Transport in quasi one-dimensional spin-1/2 systems
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n EPJ manuscript No. (will be inserted by the editor)
Transport in quasi one-dimensional spin-1/2 systems
F. Heidrich-Meisner , a , A. Honecker , and W. Brenig Materials Sciences and Technology Division, Oak Ridge National Laboratory, Tennessee 37831, andDepartment of Physics, University of Tennessee, Knoxville, Tennessee 37996, USA Institut f¨ur Theoretische Physik, Universit¨at G¨ottingen, 37077 G¨ottingen, Germany Technische Universit¨at Braunschweig, Institut f¨ur Theoretische Physik, Mendelssohnstrasse 3, 38106Braunschweig, Germany
Abstract.
We present numerical results for the spin and thermal conductivityof one-dimensional (1D) quantum spin systems. We contrast the properties ofintegrable models such as the spin-1/2
XXZ chain against nonintegrable onessuch as frustrated and dimerized chains. The thermal conductivity of the
XXZ chain is ballistic at finite temperatures, while in the nonintegrable models, thisquantity is argued to vanish. For the case of frustrated and dimerized chains, wediscuss the frequency dependence of the transport coefficients. Finally, we givean overview over related theoretical work on intrinsic and extrinsic scatteringmechanisms of quasi-1D spin systems.
Quantum magnetism in 1D is a successful example for a fruitful interplay between theory andexperiment. On the one hand, many bulk materials exist that almost perfectly realize 1D spinmodels (see [1,2] for a review) and on the other hand, powerful theoretical methods such asbosonization [3], the Bethe ansatz [4], or the density-matrix renormalization group method[5] are available. Often, excellent agreement between theoretical predictions and experimentshas been found as far as ground-state properties, excitation spectra, thermodynamic or opti-cal properties are concerned (see [1,6]). The understanding of transport properties is of greatimportance for the interpretation of transport or NMR measurements (see, e.g., Refs. [7]). Sub-stantial progress has been made in the past years, but the understanding is still incomplete,especially for systems involving many coupled degrees of freedom such as spins, orbitals, andphonons. Theoretically, the topic of transport in 1D quantum magnets is challenging. First,several experiments demand for a more complete theoretical picture as will be outlined below,and second, transport theory often requires the computation of non-trivial correlation functions[8]. Third, transport is closely related to relaxation and non-equilibrium phenomena and thusconnects to the rapidly evolving field of non-equilibrium physics of strongly correlated electronsystems.In particular, the discovery of the colossal magnetic heat transport in spin ladder materialssuch as (Sr,Ca,La) Cu O , where the magnetic contribution to the total thermal conductivity κ exceeds the phonon part substantially [9,10,11], has sparked interest in transport properties ofquasi-1D spin models. Often, a magnetic mean-free path is defined within a Boltzmann type ofdescription [9,10] used to analyze the experimental data, which in the case of spin ladders can beof the order of several hundred lattice constants [10]. This observation – originally suggested toreflect ballistic transport properties of pure spin ladders [12] – is not yet completely understood. a e-mail: [email protected] Will be inserted by the editor
Several other spin-1/2 chain materials (see, e.g., [13,14]) and 2D cuprate antiferromagnets[15,16] possess similar thermal transport properties, although typically the values measured forthe magnetic contribution are much smaller. A very interesting aspect is the strong magneticfield dependence observed in some 2D [15,17] and 1D materials [18,19]. Not all materials thatexhibit a strong dependence of κ on magnetic field are actually believed to have a significantcontribution to κ from magnetic excitations. Nevertheless, the possibility of tuning the thermalcurrent through magnetic fields is appealing and may even allow to design functional devicessuch as spin valves [15]. We refer the reader to a recent review [20] and the article by C. Hessin this volume for more details on the experimental developments.Much theoretical work has focused on intrinsic transport properties of spin systems, address-ing intriguing questions such as the different transport properties of integrable as compared tononintegrable ones. While in the remainder of this article, we restrict the discussion to theapplication of linear response theory – i.e., Kubo formulae – to spin and thermal transport ofquasi-1D spin-1/2 systems as derived in Refs. [8,21], we note that alternative approaches suchas master-equation techniques incorporating a modeling of heat baths have been pursued forquantum systems [22,23,24]. Moreover, while widely used, the derivation of Kubo formulae forheat transport may be questioned, as strictly