Skewed Non-Fermi Liquids and the Seebeck Effect
SSkewed Non-Fermi Liquids and the Seebeck Effect
Antoine Georges
1, 2, 3, 4 and Jernej Mravlje Coll`ege de France, 11 place Marcelin Berthelot, 75005 Paris, France Center for Computational Quantum Physics, Flatiron Institute, New York, NY 10010 USA CPHT, CNRS, Ecole Polytechnique, IP Paris, F-91128 Palaiseau, France DQMP, Universit´e de Gen`eve, 24 quai Ernest Ansermet, CH-1211 Gen`eve, Suisse Department of Theoretical Physics, Institute Joˇzef Stefan, Jamova 39, SI-1001 Ljubljana, Slovenia.
We consider non-Fermi liquids in which the inelastic scattering rate has an intrinsic particle-holeasymmetry and obeys ω/T scaling. We show that, in contrast to Fermi liquids, this asymmetryinfluences the low-temperature behaviour of the thermopower even in the presence of impurity scat-tering. Implications for the unconventional sign and temperature dependence of the thermopowerin cuprates in the strange metal (Planckian) regime are emphasized.
PACS numbers:
Besides its relevance to thermoelectricity, the Seebeckeffect provides invaluable insights into the fundamen-tal physics of materials[1–3]. The Seebeck coefficient S (thermopower) is sensitive to the balance between hole-like and electron-like excitations. It is negative when elec-trons dominate, positive when holes dominate and van-ishes when particle/hole symmetry holds. In many cases,the particle-hole asymmetry originates in the bandstruc-ture of the material: it is controlled by the number (den-sity of states) and velocities of the two types of exci-tations. However, it has been recognized that anothersource of asymmetry may also influence the Seebeck co-efficient: that of the lifetime (scattering rate) of theseexcitations. Although this has been discussed theoreti-cally [4–11] and put forward as a possible mechanism formaterials in which the sign of S is found to be oppositeto that predicted by bandstructure [4, 12, 13], it has re-ceived comparatively less attention. One of the reasonsis that, as detailed below, the particle-hole asymmetryof the inelastic scattering rate does not influence S atlow-temperature for metals obeying Fermi liquid theorywhen impurity scattering is also present.In this article, we show that the situation is entirelydifferent in metals which do not obey Fermi liquid theory.We consider a family of non-Fermi liquids in which the in-elastic scattering rate Γ in obeys ω/T scaling (with ω theenergy of an excitation counted from Fermi level). Wedemonstrate that in ‘skewed’ non-Fermi liquids where thescaling function has an odd frequency component, thisparticle-hole asymmetry affects the low- T behaviour ofthe Seebeck coefficient down to T = 0, even in the pres-ence of impurity scattering. The sign of S can be reversedin comparison to that expected from bandstructure. Thecase of a ‘Planckian’ metal [14–20] with Γ in ∝ ω, T turnsout to be particularly interesting. In that case, S/T ultimately diverges logarithmically at low temperature.However, the temperature dependence and sign of
S/T over an extended temperature range is strongly affectedby the particle-hole asymmetry of Γ in . As discussed be-low, this may be relevant to the understanding of the Seebeck coefficient of cuprate superconductors, especiallyclose to the critical doping where the pseudo-gap opensand a logarithmic dependence of the specific heat is ob-served [21].We recall that in metals, for non-interacting electronsand in the presence of elastic scattering only, the Seebeckcoefficient at low temperature T is given by: ST = − k B e π Φ (1)This expression involves the transport function Φ( ε ) = R d d k/ (2 π ) d v k δ ( ε − ε k ) at the Fermi level Φ = Φ( ε F )and its derivative with respect to energy Φ = Φ ( ε F ). v k = ∇ k ε k / ~ denotes the electron velocity in the direc-tion considered (we set ~ = k B = 1 in most of the follow-ing). In this simplest description, S/T does not dependon the magnitude of the scattering rate and its sign isdetermined by the particle-hole asymmetry of the band-structure encoded in the transport function:
S/T > <
0) and
S/T < > T, ω ) = γ + Γ in ( T, ω ) (2)It is convenient for our purpose to decompose the scat-tering time into components which are even and odd infrequency: τ ± ( T, ω ) = 12 (cid:20)
T, ω ) ± T, − ω ) (cid:21) (3)Note that we assume that the elastic scattering rate γ isisotropic and that the inelastic scattering rate only de-pends on frequency and not on momentum (i.e. does notvary along the Fermi surface). The isotropy assumptionallows us to keep the discussion simple and is sufficient toreveal the main effects that we wish to emphasize [22]. Itapplies for example to models with a local self-energy Σ -in that case Γ in ( T, ω ) = − ω + i + , T ) since vertex a r X i v : . [ c ond - m a t . s t r- e l ] F e b a~ = -1a~ = 0a~ = 1a~ = 2 π /312- π S / T T/T * FIG. 1: Fermi liquid:
S/T in units of | k B /e · Φ / Φ | vs. T /T ∗ for several values of the dimensionless particle-holeasymmetry parameters e a ≡ a Φ / Φ and taking a = b (seetext). This illustrates the crossover between the low- T elastic-dominated regime which does not depend on asymmetry andthe higher- T inelastic-dominated one in which the asymme-try contributes. A hole-like band contribution Φ / Φ < corrections do not contribute to transport properties insuch a case [23].Using the Kubo formula, one can derive [22] the fol-lowing expression for the Seebeck coefficient in the low-temperature regime [24]: S = − k B e I ( T ) I ( T ) (4)in which: I ( T ) = TZ ( T ) Φ Φ h x τ + i + h xτ − i (5) I ( T ) = h τ + i + TZ ( T ) Φ Φ h xτ − i (6)In these expressions, the frequency dependence of thescattering rates is expressed in terms of the scaling vari-able x = ω/T : τ ± = τ ± ( T, xT ) and we use the nota-tion: h F ( x ) i ≡ R + ∞−∞ dxF ( x ) / x . Z denotes theeffective mass renormalisation, which for a local the-ory is related to the real part of the self-energy by:1 /Z ( T, ω ) = 1 + [ReΣ(0) − ReΣ( ω )] /ω . Strictly speak-ing, the frequency dependence of Z has to be kept in theintegrals entering (5,6). As we show in [22], neglectingthis effect is actually a good approximation, and we useit here to simplify the discussion. In a local Fermi liq-uid, Z ( T ) coincides with the quasiparticle spectral weightand reaches a finite value at T = 0, while in a non-Fermiliquid Z ( T ) may vanish as T → Z = 1) and for elastic scattering only( τ + = 1 /γ, τ − = 0) we recover from (4,5,6) the simpleexpression (2) ( h i = 1 , h x i = π / Fermi liquid.
