Anomalous Hydrodynamics in One Dimensional Electronic Fluid
AAnomalous Hydrodynamics in One Dimensional Electronic Fluid
I. V. Protopopov,
1, 2
R. Samanta, A. D. Mirlin,
4, 5, 6, 2 and D.B. Gutman Department of Theoretical Physics, University of Geneva, 1211 Geneva, Switzerland Landau Institute for Theoretical Physics, 119334 Moscow, Russia Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel Institute for Quantum Materials and Technologies,Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany Institut f¨ur Theorie der Kondensierten Materie,Karlsruhe Institute of Technology, 76049 Karlsruhe, Germany Petersburg Nuclear Physics Institute, 188350 St. Petersburg, Russia (Dated: January 5, 2021)We construct multi-mode viscous hydrodynamics for one dimensional spinless electrons. De-pending on the scale, the fluid has six (shortest lengths), four (intermediate, exponentially broadregime), or three (asymptotically long scales) hydrodynamic modes. Interaction between hydro-dynamic modes leads to anomalous scaling of physical observables and waves propagating in thefluid. In four-mode regime, all modes are ballistic and acquire KPZ-like broadening with asymmet-ric power-law tails. “Heads” and “tails” of the waves contribute equally to thermal conductivity,leading to ω − / scaling of its real part. In three-mode regime, the system is in the universalityclass of classical viscous fluid [9, 24]. Self-interaction of the sound modes results in KPZ-like shape,while the interaction with the heat mode results in asymmetric tails. The heat mode is governed byLevy flight distribution, whose power-law tails give rise to ω − / scaling of heat conductivity. Understanding properties of interacting electronic sys-tems is fundamentally important across various branchesof physics. The problem is extremely non-trivial andmultifaceted due to the impact of quantum coherenceand strong interactions as well as other important in-gredients, including the underlying crystal lattice and/ordisorder. Progress has been achieved by constructing ef-fective theories for long-living modes of electronic sys-tems. Such theories are universal, i.e., insensitive to mi-croscopic details and mostly determined by qualitativeaspects such as dimensionality, symmetries, and topol-ogy. Paradigmatic examples of effective descriptions areLandau’s Fermi-liquid theory[1], the theory of superfluidliquids[2] and the theory of diffusive modes in disorderedconductors[3].Recent advances in experimental techniques have madeavailable several systems[4–9] realizing (in a certain tem-perature range) the hydrodynamic regime of electrontransport. In this regime, the dynamics is dominatedby electron-electron collisions (rather than by impu-rity or electron-phonon scattering) and can be describedby a set of equations of hydrodynamic type govern-ing the evolution of conserved densities (charge, mo-mentum, energy, etc.). Two aspects make such sys-tems spectacular. First, they exhibit electron trans-port that is profoundly different from that observedin conventional Drude conductors. It is manifested inGurzhi effect[10], spatial non-locality[11] and unconven-tional magnetoresistance[12, 13], see Refs.[14, 15] for arecent review. Second, topologically-induced qualitativediversity of underlying electronic spectra gives rise to un-conventional hydrodynamic regimes, such as relativistichydrodynamics in graphene [16].The hydrodynamics of one-dimensional (1D) interact- ing electrons is of special interest. It often involves anextended (in integrable systems even infinite) number ofconserved hydrodynamic charges[17–22]. Furthermore,the reduced dimensionality of the system greatly pro-motes hydrodynamic fluctuations[23], which can invali-date the mean-field hydrodynamic description at suffi-ciently long scales and drive the system into a fluctuation-dominated regime characterized by non-trivial scaling ofphysical observables[9, 20, 24, 26]. The relevance of fluc-tuational hydrodynamics and in particular of the cele-brated Kardar-Parisi-Zhang model in the context of 1Delectronic fluids was discussed recently in Refs. [28, 29].In this paper, we explore the full multimode fluctua-tional hydrodynamics of 1D spinless fermions with short-range interaction. Our focus is on real-time dynamicsand on thermal transport that was probed recently inseveral closely related experimental setups [30–38]. Weconfirm the frequency scaling of the thermal conductivity, κ ∝ ω − / , advocated recently based on a self-consistentkinetic theory of bosonic excitations (see Ref. [29] andreferences therein). We find, however, that the earlierkinetic treatment fails to predict the correct dependenceof the prefactor in this scaling on temperature and otherparameters of the system.To construct the non-linear hydrodynamic descriptionof the system, we employ the bosonization technique[39–42] taking into account the curvature of the electronicspectrum (i.e., finite fermion mass m ) [43–46]. Re-laxation processes in such a “non-linear Luttinger liq-uid” were analyzed in several works, see Ref. [12] fora review. It was shown in Ref. [11] (see also Refs.[1, 13, 14, 50, 51]) that at sufficiently low temperatures[ T < T FB ∼ /ml < (cid:15) F , where l is the range of theelectron-electron interaction and (cid:15) F the Fermi energy], a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n thermal excitations in a non-linear Luttinger liquid are“composite” fermions with renormalized Fermi velocity u ∼ v F , an effective mass m ∗ ∼ m and weak interactionsvanishing in the zero-momentum limit[54]. The compos-ite fermions are characterized by a long lifetime τ F , τ − F ∼ l T /m ∗ u . (1)Focusing on this low-temperature regime, we describethe dynamics of the system by the kinetic equation forfermionic quasiparticles ∂N F ( p ) ∂t + v Fp ∂N F ( p ) ∂x = ˆ I p [ N F ] . (2)Here N F is a distribution function, v Fp is the momentum-dependent velocity of fermionic quasiparticles, and ˆ I isthe collision integral. The hydrodynamic equations ariseafter the projection of the kinetic equation on the zeromodes of the collision integral and are valid at scaleslarger than fermionic mean free path uτ F . The colli-sion integral in (2) is nullified by Fermi-Dirac function n F (cid:16) (cid:15) p − vp − µT (cid:17) with chemical potential µ , temperature T and the boost velocity v . These three parameters of theequilibrium distribution correspond to the three exactlyconserved densities of the model: particle number, en-ergy, and momentum. Peculiarities of the 1D kinemat-ics give rise however to other quasi-conserved quantities(soft modes of the collision