Topological disorder triggered by interaction-induced flattening of electron spectra in solids
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Topological disorder triggered by interaction-induced flattening of electron spectra insolids
V. A. Khodel,
1, 2
J. W. Clark,
2, 3 and M. V. Zverev
1, 4 National Research Centre Kurchatov Institute, Moscow, 123182, Russia McDonnell Center for the Space Sciences & Department of Physics,Washington University, St. Louis, MO 63130, USA University of Madeira, 9020-105 Funchal, Madeira, Portugal Moscow Institute of Physics and Technology, Dolgoprudny, Moscow District 141700, Russia
We address the intervention of classical-like behavior, well documented in experimental studies ofstrongly correlated electron systems of solids that emerges at temperatures T far below the Debyetemperature T D . We attribute this unexpected phenomenon to spontaneous rearrangement of theconventional Landau state beyond a critical point at which the topological stability of this statebreaks down, leading to the formation of an interaction-induced flat band adjacent to the nominalFermi surface. We demonstrate that beyond the critical point, the quasiparticle picture of suchcorrelated Fermi systems still holds, since the damping of single-particle excitations remains smallcompared with the Fermi energy T F = p F / m e . A Pitaevskii-style equation for determination ofthe rearranged quasiparticle momentum distribution n ∗ ( p ) is derived, which applies to explanationof the linear-in- T behavior of the resistivity ρ ( T ) found experimentally. Currently,“topological” has become one of the mostcommonly used terms in condensed-matter physics, sur-passing “quantum critical point.” It is sufficient to men-tion such collocations as topological order, topologicaltransition and topological insulator. On the other hand.over decades the mathematical literature has featured,along with more traditional types of chaotic behavior, rel-evant discussions of topological entropy and topologicalchaos, which exhibit positive entropy S [1–4]. (See alsothe Supplemental Material (SM) [5] and sources [6–11]cited therein)). In the present work addressing stronglycorrelated electron systems of solids including cupratesand graphene, we investigate possible existence of a fi-nite entropy S > T much lower thanthe Debye value T D identifying the boundary betweenclassical and quantum regimes.Seemingly, this option would be obviated by the Nernsttheorem requiring S ( T ) to vanish upon reaching T =0. However, recent developments warrant a revisionof this conventional stance. The first symptoms ap-peared in measurements [12, 13] of the thermal expan-sion coefficient α ( T ) = − V − ∂V /∂T = V − ∂S/∂P of the strongly correlated heavy-fermion superconductorCeCoIn , which has a tiny critical value T c = 2 . α (0) = 0, it is neverthelessof paramount significance that at extremely low tempera-tures T > T + c = T c + 0 where the system is already in thenormal state, experiment has established the perplexingbehavior α ( T ) = α + α T. (1)The nonzero offset α ≃ . × − / K exceeds valuesfound in ordinary metals at these temperatures by a hugefactor of order 10 − . This implies that an analogous classical-like offset S , associated with α by the relation α = ∂S /∂P , is present in the entropy itself – pointingunambiguously to the presence of disorder in the regimeof extremely low T > T + c ≪ T D .Another experimental challenge is associated with thelow-temperature, non-Fermi-liquid (NFL) behavior ofthe normal-state resistivity ρ ( T ) of the same CeCoIn metal at various pressures P , which, according to FLtheory, should obey the formula ρ ( T ) = ρ + A T . In-stead, at P < P ∗ ≃ classical-like strange-metal behavior ρ ( T ) = ρ + A T, (2)shown in Fig. 1. It is as if classical physics already pre-vails at T + c < T ≪ T D . This remarkable linear-in- T behavior of ρ ( T ) is currently observed in diverse systems(see e.g. [15–17]). In some cases, the slope A experiencesa noticeable jump [18] (see below).Even more bizarre behavior has surfaced in recentstudies [19] of the resistivity of twisted bilayer graphene(TBLG) as a function of twist angle θ , as depicted inFig. 2. Profound variations of A ( θ ) are seen, especiallytoward to the so-called magic angle θ m , where the A term increases by more than three orders of magnitude,as does the residual resistivity ρ ( θ ), echoing a tenfoldvariation of ρ as a function of pressure P , as shown inFig. 