Landau's Theory of Fermi Liquids

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Landau’s theory of Fermi liquids

Florian Deman, Danil Platonov, Mobin Shakeri

Abstract

In this paper we are going to briefly review the Fermi liquid theory, which was firstly introduced as a generalization of Fermi gas theory and to explain the behaviour of 3He. Afterward, we are going through its application for a weakly-interacting metal. At the end we briefly increase the horizon by discussing breakdowns of the theory, and introducing non-Fermi liquid theories, as theories for strange metals. Introduction the Fermi gas theory, which was something that was observed experimentally for the liquid state of 3He. Historically speaking, The Fermi liquid theory was This theory, even though it was initially aimed to developed by Landau to explain the properties of liq- explain the similarities and differences seen between uid 3He [1], but soon it was expanded to a theory ex- the liquid phase of 3He and the non-interacting gas, it plaining the behaviour of normal metals with a weak was soon after realized that it could be a good model electron-electron interaction. The theory is a natural for conduction electrons inside a normal metal, since expansion of a non-interacting system of fermions, they can also be seen as weakly interacting fermions. usually called a Fermi-gas, to a system in which we This theory is usually preferred over some of the basic gradually introduce an interaction, namely, a Fermi and famous single-electron theories such as Hartree- liquid. Fock approximation, since it is not trying to explain The study of 3He has become popular, mostly be- the interaction using Slater determinants. cause of its large fusion energy, making it a possibility In the following section we will go through this the- for a future energy source [2], its strange behaviour at ory more profoundly. Evidently there are cases in low temperatures, such as superfluidity [3], and the which this Fermi liquid description of electrons break- fact that this substance stays as a liquid, even at the down1, due to strongly-interacting electrons. Some of lowest temperature, under standard pressure. The these materials can be explained by non-Fermi liquid last mentioned characteristic of 3He makes this sub- theories, which are usually called a “strange metals,” stance very special since, at its condensation temper- as opposed to a normal metals. This is an ongoing ature which is about 3K [3], where we have already area of research in many body physics. entered the quantum regime, we are dealing with a liquid, unlike the more common case of a solid. This will give rise to the phenomena of a quantum liquid. Free electron Fermi gas 3He contains two neutrons, one proton, and two We start with the foundation on which the Fermi liq- electrons, making the whole atom a fermion. At uid theory of Landau builds up, a gas of N free elec- high enough temperatures in the gas state of 3He, we trons which are confined in a cubic box with volume would expect a negligible interaction between these V = L3. Free means that the electrons do neither fermionic atoms, hence making the whole system a interact with an external potential V nor with each Fermi gas. We can slowly introduce the interaction other. The finite box allows us to use periodic bound- between the fermions, and following the adiabatic ary conditions, but of course one could make the box theorem, the ground state of the Fermi gas would infinitely large in the end by applying the thermody- transform into the ground state of the interacting namic limit. This allows us to write the wave function system. In this process, the spin, charge and momen- of the whole system as a Slater determinant tum of the fermions, remain unchanged, while their dynamical properties, such as their mass, magnetic Ψ(~r1, σ1; ... ; ~rN , σN ) = moment etc. are renormalized to a new set of values [4]. φ1(~r1, σ1) . . . φN (~r1, σ1) 1 , . .. . This was the basic idea of Landau behind the the- √ . . . N! ory of Fermi liquids. Landau introduced a quasipar- φ1(~rN , σN ) . . . φN (~rN , σN ) ticle excitations in the new interacting theory, corre- (1) sponding to the Fermi particles of the Fermi gas. This 1 idea makes the liquid theory qualitatively similar to Materials like Y1−xUxPd3

