Superdiffusive Transport of Energy in One-Dimensional Metals

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Steven by Edited and Germany; Braunschweig, a Bulchandani B. Vir one-dimensional metals in energy of transport Superdiffusive eevdnwiptswt h deto yaia measure- dynamical of and advent models many the in with studied impetus been has new state received ground the in ously 11). (10, and complex be disorder can show disentangling mate- effects to open-system chain although argued spin near-integrability, been on has of performed transport signs thermal be heat where could others of These or flux 9). rials outward (8, laser an solid generate using a to experiments in region of of small type because a this of part study irradiation in We spread state. initial energy ground an the rapid into from region expands heated energy finite when leads perturbations behavior these superdiffusive of liquids to irrelevance Luttinger The real thermalization. responsible but for Lut- are theory, that the bosonic perturbations integrability-breaking free of several contain a with limit is but low-energy liquid dimensions The tinger higher (7). in analo- differences liquid fundamental fermions, Fermi the one-dimensional to interacting gous generic of the state is liquid metallic Luttinger by The aspects response. other linear in conventional described and system thermalizing, the nonintegrable, though even is diffusion, a nonlinear or generates anomalous metals of type one-dimensional realistic in transport be energy to assumed usually diffusion. momentum is by if or system hydrodynamics, conserved, a is conventional such by approach in The either still system. state described is the thermal these of the on state based asymptotic to laws, the state conservation be thermal to few or believed condensed- a ensemble than Gibbs realistic more the most have and not quantum However, do and systems 6). matter (1–4) (5, systems models dimen- localized ensemble integrable spatial many-body one Gibbs include in examples conventional con- sion well-studied local) (sufficiently two the of laws: number Systems to servation infinite scales. an thermalize exist time there and to because length fail long at may behavior dynamical of Q transport anomalous eateto hsc,Uiest fClfri,Bree,C 94720; CA Berkeley, California, of University Physics, of Department h rbe fepnino xiain noargo previ- region a into excitations of expansion of problem The of problem simple a that argue to is work this of point The pet,udrto ospotmlil nvra classes universal multiple support to devel- understood recent to opments, thanks now, are systems many-body uantum | superdiffusion a,1 hitp Karrasch Christoph , c aeil cecsDvso,Lwec eklyNtoa aoaoy ekly A94720 CA Berkeley, Laboratory, National Berkeley Lawrence Division, Sciences Materials | utne liquids Luttinger b n olE Moore E. Joel and , | b ntttf Institut rMteaicePyi,Tcnsh Universit Technische Physik, Mathematische ur ¨ is ulse a 6 2020. 26, May published First Science Open the in Submission. y deposited 1 Direct PNAS been a is have article 1–3 This Figs. for (https://osf.io/scw75/ ).y data Framework Raw deposition: Data the under Published interest.y competing no declare authors The uhrcnrbtos ... .. n ...dsge eerh efre eerh and research, paper. performed the research, wrote designed J.E.M. and C.K., V.B.B., contributions: Author a of state ground the The into 17.) expansion ref. energy liquid; power- Luttinger of well-known a aspect into the special tunneling to electron in leads the laws dimension of variation scaling continuous operator’s the that in (Recall operators Hamiltonian. integrability-breaking scaling irrelevant the of of variation superdiffu- dimensions continuous a the the to in because originates applicable metals, behavior equally sive one-dimensional is a of that class present behavior broad then this of We model methods. density-matrix simplified (DMRG) time-dependent group using can renormalization quantitatively and behavior studied superdiffusive be shows in that law Hamiltonian lattice power a as in diverge finite rather not size. but system are limit coefficients from thermodynamic linear-response the distinct which is in systems work many-body (13–16), momentum-conserving in present the exist to that the known Note that in possibilities. standard described in three superdiffusion problems these from well-studied distinct that simple, scaling is long-time to even lead physics that condensed-matter quantum show here energy, pre- para- results or sented