Ferimi Liquid123

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CCMS Summer 2007 Lecture Series

Fermi- and non-Fermi Liquids

Lecture 3: Fermi-liquid Theory

Dmitrii L. Maslov

[email protected]

(Dated: July 22, 2007)

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Notation 1 Here and thereafter, L1 stands for Lecture 1: Basic Notions of the Many-body

Physics and L2 stands for Lecture 2: Examples of Interacting Fermi systems.

I. GENERAL CONCEPTS

All Fermi systems (metals, degenerate semiconductors, normal He3, neutron stars, etc.)

belong to the categories of either moderately or strongly interacting systems. For example,

in metals rs in the range from 2 to 5. (There are only few exceptions of this rule; for

example, bismuth, in which the large value of the background dielectric constant brings the

value of rs to 0.3 and GaAs heterostructures in which the small value of the effective mass

–0.07 of the bare mass–leads to the higher value of the Fermi energy and thus to rs < 1).

On the other hand, as we learned from the section on Wigner crystallization, the critical rs

for Wigner crystallization is very high –about 100 in 3D and about 40 in 2D. Thus almost

all Fermi systems occurring in Nature are too strongly interacting to be described by the

weak-coupling theory (Coulomb gas) but too weakly interacting to solidify. Landau put

forward a hypothesis that an interacting Fermi system is qualitatively similar to the Fermi

gas [6] . Although original Landau’s formulation refers to a translationally invariant system

of particles interacting via short-range forces, e.g., normal He3, later on his arguments were

extended to metals (which have only discrete symmetries) and to charged particles.

Experiment gives a strong justification to this hypothesis. The specific heat of almost

all fermionic systems (in solids, one need to subtract off the lattice contribution to get the

one from electrons) scales linearly with temperature: C (T ) = γ ∗T. In a free Fermi gas,

γ ∗ = γ = (π2/3) ν F = (1/3) mkF . In a band model, when non-interacting electrons move in

the presence of a periodic potential, one should use the appropriate value of the density of

states at the Fermi level for a given lattice structure. In reality, the coefficient γ ∗ can differ

significantly from the band value but the linearity of C (T ) in T is well-preserved. In those

cases when one can change number density continuously (for example, by applying pressure

to normal He3), γ ∗ is found to vary. One is then tempted to assume that the interacting

Fermi liquid is composed of some effective particles (quasi-particles) that behave as free

fermions albeit their masses and other characteristics are different from the non-interacting

values.

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Fermi liquid (or single band metal) can be written as

δE =

d3 p

(2π)3εpδnp. (1.1)

(For the sake of simplicity, I also assume that the system is isotropic, i.e., the energy and

n depend only on the magnitude but not the direction of the momentum but the argument

can be extended to anisotropic systems as well). In this formula, np is the distribution

function of quasi-particles which are elementary excitations of the interacting system. The

variation of np occurs when, e.g., a new particle is added to the system. However, these

quasi-particles are not completely free and, therefore, unlike in the phonon analogy, the

total energy is not equal to the sum of individual energies. However, the variation of the

total energy δE is related to δnp via (1.1). In fact, this relation serves as a definition of the

quasiparticle energy via the variational derivative of E :

εp = δE

δnp.

One more –and crucial assumption–is that the quasi-particles of an interacting

Fermi system are fermions, which is not at all obvious. For example, regardless of the

statistics of individual atoms which can be either fermions or bosons, phonons are always

bosons. Landau’s argument was that if quasi-particles were bosons they could accumulate

without a restriction in every quantum state. That means that an excited state of a quantumsystem has a classical analog. Indeed, an excited state of many coupled oscillators is a

classical sound wave. Fermions don’t have macroscopic states so quasi-particles of a Fermi

systems must be fermions (to be precise they should not be bosons; proposals for particles

of a statistics intermediate between bosons and fermions–anyons–have been made). On

quite general grounds, one can show that quasi-particles must have spin 1/2 regardless of

