Non-Fermi liquids and the Wiedemann-Franz law
NNon-Fermi liquids and the Wiedemann-Franz law
Raghu Mahajan, Maissam Barkeshli, Sean A. Hartnoll
Department of Physics, Stanford University,Stanford, CA 94305-4060, USA
Abstract
A general discussion of the ratio of thermal and electrical conductivities in non-Fermiliquid metals is given. In metals with sharp Drude peaks, the relevant physics is correctlyorganized around the slow relaxation of almost-conserved momenta. While in Fermi liq-uids both currents and momenta relax slowly, due to the weakness of interactions amonglow energy excitations, in strongly interacting non-Fermi liquids typically only momentarelax slowly. It follows that the conductivities of such non-Fermi liquids are obtainedwithin a fundamentally different kinematics to Fermi liquids. Among these stronglyinteracting non-Fermi liquids we distinguish cases with only one almost-conserved mo-mentum, which we term quasi-hydrodynamic metals, and with many patchwise almost-conserved momenta. For all these cases, we obtain universal expressions for the ratioof conductivities that violate the Wiedemann-Franz law. We further discuss the case inwhich long-lived ‘cold’ quasiparticles, in general with unconventional scattering rates,coexist with strongly interacting hot spots, lines or bands. For these cases, we charac-terize circumstances under which non-Fermi liquid transport, in particular a linear intemperature resistivity, is and is not compatible with the Wiedemann-Franz law. Wesuggest the likely outcome of future transport experiments on CeCoIn , YbRh Si andSr Ru O at their critical magnetic fields. a r X i v : . [ c ond - m a t . s t r- e l ] A ug The Wiedemann-Franz law
The Wiedemann-Franz law for the Lorenz ratio of thermal conductivity κ to electricalconductivity σ , at low temperature T , L ≡ κσT = π ≡ L , (1)in units with k B = e = 1, is a robust feature of Fermi liquids at low temperature [1]. Ina non-Fermi liquid metal, there is ample reason to expect the Wiedemann-Franz law to beviolated, see e.g. [2]. There may be additional gapless neutral collective degrees of freedompresent that transport heat but not charge. There may furthermore be inelastic scatteringbetween charged and neutral degrees of freedom that affects heat and charge transportdifferently. It may therefore seem surprising that in several systems exhibiting supposedlyhallmark non-Fermi liquid phenomenology, such as a linear in temperature resistivity, theWiedemann-Franz law is observed to hold at the lowest temperatures [3, 4, 5, 6, 7], whilethe interpretation of recently reported violations of the law in such materials has provedcontentious [8, 9, 10]. On the other hand, violations of the Wiedemann-Franz law in metallicregimes at low temperatures have been reported in the underdoped cuprates [11, 12, 13],upon suppressing superconductivity, and also in the c-axis conductivity of a heavy fermionnon-Fermi liquid [14]. This panoply of results clearly presents a theoretical challenge.The thermal and electrical conductivities are subtle observables because they dependcrucially on momentum relaxation in order to be finite. The interactions between chargecarriers and gapless neutral degrees of freedom causing non-Fermi liquid behavior may notin themselves cause momentum relaxation. To understand the ratio of conductivities, onemust therefore characterize the interplay between momentum-conserving and momentum-non-conserving interactions. If a quasiparticle description is apposite, this question can beaddressed via the Boltzmann equation, e.g. [1, 15]. In this paper we shall make some simpleobservations about the ratio of thermal to electric conductivity in generic non-Fermi liquids.Important differences will be found between cases in which weakly interacting quasiparticlesare present and those in which they are not. The tool we shall use to organize our discussion,which emphasizes the role of momentum relaxation, is the memory matrix formalism [16].The memory matrix formalism allows a clear-headed discussion of conductivities withoutrecourse to a quasiparticle worldview. As such it has proved useful in studying the transportof one dimensional interacting electrons [17, 18] and higher dimensional systems at quantumcriticality [19, 20, 21]. In particular, a memory matrix approach successfully predicted largeviolation of the Wiedemann-Franz law in one dimensional Luttinger liquids [22, 23]. The1nsight exploited by the memory matrix approach is the following: Even if the systemis strongly interacting, in a metallic phase the effects of momentum relaxation (due toquenched impurities or a lattice) on low energy processes can often be understood as aperturbation about an effective translation-invariant low energy theory. In many cases thiswill be synonymous with the system exhibiting a well-defined Drude peak in the opticalconductivity at the low temperatures of interest. A Drude peak certainly does not imply aquasiparticle description of transport. The memory matrix method then proceeds to isolateprecisely the quantities that may be treated perturbatively in these cases.The memory matrix formalism is to be distinguished from the earlier ‘memory function approximation’ developed in e.g. [24, 25, 26]. In that approach one works perturbatively inthe relaxation of the electrical current rather than the momentum. This is most appropriatefor non-interacting systems in which the total current (cid:126)J itself is a conserved operator (i.e.˙ (cid:126)J = 0) prior to impurity and other scatterings. For the strongly interacting non-Fermiliquids we wish to consider below, the total current itself is not an almost conserved operatorand one should instead focus on momentum conservation. The momentum and currentoperators are typically not equal in these systems. The memory matrix furthermore includesvarious projection operators, see for instance equation (7) below, that render it an exactexpression rather than an approximation.In this paper we focus on a rough classification of the different possible kinematic regimesthat can control transport in the theories of interest. Our results take the form of relation-ships between different physical quantities that pertain depending on the kinematics of thesystem. For instance, part of the power of the Wiedemann-Franz law (1) is that the righthand side is a pure number that is computable using Fermi liquid theory. Effects sensitiveto processes external to Fermi liquid theory, in particular the quasiparticle lifetime, havecancelled out of the final expression. We shall give a memory matrix description of thisfact below, and, from our more general perspective, establish conditions under which ananalogous universality may be achieved for non-Fermi liquids.Our considerations apply to the following cases:1. Non-Fermi liquids with no quasiparticles . Section 4. These may include two dimen-sional nematic quantum critical points and Fermi surfaces coupled to emergent gaugefields, in which the full Fermi surface goes critical. The kinematics of almost-conservedquantities is entirely different to Fermi liquid theory and the Wiedemann-Franz law isexpected to not hold even approximately. Currents are not approximately conserved.The only almost-conserved quantities are momenta, of which there may be one or2any (if the patches of a strongly interacting Fermi surface decouple). For the caseof only one conserved momentum, we refer to these as ‘quasi-hydrodynamic metals’,we obtain the ratios of conductivities κσT = 1 T χ QP χ JP and κσT (cid:28) . (2)Here κ is the thermal conductivity at zero electric field, while κ is the conductivityat zero electric current. The χ AB are static susceptibilities involving the operatorsfor the total momentum (cid:126)P , electric current (cid:126)J and heat current (cid:126)Q . While the specificvalues and temperature dependence of these susceptibilities will depend on details ofthe theory at all energy scales, the left hand result in (2) is universal in the sensethat the mechanism of momentum relaxation has once again cancelled from the ratioof conductivities. This universality is analogous to the Wiedemann-Franz law. Caseswith many patchwise conserved momenta are discussed in section 4.2, where we obtain κσT ∼ κσT ∼ T (cid:42) χ P Q χ P J (cid:43) . (3)While κ = κ in a Fermi liquid, κ ∼ κ showing the same temperature scaling withoutequality is a diagnostic for strongly interacting transport with patchwise conservedmomenta. The angled brackets refer to a specific average over the Fermi surface.2. Non-Fermi liquids with long-lived quasiparticles . Section 5. These are systems inwhich some or all of the charge and heat carriers remain long lived, despite the sys-tem exhibiting unconventional metallic transport. These may include finite wavevectorquantum critical points, such as spin and charge density wave transitions in metals,which have hot and cold fermionic excitations. They may also include situations whereone electron band goes critical but others do not. Finally these include cases in whichthe whole Fermi surface is critical but the fermions retain a quasiparticle character.The Wiedemann-Franz law will be obeyed in these systems at low temperatures ifthe long-lived electronic quasiparticles dominate the charge and heat transport at lowtemperatures and scatter elastically. If the electronic quasiparticles have an uncon-ventional scattering rate, the Wiedemann-Franz law will coexist with non-Fermi liquidtransport. We highlight a scenario in which the Wiedemann-Franz law can coexist atlow temperatures with a linear in temperature electrical resistivity: if the ‘cold’ de-grees of freedom scatter elastically off the ‘hot’ modes, which in turn relax momentummore efficiently than the cold excitations. In these circumstances, the hot modes actas ‘generalized phonons’. 3he above outline makes clear that the status of the Wiedemann-Franz law at low tem-peratures in a non-Fermi liquid provides immediate insight into the nature of the excitationsin the system. In section 6 we use the simple observations summarized above to organizethe existing experimental data on the Wiedemann-Franz law in heavy fermions, ruthenatesand the cuprates. See table 1 below. The law has not yet been studied in certain naturaltemperature regimes, or ambiguous results have been obtained, allowing us to predict theoutcome of future measurements.