1 where σi denotes the spin of the i’th particle, ~ri Introduction of quasiparticles the position of the i’th particle and φ (~r , σ ) the i j j Now we want to make the transition from the ideal plane wave of the i’th electron at ~r with spin σ j j free electron gas to the real Fermi liquid. The idea (i, j = 1,...,N). This wave function contains the is to make a one-to-one correspondence between the Pauli principle, since it vanishes identically if two par- eigenstates of the ideal free electron gas and the real ticles are at the same position and have equal spin. Fermi liquid2. We assume that the interaction is Each one particle wave function is a plane wave of turned on in an adiabatic way, i.e. infinitely slow. the form The eigenstates of the ideal system will progressively 1 i ~p~r φ~p(~r, σ) = √ e ~ (2) transform into certain eigenstates of the real system. V The eigenstates of the real system can be written as with momentum ~p. By imposing the periodic bound- ary conditions in x, y, z-direction we find the ground |Ψi = |Ψ0i + λ |Ψ1i , (8) state, which is called ”Fermi sea”. This means that where |Ψ0i is an eigenstate of the ideal gas, |Ψ1i is a all single particle states are filled up to the Fermi correction to the ideal eigenstate and λ is a parameter momentum pF . In momentum space one can imag- characterizing interactions, i.e. λ = 0 when we have ine this as a sphere with radius pF , where all states no interactions and λ 1 when we turned on a small within the sphere are occupied and all the others are interaction. In order to generate all real eigenstates, empty. The energy is then given by we must assume that the real ground state is gener- ated by some eigenstate of the ideal system, which X p2 E(ñ ) = n˜ , (3) is described by the distributionn ˜~pσ. This statement ~pσ 2m ~pσ ~p,σ defines a normal fermion system [7]. We imagine that once the interaction is fully turned on each 3He atom in which we introduced the distribution functionn ˜~pσ, has collected a cloud of other atoms around itself, which is equal to 1 for p ≤ pF and 0 otherwise. The through this interaction. We define the atom sur- 2 2 2 pF ~ kF rounded by its cloud as a quasiparticle [7, 3]. energy at the Fermi surface is thus F = 2m = 2m . Furthermore, we can compute the number of particles Actually, the adiabatic switching method is some- what questionable, because when quasiparticle colli- by summing over all p ≤ pF , i.e. sions excite the real system, the state describing this 3 3 situation will decay exponentially over time, accord- X 4πpF /3 V 2mF 2 N = 2 1 = 2 = , (4) ing to Fermi’s golden rule. In that case, the time over (2π~/L)3 3π2 ~2 p≤pF which the interaction is turned on is greater than the lifetime of the excited state, this state has decayed where the factor of 2 accounts for the two possible before we reach it. Hence, the switching procedure is spin states. This allows us to calculate the density of no longer reversible. Conversely, if the interaction is states turned on too quickly, the process is no longer adia- 3 batic and we would perturb the system. This problem dN V 2m 2 √ mp D() = = = . (5) does obviously not arise, if the state under consider- d 2π2 2 π2 3 ~ ~ ation is the ground state, because it is stable and can We obtain the specific heat (per volume) at low tem- therefore be precisely defined. The only region where peratures by first calculating the energy and then us- this approach is valid is near the Fermi surface, be- ing cause quasiparticle lifetimes become√ sufficiently large there. This can be seen by the ∝ behaviour of ∂E C = (6) the density of states in Fermi’s golden rule. That re- ∂T V gion can be imagined as a transitional zone, where which eventually yields quasiparticles get excited above the Fermi level, due to collisions. At very low temperatures, the excited 2 π 2 mpF 2 states are very close to the Fermi momentum, which C = D(F )kBT = kBT, (7) 3 3~3 means that damping is negligible. Thus, we conclude that our approach is physically meaningful. where kB is Boltzmann’s constant. For a more de- tailed calculation please refer to [5]. Energy of the quasiparticles The total energy of the system is not the sum of the Normal Fermi liquid single particle energies anymore, since we have interactions. In this case the energy is a functional of the

In the following we want to make the transition from 2For a more sophisticated approach with Green’s functions the ideal Fermi gas to the real Fermi liquid. c.f. [6]