integral). First, equilibrationof the particle number between the left and right moversrequires processes involving a deep hole near the bottomof the band. In the bosonic description of the Luttingerliquid, such processes correspond to the Umklapp scat-tering and manifest themselves only at exponentially longlength scale[15–17] L U ∼ uT − / (cid:15) / F e (cid:15)FT . (3)Thus, at scales shorter than L U the system possesses fourconserved quantities (total energy, total momentum, andtwo chiral number densities). Second, a detailed analy-sis of collision processes leading to Eq. (1) shows thatin such a collision the energy and momentum exchangebetween the chiral sectors is parametrically suppressed(compared to the thermal energy or momentum) by a fac-tor ( T /(cid:15) F ) (cid:28) L ∼ (cid:16) (cid:15) F T (cid:17) m ∗ u l T (4)the chiral sectors are effectively decoupled and six hydro-dynamic modes exist in the system.In the six-mode regime the particle densities, mo-mentum, and energies of each chiral sector are sepa-rately conserved and we combine them into two chi-ral vectors q Tη = ( ρ η , π η , (cid:15) η ), η = R, L . We denote by φ Tη = T − η ( µ η , v η , −
1) the vector of the correspond-ing conjugate thermodynamic variables. The conservedquantities obey the continuity equations ∂ t q iη + ∂ x J iη = 0 , (5)with index i specifying the conserved charge and the cor-responding flux, J η = ( J ρη , J πη , J (cid:15)η ).On the linear level, one relates q η ( ω, k ) = χ ret η ( ω, k ) φ η ( ω, k ) , (6)via the polarisation operator χ ret i,j ; η ( x, t ) = − iθ ( t ) (cid:104) [ˆ q i,η ( x, t ) , ˆ q j,η (0 , (cid:105) . Similarly, currents can berepresented in terms of current response function M , J η ( ω, k ) = M η ( ω, k ) φ η ( ω, k ) /ik . (7)In the ω = 0, small- k limit, the matrix M η ( k ) =( ikA + k D ) χ η is build out of matrices of velocities( A ), diffusion coefficients ( D ), and static susceptibilities χ η ≡ χ ret η ( ω = 0 , k → A andthe matrix of static susceptibilities are thermodynamicquantities and can be computed straightforwardly in theapproximation neglecting the composite-fermion interac-tion. The matrix of diffusion coefficients D requires morework; it can be obtained from the linearized kinetic equa-tion (2). See Supplemental Material (SM) [58] for explicitexpressions for χ and M .To incorporate non-linear effects into the hydrody-namic description, we extend the expressions for hydro-dynamic currents by terms of second order in the con-served densities: J η = ( M η /ik ) χ − η q η + 12 (cid:88) i,j H η ; i,j q iη q jη . (8)Here, we have take the static limit ω = 0 and the (vector-valued) coefficients H η ; i,j can be computed neglecting theinteraction of composite fermions[58].Equations (s87), (6), and (8) describe the six-mode hy-drodynamics that exist at short length scales, L < L . Atlonger scales, the collisions equilibrate the temperaturesand the boost velocities in the two chiral sectors. Thehydrodynamic theory of the four-mode regime can be ob-tained through the reduction of the six-modes equationsby setting T L = T R = T , v L = v R = v and working withthe total energy and momentum densities, (cid:15) = (cid:15) R + (cid:15) L and π = π R + π L .At still larger length scales, L > L U , the systemreaches equilibrium with respect to particle exchange be-tween the chiral sectors. The corresponding three-modeshydrodynamics can be obtained through the reduction ofthe four-mode theory by setting µ L = µ R = µ .In the linear hydrodynamic approximation, the conti-nuity equations dictate that χ ret η ( ω, k ) = M η ( iωχ η − M η ) − χ η . (9)The information encoded in the polarization operator en-ables one to compute the full set of kinetic coefficients,accessible via linear-response measurements. At firstglance, the non-linear terms in hydrodynamic equationsare irrelevant for the discussion of such linear-responsequantities. This conclusion is, however, invalidated byhydrodynamic fluctuations that were so far neglected.Once the fluctuations are taken into account, the non-linear hydrodynamic couplings induce strong renormal-izations of bare kinetic coefficients, totally modifying thelinear-response characteristics of the system.To account for fluctuations we promote the hydrody-namic equations (s87) to the Keldysh action (of Martin-Siggia-Rose type)[58]. Since at hydrodynamic scalesthe system is locally at equilibrium, the fluctuation-dissipation theorem holds. Therefore, the retarded partof the polarization operator χ ret determines also theKeldysh components and thus the entire action at theGaussian level. The quadratic terms in the hydrody-namic currents (8) correspond to cubic vertices in theaction.Following Ref. [26], we analyze the resulting Keldyshaction of fluctuational hydrodynamics within the mode-coupling approximation[59]. To perform the calculationit is convenient to pass to the eigenmodes of the lin-earized hydrodynamic theory. We define a new basis Ψ = R q , where R diagonalizes the velocity matrix A , RAR − = diag( v , . . . , v N ). Because of the mode sep-aration caused by different mode velocities, only diago-nal correlations survive in the long-time limit, and theKeldysh pair-correlation functions of the eigenmodes f j ( x, t ) = (cid:104) Ψ j ( x, t )Ψ j (0 , (cid:105) (10)satisfy the self-consistent Dyson equations[58] (cid:16) ∂ t + v j ∂ x − ˜ D j ∂ x (cid:17) f j ( x, t ) = (cid:90) ∞−∞ dy (cid:90) t ds × f j ( x − y, t − s ) ∂ y R j ( y, s ) . (11)Here R j ( y, s ) = 1 T N (cid:88) l,m =1 λ jlm f l ( y, s ) f m ( y, s ) , (12)˜ D j are diagonal elements of the effective diffusion ma-trix ˜ D describing broadening of eigenmodes, and couplingconstants λ jlm account for the mode interaction. Theseconstants are computed from microscopis parameters ofthe original fermionic model[58].We now employ this theory to study pulses propaga-tion in an electronic fluid as well as its linear-responseproperties.We consider the time evolution of a generic distur-bance created in a limited region of the fluid. Due to energy relaxation for times longer than fermionic en-ergy relaxation time τ F , any disturbance is fully pro-jected onto eigenmodes of the collision integral. At timesshorter than L /u , this yields six hydrodynamic modesΨ j . The degree to which the modes are excited dependson the overlap of the disturbance with Ψ j . These modesgive rise to six ballistic pulses propagating through thefluid. Due to differences between the mode velocities,∆ u ij ≡ u i − u j , the separation between the peaks growthslinearly with time, L ij = ∆ u ij t . The width of each peakis broadened, within the