The particles a coefficients. transport either implies linear-response of finite behavior, spreading with Diffusion of diffusive particles. rate covers slower Brownian metrically class of have third also example, A that for models property. integrable interac- ballistic to of lead this kinds dimension some one and in behavior, of tions ballistic Both to fronts. interactions lead wave which cases propagating dis- these in and to limit behavior lead nonlinear fluid to velocities classic lead different the second, whose first, and particles physics: persion, free classic of considering of case classes from the illustrative come Two behaviors (12). gases possible atomic ultracold on ments owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To a,c nmtrassc ssi hisadcro nanotubes. carbon and problem chains spin a as experiments such in transport materials in heat and time-resolved possibilities to relevant standard long-time directly to the lead outside can scaling physics problems condensed-matter response. well-studied quantum simple, linear in even conventional that illustrate by results and aspects Our thermalizing, other nonintegrable, in is diffusion, described sim- system nonlinear the or a anomalous though metals, for of even one-dimensional type down in a breaks transport generates which paradigm energy of this problem that ple show We con- Fourier’s by law. to according either diffusion described by or be hydrodynamics to ventional assumed usually is equilibrium thermal to approach the systems, condensed-matter typical For Significance u rsnainsat iha xlcteapeo local a of example explicit an with starts presentation Our y PNAS NSlicense.y PNAS | ue9 2020 9, June | o.117 vol. tBancwi,38106 Braunschweig, at ¨ | o 23 no. | 12713–12718 electron

PHYSICS realistic Luttinger liquid is that the system is never fully in the (18), using time-dependent, finite-temperature DMRG simula- linear-response regime because of the singular zero-temperature tions (21, 27–30). It was found that for h 6= 0, the AC thermal thermal conductivity. The result is an anomalous diffusion equa- conductivity exhibits an O(h2) broadening of the Drude peak tion with solutions of Barenblatt–Pattle type, which exhibit arising from integrability at h = 0. This broadening yields a finite superdiffusive space–time scaling. DC thermal conductivity σh (T ) for h 6= 0. (Obtaining the value of σh (T ) numerically for low temperatures T . 0.2 appears to The Microscopic Model be beyond the present state of the art.) In the same work, it was For a microscopic realization of universal Luttinger liquid argued that in the gapless phase of the Hamiltonian Eq. 1, the physics that is amenable to numerical simulation, we consider DC charge conductivity should scale with temperature as a spin-1/2 XXZ chain in the presence of a staggered magnetic PN ν(K ) field, with Hamiltonian H = i=1 hi , where σc (T ) ∼ T , T → 0, [7] x x y y z z i z ν(K ) hi = JSi Si+1 + Si Si+1 + ∆Si Si+1 + (−1) hSi [1] at low temperatures, with some universal function depend- ing only on the Luttinger parameter K characterizing the effec- (in the following, we set J = a = ~ = 1, where a denotes the lat- tive Hamiltonian Eq. 2. It was further verified that for several tice length scale). This model was studied in previous work (18, values of ∆ and h in the gapless phase, the numerical values 19), and for ∆ 6= 0, the staggered field can be verified to break of the Luttinger parameter K (∆, h) and the scaling exponent integrability of the spin-1/2 XXZ chain by a level-statistics anal- ν(K ) obtained from DMRG are consistent with the analytical ysis. Meanwhile, the effect of the staggered field perturbation bosonization prediction (31, 32) on the low-energy physics of the system can be determined via bosonization. For infinitesimal h, the bosonized Hamiltonian can ν(K ) = 3 − 2K . [8] be written as This is an instance of a very general scenario whereby perturb- Z L Z L √ u 2 2 ing a Luttinger liquid with an irrelevant vertex operator leads H = dx Π + (∂x φ) + ch dx cos 2 πK φ 2 0 0 to a nontrivial power-law dependence on temperature in the low-T charge conductivity, which scales continuously with the + HUmklapp + Hband curvature + Hhigher terms in h , [2] Luttinger parameter K . This result follows by nonperturbative where L = Na and the momentum and phase degrees of freedom resummation of the charge–current autocorrelation function, satisfy canonical commutation relations [Π(x), φ(y)] = iδ(x − combined with a low-order Taylor expansion of the perturbation