Chapter 6: the Fermi Liquid

Home , Density of states, Electronic specific heat, Fermi gas, Fermi liquid theory, Fermi surface

Chapter 6: The Fermi Liquid

L.D. Landau

February 7, 2017

Contents

1 introduction: The Electronic Fermi Liquid 3

2 The Non-Interacting Fermi Gas 5 2.1 Inﬁnite-Square-Well Potential ...... 5 2.2 The Fermi Gas ...... 9 2.2.1 T = 0, The Pauli Principle ...... 9 2.2.2 T 6= 0, Fermi Statistics ...... 12

3 The Weakly Correlated Electronic Liquid 22 3.1 Thomas-Fermi Screening ...... 22 3.2 The Mott Transition ...... 25 3.3 Fermi liquids ...... 28 3.4 Quasi-particles ...... 29 3.4.1 Particles and Holes ...... 29 3.4.2 Quasiparticles and Quasiholes at T = 0 ...... 33 3.5 Energy of Quasiparticles...... 38

4 Interactions between Particles: Landau Fermi Liquid 41 4.1 The free energy, and interparticle interactions ...... 41

1 4.2 Local Energy of a Quasiparticle ...... 45 4.2.1 Equilibrium Distribution of Quasiparticles at Finite T . . . . . 47 4.2.2 Local Equilibrium Distribution ...... 49 4.3 Eﬀective Mass m∗ of Quasiparticles ...... 51

2 1 introduction: The Electronic Fermi Liquid

As we have seen, the electronic and lattice degrees of freedom decouple, to a good approximation, in solids. This is due to the diﬀerent time scales involved in these systems. h¯ τion ∼ 1/ωD τelectron ∼ (1) EF where EF is the electronic Fermi energy. The electrons may be thought of as instantly reacting to the (slow) motion of the lattice, while remaining essentially in the electronic ground state. Thus, to a good approximation the electronic and lattice degrees of freedom separate, and the small electron-lattice (phonon) interaction (responsible for resistivity, superconductivity etc) may be treated as a perturbation (with ωD/EF as an expansion pa- rameter); that is if we are capable of solving the problem of the remaining purely electronic system. At ﬁrst glance the remaining electronic problem would also appear to be hopeless since the (non-perturbative) electron- electron interactions are as large as the combined electronic kinetic energy and the potential energy due to interactions with

3 the static ions (the latter energy, or rather the corresponding part of the Hamiltonian, composes the solvable portion of the problem). However, the Pauli principle keeps low-lying orbitals from being multiply occupied, so is often justiﬁed to ignore the electron-electron interactions, or treat them as a renormaliza- tion of the non-interacting problem (eﬀective mass) etc. This will be the initial assumption of this chapter, in which we will cover

• the non-interacting Fermi liquid, and

• the renormalized Landau Fermi liquid (Pines & Nozieres).

These relatively simple theories resolve some of the most important puzzles involving metals at the turn of the century. Per- haps the most intriguing of these is the metallic speciﬁc heat. Except in certain “heavy fermion” metals, the electronic contribution to the speciﬁc heat is always orders of magnitude smaller than the phonon contribution. However, from the classical theo- rem of equipartition, if each lattice site contributes just one electron to the conduction band, one would expect the contributions from these sources to be similar (Celectron ≈ Cphonon ≈ 3NrkB).

4 This puzzle is resolved, at the simplest level: that of the non- interacting Fermi gas.

2 The Non-Interacting Fermi Gas

2.1 Inﬁnite-Square-Well Potential

We will proceed to treat the electronic degrees of freedom, ignoring the electron-electron interaction, and even the electron- lattice interaction. In general, the electronic degrees of freedom are split into electrons which are bound to their atomic cores with wavefunctions which are essentially atomic, unaﬀected by the lattice, and those valence (or near valence) electrons which react and adapt to their environment. For the most part, we are only interested in the valence electrons. Their environment described by the potential due to the ions and the core electrons– the core potential. Thus, ignoring the electron-electron interactions, the electronic Hamiltonian is P 2 H = + V (r) . (2) 2m

5 As shown in Fig. 1, the core potential V (r), like the lattice, is periodic V(r)

a V(r+a) = V(r)

Figure 1: Schematic core potential (solid line) for a one-dimensional lattice with lattice constant a.

For the moment, ignore the core potential, then the electronic wave functions are plane waves ψ ∼ eik·r. Now consider the core potential as a perturbation. The electrons will be strongly eﬀected by the periodicity of the potential when λ = 2π/k ∼ a 1. However, when k is small so that λ a (or when k is large, so λ a) the structure of the potential may be neglected, or we can assume V (r) = V0 anywhere within the material. The potential still acts to conﬁne the electrons (and so maintain charge neutrality), so V (r) = ∞ anywhere outside the material.

1Interestingly, when λ ∼ a, the Bragg condition 2d sin θ ≈ a ≈ λ may easily be satisﬁed, so the electrons, which may be though of as DeBroglie waves, scatter oﬀ of the lattice. Consequently states for which λ = 2π/k ∼ a are often forbidden. This is the source of gaps in the band structure, to be discussed in the next chapter.

6 Figure 2: Infinite square-well potential. V (r) = V0 within the well, and V (r) = ∞ outside to confine the electrons and maintain charge neutrality. Thus we will approximate the potential of a cubic solid with linear dimension L as an infinite square-well potential.   V0 0 < ri < L V (r) = (3)  ∞ otherwise The electronic wavefunctions in this potential satisfy h¯2 − ∇2ψ(r) = (E0 − V ) ψ(r) = Eψ(r) (4) 2m 0 The normalize plane wave solution to this model is 3 1/2 Y 2 ψ(r) = sin k x where i = x, y, or z (5) L i i i=1 and kiL = niπ in order to satisfy the boundary condition that ψ = 0 on the surface of the cube. Furthermore, solutions with ni < 0 are not independent of solutions with ni > 0 and may

7 be excluded. Solutions with ni = 0 cannot be normalized and are excluded (they correspond to no electron in the state). The