linear hydrodynamics, by thecorresponding diffusion process as ( ˜ D j t ) / . The non-linear couplings further broaden the shape of the pulsesand modify their shape. Comparing the linear and non-linear terms, one can show that non-linear broadeningdominates over the normal diffusion at scales beyond L ∗ = L ( T BF /T ) (cid:29) L . Therefore, when fluid entersthe four-mode regime, it is still governed by essentiallylinear theory, with conventional diffusive scaling.At L ∼ L , the number of hydrodynamic modes is re-duced to four and the pulses are reshaped into four peaks.The evolution in earlier stages of the four-mode regimeis well described by linearized hydrodynamic. But for L > L ∗ the non-linear terms start to dominate and thenormal diffusion process is replaced by the anomalousone. Essentially, at this stage, one can drop the barediffusion terms in Eq. (11). In the four-mode regimeall non-linear coupling constants are of the same order, λ ijk ∼ λ ≡ T u / . However, only the interaction be-tween modes propagating in the same direction is signif-icant. Therefore, Eq. (11) splits into two sets of chiralequations. Near the maximum of any given mode, thecoupling to other modes is exponentially small and can beneglected. Equation (11) is mathematically equivalent topair velocity correlation function in the stochastic Burg-ers equation and the corresponding KPZ problem[60].Thus, nears the maximum f i ( x, t ) ∼ T ( λt ) / f KPZ (cid:18) T ( x − u i t )( λt ) / (cid:19) . (13)Here f KPZ ( x ) is the universal dimensionless KPZ func-tion, with f KPZ ( x ) ∼ | x | ≤ f KPZ ( x ) ∼ e − . | x | for | x | (cid:29) f i ( x, t ) ∼ (cid:88) j θ [( x − u i t )sgn(∆ u ji )] u (cid:18) T | ∆ u ji | (cid:19) / × t | x − u i t | − / for | x − u i t | (cid:29) ut / T / . (14)One may interpret this as propagation of one degree offreedom away from its light cone via the interaction with x f ( x , t ) t - tx - tx - x f ( x , t ) t - t - tx - FIG. 1: Schematic shape of pulse evolution through four andthree mode regimes. The scaling of the heads and tails of thepeaks with time is depicted, see text for details. a faster or slower degree of freedom. In Eqs. (13), (s109)and below we omit numerical coefficients of order unity,as emphasized by the sign ∼ replacing the equality sign.At distances larger than L U , the fluid is described bythree hydrodynamic modes. This is a universal regimerepresenting the ultimate infrared fixed point of any non-integrable system. It is characterized by two ballisticsound modes (index j = 2 ,
3) and one static (i.e., zero-velocity) heat mode ( j = 1). The pulse propagation insuch a regime was analyzed in the context of classical flu-ids in Refs. [9, 26, 63]. The sound mode acquires the KPZshape, Eq.(13). For the corresponding self-coupling con-stant we find λ ≡ λ ∼ T /m u / . Due to the time-reversal symmetry, the self-coupling of the heat mode isidentically zero. Therefore, in the absence of the inter-mode coupling, the spread of the heat mode would bediffusive. The non-linear interaction between the heatand sound mode, which is characterized by a coupling λ ∼ T /m u / , leads to the formation of power-lawtails for the heat and sound modes[58]. It transforms theheat mode into symmetric Levy-flight distribution with α = 5 / f ( x, t ) ∼ Tδx ( t ) f Levy ,α =5 / (cid:18) xδx ( t ) (cid:19) . (15)The heat mode has a maximum at x = 0 and the width δx ( t ) ∼ t / T / m − / u − / . The t / scaling of thewidth was also obtained in the context of classical an-harmonic chains[64, 65]. The value at the maximum is f (0 , t ) ∼ T /δx . Away from the maximum (for x (cid:29) δx )the heat mode has power law tails[66] that scale as f ( x, t ) = T / m u / tx − / , implying anomalous heat dif-fusion.We now consider the linear response properties of theelectronic fluid. Generally speaking, an N -componentliquid has N ( N − / J T forfluids is local and can be computed by subtracting anadvective contribution from the energy current J E , [23] J T = J E − ¯ wJ ρ , (16)where ¯ w is the enthalpy of the fluid per one electron and J ρ the particle current. The Kubo formula for thermalconductivity reads[68] σ T ( ω, k ) = 1 − iωT (cid:2) K TT ( ω, k ) − K TT (0 , (cid:3) , (17)where K TT ( ω, k ) = − i (cid:104) [ ˆ J T ( x, t ) , ˆ J T (0 , (cid:105) ret ( ω, k ). Em-ploying Eq.(17) and setting k = 0, we find [58] that inthe 6-mode regime σ T ( ω ) = − π uT iω + π u m l T . (18)The Drude peak corresponds to the ballistic propagationof heat[69, 70], while the real part of conductivity is dueto heat diffusion. As the system enters the four-moderegime, the propagation of all modes remains ballistic.Hence, the imaginary part of the heat conductivity isunchanged, Im σ T ( ω ) = π uT / ω . The real part of theheat conductivity, on the other hand, is renormalized.The effects of the renormalization are associated with ananomalous broadening of the pulses, with two contribu-tions coming from the head and the tails of the peak.Both happen to be of the same order and lead toRe σ T ( ω ) ∼ uT / ω − / . (19)Finally, we discuss the three-mode regime, where theheat conductivity is determined solely by the static mode.Therefore, the ballistic contribution is suppressed, givingrise to an exponentially large constant: i/ω (cid:55)→ τ U . Thereal part of the thermal conductivity thus scales asRe σ T ( ω ) ∼ uT τ U + T / m u ω − / . (20)To summarize, we have developed a multi-mode hydro-dynamic approach for the electronic fluid. Depending onthe number of conserved charges, the fluid has six, four,or three hydrodynamic modes. Though the three-moderegime is an ultimate long-distance fixed point, it is onlyreached at exponentially long distances, leaving room toan exponentially long four-mode viscous hydrodynamicregime.The interaction between the hydrodynamic modesleads to the renormalization of transport coefficients, giv-ing rise to universal scaling behavior, and shapes pulsespropagating through the fluid. In the six- and four-moderegimes, all pulses propagate ballistically. The “head” ofevery pulse is controlled by self-interaction, resulting in aKPZ scaling of pulse width ( t / ) and amplitude ( t − / )with time. Interaction between the modes propagatingwith different velocities results in power-law tails scalingas x − / with distance x from the mode center and di-rected towards another mode. As the system reaches athree-mode regime, the pulses redistribute, and a staticheat and two ballistic sound peaks are formed. The widthof the ballistic modes has KPZ scaling with time. Theinteraction between the sound waves and the heat modegives rise to power-law tails for all peaks. Each soundmode acquires a rear tail. The static heat mode is aLevy flight function with α = 5 /