Let us briefly introduce the memory matrix. To follow the present paper, only the logicalstructure of equation (6) below need be understood. We will avoid actual evaluation of thememory matrix. The starting point are the correlators χ ( z ) and the static susceptibility χχ AB ( z ) = i (cid:90) ∞ dte izt (cid:104) [ A ( t ) , B (0)] (cid:105) , χ AB = lim z → i + χ AB ( z ) , (4)with the operators A, B being taken from the set of (i) the operators whose two point func-tion we wish to compute, these are the total electric and thermal currents { (cid:126)J , (cid:126)Q } , and (ii)any almost-conserved operators that have an overlap with the currents of interest, for in-stance the total momentum (cid:126)P . The quantity of interest to us is the matrix of conductivities σ ( ω ) = χ ( ω ) − χiω . (5)The limit of real frequencies is z → ω + i + , as this is where the integrals in (4) converge.The memory matrix formalism expresses the matrix of conductivities as σ ( ω ) = 1 − iω + M ( ω ) χ − χ , (6)with the memory matrix being given by [16] M AB ( ω ) = (cid:90) /T dλ (cid:28) ˙ A (0) Q iω − Q L Q Q ˙ B ( iλ ) (cid:29) . (7)Here L is the Liouville operator L = [ H, · ] and Q projects onto the space of operatorsorthogonal to { (cid:126)P , (cid:126)J , (cid:126)Q } . The reason it is useful to introduce the memory matrix is that ifthe effects of momentum or current non-conservation are small at low energies, this quantitycan be treated perturbatively because the ˙ A and/or ˙ B appearing in (7) will be small. Thed.c. conductivities are σ d.c. = lim ω → χM ( ω ) − χ ≡ Γ − χ . (8)4ere we introduced the matrix of relaxation rates Γ. While χ and M will be symmetric,Γ need not be. Equation (8) expresses the d.c. conductivities as a combination of ‘fast’processes described by χ , in which currents and momenta source each other, and ‘slow’processes described by M . The inverse memory matrix M − will be dominated by thedecay of any almost-conserved quantities in the system that overlap with the currents. Thequantities of interest are then the electrical, heat and thermoelectric conductivities σ ≡ σ d.c. JJ , κ ≡ σ d.c. QQ T , α ≡ σ d.c. JQ T . (9)Equation (6) shows that when some of the relaxation rates can be treated pertubatively– when the memory matrix approach is useful – then there will be a sharp Drude peak inthe frequency dependent conductivity.
As a first application of the memory matrix formalism, we can give a re-derivation of theWiedemann-Franz law (1). The effective low energy and momentum-conserving theory inthis case is the patchwise description of the excitations of a Fermi surface [27, 28]. Letus label the patches by θ and the momentum locally perpendicular to the Fermi surfaceas k . Fermi liquids have the rather special property that to lowest order in energies thedifferent patch theories are decoupled and furthermore free. This last statement requiresqualification in two dimensions due to the logarithmic growth in the strength of scatteringby disorder and by soft collective particle-hole excitations at low temperatures. We returnto this point at the end of the section. Decoupled free patches leads to the fact that thequasiparticle densities δn θk = c † θk c θk are all independently conserved. The electron-electroninteractions that can relax the density within patches, such as normal forward scatteringand umklapp processes, are negligible at the lowest temperatures and energy scales [1].The leading density (and, shortly, momentum) relaxing interaction is elastic scattering byquenched impurities. Thus we write ddt δn θk = − (cid:90) dθ (cid:48) dk (cid:48) Γ θθ (cid:48) kk (cid:48) δn θ (cid:48) k (cid:48) . (10)An explicit expression can be obtained for the relaxation rates per unit perpendicular mo-mentum Γ θθ (cid:48) kk (cid:48) from e.g. the appropriate Boltzmann equation [1]. To implement thememory matrix method, we need to assume that the scattering rate Γ is small in units ofthe Fermi energy µ . The Wiedemann-Franz law can in fact hold without this assumption529]. The large number of almost conserved quantities δn θk means that the memory matrixmethod is applied in a particular way for Fermi liquids. Our derivation of Wiedemann-Franzwill parallel very closely the usual derivation [1]. We are emphasizing the role of variousalmost conserved quantities from the perspective of the effective low energy theory, with aview to generalizing to the non-Fermi liquid case in the following.The momentum and heat and electrical currents may be constructed patchwise out ofthe almost conserved densities. To lowest order in energies (cid:126)P θ = (cid:126)k F θ (cid:90) dk δn θk , (cid:126)Q θ = (cid:126)v F θ (cid:90) dk ε θk δn θk , (cid:126)J θ = (cid:126)v F θ (cid:90) dk δn θk . (11)The quasiparticle energy ε θk vanishes at the Fermi surface and so must be kept inside the k integral. The total momenta and currents are obtained by integrating these expressionsover the whole Fermi surface.From (11) we that, for this Fermi liquid case, the rate at which the total almost-conservedquantities relax will be controlled by the relaxation of the quasiparticles densities (10). Wemust therefore consider the memory matrix for these densities directly. We can then write,picking some spatial direction (cid:126)n and setting v F θ = (cid:126)v F θ · (cid:126)n , κ = 1 T (cid:90) dθdθ (cid:48) dkdk (cid:48) v F θ v F θ (cid:48) ε θk ε θ (cid:48) k (cid:48) σ d.c. δn θk δn θ (cid:48) k (cid:48) = 1 T (cid:90) dθdθ (cid:48) v F θ v F θ (cid:48) (cid:90) dkdk (cid:48) (cid:0) Γ − (cid:1) θθ (cid:48) kk (cid:48) ε θk ε θ (cid:48) k (cid:48) χ δn θ (cid:48) k (cid:48) δn θ (cid:48) k (cid:48) . (12)In the first line σ d.c. δnδn is defined by the general formula (8) for d.c. ‘conductivities’. In thesecond line we have used the fact that to lowest order in energies the quasiparticle suscep-tibility is only nonzero for fluctuations on the same patch and with the same momentum.At the end of this section we mention the role of scattering by collective particle-hole ex-citations that can couple patches. We have also introduced the matrix of relaxation ratesΓ, as defined in (8), which is the same as the matrix appearing in (10). These matricesare seen to be the same upon Fourier transforming the conductivity matrix (6) and takingthe late time limit [16]. Elasticity of the collisions, together with the fact that Γ becomesindependent of the perpendicular momentum to lowest order in energies, allows us to write (cid:0) Γ − (cid:1) θθ (cid:48) kk (cid:48) = (cid:0) Γ − (cid:1) θθ (cid:48) δ ( ε θk − ε θ (cid:48) k (cid:48) ). We may therefore use the energy conservation deltafunction to replace the ε θk in (12) with ε θ (cid:48) k (cid:48) to obtain κ = 1 T (cid:90) dθdθ (cid:48) v F θ v F θ (cid:48) (cid:0) Γ − (cid:1) θθ (cid:48) (cid:90) dk (cid:48) ε θ (cid:48) k (cid:48) χ δn θ (cid:48) k (cid:48) δn θ (cid:48) k (cid:48) (cid:90) dk δ ( ε θk − ε θ (cid:48) k (cid:48) )= T π (cid:90) dθdθ (cid:48) v F θ v F θ (cid:48) (cid:0) Γ − (cid:1) θθ (cid:48) (cid:90) dk (cid:48) χ δn θ (cid:48) k (cid:48) δn θ (cid:48) k (cid:48) (cid:90) dk δ ( ε θk − ε θ (cid:48) k (cid:48) )= T π σ . (13)6o obtain the second line we have used the