y). Here, the Luttinger parameter K is given by the Bethe ansatz self-energy in frequency (33–35). Although such nonperturba- result 2K cos−1(−∆) = π, and various other coupling constants tive resummation techniques are not directly applicable to the can be determined exactly (20). From the scaling dimension thermal conductivity σh (T ), it is natural to expect that the same phenomenon occurs, with [h] = 2 − K , it follows that the√ staggered field is relevant and opens a gap for K < 2 or − 2/2 < ∆ ≤ 1. However, for K > √ λ(K ) 2, or −1 < ∆ < − 2/2, this perturbation is irrelevant and the σh (T ) ∼ T , T → 0, [9] model remains in a gapless Luttinger liquid phase. In this paper, we focus on the latter regime. for some exponent λ(K ) < 0 that depends on K and the In what follows, it is to be understood that the effective scaling dimension of the irrelevant perturbation. For exam- Luttinger parameter in the gapless regime, K (∆, h), varies con- ple, the assumption that σc (T ) and σh (T ) are related by tinuously with ∆ and h and, in particular, will differ from the Wiedemann–Franz scaling σh (T ) ∼ T σc (T ) would imply that Bethe ansatz prediction for h 6= 0. For this reason, the values of λ(K ) = 1 + ν(K ). Indeed, this holds for the tunneling electri- K that we quote for the specific values of ∆ and h considered cal and thermal conductances through a single impurity in a below are obtained from an independent ground-state DMRG Luttinger liquid (36), although the Lorenz number (the coeffi- calculation (21, 22). cient of the Wiedemann–Franz ratio) is modified from its Fermi liquid value. Low-Temperature Hydrodynamics In general, one should not assume that λ(K ) and ν(K ) are We now consider linear-response transport in the system Eq. 1. always so simply related (at least it is not clear to us that this must The direct current (DC) charge and heat conductivities may be be the case for all integrability-breaking perturbations), but even λ(K ) defined by the Kubo formulae (23–26) without a specific value for , the ansatz Eq. 9 has striking consequences for the problem, mentioned in the Introduction, 1 Z tM of expansion of a small high-temperature region into a large low- λ σc = lim lim Re dt hJc (t)Jc (0)i, [3] temperature background. To see this, let us write σh (T ) = CT , tM →∞ N →∞ NT 0 where C is a nonuniversal, temperature-independent prefactor. Z tM 1 Then, in the linear-response regime and to leading order in tem- σh = lim lim Re dt hJh (t)Jh (0)i, [4] tM →∞ N →∞ N 0 perature, Eq. 9 implies that temperature gradients give rise to λ heat currents according to jQ (x) ∼ −CT ∂x T (x). We now con- PN where the respective current operators J = i=1 ji are given by sider states of the model Eq. 2 that are in local thermodynamic the continuity equations equilibrium, in the sense that they are well described by smoothly varying local temperature distribution, T (x, t). For flows in such N X states that are driven purely by temperature gradients, the heat ∂t hi + jh,i+1 − jh,i = 0 =⇒ Jh = i [hi−1, hi ], [5] current coincides with the energy current, and we can write down i=2 a hydrodynamic equation N z X z ∂t Si + jc,i+1 − jc,i = 0 =⇒ Jc = i [hi−1, Si ]. [6] λ ∂t ρE = ∂x CT ∂x T [10] i=2

A numerical study of the heat and charge conductivities σh (T ) for small perturbations ρE (x, t) of energy density relative to the and σc (T ) in the model Eq. 1 was performed in previous work ground state energy density, which is expected to hold to leading

12714 | www.pnas.org/cgi/doi/10.1073/pnas.1916213117 Bulchandani et al. 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PHYSICS robust to the details of the localized initial wave packet, as shown in Fig. 3. This suggests that a more refined model than Eq. 11 is required to capture the precise shape of the superdiffusing wave packet. It is expected that using other perturbations to break integrability, or considering multicomponent Luttinger liquids, will lead to different scaling functions and exponents, but the analytical model above suggests that superdiffusive or ballistic behavior should be expected as long as the scaling of linear- response thermal conductivity with temperature is larger than linear in T as T → 0. Our results are therefore consistent with a generic scenario of superdiffusive low-temperature heat transport in one- dimensional metals. A natural question is whether the same phenomenon could arise in spatial dimension d > 1. We claim that this phenomenon can occur in higher dimensions, but it is no longer as generic. In