3 (π/L) kx π/L

Figure 3: Allowed k-states for an electron confined by a infinite-square potential. Each state has a volume of (π/L)3 in k-space. eigenenergies of the wavefunctions are h¯2∇2 h¯2 X h¯2π2 − ψ = k2 = n2 + n2 + n2 (6) 2m 2m i 2mL2 x y z i and as a result of these restrictions, states in k-space are confined to the first quadrant (c.f. Fig. 3). Each state has a volume (π/L)3 of k-space. Thus as L → ∞, the number of states with energies E(k) < E < E(k) + dE is (4πk2dk)/8 dZ0 = . (7) (π/L)3

8 2 2 q Then, since E = h¯ k , so k2dk = m 2mE/h¯2dE 2m h¯2 1 2m3/2 dZ = dZ0/L3 = E1/2dE . (8) 4π2 h¯2 or, the density of state per unit volume is dZ 1 2m3/2 D(E) = = E1/2 . (9) dE 4π2 h¯2 Up until now, we have ignored the properties of electrons. How- ever, for the DOS, it is useful to recall that the electrons are spin-1/2 thus 2S + 1 = 2 electrons can ﬁll each orbital or k- state, one of spin up the other spin down. If we account for this spin degeneracy in D, then 1 2m3/2 D(E) = E1/2 . (10) 2π2 h¯2

2.2 The Fermi Gas

2.2.1 T = 0, The Pauli Principle

Electrons, as are all half-integer spin particles, are Fermions. Thus, by the Pauli Principle, no two of them may occupy the same state. For example, if we calculate the density of electrons

9 per unit volume Z ∞ n = D(E)f(E,T )dE , (11) 0 where f(E,T ) is the probability that a state of energy E is occupied, the factor f(E,T ) must enforce this restriction. How- ever, f is just the statistical factor; c.f. for classical particles

f(E,T ) = e−E/kBT for classical particles , (12) which for T = 0 would require all the electrons to go into the ground state f(0, 0) = 1. Clearly, this violates the Pauli principle. At T = 0 we need to put just one particle in each state, starting from the lowest energy state, until we are out of particles. Since E ∝ k2 in our simple square-well model, will ﬁll up all k-states until we reach some Fermi radius kF , corresponding to some Fermi Energy EF h¯2k2 E = F , (13) F 2m thus,

f(E,T = 0) = θ(EF − E) (14)

10 kz

ky 2 2 h k k f f = E 2m f

occupied states

kx k f D(E)

Figure 4: Due to the Pauli principle, all k-states up to kF , and all states with energies up to Ef are ﬁlled at zero temperature. and

Z ∞ Z EF n = D(E)f(E,T )dE = D(E)DE 0 0 3/2 Z EF 2m 1 1/2 = 2 2 E DE h¯ 2π 0 2m3/2 1 2 = E3/2 , (15) h¯2 2π2 3 F or h¯2 E = 3π2n2/3 = k T (16) F 2m B F which also deﬁnes the Fermi temperature TF . Thus for metals, 23 3 −11 5 in which n ≈ 10 /cm , EF ≈ 10 erg ≈ 10eV ≈ kB10 K. Notice that due to the Pauli principle, the average energy of

11 the electrons will be ﬁnite, even at T = 0! Z EF 3 E = D(E)EdE = nEF . (17) 0 5

However, it is the electrons near EF in energy which may be excited and are therefore important. These have a DeBroglie wavelength of roughly ◦ 12.3 A ◦ λe = ≈ 4 A (18) (E(eV))1/2 thus our original approximation of a square well potential, ignoring the lattice structure, is questionable for electrons near the Fermi surface, and should be regarded as yielding only qual- itative results.

2.2.2 T 6= 0, Fermi Statistics

At ﬁnite temperatures some of the states will be thermally excited. The energy available for these excitations is roughly kBT , and the only possible excitations are from ﬁlled to un-

ﬁlled electronic states. Therefore, only the states within kBT

(EF − kBT < E < EF + kBT ) of the Fermi surface may be excited. f(E,T ) must be modiﬁed accordingly.

12 What we need is then f(E,T ) at finite T which also satisfies the Pauli principle. Lets return to our model of a periodic solid which is constructed by bringing individual atoms together from an infinite separation. First, just consider a solid constructed from only two atoms, each with a single orbital (Fig. 5). For

1 2 δ δ n = +1 n1 = -1 1

Figure 5: Exchange of electrons in a solid composed of two orbitals. this system, in equilibrium, X ∂F 0 = δF = δn (19) ∂n i i i P electrons are conserved so i δni = 0. Thus, for our two orbital system ∂F ∂F δn1 + δn2 = 0 and δn1 + δn2 = 0 (20) ∂n1 ∂n2 or ∂F ∂F = (21) ∂n1 ∂n2

13 A similar relation holds for an arbitrary number of particles. Apparently this quantity, the increased free energy needed to add a particle to the system, is a constant ∂F = µ (22) ∂ni for all i. µ is called the chemical potential. Now consider an ensemble of orbitals. We will treat the thermodynamics of this system within the canonical ensemble (i.e., the system is in contact with a thermal bath, and the particle number is conserved) for which F = E − T S is the appropriate potential. The system energy E and Entropy S may be written as functions of the orbital energies Ei and occupancies ni and the degeneracy gi of the state of energy Ei. For example, g = 4 n = 2 E4 4 4 g = 4 n = 4 E3 3 3 g = 2 n = 1 E2 2 2 g = 2 n = 2 E1 1 1

Figure 6: states from an ensemble of orbitals.

X E = niEi . (23) i

14 The entropy S requires a bit more thought. If P is the number of ways of distributing the electrons among the states, then

S = kB ln P. (24)

Consider a set of gi states with energy Ei. The number of ways of distributing the ﬁrst electron in these states is gi. For a second electron we then have gi − 1 ways... etc. So for ni electrons there are g ! i (25) ni!(gi − ni)! possible ways of accommodating the ni (indistinguishable) electrons in gi states. The number of ways of making the whole system (ie, ﬁlling energy levels with Ei =6 Ej) is then

Y gi! P = , (26) n !(g − n )! i i i i and so, the entropy X S = kB ln gi! − ln ni! − ln(gi − ni)! . (27) i For large n, ln n! ≈ n ln n − n, so X S = kB gi ln gi − ni ln ni − (gi − ni) ln(gi − ni) (28) i

15 and X X F = niEi −kBT gi ln gi −ni ln ni −(gi −ni) ln(gi −ni) i i (29) We will want to use the chemical potential µ in our thermo- dynamic calculations ∂F µ = = Ek + kBT (ln nk + 1 − ln(gk − nk) − 1) , (30) ∂nk where β = 1/kBT . Solving for nk

gk nk = . (31) 1 + eβ(Ek−µ) Thus the probability that a quantum state with energy E is occupied, is (the Fermi function) 1 f(E,T ) = . (32) 1 + eβ(Ek−µ) At T = 0, β = ∞, and f(E, 0) = θ(µ − E). Thus µ(T = 0) =

EF . However in general µ is temperature dependent, since it must be adjusted to keep the particle number ﬁxed. In addition, when T =6 0, f becomes less sharp at energies E ≈ µ. This reﬂects the fact that particles with energies E − µ ≈ kBT may be excited to higher energy states.