speaking, no analogue to the voltage or magne-tization gradients driving electrical and spin currents exists in the case of thermal transport.We refer the reader to recent work on this issue [24,25,26,27]. For brevity, we also concen-trate on spin-1/2 systems and refer to the literature for more details on Haldane systems [28].Note, though, that due to similar low-energy properties [29], the transport behavior of gappedquantum systems such as spin ladders and spin-1 chains can be expected to be generic at lowtemperatures. Analogous questions, i.e., the properties of integrable vs nonintegrable systems,the validity of Fourier’s law, and the modeling of heat baths are timely subjects in the studyof transport of classical systems (see Ref. [30] for a review).In linear response theory, ballistic transport is defined by the existence of a finite Drudeweight D [31,32], which is the zero-frequency contribution to the real part of the conductivity:Re κ [ σ ]( ω ) = D th[s] δ ( ω ) + κ [ σ ] reg ( ω ) , (1)where κ denotes the thermal and σ the spin conductivity. δ ( ω ) is a δ -function and κ [ σ ] reg ( ω )is assumed to be regular at ω = 0. Generally, the transport coefficients are computed fromcurrent-current correlation functions: κ [ σ ]( ω ) = − β r N Z ∞ dt e i ( ω + i + ) t Z β dτ h j th[s] j th[s] ( t + iτ ) i . (2)Here and in all succeeding equations, r = 0 for spin transport (labeled by ’s’) and r = 1 forthermal transport (labeled by ’th’). β = 1 /T is the inverse temperature and h . i denotes thethermodynamic expectation value. A finite Drude weight implies a divergent dc conductivity. If D vanishes, then either a finite dc conductivity σ dc = lim ω → σ reg ( ω ) can result, or, if σ reg ( ω )exhibits an anomalous frequency dependence for ω → σ dc may still diverge [33]. Note thathere, we mainly consider finite temperatures, while the Drude weight was original introducedby Kohn to characterize a metal at T = 0 [31,32].Trivially, if the respective current operator commutes with the Hamiltonian, the Drudeweight is finite at any temperature. It has long been known that the energy current operatorof the spin-1/2 XXZ chain is a conserved quantity [34], but only later, a deeper connectionbetween the existence of finite Drude weights at finite temperatures and the integrability of amodel system has been made [35].As a main objective of this paper, we wish to summarize recent theoretical progress, concen-trating on one-dimensional systems and their intrinsic spin and heat transport properties (seealso Refs. [33,36] for recent reviews). We will contrast the properties of integrable systems suchas the spin-1/2
XXZ chain discussed in Sec. 2 against nonintegrable ones. As an example forthe latter class of systems, we present numerical results for the spin and thermal conductivityof the frustrated and dimerized spin-1/2 chain in Sec. 3. With respect to the experimental find-ings, obviously, both intrinsic as well as extrinsic scattering processes are of relevance. Recenttheoretical results on extrinsic scattering channels are summarized in Sec. 4. ill be inserted by the editor 3
XXZ chain
We now turn to the nearest-neighbor spin-1/2
XXZ chain. The Hamiltonian is: H XXZ = X l h l = J X l (cid:20)
12 ( S + l S − l +1 + h.c. ) + ∆S zl S zl +1 (cid:21) . (3)We set J = 1 in the following and periodic boundary conditions are imposed throughout thiswork. The current operators corresponding to the local energy density d l = h l defined in Eq. (3)and local spin density d l = S zl are obtained from the equations of continuity: j th[s] ,l +1 − j th[s] ,l = − i [ H, d th[s] , l ] ⇒ j th[s] = i N X l =1 [ h l − , d l ] . (4)It turns out that the energy current of the spin-1/2 XXZ chain is a nontrivial conservedquantity of this integrable model [35]. Hence in the case of thermal transport, Re κ ( ω ) = D th δ ( ω ) for any exchange anisotropy ∆ of this model. Although the spin current is not ingeneral conserved in the case of the spin-1/2 XXZ chain, it has nevertheless been conjecturedthat D s should be finite at T >
As for the Drude weights of the spin-1/2
XXZ chain, the following picture has emerged: thethermal transport is ballistic for any exchange anisotropy and at all non-zero temperatures. Itsdependence on T and ∆ has been studied by means of Bethe-ansatz (BA) techniques [41,42],exact diagonalization (ED) [12,43,44], and with mean-field theory [43,45]. An example is shownin Fig. 1(a), where we display ED data for the thermal Drude weight of the XXZ chain at ∆ = 1 vs temperature [43], in comparison with BA results from Ref. [41]. Note that numerically,the Drude weight can be computed from [35] D th[s] ( T ) = πβ r +1 Z N X m,nEm = En e − E n /T |h m | j th[s] | n i| . (5)Here, | n i and E n are eigenstates and -energies of H , respectively, and Z = P n e − E n /T denotesthe partition function. Using system sizes as large as N = 20, the ED agrees with BA down totemperatures of T /J ∼ .
