Landau s Fermi Liquid Theory - PDF Free Download

Thors Hans Hansson Stockholm University

Outline 1 Fermi Liquids Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids How? 2 Landau s Phenomenological Approach The free Fermi gas Landaus basic idea The Energy Functional E[n(p)] The effective mass The quasiparticle Fermi surface When can Fermi Liquid theory be used? Quasiparticle interactions 3 Renormalization Group approach to FL theory An RG primer Fermi Liquids as a RG fixed point

Next subject 1 Fermi Liquids Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids How? 2 Landau s Phenomenological Approach The free Fermi gas Landaus basic idea The Energy Functional E[n(p)] The effective mass The quasiparticle Fermi surface When can Fermi Liquid theory be used? Quasiparticle interactions 3 Renormalization Group approach to FL theory An RG primer Fermi Liquids as a RG fixed point

Fermi liquids - Why do we want it? The free, i.e. non-interacting, Fermi gas give basic understanding of both cold Fermi systems. In particular, adding neutralizing or confining potentials it gives a qualitative understanding of Specific heat of (many) metals at low temperature The formation of neutron stars Adding a periodic potential, the band theory of non-interacting electrons also explains Conductors semiconductors metals Shape of Fermi surface and related spectroscopy The success of this ridiculously oversimplified model is not a coincidence. The theory of Fermi Liquids provides an explanation.

Fermi liquids - Why do we want it? The concept of a Fermi Liquid describes (strongly) interacting fermions using concepts that naively would only be applicable to a very weakly interacting gas. That this is at all possible makes Fermi Liquid theory a very versatile, but at the same time its very success is puzzling. Fermi Liquid theory is: Is simple to use

Fermi liquids - What is it? Fermi liquid theory describe a strongly interacting Fermi system, such as nuclear matter, liquid Helium III, or electrons interacting via Coulomb forces. The basic concepts/ideas in Fermi liquid theory (of electrons) are: Weakly interacting quasielectrons and quasiholes The quasiparticles have the same quantum numbers as electrons and holes, i.e. momentum k, spin 1 2 and charge ±e. The state of the Fermi liquid is described as a collection of n quasiparticles, k 1, α 1... k N, α N The interaction between the quasiparticles is described by a few Fermi liquid parameters f l.

Fermi Liquids How?

Fermi Liquids How? I will now describe Landaus intuitive approach to Fermi liquids, based on adiabatic evolution from the free Fermi gas.

Fermi Liquids How? I will now describe Landaus intuitive approach to Fermi liquids, based on adiabatic evolution from the free Fermi gas. Later a diagrammatic approach was used to derive Fermi Liquid theory, but this is rather complicated and is based on resuming infinite classes of diagrams. The modern Renormalization Group approach to Fermi Liquid theory, due primarily to Shankar and Polchinski, defines the Fermi Liquid as a fixed point of the RG flow. This roughly means that the FL description is what is left when all the high energy degrees of freedom are integrated out.

The free Fermi gas For a gas of free fermions with mass m e, the zero temperature ground state is obtained by filling all single particle (plane wave) states up to the Fermi energy ɛ F. With the T = 0 distribution function n 0 ( p) = θ( p p F ) we get, Total energy ˆ E = V Tr ˆ Tr d 3 p (2π) 3 ɛ(p)n 0(p) dτ p2 n 0 (p) = V pf 5 2m e 5π 2 = Vp2 F 2m e 5π 2 ɛ F where V is the, ɛ(p) = p2 2m e, ɛ F = ɛ(p F ), and ˆ ˆ d 3 p dτ = V (2π) 3 = ki

The free Fermi gas Total number of particles ˆ N = dτ n 0 (p) = V p3 F 3π 2 Excitations from the ground state is obtained by changing the occupation numbers of the single particle levels, i.e. by changing the distribution function, n(p) = θ(p p F ) + δn(p) For discrete momentum states, δn can only take the values ±1 because of the Pauli principle. The distribution function at finite temperature T is determined by maximizing the entropy, ˆ S = dτ[n(p) ln n(p) + (1 n(p)) ln(1 n(p))] under the constraints δe = δn = 0 to get:

The free Fermi gas Finite T Fermi-Dirac distribution function n β (p) = 1 e β(ɛ(p) ɛ F ) + 1, β = 1 kt Note that no dynamical information was needed - only that the particles are fermions. These considerations can be generalized to include an external potential, but in those cases the integrals can in general not be calculated on a closed form and there are no analytical expression for energy, entropy etc.