We first consider a (local) Fermi liquidwith an inelastic scattering rate:Γ in ( T, ω ) = λ (cid:2) ω + ( πT ) + aω + bωT (cid:3) + · · · (7) The key point is that the odd part of the scattering ratescales as ω , ωT ∼ xT and hence is subdominant ascompared to the even-frequency part: the inelastic scat-tering rate of conventional Fermi liquids is asymptotical-lly particle-hole symmetric at low energy. Adding theelastic scattering rate, we see that there are two regimes.In the ‘elastic regime’ γ dominates over the inelasticscattering rate: this holds for γ & λ ( πT ) or alterna-tively for T . T ∗ with T ∗ ∼ ( γ/π λ ) / a crossovertemperature. In the ‘inelastic’ regime ( γ . λ ( πT ) or T & T ∗ ), inelastic scattering dominates [24]. Let us con-sider first the ‘inelastic’ regime, in which τ + is of order1 /T , and τ − of order 1 /T . Hence, in the denominator I of (4), h τ + i ∼ /T dominates over T h xτ − i ∼ const . .In contrast, in the numerator I , the odd term h xτ − i hasthe same 1 /T temperature dependence as the even one T h x τ + i . Taking into account that, in a Fermi liquid,the effective mass enhancement 1 /Z = m ∗ /m reaches aconstant at low temperature, we obtain in the inelasticlimit T (cid:29) T ∗ : ST (cid:12)(cid:12)(cid:12)(cid:12) F L in ’ − k B e (cid:20) (12 − π ) 1 Z Φ Φ − π ( c a a + c b b ) (cid:21) (8)with c a = h x / (1 + x /π ) i ’ . c b = h x / (1 + x /π ) i ’ .
09. Remarkably, the odd-frequency terms of Γ in directly contribute to S in thislimit, on equal footing with the bandstructure term. Thiswas, to our knowledge, first emphasized in Ref. [5] inwhich expression (8) was derived, and further discussedin Refs [6–11]. We also note that the prefactor of thebandstructure term is modified (from π / ’ .
29 to12 − π ’ .
13) as compared to the elastic (low- T ) limit.However, in the low- T ‘elastic’ limit ( γ (cid:29) λπ T or T (cid:28) T ∗ ), this interesting effect disappears. Indeed, τ + ∼ /γ is constant in this limit, while τ − vanishes as T ,leading to: ST (cid:12)(cid:12)(cid:12)(cid:12) F L el ’ − k B e π Z Φ Φ (9)Hence, odd-frequency scattering does not contribute atlow temperature. The conventional elastic value (2) isrecovered, with the notable difference that the prefactoris enhanced by the effective mass 1 /Z = m ∗ /m . Indeed,it was emphasized in Ref. [26] that S/T is proportionalto the linear term in the specific heat ( ∼ m ∗ /m ) in manymaterials. The crossover between the low- T elastic limitand the high- T inelastic limit, described by (4), is illus-trated on Fig. 1. ‘Skewed’ Non-Fermi liquids. In the Fermi liquid case,the odd-frequency terms in the inelastic scattering rateare subdominant in comparison to the even ones, andhence do not contribute to S at low- T . We consider herea class of non-Fermi liquids in which, in contrast, theodd-frequency terms are of the same order as the even-frequency ones, such that the inelastic scattering rate -0.15-0.1-0.0500.050.1 0 0.5 1 1.5 2 2.5 3 3.5 4 S T/T * α =0.0 α =-0.1 α =-0.2 η =0.05, α =-0.3 η =0.0, α =-0.3 FIG. 2: ‘Skewed’ non-Fermi liquid with ν = 1 /