well-known result relating the heat and electriccurrent susceptibilities for free fermions. Namely (cid:82) dkε χ δnδn = T π / (cid:82) dkχ δnδn . Thisfollows from performing integrals of the Fermi-Dirac distribution, because the susceptibili-ties χ δnδn = f (cid:48) FD ( ε ), and holds patchwise in a Fermi liquid.The thermal conductivity at zero electric current is κ = κ − α T /σ , with α the thermo-electric conductivity. The thermoelectric conductivity is given by adapting the formulae in(12) in the obvious way. The susceptibility integral that appears is now (cid:82) dkεχ δnδn which,again using the Fermi-Dirac distribution, is seen to go like T at low temperatures. Itfollows that κ (cid:29) α T /σ at low temperatures, and hence κ = κ in this regime. The result(13) therefore implies the Wiedemann-Franz law (1).In two space dimensions, Fermi liquid theory suffers from marginally relevant perturba-tions due to disorder and scattering by collective particle-hole excitations. The collectivemodes of an interacting disordered Fermi liquid lead, in two dimensions, to logarithmic cor-rections to the electric and thermal conductivities that grow as the temperature is lowered.These corrections violate the Wiedemann-Franz law, e.g. [30, 31]. In the good metal-lic regimes we are focussing on, away from metal-insulator transitions, these effects onlybecome important at exponentially small temperatures. In a Fermi liquid, the small effects of interactions at low energies implied the existence ofan infinite collection of conserved densities δn θk in the effective low energy theory. Thenon-Fermi liquids that we consider in the remainder are characterized by the persistenceof strong interactions in the low energy theory. Therefore, the large number of conserveddensities δn θk are not present. It follows that generically the total electrical and heatcurrents, (cid:126)J and (cid:126)Q , are not conserved (unlike in a Fermi liquid).We will require an essential additional feature of the non-Fermi liquids. Namely, thatin the strongly interacting effective low energy theory, the total momentum is conserved, sothat ˙ (cid:126)P = 0 up to the effect of irrelevant or weak marginal operators. This will allow thesesystems to be ‘good’ metals with a sharp Drude peak, despite the absence of quasiparti-cles. The kinematics of conserved quantities underlying such non-Fermi liquid transport issignificantly different to that of a Fermi liquid. In these non-quasiparticle based circum-stances, overly na¨ıve analogies such as considering scattering by quantum critical bosons tobe similar to scattering by phonons in a Fermi liquid will lead to incorrect conclusions.7he conservation of momentum up to effects that are small at low energies is a keyassumption that allows the memory matrix method to work. It can be expected to betrue in a metal exhibiting a well-defined Drude peak at low temperatures. This is becausea perturbative approach to the memory matrix will lead to a sharp Drude peak in (6).Our discussion in this paper does not apply to cases in which momentum-non-conservinginteractions have strong effects at low energies. For instance, experimentally speaking, atthe boundary of metal-insulator transitions one can encounter ‘bad’ metals that violate theMott-Ioffe-Regal resistivity bound [32] and do not exhibit Drude peaks [33]. In appendixA we briefly discuss theoretical circumstances in which such strong momentum-violatinginteractions arise.While interactions in a non-Fermi liquid mean that the low energy theory is not free,it is still possible that there may be a decoupling of excitations into patches in momentumspace, analogous to the patches of a Fermi surface in Fermi liquid theory. In these cases therewill not just be one conserved momentum (cid:126)P , but rather a family of conserved momenta (cid:126)P θ , labelled by the patch θ . We will consider these two cases, with one and with manyconserved momenta, separately. We will refer to metallic systems in which there is only one almost-conserved vector operatorin the effective low energy theory, typically the total momentum, as ‘quasi-hydrodynamic’non-Fermi liquids. For such non-Fermi liquids we can obtain general results concerning theratio of conductivities. The total momentum is relaxed on a much longer timescale than allother quantities, including the currents; schematically (cid:104) ˙ (cid:126)P (cid:105) ∼ (cid:15) µ (cid:104) (cid:126)P (cid:105) , while, for instance, (cid:104) ˙ (cid:126)J (cid:105) ∼ µ (cid:104) (cid:126)J (cid:105) , with (cid:15) (cid:28) µ a microscopic scale. Our discussion is easily adapted to casesin which the electric current is equal to the momentum. The only general requirement forour results is that there be only one almost-conserved vector operator. From the hierarchyof relaxation rates, we can anticipate that in the ω = 0 memory matrix M P P ∼ (cid:15) and M P J ∼ M P Q ∼ (cid:15) , while the remaining components of the memory matrix (7) are order one.It follows that the inverse of the memory matrix will be dominated by (cid:0) M − (cid:1) P P ∼ (cid:15) − (cid:29) O at the latticemomentum k L : H = H − (cid:15) O ( k L ). It follows that ˙ (cid:126)P = i [ H, (cid:126)P ] = (cid:15) (cid:126)k L O ( k L ). In this case one finds that due to the projections Q in the definition of the memory matrix (7), then M PJ ∼ M PQ ∼ (cid:15) are of the same order as M PP , but our general statement above that (cid:0) M − (cid:1) PP ∼ (cid:15) − (cid:29) (cid:0) M − (cid:1) P P dominating the inverse memory matrix, the d.c. conductivities (8) are σ = χ JP (cid:0) M − (cid:1) P P , κ = 1
T χ QP (cid:0) M − (cid:1) P P . (14)Taking the ratio of the conductivities in (14), we obtain the advertised (2) κσT = 1 T χ QP χ JP . (15)In this formula for the ratio of conductivities the mechanism of momentum relaxation hascancelled out, leaving an expression in terms of purely thermodynamic quantities. In thissense (15) captures a universality analogous to that of the Wiedemann-Franz law. The valueand indeed temperature dependence of this ratio will however depend on the low energytheory describing the system.Fast relaxation of currents but slow relaxation of momentum is characteristic of hydro-dynamic transport. For this reason we call these theories quasi-hydrodynamic. It has beensuggested that a similar notion applies to highly correlated electron gases in semiconductorheterostructures [34, 35]. Indeed our formula (15) can also be derived, in the relativistic caseat least and now at sufficiently high temperatures, using standard hydrodynamics [19]. Weare, however, in a low temperature regime that is not that of conventional hydrodynamics.The memory matrix is the appropriate theoretical framework for us. See also [36].An interesting result is obtained if we consider the thermal conductivity κ = κ − α T /σ inthis case. Extending the dominance of (cid:0) M − (cid:1) P P in (14) to the thermoelectric conductivity α , one finds that the two terms in the expression for κ exactly cancel. We explain thesimple physics of this cancellation at the end of the following subsection. The leadingnonvanishing term in the thermal conductivity is a universal quantity computable in thelow energy effective theory. This universal term does not benefit from the enhancement by (cid:0) M − (cid:1) P P (cid:29)