particular, physical systems in d > 1 that are close to a noninteracting fixed point will exhibit superdiffusion of heat as T → 0, for the same reason as the Luttinger liquid; however, in higher dimensions, such systems represent the exception when there is emergent conformal invariance rather Decrease of the effective superdiffusion exponent as the strength Fig. 2. than the rule, since the conductivity of an interacting, confor- of the integrability-breaking staggered field h is increased. The initial wave mally invariant quantum critical point above one dimension is packet is given by Eq. 16 with βJ = 12, βMJ = 8, l = 2, the model anisotropy is set to ∆ = −0.99, and only h is varied. Effective exponents are com- generally finite, rather than divergent as in one dimension. puted from logarithmic time derivatives of absolute moments n = 2, 3, 4. If particle-hole symmetry is abandoned, the clean Fermi liq- (Inset) Time evolution of a higher-temperature wave packet with βJ = 1, uid with interactions on a lattice and a nonzero Fermi surface, βMJ = 0.2, l = 2, at anisotropy ∆ = −0.99, and staggered field h = 0.49. which is not conformally invariant, is an example of how there Absolute moments n = 2, 3, 4, 5 demonstrate near-diffusive exponent α ≈ can be a diverging conductivity as T → 0 above one dimension. 0.58 (dashed line). Indeed, the conductivities diverge fast enough that an analysis of the expansion of a lump of charge and energy into the vacuum in terms of superdiffusion fails to be self-consistent. This could indi- wave-packet spreading will transition from superdiffusive to dif- cate that the ultimate behavior is ballistic, but a different method fusive behavior after some characteristic time scale tD (T ) ∼ is needed for a reliable answer. T λ−1, that diverges faster than T −1 as T → 0. The temperature dependence of this time scale follows by linearizing the nonlinear diffusion model Eq. 11 about a constant bulk temperature. To corroborate this picture, we have checked numerically that by increasing the bulk temperature T , the time scale tD (T ) can be brought down until the effective exponent begins to decrease t = 0 toward α = 0.5 on the numerically accessible time scale. An example for this is shown in Fig. 2, Inset. (One expects that the same holds true for sufficiently shallow wave packets, but ver- ifying this is beyond the reach of our numerics.) For the low bulk temperature βJ = 12 considered in Fig. 1 and the main plot of Fig. 2, our results indicate that the numerically accessible time scale (t ∼ 50) is in a regime t tD (T ) during which the dynamics is superdiffusive. That this dynamics represents genuine anomalous diffusion, rather than a generic transient en route to diffusion, is demonstrated by the numerical observa- tion that effective exponents obtained from different moments of the wave packet converge to the same superdiffusive value, as in Eq. 14. We have additionally checked that in the limit of t = 30 bulk temperature T = 0, for which we expect tD → ∞, superdiffusion is observed on accessible time scales. This was simulated by initializing the system in the ground state of the Hamiltonian H 0 = H + δH , with H given by Eq. 1 and δH a localized inho- mogeneity near x = 0, before time-evolving numerically under H using pure-state time-dependent DMRG (21, 27). Thus, the numerical collapse to a single exponent depicted in Figs. 1 and 2 indicates that our simple hydrodynamic model for propagation of heat in weakly perturbed Luttinger liquids, Eq. 11, is at least qualitatively correct, since it predicts that spreading of localized initial wave packets at low temperature should Fig. 3. Comparison of long-time shape of two different, localized initial be controlled a single superdiffusive exponent, α. On the other profiles with the same total energy, in a model with ∆ = −0.85 and h = 0.2. hand, the splitting of the wave packet into a doubly-peaked struc- The Gaussian initial profile (red dash) is as in Fig. 1. The quartic initial pro- 4 2 ture, as depicted in Fig. 1, is markedly different from the shape file (blue dash) has the form β(x) ∝ e−cx +dx , with c and d chosen to yield of the Barenblatt–Pattle fundamental solution to the fast diffu- approximately the same total energy as the Gaussian profile. Qualitatively sion equation (37), which exhibits a single maximum for all time. different initial profiles (Top) lead to a similar scaling form for the wave Moreover, the doubly-peaked structure appears to be somewhat packets at long times (Bottom).