3, with symmetric tails.The anomaly in peak shapes leads to anomalous kineticcoefficients, in particular, the thermal conductivity.We conclude by comparing the results of the presentanalysis for σ T with earlier calculations performed withinthe self-consistent kinetic approach[29]. Reassuringly,both approaches yield two regimes of anomalous scal-ing of σ T separated by the scale L U . Further, the ki-netic approach yields for these regimes results analo-gous to Eqs. (20) and (19), with the same ω − / scaling.Such agreement in scaling resulting from self-consistentkinetic[23, 28, 71] and classical renormalization-group[24] approaches has been known for a long time. Thisagreement is highly non-trivial and perhaps even puz-zling. Indeed, although our starting point here is atransport coefficient computed within fermionic kinetictheory[29, 72], the subsequent analysis in this work andRef. [29] is very different. In the kinetic frameworkof Ref. [29], the ω − / scaling results from subthermalbosons with wave vectors k (cid:28) T /u that can propagateanomalously large distances without scattering. At thesame time, in the present framework, this enhancementof thermal conductivity results from the interaction be-tween bosonic (hydrodynamic) modes leading to anoma-lous hydrodynamics. Importantly, while the frequencyscaling agrees in two approaches, the prefactors (in par-ticular, the temperature scaling) are essentially differ-ent. Effects of renormalization controlling results of thepresent work turn out to be dominant for σ T , in bothfour-mode and three-mode regimes.Note added: While preparing the paper for publica-tion, we learnt about a recent paper [73] that has someoverlap with the present work. Acknowledgements
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I.V. Protopopov, R. Samanta, A.D. Mirlin and D.B. Gutman
I. BOLTZMANN EQUATION AND KINETIC COEFFICIENTS
In this section of SM we derive the hydrodynamic description of the fluid in the 6-mode regime from the kinetictheory, establish the relation between currents and the zero modes of the collision integral, needed for Eq.(6)in themain text; express currents via thermodynamic variables, see Eq.(8) in the main text and evaluate the bare diffusioncoefficients.Our starting point is a kinetic equation formulated in terms of ”composite” electrons[s1–3]. ∂f ( p, x, t ) ∂t + v p ∂f ( p, x, t ) ∂x = ˆ I [ f ] . (s1)We now outline this approach to compute linear response. Close to local equilibrium the distribution function can bewritten as f ( p, x, t ) = n F (cid:18) (cid:15) p − v ( x, t ) p − µ ( x, t ) T ( x, t ) (cid:19) + δf ( p, x, t ) , (s2)where n F is the Fermi-Dirac distribution. Here the first term corresponds to the zero mode of the collision integraland the second to “massive” excitations. On scales longer than the inelastic collision length, the two parts are relatedvia δf ( p, x, t ) = 14 T (cid:90) ( dp (cid:48) ) ˆ I − p,p (cid:48) v p (cid:48) g ( p (cid:48) ) (cid:16) ∂ x µ − µT ∂ x T + p (cid:48) ∂ x v (cid:17) . (s3)Here we define g ( p ) = cosh − (cid:16) (cid:15) p − µ T (cid:17) , and ˆ I − p,p (cid:48) is an operator that is inverse to a linearised collision integral. Theˆ I − p,p (cid:48) is not symmetric with respect to p ↔ p (cid:48) . However by a transformationˆ I − p,p (cid:48) = g ( p )ˆ I − p,p (cid:48) g − ( p (cid:48) ) (s4)it is related to a symmetric operator ˆ I − p,p (cid:48) . We now use the eigenfunctions of ˆ I − p,p (cid:48) to classify the spectrum of kineticequation, and separate the currents into ideal and dissipative parts J α ( k ) = J ideal α ( k ) + J diss α ( k ) . (s5)The dissipationless (ideal) part of the current is carried by gapless excitations that are parametrized by time andspace-dependent zero modes ( T, µ, v ). These excitations give rise to particle, momentum and energy currents that onthe linear level are given by J ideal α ( k ) = 14 (cid:90) ( dp )ˆ j α ( p ) g ( p ) (cid:20) φ ( k ) + p φ ( k ) + ( (cid:15) p − µ ) φ ( k ) (cid:21) . (s6)Here α = { ρ, π, (cid:15) } and we denote ˆ j ρ ( p ) = v p , ˆ j π = pv p , ˆ j (cid:15) ( p ) = ( (cid:15) p − µ ) v p . The zero modes are combined into φ Tη = T − η ( µ η , v η , − . (s7)The currents above rely solely on the thermodynamic properties of the system. For this reason, they are identicalto corresponding currents in the ideal fermionic hydrodynamics[s4], with identical conserved quantities. In particular,in the six-mode regime, the chiral currents agree with chiral currents of hydrodynamics on one-dimensional edge of aquantum Hall sample.In addition to the zero modes, there are finite-energy modes of the collision integral that lead to dissipative processes.They correspond to following currents J dis α ( k ) = ik (cid:90) ( dp )( dp (cid:48) ) j α ( p ) g ( p ) I − p,p (cid:48) g ( p (cid:48) ) v p (cid:48) (cid:20) φ ( k ) + p (cid:48) φ ( k ) + ( (cid:15) p (cid:48) − µ ) φ ( k ) (cid:21) == ik (cid:2) d α, φ ( k ) + d α, φ ( k ) + d α, φ ( k ) (cid:3) . (s8)2Here we define d α,β = − (cid:90) ( dp )( dp (cid:48) )ˆ j α ( p ) g ( p )ˆ I − p,p (cid:48) g ( p (cid:48) )ˆ j β ( p (cid:48) ) . (s9)By construction, the operator ˆ I − is computed in the space orthogonal to zero modes. Therefore, to compute thedissipative contribution one needs to project the current vertices j α ( p ) onto a subspace that is orthogonal to zeromodes. In the six-mode regime, the zero modes correspond to the conservation of chiral densities, momenta, andenergies. This corresponds to functions g ( p ), pg ( p ) and ( (cid:15) ( p ) − µ ) g ( p ). We now focus on the dc limit ω → πJ Rρ ( k ) = (cid:2) T − ikd (cid:3) φ ( k ) + (cid:2) muT − π T mu − ikd (cid:3) φ ( k ) + (cid:2) − ikd (cid:3) φ ( k ) , πJ Rπ ( k ) = (cid:2) muT − π T mu − ikd (cid:3) φ ( k ) + (cid:2) m u T − ikd (cid:3) φ ( k ) + (cid:2) π T u − ikd (cid:3) φ ( k ) , πJ R(cid:15) ( k ) = (cid:2) − ikd (cid:3) φ ( k ) + (cid:2) π T u − ikd (cid:3) φ ( k ) + (cid:2) π T − ikd (cid:3) φ ( k ) . (s10)It is worth noting that for strictly parabolic spectrum only the thermal conductivity d ∼ um l is finite, while therest of the elements d α,β = 0. In the kinetic theory of nearly ideal gas, it is well known that its viscosity coefficientvanishes for particles with a strictly parabolic spectrum [s5]. For d = 1 we find that the real part of all kineticcoefficients, except thermal conductivity, also vanishes.The relation above yield current J α as linear functions of the external forces φ , thus defining a matrix of linearresponse