2 particle distribution E[n~pσ], what is a direct conclu- [5]. Including the interaction part in the expansion, sion of the Hohenberg-Kohn theorem [8]. Remember yields a functional is an object which takes a function as an 0 X input and returns a number. Therefore, in order to E[n~pσ] = E[n~pσ] + (~p)δn~pσ get the total energy of the system, we need to know ~pσ (13) the particle distribution. If n~pσ is sufficiently close 1 X 0 3 + f(~p,~p )δn δn 0 0 + O(δn ), to the ground state distribution n0 , we can Taylor 2 ~pσ ~p σ ~pσ ~p~p0;σσ0 expand the energy functional 0 0 X 2 where f(~p,~p ) is called the interaction function of the E[n~pσ] = E[n~pσ] + (~p)δn~pσ + O(δn ). (9) quasiparticles and describes the interaction of two ~pσ particles with momentum ~p and ~p0. In the case of 0 δE[n ] a Fermi gas we have f(~p,~p ) = 0. By differentiating In the first order (~p) = ~pσ denotes the func- δn~pσ (13) with respect to n we find tional derivative of the total energy with respect to ~pσ 0 the distribution function and δn~pσ = n~pσ − n is the X 0 ~pσ ˜(~p) = (~p) + f(~p,~p )δn~p0,σ0 (14) deviation of n from the ground state distribution ~pσ ~p0;σ0 0 function n~pσ. We can imagine (~p) as the change of energy of the system when we add a single quasipar- what is called the ”true” quasiparticle energy [3]. ticle with momentum ~p to the system. Note that we This allows us to rewrite (13), such that actually should expand the expression to second or- 0 X der, in order to account for interactions between the E[n~pσ] = E[n~pσ] + ˜(~p)δn~p,σ (15) particles, but for simplicity we neglect it for now and ~p0;σ0 will include it in the next section. and expanding this in the vicinity of the Fermi surface If our distribution is close to a step function θ(x), i.e. gives us close to a Fermi-Dirac distribution at T = 0, we can p replace n~pσ in (~p) by θ(pF − |~p|). We can therefore F X 0 (~p)−( ~pF ) = (p−pF )+ f(~p,~p )δn~p0,σ0 . (16) approximate (~p) in the vicinity of the Fermi surface m∗ ~p;σ in powers of p − pF [6]. This procedure yields It is conventional to define the dimensionless quantity pF (~p) − ( ~pF ) ≈ vF (p − pF ) = (p − pF ), (10) m∗ dn X F (~p,~p0) ≡ f(~p,~p0) = F P (cos θ), (17) d l l whereby vF is the velocity of the quasiparticle on the l Fermi surface. Furthermore, in analogy to the ideal ∗ pF N gas we defined the effective mass m = of the where n = , Pl(cos θ) are the Legendre polynomi- vF V quasiparticle. The effective mass thus determines the als and θ is the angle between the momenta ~p and density of state and the specific heat of the real Fermi ~p0. This procedure is justified, because ~p and ~p0 are liquid. The expressions look the same as in the free momenta close to the Fermi surface. Hence, the sys- Fermi gas, the only thing we have to change is replac- tem is invariant under rotations in momentum space ing the mass m by the effective mass m∗, which gives and we may expand F (~p,~p0) in Legendre polynomials. us Thus, the two particle interaction is completely deter- m∗p F mined by the Fl, which are sometimes called Landau D(F ) = (11) π2~3 parameters. Now we want to find a relation between and the real and the effective mass. For that we make a m∗p F 2 Galileo boost to a system I0, which moves with re- C = 3 kBT. (12) 3~ spect to the original system I with velocity ~u. There- Thus, we also see the the linear behaviour of the heat fore, the distribution function becomes capacity at very low temperatures, but with a different slope than free electron gas. n~pσ → n~p−m~u,σ = θ(pF − |~p − m~u|). (18)