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Fanton, T. J. Kino, in S. phonons G. ballistic Wu, with D. drops X. electron-hole of 9. Interaction Dynes, C. R. Hensel, C. J. 8. generic Giamarchi, T. for 7. mechanism its and Schollw U. Thermalization Barthel, T. Olshanii, 6. M. dynamical Dunjko, a V. chains. Rigol, as spin M. quantum dimension for 5. one localization many-body in On Imbrie, localization Z. J. Many-body many-body- 4. fully Altman, of E. Phenomenology Vosk, Oganesyan, R. V. 3. Nandkishore, R. Huse, A. D. 2. Papi Z. Serbyn, M. 1. nte ieto o uuewr st xedtecurrent the extend to is work future for direction Another prox- of importance the concerns question interesting An the of calculation direct a is work future for goal desirable One materials. waves. density charge (1993). and superconductors Tc wind. phonon The pulses: heat systems. systems. quantum isolated (2016). 998–1048 point. fixed group renormalization systems. localized states. localized many-body hs e.Lett. Rev. Phys. hs e.B Rev. Phys. unu hsc nOeDimension One in Physics Quantum hs e.B Rev. Phys. ,D .Aai,Lclcnevto asadtesrcueo the of structure the and laws conservation Local Abanin, A. D. c, ´ c,Dpaigadtesed tt nqatmmany-particle quantum in state steady the and Dephasing ock, ¨ 043(2005). 104423 72, 061(2008). 100601 100, Nature hs e.Lett. Rev. Phys. 722(2014). 174202 90, hs e.Lett. Rev. Phys. 5–5 (2008). 854–858 452, hs e.Lett. Rev. Phys. σ h (T 221(2013). 127201 111, ) 6–7 (1977). 969–972 39, ∼ Ofr nvriyPes 2003). Press, University (Oxford T steband-curvature the is 2, e.Si Instrum. Sci. Rev. 624(2013). 067204 110, λ htdvre tlow at diverges that , .Sa.Phys. Stat. J. 3321–3327 64, σ h (T 163, ). 9 .J edz-rns .R lr,D ash oxsec feeg ifso n local and diffusion energy of Coexistence Jaksch, D. Clark, in R. conductivities S. thermal Mendoza-Arenas, J. and trans- J. electrical of 19. Scaling Thermal Moore, E. J. Gutman, Karrasch, liquid. C. Luttinger Huang, Y. one-channel B. D. a 18. in Transport Mirlin, Fisher, M. one- Kane, D. A. C. degenerate 17. the Protopopov, of V. I. conductivity Samanta, Thermal R. Ristivojevic, 16. Z. Matveev, one-dimensional A. K. in functions. deviation 15. large conduction from coefficients Transport heat Limmer, T. D. Gao, Anomalous Y. C. Ramaswamy, 14. S. Narayan, gases. ultracold O. with physics 13. Many-body Zwerger, integrability. W. of Dalibard, Relevance J. chains: Bloch, spin I. quantum 12. in transport Heat Berciu, M. Wu, J. 11. matrix width acteristic (β bulk by parameterized inverse-temperature Gaussian a by specified is cases all profile, in state initial couplings 1–3, The and Figs. 3, and For 1 28–30. Figs. in refs. of method set the we following temperatures, finite at ueia eut r bandfo MGsmltos(1 7 fthe of 27) (21, simulations DMRG from obtained model microscopic are results Numerical of Methods and Materials richness be to full well- remains the systems the that quantum to explored. low-dimensional indicates perturbations in liquid generic transport Luttinger from heat anomalous arise many- studied that nonintegrable the fact can as near The transport (55). well systems transition quantum (47, as localization disordered body (49–53) systems and in integrable models one-dimensional down, (54) quantum break classic to can from understanding 48) transport ranging emerging diffusive dichotomy contexts and an usual the ballistic with systems, between physical consistent low-dimensional for are that results ical for possibility intriguing an heat. suggests of (46) diffusion ultrafast regime this in energy where a upre yteU eateto nry fc fSine Basic Science, Investigatorship. Simons