16 1.5

1.0 +1) ) ω−µ ( β 0.5 1/(e

0.0 0.0 0.5 1.0 1.5 2.0 ω

−β(ω−µ) Figure 7: Plot of the Fermi function 1/ e + 1 when β = 1/kBT = 20 and µ = 1. Not that at energies ω ≈ µ the Fermi function displays a smooth step of width

≈ kBT = 0.05. This allows thermal excitations of particles near the Fermi surface.

Specific Heat The form of f(E,T ) also clarifies why the electronic specific heat of metals is so small compared to the clas- 3 sical result Cclassical = 2nkBT . The reason is simple: only the electrons with energies within about kBT of the Fermi surface may be excited (about kBT of the electron density) each with EF excitation energy of about kBT . Therefore,

kBT T Uexcitation ≈ kBT n = nkBT (33) EF TF so T C ≈ nkB (34) TF

17 5 2 Then as T TF (TF is typically about 10 K in most metals ) T C ≈ nkB Cclassical ≈ nkB. Thus at temperatures where TF the phonons contribute essentially a classical result to the spe- ciﬁc heat, the electronic contribution is vanishingly small. In general this holds except at very low T where the phonon con- 3 tribution Cphonon ∼ T goes to zero faster than the electronic contribution to the speciﬁc heat.

Specific Heat Calculation Of course, since we know the free energy of the non-interacting Fermi gas, we can calculate the form of the specific heat. Here we will follow Ibach and Lüthand Kittel; however, since the chemical potential does depend upon the temperature, I would like to make the approximations we make a bit more explicit. Upon heating from T = 0 to finite T , the Fermi gas will gain energy

Z ∞ Z EF U(T ) = dE ED(E)f(E,T ) − dE ED(E) (35) 0 0

2 Heavy Fermion systems are the exception to this rule. There TF can be as small as a fraction of a degree Kelvin. As a result, they may have very large electronic speciﬁc heats.

18 so dU Z ∞ df(E,T ) CV = = dE ED(E) . (36) dT 0 dT Then since at constant volume the electronic density is constant, dn R ∞ so dT = 0, and n = 0 dED(E)f(E,T ), dn Z ∞ df(E,T ) 0 = EF = dE EF D(E) (37) dT 0 dT so we may write Z ∞ df CV = dE (E − EF ) D(E) . (38) 0 dT

In f, the temperature T enters through both β = 1/kBT and µ df ∂f ∂β ∂f ∂µ = + dT ∂β ∂T ∂µ ∂T βeβ(E−µ) E − µ ∂µ = − (39) eβ(E−µ) + 12 T ∂T

∂µ However, ∂T depends upon the details of the density of states near the Fermi surface, which can diﬀer greatly from material to material. Furthermore, ∂µ < 1 especially in common metals ∂T E−µ at temperatures T TF , and the ﬁrst term T is of order ∂µ one (c.f. Fig. 7). Thus, for now we will neglect ∂T relative to E−µ T (you will explore the validity of this approximation in your

19 homework), and, consistent with this approximation, replace µ by EF , so df βeβ(E−EF ) E − E ≈ F , (40) 2 dT eβ(E−EF ) + 1 T and 2 1 Z ∞ (E − E ) D(E)βeβ(E−EF ) C ≈ dE F let x = β(E − E ) V 2 2 F kBT 0 eβ(E−EF ) + 1 Z ∞ x x 2 e ≈ kBT dx D + EF x (41) x 2 −βEF β (e + 1) x As shown in Fig. 8, the function x2 e is only large in the (ex+1)2 region −10 < x < 10. In this region, and for temperatures 0.5 0.4 2 0.3 +1) x /(e

x 0.2 e 2 x 0.1 0.0 -10 -5 0 5 10 x

Figure 8: Plot of x2 ex vs. x. Note that this function is only ﬁnite for roughly (ex+1)2 5 −10 < x < 10. Thus, at temperatures T TF ∼ 10 K, we can approximate x D β + EF ≈ D (EF ) in Eq. 42.

x T TF , D β + EF ≈ D (EF ), since the density of states

20 usually does not have features which are sharp on the energy scale of 10kBT . Thus Z ∞ x 2 e CV ≈ kBTD (EF ) dx x x 2 −βEF (e + 1) π2 ≈ k2 TD(E ) . (42) 3 B F Note that no assumption about the form of D(E) was made other than the assumption that it is smooth within kBT of the Fermi surface. Thus, experimental measurements of the speciﬁc heat at constant volume of the electrons, gives us information about the density of electronic states at the Fermi surface. Now let’s reconsider the DOS for the 3-D box potential. 3/2 1/2 1 2m 1/2 E D(E) = 2 E = D(EF ) (43) 2π h¯ EF

R EF 2 For which n = 0 D(E)dE = D(EF )3EF , so π2 T 3 CV = nkB nkB (44) 2 TF 2 where the last term on the right is the classical result. For room temperatures T ∼ 300K, which is also of the same order of magnitude as the Debye temperatures θD, 3 C ∼ nk C (45) V phonon 2 B V electron 21 So, the only way to measure the electronic speciﬁc heat in most materials is to go to very low temperatures T θD, for which 3 CV phonon ∼ T . Here the total speciﬁc heat

3 CV ≈ γT + βT (46)

We will see that gives us some measurement of the electronic ef- fective mass for our Fermi liquid theory. I.e. it tell us something about electron- electron interactions.

