Classical behavior of strongly correlated Fermi systems near a quantum critical point. Transport properties
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Classical behavior of strongly correlated Fermi systems near a quantum critical point.Transport properties
V. A. Khodel,
1, 2
J. W. Clark, and M. V. Zverev
1, 3 Russian Research Centre Kurchatov Institute, Moscow, 123182, Russia McDonnell Center for the Space Sciences & Department of Physics,Washington University, St. Louis, MO 63130, USA Moscow Institute of Physics and Technology, Moscow, 123098, Russia (Dated: August 15, 2018)The low-temperature kinetics of the strongly correlated electron liquid inhabiting a solid is an-alyzed. It is demonstrated that a softly damped branch of transverse zero sound emerges whenseveral bands cross the Fermi surface simultaneously near a quantum critical point at which thedensity of states diverges. Suppression of the damping of this branch occurs due to a mechanismanalogous to that affecting the phonon mode in solids at room temperature, giving rise to a classicalregime of transport at extremely low temperatures in the strongly correlated Fermi system.
PACS numbers: 71.10.Hf, 71.10.Ay 67.30.E- 67.30.hr
After a decade of comprehensive studies, the preva-lence of non-Fermi-liquid (NFL) behavior in strongly cor-related Fermi systems is no longer a revelation. How-ever, various features of NFL phenomena still await sat-isfactory explanation, especially the puzzling observa-tions pointing to characteristic classical behavior in aquantum-critical regime. For example, at extremely lowtemperatures around 1 mK, experimental measurementsof the specific heat of two-dimensional (2D) He, as real-ized in dense He films, are described by the classicalformula C ( T ) = β + γT , where β and γ are constants.Also in contrast to FL theory, the low-temperature resis-tivity ̺ ( T ) of many high- T c compounds in strong mag-netic fields, as well as certain heavy-fermion metals, isobserved to vary linearly with T . Hence these systemsbehave as if a major contribution to the collision termcomes from the electron-phonon interaction, in spite ofthe fact that the phonon Debye temperature T D exceedsmeasurement temperatures by a factor 10 − .Previously, we have attributed the presence of theclassical term β in the specific heat C ( T ) of 2D liquid Heto softening of the transverse zero-sound mode (TZSM),occurring in the region of the quantum critical point (QCP) where the density of states N (0), proportional tothe effective mass M ∗ , diverges. Here we shall addressthe impact of the TZSM on transport properties in theQCP regime. The TZSM exists only in those correlatedFermi systems where the effective mass M ∗ exceeds thebare mass M by a factor more than 3, a requirementthat is always met on approaching the QCP. In conven-tional Fermi liquids, the Fermi surface consists of a singlesheet, so the TZSM has a single branch with velocity c t exceeding the Fermi velocity v F . Consequently, there is aban against emission and absorption of sound quanta byelectrons, and the role of the TZSM in kinetics is of littleinterest. However, in heavy-fermion metals, it is usual forseveral bands to cross the Fermi surface simultaneously,thereby generating several zero-sound branches. For allbranches but one the sound velocities are less than thelargest Fermi velocity. Hence the aforementioned ban is lifted, and these branches of the TZSM spectrum expe-rience damping, in a situation similar to that for zero-spin sound. In the latter instance, Landau damping isso strong that the mode cannot propagate through theliquid. It will be seen, however, that this is not thecase for damping of the TZSM, because of the soften-ing of this mode close to the QCP. Due to the softeningeffect, the contribution of the damped TZSM to the col-lision integral has the same form as the electron-phononinteraction at room temperature. On the other hand, wewill also find that in heavy-fermion metals, analogouslyto the case of liquid He films, softening of the TZSMacts to lower a characteristic temperature Ω t that playsthe role of a Debye temperature, setting the stage for theexistence of a classical transport regime at extremely lowtemperatures.In the canonical view of quantum phase transitions,the QCP has been identified with an end point of theline T N ( ρ ) of a second-order phase transition associatedwith violation of some Pomeranchuk stability condition.In turn this violation is associated with divergence ofthe energy derivative ∂ Σ( p, ε ) /∂ε of the self-energy andconsequent vanishing of the quasiparticle weight z =(1 − ∂ Σ( p, ε ) /∂ε ) − in single-particle states at the Fermisurface, thus triggering divergence of the effectivemass M ∗ defined by M/M ∗ = z (1+ (cid:0) ∂ Σ( p , ε ) /∂ǫ p (cid:1) | p = p F .A number of key experimental studies performed re-cently fail to support the canonical view of the QCP.In 2D liquid He, experiment has not identified anyphase transition that can be so associated with the pointof the divergence of N (0). It has been acknowledged that a similar situation (cf. Ref. 13,14) also prevails forthe QCPs of heavy-fermion metals. In essence, the pointwhere the density of states diverges is separated by anintervening NFL phase from points where lines of somesecond-order phase transition terminate. Furthermore,these transitions possess unusual properties such as hid-den order parameters; therefore within the standard col-lective scenario they can hardly qualify as triggers of theobserved rearrangements.We are therefore compelled to interchange the horseand the cart, relative to the canonical scenario. Follow-ing Refs. 15,16, we attribute the QCP to vanishing of theFermi velocity v F at a critical density ρ ∞ , which occursif 1 + (cid:0) ∂ Σ( p , ε ) /∂ǫ p (cid:1) | p = p F = 0. Accordingly, in this sce-nario for the QCP, it is the momentum-dependent part ofthe mass operator that plays the decisive role.The authors of most theory articles devoted to thephysics of the QCP claim that switching on the interac-tions between particles never produces a significant mo-mentum dependence in the effective interaction function f , and hence that the option we propose and develop is ir-relevant. This assertion cannot withstand scrutiny. Thenatural measure of the strength of momentum-dependentforces in the medium is provided by the dimensionlessfirst harmonic F = f p F M ∗ /π of the interaction func-tion f ( p , p ) of Landau theory. In a system such as3D liquid He where the correlations are of moderatestrength, the result F ≥ .