Landaus basic idea Landau s basic idea was that the interacting system can be thought of as connected to the free fermi gas by an adiabatic switching process. In particular, this means

Landaus basic idea Landau s basic idea was that the interacting system can be thought of as connected to the free fermi gas by an adiabatic switching process. In particular, this means there is an one-one correspondence between the excitations in the free and the interacting system. For a state with one fermion added to the ground state, this means p, α free p, α int. The quantum numbers charge q, momentum p, and spin α of the quasiparticle remain the same, but the energy is changed There is a Fermi surface also in the interacting system

The Energy Functional E[n(p)] When interactions are present the total energy still depends on the distribution function, n(p) but this functional dependence is very complicated. We are however only interested in the change in energy as we vary the occupation numbers: Variation of E[n] ˆ δe[n(p)] = dτɛ 0 (p)δn(p) + 1 2V ˆ dτdτ f ( p, p )δn(p)δn(p ) +... where V is the volume of the system, and ɛ 0 (p) = δe[n] δn(p) n=n 0 f ( p, p ) = δ2 E[n] δn(p)δn(p ) n=n 0

The quasiparticle distribution function Since the entropy follows from a counting argument, and since there is an one-one correspondence between the states in the interacting and the non-interacting system, the calculation of the entropy will look precisely the same, except that the constraint δe = 0 now becomes, ˆ dτ ɛ[n, p]δn(p) = 0, Minimizing the entropy under this constraint gives, Functional equation for the quasiparticle distribution function n β (p) = 1 e β(ɛ[n,p] ɛ F ) + 1. which is a very complicated functional equation. From the previous arguments we however know that it will approach n 0 (p) = θ(p p F ) as T goes to zero.

The effective mass m I - definition Now we assume that the temperature is low enough that we can put n(p) = θ(p p F ) and expand around the Fermi momentum, ɛ[n(p ), p] ɛ[θ(p p F ), p] ɛ F + v F (p p F ) +... To understand the meaning of ɛ F and v F, recall that p F is the same as in the free theory, but the Fermi energy is not. But ɛ[n, p] is the energy cost for adding a single quasiparticle, so ɛ F = µ = chemical potential For the non-interacting gas, From this we conclude ɛ(p) = ɛ F + p F m e (p p F ) +.... v F = Fermi velocity = p F m p F m e This relation defines the effective mass m.

The effective mass m II - how to measure One of the simplest observables that can be measured for a Fermi liquid is its specific heat, c V. It is the same as for the free electron gas with the substitution m e m, c V = m p F 3 k 2 B T, The effective mass can vary a lot: 3 He m /m e 3 Heavy fermion compounds such as UPt 3 m /m e 100 1000

Meaning of the quasiparticle Fermi surface At zero temperature, the quasiparticle distribution function n(p) is sharp (dotted line). The momentum distribution of the electrons, N(p) = Ψ FL c pc p Ψ FL Z is the wave function renormalization constant, or the strength of the single particle pole.

When is FL theory applicable? A quasielectron at a momentum p 1 it can decay into two quasielectrons and one quasihole, p 1 p 2 + p 3 + p 2 p 1 p 3 + This makes the quasiparticles unstable!!! Landau s fundamental observation Close to the Fermi surface, the phase space for decay vanish p p F 2 and the quasiparticles become almost stable!! In fact, the life time τ becomes τ = 1 Γ p p F 2 For FL theory to be applicable, we must have T T F T F for metallic sodium is 40, 000K but for liquid He 3 only 7K!!

Interactions - definition of the FL parameters Writing out the spin indices ɛ[n(p ), p] αβ = ɛ 0,αβ (p) + 1 V ˆ dτ f ( p, p ) αγ,βδ δn γδ (p ) +.... f only depends on the angle θ given by p p = pp cos θ pf 2 cos θ Symmetries (rotation, TR) and Fermi statistics implies, f ( p, p ) αγ,βδ = f (cos θ)δ αβ δ γδ + g(cos θ) S αβ S γδ. For an isotropic liquid (no B fields), we have n αβ = nδ αβ, and ˆ ɛ[n(p ), p] = ɛ F + v F (p p F ) + dτ f (cos θ)δn(p ) F-parameters F (θ) = p F m π 2 f (cos θ) = (2L + 1)F L P L (cos θ) L=0

Important relations The effective mass m is not independent of the parameters F L. In fact follows from Gallilean invariance. The compressibility κ is given by, where ρ is the density. κ = m = 1 + F 1 m e 3 m p F π 2 ρ 2 (1 + F 0 ) The response to a magnetic field involves the function g(cos θ), and the susceptibility is determined by the parameter G 0, where G L is defined analogously to F L

Momentum cutoff From a microscopic eucledian action S = S Λ, construct a sequence of actions S Λn where Λ n is a cutoff and the action S Λn describes the physics for momenta p < Λ n. The first Λ can be thought of as a physical cutoff. Study the Eucledian partition function of a φ 4 theory ˆ Z[T, µ,... ] Z = D[φ, φ ] e S[φ] (1) ˆ [ ] 1 S = d d x 2 φ ( 2 + g 2 )φ + g 4 φ 4 = S 0 + S int (2)