3 The Weakly Correlated Electronic Liquid

3.1 Thomas-Fermi Screening

As an introduction to the effect of electronic correlations, consider the effect of a charged oxygen defect in one of the copper- oxygen planes of a cuprate superconductor shown in Fig. 9. Assume that the oxygen defect captures two electrons from the metallic band, going from a 2s22p4 to a 2s22p6 configuration. The defect will then become a cation, and have a net charge of two electrons. In the vicinity of this oxygen defect, the electrostatic potential and the electronic charge density will be re-

22 O

Cu O Cu O Cu O Cu O Cu O Cu O

O O O O O O

Cu O Cu O Cu O Cu O Cu O Cu O q=2e- O O O O O O O

Cu O Cu O Cu O Cu O Cu O Cu O

O O O O O O

Figure 9: A charged oxygen defect is introduced into one of the copper-oxygen planes of a cuprate superconductor. The oxygen defect captures two electrons from the metallic band, going from a 2s22p4 to a 2s22p6 conﬁguration. duced. If we model the electronic density of states in this material with our box-potential DOS, we can think of this reduction in the local charge density in terms of raising the DOS parabola near the defect (cf. Fig. 10). This will cause the free electronic charge to ﬂow away from the defect. Near the defect (since e < 0 and hence eδU(rnear) < 0)

Z EF +eδU(rnear) n(rnear) ≈ D(E)DE (47) 0

While away from the defect, δU(raway) = 0, so

Z EF n(raway) ≈ D(E)DE (48) 0

23 near charged Away from defect charged defect

-eδU

Figure 10: The shift in the DOS parabola near a charged defect. or Z EF +eδU(r) Z EF δn(r) ≈ D(E)DE − D(E)DE (49) 0 0

If |eδU| EF , then

δn(r) ≈ eδUD(EF ) . (50)

We can solve for the change in the electrostatic potential by solving Poisson equation.

2 2 ∇ δU = 4πδρ = 4πeδn = 4πe D(EF )δU . (51)

2 2 2 2 3 Let λ = 4πe D(EF ), then ∇ δU = λ δU has the solution qe−λr δU(r) = (52) r 3 −λr The solution is actually Ce /r, where C is a constant. C may be deterined by letting D(EF ) = 0, so the medium in which the charge is embedded becomes vacuum. Then the potential of the charge is q/r, so C = q.

24 The length 1/λ = rTF is known as the Thomas-Fermi screening length. 2 −1/2 rTF = 4πe D(EF ) (53)

Lets estimate this distance for our square-well model, a π a r2 = 0 ≈ 0 TF 3(3π2n)1/3 4n1/3 1 n −1/6 rTF ≈ 3 (54) 2 a0 ◦ 23 −3 In Cu, for which n ≈ 10 cm (and since a0 = 0.53 A) −1/6 23 ◦ 1 10 −8 rT F Cu ≈ ≈ 0.5 × 10 cm = 0.5 A (55) 2 (0.5 × 10−8)−1/2 Thus, if we add a charge defect to Cu metal, the eﬀect of the ◦ 1 defect’s ionic potential is screened away for distances r > 2 A!

3.2 The Mott Transition

Now consider an electron bound to an ion in Cu or some other metal. As shown in Fig. 11 the screening length decreases, and bound states rise up in energy. In a weak metal (i.e., something like YBCO), in which the valence state is barely free, a reduction in the number of carriers (electrons) will increase the screening

25 egh since length, bound. become free, were 11: Figure iinwsﬁs rpsdb .Mt,adi aldteMott the called is and Mott, N. by proposed ﬁrst was sition oxides, transition-metal tran- metal-insulator some etc.This semiconductors, in amorphous glasses, transition believed MI is the and explain insulator, an to to metal a an from causing transition bound, abrupt become to length, free screening were the that states increase some turn, causing in will This pressure). the modulating by even or dopants, compensating done certain is doping this by semiconductors, Si-based in (i.e., the carriers decrease of we density that suppose and ions, such of system centrated bound. state valance trap free to one it the causing states–making potential, more the bind or of range the extend will This o mgn htisedo igedfc,w aeacon- a have we defect, single a of instead that imagine Now cenddfc oetas stesreiglnt nrae,sae that states increases, length screening the As potentials. defect Screened -e-r/rTF/r r TF =1/4 r TF r = n r free states bound states F T -1/6 ∼ 26 n −

1 -r/rTF / -e /r 6 . r TF =1 r (56) transition, more signiﬁcantly Mott proposed a criterion based on the relevant electronic density for when this transition should occur. In Mott’s criterion, a metal-insulator transition occurs when this potential, in this case coming from the addition of an ionic impurity, can just bind an electronic state. If the state is bound, the impurity band is localized. If the state is not bound, then the impurity band is extended. The critical value of λ = λc may be determined numerically[1] with λc/a0 = 1.19, which yields the Mott criterion of

−1/3 2.8a0 ≈ nc , (57) where a0 is the Bohr radius. Despite the fact that electronic interactions are only incorporated in the extremely weak coupling limit, Thomas-Fermi Screening, Mott’s criterion even works for moderately and strongly interacting systems[2].

27 3.3 Fermi liquids

The purpose of these next several lectures is to introduce you to the theory of the Fermi liquid, which is, in its simplest form, a collection of Fermions in a box plus interactions. In reality , the only physical analog is a gas of 3He, which due its nuclear spin (the nucleus has two protons, one neutron), obeys Fermi statistics for suﬃciently low energies or temperatures. In addition, simple metals, from the ﬁrst or second column of the periodic table, for which we may approximate the ionic potential

V (R) = V0 (58) are a close approximant to Fermi liquids. Moreover, Fermi Liquid theory only describes the ”gaseous” phase of these quantum fermion systems. For example, 3He also has a superﬂuid (triplet), and at least in 4He-3He mixtures, a solid phase exists which is not described by Fermi Liquid The- ory. One should note; however, that the Fermi liquid theory state does serve as the starting point for the theories of superconductivity and super ﬂuidity.

28 One may construct Fermi liquid theory either starting from a many-body diagrammatic or phenomenological viewpoint. We, as Landau, will choose the latter. Fermi liquid theory has 3 basic tenants:

1. momentum and spin remain good quantum numbers to de- scribe the (quasi) particles.

2. the interacting system may be obtained by adiabatically turning on a particle-particle interaction over some time t.

3. the resulting excitations may be described as quasi-particles with lifetimes t.