coefficients ˆ L (cid:104) J n (cid:105) ( ω, k ) = ikT (cid:20) L n, ( k ) φ ( ω, k ) + L n, ( k ) φ ( ω, k ) + L n, ( k ) φ ( ω, k ) (cid:21) . (s11)It is useful to express the current operators in terms of thermodynamic variables. Resolving the relation between thezero modes of the collision integral φ and thermodynamic variables ρ R , π R , (cid:15) R one extends Eq. (s10) onto non-linearlevel J Rρ = 1 m π R , (s12) J Rπ = 2 (cid:15) R + mu ρ R , (s13) J R(cid:15) (cid:39) mu ρ R − u π R + 3 u(cid:15) R − ikD (cid:15) (cid:0) (cid:15) R + mu ρ R − uπ R (cid:1) + ∆ J R(cid:15) . (s14)Here we have defined the energy diffusion constant D (cid:15) = u /l T . It is worth mentioning that the results for theparticle J ρ and momentum J π currents are exact. There are no non-linear in density correction to these currents,neither T /mu corrections. The results for the energy current contains only the leading in T /mu part. There is alsoa non-linear in density-field part, which is given by∆ J R(cid:15) (cid:39) − m u π T ( (cid:15) R + mu ρ R − uπ R ) . (s15)The linear-response coefficients L ij computed above on the linear level, see Eq.(s10), are affected by non-linearitiesthat have been so far ignored. The latter affect the character of heat propagation, or on a more technical level inducea renormalization of the theory. This necessitates development of a non-linear field theory of multi-component fluid.We now promote the equations of motion to the level of action, formulating an effective classical field theory[s6, 7]. II. MARTIN-SIGGIA-ROSE ACTION
In this section we describe the classical field theory that corresponds to the fluctuation hydrodynamics of electronfluid, and derive mode-coupling equation, see Eq.(11) in the main text.We start with Gaussian part of the action S = ( q T , ¯ q T ) − ω, − k T ˆ χ − ω,k (cid:18) q ¯ q (cid:19) ω,k . (s16)3Here we denote ˆ χ − ω,k = (cid:32) χ a − ω,k ˆ χ r − ω,k ( ˆ χ r − ω,k − ˆ χ a − ω,k ) B ω (cid:33) , (s17)where B ω = coth (cid:0) ω T (cid:1) . The action is encoded by χ r − η ( ω, k ) = iωM − η ( k ) + χ − ,η ( k ) . (s18)In the limit ω → M η ( k ) χ − ,η ( k ) = ikA η + k D η (s19)yield the generalised velocity A and diffusion D matrices. Their size is equal to the number of conserved modes.Because the system is in local equilibrium, analytic properties and fluctuation-dissipation theorem allow to restoreadvance and Keldysh components, χ K ( ω, k ) = [ χ ret ( ω, k ) − χ adv ( ω, k )] coth ω T . The interaction part of the action S int = − iT (cid:88) p,l,m ¯ q p Γ p,l,m q l q m , (s20)and the interaction vertex Γ p,l,m ( k ) = k (cid:88) p ˆ M − p,p ( k ) H p l,m ( k ) . (s21)Here H pl,m ( k ) = ∂ J p ( k ) ∂q l ∂q m . (s22)Because matrices M and χ are symmetric, the Keldysh action (s16) can be diagonalized as a quadratic form by thelinear transformation q = R − Ψ , ¯ q = R − ¯ Ψ . (s23)Since the compressibility matrix has positive eigenvalues, one can choose1 T Rχ R T = ˆ1 (s24)and RAR − = ˆ v . (s25)Here ˆ v = diag( v , v , . . . ) is the diagonal velocity matrix.After the rotation (s23), the retarded part of the Gaussian action takes the form S ret0 [ Ψ ] = (cid:88) m,n ¯ Ψ Tm ( − ω, − k ) (cid:20) iω (cid:16) i ˆ vk + k ˜ D (cid:17) − + ˆ1] m,n Ψ n ( ω, k ) , (s26)where ˜ D = RDR − is the diffusuion matrix in the eigenmode basis. The corresponding retarded propagator reads i (cid:104) ¯ Ψ m Ψ n (cid:105) ω,k = (cid:16) iω ( i ˆ vk + ˜ Dk ) − + 1 (cid:17) − m,n . (s27)The advanced propagator is related to the retarded one via i (cid:104) Ψ m ¯ Ψ n (cid:105) ω,k = − i (cid:104) ¯ Ψ m Ψ n (cid:105) ∗ ω,k . (s28)The Keldysh part of the propagator follows from FDT theorem[s8] (cid:104) Ψ m Ψ n (cid:105) ( ω, k ) = (cid:20) (cid:104) ¯ Ψ m Ψ n (cid:105) ( ω, k ) − (cid:104) Ψ m ¯ Ψ n (cid:105) ( ω, k ) (cid:21) m,n coth ω T . (s29)4From here one infers equal time correlation function in x-representation (cid:104) Ψ k ( x, Ψ k (0 , (cid:105) = T δ ( x ).Next, we discuss the non-linear part of the action. In terms of eigenmodes Ψ it reads S int = − i (cid:88) k,l,m γ k,l,m ¯ Ψ Tk Ψ l Ψ m , (s30)where γ k,l,m = T (cid:88) k ,l ,m R − T k,k Γ k ,l ,m R − l ,l R − m ,m . (s31)By varying the action with respect to the field ¯ Ψ k , one derives the equation of motion (cid:16) ∂ t − v k ∂ x − ˜ D k ∂ x (cid:17) Ψ k ( x, t ) − T (cid:88) l,m λ k,l,m ∂ x ( Ψ l ( x, t ) Ψ m ( x, t )) = 0 . (s32)Here λ p,l,m = − iT (cid:88) R p,p H p l ,m ( R − ) l ,l ( R − ) m ,m . (s33)Multiplying this equation by Ψ k (0 ,
0) and averaging over the action, one gets an equation of motion for the correlationfunction (cid:16) ∂ t − v k ∂ x − ˜ D k ∂ x (cid:17) (cid:104) Ψ k ( x, t ) Ψ k (0 , (cid:105) − T (cid:88) l,m λ k,l,m ∂ x (cid:104) Ψ l ( x, t ) Ψ m ( x, t ) Ψ k (0 , (cid:105) = 0 . (s34)Employing the Wick’s theorem, one expresses the triple correlation function in terms of the pair correlation function.Performing the self-consistent approximation, one finds the mode-coupling equation[s9]. III. RESPONSE FUNCTIONS FROM MICROSCOPICS
In this part of SM we derive the response coefficients that appear in Eqs. (7) and (8) of the main text for the 6,4and 3-mode regimes. We also find the eigen-modes Ψ of linearised hydrodynamics in all the regimes, defined in themain text, compute their velocities ( v j )and diffusion matrices ( ˜ D ), and coupling constants ( λ ijk ) needed for Eq.(10)in the main text. A. 6-mode regime
In this section we derive the hydrodynamic model from the kinetics (s2) for the 6-modes regime. We start with thesusceptibility matrix, that connects q and φ variables on the linear level. Multiplying the distribution function by1 , p, (cid:15) p and integrating over momentum, we find that for the right-moving electrons χ = T π π T m u + π T m u + u m − π T m u − π T mu m − π T m u + m u − π T u − π T m u − π T mu π T m u + π T u . (s35)Integrating the distribution function (s2) with current operators j α ( p ) over the momentum, we compute the matrixof currents M . The low- k expansion of M reads M (cid:39) ikT π − π T m u − π T mu + mu − π T m u − π T mu + mu m u π T m u + π T u π T m u + π T u π T . (s36)We next find the eigenmodes of linearised hydrodynamic (Ψ), and compute the rotation matrix R (cid:39) mu / T √ mu π / + (cid:113) π T − √ π T mu − √ uπ / √ π T mu + √ π / (cid:113) π T − √ mu π / √ uπ / − √ π T mu √ π T mu − √ π / − √ mu π / √ uπ / − √ π / . (s37)5The velocities of eigenmodes in this regime are given by u (cid:39) u + (cid:113) πTmu , (s38) u (cid:39) u − (cid:113) πTmu , (s39) u (cid:39) u + 49 π T m u . (s40)For the strictly parabolic spectrum the diffusion matrix reads D (cid:39) D (cid:15) mu u . (s41)In terms of eigenmodes the diffusion matrix ˜ D = R D R − reads˜ D (cid:39) π D (cid:15) − − (cid:113) − (cid:113) − (cid:113) (cid:113) − . (s42)The right- and left-moving heat modes can be expressed in terms of hydrodynamic eigenmodes as (cid:15) R − ¯ wρ R = πT √ u (Ψ + Ψ ) , (cid:15) L − ¯ wρ L = πT √ u (Ψ + Ψ ) . (s43) B. 4-mode regime