25 for this measure extractedfrom specific-heat data is already rather large. The dataon 2D liquid He are yet more damaging to the claim ofminimal momentum dependence, since the effective massis found to diverge in dense films.
In the case of QCPphenomena occurring in strongly correlated systems of ionic crystals, it should be borne in mind that the elec-tron effective mass is greatly enhanced due to electron-phonon interactions that subserve polaron effects.
The change in sign of v F at the QCP results not onlyin a divergent density of states, but also in a rearrange-ment of the Landau state beyond the QCP. As a rule,however, such a rearrangement already occurs before thesystem attains a QCP. This may be understood fromsimple arguments based on the Taylor expansion of thegroup velocity v ( p ) = ∂ǫ ( p ) /∂p , which has the form v ( p ) = v F ( ρ ) + v ( ρ ) p − p F p F + 12 v ( p − p F ) p F (1)in the vicinity of the QCP. We assert that the last coef-ficient v is positive, to ensure that the spectrum ǫ ( p ) = v F ( ρ )( p − p F )+ 12 p F v ( ρ )( p − p F ) + 16 p F v ( p − p F ) (2)derived from Eq. (1) exhibits proper behavior at largeseparations from the Fermi surface.By its definition, the QCP is situated at a density ρ ∞ where v F ( ρ ) vanishes. The QCP must in fact correspondto an extremum of the function v ( p, ρ ∞ ), which vanishesfor the first time at p = p F . Thus, the simultaneous van-ishing of the coefficient v ( ρ ∞ ) = ( dv ( p, ρ ∞ ) /dp ) | p = p F ofthe second term in the Taylor series is crucial to the oc-currence of the QCP. Generally, v does not meet thisadditional requirement. However, in relevant cases its fi-nite value remains extremely small, making it possible totune the QCP by imposing an external magnetic field.When v = 0 in Eq. (2), this equation unavoidablyacquires two additional real roots at a critical density ρ t where v F ( ρ t ) = 3 v / (8 v ), namely p , − p F = − p F v v ± s − v F ( ρ ) v v ! . (3)Clearly, this transition, identified as a topological phasetransition (TPT), takes place already on the disor-dered side of the QCP regime where v F ( ρ ) is still posi-tive. Accordingly, a new hole pocket opens and the Fermisurface gains two additional sheets, the new T = 0 quasi-particle momentum distribution being given by n ( p ) = 1for p < p and p < p < p F , and zero elsewhere. Theemergence of new small pockets of the Fermi surface is anintegral feature of the QCP phenomenon, irrespective ofwhether the strongly correlated Fermi system is 2D liq-uid He, a high- T c superconductor, or a heavy-fermionmetal.At the TPT point ρ t , the density of states, given by N ( T ) = Z ∂n ( p , T ) ∂ǫ ( p ) dυ (4)with dυ denoting an element of momentum space, is alsodivergent. Inserting the spectrum (2), straightforwardcalculation yields N ( T → ∝ T − / , in contrast to thebehavior N ( T → , ρ ∞ ) ∝ T − / obtained in the casewhere v ( ρ ∞ ) = 0.Having tracked the initial evolution of the topology ofthe Fermi surface in the QCP region, our analysis turnsnext to the salient features of the TZSM spectrum in sys-tems having a multi-connected (i.e., multi-sheet) Fermisurface. We first examine how the TZSM softens in 3Dsystems with a singly-connected Fermi surface, where thedispersion relation has the well-known form F (cid:20) − s − (cid:18) s s + 1 s − − (cid:19)(cid:21) (5)with s = c/v F and F = f p F M ∗ /π . The TZSM is seento propagate only if F >
6, i.e., M ∗ > M . Near theQCP where M ∗ ( ρ ) → ∞ , one has v F /c →