3.4 Quasi-particles

The last assumption involves a new concept, that of the quasiparticles which requires some explanation.

3.4.1 Particles and Holes

Particles and Holes are excitations of the non-interacting system at zero temperature. Consider a system of N free Fermions

29 each of mass m in a volume V . The eigenstates are the anti- symmetrized combinations (Slater determinants) of N diﬀerent single particle states.

1 ip·r/h¯ ψp(r) = √ e (59) V

The occupation of each of these states is given by np = θ(p−pF ) where pF is the radius of the Fermi sphere. The energy of the system is X p2 E = n (60) p 2m p and pF is given by N 1 p 3 = F (61) V 3π2 h¯

Now lets add a particle to the lowest available state p = pF then, for T = 0, ∂E p2 µ = E (N + 1) − E (N) = 0 = F . (62) 0 0 ∂N 2m If we now excite the system, we will promote a certain number of particles across the Fermi surface SF yielding particles above and an equal number of vacancies or holes below the Fermi surface. These are our elementary excitations, and they are

30 0 quantiﬁed by δnp = np − np  0  δp,p0 for a particle p > pF δnp = . (63) 0  −δp,p0 for a hole p < pF If we consider excitations created by thermal ﬂuctuations, then E particle excitation δn = 1 p' E δn = -1 F hole p excitation

D(E)

Figure 12: Particle and hole excitations of the Fermi gas.

δnp ∼ 1 only for excitations of energy within kBT of EF . The energy of the non-interacting system is completely characterized as a functional of the occupation X p2 X p2 E − E = (n − n0) = δn . (64) 0 2m p p 2m p p p Now lets take our system and place it in contact with a particle bath. Then the appropriate potential is the free energy,

31 E 2 δF = p' /2m - µ µ 2 E F = = p /2m 2 F δF = µ - p /2m

2 δF = | µ - p /2m |

D(E)

2 Figure 13: Since µ = pF /2m, the free energy of a particle or a hole is δF = |p2/2m − µ| > 0, so the system is stable to these excitations. which for T = 0, is F = E − µN, and X p2 F − F = − µ δn . (65) 0 2m p p

The free energy of a particle, with momentum p and δnp0 = p2 δp,p0 is 2m − µ and it corresponds to an excitation outside SF . p2 The free energy of a hole δnp0 = −δp,p0 is µ − 2m, which corre- 2 sponds to an excitation within SF . However, since µ = pF /2m, the free energy of either at p = pF is zero, hence the free energy of an excitation is 2 p /2m − µ , (66) which is always positive; ie., the system is stable to excitations.

32 2 -a/r U ≈ e e TF a a

Figure 14: Model for a fermi liquid: a set of interacting particles an average distance a apart bound within an inﬁnite square-well potential.

3.4.2 Quasiparticles and Quasiholes at T = 0

Now let’s consider a system with interacting particles an average distance a apart, so that the characteristic energy of inter- 2 e −a/rTF action is a e . We will imagine that this system evolves slowly from an ideal or noninteracting system in time t (i.e., 2 e −a/rTF the interaction U ≈ a e is turned on slowly, so that the non-interacting system evolves while remaining in the ground state into an interacting system in time t). 0 If the eigenstate of the ideal system is characterized by np, then the interacting system eigenstate will evolve quasistatis- 0 tically from np to np. In fact if the system is isotropic and 0 remains in its ground state, then np = np. However, clearly in some situations (superconductivity, magnetism) we will neglect

33 some eigenstates of the interacting system in this way. Now let’s add a particle of momentum p to the non-interacting ideal system, and slowly turn on the interaction. As U is

p p p

time = 0 time = t 2 U = 0 U = (e /a) exp(-a/rTF )

Figure 15: We add a particle with momentum p to our noninteracting (U = 0) Fermi liquid at time t = 0, and slowly increase the interaction to its full value U at time t. As the particle and system evolve, the particle becomes dressed by interactions with the system (shown as a shaded ellipse) which changes the eﬀective mass but not the momentum of this single-particle excitation (now called a quasi-particle). switched on, we slowly begin to perturb the particles close to the additional particle, so the particle becomes dressed by these interactions. However since momentum is conserved, we have created an excitation (particle and its cloud) of momentum p. We call this particle and cloud a quasiparticle. In the same way, if we had introduced a hole of momentum p below the Fermi surface, and slowly turned on the interaction, we would have produced a quasihole. Note that this adiabatic switching on procedure will have

34 diﬃculties if the lifetime of the quasi-particle τ < t. If so, then the the quasiparticle could decay into a number of other quasiparticles and quasiholes. If we shorten t so that again τ t, then switching on U could excite the system out of its ground state). We will show that such diﬃculties do not arise so long as the energy of the particle is close to the Fermi energy. Here there are few states accessible for creating particle-hole excitations so collisions are rare. p 4 p 3 p 1 p 2

Figure 16: A particle with momentum p1 above the Fermi surface (p1 > pF ) interacts with one of the particles below the Fermi surface with momentum p2. As a result, two new particles appear above the Fermi surface (all other states are full) with momenta p3 and p4..

To estimate this lifetime consider the following argument from AGD: A particle with momentum p1 above the Fermi

35 surface (p1 > pF ) interacts with one of the particles below the

Fermi surface with momentum p2. As a result, two new particles appear above the Fermi surface (all other states are full) with momenta p3 and p4. This may also be interpreted as a particle of momentum p1 decaying into particles with momenta p3 and p4 and a hole with momentum p2. By Fermi’s golden rule, the total probability of such a process if proportional to 1 Z ∝ δ (ε + ε − ε − ε ) d3p d3p (67) τ 1 2 3 4 2 3

2 p1 where ε1 = 2m − EF , and the integral is subject to the constraints of energy and momentum conservation and that

p2 < pF , p3 > pF , p4 = |p1 + p2 − p3| > pF (68)

It must be that ε1 + ε2 = ε3 + ε4 > 0 since both particles 3 and 4 must be above the Fermi surface. However, since ε2 < 0, < if ε1 is small, then |ε2| ∼ ε1 is also small, so only of order

ε1/EF states may scatter with the state k1, conserve energy, and obey the Pauli principle. Thus, restricting ε2 to a narrow shell of width ε1/EF near the Fermi surface, and reducing the scattering probability 1/τ by the same factor.