2. Seebeckcoefficient vs.
T /T ∗ in units of k B /e , for πT ∗ /γ = 0 . α . At low- T , thesign of S is seen to depend on α , while at high- T all curveshave the same sign, set by the bandstructure η = γ Φ / Φ > η = 0 in which the sign of S is determined solelyby the scattering rate asymmetry (see text). obeys a scaling form:Γ in ( T, ω ) = λ ( πT ) ν g (cid:16) ωT (cid:17) (10)Here, ν is an exponent (we focus on ν ≤ g ( x ) contains both aneven and an odd component. It has a regular expan-sion at small x , so that for ω . T : Γ in ∼ λ ( πT ) ν g (0) + g (0) T ν − ω + · · · , while g ( x ) ∼ c ∞± · | x | ν at large x . Theeffects discussed in this article do not depend on the spe-cific form of the scaling function g ( x ). Systems obeyingthis scaling form with a non-even scaling function can becalled ‘skewed non-Fermi liquids’.The scaling form (10) has been shown to apply inmicroscopic models such as overscreened Kondo mod-els [27]. It is also relevant to Sachdev-Ye-Kitaev (SYK)models [28, 29], see Refs. [30–37]. For such models obey-ing conformal invariance at low-energy, g ( x ) is a universalscaling function that depends only on the exponent ν andon a ‘spectral asymmetry’ parameter α . Its exact formwas derived in Ref. [27] and reads (with the normalisation g α =0 (0) = 1): g ( x ) = (cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:20) ν i x + α π (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) cosh( x/ α/ ν ) / (11)The ω/T scaling form is also relevant to the proximityof a quantum critical point associated with a strong-coupling fixed point [38].The crossover temperature T ∗ separating the ‘elas-tic’ and ‘inelastic’ limits now reads ( πT ∗ ) ν = γ/λ .In the low- T elastic limit, τ + ∼ /γ and τ − ∼ − λγ − ( πT ) ν g − ( x ) ∼ − γ − ( T /T ∗ ) ν g − ( x ). In this ex-pression, g − ( x ) = [ g ( x ) − g ( − x )] / g . For ν <
1, thequasiparticle weight vanishes at low- T as ∼ T − ν , and itcan be shown [22] that T /Z ( T ) ∼ λ ( πT ) ν /c Z + T with c Z a universal constant depending only on the scaling func-tion g . Hence, we see that, remarkably, the two termsin the numerator I of (4) have the same T -dependence: T h x τ + i /Z ∼ T ν and h xτ − i ∼ T ν . One therefore ob-tains for T (cid:28) T ∗ : S (cid:12)(cid:12) NF L el = − k B e λγ ( πT ) ν (cid:20) π c Z γ Φ Φ − c − (cid:21) = − k B e (cid:18) TT ∗ (cid:19) ν (cid:20) π c Z η − c − (cid:21) (12)in which c − = h xg − ( x ) i is again a universal constantdepending only on g [22]. The dimensionless parameter η ≡ γ Φ / Φ is the ratio of the elastic scattering rate tothe characteristic energy scale associated with the band-structure asymmetry. Hence, in this case, S behaves as T ν at low- T , corresponding to a divergent slope S/T .Expression (12) has several remarkable features.Firstly, we see that the odd-frequency inelastic scatteringcontributes to the low- T Seebeck on equal footing withthe even-frequency/elastic contribution. Both terms in(12) have the same T -dependence ∼ T ν but for differ-ent reasons: the first one because of the vanishing of Z ( T ), and the second one because of the T -dependenceof Γ in . Secondly, in contrast to the former, this odd-frequency contribution is completely independent of theband-structure asymmetry: its sign is dictated by that ofthe constant c − , and thus by the intrinsic asymmetry ofthe inelastic rate scaling function [22]. If this term dom-inates, the overall sign of the Seebeck can be opposite tothat predicted by band-structure considerations in thelow- T limit, even when elastic scattering is present. Thisis one of the main results of this work. The odd-frequencycontribution dominates over the first term when the di-mensionless ratio η = γ Φ / Φ is small, i.e. for clean-enough systems. Thirdly, we note that S depends onboth the inelastic constant λ and the elastic rate γ . Thisis an unusual situation in which the strength of the scat-tering does not drop out of the value of S at low- T .In order to discuss the T -dependence of S , we define θ = T /T ∗ and rewrite Eq. (4) as: S ( T ) = − k B e (cid:20) η (cid:18) c Z θ ν + T ∗ γ θ (cid:19) F ( θ ) F ( θ ) + F ( θ ) F ( θ ) (cid:21) F n ( θ ) ≡ h x n g ( x ) θ ν i , (13)where we have omitted the term involving τ − in I whichis negligible in practice [22]. At high- T , we obtain S ∼− k B e h η c c Tγ + η c c c Z (cid:0) TT ∗ (cid:1) ν + c c i with c n = h x n /g i [22].Hence, as long as η = γ Φ / Φ = 0, the sign of S at hightemperature is fixed by the bandstructure term, in con-trast to the low- T limit. Remarkably, in the absence ofelastic scattering or band asymmetry ( η = 0), S insteadtends to a constant value − c /c k B /e which dependsonly on the universal scaling function and its asymme-try. A related finding was reported in Ref. [33] (see also[36]) in the context of SYK models, where S was shownto be constant and determined by the spectral asymme-try α (related by holography to the electric field/chargeof the black hole). On Fig. 2, we display the tempera-ture dependence of S vs. T /T ∗ in the non-Fermi liquidcase for several values of the asymmetry parameter α inEq. (11). We see that at low- T the sign of S can bechanged by the scattering rate asymmetry, while it is setby the bandstructure term only at high- T . Skewed Planckian metal.