In this section we derive the hydrodynamic model from the kinetics for the 4-modes regime. In this case, the set ofzero modes of the collision integral is described by φ T = T − ( µ R , µ L , v, − . (s44)Repeating the steps analogous to the 6-mode regime, we find the susceptibility matrix χ = T π π T m u + π T m u + u m − π T m u − π T mu π T m u + π T m u + u − m − π T m u − π T mu m − m − π T m u + 2 m u − π T u − π T m u − π T mu − π T m u − π T mu π T m u + π T u (s45)and the matrix of currents M = ik T π − π T m u − π T mu + mu − − π T m u − π T mu + mu − π T m u − π T mu + mu − π T m u − π T mu + mu π T m u + π T u π T m u + π T u . (s46)The rotating matrix R (cid:39) √ u π / T mu − mu − u − mu mu u mu − mu − u − mu mu u . (s47)6The velocities in this regime split linearly with temperature, in agreement with a general argument given in Ref. [s10]: u = − u + πT √ mu , (s48) u = u − πT √ mu , (s49) u = − u − πT √ mu , (s50) u = u + πT √ mu . (s51)We next compute the diffusion matrix D (cid:39) D (cid:15) mu mu . (s52)After the rotation into eigen-mode basis ˜ D = R D R − , one finds˜ D (cid:39) π D (cid:15) . (s53)The density of the heat mode in the 4-mode regime is expressed in terms of hydrodynamic eigen modes as (cid:15) − ¯ w ( ρ R + ρ L ) = T (cid:114) π u (Ψ + Ψ + Ψ + Ψ ) . (s54)All the coupling constants in this regime are of the same order λ = iT u / (cid:112) /π − − − − −
11 1 1 1 − − − , (s55) λ = iT u / (cid:112) /π − − −
11 1 3 − − − − − , (s56) λ = iT u / (cid:112) /π − − − − − − − − , (s57) λ = iT u / (cid:112) /π − − − − − − − − . (s58) C. 3- mode regime
In the three mode regime zero modes of the collision integral are given by φ T = T − ( µ, v, − . (s59)7The response function χ = Tπu π T m u + π T m u + 1 0 − π T m u − π T mu − π T m u + m u − π T u − π T mu π T . (s60)The dissipationless part of the current matrix is given by M = ikTπ − π T m u − π T mu + mu − π T m u − π T mu + mu π T m u + π T u π T m u + π T u . (s61)The rotating matrix R (cid:39) − π / T √ mu / √ π √ uT − √ π / T m u / (cid:112) π √ u − √ π m √ u √ πmu / (cid:112) π √ u √ π m √ u √ πmu / . (s62)The velocities of the eigenmodes in 3-mode regime are u = 0 , u (cid:39) − u − π T m u , u (cid:39) u + π T m u . (s63)The heat current (cid:15) − ¯ wρ = (cid:114) π u T Ψ . (s64)The coupling constants λ = i √ π T m √ u − i √ π T m √ u i √ π T m √ u i √ π / T m u / − i √ π T m √ u − i √ π / T m u / , (s65) λ = i √ π / T m u / iπ / T √ m u / − iπ / T √ m u / iπ / T √ m u / iπ / T √ m u / iπ / T √ m u / − iπ / T √ m u / iπ / T √ m u / − iπ / T √ m u / , (s66) λ = − i √ π / T m u / iπ / T √ m u / − iπ / T √ m u / iπ / T √ m u / iπ / T √ m u / − iπ / T √ m u / − iπ / T √ m u / − iπ / T √ m u / − iπ / T √ m u / . (s67)Note that self coupling of the heat mode vanishes λ , , = 0 . (s68) IV. HIERARCHY OF TIME AND LENGTH SCALES
In this section we evaluate the length scales that appear in the problem. We divide them into two groups. The firstone are the scales that appear on the level of Bolztmann equation, such as τ F , L and L U , see Eqs(1), (3) and (4) inthe main text. The second scale is L ∗ and it appear due to renormalization of the bare parameters by fluctuations,see the discussion bellow Eq(12) in the main text.8 A. Scales of Boltzmann equation
In this section we summarise different time and length scales in the problem. The shortest time scale is a scale isdetermined by three-particle collisions. The corresponding matrix element of three fermion collision[s11] is given by W k (cid:48) k (cid:48) k (cid:48) kk k = γl m ∗ u ( k − k ) ( k (cid:48) − k (cid:48) ) δ ( k + k − k (cid:48) − k (cid:48) ) δ ( k − k (cid:48) ) . Here γ = α (1+ α ) and α = − K K and l is a radius of short range interaction. On the level of diagonal approximation,we find the decay rate 1 τ F ( k ) = γl T k m ∗ u , k > Tu , γl T m ∗ u , k < Tu . (s69)reproducing the results of Ref. [s11–14]. For spatial scales longer than uτ F ( T /u ) electrons form a fluid with sixchiral modes. The separation between the chiral sectors is not exact and breaks and scale L ∼ uτ F ( p )( pδp ) where p ∼ T /u, δp ∼ T (cid:15) F u is the momentum transfer between left and right branch. Thus, left and right momentum andenergies are no longer conserved hydrodynamic variable beyond the scale L = uτ F ( p ) (cid:18) m ∗ u T (cid:19) = m ∗ u γl T . (s70)Finally, if one accounts for merger of the chiral branches at the bottom of the energy, one find a time of the equilibrationfor the number of particles between the left and right fermions. Within the bosonic description this process correspondsto the Umklapp scattering, with a length scale [s15–18] L U ∼ uT − / (cid:15) / F e (cid:15)FT . (s71) B. From weak to strong coupling fixed point using MSR formalism
To determine the scale L ∗ we analyse the RG flow and estimate the value of dimensionless coupling constants. Toperform weak coupling RG we focus on the action (s26) and (s30). Ignoring the interaction between different modesthe action in 6-mode and 4-mode regimes can be mapped onto KPZ model[s20]. By defining ψ = − i∂ x h and p = ¯ ψ/u ,one finds the standard action of KPZ model [s6] iS = p (cid:16) ∂ t − u∂ x + ˜ D∂ x (cid:17) h − T ˜ Dp + λT p ( ∂ x h ) . (s72)The RG equation of this model can be written in terms of dimensionless coupling constant g = λ uτ F T ˜ D , (s73) ∂ l g = g − g , (s74)where l = ln( L/L ), and L is an ultraviolet length cutoff. The solution reads g ( l ) = e l g g ( e l − , (s75)where g is a bare value of dimensionless coupling constant. Estimating (s73) for the 4-mode regime, we find g (cid:39) (cid:16) TT BF (cid:17) (cid:28)
1. This means that the fermionic hydrodynamics is in the weak coupling regime at spatial scales ofthe order of L . This implies that computations performed within the linearized hydrodynamics in this regime arejustified.The system enters into strong coupling limit at g ( L ∗ ) ∼
1, i.e. at L ∗ ∼ L /g ∼ L (cid:0) T BF T (cid:1) (cid:29) L .9 V. ASYMPTOTICS OF THE DISTRIBUTION FUNCTIONS
In this section we study the asymptotic form of the pulses in different regimes, analysing the self-consistent mode-coupling equation. The results of this section are used in Eqs.(13), (14),(15) of the main text. We start with the4-mode regime.