1. It follows that κσT (cid:28) , (16)for these systems. A small Lorenz number therefore seems to be a characteristic, model-independent, feature of quasi-hydrodynamic non-Fermi liquid metals. This is perhaps themost dramatic of the kinematically-driven results we will find. It illustrates the limitationsof a weakly interacting intuition; the result (16) cannot be understood starting from a Fermiliquid and then perturbatively adding the effects of additional neutral degrees of freedom(more heat conduction) or additional inelastic scatterings (more heat relaxation). In theFermi liquid discussion of section 3, all relaxation was controlled by the matrix Γ of density dominates the inverse of the memory matrix remains true [21]. If the low energy degrees of freedom of the non-Fermi liquid decouple into patches in mo-mentum space, then there will be many conserved momenta. The difference with the Fermiliquid case will be that the individual patch theories will not be free. Thus the only almostconserved quantity in each patch will be the momentum (cid:126)P θ . The ratio of conductivities(15) becomes κσT = 1 T (cid:82) dθdθ (cid:48) χ Q θ P θ (cid:0) M − (cid:1) P θ P θ (cid:48) χ P θ (cid:48) Q θ (cid:48) (cid:82) dθdθ (cid:48) χ J θ P θ ( M − ) P θ P θ (cid:48) χ P θ (cid:48) J θ (cid:48) . (17)Part of the assumption of decoupled patches here is that the interpach susceptibilities vanishto leading order at low energies. This requires that the Landau interpatch interactions areirrelevant in these non-Fermi liquids or that they can be ‘diagonalized’ to give decoupledpatchwise theories. We see immediately that the patchwise susceptibilities entering theabove formula are distinct from those appearing in the Fermi liquid case (13). Transportin non-Fermi liquids is controlled by a different underlying kinematical structure.In appendix B we describe the Ising-nematic quantum phase transition in two dimen-10ional metals as an example of theory [41] with a strongly interacting patchwise descriptionat low energies. The almost conserved quantities are the momenta in each patch.The expression (17) for the ratio of conductivities is less universal than our result (15) forquasi-hydrodynamic metals. Assuming that the irrelevant interpatch scattering is controlledby one scale to leading order at low energies, we can write (cid:0) M − (cid:1) P θ P θ (cid:48) = λ F ( θ, θ (cid:48) ). Here λ is a rate of momentum relaxation whereas F ( θ, θ (cid:48) ) is a dimensionless ‘kinematic’ functionof pairs of points on the Fermi surface that does not contain another scale. Then we canwrite, at least in terms of extracting the temperature dependence of the ratio, κσT ∼ T (cid:42) χ P Q χ P J (cid:43) ∼ κσT . (18)The angled brackets here denote a schematic average over the Fermi surface in which therelaxation rate λ has cancelled out.To obtain the second relation in (18), we can first explain why the cancellation we foundabove for quasi-hydrodynamic metals in the computation of κ = κ − α T /σ does not occurhere. Recall that κ is defined as the heat conductivity at vanishing electric current. Because χ JP (cid:54) = 0 for the metallic states we are considering, the no-current boundary conditionrequires that the total momentum also vanish. For the quasi-hydrodynamic metals, thetotal momentum was the only conserved quantity. Therefore, in states with vanishing totalmomentum, the heat current can relax and heat conduction is universal. In the metals withpatchwise conserved momenta, however, the vanishing of the total current does not implythat all of the patch momenta must independently vanish. The nonvanishing patch momentawill then not allow the total heat current to relax within the momentum-conserving lowenergy effective theory. This can be seen explicitly by verifying that because of the integralsover patches on the Fermi surface in expressions like (17), the (cid:0) M − (cid:1) P θ P θ (cid:48) dependence nolonger cancels out in κ = κ − α T /σ . In addition to the absence of a cancellation, we canargue that κ ∼ κ have the same temperature scaling: In these non-Fermi liquids where onlythe patchwise momenta are conserved, only χ P J and χ P Q appear in the conductivities. Itis then easily seen that, unlike in the Fermi liquid case, α T /σ and κ contain exactly thesame susceptibilities and therefore have the same temperature dependence. Thus we finallyobtain the second relation in (18).Universal heat conduction has also been obtained theoretically in the past by dividingthe system into two sets of modes. Among the first set of modes, interactions degrade allcurrents while the momentum can be transferred to the second set of modes. The secondset of modes is then assumed to dissipate momentum very quickly. In these circumstances11 universal heat conduction can be associated to the first set of modes. See e.g. [42, 43]. In many experimental systems believed to be close to quantum critical points, the orderparameter carries a finite wavevector. For instance in metallic spin and charge density wavetransitions. In such cases, fluctuations of the order parameter are most efficient at scatteringlow energy fermions in the vicinity of hot spots or hot lines on the Fermi surface. The hotloci are connected in momentum space by an ordering wavevector while the remainingpatches on the Fermi surface are referred to as ‘cold’. The cold fermions can in turn scatteroff excitations at the hot loci [44]. In this section we discuss the case in which at least somelong-lived ‘cold’ quasiparticles survive in the regime showing non-Fermi liquid transport.The possibility of long-lived quasiparticles coupled to strongly interacting critical exci-tations also arises in systems with multiple bands. One band can become ‘hot’ while theother ‘cold’ bands retain a quasiparticle character. Evidence for this phenomenon can beseen in quantum oscillation experiments in e.g. Sr Ru O [45]. Interband scattering hasthe potential to lead to unconventional lifetimes for the stable quasiparticles.In terms of our effective field theory approach to quantum transport, in the cases wherequasiparticle and non-quasiparticle excitations coexist, one must consider separately thestrongly interacting patches at or close to the hot loci and the quasiparticle patches awayfrom the hot loci. For the long-lived cold fermions, we can apply the memory matrix methodmuch as for Fermi liquids. As we noted in appendix A, in spin and charge density wavetransitions in two space dimensions, the theory of the hot loci involves strong scattering witha nonzero momentum transfer [44]. Momentum is therefore not approximately conservedand the memory matrix method is not applicable to the hot excitations. On the other hand,this fact also suggests that once momentum is transferred from the cold patches to the hotloci via weak interpatch scattering, then it is relaxed very quickly.The interplay of hot and cold patches has immediate consequences for transport. Animportant likelihood is that any anomalously large resistivity from the hot patches will beshort circuited by the