We finally discuss the case ofa ‘Planckian’ metal [14–20] ( ν = 1) with a particle-holeasymmetry scaling with ω/T , so that Γ in = πλg (0) T + πλg (0) ω + · · · at low ω and T (‘skewed’ Planckianmetal). Naively setting ν = 1 in expression (12) wouldlead to a finite S/T at low- T . However, the quasiparti-cle weight actually vanishes logarithmically: 1 /Z ( T ) ∼ πλc − Z ln Λ /T , with Λ a high-energy cutoff and c Z a universal constant depending on the scaling function g [22] (note that λ is dimensionless in the Planckian case).Hence, in the low- T ‘elastic’ limit T (cid:28) T ∗ = γ/πλ : ST (cid:12)(cid:12)(cid:12)(cid:12) SP M el ∼ − k B e (cid:20) πλ π c Z Φ Φ ln Λ T + π Φ − c − πλγ + · · · (cid:21) (14)Thus, in a Planckian metal, S/T ultimately diverges log-arithmically at very low- T . This corresponds to the log-arithmic divergence of the effective mass (specific heatcoefficient). The sign of the logarithmic term in S isdictated by bandstructure. In contrast, the last termin (14) is controlled by the odd-frequency part of theinelastic scattering, and has a sign which can counter-act the conventional bandstructure effect correspondingto the second term in (14). Interestingly, we note thatthe ‘skewed’ term dominates over the conventional onefor cleaner systems η/λ . c − /π . The full temperaturedependence is again given by (13), setting ν = 1 andreplacing c Z by c Z / ln Λ /T .To illustrate these effects, we display on Fig 3 the T -dependence of S/T for a ‘skewed’ Planckian metal.Two opposite signs of the bandstructure term Φ / Φ areconsidered. In the absence of a scattering rate asym-metry ( α = 0) it is seen that S/T has a rather weak T -dependence (except at low- T where the logarithmicterm becomes relevant), while it acquires significant T -dependence for α = 0. Importantly, in the presence ofan asymmetry, the sign of the Seebeck coefficient can bereversed in comparison to its bandstructure value over awide range of temperature. Discussion.
A number of materials display non-Fermiliquid behaviour and an unconventional T -dependence -0.500.51 0 0.5 1 1.5 2 2.5 3 3.5 4 S / T T/T * η =0.02, α =0.0 η =-0.02, α =0.0 η =0.02, α =-0.4 η =-0.02, α =-0.4 00.40.80.001 0.1 1001234 0 1 2 3 4 ρ / ρ FIG. 3: Planckian metal. Temperature dependence of
S/T (in units of k B / ( eγ )) for two opposite values of the band-structure term η = γ Φ / Φ : electron-like η > η <
0. Plain (resp. dahed) lines are for a particle-hole asym-metric (resp. symmetric) scattering rate ( α = 0, resp. α = 0). πT ∗ /γ = 1. Top inset: S/T on a logarithmic scale, emphasiz-ing the low- T behaviour. Bottom inset: Linear dependenceof the resistivity ρ/ρ ( T = 0) = I (0) /I ( T ) on T /T ∗ . of the Seebeck coefficient. This has been addressed inprevious theoretical work, such as Refs [39] and [40]which have emphasized the logarithmic divergence of S/T in metals with a T -linear scattering rate. The T -dependence of S close to a Fermi surface Lifshitz tran-sition has been considered in [41, 42]. However, to ourknowledge, the key role of a particle-hole asymmetry ofthe inelastic scattering rate in non-Fermi liquid metalswith ω/T scaling has not been discussed before. Thetheory presented here may be relevant when both thetemperature dependence and the sign of the Seebeck co-efficient are found to be unconventional.Cuprate superconductors in the ‘strange metal’ regimedisplay clear signatures of quantum criticality: T-linearresistivity (for reviews see e.g.[20, 43, 44]), a logarith-mic divergence of the specific heat coefficient C/T [21],and ω/T scaling observed in optics [45] and angular-resolved photoemission (ARPES) [46] spectroscopies. Anincrease of the in-plane
S/T at low- T reminiscent ofFig. 3 has been reported for La . − x Eu . Sr x CuO (Eu-LSCO) [47] at hole doping p ’ .
21 and p ’ .