By using (18) while equating (~p) − ( ~p ) = pF (p − Quasiparticle interaction F m pF ) and (16), we get Another important part of the theory is an eﬀective pF pF X m~ueˆ = m~ueˆ + f(~p,~p0)(n − n0 ), interaction between the quasiparticles. For simplicity m ~p m∗ ~p ~p−m~u,σ ~pσ we will only consider spin independent interactions in ~pσ the following. A generalization to the spin dependent (19) case can be found in [6] or [7]. In the previous sec- wheree ˆ~p is the unit vector in ~p direction. Combining tion we neglected the interaction, but in order for this result with (17) ﬁnally yields the theory to be physically meaningful, i.e. satisfy- m∗ F = 1 + 1 . (20) ing Galilean invariance, we must include interactions m 3

3 The first two coefficients F0 and F1 can be determined in experiments and we are able to calculate the effective mass. Another quantity which can be determined with these coefficients is the sound velocity

r1 c = v (1 + F )(1 + F ). (21) F 3 0 1 In order to determine other quantities, such as the magnetic susceptibility, one has to take spin into account. This will introduce another set of coeﬃcients, which can also be determined experimentally [5, 3]. For 3He, the eﬀective mass ratio and the Landau parameters are [9]:

∗ m /m F0 F1 melting pressure3 6.2 94.0 15.7 zero pressure 3.0 10.1 6.0

Specific Heat It is worth noting that even with quasiparticle interaction turned on, the specific heat would still vary linearly at low temperatures. Furthermore, the slope 3 will not change from the first order calculations. Figure 1: Specific heat of He at two different densi- Since there is one-to-one mapping between non- ties [10]. At low temperatures heat capacity interacting quasiparticle and interacting quasiparti- grows linearly. cle situations, equilibrium distribution still remains the Fermi distribution References 1 δn (T ) = ~pσ eβ(˜(~p)−µ) + 1 [1] LD Landau. “On the theory of the Fermi liquid”. In: Sov. Phys. JETP 8.1 (1959), p. 70. Thus, interaction term is O(T 4) in energy and that will not affect heat capacity at second order [9]. See [2] Thomas Simko and Matthew Gray. “Lunar (6). Helium-3 Fuel for Nuclear Fusion: Technology, Predicted scaling of heat capacity was observed ex- Economics, and Resources”. In: World Future perimentally, see Figure 1. Review 6.2 (2014), pp. 158–171. [3] Anthony J Leggett. “A theoretical description of the new phases of liquid He 3”. In: Reviews Conclusion of Modern Physics 47.2 (1975), p. 331. We have extended the Fermi gas theory by adding [4] Piers Coleman. Introduction to many-body weak interactions. By the virtue of adiabatic theo- physics. Cambridge University Press, 2015. rem, these can be though to lift the ground state of [5] Erkki Thuneberg. “Introduction to Landau’s the Fermi gas to Fermi liquid, where we have a one- Fermi Liquid Theory”. In: Dept. of Physical to-one mapping between the particles in both mod- Sciences, University of Oulu (2009). els. Due to the interactions, energy now depends on [6] LD Landau, EM Lifshitz, and LP Pitaevskij. the relative position of the particles. To the first or- “Statistical Physics, Part 2: Theory of the Con- der, with no interactions between quasiparticles, this densed State”. In: Course of theoretical physics simply changes mass of the particles by the effective 9 (1987). mass m∗ = pF . We then added interactions (spin in- vF dependent) by considering second order terms in the [7] David Pines. Theory of Quantum Liquids: Nor- energy functional. This allowed us to express the en- mal Fermi Liquids. CRC Press, 2018. ergy of the system in terms of Legendre polynomials [8] Pierre Hohenberg and Walter Kohn. “Inho- with some free parameters which can be determined mogeneous electron gas”. In: Physical review experimentally. 136.3B (1964), B864.

329 atm at 0.3 K

4 [9] Michael Cross. “Physics 127 c : Statistical Me- chanics Fermi Liquid Theory : Thermodynam- ics”. In: Caltech, 2004. [10] Dennis S. Greywall. “Speciﬁc heat of normal liquid 3He”. In: Phys. Rev. B 27 (5 1983), pp. 2747–2766. doi: 10.1103/PhysRevB.27. 2747. url: https://link.aps.org/doi/10. 1103/PhysRevB.27.2747.