A. 4-mode regime
To compute the asymptotic, we first analyse the impact of the interaction between two modes. The ”slow” modepropagating with velocities u k and a ”fast” mode moving a velcity u l in the same direction. To be concrete, let usfocus on the right (fron) tail of the mode k . In the reference frame that moves with velocity u k , the tail of this modecontrolled by the coupling to the mode l is governed by the equation ∂ t f k ( x, t ) (cid:39) λ kll T (cid:90) ∞−∞ dy (cid:90) t dsf k ( x − y, t − s ) ∂ y f l ( y, s ) . (s76)Here we omit the self-coupling and diffusion terms, which play no role far from the maximum. Due to the scaleseparation, this equation can be further simplified as follows: ∂ t f k ( x, t ) (cid:39) λ kll T λ / lll ∂ x (cid:90) t dss / f k ( x − ∆ u kl s, t − s ) . (s77)Here we used the fact that the integration over spatial coordinate is limited to a region much smaller than theseparation between the peaks, yielding (cid:82) dyf l ( y, s ) = T λ / lll s / . The slow mode can be approximated by its tailestimated at the peak of the fast mode. The latter is located at the point y ∼ ∆ u kl s . As distance x is smaller thatthe separation between the pulses s ∆ u kl , one may neglect s ∼ x/ (∆ u kl ) compared with t . Therefore, Eq.(s77) can befurther simplified, yileding ∂ t f k ( x, t ) (cid:39) λ kll λ / lll T ∂ x (cid:90) t dss / f k ( x − ∆ u kl s, t ) . (s78)In the Fourier space, this reads ∂ t f k ( q, t ) = − λ kll λ / lll T q (cid:90) t dss / e − ik ∆ u kl s f k ( q, s ) . (s79)Therefore one can look for a solution of the form f k ( q, t ) = T h ( q γ t ) . (s80)Plugging this anzats into Eq.(s109), one finds h ( z ) (cid:39) exp (cid:32) − λ kll zλ / lll T (∆ u kl ) / (cid:33) . (s81)Fourier transforming back, we get the result for the tail of the mode f k : f k ( x, t ) (cid:39) λ kll tx − ( γ +1) T λ / lll (∆ u kl ) / . (s82)Substituting γ = 5 / f k ( x, t ) ∼ ( mu ) / tx − / , (s83)where x is the distance from the center of the peak. For each of the modes, the tail is on the side directed to theother mode. A similar analysis yields the tail between oppositely moving modes: f k ( x, t ) ∼ ( T u ) / tx − / . (s84)10 B. 3-mode regime
We now apply similar arguments for the 3-mode regime. In this case the tails of the static heat mode ( j = 1) isgoverned by Eq.(s82). Substituting the coupling constant for this regime, we find f ( x, t ) (cid:39) T / m u / tx − / for x (cid:29) (cid:18) T t m u (cid:19) / . (s85)The rear tails of sound modes ( j = 2 ,
3) in this regime are formed due to interaction between sound modes, with theresult f / ( x, t ) ∼ T / m u / tx − / . (s86) VI. THERMAL CONDUCTIVITY
In this section we compute thermal conductivity via Kubo formula in 6,4, and 3-modes regimes. We establish theformal relation between the linear response thermal conductivity and the pulse propagation problem. The results ofthis section are used in Eqs.(18), (19) and (20) of the main text.
A. Kubo formula for thermal conductivity
Here we review the Kubo formula approach for multi-component fluid, focusing in more details on the thermalconductivity. The starting point is that the fluid state assumes a local equilibrium, therefore the response functionto external forces can be presented in the linear response formalism[s21, 22]. For multi-component fluid this refers toequilibration of the corresponding modes. For the brevity of notation, we suppress the chirality indexes, and restorethem when needed. In the presence of the external time-dependent perturbation ˆ V the Hamiltonian of the fluid isgiven by ˆ H ( t ) = ˆ H + ˆ V ( t ) . (s87)The perturbation can be expressed as time and space dependent thermodinamic potentialsˆ V ( t ) = − (cid:90) dx (cid:26) δT ( x, t ) T [ˆ (cid:15) ( x, t ) − µ ˆ ρ ( x, t )] + δµ ( x, t )ˆ ρ ( x, t ) + v ( x, t )ˆ g ( x, t ) (cid:27) . (s88)The expectation value of a generic operator J i at a time t is given as an average with respect to equilibrium densitymatrix: (cid:104) ˆ J i (cid:105) ( t ) = i (cid:126) (cid:90) t −∞ dt (cid:48) Tr (cid:8) ˆ ρ [ ˆ V I ( t (cid:48) ) , ˆ J Ii ( t )] (cid:9) . (s89)We now define the retarded current-current correlation function K ij ( ω, k ) = i (cid:126) (cid:90) ∞−∞ dx (cid:90) ∞ dte − ikx + iωt (cid:104) [ ˆ J i ( x, t ) , ˆ J j (0 , (cid:105) . (s90)By using (s89) one can show that linear-response coefficients L ij can be expressed as L ij ( k ) = 1 − iω (cid:20) K ij ( ω, k ) − K ij ( ω = 0 , k → (cid:21) . (s91)The Kubo framework can be used for computing any linear response coefficients, and in particular thermal conductiv-ity. To do it, one needs to define a thermal current. In a general many-body problem it can be computed by couplingthe system to the gravitational field. For the fluid, this reduces to a much simpler expression[s23]ˆ J T = ˆ J E − ¯ w ˆ J ρ . (s92)11Here J E and J ρ are energy current and particle currents that are determined by energy conservation and particleconservation, ¯ w (cid:39) π T / mu is enthalpy of the fluid per one electron (without the Fermi energy part). The Kuboformula for thermal conductivity reads σ T ( ω, k ) = 1 iωT (cid:90) L dx (cid:90) ∞ dte − ikx + iωt (cid:104) [ ˆ J T ( x, t ) , ˆ J T (0 , (cid:105) . (s93)Using the continuity equation for energy, ∂(cid:15)∂t = − div J E , (s94)and particle number, ∂ρ∂t = − div J ρ , (s95)one can expressed the heat-conductivity as σ T ( ω, k ) = 1 T ωk (cid:104) [ˆ (cid:15) − ¯ w ˆ ρ, ˆ (cid:15) − ¯ w ˆ ρ (cid:105) ret ω,k . (s96)This can be cast in terms of the response function χ ret i,j ; η ( x, t ) = − iθ ( t ) (cid:104) [ˆ q i,η ( x, t ) , ˆ q j,η (0 , (cid:105) . (s97)Specifically, for the six-mode regime σ T ( ω, k ) = iωk T (cid:88) η (cid:20) χ ret33; η ( ω, k ) + ¯ w χ ret11; η ( ω, k ) − wχ ret31; η ( ω, k ) (cid:21) , (s98)and similarly for 4-mode and 3-mode regimes. On the Gaussian level, the correlation function (s98) can be easilycomputed. Indeed, in this case χ ret ( ω, k →
0) = 1 iωT M ( k ) . (s99) B. Analysis of thermal conductivity in different regimes
In this section we employ the Kubo formula for the thermal conductivity, and compute it for all possible regimes.We start with 6-modes case.