25, which can be improved by employing extrapolation methods [43].Spin transport in the
XXZ chain is a more involved problem as [
H, j s ] = 0, and has been theobjective of many studies [35,44,45,38,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61]. Whilethe spin Drude weight at T = 0 is known exactly [62], for the massless regime | ∆ | <
1, whereagreement exists that D s ( T > >
0, several BA calculations arrive at contradicting resultsfor the temperature dependence [49,57,63]. The same holds for the question whether D s ( T > ∆ = 1. We refer the reader to Refs. [36,57] for adiscussion of this issue and the conceptual problems that BA approaches face. Recent numericalstudies are consistent with D s ( T > > D s ( T > > C s = lim T →∞ [ T · D s ] in Fig. 1(c). Withinnumerical precision and under the assumption that C s ∝ /N does not change at very large N , the extrapolation results in finite values for | ∆ | ≤
1. We refer to Ref. [44] for a detaileddiscussion of the finite-size scaling and to Refs. [44,50,56] for recent work on the massive,antiferromagnetic regime ∆ >
1. Numerical results for the ferromagnetic phase ∆ ≤ − Trivial conserved quantities are, e.g., the total energy. Will be inserted by the editor
T/J D t h ( h = , T ) / J ED, N=8,...,20ED: N=7,...,19BA (a) ED vs BA, ∆ =1 T/J D s ( h = , T ) / J (b) ∆ =1, exact diagonalization N=9,11,17N=8,10,14,20 C s ( N ) / J ∆ =0 ∆ =0.5 ∆ =0.6 ∆ =1 ∆ =1.5(c) Fig. 1.
Spin-1/2
XXZ chain. (a): Thermal Drude weight, ED (dashed lines: even N , dot-dashedlines: odd N ; see [43]) vs BA results (solid line, [41]) at ∆ = 1. (b): Spin Drude weight as a functionof temperature and for several system sizes at ∆ = 1. (c) Finite-size scaling at high temperatures: C s = lim T →∞ [ T · D s ] for ∆ = 0 , . , . , , . N = 20 at ∆ = 1). In the presence of magnetic fields H Z = − h P i S zi , the energy current j th and the spin current j s couple since particle-hole symmetry is broken and hence h j th j s i 6 = 0 [35]. This gives rise tomagnetothermal effects, similar to the Seebeck effect of conduction electrons [8]. Transport inthe
XXZ chain in the presence of magnetic fields has been studied theoretically in [65,66,67,68],focusing on the magnetothermal effect and the thermomagnetic power. To describe transport,now a 2 × (cid:18) J J (cid:19) = (cid:18) L L L L (cid:19) (cid:18) ∇ h −∇ T (cid:19) , (6)where ∇ h and ∇ T are the gradient in field and temperature, respectively. The conservation of j th is sufficient to show that all four Drude weights D ij corresponding to the transport coeffi-cients L ij are finite at all temperatures [35]. Then, assuming the condition of zero magnetizationcurrent flow, the thermal Drude weight K th is obtained as K th ( h, T ) = D ( h, T ) − D ( h, T ) T D ( h, T ) , (7)where now the magnetothermal correction D ( h, T ) / [ T D ( h, T )] contributes as well. Some ofthe main results of our work Ref. [66] are (i) a reduction of K th ( h, T ) due to the magnetothermalcorrection and (ii) expressions for the leading contributions to K th ( h, T ) at low temperatures.The latter has been obtained from mean-field theory and bosonization. In the gapless phase ofthe XXZ chain (see, e.g., [69] for details on the phase diagram), K th ( h, T ) ∝ T , but at thequantum critical line separating the gapless from the ferromagnetic regime, we find K th ( h, T ) = AT / with A independent of the exchange anisotropy ∆ . Note that K th ( h, T ) has not beencalculated yet by means of BA since D s ( h, T ) escapes an analytical treatment [57,67]. Mean-field theory, as outlined in Refs. [43,44,66], proves useful as it provides a quantitatively goodapproximation to the thermal Drude weight both in zero [43] and finite magnetic fields [66].In most cuprate based spin chain and ladder materials, exchange