24 and forLa . − x Nd . Sr x CuO (Nd-LSCO) [48, 49] at p ’ . p ∗ at which the pseudo-gap phase terminates. Interestingly, S was found to bepositive at those doping levels, while simple considera-tions based on band-structure and isotropic elastic scat-tering yield a negative value [50]. It is thus tempting toinfer from these observations that the quantum critical(strange metal) regime of those cuprate superconductorsmay be described as a ‘skewed Planckian metal’. We em-phasize that experiments involving different controlledlevels of disorder would play a decisive role in assess-ing the relevance of the mechanism proposed here, sincethe asymmetric term in (14) becomes larger for cleanersystems. In Eu-LSCO, Nd-LSCO and also Bi-2201 [51],the increase of S/T at low- T observed experimentally ap-pears consistent with a logarithmic dependence. Giventhe sign of the bandstructure term for Nd-LSCO, onewould expect from (14) a negative coefficient of this log-arithmic term, in contrast to the experimental observa-tion. This may suggest that the asymmetry term is actu-ally dominant (as on Fig. 3) in the range of temperatureof the measurement, in which the increase of C/T is mod-erate [21].In the overdoped regime, the Seebeck coefficient of theLSCO family remains positive [52, 53]. These overdopedcompounds behave as Fermi liquids at low enough tem-perature however [54–56]. Hence, in view of the discus-sion above (Fig. 1), a particle-hole asymmetry of the in-elastic scattering rate is unlikely to be responsible forthis unexpected sign of S . A more realistic explanationmay be that the elastic scattering rate has unconven-tional dependence on momentum and energy [43, 57, 58].We also note that the Seebeck coefficient of other single-layer cuprates such as Hg1201 [59] and Bi2201 [60, 61] hasbeen reported to be negative in the overdoped regime.On a more theoretical level, our results are relevantin the context of the non-Fermi liquid quantum criti-cal point separating a metallic spin-glass phase and aFermi liquid metal in doped SYK models, which has at-tracted a lot of attention recently [34, 35, 37, 62–64].Indeed, numerical results show strong spectral asymme-try in the quantum critical regime, possibly signallinga skewed non-Fermi liquid at the critical point [37]. Wealso note that in the context of transport in SYK models,the Seebeck coefficient has been emphasized as a probeof the universal ground-state entropy [33, 36].Our work may also have relevance to twisted bilayergraphene and related systems, in which linear resistiv-ity and Planckian behaviour have been observed, seee.g. [65]. A systematic investigation of thermoeletric ef-fects and heat transport in these materials would be ofgreat interest. Note added.
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Shin, Physical Review B - CondensedMatter and Materials Physics , 024533 (2005).[62] D. G. Joshi, C. Li, G. Tarnopolsky, A. Georges, andS. Sachdev, Phys. Rev. X , 021033 (2020).[63] P. Cha, N. Wentzell, O. Parcollet, A. Georges,and E.-A. Kim, Proceedings of the NationalAcademy of Sciences [65] J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, andP. Jarillo-Herrero, Flavour hund’s coupling, correlatedchern gaps, and diffusivity in moir´e flat bands (2020),arXiv:2008.12296 [cond-mat.mes-hall] .[66] H. Jin, A. Narduzzo, M. Nohara, H. Takagi, N. Hussey,and K. Behnia, Positive Seebeck coefficient in highlydoped La − x Sr x CuO ( x =0.33); its origin and implica-tion (2021), arXiv:arXiv:2101.10750 [cond-mat.supr-con]. kewed Non-Fermi liquids and the Seebeck EffectSupplemental Material Antoine Georges ∗ Coll`ege de France, 11 place Marcelin Berthelot, 75005 Paris, FranceCenter for Computational Quantum Physics, Flatiron Institute, New York, NY 10010 USACPHT, CNRS, Ecole Polytechnique, IP Paris, F-91128 Palaiseau, France andDQMP, Universit´e de Gen`eve, 24 quai Ernest Ansermet, CH-1211 Gen`eve, Suisse
Jernej Mravlje † Department of Theoretical Physics, Institute Joˇzef Stefan, Jamova 39, SI-1001 Ljubljana, Slovenia.
I. KUBO FORMULA: THERMOPOWER AND CONDUCTIVITY