1. 6-mode regime
In 6-mode case Eq. (s98) combined with Eq.(s99) yields σ T ( ω, k ) = 1 T (cid:20) L ( k ) + ¯ w L ( k ) − ¯ wL ( k ) − ¯ wL ( k ) (cid:21) . (s100)in agreement with linear response computations done within Boltzmann equation, see Eq. (s11).After computing the response matrix one finds σ T ( ω = 0 , k ) = π T ik + π u l m T . (s101)Here the first term correspond to the ballistic transport and yield the σ T = πT L upon a replacement k → π/L forthe ideal quantum wire of the length L . The second term to the diffusion spreading of the heat.12To discuss effects of renormalization is convenient to connect the thermal conductivity to the problem of pulsepropagation. In the 6-mode regime, the thermal conductivity can be written in terms of Keldysh correlation function f ij as σ T ( ω, k ) = π ω uk (cid:88) j ,j =1 f j ,j ( ω, k ) + (cid:88) j ,j =4 f j ,j ( ω, k ) . (s102)By emploing Eq.(s29) one readily reproduces the d.c. limit the real part of the conductivity Eq.(s101).Re σ ( ω → , k ) = π T u [ ˜ D + ˜ D + ˜ D + ˜ D ] = π u l m T . (s103)Because the renormalization is weak in this regime, this is a good (up to a small correction) estimate of thermalconductivity.
2. 4-mode regime
In this regime the bare thermal conductivity is given by σ T ( ω = 0 , k ) ∼ π T ik + u l T . (s104)The imaginary (ballistic) part is protected by momentum conservation and therefore is uncahnged, compared with6-mode regime. The real part of the thermal conductivity is parametrically bigger. We now cast the Kubo formulain terms of Keldysh correlation functions, σ ( ω, k ) = π ω uk (cid:88) j ,j =1 f j ,j ( ω, k ) . (s105)Computed on the Gaussian level, this yieldsRe σ ( ω = 0 , k ) = π T u (cid:88) j ,j =1 ˜ D j ,j ∼ u l T . (s106)in agreement with Eq.(s104).We now take into account the self-renormalization effects, by employing Eq.(s105) with the pulse shape found fromthe self-consistent equation (8). Both the heads and tails of pulses are modified by interaction, and contribute to thethermal conductivity. The head of each pulse is governed by KPZ function f ( x, t ) (cid:39) T ( λt ) / f KPZ (cid:16) T ( x − ut )( λt ) / (cid:17) . Thisyields the contribution to thermal conductivity that scales with frequency asRe σ ( ω, k = 0) ∼ λ / T u ω − / (cid:39) uT / ω − / . (s107)In addition, the tails of the distribution function(s83) contributeRe σ ( ω, k ) ∼ ( mu ) / k − / . (s108)Substituting k = ω/ ∆ u one gets Re σ ( ω, k = 0) ∼ uT / ω − / . (s109)Thus, we see that the contribution to the thermal conductivity from the head (s107) and the tails (s109) in four moderegime have the same order.13
3. 3-mode regime
The value of the thermal conductivity in the whole 3-mode regime is strongly renormalized by interaction betweenthe modes, so that its bare value has no significance. We thus express the conductivity in terms of pulse correlationfunctions σ ( ω, k ) = π ω uk f ( ω, k ) . (s110)Due to the lack of self interaction there is no anomalous peak broadening of the heat mode in the three mode regime,and the thermal conductivity is determined solely by the tails of the pulse. Substitution of the asymptotics Eq.(s85)into Eq.(s110) yields Re σ ( ω, k ) ∼ T / u / m k − / . (s111)Substituting k = ω/u , we find Re σ ( ω, k = 0) ∼ T / u m ω − / . (s112) [s1] A.V. Rozhkov, ” Density-density propagator for one-dimensional interacting spinless fermions with nonlinear dispersionand calculation of the Coulomb drag resistivity”, Phys. Rev. B , 125109 (2008); ” Class of exactly soluble modelsof one-dimensional spinless fermions and its application to the Tomonaga-Luttinger Hamiltonian with nonlinear disper-sion”, Phys. Rev. B , 245123 (2006); ”Fermionic quasiparticle representation of Tomonaga-Luttinger Hamiltonian”,Eur.Phys.J. , 193 (2005).[s2] R. Samanta, I.V. Protopopov, A.D. Mirlin, D.B. Gutman, ”Thermal Transport in One Dimensional Electronic Fluid”,Phys. Rev. Lett. , 206801 (2019)[s3] I.V. Protopopov, D.B. Gutman, M.Oldenburg, and A.D. Mirlin, ”Dissipationless kinetics of one dimensional interactingfermions”, Phys. Rev. B , 161104 (2014).[s4] I. V. Protopopov, D. B. Gutman, P. Schmitteckert, A. D. Mirlin, ”Dynamics of waves in 1D electron systems: Densityoscillations driven by population inversion”, Phys. Rev. B , 045112, (2013).[s5] E.M. Lifshits and L.P. Pitaevskii Physical Kinetics: Volume 10 (Pergamon Press 1989).[s6] A. Kamenev,
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