36 E ky k1 k4 k3 E 3 F k - k 1 3 k2 2 1 k - k 4 2 4 kx

N(E)

Figure 17: A quasiparticle of momentum p1 decays via a particle-hole excitation into a quasiparticle of momentum p4. This may also be interpreted as a particle of momentum p1 decaying into particles with momenta p3 and p4 and a hole with < momentum p2. Energy conservation requires |ε2| ∼ ε1. Thus, restricting ε2 to a narrow shell of width ε1/EF near the Fermi surface. Momentum conservation k1 − k3 = k4 − k2 further restricts the available states by a factor of about ε1/EF . Thus −2 the lifetime of a quasiparticle is proportional to ε1 . EF

Now consider the constraints placed on states k3 and k4 by momentum conservation

k1 − k3 = k4 − k2 . (69)

Since ε1 and ε2 are conﬁned to a narrow shell around the Fermi surface, so too are ε3 and ε4. This can be seen in Fig. 17, where the requirement that k1 − k3 = k4 − k2 limits the allowed states for particles 3 and 4. If we take k1 ﬁxed, then the allowed states for 2 and 3 are obtained by rotating the vectors k1 −k3 = k4−k2; however, this rotation is severely limited by the fact that

37 particle 3 must remain above, and particle 2 below, the Fermi surface. This restriction on the final states further reduces the scattering probability by a factor of ε1/EF . 2 Thus, the scattering rate 1/τ is proportional to ε1 so EF that excitations of sufficiently small energy will always be sufficiently long lived to satisfy the constraints of reversibility. Fi- nally, the fact that the quasiparticle only interacts with a small number of other particles due to Thomas-Fermi screening (i.e., those within a distance ≈ rTF ), also significantly reduces the scattering rate.

3.5 Energy of Quasiparticles.

As in the non-interacting system, excitations will be quanti- ﬁed by the deviation of the occupation from the ground state 0 occupation np 0 δnp = np − np . (70)

At low temperatures δnp ∼ 1 only for p ≈ pF where the particles are suﬃciently long lived that τ t. It is important to 0 emphasize that only δnp not np or np, will be physically rele-

38 vant. This is important since it does not make much sense to talk about quasiparticle states, described by np, far from the Fermi surface since they are not stable and cannot be excited by the perturbations we are considering (i.e., thermal, or in a transport experiment). For the ideal system X p2 E − E = δn . (71) 0 2m p p

For the interacting system E[np] becomes much more compli- cated. If however δnp is small (so that the system is close to its ground state) then we may expand:

X 2 E[np] = Eo + pδnp + O(δnp) , (72) p where p = δE/δnp. Note that p is intensive (ie. it is independent of the system volume). If δnp = δp,p0, then E ≈ E0 + p0; 0 i.e., the energy of the quasiparticle of momentum p is p0.

In practice we will only need p near the Fermi surface where

δnp is ﬁnite. So we may approximate

p ≈ µ + (p − pF ) · ∇p p| (73) pF

39 where ∇pp = vp, the group velocity of the quasiparticle. The ground state of the N +1 particle system is obtained by adding

∂E0 a particle with p = F = µ = ∂N (at zero temperature); which deﬁnes the chemical potential µ. We make learn more about

p by employing the symmetries of our system. If we explicitly display the spin-dependence,

p,σ = −p,−σ under time-reversal (74)

p,σ = −p,σ under BZ reﬂection (75)

So p,σ = −p,σ = p,−σ; i.e., in the absence of an external magnetic ﬁeld, p,σ does not depend upon σ if. Furthermore, for an isotropic system p depends only upon the magnitude of

p dp(|p|) p, |p|, so p and vp = ∇p(|p|) = |p| d|p| are parallel. Let us deﬁne m∗ as the constant of proportionality at the fermi surface

∗ vpF = pF /m (76)

Using m∗ it is useful to deﬁne the density of states at the fermi surface. Recall, that in the non-interacting system, 3/2 1 2m 1/2 mpF D(EF ) = E = (77) 2π2 h¯2 F πh¯3

40 where p =hk ¯ , and E = p2/2m. Thus, for the interacting system at the Fermi surface ∗ m pF Dinteracting(EF ) = , (78) πh¯3 where the m∗ (generally > m, but not always) accounts for the fact that the quasiparticle may be viewed as a dressed particle, and must “drag” this dressing along with it. I.e., the eﬀective mass to some extent accounts for the interaction between the particles.

4 Interactions between Particles: Landau Fermi Liquid

4.1 The free energy, and interparticle interactions

The thermodynamics of the system depends upon the free energy F , which at zero temperature is

F − F0 = E − E0 − µ(N − N0) . (79)

Since our quasiparticles are formed by adiabatically switching on the interaction in the N + 1 particle ideal system, adding

41 one quasiparticle to the system adds one real particle. Thus, X N − N0 = δnp , (80) p and since X E − E0 ≈ pδnp , (81) p we get X F − F0 ≈ (p − µ) δnp . (82) p As shown in Fig. 18, we will be interested in excitations of the system which distort the Fermi surface by an amount proportional to δ. For our theory/expansion to remain valid, we must

Figure 18: We consider small distortions of the fermi surface, proportional to δ, so 1 P that N p |δnp| 1.

42 have 1 X |δn | 1 . (83) N p p

Where δnp =6 0, p − µ will also be of order δ. Thus,

X 2 (p − µ) δnp ∼ O(δ ) , (84) p so, to be consistent we must add the next term in the Taylor series expansion of the energy to the expression for the free energy.

X 1 X 3 F − F = ( − µ) δn + f 0δn δn 0 + O(δ ) (85) 0 p p 2 p,p p p p p,p0 where δE fp,p0 = (86) δnpδnp0

The term, proportional to fp,p0, was added (to the Sommerfeld theory) by L.D. Landau. Since each sum over p is proportional to the volume V , as is F , it must be that fp,p0 ∼ 1/V . However, it is also clear that fp,p0 is an interaction between quasiparticles, each of which is spread out over the whole volume V , so the 3 probability that they will interact is ∼ rTF /V , thus

3 2 fp,p0 ∼ rTF /V (87)

43 In general, since δnp is only of order one near the Fermi surface, we will only care about fp,p0 on the Fermi surface (as- suming that it is continuous and changes slowly as we cross the Fermi surface.