of a Ultrafast Office and the (KC2203); Energy, within Program J.E.M. of DE-AC02-05-CH11231 Science Agency. Materials Contract Department Projects under US Sciences, Research the V.B.B.Energy Advanced by Defense KA3360/2-2. Sys- supported the Quantum was Grant Nonequilibrium of and Program Program Driven tems the Noether from support Emmy acknowledges through meinschaft Science ACKNOWLEDGMENTS. Open the in deposited been have 1–3 (https://osf.io/scw75/). Figs. Framework for data Raw paper. this plot. Availability. corresponding Data the of scale the step on 1% Trotter most a at use is We quantities 29. ref. of in size proposed scheme time DMRG in time-dependent evolved then is matrix density uhrfieet fteter owtsadn,ornumer- our notwithstanding, theory the of refinements Such hraiaini nonequilibrium in thermalization chain. integrable almost an fluids. (1992). 1220–1223 electronic one-dimensional (2019). in port gas. fermi dimensional (2017). 571 19, systems. conserving momentum Phys. B Rev. Phys. e.E Rev. ∆t h J 8–6 (2008). 885–964 80, j = 419(2015). 042129 91, eoe the denotes = ,take 1, H 0.2/J 146(2011). 214416 83, = l h eutn rfiei sdt en h nta density initial the define to used is profile resulting The . J h icre egti hsnsc htteerro all of error the that such chosen is weight discarded the ; l aawr eeae yteagrtmdsrbdin described algorithm the by generated were data All i N =−N N X /2−1 = ρ(0) 0 ie n osdrcouplings consider and sites 200 hs e.B Rev. Phys. j /2 hsmadi h aitna Eq. Hamiltonian the in summand th PNAS β ..i upre yteDush Forschungsge- Deutsche the by supported is C.K. n eta (β central and ) (x S i x = hs e.B Rev. Phys. ) S = i x +1 tr = ∆ | β hs e.Lett. Rev. Phys. ( + e − xxz ue9 2020 9, June 548(2019). 155428 99, − e S − (β −0.99, i y P pncan ihitgaiiybreaking. integrability with chains spin S P j N 116(2013). 115126 88, − i y =−N +1 /2−1 ρ(t j N =−N /2−1 β M ∆S + M ) nes eprtrsadachar- a and temperatures inverse ) /2 h = )e /2 061(2002). 200601 89, ,0.49} 0 .2, 0.1, {0.05, ∈ β −(x e β | (j hs e.Lett. Rev. Phys. −iHt i z )h (j S /l )h o.117 vol. j i z +1 ) j 2 ρ(0)e ) , + , (−1) iHt = ∆ | codn othe to according , i hs e.Lett. Rev. Phys. hS o 23 no. −0.85, hsinitial This 15. i z 206801 122, e.Mod. Rev. nFg 2. Fig. in | h Entropy 12717 = Phys. [15] [16] [17] 0.2 68,

PHYSICS 20. S. Lukyanov, Low energy effective Hamiltonian for the XXZ spin chain. Nucl. Phys. B 38. A. Rosch, N. Andrei, Conductivity of a clean one-dimensional wire. Phys. Rev. Lett. 85, 522, 533–549 (1998). 1092–1095 (2000). 21. S. R. White, Density-matrix formulation for quantum renormalization groups. Phys. 39. P. Jung, A. Rosch, Lower bounds for the conductivities of correlated quantum systems. Rev. Lett. 69, 2863–2866 (1992). Phys. Rev. B 75, 245104 (2007). 22. C. Karrasch, J. E. Moore, Luttinger liquid physics from the infinite-system density 40. O. A. Castro-Alvaredo, B. Doyon, T. Yoshimura, Emergent hydrodynamics in inte- matrix renormalization group. Phys. Rev. B 86, 155156 (2012). grable quantum systems out of equilibrium. Phys. Rev. X 6, 041065 (2016). 23. R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and 41. B. Bertini, M. Collura, J. De Nardis, M. Fagotti, Transport in out-of-equilibrium simple applications to magnetic and conduction problems. J. Phys. Soc. Jpn. 12, 570– XXZ chains: Exact profiles of charges and currents. Phys. Rev. Lett. 117, 207201 586 (1957). (2016). 24. R. Kubo, M. Yokota, S. Nakajima, Statistical-mechanical theory of irreversible 42. V. B. Bulchandani, R. Vasseur, C. Karrasch, J. E. Moore, Solvable hydrodynamics of processes. II. Response to thermal disturbance. J. Phys. Soc. Jpn. 12, 1203–1211 (1957). quantum integrable systems. Phys. Rev. Lett. 119, 220604 (2017). 