1. Since ρ must be a parameter-independent quan-tity [20] if the impurity population remains unchanged,its documented behavior defies explanation within thestandard FL approach.Moreover, in high-temperature superconducting, over-doped copper oxides, where T c ( x ) terminates at criticaldoping value x c with nearly linear dependence on x c − x (see Fig. 3), the quite remarkable doping independence A ( x ) /T c ( x ) = const , (3) ! " AT n A [ $ % c m K - n ] P [GPa] & [ m J / m o l C e K ] b [ $ % c m ] CeCoIn n a FIG. 1: Upper panel: Values of the residual resistivity ρ (left axis, open squares) and the index n in the fit ρ ( T ) = ρ + AT n (right axis, solid squares) versus pressure P . Bot-tom panel: Temperature coefficient of resistivity A (left panel,open squares) and specific-heat coefficient γ (right panel, solidsquares and solid triangles). The authors thank J. D. Thomp-son for permission to present data published in Ref. [14]. has been discovered [21, 22], a feature shared with theBechgaard salts [23]. As emphasized in Ref. [22], this fea-ture points to the presence of a hidden phase, emergentat x c simultaneously with the superconducting state.Explanation of the strange-metal behavior (2) ob-served ubiquitously at low T has become one of the mostintensely debated theoretical problems of the moderncondensed-matter theory. Analysis of proposed scenariosin a recent review article [24] has concluded that noneof these is capable of explanation of all the relevant ex-perimental findings. In particular, candidates based ona quantum-critical-point (QCP) scenario fall short. Aswitnessed by the phase diagrams of CeCoIn , cuprates,and graphene, there are no appropriate ordered phasesadjacent to the strange-metal region; the effects of asso-ciated quantum fluctuations are small.In this situation, we turn to a different scenario, basedon the formation of a fermion condensate (FC) [25–33].Analogy with a boson condensate (BC) is evident in therespective densities of states ρ FC ( ε ) = n FC δ ( ε ) [25] and ρ BC ( ε ) = n BC δ ( ε ), where n FC and n BC are the FC andBC densities. To be more specific, the essence of thephenomenon of fermion condensation lies in a swelling ofthe Fermi surface , i.e. in emergence of an interaction-induced flat portion ǫ ( p ) = 0 of the single-particle spec-trum ǫ ( p ) that occupies a region p ∈ Ω where thereal quasiparticle momentum distribution (hereafter de-noted n ∗ ( p )) departs drastically from the Landau step n L ( p ) = θ ( − ǫ ( p )).The trigger for such a profound rearrangement of the T[K] !" $ % !!&!’! !!&!’! !!!&!’! !!&!’! !!&!’! ! FIG. 2: NFL resistivity ρ ( T ) measured in TBLG devices atdifferent twist angles. The authors express their gratitude toA. F. Young for permission to present data published as Fig.3i of Ref. [12] and providing the corresponding file. T c [ K ] A ! " c m / K ] x c -x FIG. 3: Dependence of the factor A in the resistivity ρ ( T )(red circles, right axis) and the critical temperature T c (bluesquares, left axis) of overdoped La − x Sr x CuO films on thedoping x measured from its critical value x c = 0 .
26 [21]. Redand blue lines show the best linear fits to the data, which sup-port the conclusion that A ( x ) ∝ T c ( x ), indicative of behaviorinconsistent with conventional theory. Landau state lies in violation of its necessary stabilitycondition (NSC), which requires positivity of the change δE = P p ǫ ( p ) δn L ( p ) of the ground state energy E un-der any variation of the n L ( p ) compatible with the Pauliprinciple [28]. In Landau theory with ǫ ( p ) = v F ( p − p F ),this NSC is known to hold as long as the Fermi veloc-ity v F remains positive. Beyond a critical point where itbreaks down, the Fermi surface becomes multi-connected.This aspect is a typical topological signature. Accord-ingly, the word topological in the term topological chaoshas a twofold meaning, such that the associated bifur-cation point can be called a topological critical point(TCP). Frequently, as in a neck-distortion problem ad-dressed by I. M. Lifshitz in his seminal article [34], thecorresponding topological rearrangement of the Fermisurface is unique. However, this is not the case in deal-ing with the TBLG problem, where nearly-flat-band so-lutions are found, a distinctive feature of those being re-lated to the passage of the Fermi velocity through zeroat the first magic twist angle θ (1) m [35–38]. A variety ofoptions for violation of the topological stability of theTBLG Landau state then arise. In contrast, within theFC scenario, introduction of e − e interactions leads tothe advent of interaction-induced flat bands which re-place the nearly-flat bands found in Refs. [35–38]. Tech-nically, this procedure is reminicent of the Maxwell con-struction in statistical physics, where the isotherm in theVan der Waals pressure-volume phase diagram is in re-ality replaced by a horizontal line. An analogous situa-tion is inherent in cuprates and other strongly correlatedelectron systems of solids. Importantly, in the familiartemperature-doping phase diagram, it is the TCP thatseparates the well-understood FL behavior from the be-havior associated with topological chaos, which is respon-sible for the strange-metal regime.We begin analysis with the reminder that in su-perconducting alloys that obey Abrikosov-Gor’kov the-ory [39, 40], the damping γ acquires a finite value due toimpurity-induced scattering, implying failure