(half-integer) spin of original particles [3],[4]. (Thus, quasi-particles of a system composed

of fermions with spin S = 3/2 still have spin 1/2.) Therefore, the quasiparticle energy andthe occupation number are the operators in the spin space represented by 2 × 2 matrices ε̂and n̂. As long as one has a system of fermions with doubly (S z = ±1/2) degenerate states,the entropy on purely combinatorial grounds is

S/vol = −Tr

d3 p

(2π)3 [n̂p ln n̂p − (1 − n̂p)ln(1 − n̂p)] ,

which is the same expression in for a Fermi gas. The equlibrium occupation numbers are

obtained by equating the variation of S at fixed volume to zero. As for a Fermi gas, this

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gives

n̂ = 1

expε̂−µT

+ 1

, (1.2)

In contrast to the Fermi-gas, however, ε̂ is a functional of n̂ itself.

If the system is not in the presence of the magnetic field and not ferromagnetic,

ε̂αβ = εδ αβ , n̂αβ = nδ αβ .

In a general case, instead of (1.1) we have

δE =

d3 p

(2π)3Trε̂ ( p) δ ̂n ( p) ,

which, for a spin-isotropic liquid, reduces to

δE = 2

d3 p

(2π)3Trε ( p) δn ( p) .

The occupation number is normalized by the condition

δN =

d3 p

(2π)3Trδn ( p) = 2

d3 p

(2π)3δn ( p) = 0,

where N is the total number of real particles.

For T=0, the chemical potential coincides with energy of the topmost state

µ = ε ( pF ) ≡ E F .

Another important property (known as Luttinger theorem) is that the volume of the Fermi

surface is not affected by the interaction. For an isotropic system, this means the Fermi

momenta of free and interacting systems are the same. A simple argument is that the

counting of states is not affected by the interaction, i.e., the relation

N = 24πp3

F /3(2π)3

holds in both cases. A general proof of this statement is given in Ref. [ [3]].

B. Interaction of quasi-particles

Phonons in a solid do not interact only in the first (harmonic) approximation. Anhar-

monism results in the phonon-phonon interaction. However, the interaction is weak at low

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energies not really because the coupling constant is weak but rather because the scattering

rate of phonons on each other is proportional to a high power of their frequency: τ −1ph-ph ∝ ω5.As a result, at small ω phonons are almost free quasi-particles. Something similar happens

with the fermions. The nominal interaction may as well be strong. However, because of the

Pauli principle, the scattering rate is proportional to (ε− εF )2 and weakly excited statesinteract only weakly. In the Landau theory, the interaction between quasi-particles is intro-

duced via a phenomenological interaction function defined by the proportionality coefficient

(more precisely, a kernel) between the variation of the occupation number and the corre-

sponding variation in the quasi-particle spectrum

δεαβ = d3 p

(2π)3f αγ,βδ ( p, p

) δnγδ ( p) .

Function f αγ,βδ ( p, p) describes the interaction between the quasi-particles of momenta p and

p (notice that these are both initial states of the of the scattering process). Spin indices

α and β correspond to the state of momentum p whereas indices γ and δ correspond to

momentum p. In a matrix form,

δ ̂ε = Tr d3 p

(2π)3f̂ ( p, p) δ ̂n ( p) , (1.3)

where Tr

denotes trace over spin indices γ and δ. For a spin-isotropic FL, when δ ̂ε = δ αβ δεand δ ̂n = δ αβ δn external spin indices (α and β ) can also be traced out and Eq.(1.3) reduces

to

δε =

d3 p

(2π)3f ( p, p) δn ( p) ,

where

f ( p, p) ≡ 12

TrTr f̂ ( p, p) .

For small deviations from the equilibrium, δn ( p) is peaked near the Fermi surface. Function

f̂ ( p, p) can be then estimated directly on the Fermi surface, i.e., for | p| = | p| = pF . Then f̂ depends only on the angle between p and p. The spin dependence of f̂ can be established on

quite general grounds. In a spin-isotropic FL, f̂ can depend only on the scalar product of

spin operators but on the products of the individual spin operators with some other vectors.