The conductivity σ , thermopower S and (electronic) heat conductivity κ are given by: σ = e L , S = − L eL , κ = T (cid:20) L − L L (cid:21) (1)in which the L s are Onsager’s coefficient and e is the absolute magnitude of the electron charge. In the following, weset for simplicity ~ = k B = e = 1 (except when restored in final results). Within the Kubo formalism, and neglectingvertex corrections, the Onsager coefficients are given by (denoting for simplicity L ≡ L , L ≡ L , L ≡ L : L n = 1 T n Z dω (cid:18) − ∂f∂ω (cid:19) ω n T ( ω ) (2)in which f ( ω ) = 1 / [1 + e ω/T ] is the Fermi function and T ( ω ) = 2 π Z d d k (2 π ) d v k A ( k , ω ) (3)In this expression, v k = ∇ k ε k / ~ denotes the band velocity in the direction being considered, and A ( k , ω ) is theelectronic spectral function, related to the self-energy Σ = Σ( ω + i + , k ) by: A ( k , ω ) = 1 π Γ / ω + µ − ε k − ReΣ) + (Γ / (4)In this expression, Σ = Σ( ω + i + , k ) is the self-energy due to inelastic interactions between electrons and Γ is thefull scattering rate including both elastic and inelastic terms:Γ( k , ω ) = γ k − k , ω + i + ) (5)These expressions can be further simplified when the elastic scattering rate γ k and electron-electron self-energy Σ donot depend on momentum, in which case neglecting the vertex corrections also becomes exact. We define the bandtransport function (density of states weighted by velocities) as:Φ( ε ) = 2 Z d d k (2 π ) d v k δ ( ε − ε k ) (6)so that: T ( ω ) = π Z dε Φ( ε ) A ( ε, ω ) (7) ∗ ageorges@flatironinstitute.org † [email protected] a r X i v : . [ c ond - m a t . s t r- e l ] F e b We change the integration variable by setting ε = ω + µ − ReΣ + Γ2 y , so that: T ( ω ) = π Z + ∞−∞ dy Φ (cid:18) ω + µ − ReΣ + Γ2 y (cid:19) (cid:18) /πy + 1 (cid:19) (8)We now perform an expansion of this expression for small Γ and retain only the most singular term, yielding: T ( ω ) = 1Γ Φ ( ω + µ − ReΣ) + · · · (9)where we have used R + ∞−∞ dy /π ( y + 1) = 1 / (2 π ). It is convenient to introduce the notations: ε F ( T ) ≡ µ − ReΣ( ω = 0 , T ) , − Z ( T, ω ) ≡ ω [ReΣ( ω, T ) − ReΣ(0 , T )] (10)so that, to dominant order in Γ: T ( ω ) = 1Γ( T, ω ) Φ (cid:20) ε F ( T ) + ωZ ( T, ω ) (cid:21) + · · · (11)Inserting this expression in the Onsager coefficients L n above, and changing variable to x = ω/T in the integral overfrequency, we obtain, with τ = 1 / Γ: L n = Z dx x n x Φ (cid:20) ε F ( T ) + x TZ ( T, xT ) (cid:21) τ ( T, xT ) (12)We now perform a low- T expansion of this expression. We note that for both the Fermi liquid and non-Fermi liquidcases considered in this article, T /Z ( T, xT ) vanishes at low- T (as ∼ T and ∼ T ν , respectively). We assume furthermorethat the T -dependence of ε F ( T ) is subdominant and can be neglected. We thus obtain: L n = Φ( ε F ) h x n τ ( T, xT ) i + T Φ ( ε F ) h x n +1 τ ( T, xT ) Z ( T, xT ) i + · · · (13)with the notation: h F ( x ) i = Z + ∞−∞ dx F ( x )4 cosh x (14)We have checked furthermore that neglecting the frequency ( x ) dependence of Z in these equations is a good approx-imation, so that we finally obtain the expressions used in the main text, with the notation Φ = Φ( ε F ) , Φ = Φ ( ε F ): S = − k B e I ( T ) I ( T ) (15)in which: I ( T ) = TZ ( T ) Φ Φ h x τ + i + h xτ − i (16) I ( T ) = h τ + i + TZ ( T ) Φ Φ h xτ − i (17)in which τ ± ( T, ω ) = [ τ ( T, ω ) ± τ ( T, − ω )] / τ . We also note the expression ofthe conductivity: σ = e Φ I ( T ) ’ e Φ h τ + i (18)We note that, although the non-Fermi liquid case does not have conventional quasiparticles, the final expressions for S , σ and the Onsager coefficient L n ’s at low- T are formally identical to the ones obtained by applying a Boltzmannequation formalism to excitations with a lifetime τ ex = τ /Z and dispersing as ε ex k = Zε k , corresponding to a transportfunction of these excitations Φ ex ( ε ) = Z Φ( ε/Z ). -0.2-0.15-0.1-0.0500.050.1 0 0.5 1 1.5 2 2.5 3 γ imp =1 S T/T * Boltzmann with Re Σ Kubo α =0.0 α =-0.1 α =-0.2 η =0.05, α =-0.3 η =0.0, α =-0.3 FIG. 1. Seebeck coefficient in units k B /e for the ν = 1 / πT ∗ /γ = 1. The results of the Boltzmann calculation with Z = Z ( T ) (full lines, except η = 0 . , α = − . Z = Z ( ω, T ) (dashed). The results of the latter two cannotbe distinguished on the scale of this plot. Similar agreement between the Boltzmann and Kubo results is found for all the datain the main text. On Fig. 1, we compare the Seebeck coefficient evaluated with the full Kubo formula and with the simplifiedBoltzmann-like expressions. The data is for the subplanckian ν = 1 / Z ( ω, T ) by its zero-frequency value Z ( ω, T ) → Z (0 , T ) (full lines). The result of such Boltzmann calculations are also the data given inthe main text. Most of the discrepancy between Kubo and Boltzmann can be remedied if one retains the frequencydependence of Z ( ω, T ) (dashed lines that overlap with the dotted ones).As far as the momentum dependence is concerned, which is relevant for instance in cuprates, we verified that forthe momentum dependence recently reported in an ADMR study [1] our considerations still apply and go throughlargely unchanged, which will be discussed in a separate publication. II. SKEWED NON-FERMI LIQUIDS: ω/T