of the basicpostulate γ/ǫ ( p ) < G q ofthe single-particle Green function G = ( ǫ − ǫ p − Σ) − hasthe form G q ( p , ε ) = 1 − n L ( p ) ε − ǫ ( p ) + iγ + n L ( p ) ε − ǫ ( p ) − iγ , (4)is still applicable [41].Beyond the TCP where interaction-induced flat bandsemerge, further alteration of the pole part G q occurs, itsform becoming [25–29]: G q ( p , ε ) = 1 − n ∗ ( p ) ε − ǫ ( p ) + iγ ( ε ) + n ∗ ( p ) ε − ǫ ( p ) − iγ ( ε ) , (5)with γ > < n ∗ ( p ) < n L ( p ), which re-sides solely in the Ω region, is to be determined throughsolution of a nonlinear integral Landau-Pitaevskii styleequation (cf. Refs. [42–44]) of the theory of fermion con-densation, viz. ∂ǫ ( p ) ∂ p = ∂ǫ ( p ) ∂ p + 2 Z f ( p , p ) ∂n ∗ ( p ) ∂ p d p (2 π ) . (6)Here f ( p , p ) is the spin-independent part of the Landauinteraction function. The free term includes all contribu-tions to the group velocity that remain in the f = 0limit. A salient feature of the T = 0 FC solutions is the identical vanishing of the dispersion of the spectrum ǫ ( p )in the Ω region. At T >
0, the FC spectrum acquires asmall dispersion, linear in T [27], ǫ ( p , T ) = T ln 1 − n ∗ ( p ) n ∗ ( p ) , p ∈ Ω . (7)Experimental verification of this effect through ARPESmeasurements is crucial for substantiation of the FC con-cept under consideration.Eq. (6) is derived from the formal relation δ Σ = (cid:0) U δG (cid:1) (with δG ( p, ε ) = G ( p − e A , ε ) − G ( p , ε )) of variationalmany-body theory for the self-energy in terms of the sub-set of Feynman diagrams U of the two-particle scatter-ing amplitude that are irreducible in the particle-holechannel, hence regular near the Fermi surface. Assuminggauge invariance of the theory, one finds [44, 45] − ∂G − ( p , ε ) ∂ p = p m e − (cid:18) U ( p , ε ; k , ω ) ∂G ( k , ω ) ∂ k (cid:19) . (8)The round brackets in this equation imply integrationand summation over intermediate momenta and spinswith a proper normalization factor. Implementation ofa slightly refined universal quantitative procedure [23,24]for renormalization of this equation allows it, irrespectiveof correlations, to be recast in closed form, as if one weredealing with a gas of interacting quasiparticles . (Theword “gas” is appropriate, since Eq. (6) contains onlythe single phenomenological amplitude f of quasiparticle-pair collisions). A salient feature of this procedure is thatEq. (6) holds both in conventional Fermi liquids and inelectron systems of solids moving in the periodic exter-nal field of the crystal lattice. This follows because solely gauge invariance was assumed in its derivation, whichtherefore holds for crystal structures as well. In short,the widespread impression that the FL approach is inap-plicable to crystal structures is groundless. We emphasizeonce more that the FL renormalization procedure worksproperly irrespective of the magnitude of the ratio γ/ǫ ( p )(see the SM [5] for specifics, and especially references[39, 41–43, 46]).We are now in position to consider the connection be-tween the customary iterative procedure for solving thebasic FC equation (6) and the topological chaos problemaddressed in many mathematical articles (see especially[1, 4]). In the standard iterative scheme, the j th iter-ation n ( j ) ( p ), with j = 0 , , , ... , is inserted into theright side of Eq. (6) to generate the next iteration of thesingle-particle spectrum, ǫ ( j +1) ( p ), and this process isrepeated indefinitely to finally yield a convergent resultwhose topological entropy (TE) is equal to 0. However,beyond the TCP, such a procedure fails, since the it-erations n ( j ) ( p ) then undergo chaotic jumps from 0 to1 and vice-versa, generating noise, identified with someTE. To evaluate the spectrum quantitatively, in Ref. [31],the iterative discrete-time map was reconstructed in sucha way that the discrete time t j replaces the iterationnumber j . Subsequent time-averaging of relevant quan-tities, adapted from formulas of classical theory, allowsone to find a specific self-consistent solution. Its promi-nent feature is the development of an interaction-inducedflat portion in the single-particle spectrum ǫ ( p ) that em-braces the nominal Fermi surface (for exemplification, seethe SM [5]). Another distinctive signature of the set ofspecific FC solutions of Eq. (6) lies in the occurrence of anonzero entropy excess S ∗ , emergent upon their substitu-tion into the familiar combinatoric formula for evaluationof the entropy. This yields [25, 31, 32, 47] S ∗ = − X p [ n ∗ ( p ) ln n ∗ ( p ) + (1 − n ∗ ( p )) ln(1 − n ∗ ( p ))] , (9)where summation is running over the FC region, and,in turn, a NFL nonzero value α ∗ of the coefficient ofthermal expansion. Because the presence of a nonzero S ∗ would contradict the Nernst theorem S ( T = 0) = 0 if itsurvived to T = 0, the FC must inevitably disappear [25,32, 47] at some very low T . One well-elaborated scenariofor this metamorphosis is associated with the occurrenceof phase