The effective action The action becomes S = 1 2 (p2 + g 2 ) φ ( p) φ( p) + 0 p Λ 1 = S 0 [φ < ] + S 0 [φ > ] + S int [φ <, φ > ] Λ 1 p Λ 1 2 (p2 + g 2 ) φ ( p) φ( p) + S int

The effective action The action becomes S = 1 2 (p2 + g 2 ) φ ( p) φ( p) + 0 p Λ 1 = S 0 [φ < ] + S 0 [φ > ] + S int [φ <, φ > ] Λ 1 p Λ 1 2 (p2 + g 2 ) φ ( p) φ( p) + S int We define the effective action S eff at scale Λ 1 = Λ/s by ˆ e Seff [φ <] = e S 0[φ <] D[φ > ] e S 0[ p >] S int [φ <,φ >] Rescale : p < 1 s p φ < ζ φ which resets the cutoff to Λ and the kinetic term to p 2. We then get (suppressing the s), ˆ [ S eff = d d x φ ( 2 + g 2)φ + g 4 φ 4 + g 6 φ 6 + g 22 ] 2 φ 2 +....

Dimensional analysis and flow equations Mass dimensions: [g 2 ] = 2 and [g 4 ] = 4 d (recall that [φ] = d 2 under the scale transformation, 1) so g 2 g 2 = s 2 g 2 and g 4 g 4 = s 4 d g 4 We expect g 2 to be relevant and g 4 relevant or irrelevant depending on whether d < 4 or d > 4. The RG equations are s dg i ds = dg dt = β i(g 1, g 2,... ) The d = 4 β-functions can be obtained from perturbation theory, dg 2 dt = 2g 2 + ag 4 dg 4 dt = bg 2 4

The RG flow for φ 4 theory in d = 4

Why fermions are different For bosons the RG transformation restricts to momenta to a smaller and smaller ball around p = 0. An RG fixed point is a set of coupling constants {g i }. For fermions the momenta are restricted to a smaller ands smaller shell around the Fermi momentum p F. An RG fixed point is a set of coupling functions {f i ( Ω)} defined on the Fermi surface, p = p F Ω.

Two body scattering Consider scattering between two particles in d = 3 close to the Fermi surface, 1 + 2 3 + 4 Parametrize: k i = (p F + k i ) Ω i, i = 1,... 4 where Ω = (cos θ, sin θ). Ω 1 = Ω 3 and Ω 2 = Ω 4 (I ) Ω 1 = Ω 4 and Ω 2 = Ω 3 (II ) Ω 1 = Ω 2 and Ω 3 = Ω 4 (III )

Fixed point functions A four particle interaction in momentum space can be written as, ˆ 4 S int = dω i ψ p 3 ψ p 4 ψ p2 ψ p1 f (ω i, k i, θ i ) δ 2 ( p i )δ( i=1 p i i i ω i ) Processes I and II are allowed for arbitrary angles θ 1 and θ 2, and because of rotational invariance, it is characterized by a fixed point function: F (θ 1 θ 2 ) = f (θ 1, θ 2, θ 1, θ 2 ) = f (θ 1, θ 2, θ 2, θ 1 ) where the sign is due to fermi statistics. The Process III is scattering between particles at opposite positions on the Fermi circle. Here the coupling function can depend only on the angle θ 1 θ 3, V (θ 1 θ 3 ) = f (θ 1, θ 1, θ 3, θ 3 ).

Flow equations Here we just used geometric intuition to put in the constraints on the angles by hand, but a carful evaluation of the diagrams taking the cutoff into account, will give the same result, and the flow eqs. df (θ) = 0 dt dv (θ) = 1 ˆ dθ dt 4π 2π V (θ θ )V (θ ) which has the solution: V L (0) V L (t) = 1 + V L (0)t/(4π) where V L (0) are the starting values for the RG evolution.

Physical interpretation V L (t) = V L (0) 1 + V L (0)t/(4π) Take all V L (0) > 0, as would be the case for a reasonable repulsive potential, then

Physical interpretation V L (t) = V L (0) 1 + V L (0)t/(4π) Take all V L (0) > 0, as would be the case for a reasonable repulsive potential, then V (θ) will renormalize to zero.

Physical interpretation V L (t) = V L (0) 1 + V L (0)t/(4π) Take all V L (0) > 0, as would be the case for a reasonable repulsive potential, then V (θ) will renormalize to zero. F (θ) remains marginal and defines a fixed point theory which is a Fermi liquid!! If (at least) one V L (0) < 0 is, as would be the case for any attractive interaction, the RG flow will hit a singularity for some t, and The coupling constant, V L grows out of control, and the perturbative treatment can no longer be trusted. The renormalization of V (θ) comes from the BCS diagram c.