SCALINGA. Scaling function and conformally invariant case
In the main text, we consider (local) non-Fermi liquids with a scattering rate obeying ω/T scaling ( ν ≤ in ( T, ω ) = λ ( πT ) ν g (cid:16) ωT (cid:17) (19)The scaling function g ( x ) has a regular expansion at small x : g ( x ) = g (0) + x g (0) + · · · , while at large x it obeys g ( x → + ∞ ) ∼ c ∞ + x ν and g ( x → −∞ ) ∼ c ∞− | x | ν . The ‘skewed’ case with a particle-hole asymmetry will in generalhave g (0) = 0 and c ∞ + = c ∞− . Hence:Γ in ( T (cid:29) ω ) ∼ λg (0) ( πT ) ν + λg (0) πω ( πT ) − ν + · · · (20)Γ in ( T (cid:28) | ω | ) ∼ c ∞± λ | πω | ν + · · · (21)The physical results discussed in our article do not depend on the specific form of the scaling function g ( x ) providedit obeys these general properties.We emphasize in the main text the following family of scaling functions: g α,ν ( x ) = (cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:20) ν i x + α π (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) cosh( x/ α/ ν ) / (22) - -
10 10 20123456 - -
10 10 2051015
FIG. 2. Scaling function g α,ν ( x ) for ν = 1 / ν = 1 (right), for different values of the asymmetry parameter α = 0 , . , , . It has been shown in [2], in the context of overscreened Kondo impurity models, that this specific form of the scalingfunction holds in models which have conformal invariance at low energy/temperature. It is also relevant in the contextof SYK models [3–7]. The parameter α controls the particle-hole asymmetry (see Fig. 2), with α = 0 correspondingto particle-hole symmetry g ( x ) = g ( − x ) and g − α,ν ( x ) = g α,ν ( − x ). The normalisation g α =0 ,ν (0) = 1 was chosen inthe above expression. In the Planckian case ν = 1, this can be cast in the explicit form: g α,ν =1 ( x ) = ( x + α ) / x + α ) /
2] cosh( x/ α/ , Γ in ( T, ω ) = λπT ( ω/T + α ) / ω/T + α ) /
2] cosh( ω/ T )cosh( α/
2) (23)The connection between this scaling form and conformal invariance in local models is the following. This invarianceimplies that any imaginary time ( τ ) fermionic correlation function (for example the self-energy) takes the followingform in the limit where τ and inverse temperature β = 1 /T are both large compared to microscopic scales, but witharbitrary τ /β [2]: Σ( τ ) ∝ e α ( τ/β − / (cid:18) π/β sin πτ /β (cid:19) ν (24)and this function has the following spectral representation ( τ = τ /β ): e α ( τ − / (cid:16) π sin πτ (cid:17) ν = − C Z + ∞−∞ dx e − xτ e − x g α,ν ( x ) (25)with C = cosh( α/ π ) ν Γ[(1 + ν ) / / ( π Γ[1 + ν ]).As mentioned in the main text, our results do not depend on the specific form of the scaling function g . In theabsence of a particle-hole asymmetry, the following phenomenological scaling function has sometimes considered inthe literature (see e.g [8]): g α =0 ( x ) = (cid:2) x /π (cid:3) ν/ (26)Note that ν = 2 reduces to the Fermi liquid scaling function 1 + x /π - this is also the case of (22) for ν = 2 , α = 0.Starting with this form, one can introduce an asymmetry for example by deforming the scaling function in an analogousway to (22) [2]: g α ( x ) = g ( x + α ) cosh( x/ x + α ) /
2] cosh( α/
2) (27)A simple calculation using the spectral representation of Σ then shows that:Σ α ( τ ) = e α ( τ/β − / cosh α/ α =0 ( τ ) (28) B. Scaling form of Z ( T, ω ) Here, we discuss the scaling form of the real part of the self-energy. We start from the spectral representation:ReΣ( ω ) = P Z d(cid:15) σ ( (cid:15) ) ω − (cid:15) , σ ( ω ) = − π ImΣ( ω + i + ) = 12 π Γ in ( T, ω ) (29)in which ’P’ denotes the principal part of the integral and we have recalled the relation (factor of two) between theimaginary part of the self-energy and the inelastic scattering rate for a local theory. The frequency and temperaturedependent effective mass enhancement 1 /Z ( T, ω ) is given by:1 − Z ( T, ω ) ≡ ω [ReΣ( ω ) − ReΣ(0)] = P Z d(cid:15) σ ( (cid:15) ) (cid:15) ( ω − (cid:15) ) (30)Because we have subtracted ReΣ(0), we can substitute the scaling form of σ = Γ in / ν <
1, and thus obtain: TZ ( T, ω = xT ) = T − λ π ( πT ) ν P Z dy g ( y ) y ( x − y ) (31)Hence, remarkably, T /Z obeys a universal scaling form which depends only on the scaling function g . This is incontrast to a Fermi liquid in which Z depends on all energy scales (as signalled by the fact that inserting the low-energy expression σ ( ω ) ∝ ω + ( πT ) in (29) would lead to a divergent integral).In the main text, we ignore the ω (i.e. x ) dependence of Z and replace Z ( T, ω ) by Z ( T,