transitions, such as the BCS superconductingtransition emergent in the case of attraction forces inthe Cooper channel, or an antiferromagnetic transition,typically replacing the superconducting phase in externalmagnetic fields H exceeding the critical field H c .At H < H c , a nonzero BCS gap ∆(0) in the single-particle spectrum E ( p ) = p ǫ ( p ) + ∆ does provide fornullification of S ( T = 0). This scenario applies in sys-tems that host a FC as well, opening a specific route tohigh- T c superconductivity [25, 48, 49]. Indeed, considerthe BCS equation for determining T c : D ( p ) = − Z V ( p , p ) tanh ǫ ( p ,T c )2 T c ǫ ( p , T c ) D ( p ) dv . (10)Here D ( p ) = ∆( p , T → T c ) / √ T c − T plays the role ofan eigenfunction of this linear integral equation, while V ( p , p ) is the block of Feynman diagrams for the two-particle scattering amplitude that are irreducible in theCooper channel. Upon insertion of Eq. (7) into this equa-tion and straightforward momentum integration over theFC region, one arrives at a non-BCS linear relation T c ( x ) = c ( x ) T F , (11)where c ( x ) = λ η ( x ), with λ denoting the effective pairingconstant and η ( x ) the FC density. This behavior is inaccord with the experimental Uemura plot [38, 50].As discussed above, the entropy excess S ∗ ∝ η comesinto play at temperatures T + c < T ≪ T D so as to invokea T -independent term α in the coefficient of thermalexpansion, which, in that regime, serves as a signatureof fermion condensation [32, 47]. Accordingly, execution of extensive low- T measurements of the thermal expan-sion coefficients in candidate materials would, in princi-ple, provide means (i) to distinguish between flat bandsthat do not entail excess entropy S ∗ and the interaction-induced exemplars, and (ii) to create a database of sys-tems that exhibit pronounced NFL properties, in aid ofsearches for new exotic superconductors.Very recently, the FC scenario has gained tentativesupport from ARPES measurements performed in mono-layer graphene intercalated by Gd, which have revealedthe presence of a flat portion in the single-particle spec-trum [51]. However, verification of the correspondencebetween the flat bands detected in the bilayer systemTBLG [19, 35, 36, 38, 52–54] and the interaction-drivenvariety considered here requires a concerted analysis ofkinetic properties, especially of comprehensive experi-mental data on the low T resistivity ρ ( T ) = ρ + A T + A T .Numerous theoretical studies of the NFL behavior of ρ ( T ) based on the FC concept have been performed. Di-recting the reader to Refs. [47, 55, 56] for details, wesummarize their pertinent results in the relation ρ ( x, P, θ ) = ρ i + a η ( x, P, θ ) , A ( x, P, θ ) = a η ( x, P, θ ) , (12)where ρ i is the impurity-induced part of ρ and a , a arefactors independent of input parameters. This expressionproperly explains the data shown in Figs. 1 and 2. In-deed, we see that in systems having a FC, the residual re-sistivity ρ depends critically on the FC density η , whichchanges under variation of input parameters such as dop-ing x , pressure P , and twist angle θ – an effect that ismissing in the overwhelming majority of extant scenariosfor the NFL behavior of the resistivity ρ ( T ). Comparisonof Eq. (12) with Eq. (11) shows that the theoretical ratio A ( x ) /T c ( x ) is indeed doping-independent, in agreementwith the challenging experimental results shown in Fig. 3.Moreover, assuming that the FC parameter η ( x ) varieslinearly with x c − x , which is compatible with model nu-merical calculations based on Eq. (6), the correspondingresult obtained from Eq. (12) is consistent with availableexperimental data [56].Turning to the issue of classical-like Planck dissipa-tion [57], we observe that such a feature is inherent insystems that possess a specific collective mode, trans-verse zero-sound (TZS), which enters provided m ∗ /m e > it is of order 10 [60]). In the com-mon case where the Fermi surface is multi-connected,some branches of the TZS mode turn out to be damped,thereby ensuring the occurrence of a linear-in- T term inthe resistivity ρ ( T ). This is broadly analogous to the sit-uation that arises for electron-phonon scattering in solidsin the classical limit T > T D . (See also the SM [5] andRefs. [58, 60–62].) As a result, FC theory predicts thata break will occur in the straight line ρ ( T ) = ρ + A T at some characteristic Debye temperature T TZS [55, 63],in agreement with experimental data on Sr Ru O [60].However, in the case T TZS < T c , often inherent in exoticsuperconductors such as CeCoIn [14] and TBLG (M4device) [54], this break disappears, and the behavior of ρ ( T ) is fully reminiscent of that in classical physics.In contrast, a current scenario of Patel and Sachdev(PS) for Planckian dissipation [64] attributes the NFL be-havior (2) of ρ ( T ) to the presence of a significant randomcomponent in the amplitude of the interaction betweenquasiparticles near the