Thus the most general form of f̂ for a spin-isotropic system is

ν ∗ f̂ ( p, p) = F s (θ) Î + F a (θ) σ̂ · σ̂,

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where Î is the unity matrix, σ̂ is the vector of three Pauli matrices, θ is the angle between

p and p, and the density of states was introduced just to make functions F (θ) and G (θ)

dimensionless. (Star in ν ∗ means that we have used a renormalized value of the effective

mass so that ν

∗

= m

∗

kF /π

2

, but this is again just a matter of convenience.) Explicitly,

ν ∗f αγ,βδ ( p, p) = F s (θ) δ αβ δ γδ + F

a (θ) σ̂αβ · σ̂γδ . (1.4)

In general, the interaction function is not known. However, if the interaction is weak,

one can relate f̂ to the pair interaction potential. The microscopic theory of a Fermi

liquid [3] shows that the Landau interaction function is related to the interaction vertex

Γαβ,γδ ( pε, pε| p + q, ε + ω, p − q, ε − ω), which describes scattering of two fermions in ini-

tials states with momenta, energies, and spin projections p, ε, α and p

, ε

, β, respectivelyinto the final states p + q, ε + ω, γ and p, ε + ω, δ via

f αβ,γδ ( p, p) = Z 2 lim

q/ω→0Γαβ,γδ ( pε, pε

| p + q, ε + ω, p − q, ε− ω) ,

where Z is the renormalization factor of the Green’s function.

Example 2 At first order in the interaction, Γ is represented by two diagrams, one of which

is obtained from the other by a permutation of outgoing lines. If the interaction conserves

spin,

Γαβ,γδ ( pε, pε| p + q, ε + ω, p − q, ε− ω) = δ αγ δ βδU (q ) − δ αδδ βγ U ( p− p + q ) .

Using an SU(2) identity

δ αδδ βγ = 1

2 (δ αγ δ βδ + σ̂αγ ·σ̂βδ) ,

this expression for Γ reduces to

Γαβ,γδ ( pε, pε| p + q, ε + ω, p − q, ε − ω) = δ αγ δ βδ

U ( q ) − 1

2U ( p− p + q )

−12

σ̂αγ ·σ̂βδU ( p − p + q ) .

Exchanging indices β and γ and taking the limit q → 0 (at this order the vertex does not depend on the energy), we obtain for f̂

f αγ,βδ ( p, p) = δ αβ δ γδ

U (0) − 1

2U (| p− p|)

− 1

2σ̂αβ · σ̂γδU (| p− p|) .

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[Because this relation holds only at first order of the perturbation theory, Z still equals to 1.]

On the Fermi surface, | p − p| = 2 pF sin θ/2. Comparing this expression with Eq.(1.4), we find that

F s (θ) = ν −1

U (0) − 12U (2 pF sin θ/2)

(1.5)

F a (θ) = −12

U (2 pF sin θ/2) .

Notice that a repulsive interaction (U > 0) corresponds to the attraction in the spin-exchange

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In the second term re-label the variables p → p, p → p , use the fact that f ( p, p) = f ( p, p)and integrate by parts

d3 p

(2π)3 pδn =

d3 p

(2π)3m

∂ε

∂ pδn −

d3 p

(2π)3

d3 p

(2π)3f ( p, p)

∂n ( p)

∂ p δn ( p) .

Because this equation should be satisfied for an arbitrary variation δn ( p) ,

p

m =

∂ε

∂ p −

d3 p

(2π)3f ( p, p)

∂n ( p)

∂ p .

Near the Fermi surface, the quasi-particle energy can be always written as

ε ( p) = v∗F ( p − pF )

so that

∂ε∂ p = v∗F ˆ p = pF m∗ ˆ p,

where ˆ p is the unit vector in the direction of p, and thus

p

m =

pF ˆ p

m∗ −

d3 p

(2π)3f ( p, p)

∂n ( p)

∂ p

Now∂n ( p)

∂ p =

∂ε ( p)

∂ p∂n

∂ε .