0) = Z ( T ). Comparisonwith the full Kubo formula calculations (see above) justify this approximation for transport calculations. The x → TZ ( T, ω = 0) = T + λ π ( πT ) ν Z dy g ( y ) y (32)so that, with the notation introduced in the text: TZ ( T ) = T + λ ( πT ) ν c Z , c Z = 12 π Z dy g ( y ) y (33)These expressions are valid for ν <
1. The Planckian case ν = 1 requires a slightly different analysis. In that case,the scaling function cannot be substituted in (29) since the integral would diverge logarithmically at large frequency.Hence, a cutoff Λ must be kept. We focus here on the zero-frequency limit Z ( T ) which reads, after an integration bypart: 1 − Z ( T ) = − P Z d(cid:15) σ ( (cid:15) ) (cid:15) (34)Substituting the scaling form σ ( (cid:15) ) = Γ in / π = λ/ g ( (cid:15)/T ), we obtain:1 − /Z ( T ) = − λ P Z +Λ /T − Λ /T dx g ( x ) x (35)At large x , g tends to a constant in the Planckian case g ( x ∼ ±∞ ) → g ( ±∞ ), so that we obtain:1 Z ( T ) = 1 + λ g (+ ∞ ) − g ( −∞ )] ln Λ T (36)Hence, using the notation in the main text: 2 π/c Z = [ g (+ ∞ ) − g ( −∞ )]. In the Planckian case, the high-energycutoff does not entirely disappear from the expression of Z , but we note that the prefactor of the ln T term does notdepend on the cutoff. III. TEMPERATURE DEPENDENCE OF THE SEEBECK COEFFICIENT
Setting θ = T /T ∗ and η = γ Φ / Φ , expressions (16,17) above can be written as: I ( θ ) = F ( θ ) + η (cid:20) T ∗ γ θ + 1 c Z θ ν (cid:21) F ( θ ) (37) I ( θ ) = F ( θ ) + η (cid:20) T ∗ γ θ + 1 c Z θ ν (cid:21) F ( θ ) (38)with: F n ( θ ) = h x n θ ν g ( x ) i (39)The low-temperature expansion ( θ →
0) of the functions F n reads: F ( θ ) = 1 − h g i θ ν + h g i θ ν + · · · F ( θ ) = − h xg i θ ν + h xg i θ ν + · · · F ( θ ) = π − h x g i θ ν + h x g i θ ν + · · · (40)The key difference being that F ∼ c − θ ν ( c − = h xg i ) while F , ∼ const. . As a result, θ ν F and F have the sametemperature dependence in this regime and hence both contribute to I and to the thermopower. In contrast, in I ,the term θ ν F ∼ θ ν can be neglected in comparison to F in both the conductivity and the Seebeck coefficient. F is however much smaller in magnitude than θ ν F , as shown on Fig. 4. Hence, the odd-frequency contribution F in I is comparable to the even one ηθ ν F only when the parameter η = γ Φ / Φ is small (of order 10 − ). Thisis typically the case since the elastic scattering rate is usually much smaller than electronic energy scales. Hence, atlow- T the Seebeck coefficient reads for ν < S ∼ − k B e (cid:18) TT ∗ (cid:19) ν (cid:20) π c Z η − c − (cid:21) (41)The dependence of the coefficient c − on the asymmetry parameter α for the choice (22) of scaling functions is displayedon Fig. 3.In the high-temperature limit ( θ → ∞ ) , all the functions F n have the same temperature dependence ∼ c n /θ ν : F n ( θ ) = 1 θ ν h x n g i − θ ν h x n g i + · · · (42)Hence formally, at very high T , S tends to a constant = − c /c k B /e in the presence of odd frequency scattering.However, the term involving F in I remains small for most temperatures of interest as shown on Fig. 4. Hence, inpractice, the relevant high- T behaviour of S is: S ∼ − k B e (cid:20) c c Φ Φ T + η c c c Z (cid:18) TT ∗ (cid:19) ν + c c (cid:21) (43) [1] G. Grissonnanche, Y. Fang, A. Legros, S. Verret, F. Lalibert´e, C. Collignon, J. Zhou, D. Graf, P. Goddard, L. Taillefer, andB. J. Ramshaw, arXiv e-prints , arXiv:2011.13054 (2020), arXiv:2011.13054 [cond-mat.str-el].[2] O. Parcollet, A. Georges, G. Kotliar, and A. Sengupta, Phys. Rev. B , 3794 (1998).[3] A. Georges, O. Parcollet, and S. Sachdev, Phys. Rev. B , 134406 (2001).[4] O. Parcollet and A. Georges, Phys. Rev. B , 5341 (1999).[5] S. Sachdev, Phys. Rev. X , 041025 (2015).[6] R. A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen, and S. Sachdev, Phys. Rev. B , 155131 (2017).[7] P. Dumitrescu, N. Wentzell, A. Georges, and O. Parcollet, unpublished, in preparation.[8] T. J. Reber, X. Zhou, N. C. Plumb, S. Parham, J. A. Waugh, Y. Cao, Z. Sun, H. Li, Q. Wang, J. S. Wen, Z. J. Xu, G. Gu,Y. Yoshida, H. Eisaki, G. B. Arnold, and D. S. Dessau, Nature Communications , 5737 (2019). - - - - FIG. 3. Coefficient c − vs. α for ν = 0 . ν = 1 (red). FIG. 4. θ ν F ( θ ) /F ( θ ) vs. θ (Red) and F ( θ ) /θ ν F ( θ ) vs. θ (Blue) for ν = 0 . α = 0 ..