Fermi surface. However, such amechanism is hardly relevant to the physics of cuprates.Indeed, in their phase diagrams, the strange-metal re-gions located above respective high- T c domains are com-monly adjacent to the familiar FL ones, whose propertiesobey Landau FL theory, in which the interaction ampli-tudes are free from random components. Hence the PSmodel can be a toy model at best. Otherwise, boundariesof the high- T c regions must simultaneously be points ofphase transitions between FL phases and phases withrandom behavior of the interaction amplitudes, which isunlikely.In conclusion, we have demonstrated that the conceptof topological chaos is capable of explaining the non-Fermi-liquid, classical-like behavior of strongly correlatedelectron systems that is emergent at temperatures T farbelow the Debye value T D , where such behavior hithertoseemed impossible. 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V. A. Khodel, J. W. Clark, M. V. Zverev
1. Topological entropy and fermion condensation:an elemental example
The relationship of fermion condensation to the math-ematical concept of topological entropy [1–4] and oper-ations performed in treating classical chaos [6–9] can bedemonstrated in the following example from high-energyphysics.Superdense quark-gluon plasma (QGP) is uniqueamong quantum many-body systems in that the topo-logical rearrangement of the Landau state leading to afermion condensate already occurs in first-order pertur-bation theory, which yields a schematic single-particlespectrum ǫ ( p ) − µ = v F ( p − p F ) + a ( p − p F ) ln( p − p F ) . (S1)This result is reminiscent of the single-particle spectrumof a homogeneous Coulomb plasma, with a crucial dif-ference: the QGP parameter a carries a positive sign,due to the attractive character of quark-gluon exchange,leading to an infinite negative slope of the spectrum ǫ ( p )at the Fermi momentum p = p F . As demonstrated inthe iterative procedure applied by Pethick, Baym, andMonien (PBM) [10], this difference has phenomenal con-sequences, giving rise to unlimited breeding of new sheetsof the Fermi surface.The same PBM Fermi-sheet proliferation persists in amore accurate evaluation of the QGP spectrum based onthe Dyson equation [31] ǫ ( p ) = ǫ p + g Z ln 1 | p − p | n ( p ) dp , (S2)where ǫ p ≃ cp is the bare spectrum with c the velocity oflight, g > n ( p ) isthe corresponding quasiparticle momentum distribution.Since an attempt at straightforward iterative solutionof Eq. (S2) fails, an alternative approach was appliedin Ref. [31]. As outlined below, the problem was re-formulated in terms of an iterative discrete-time map,in analogy with basic treatments of classica dynamicalchaos (e.g., [7]), with subsequent time-averaging of rel-evant quantities. The salient feature of the resultingself-consistent solution of Eq. (S2) is the development ofan interaction-induced flat portion in the single-particlespectrum ǫ ( p ), embracing the nominal Fermi surface.In the standard iterative scheme, the j th iteration n ( j ) ( p ), with j = 0 , , ... , is inserted into the right side of Eq. (S2) to generate the next iteration of thesingle-particle spectrum, ǫ ( j +1) ( p ), measured now fromthe chemical potential µ , and this process is repeated in-definitely. As seen in Fig. S1, the divergence of the slopeof ǫ (0) ( p ) leads originally to a specific 2-cycle in whichall even iterations coincide with the Landau momentumdistribution n L ( p ) = θ ( p F − p ). Coincidence occurs inall odd iterations as well; however, their structure differsfrom that of the Landau state by the presence of a breakin n F L ( p ). In principle, this cycle is eliminated in a moresophisticated iterative scheme in which the neighboringiterations are mixed with each other [11, 31]. However,such a refinement does not lead to convergence.To overcome this difficulty, it is beneficial to intro-duce discrete time-steps numbered j at which the func-tions ǫ ( p, t ) and n ( p, t ) are updated, the latter undergo-ing chaotic jumps from 0 to 1 and vise-versa as j andhence t increases. A further adjustment makes the cru-cial difference. Aided by formulas adapted from classicalmechanics, ǫ ( p, t ) and n ( p, t ) are replaced, respectively,by an averaged single-particle spectrum ǫ ( p ) and a cor-responding averaged occupation number n ( p ): ǫ ( p ) = lim τ →∞ τ τ Z ǫ ( p, t ) dt = lim N →∞ N N X ǫ ( j ) ( p ) ,n ( p ) = lim τ →∞ τ τ Z n ( p, t ) dt = lim N →∞ N N X n ( j ) ( p ) . (S3)Importantly, Eq. (S2) holds if one inserts the time-averaged quantities ǫ ( p ) and n ( p ) instead of the originalones.Results from calculations demonstrating emergence ofthe interaction-induced flat bands of the QGP are dis-played in Fig. S2. As seen, the function ǫ ( p ) does vanishidentically in a momentum region Ω where the 2-cycleoriginally sets in, while the quasiparticle momentum dis-tribution, denoted by n ∗ ( p ), emerges as a continuous function of p in this region.
