applies also to systems without crystalline symmetry, although it is expressed in a form involving

quantities that are not easy to observe.

Let us consider a lattice of sites i, with a bare electron hopping tij which has no particular

symmetry. Then the electron Green’s function Gij(iω) is a matrix indexed by the lattice sites, as

is the self-energy Σij(iω). These are related by Dyson’s equation, which now has a matrix form

[(iω + µ)δij − tij − Σij(iω)]Gjk = δik . (52)

The low-lying quasiparticles are no longer plane-wave eigenstates, but the arguments leading to

(38) still apply, and we have

Im[Σij(Ω + i0+)

]→ 0 as Ω→ 0 at T = 0 (53)

We now proceed with the computation of the average density, which generalizes (43) to

ρe =1

N

∫ ∞−∞

dω

2πTr [G(iω)] eiω0

+

. (54)

where N is the number of sites, and the trace is over the site indices. The analysis is then a close

parallel of that carried out for clean systems. The existence of the Luttinger-Ward functional now

replaces (45) by ∫ ∞−∞

dω

2πTr

[G(iω)

∂

∂ωΣ(iω)

]= 0 , (55)

where a matrix multiplication is implied between G and Σ. The analysis from (49) to (51) is

replaced by

ρe =i

N

∫ ∞−∞

dω

2π

∂

∂ωTr ln [G(iω)] eiω0

+

=1

N

∑α

θ(−εα) , (56)

17

where εα are the eigenvalues of the matrix tij − µδij + Σij(i0+). In general, we do not know the

values of Σij(i0+), and so this result is not easy to apply. However, it does yield information on

the nature of the quasiparticles, which are the eigenstates with small |εα|.

[1] H. Bruus, K. Flensberg, and O. U. Press, Many-Body Quantum Theory in Condensed Matter Physics:

An Introduction, Oxford Graduate Texts (OUP Oxford, 2004).

[2] S. Powell, S. Sachdev, and H. P. Buchler, “Depletion of the Bose-Einstein condensate in Bose-Fermi

mixtures,” Phys. Rev. B 72, 024534 (2005), cond-mat/0502299.

[3] P. Coleman, I. Paul, and J. Rech, “Sum rules and Ward identities in the Kondo lattice,” Phys. Rev.

B 72, 094430 (2005), cond-mat/0503001.

[4] M. Potthoff, “Non-perturbative construction of the Luttinger-Ward functional,” Condens. Mat. Phys

9, 557 (2006), arXiv:cond-mat/0406671.

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