cold fermions [46]. This is not necessarily the case in all temperatureregimes [47]. We proceed to consider situations in which the cold fermions dominate thetransport, but acquire non-Fermi liquid lifetimes through scattering off hot excitations, whileremaining well-defined quasiparticles. We will see that such scenarios allow the Wiedemann-Franz law to coexist at low temperatures with non-Fermi liquid transport.12 .1 Linear in temperature resistivity and Wiedemann-Franz In section 6 below we will see that several materials exhibit non-Fermi liquid transportwhile simultaneously satisfying the Wiedemann-Franz law. In this section we show thatthe coexistence of the Wiedemann-Franz law with unconventional transport places strongconstraints on the dissipative process controlling the transport.It is well known that for temperatures above the Debye temperature but below the Fermienergy, scattering of electrons by phonons leads to a linear in temperature resistivity and alsoto the Wiedemann-Franz law [1]. There are four important facts in this temperature rangethat allow these two phenomena to coexist: (i) the rate at which the electronic quasiparticleslose momentum to the phonons is given by Γ ∼ T , (ii) the phonons lose momentum viaUmklapp scattering at a faster rate Γ U (cid:29) Γ, (iii) the electronic quasiparticles remain longlived and (iv) because the average energy of the fermions involved in the scattering is muchlarger than that of the phonons (whose energy is bounded above by the Debye energy),the electrons effectively experience elastic scattering. It is clear that points (iii) and (iv)allow the derivation of Wiedemann-Franz in section 3 to go through. Points (i) and (ii)are additionally required because they ensure that electron scattering immediately leads tomomentum dissipation, allowing currents to relax. Point (ii) is furthermore related to thefact that the electronic contribution to thermal conduction will be much bigger than thephonon contribution.We wish to know what general types of emergent collective low energy excitations couldsatisfy conditions (i) to (iv). The experimental case has been made that some linear intemperature resistivity is due to scattering of well-defined quasiparticles by critical modesthat are classicalized by being above their effective Debye temperature [48]. To substantiatethis picture a framework for very low or even vanishing effective Debye scales is necessary.We can take a general approach. Suppose we are given a critical bosonic mode O witha retarded Green’s function D R ( ω, k ). Let the scattering of the cold fermions ψ with thebosonic mode be described by the coupling S int = λ (cid:82) dtd d xψ † ψ O . To describe scatteringby hot fermions, we can imagine that the operator O is a fermion bilinear. We assume thatthis coupling can be treated perturbatively at low energies, consistent with the survival ofthe cold fermions as quasiparticles. This kind of coupling between cold fermions and hotoperators was considered in [44] for the metallic spin density wave transition. If we assumethat the hot excitations are able to dissipate momentum very efficiently, as seems to be thecase for the spin density wave transition in two dimensions [44], then the transport will be13ominated by cold quasiparticles. This transport will be controlled by the decay rateImΣ( ω, p ) = λ (cid:90) d d k (2 π ) d d Ω π Im G R ( ω − Ω , p − k ) Im D R (Ω , k ) f FD ( ω − Ω) f BE (Ω) f FD ( ω ) . (19)This formula is essentially Fermi’s golden rule allowing for decay into non-quasiparticlemodes and is easily derived using standard thermal field theory methods. Here f FD and f BE are the Fermi-Dirac and Bose-Einstein distributions, respectively. In general we shouldallow for the coupling λ to be k dependent. For the free fermionIm G R ( ω, k ) = π δ ( ω − (cid:15) ( k )) , (20)where (cid:15) ( k ) is the free fermion dispersion and vanishes on the Fermi surface. We can nowask: for what Green’s functions D R ( ω, k ) is a linear in temperature resistivity obtained?When is this scattering sufficiently elastic that the Wiedemann-Franz law is true?Elasticity will hold when the energy transfer is much less than the typical energy of thecold electrons, namely when Ω (cid:28) ω ∼ T throughout the integrand in (19). The fermionenergy ω is measured from the chemical potential. From (19), expanding the factor of f BE ≈ T /
Ω and noting the cancellation of the two factors of f FD in this regime of energies,we obtain ImΣ( ω, p ) = λ T (cid:90) d d k d Ω(2 π ) d Im D R (Ω , k )Ω δ (Ω − ω + (cid:15) ( p − k )) . (21)If the bosonic spectral weight Im D R does not have a strong temperature dependence, anddies off sufficiently quickly at large energies so that there is no contribution to the integralfrom Ω ∼ T , then we immediately obtain a linear in temperature scattering rate. Forinstance, for Debye phonons, Im D R phon. (Ω , k ) = δ (Ω − c s k ) . (22)The lattice cutoff on the momentum k < k L therefore implies that | Ω | < c s k L . It followsthat if the temperature is above the Debye temperature, c s k L (cid:28) T , we consistently findelastic scattering with a linear in temperature decay rate. This is of course a textbookresult for high temperature scattering of electrons by phonons.Given that the boson in the cases of interest is emerging from a strongly correlatedsector, we do not expect its spectral weight to take the free form (22). We will call abosonic mode with a spectral weight Im D R (Ω , k ) such that there exists an effective Debyescale Ω D , so that for T (cid:29) Ω D equation (21) holds, a ‘generalized phonon’.If classicalized phonons provide one canonical way to obtain a linear in temperaturescattering rate via a bosonic mode, the other prototypical framework is the marginal Fermi14iquid [49]. There the bosonic spectral weight is postulated to have the formIm D R MFL (Ω , k ) ∼ Ω /T Ω (cid:46) T sgn(Ω) Ω (cid:38) T , (23)over some large range of momentum. Setting Ω = T ˆΩ and ω = T ˆ ω in the formula for thelifetime (19) and using (20) we again obtain a linear in temperature relaxation rate. Thisis a completely different regime from the phonon scattering. The energy transferred to thebosonic mode saturates the temperature scale and hence the scattering is manifestly notelastic. Independently of how efficiently the boson can lose its momentum and the extentto which the boson contributes to thermal transport, the mechanism is not compatible withthe Wiedemann-Franz law. This is because inelasticity means that we have lost the simplerelation between current and heat relaxation that played a key role the derivation of thelaw in section 3.The Wiedemann-Franz law therefore is a diagnostic that can differentiate marginal Fermiliquid-like from phonon-like linear in temperature relaxation rates.As is well known, the fermion relaxation rate and the transport relaxation rate can differif small-angle scattering dominates transport. At temperatures above the effective Debyetemperature of the ‘generalized phonon’ mode, typical scatterings involve a momentumtransfer of order the lattice momentum. These large momentum scatterings relax currentsand momenta at the same rate as the fermions relax. It