2. Equality of quasiparticle and particle numbersin strongly correlated Fermi systems
In this section we outline a quasiparticle formalismfree of the Landau restriction γ/ǫ ( p ) ≪
1, where γ rep-resents the damping of single-particle excitations and ǫ F n ( p ) n ( p ) j=1(b) p p p ( p ) / c p F j=2(c) ( p ) / c p F p F j=0(a) p/p F ( p ) / c p F n ( p ) j=3p/p F (d) p p p ( p ) / c p F p/p F n ( p ) FIG. S1: Iterative maps for Eq. (S2) with the dimensionlessparameter g/c set to unity. The left-hand panels show itera-tions for the spectrum ǫ ( j ) ( p ) in units of cp F for j = 0 , , , n ( j ) ( p ). their energy. In strongly correlated electron systems suchas graphene, the Landau quasiparticle picture is puta-tively inapplicable, since the ratio γ/ǫ is known not to besmall, in contrast to requirements of the original Landautheory [42, 43, 46]. However, we shall find that smallnessof this ratio is a sufficient, but not necessary, conditionfor validity of the quasiparticle pattern. Indeed, in super-conducting alloys, the quasiparticle formalism is opera-tive (cf. the textbook [44] and/or Ref. [41]), despite thefact that γ/ǫ greatly exceeds unity due to the presence,in γ , of a finite term arising from energy-independentimpurity-induced scattering. Moreover, the actual re-quirement for validity of the quasiparticle method hingeson the smallness of the damping γ compared with the Fermi energy T F , which is met in the overwhelming ma-jority of electron systems of interest. Consequently, incorrelated homogeneous Fermi liquids, the particle num-ber always coincides with the quasiparticle number, ir-respective of the magnitude of the damping of single-particle excitations, as long as γ ≪ T F . F n ( p )
100 10 3 1 0.3 _ ( p ) / c p F p/p F FIG. S2: Single-particle spectrum (upper panel) and momen-tum distribution (lower panel) averaged according to Eq. (S3).Correspondence between the color of lines and the number ofeffective iterations Nζ , the product of the number of realiterations N , and the parameter of mixing of neighboring it-erations ζ , is indicated. Proof of this statement is based on the dichotomycharacterizing the impact of long-wave external fields V ( k → , ω →
0) on correlated Fermi systems, whichdepends crucially on the ratio ω/k . Indeed, due to thefictitious character of coordinate-independent externalfields V ( k = 0 , ω ), no physical change of the system oc-curs upon their imposition. On the contrary, changedoes ensue in the complementary case of static fields V ( k, ω = 0), its principal effect being expressed in thepole parts G q of the Green functions G .By way of illustration, in what follows we adopt a polepart of the form G q ( p, ε ) = (cid:0) ε − ǫ ( p ) + iγ sgn ( ε ) (cid:1) − , (S4)with γ >
0, as introduced by Abrikosov and Gor’kov intheir theory of superconducting alloys [39]. However, thefinal results are invariant with respect to the explicit formof G q .Further, in the ensuing analysis it is instructive to rep-resent the quasiparticle density n as an integral n = − ZZ p n ∂G q ( p, ε ) ∂p n d p dε (2 π ) i , (S5)where p n is the momentum component normal to theFermi surface. Here, integration over energy is assumedto be performed before differentiation with respect to mo-mentum p . The correct result is also obtained providedthe derivative ∂G q /∂p n in the integrand of Eq. (S5) isrewritten in the form ∂G q ( p, ε ) ∂p n = − lim k → G q ( p , ε ) G q ( p + k , ε ) ∂G − q ( p, ε ) ∂p n , (S6)yielding n = 2 Z Z p n lim k → G q ( p , ε ) G q ( p + k , ε ) ∂G − q ( p, ε ) ∂p n d p dε (2 π ) i . (S7)Evidently, integration over energy in Eq. (S7) produces anonzero result only if the poles of G ( p , ε ) and G ( p + k , ε )lie on opposite sides of the energy axis. This require-ment is met provided the energies ǫ ( p ) and ǫ ( p + k )have opposite signs, so as to generate the relevant factor( dn ( p ) /dǫ ( p ))( dǫ ( p ) /dp n ) ≡ dn ( p ) /dp n in the integrationover energy. The analogous relation ρ = 2 Z Z p n lim k → G ( p , ε ) G ( p + k , ε ) ∂G − ( p, ε ) ∂p