Near the Fermi surface,

∂n∂ε

= −δ (ε − E F )

and p

m =

pF ˆ p

m∗ +

1

2ν ∗F

dΩ

4π f ( p, p)

pF ˆ p

m∗

or p

m =

pF ˆ p

m∗ +

1

2ν ∗F

dΩ

4π f ( p, p)

pF ˆ p

m∗ =

pF ˆ p

m∗ +

1

2

pF π2

dΩ

4π f ( p, p) pF ˆ p

.

Setting | p| = pF and multiplying both sides of the equation by ˆ p, we obtain

1m

= 1m∗

+ 12

pF π2

dΩ

4πf (θ)cos θ

or1

m∗ =

1

m − pF

2π2

dΩ

4πf (θ)cos θ. (1.7)

This is the Landau’s formula for the effective mass. Noticing that f (θ) = (1/2)TrTr f̂ (θ) =

(2/ν ∗F ) F (θ) = (2π2/m∗ pF ) , the last equation can be reduced to

m∗

m = 1 +

dΩ

4πF (θ)cos θ ≡ 1 + F 1.

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Although we do not know the explicit form of f (θ) , some useful conclusions can be made

already from the most general form [Eq.(1.7)]. If f (θ)cos θ is negative, 1/m∗ > 1/m →m∗ < m; conversely, if f (θ)cos θ is positive, m∗ > m. For the weak short-range interaction,we know that m∗ > m, whereas for a weak long-range (Coulomb) interaction m∗ < m. Now

we see that both of these cases are described by the Landau’s formula. Recalling the weak-

coupling form of F (θ) [Eq.(1.5)], we find

F 1 =

dΩ

4πF (θ)cos θ =

dΩ

4π

U (0) − 1

2U

2 pF sin

θ

2

cos θ

= − dΩ

4π

1

2U

2 pF sin

θ

2

cos θ.

If U is repulsive and picked at small θ where cos θ is positive, F 1 < 0 and m∗ < m. This

case corresponds to a screened Coulomb potential. If U is repulsive and depends on θ onlyweakly (short-range interaction), the angular integral is dominated by values of θ close to

π, where cos θ m. In general, forward scattering reduces the effective mass,

whereas large-angle scattering enhances it.

E. Spin susceptibility

1. Free electrons

We start with free electrons. An electron with spin s has a magnetic moment µ = 2µBs,

where µB = eh̄/2mc is the Bohr magneton. Zeeman splitting of energy levels is ∆E =

E ↑ − E ↓ = µBH − (−µBH ) = 2µBH. The number of spin-up and -down electrons is foundas an integral over the density of states

n↑,↓ = 1

2

E F ±µBH 0

dεν (ε) .

The Fermi energy now also depends on the magnetic field but the dependence is only

quadratic (why?), whereas the Zeeman terms are linear in H. For weak fields, one can ne-

glect the field-dependence of E F . Total magnetic moment per unit volume–magnetization–is

found by expanding in H

M = µB (n↑ − n↓) = µB2

E F +µBH 0

dεν (ε) − E F −µBH 0

dεν (ε)

= µB

2

E F 0

dεν (ε) + µBHν F − E F 0

dεν (ε) + µBHν F

= µ2BHν F .

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Spin susceptibility

χ = ∂M

∂H = µ2Bν F (1.8)

is finite and positive, which means that spins are oriented along the external magnetic field.

Notice that if, for some reason the bare g- factor of electrons is not equal to 2, thenEq.(1.8) changes to

χ = g

2µ2Bν F (1.9)

2. Fermi liquid

In a Fermi liquid, the energy of a quasi-particle in a magnetic field is changed not only

due to the Zeeman splitting (as in the Fermi gas) but also because the occupation number

is changed. This effect brings in an additional term in the energy as the energy is related

to the occupation number. This is expressed by following equation

δ ̂ε = −µBH · σ̂ + Tr

d3 p

(2π)3f̂ ( p, p) δn ( p) .