is therefore sufficient to considerthe quasiparticle decay rate (19) to obtain the d.c. resistivity.Several materials showing a linear in temperature resistivity at low temperatures alsoshow a specific heat with a temperature dependence of c ∼ − T log T , e.g. [50, 51]. A freephonon, with spectral weight (22), is well known to contribute a constant specific heat,contradicting the observations. However, we do not expect the critical mode to be a freeboson. The contribution of a ‘generalized phonon’ with retarded Green’s function D R ( ω, k )to the specific heat is computed in appendix C. It is found that the simplest generalizedphonons of (38) again require an (unobserved) constant contribution to the specific heatat low temperatures, as we might have anticipated from energy equipartition. This factmotivates experimental probes that are able to resolve a small constant contribution tothe specific heat at low temperatures in materials showing a linear in T resistivity withWiedemann-Franz simultaneously satisfied. This point was emphasized to us by Andy Mackenzie. Revisiting the experiments
Following the discussions above of various flavors of non-Fermi liquids with and withoutquasiparticles, it is instructive to revisit the experimental results.We primarily discuss the heavy fermions YbRh Si , CeCoIn , CeRhIn and the ruthen-ate Sr Ru O . These materials exhibit non-Fermi liquid behavior at, or in the vicinity of,a metallic critical point. In applying our considerations to these materials we are implicitlyassuming that they show well-defined Drude peaks. In the absence of magnetic fields orpressure, the heavy fermion optical conductivities have been studied in e.g. [52, 53]. Asharp Drude peak was observed over temperature ranges showing non-Fermi liquid trans-port. Our discussion ignores the direct effects of magnetic fields on transport. Magneticfields can be incorporated into the memory matrix formalism and have a strong effect onthe d.c. conductivities if they dominate over other sources of momentum relaxation [19].The ratio of electric and thermal Hall conductivities in CeCoIn was studied in [54]. L/L T1 I II III
Figure 1: Ratio of conductivities as a function of temperature in conventional metals andin certain non-Fermi liquids. Wiedemann-Franz is satisfied in regions I and III, but not inthe intermediate region II. In conventional metals the crossover from II to III occurs at theDebye temperature. The non-Fermi liquids under discussion exhibit the same structure atvery low temperatures.A plot of the ratio of the electronic contribution to the conductivities
L/L ≡ κ/σT × /π as a function of low temperature in CeCoIn , CeRhIn and Sr Ru O [5, 14, 7, 55]shows the same shape as is observed in conventional metals due to scattering off phonons andimpurities [1]. That is, there are three regimes: At the highest temperatures, Wiedemann-16ranz is obeyed due to scattering by phonons above their Debye temperature, as we recalledin the previous section. Below the Debye temperature, Wiedemann-Franz is violated due tothe onset of inelastic scattering by phonons. At the lowest temperatures, Wiedemann-Franzis again recovered due to elastic impurity scattering dominating. We sketch this situationin figure 1. For the non-Fermi liquids, however, the temperature scales at which thesetransitions occur are well below the Debye scale of the metals and therefore are not relatedto actual phonon scattering.The relevant experimental information is summarized in table 1 below. In this table wesee that the materials fall into two distinct classes.Material region I: WFvery low T region II: (cid:8)(cid:8)(cid:8)(cid:72)(cid:72)(cid:72) WFintermediate low T region III: WFhigher low T CeCoIn (a-axis, critical field) [5, 14] yes crossover to ρ ∼ T / ρ ∼ T CeRhIn (non-critical field, AFM) [55] yes ρ ∼ T ρ ∼ T Sr Ru O (non-critical field) [7, 48, 51] yes ρ ∼ T ρ ∼ T Sr Ru O (critical field) [7, 48, 51] yes ρ ∼ T ρ ∼ T YbRh Si (critical field) [56, 8, 9, 10] disputed ρ ∼ T no measurementsCeCoIn (c-axis, critical field) [14] no ρ ∼ T no measurementsTable 1: Schematic electrical resistivity in the three low temperature regimes of figure1, characterized by whether or not Wiedemann-Franz holds, for several non-Fermi liquidmaterials. In the description of the materials, ‘critical field’ means that the material hasbeen tuned to a quantum critical point by an external magnetic field. ‘No measurements’means that a higher temperature region III with Wiedemann-Franz satisfied is not foundat the temperatures currently reported. The materials can be divided into those in whichthe linear resistivity is observed simultaneously with Wiedemann-Franz holding and thosein which it is not. This suggests a division into non-Fermi liquids that do not and do,respectively, admit a quasiparticle description of transport, as we discuss in the text. CriticalSr Ru O is a confusing exception here, as we also discuss in the text. Non-critical Sr Ru O , CeRhIn and critical a-axis CeCoIn In table 1, three cases show a crossover from Wiedemann-Franz together with linear intemperature resistivity above some temperature to violation of Wiedemann-Franz togetherwith a resistivity scaling like ρ ∼ T x , with x >
1, below that temperature (Sr Ru O away from criticality and a-axis transport in CeCoIn at criticality). Thisis exactly what one expects for scattering of well-defined quasiparticles by a ‘generalizedphonon’ mode going through its effective ‘Debye temperature’. Restoration of Wiedemann-Franz at the lowest temperatures in these materials is then also to be expected because theelectronic quasiparticle has not been destroyed.The behavior of these three materials is consistent with the kinematical framework ofquantum criticality combined with long-lived particles described in section 5 above. Critical YbRh Si , c-axis CeCoIn and Sr Ru O In critical YbRh Si and Sr Ru O and c-axis transport in critical CeCoIn , in contrast, alinear in temperature resistivity is found in temperature regimes in which Wiedemann-Franzis not satisfied. This transport cannot be explained by a Debye scale collapsing to zero atthe critical point. The existing data for YbRh Si and c-axis CeCoIn in this regime iscompatible with strongly interacting transport without quasiparticles of the sort describedin section 4 above. If this is the case we can expect formulae like (18) to hold and describe theviolation of the Wiedemann-Franz law. In both these materials, over the low temperaturerange showing a linear resistivity, the Lorenz ratio has a simple temperature dependence. Itappears either almost constant or weakly linear in temperature [14, 8, 9]. This is suggestiveof the notion that these dependences could be obtained from a ratio of susceptibilities suchas (18). If this is true, we can predict that L/L will continue to be constant or linearat larger temperatures, while the linear in temperature resistivity holds, and Wiedemann-Franz will not be recovered. It would be interesting to measure κ in these materials,where any universality may be expected to be more pronounced. In fact, the temperaturescaling κ ∼ κ without strict equality (recall that κ = κ for quasiparticles) is a diagnostic forstrongly correlated transport with patchwise conserved momenta. Confirmation of κ ∼ κ would rule out a quasiparticle based marginal Fermi liquid-type explanation for the linearin temperature resistivity in these materials.The possible restoration of Wiedemann-Franz as T → Si at the critical field As this paper was completed, measurements of the Lorenz ratio in YbAgGe were announced [57]. At thecritical magnetic field, these showed