couplings are of the orderof 1000K [1,2] hence little effects of a magnetic field on the thermal conductivity have beenobserved. In a recent experiment, the thermal conductivity of copper pyrazine dinitrate has beenstudied [19]. Since J ∼ . K [70], a significant field dependence is found at low temperatures.The analysis of the experimental data employing the mean-field theory description of Ref. [66]yields a constant mean-free path. The origin of this result especially at the quantum critical lineremains to be elucidated. Magnetothermal effects do not seem to be present in this material. We invoke the notion of particle hole symmetry as 1D spin models can be mapped onto spinlessfermions with local interactions using the Jordan-Wigner transformation [8]. Note that J = j s and J = j th − hj s and therefore, D s = D .ill be inserted by the editor 5 ω /J κ r e g ( ω ) / J T/J=1T/J=2 ω /J I t h ( ω ) / I C t h ( N ) / C t h ( N = ) α =0.1 α =0.2 α =0.35 α =1 (a) α =0.2T/J=0.5,1,2(b) (c) ω /J κ r e g ( ω ) / J N=18N=20 0 1 2 3 4 ω /J00.20.40.60.81 I( ω ) / I T/J=5T/J=1T/J=0.3(d) (e)T/J=1T/J=2
Fig. 2. (a): Regular part of the thermal conductivity κ reg ( ω ) for a frustrated chain with α = 0 . N = 18 sites; T /J = 1 , C th = lim T →∞ [ T D th ] for α = 0 . , . , . , N = 8 , , . . . ,
20 sites (see also Ref. [44]). (c): Integrated spectral weight I th for α = 0 . T /J = 0 . , ,
2. (d): κ reg ( ω ) as a function of frequency ω for α = 1 [ N = 18 , T /J = 0 . , , While originally conjectured to exhibit diffusive transport properties [37,38,71], upon the exper-imental observation of large thermal conductivities in spin ladder, nonintegrable models havebeen discussed controversially in the literature [12,40,43,44,46,51,54,60,61,72,73,74,75,76]. As ofnow, many studies point at a vanishing of both Drude weights [40,43,44,61,72,77,78] in massivephases of nonintegrable models, including spin ladders. The massless regime of the frustratedchain remains a controversial issue [40,44]. Here we illustrate some numerical results for thefinite-size scaling of the Drude weights [44] taking the example of the frustrated and dimerizedchain and in particular, we also discuss the frequency dependence of the transport coefficients.The Hamiltonian of the dimerized and frustrated spin-1 / H = N X l =1 h l = J N X l =1 [ λ l S l · S l +1 + α S l · S l +2 ] ; (8)where α parameterizes the frustration and dimerization is introduced through λ l = 1( λ ) for aneven(odd) site index l ( λ ≤ κ [ σ ]( ω ) appearing in Eq. (1) can be written as: κ [ σ ] reg ( ω ) = πβ r Z N − e − βω ω X m,nEm = En e − E n /T |h m | j th[s] | n i| δ ( ω − ( E m − E n )) . (9) As a result of preceding studies of the finite-size scaling of the thermal Drude weight [44], weconcluded that no indications for a finite Drude weight are evident from the system sizes acces-sible by ED. This result is illustrated in Fig. 2(b), where we show the leading coefficient of anexpansion of the thermal Drude weight D th ( T ) in powers of 1 /T , i.e., C th = lim T →∞ [ T D th ( T )],as a function of system size (including new data for N = 20 as compared to Ref. [44]). Thedecrease of C th with N is evident for all α as soon as the system size N becomes large enough(see Ref. [44] for details). The same picture arises for spin transport [44].Let us mention the main features of Re κ ( ω ) as found for the case of α = 1, i.e., in themassive regime, shown in Fig. 2(d) [72]: (i) κ reg ( ω ) is a broad, featureless function extendingup to frequencies ω/J .
4; (ii) at
T /J = 1 and N = 20, the thermal Drude weight only givesa small contribution to the total weight of less than 3%.We now proceed by a discussion of the frequency dependence of the thermal conductivity offrustrated chains in the massless regime, i.e., α . .