The first term is just the Zeeman splitting, the second one comes from the interaction. Now,

δ ̂n = ∂ ̂n

∂εδ ̂ε,

and we have an equation for δ ̂ε

δ ̂ε = −µBH · σ̂ + Tr

d3 p

(2π)3f̂ ( p, p)

∂ ̂n

∂ε δ ̂ε ( p) .

Replacing ∂ ̂n∂ε

by the delta-function and setting | p| = pF

δ ̂ε ( pF ˆ p) = −µBH · σ̂ − Trν ∗F dΩ

4πf̂ ( pF ˆ p, pF ˆ p

) δ ̂ε ( pF ˆ p) . (1.10)

Let’s try a solution in the following form

δ ̂ε = −µB2

gH · σ̂, (1.11)

where g has the meaning of an effective g− factor. For free electrons, g = 2. Recalling thatν ∗F f̂ = F (θ)

Î + G (θ) σ̂ · σ̂ and substituting (1.11) into (1.11), we obtain

−µB2

gH · σ̂ = − µBH · σ̂ − Tr

dΩ

4π

F s (θ) Î + F a (θ) σ̂ · σ̂

−µB

2 gH · σ̂

.

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The term containing F in the integral vanishes because Pauli matrices are traceless: Trσ = 0.

The term containing G is re-arranged using the identity Tr(σ̂ · σ̂) σ̂ = 2σ̂ upon which weget

g

2 = 1

− g

2F a0

,

where

F a0 =

dΩ

4πF a (θ) .

Therefore,

g = 2

1 + F a0.

Substituting this result into Eq.(1.9) and replacing ν F by its renormalized value ν ∗F , we

obtain the spin susceptibility of a Fermi liquid

χ∗ = µ2Bν

∗F

1 + F a0= χ

1 + F s11 + F a0

,

where χ is the susceptibility of free fermions. Notice that χ∗ differs from χ both because

the effective mass is renormalized and because the g-factor differs from 2. When F a0 = −1,the g-factor and susceptibility diverge signaling a ferromagnetic instability. However, even

if G0 = 0, χ∗ is still renormalized in proportion to the effective mass. If m∗ diverges (which

is a signature of a metal-insulator transition of Mott type), χ∗ diverges as well. Recall

that [Eq.(1.5)] in the weak-coupling limit F a (θ) = − 12

U (2 pF sin θ/2) . Thus, for repulsive

interactions g > 2 which signals a ferromagnetic tendency. This is another manifestation of

general principle that repulsively interacting fermions tend to have spin aligned to minimize

the energy of repulsion. For normal He3, G0 ≈ −2/3.

F. Zero sound

All gases and liquids support sound waves. Even ideal gases have finite compressibilities

and therefor finite sound velocities

s =

∂P

∂ρ .

For example, in an ideal Boltzmann gas P = nT = ρT/m and

s =

T

m =

1√ 3

vT ,

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at xxx.lanl.gov/abs/comd-mat/9503150.

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statistical physics , (Dover Publications, New York, 1963).

[4] E.M. Lifshitz and L.P. Pitaevskii, Statistical Physics II, (Course of Theoretical Physics, v.

IX), Pergamon Press, Oxford, 1980.

[5] D. Pines and P. Nozieres, The Theory of Quantum Liquids , Benjamin, New York, 1969.

[6] L. D. Landau, Zh. Eksp. Teor. Fiz. 30, 1058 (1956)–[Sov. Phys. JETP 3, 920 (1957)]. Reprinted

in: D. Pines, The Many-Body Problem , Benjamin, Reading, MA.

[7] B. L. Altshuler and A. G. Aronov, in Electron-electron interactions in disordered conductors ,

edited by A. L. Efros and M. Pollak (Elsevier, 1985), p. 1.

[8] G. Zala, B. N. Narozhny, and I. L. Aleiner, Phys. Rev. B 64, 214204 (2001).

[9] E. Abrahams, S. V. Kravchenko, and M. P. Sarachik, Rev. Mod. Phys. 73, 251 (2001).

[10] B. L. Altshuler, D. L. Maslov, and V. M. Pudalov,

Physica E 9, 209 (2001).

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