n d p dε (2 π ) i (S8)applies for the total density ρ .Hereafter we adopt symbolic notations frequently em-ployed in Fermi Liquid (FL) theory. With round brack-ets implying summation and integration over all inter-mediate variables and the normalization factor 1 / (2 π ) i ,Eqs. (S5) and (S8) then become n = (cid:18) p n G q G q ∂G − q ∂p n (cid:19) , ρ = (cid:18) p n GG ∂G − ∂p n (cid:19) . (S9)Following Pitaevskii [43], we exploit two generic iden-tities of many-body theory. The first of these, − ∂G − ( p, ε ) ∂p n = p n m + (cid:18) U ( p, k ) ∂G ( k, ω ) ∂k n (cid:19) , (S10)where U represents the block of Feynman diagramsfor the scattering amplitude that are irreducible in theparticle-hole channel, is derived assuming gauge invari-ance of the theory [44]. The second, of the form ∂G − ( p, ε ) ∂ε p n = p n + (cid:18) U ( p, k ) ∂G ( k, ε ) ∂ε k n (cid:19) , (S11)stems from the commutativity of the momentum opera-tor with the total Hamiltonian of the system [45].The first step in the proof of the equality ρ = n relieson Landau’s decomposition of the product of two single-particle Green functions into a sum of terms,lim k → G ( p , ε ) G ( p + k , ε ) = z A ( p, ε ) + B ( p, ε ) , (S12)in which B is a part of the limit regular near the Fermisurface, while the remaining pole part is a product of thequasiparticle propagator A ( p, ε ) = lim k → G q ( p , ε ) G q ( p + k , ε ) , (S13) and a square of the quasiparticle weight z =(1 − ∂ Σ( p, ε ) /∂ε ) − in the single-particle state.It should be emphasized that there exists an importantformula [42, 44] analogous to Eq. (S13), namely ∂G ( p, ε ) ∂ε = − lim ω → G ( p, ε ) G ( p, ε + ω ) ∂G − ( p, ε ) ∂ε ≡ − B ( p, ε ) ∂G − ( p, ε ) ∂ε . (S14)Evidently, the result of integration of this expression overenergy vanishes identically, since poles of the product G ( p, ε ) G ( p, ε + ω ) lie on the same side of the energy axis.Further, one finds that regular contributions to the keyrelations involved come from both the regular parts B ofthe product GG and the block U itself.The key step in the regularization procedure developedby Landau then lies in the introduction of a specific in-teraction amplitude Γ ω determined by the equationΓ ω = U + (cid:16) U B Γ ω (cid:17) ≡ U + (cid:16) Γ ω B U (cid:17) , (S15)which is capable of absorbing all the regular contribu-tions, irrespective of the explicit form of the propagator A . Indeed, let us multiply both members of Eq. (S10)from the left by the product Γ ω B , integrate over all vari-ables, and eliminate the expression Γ ω B U in the last termwith the aid of Eq. (S15), yielding finally − (cid:18) Γ ω ( p, k ) B ( k, ε ) ∂G − ( k, ε ) ∂k n (cid:19) = (cid:18) Γ ω ( p, k ) B ( k, ε ) k n m (cid:19) + (cid:18) Γ ω ( p, k ) ∂G ( k, ε ) ∂k n (cid:19) − (cid:18) U ( p, k ) ∂G ( k, ω ) ∂k n (cid:19) . (S16)Simple algebraic transformations then lead to the re-lation − ∂G − ( p, ε ) ∂p n = ∂G − ( p, ε ) ∂ε p n m − (cid:18) z Γ ω A ∂G − ( k, ω ) ∂k n (cid:19) . (S17)In obtaining this result, we have employed the equa-tion [43–45] ∂G − ( p, ε ) ∂ε p n = p n + (cid:16) Γ ω Bk n (cid:17) . (S18)Near the Fermi surface, the relevant derivatives of G − are evaluated in terms of the corresponding derivativesof the pole part G − q to obtain − ∂G − q ( p, ε ) ∂p n = ∂G − q ( p, ε ) ∂ε p n m + (cid:18) z Γ ω ∂G q ( k, ω ) ∂k n (cid:19) . (S19)Significantly, integration over energy is obviated due tothe presence of the Fermi surface, providing the structureof the pole term G q is given by Eq. (S4).0Thus, in homogeneous matter, we are led to aPitaevskii-style equation ∂ǫ ( p ) ∂p n = p n m + (cid:18) f ∂n ∗ ( k ) ∂k n (cid:19) , (S20)where the Landau notation f = z Γ ω is introduced. Weemphasize that the sole condition employed has been thesmallness of the ratio γ/T F ≪ n ∗ ( p ) acquires the non-Fermi-Liquid (NFL)form found from numerical solution of this equation.With regard to TBLG and similar electron systems, thefirst term on the right side of this equation must be im-proved, as has been done for example in Ref. [36].To