L/L depending linearly on temperature over a range of low tempera-tures, violating Wiedemann-Franz and furthermore crossing straight through the Wiedemann-Franz value.If the thermal transport is not dominated by phonons – see the caveats described in [57] – then these resultsare potentially a striking realization of strongly correlated transport as we have characterized it in section4: in the critical regime, showing a linear in temperature electrical resistivity, the Wiedemann-Franz valuefor the ratio of conductivities does not appear as a useful reference point for understanding the transport. Ru O tuned to criticality Wiedemann-Franz can be restored at higher temperatures without even the slope in the linear resistivitychanging. This is because the linear resistivity would seem to be necessarily due to differ-ent scattering mechanisms in the two regimes (i.e. with and without Wiedemann-Franzholding). The observations of thermal conductivity in Sr Ru O in [7] predated the moreprecise measurements, on purer samples, of electrical conductivity in [51, 48]. Perhaps thethermal conductivity in the critical regime should be revisited. Our discussion suggests thateither the Wiedemann-Franz law holds down to low temperatures at the critical magneticfield, or there is a change in the temperature dependence of the resistivity. Cuprates
The existing experimental discussion of Wiedemann-Franz in the cuprates is mostly con-cerned with very low temperatures in which the electrical resistivity is either constant orincreasing as the temperature is lowered, the latter case due to the proximity of localizedphases. These results do not probe the linear in temperature resistivity.The observation of the Wiedemann-Franz law in the overdoped cuprates [4] is consistentwith well-defined quasiparticles scattering elastically off impurities at zero temperature.Several explanations have been given for the reported violation of Wiedemann-Franz inunderdoped and optimally doped cuprates [11, 12, 13] at very low temperatures (suppressingsuperconductivity as necessary). We limit ourselves to the comment that, if violation ofWiedemann-Franz is due to strong interactions and no quasiparticles, then there is no reasonto prefer
L/L larger [11, 12] or smaller [13] than unity. As we have repeatedly stressedthroughout, L is not a legitimate reference point in strongly interacting circumstances thatare governed by different kinematics than those underlying the Wiedemann-Franz law.The Hall Lorenz ratio has been measured in optimally doped YBCO at temperatureswith a linear resistivity [58]. The use of Hall conductivities removes the phonon contributionto heat transport. Wiedemann-Franz does not hold, and the Lorenz ratio shows a cleanlinear dependence on temperature. This behavior is consistent with the Lorenz ratio beinggiven by a ratio of thermodynamic susceptibilities like (18) in the absence of quasiparticles.19 Discussion: The two faces of quantum criticality
In this work we have focussed on the kinematics of non-Fermi liquid transport. The startingpoint for any discussion of transport in a metal, in particular if strong interactions areinvolved, needs to be: What are the almost-conserved quantities governing the low energydynamics?The focus on almost-conserved quantities split our discussion of non-Fermi liquid trans-port into two cases. In the first case of ‘total quantum criticality’, all quasiparticles aredestroyed by strong interactions at low energies. Therefore all transport processes occur viastrongly interacting modes and the only almost conserved quantities are momenta. In thesecond case of ‘backseat quantum criticality’, only a subset of the degrees of freedom arestrongly interacting at low energies. The long lived degrees of freedom can acquire uncon-ventional lifetimes due to scattering off the critical modes, but the critical modes themselvesdo not participate directly in transport as they e.g. relax their momentum too quickly.In a ‘total quantum criticality’ scenario the Wiedemann-Franz law is completely off themap. Because the kinematics of almost-conserved quantities is not related to that of aFermi liquid, one cannot understand these cases by starting with a Fermi liquid and thenimagining adding perturbatively additional neutral heat carriers (to increase the thermalconductivity) or additional inelastic scattering (to increase thermal resistivity).In our discussion of totally quantum critical transport it was interesting to distinguishthe theoretically natural ratio κ/σT , involving the thermal conductivity at vanishing elec-tric field κ , from the usual κ/σT . For long-lived fermionic quasiparticles κ = κ (cid:29) α T /σ ,for hydrodynamic non-Fermi liquids κ (cid:28) κ , while for non-Fermi liquids with patchwise con-served momenta κ ∼ κ ∼ α T /σ . The ratio κ/σT in totally critical non-Fermi liquids wasuniversally related to a ratio of thermodynamic susceptibilities in (15) or, slightly less pow-erfully, (18). Perhaps these susceptibilities can be independently extracted experimentally.We noted in section 6 that the ratio of conductivities does show a fairly regular temperaturedependence in the relevant temperature regimes of the candidate totally quantum criticalmaterials YbRh Si , c-axis CeCoIn and cuprates.‘Backseat quantum criticality’ is compatible with the Wiedemann-Franz law if certaincircumstances hold. In particular, a linear in temperature resistivity can coexist with theWiedemann-Franz law if it is caused by scattering off a ‘generalized phonon’ mode aboveits effective ‘Debye temperature’. Crucially, in these cases, Wiedemann-Franz is expectedto hold above a certain low temperature. This behavior has been observed in CeRhIn andSr Ru O away from the critical magnetic fields and in a-axis transport in CeCoIn at the20ritical field. The linear in temperature resistivity observed in these materials would seemtherefore to be of a fundamentally different nature to that in the materials mentioned in theprevious paragraph, where the linear resistivity did not coexist with the Wiedemann-Franzlaw. The Wiedemann-Franz law is therefore an interesting diagnostic of quantum criticalphysics not just at the lowest possible temperature scales, but also higher temperatures. Acknowledgements
In writing this paper we have benefitted greatly from discussions and correspondence withDiego Hofman, Nigel Hussey, Andy Mackenzie, John McGreevy, Max Metlitski, Joe Polchin-ski, Jean-Phillipe Reid, Subir Sachdev, Todadri Senthil, Brian Swingle, Louis Taillefer,William Witczak-Krempa, Jan Zaanen and especially Steve Kivelson. SAH is partiallyfunded by a DOE Early Career Award and by a Sloan fellowship. RM is supported by aGerhard Casper Stanford Graduate Fellowship. MB is supported by the Simons Foundation.