Interested only in fp,p0 ! (88) p=p0=µ

0 Thus, fp,p0 only depends upon the angle between p and p .

We can also reduce the spin dependence of fp,p0 to a symmetric and anti symmetric part. First in the absence of an external ﬁeld, the system should be invariant under time-reversal, so

fpσ,p0σ0 = f−p−σ,−p0−σ0 , (89) and, in a system with reﬂection symmetry

fpσ,p0σ0 = f−pσ,−p0σ0 . (90)

Then

fpσ,p0σ0 = fp−σ,p0−σ0 . (91)

It must be then that f depends only upon the relative orienta- tions of the spins σ and σ0, so there are only two independent components fp↑,p0↑ and fp↑,p0↓. We can split these into sym-

44 metric and antisymmetric parts.

a 1 s 1 f 0 = f 0 − f 0 f 0 = f 0 + f 0 . p,p 2 p↑,p ↑ p↑,p ↓ p,p 2 p↑,p ↑ p↑,p ↓ (92) a fp,p0 may be interpreted as an exchange interaction, or

s 0 a fpσ,p0σ0 = fp,p0 + σ · σ fp,p0 (93) where σ and σ0 are the Pauli matrices for the spins. a Our ideal system is isotropic in momentum. Thus, fp,p0 and s 0 fp,p0 will only depend upon the angle θ between p and p , and a s so we may expand either fp,p0 and fp,p0 ∞ α X α fp,p0 = fl Pl(cos θ) . (94) l=0 Conventionally these f parameters are expressed in terms of reduced units. ∗ α V m pF α α D(EF )f = f = F . (95) l π2h¯3 l l

4.2 Local Energy of a Quasiparticle

Now consider an interacting system with a certain distribution of excited quasiparticles δnp0. To this, add another quasiparticle

45 p

Figure 19: The addition of another particle to a homogeneous system will yeilds in forces on the quasiparticle which tend to restore equilibrium.

0 0 of momentum p (δnp → δnp + δp,p0). From Eq. 85 the free energy of the additional quasiparticle is X ˜p − µ = p − µ + fp0,pδnp0 , (96) p0

(recall that fp,p0 = fp0,p). Both terms here are O(δ). The second term describes the free energy of a quasiparticle due to the other quasiparticles in the system (some sort of Hartree-like term).

The term ˜p plays the part of the local energy of a quasiparticle. For example, the gradient of ˜p is the force the system exerts on the additional quasiparticle. When the quasiparticle is added to the system, the system is inhomogeneous so that

δnp0 = δnp0(r). The system will react to this inhomogeneity

46 by minimizing its free energy so that ∇rF = 0. However, only the additional free energy due the added particle (Eq. 96) is inhomogeneous, and has a non-zero gradient. Thus, the system will exert a force X −∇r˜ = −∇r fp0,pδnp0(r) (97) p0 on the added quasiparticle resulting from interactions with other quasiparticles.

4.2.1 Equilibrium Distribution of Quasiparticles at Finite T

˜p also plays an important role in the ﬁnite-temperature properties of the system. If we write X 1 X E − E = δn + f 0 δn 0δn (98) 0 p p 2 p ,p p p p p,p0 P Now suppose that p |hδnpi| N, as needed for the expansion above to be valid, so that

δnp = hδnpi + (δnp − hδnpi) (99) where the ﬁrst term is O(δ), and the second O(δ2). Thus,

δnpδnp0 ≈ −hδnpihδnp0i + hδnpiδnp0 + hδnp0iδnp (100)

47 We may use this to rewrite the energy of our interacting system X 1 X X E − E ≈ δn − f 0 hδn ihδn 0i + f 0 hδn iδn 0 0 p p 2 p ,p p p p ,p p p p p,p0 p,p0   X X 1 X ≈ + f 0 hδn 0i δn − f 0 hδn ihδn 0i  p p ,p p  p 2 p ,p p p p p0 p,p0

X 1 X 4 ≈ h˜ iδn − f 0 hδn ihδn 0i + O(δ ) (101) p p 2 p ,p p p p p,p0 At this point, we may repeat the arguments made earlier to de- termine the fermion occupation probability for non-interacting Fermions (the constant factor on the right hand-side has no eﬀect). We will obtain 1 np(T, µ) = , (102) 1 + exp β(h˜pi − µ) or 1 δnp(T, µ) = − θ(pf − p) . (103) 1 + exp β(h˜pi − µ) However, at least for an isotropic system, this expression bears closer investigation. Here, the molecular ﬁeld (evaluated within kBT of the Fermi surface) X h˜p − pi = fp0,phδnp0i (104) p0

48 must be independent of the location of p on the Fermi surface (and of course, spin), and is thus constant. To see this, reconsider the Legendre polynomial expansion discussed earlier X h˜p − pi = fp0,phδnp0i p0 Z X 3 ∝ d pflPl(cos θ)hδnp0i l Z 3 ∝ f0 d phδnp0i = 0 (105)

In going from the second to the third line above, we made use of the isotropy of the system, so that hδnp0i is independent of the angle θ. The evaluation in the third line, follows from particle number conservation. Thus, to lowest order in δ

1 4 np(T, µ) = + O(δ ) (106) 1 + exp β(p − µ)

4.2.2 Local Equilibrium Distribution

Now suppose we introduce a local weak perturbation such as the type discussed in Sec. 4.2 to an isotropic system at zero temperature. Such a perturbation could be caused by, eg. a

49 sound wave or a weak magnetic ﬁeld (which you will explore in your homework, to calculate the sound velocity and susceptibility of a Landau Fermi liquid). This pertubation will cause a small deviation of the equilibrium distribution function, leading to a new ”local equilibrium” distribution

n¯p = np(˜p − µ) (107) where the argument of the RHS indicates that this is the distribution corresponding to the local energy discussed above. The gradient of the local energy yields a force which tries to restore the equilibrium distribution np(p − µ) derived above). The deviation from true equilibrium is

δnp = np − n¯p (108) ∂np(p − µ) = δn¯p + (˜p − p) (109) ∂p Using Eq. 96, we ﬁnd

∂np X δnp = δn¯p − fp,p0δnp0 (110) ∂p p0 At zero temperature, the factor ∂np = −δ( − µ), so both δn ∂p p p and δn¯p are restricted to the fermi surface. Since the pertubation of interest is small, we may expand both δnp and δn¯p in