0, and Eq. (5)simplifies to 1 = F v F c , (6)which implies c ( ρ → ρ ∞ ) → r p F v F ( ρ ) M ∝ p F M s MM ∗ ( ρ ) → , (7)an analogous formula being obtained for a 2D system.To facilitate analysis of damping of the TZSM in sys-tems having a multi-connected Fermi surface, we restrictconsideration to the case of two electron bands. The TPTis assumed to occur at one of the bands, so that its Fermivelocity, denoted again by v F , tends to zero, while theFermi velocity v o of the other band remains unchangedthrough the critical density region. The model disper-sion relation for the complex sound velocity c = c R + ic I becomes1 = F (cid:20) − (cid:18) c v F − (cid:19) (cid:18) c v F ln c + v F c − v F − (cid:19)(cid:21) ++ F v F v o (cid:20) − (cid:18) c v o − (cid:19) (cid:18) c v o ln c + v o c − v o − (cid:19)(cid:21) . (8)It can easily be verified that the contribution of thesecond term to the real part of the right-hand side ofEq. (8) is small compared to that of the first term, since v F /v o → c R + ic I + v o ) / ( c R + ic I − v o )] ≃ − iπ , thecorresponding contribution iπF v F c R / (4 v o ) to the imag-inary part of the right-hand side cannot be ignored, else c I = 0. By this reasoning, Eq. (8) assumes the simplifiedform 1 = F v F ( c R + ic I ) − i π v o F v F c R (9)analogous to Eq. (6). Its solution obeys c R ∝ s MM ∗ ( ρ ) , c I ∝ MM ∗ ( ρ ) . (10)Importantly, we see then that the ratio c I /c R ∝ p M/M ∗ ( ρ ) is suppressed in the QCP regime, which al-lows us to analyze the contribution of the TZSM to thecollision term entering the resistivity along the same linesas in the familiar case of the electron-phonon interaction.By contrast, the group velocities of the damped branchesof longitudinal zero sound are found to be insensitive tovariation of the effective mass in the QCP region. Nosuch quenching by a small parameter
M/M ∗ arises, sothese modes cannot propagate in the Fermi liquid.It should now be clear that toward the QCP, the ef-fective Debye temperature Ω t = ω ( k max ) = c R k max goesdown to zero , independently of the value of the wave num-ber k max characterizing the cutoff of the TZSM spectrum.Thus, the necessary condition Ω t < T for emergence ofa regime of classical behavior is always met. However,another condition must also be satisfied if there is to ex-ist a well-pronounced classical domain at extremely lowtemperature. Consider that the boson contribution F B = T Z ln (cid:16) − e − ck/T (cid:17) θ ( k − k max ) dυ (11)to the free energy is proportional to some power of k max ,depending on the dimensionality of the problem. Thesame is true of the corresponding contributions to ki-netic phenomena. Therefore the extra condition neededis that k max must not to be too small. We identify k max with the new characteristic momentum arising beyondthe point of the TPT, namely the distance d = p − p between the new sheets of the Fermi surface. Indeed, formomenta p situated at distances from the Fermi surfacesignificantly in excess of d , the single-particle spectrum ǫ ( p ) is no longer flat. The collective spectrum ω ( k ) deter-mined from the corresponding Landau kinetic equation isno longer soft, and consequently the imaginary part c I of c becomes of the same order as c R , barring propagationof the TZSM.This scenario is illustrated for the key property of re-sistivity in electron systems of solids. The kernel of theelectron-TZSM collision integral underlying the resistiv-ity ̺ ( T ) then contains terms n o ( p + k )(1 − n o ( p )) N ( k ) − n o ( p )(1 − n o ( p + k ))(1 + N ( k )) and n o ( p + k )(1 − n o ( p ))(1 + N ( k )) − n o ( p )(1 − n o ( p + k )) N ( k ), in which N ( k ) denotes a nonequilibrium TZSM momentum distri-bution and n o ( p ), a nonequilibrium electron momentumdistribution of the band that can absorb and emit theTZSM. The explicit linearized electron-phonon-like formof the corresponding component of the collision integralis I e,ph ∝ Z w ( p , k ) ω ( k ) ∂N ( ω ) ∂ω ( δn i − δn f )) δ ( ǫ i − ǫ f )) dυ. (12)In this expression, w is the collision probability, N ( ω ) =[exp( ω ( k ) /T ) − − is the equilibrium TZSM momentumdistribution, and δn i,f stands for the deviation of thereal momentum distribution of the electron band labeled o from its nonequilibrium counterpart, with n i = n ( p )and n f = n ( p + k ). In the classical case of the electron-phonon interaction, at T D < T one has ∂N ( ω ) /∂ω ∝− T /ω while all the other factors are T -independent,resulting in linear variation of the resistivity ̺ ( T )with T . Based on the analogy we have established be-tween the roles of phonons and the TZSM, the resistivityof the strongly correlated electron system must obey theFL law ̺ ( T ) ∝ T only at T < Ω t . In the opposite caseΩ t < T , the resistivity exhibits a linear dependence on T .Imposition of a magnetic field cannot kill the soft modeof transverse zero sound as long as the flattening of thesingle-particle spectrum responsible for strong depressionof the effective Debye temperature Ω t persists.These results and conclusions are in agreement withexperimental data on the low-temperature resistivityof the doped heavy-fermion metal YbRh (Si . Ge . ) ,data which indicates that the linear-in- T dependence ofthe resistivity is robust down to temperatures as low as20 mK. Remarkably, the linearity of ̺ ( T ) continues tohold in external magnetic fields up to B ≃ T , far inexcess of the critical value B c ≃ . T at which this com-pound undergoes some phase transition with a hidden or-der parameter. A linear T dependence of ̺ ( T ) is presentas well in the Hertz-Millis spin-density-wave (SDW) sce-nario for the QCP in 2D electron systems. However,critical spin fluctuations die out at
B > B c , since theSDW transition is suppressed. Thus the observed behav-ior of ̺ ( T ) contradicts the SDW scenario. Our scenariois in fact compatible with this behavior, since the TZSMspectrum is less sensitive than the structure of criticalspin fluctuations to the magnitude of the magnetic field.A linear T dependence of ̺ ( T ) can also emerge if lightcarriers are scattered by heavy bipolarons. However,there is no evidence for the presence of these quasiparti-cles in heavy-fermion metals.Let us now turn to the analysis of the soft TZSM con-tribution to the thermopower. One may recall that in theclassical situation, the phonon-drag thermopower S d ( T )associated with nonequilibrium phonons is known to account for a substantial part of the Seebeck coefficient S ( T ). The same is true for the class of quantum-criticalsystems considered here, except that the domain in whichthe drag term S d ( T ) contributes appreciably extendsdown to extremely low temperatures. Significantly, at T → T , whereas at T > Ω t it falls off as T − , producing a bell-like shape of S d ( T ) with a sharp maximum at T ≃ Ω t . Since the re-maining contributions to S are rather smooth, this salientfeature of S d appears to be responsible for the changeof sign of the full Seebeck coefficient S ( T ) at extremelylow T observed experimentally in the heavy-fermionmetal YbRh Si , as well as the nonregular behavior of S ( T →
0) found in several heavy-fermion compounds. The TZSM scenario proposed here predicts that the See-beck coefficient in YbRh (Si . Ge . ) will exhibit thesame anomalous behavior at magnetic fields substantiallyexceeding a corresponding critical value B c .In summary, we have investigated the conditions pro-moting the formation of soft damped collective modesthat play the same role in kinetic phenomena as phonons.We have shown that such a damped soft branch belongingto the transverse zero-sound mode emerges when severalbands cross the Fermi surface simultaneously, with one ofthe bands subject to a divergence of the effective mass ofcarriers. We have discussed prerequisites for lowering thecorresponding Debye temperature Ω t and demonstratedthat the inequality Ω t < T is met near the quantum crit-ical point, where a classical regime sets in at extremelylow temperatures.We thank A. Alexandrov and V. Shaginyan for stim-ulating discussions. This research was supported by theMcDonnell Center for the Space Sciences, by GrantsNo. 2.1.1/4540 and NS-7235.2010.2 from the RussianMinistry of Education and Science, and by Grants No. 09-02-01284 and 09-02-00056 from the Russian Foundationfor Basic Research. H. v. L¨ohneysen, A. Rosch, M. Vojta, P. W¨olfle, Rev. Mod.Phys. , 1015 (2007). P. Gegenwart, Q. Si, F. Steglich, Nature Phys. , 186(2008). A. Casey, H. Patel, J. Nyeki, B. P. Cowan, J. Saunders,Phys. Rev. Lett. , 115301 (2003). M. Neumann, J. Nyeki, B. P. Cowan, J. Saunders, Science , 1356 (2007). P. Gegenwart, J. Custers, T. Tayama, K. Tenya, C. Geibel,O. Trovarelli, F. Steglich, K. Neumaier, Acta Phys. Pol.
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