A Strong momentum relaxation
This appendix describes circumstances in which our discussion of transport in stronglyinteracting systems – organized around the existence of almost conversed momentum ormomenta in the effective low energy theory – does not apply.Strong momentum-violating interactions can manifest themselves in two ways. Thefirst occurs when an emergent particle-hole symmetry in the low energy theory results insusceptibilities such as χ JP going to zero. Then the current can relax quickly without being‘dragged along’ by the emergent almost-conserved momentum. The second possibility isthat the effects of scattering become so strong that the momentum relaxation rate Γ → ∞ .These two scenarios are distinguished by the fact that in the first case the spectral weight ofthe Drude peak vanishes while in the second the weight is conserved but the peak becomesinfinitely broad. We briefly discuss these two cases in turn.Emergent particle-hole symmetry arises in the metallic spin density wave quantum phasetransition in two dimensions [59] and in certain theories of continuous Mott transitions[60, 61]. In the spin density wave case, umklapp scattering by finite momentum criticalmodes connecting the hot spots contributes a universal critical conductivity [44]. We shallreturn to spin density wave transistions in section 5, as transport is dominated by exci-tations away from the hot spots. For the Mott transitions, universal charge transport isobtained from the particle-hole symmetric dynamics of fluctuations about the half-filled21tate [60, 61]. Particle-hole symmetry is insufficient to obtain universal heat transport, as χ QP will typically not vanish. Therefore heat transport will be tied to the non-universalfate of momentum conservation. We will qualify this statement in section 4.2, as there arecircumstances in which κ is universal even while κ is not.A divergent momentum relaxation rate in a theory without disorder requires that anumklapp-like operator become relevant in the low energy theory. A localization transitiondriven via this mechanism has recently been realized in a holographic context [62]. B Patchwise conserved momenta: Ising-nematic example
In this appendix we consider the theory of the Ising-nematic quantum phase transition intwo dimensional metals, developed recently in [41], as an example of a strongly interactingtheory with a patchwise description. Earlier work on this model includes [63, 64, 65]. Whilethe loop expansion is ultimately not controlled in this model, it will serve to illustrate thedifferent ways in which various quantities relax. The theory is obtained by zooming inon a pair of antipodal patches of a Fermi surface and maintaining a scaling regime whereinteractions with a collective boson are strong. Because the fermions only interact efficientlywith the boson when the boson momentum is parallel to the Fermi surface, due to the Fermisurface curvature, to capture this process it is necessary to ‘thicken’ the patches. Thus,unlike in the Fermi liquid case, the patchwise theory describes propagation in two spacedimensions [41]. The effective patchwise low energy theory of [41] describes two antipodalfermions ψ s , with s = ± , interacting with a gapless boson φ L = (cid:88) s = ± ψ † s (cid:0) iη∂ t + is∂ x + ∂ y (cid:1) ψ s − (cid:88) s = ± λ s φψ † s ψ s + N e (cid:104) η (cid:48) ( ∂ t φ ) − ( ∂ y φ ) (cid:105) . (24)The spin flavor index running from 1 to N has been suppressed. In the Ising-nematic model λ + = λ − . This theory can also describe a spin liquid when λ + = − λ − . The renormalizationgroup (RG) scaling that one considers for this model is: ∂ x → b ∂ x , ∂ y → b∂ y , ∂ t → b ∂ t together with ψ → b ψ, φ → b φ . The couplings λ ± and e are dimensionless under thisscaling. We see that both of the time derivative terms we have included in the aboveLagrangian are in fact irrelevant under this scaling. Thus we should consider η, η (cid:48) →
0. Nontrivial frequency dependence compatible with the RG scaling will be generatedradiatively [41]. We have included the time derivative terms above in order to be ableto easily define tree level operators for e.g. the momentum. The fact that the classicalfrequency dependence will be swamped at low energies by terms generated in the RG flow22ells us that the momentum operators we will shortly define in fact undergo very significantvertex corrections, which we will ignore. The comments that follow should be taken at thelevel of a qualitative discussion of momentum versus current relaxation in these theories.We proceed to compute the time derivatives of currents and momenta in the patchtheory (24). Our modest goal here is to use these expressions as an explicit example of howpatchwise momenta but not currents are conserved in a strongly interacting theory. Giventhe Lagrangian (24) we can write down the patch momenta and current densities using theNoether procedure and the equations of motion (cid:126)p = (cid:88) s iη (cid:16) (cid:126) ∇ ψ † s ψ s − ψ † s (cid:126) ∇ ψ s (cid:17) + Ne η (cid:48) ∂ τ φ(cid:126) ∇ φ , (25) j x = (cid:88) s sψ † s ψ s , (26) j y = i (cid:88) s (cid:16) ∂ y ψ † s ψ s − ψ † s ∂ y ψ s (cid:17) , (27) q x = − η (cid:88) s s (cid:16) ψ † s ( is∂ x + ∂ y ) ψ s + c.c (cid:17) + 1 η (cid:88) s λ s φ ψ † s ψ s , (28) q y = − η (cid:88) s (cid:16) ∂ y ψ † s ( − ∂ x + i∂ y ) ψ + c.c (cid:17) − Ne ∂ y φ ˙ φ + 1 η (cid:88) s λ s φ (cid:16) i∂ y ψ † s ψ s + c.c (cid:17) , (29)and the Hamiltonian density is h = N e (cid:104) η (cid:48) ( ∂ τ φ ) + ( ∂ y φ ) (cid:105) + (cid:88) s (cid:18) is (cid:104) ∂ x ψ † s ψ s − ψ † s ∂ x ψ s (cid:105) + ∂ y ψ † s ∂ y ψ s + λ s φψ † s ψ s (cid:19) . (30)The total patch quantities { (cid:126)P , (cid:126)J , (cid:126)Q, H } are obtained by integrating the densities over space.This allows us to compute the time derivatives i ˙ (cid:126)P = [ P, (cid:126)H ] = 0 , (31) i ˙ J x = [ J x , H ] = 0 , (32) i ˙ J y = [ J y , H ] = − iη (cid:88) s λ s (cid:90) d x ∂ y φ ψ † s ψ s , (33) i ˙ Q x = [ Q x , H ] = iη (cid:88) s λ s (cid:90) d x ˙ φψ † s ψ s , (34)and i ˙ Q y = (cid:88) s (cid:90) d x (cid:18)(cid:20) − iλ s η (cid:48) φ + iλ s η φ (cid:21) ∂ y ( ψ † ψ ) + iλ s φη (cid:104)(cid:16) ∂ y ψ † ψ − ∂ y ψ † ∂ y ψ (cid:17) + c.c. (cid:105) + 2 ie λ s N ηη (cid:48) ˙ φ (cid:104) i∂ y ψ † ψ + c.c. (cid:105) + isλ s η φ ∂ x (cid:104) i∂ y ψ † ψ + c.c. (cid:105)(cid:19) . (35)23his exercise shows explicitly how interactions cause currents to relax while conservingmomentum. The coupling constants are order one at the fixed point. C Specific heat of generalized phonons
The contribution of a ‘generalized phonon’ with retarded Green’s function D R ( ω, k ) to thespecific heat may be computed from the following general expression for the entropy density s = (cid:90) d d k (2 π ) d d Ω π Ω8 T Im log D R (Ω , k )sinh T , (36)= (cid:90) d d k (2 π ) d (cid:90) ∞ d Ω π Ω4 T arg D R (Ω , k )sinh T . (37)This formula is obtained from standard thermal field theory manipulations, using a spectralrepresentation for log D R ( ω, k ). The specific heat is then given as usual by c = T ∂s/∂T .To obtain the second line we used the fact that D R ( − Ω , k ) = D R (Ω , k ).Now consider a generalized phonon with spectral weight satisfyingIm D R g-phon. (Ω , k ) = 0 for | Ω | > Ω (cid:63) ( k ) . (38)This is the simplest way to implement a Debye scale Ω D = max k Ω (cid:63) ( k ). We are assum-ing nothing about the distribution of spectral weight at energies below Ω (cid:63) ( k ). From theKrammers-Kronig relation we obtain the following result for the real partRe D R g-phon. (Ω , k ) < > Ω (cid:63) ( k ) . (39)This result uses only positivity of the spectral weight (for Ω > > Ω (cid:63) ( k ) the argument in (37) satisfies arg D R (Ω , k ) = π . In the regime of temperatureswhere we obtain a linear in temperature scattering rate, T (cid:29) Ω D , we can now isolate thefollowing universal contribution to the entropy density s univ. = (cid:90) d d k (2 π ) d (cid:90) T Ω (cid:63) ( k ) d ΩΩ = (cid:90) d d k (2 π ) d log T Ω (cid:63) ( k ) . (40)This gives the constant specific heat c = (cid:90) d d k (2 π ) d , (41)as we might have anticipated from energy equipartition.24 eferences [1] J. M. Ziman, Electrons and phonons , OUP, 1960.[2] K-S. Kim, and C. P´epin, “Violation of the Wiedemann-Franz Law at the Kondo Break-down Quantum Critical Point,” Phys. Rev. Let. , 156404 (2009) [arXiv:0811.0638[cond-mat.str-el]].[3] S. Kambe, H. Suderow, T. Fukuhara, J. 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