50 a series of Legendre polynomials, and we will also split them into symmetric and antisymmetric parts (as we did with fp,p0 previously). For example,

s X s δnp = δ(p − µ)δnl Pl (111) l If we make a similar expansion for the antisymmetric and symmetric parts of δn¯p, and substitute this back into Eq. 109, then we ﬁnd F a δn¯a = 1 + l δna (112) l 2l + 1 l F s δn¯s = 1 + l δns (113) l 2l + 1 l

4.3 Eﬀective Mass m∗ of Quasiparticles

This argument most closely follows that of AGD, and we will follow their notation as closely as possible (without introducing any new symbols). In particular, since an integration by parts is necessary, we will use a momentum integral (as opposed to a momentum sum) notation X Z d3p → V . (114) (2πh¯)3 p

51 The net momentum of the volume V of quasiparticles is Z d3p P = 2V pn net quasiparticle momentum (115) qp (2πh¯)3 p which is also the momentum of the Fermi liquid. On the other hand since the number of particles equals the number of quasiparticles, the quasiparticle and particle currents must also be equal Z d3p J = J = 2V v n net quasiparticle and particle current qp p (2πh¯)3 p p (116) or, since the momentum is just the particle mass times this current Z d3p P = 2V m v n net quasiparticle and particle current p (2πh¯)3 p p (117) where vp = ∇p˜p, is the velocity of the quasiparticle. So Z d3p Z d3p pn = m ∇ ˜ n (118) (2πh¯)3 p (2πh¯)3 p p p

Now make an arbitrary change of np and recall that ˜p depends upon np, so that X Z d3p δ˜ = V f 0δn 0 . (119) p (2πh¯)3 p,p p σ0

52 For Eq. 118, this means that Z d3p Z d3p pδn = m ∇ ˜ δn (120) (2πh¯)3 p (2πh¯)3 p p p Z 3 Z 3 0 d p X d p +mV ∇ f 0δn 0 n , (2πh¯)3 (2πh¯)3 p p,p p p σ0 or integrating by parts (and renaming p → p0 in the last part), we get Z d3p p Z d3p δn = ∇ ˜ δn (121) (2πh¯)3 m p (2πh¯)3 p p p X Z d3p0 Z d3p −V δn f 0∇ 0n 0 , (2πh¯)3 (2πh¯)3 p p,p p p σ0

Then, since δnp is arbitrary, it must be that the integrands themselves are equal p X Z d3p0 = ∇ ˜ − V f 0∇ 0n 0 (122) m p p (2πh¯)3 p,p p p σ0

p0 0 The factor ∇p0np0 = − p0 δ(p − pF ). The integral may be evaluated by taking advantage of the system isotropy, and setting p parallel to the z-axis, since we mostly interested in the properties of the system on the Fermi surface we take p = pF , let θ be the angle between p (or the z-axis) and p0, and ﬁnally note

53 ∗ that on the Fermi surface ∇p˜p| = vF = pF /m . Thus, p=pF Z 02 0 pF pF X p dpdΩ p 0 = + f 0 0 δ(p − p ) (123) m m∗ (2πh¯)3 pσ,p σ p0 F σ0 However, since both p and p0 are restricted to the Fermi surface p0 p0 = cos θ, and evaluating the integral over p, we get Z 1 1 V pF X dΩ = + f 0 0 cos θ , (124) m m∗ 2 (2πh¯)3 pσ,p σ σ,σ0 1 where the additional factor of 2 compensates for the additional spin sum. If we now sum over both spins, σ and σ0, only the symmetric part of f survives (the sum yields 4f s), so 1 1 4πV p Z = + F d (cos θ) f s(θ) cos θ , (125) m m∗ (2πh¯)3 We now expand f in a Legendre polynomial series

α X α f (θ) = fl Pl(cos θ) , (126) l and recall that P0(x) = 1, P1(x) = x, .... that Z 1 2 dxPn(x)Pm(x)dx = δnm (127) −1 2n + 1 and ﬁnally that V m∗p D(0)f α = F f α = F α , (128) l π2h¯3 l l 54 we ﬁnd that 1 1 F s = + 1 , (129) m m∗ 3m∗ ∗ s or m /m = 1 + F1 /3.

Quantity Fermi Liquid Fermi Liquid/Fermi Gas ∗ ∗ m pF 2 CV m s Speciﬁc Heat Cv = 3 k T = = 1 + F /3 3¯h B CV 0 m 1 s κ 1+F0 Compressibility = s κ0 1+F0 /3 2 2 s 2 pF s c 1+F0 Sound Velocity c = ∗ (1 + F0 ) = s 3mm c0 1+F1 /3 ∗ 2 s m pF β χ 1+F1 /3 Spin Susceptibility χ = 2 3 a = a π ¯h 1+F0 χ0 1+F0 α Table 1: Fermi Liquid relations between the Landau parameters Fn and some experimentally measurable quantities. For the latter, a zero subscript indicates the value for the non-interacting Fermi gas.

The eﬀective mass cannot be experimentally measured di- rectly; however, it appears in many physically relevant measurable quantities, including the speciﬁc heat ∂E/V 1 ∂ X C = = ˜ n . (130) V ∂T V ∂T p p VN p

To lowest order in δ, we may neglect fp,p0 in both ˜p and np, so 1 X ∂np C = . (131) V V p ∂T p P Recall that the density of states D(E) = p δ(E − p), and making the same assumption that we made for the non-interacting

55 ∂µ system, that ∂T is negligible, we get, 1 Z ∂ 1 C = dD() . (132) V V ∂T exp β( − µ) + 1 This integral is identical to the one we had to evaluate for the non-interacting system, and yields the result π2 C = k2 TD(E ) V 3V B F k2 T m∗p = B F . (133) 3¯h3 Thus, measuring the electronic contribution to the speciﬁc heat ∗ CV yields information about the eﬀective mass m , and hence s F1 . Other measurements are related to some of the remaining Landau parameters, as summarized in table 1.

References

[1] https://arxiv.org/pdf/hep-ph/0407258v2.pdf

[2] https://arxiv.org/pdf/1411.4372.pdf