prove the coincidence ρ = n , we multiply both sidesof Eq. (S17) from the left by the product p n B and inte-grate over all intermediate variables. Exploiting the factthat the integral (cid:16) B ∂G − ( p, ε ) /∂ε (cid:17) ≡ − (cid:16) ∂G ( p, ε ) /∂ε (cid:17) vanishes identically in integrating over the energy, whileobserving that the remaining term involving the product p n B Γ ω simplifies with the aid of relation (S18), we areled to (cid:18) p n B ∂G − ( p, ε ) ∂p n (cid:19) = (cid:18) p n Bz Γ ω A ∂G − ( k, ε ) ∂k n (cid:19) = z (cid:18) p n A ∂G − ( p, ε ) ∂ε ∂G − ( p, ε ) ∂p n (cid:19) − z (cid:18) p n A ∂G − ( p, ε ) ∂p n (cid:19) . (S21)Transfer of the last term in Eq. (S21) to the left side ofthis equation and further straightforward manipulationof the relation obtained leads us finally to ρ = 23 (cid:18) p n G q ( p, ε ) G q ( p, ε ) ∂G − q ( p, ε ) ∂p n (cid:19) = n, (S22)thereby establishing the coincidence of particle and quasi-particle numbers, independently of the magnitude of thedamping of single-particle excitations. In the end, allthat matters is the presence of the Fermi surface.The final result ρ = n is not sensitive to the explicitform of the quasiparticle propagator A . Consequently,it can be understood that the equality between particleand quasiparticle numbers generalizes to Fermi systemshaving Cooper pairing with a BCS gap function ∆( p ).In that case it takes the form ρ = n = 2 X p v ( p ) ≡ X p (cid:18) − ǫ ( p ) p ǫ ( p ) + ∆ ( p ) (cid:19) (S23)irrespective of the structure of the single-particle spec-trum, thereby allowing for the presence of flat bands and/or damping of normal-state single-particle excita-tions. This statement, best proved within the frame-work of the Nambu formalism along the same lines asbefore, provides the hitherto missing element of the FLapproach to the theory of strongly correlated supercon-ducting Fermi systems.
3. New branches of the collective spectrum andlow- T kinetic properties of strongly correlatedelectron systems Here we discuss consequences stemming from the oc-currence of additional branches of collective excitationsin strongly correlated electron systems of solids [61, 62]whose presence is associated with a substantial enhance-ment of the effective mass m ∗ – often exceeding 10 m e asin heavy fermion metals, notably CeCoIn [60]. Primar-ily, one is dealing with the transverse zero sound (TZS)emergent when the first dimensionless harmonic of theLandau interaction function satisfies F = f p F m ∗ /π > v L ≃ p F /m e greatly exceeds that of the heavy carrierspopulating the second band, v H = p F /m ∗ . In such sys-tems, there exist several branches of TZS. Here we focuson a mode whose velocity is smaller than the Fermi value v L . In this case, the dispersion relation yielding the com-plex value of its velocity c = c R + ic I takes the form [62]1 = F (cid:20) − (cid:18) c v H − (cid:19)(cid:18) c v H ln c + v H c − v H − (cid:19)(cid:21) + F v H v L (cid:20) − (cid:18) c v L − (cid:19)(cid:18) c v L ln c + v L c − v L − (cid:19)(cid:21) . (S24)The imaginary part of the expression on the right side ofthis equation vanishes identically. As is easily verified, itsreal part comes primarily from the first term in the squarebrackets on the right-hand side, since the second term issuppressed by presence of the small factor v H /v L . In therealistic case F ≫
1, we arrive after some algebra [62]at the result c R ∝ v L r m e m ∗ , c I ∝ v L m e m ∗ . (S25)1Thus we see that in strongly correlated electron systems,there is no ban on emission and absorption of the TZSquanta, by virtue of the condition c R /v L <
1. One isthen allowed to study the associated collision term alongthe same lines as in the familiar case of electron-phononinteractions in solids, where the resistivity ρ ( T ) varies lin- early with T provided T > T D . Additionally, the velocity c R typically turns out to be smaller than the phonon ve-locity. This becomes the essential factor, especially inthe case where the relevant Debye temperature becomeslower than T cc