25. J. M. Luttinger, Theory of thermal transport coefficients. Phys. Rev. 135, A1505–A1514 43. A. J. Friedman, S. Gopalakrishnan, R. Vasseur, Diffusive hydrodynamics from (1964). integrability breaking. Phys. Rev. B 101, 180302(R) (2020). 26. A. Kapustin, L. Spodyneiko, Absence of energy currents in an equilibrium state and 44. R. G. Pereira et al., Dynamical structure factor at small q for the XXZ spin-1/2 chain. J. chiral anomalies. Phys. Rev. Lett. 123, 060601 (2019). Stat. Mech. Theor. Exp. 2007, P08022 (2007). 27. U. Schollwock,¨ The density-matrix renormalization group in the age of matrix product 45. H. Schulz, Correlated fermions in one dimension. Int. J. Mod. Phys. B 05, 57–74 (1991). states. Ann. Phys. 326, 96–192 (2011). 46. K. A. Matveev, A. Furusaki, Decay of fermionic quasiparticles in one-dimensional 28. S. R. White, A. Feiguin, Real-time evolution using the density matrix renormalization quantum liquids. Phys. Rev. Lett. 111, 256401 (2013). group. Phys. Rev. Lett. 93, 076401 (2004). 47. H. van Beijeren, Exact results for anomalous transport in one-dimensional Hamilto- 29. C. Karrasch, J. Bardarson, J. Moore, Finite-temperature dynamical density matrix nian systems. Phys. Rev. Lett. 108, 180601 (2012). renormalization group and the drude weight of Spin-1/2 chains. Phys. Rev. Lett. 108, 48. H. Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. 227206 (2012). 154, 1191–1227 (2014). 30. D. Kennes, C. Karrasch, Extending the range of real time density matrix renormaliza- 49. M. Znidariˇ c,ˇ Spin transport in a one-dimensional anisotropic Heisenberg model. Phys. tion group simulations. Comput. Phys. Commun. 200, 37–43 (2016). Rev. Lett. 106, 220601 (2011). 31. A. Luther, I. Peschel, Fluctuation conductivity and lattice stability in one dimension. 50. M. Ljubotina, M. Znidari,ˇ T. Prosen, Spin diffusion from an inhomogeneous quench in Phys. Rev. Lett. 32, 992–995 (1974). an integrable system. Nat. Commun. 8, 16117 (2017). 32. J. Sirker, R. G. Pereira, I. Affleck, Conservation laws, integrability, and transport in 51. E. Ilievski, J. De Nardis, M. Medenjak, T. Prosen, Superdiffusion in one-dimensional one-dimensional quantum systems. Phys. Rev. B 83, 035115 (2011). quantum lattice models. Phys. Rev. Lett. 121, 230602 (2018). 33. A. Luther, I. Peschel, Single-particle states, Kohn anomaly, and pairing fluctuations in 52. J. De Nardis, D. Bernard, B. Doyon, Hydrodynamic diffusion in integrable systems. one dimension. Phys. Rev. B 9, 2911–2919 (1974). Phys. Rev. Lett. 121, 160603 (2018). 34. H. J. Schulz, Phase diagrams and correlation exponents for quantum spin chains of 53. S. Gopalakrishnan, R. Vasseur, Kinetic theory of spin diffusion and superdiffusion in arbitrary spin quantum number. Phys. Rev. B 34, 6372–6385 (1986). XXZ spin chains. Phys. Rev. Lett. 122, 127202 (2019). 1 35. M. Oshikawa, I. Affleck, Electron spin resonance in s = 2 antiferromagnetic chains. 54. J. D. Nardis, M. Medenjak, C. Karrasch, E. Ilievski, Anomalous spin diffusion in one- Phys. Rev. B 65, 134410 (2002). dimensional antiferromagnets. Phys. Rev. Lett. 123, 186601 (2019). 36. C. L. Kane, M. P. A. Fisher, Thermal transport in a Luttinger liquid. Phys. Rev. Lett. 76, 55. R. Vasseur, J. E. Moore, Nonequilibrium quantum dynamics and transport: From 3192–3195 (1996). integrability to many-body localization. J. Stat. Mech. Theor. Exp. 2016, 064010 37. J. L. Vazquez, The Porous Medium Equation (Oxford University Press, 2006). (2016).

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