241 [79]. Our numerical results for κ reg ( ω ) Will be inserted by the editor κ r e g ( ω ) / J T/J=0.3T/J=0.4T/J=0.5 ω /J I t h ( ω ) / I T/J I t h ( ω = . J ) N=16N=18(a) λ =0.1, α =0, N=18(b) N=18 T/J=0.2,0.5,1,2 σ r e g ( ω ) T/J=0.4T/J=0.5T/J=1 ω /J I s ( ω ) / I ω /J σ r e g ( ω ) (c) (d) N=18 T=0.4,0.5,1,2 Fig. 3.
Dimerized chain with λ = 0 .
1. (a): Regular part of the thermal conductivity as a functionof frequency ω ( N = 18 sites; T /J = 0 . , . , .
5; solid, dotted, dashed line). (b): Integrated weight I th vs ω for T /J = 0 . , . , ,
2. Inset of (a): Integrated weight I th ( ω ) for ω/J = 0 . ω ( N = 18 sites; T /J = 0 . , . ,
1; solid,dotted, dashed line). (d): Integrated weight I s vs ω for T /J = 0 . , . , ,
2. Inset of (c): Enlarged viewof the low-frequency region of panel (c). Vertical, dotted lines mark the position of the spin gap. and α = 0 . N = 18 sites and T /J = 1 , κ reg ( ω ) consists of anarrow peak centered around ω = 0, extending up to ω/J . .
05. This is reflected in theintegrated spectral weight I th ( ω ) = R ω dω ′ Re κ ( ω ′ ), depicted in Fig. 2(c). As this quantity alsoincludes the contribution from the Drude weight, the figure reveals that the thermal Drudeweight on the system sizes considered here amounts to more than 50% of the total spectralweight. This observation is in stark contrast to the behavior of κ ( ω ) in the massive regime onchains of a comparable length as summarized above. While on the one hand, the analysis ofthe finite-frequency properties of κ for α > .
241 supports the conclusion of a vanishing Drudeweight, the question arises on the other hand whether the conclusion of D th → D th with system size N for N large enough. Moreover,bosonization studies on general grounds predict a vanishing Drude weight for thermal trans-port, irrespective of the presence of a gap [43,44,71,77]. In a recent work [78], Jung et al. haveshown that, to first order in α , the commutator [ H, j th ] between the Hamiltonian and the energycurrent operator vanishes, preserving the exact conservation of the energy current operator ofthe nearest-neighbor XXZ chain. While this feature explains the peculiar behavior of κ ( ω )for small system sizes as observed here, a vanishing of the Drude weight can still be expectedin the thermodynamic limit where all terms of [ H, j th ] in powers of α become relevant andcause a finite dc conductivity. The results of a quantum Monte-Carlo (QMC) study, however,seem to indicate that in massless phases of nonintegrable models, finite Drude weights mayexist [40]. Note though that the interpretation of Monte-Carlo data at finite frequencies is quiteinvolved as an analytic continuation from Matsubara to real frequencies needs to be performed[39,40,51].In summary, the peculiar feature of κ ( ω ) of frustrated chains in the massless regime, i.e. α < . ω = 0 only. ill be inserted by the editor 7 We next address finite-frequency transport properties of the dimerized chain ( α = 0). In thefollowing, we choose λ = 0 .
1, i.e., we focus on the limit of strong dimerization λ ≪
1. Tofirst order in λ , the dispersion relation of the elementary triplet excitation is described by ǫ k /J = 1 + ( λ/
2) cos( k ) [80], where k denotes the momentum. The spin gap G is quite largeand roughly given by G/J = 0 .
95, while triplet-triplet interactions are suppressed by decreas-ing λ . One may therefore on the one hand expect both the spin and heat conductivity to besmall due to the large spin gap, but on the other hand, the transport properties should be wellapproximated by considering a weakly interacting gas of hardcore bosons [81], which may, as afuture project, allow for a comparison between numerical and analytical results.Our numerical results for the conductivities κ ( ω ) and σ ( ω ) are presented in Figs. 3(a) and3(c), respectively. The computations were performed for N = 18 sites, λ = 0 .
1, and severalfinite temperatures as listed in the figure’s caption. The distinctive features of both conductivi-ties visible in Figs. 3(a) and 3(c) are: (i) Significant spectral weight is only found around ω = 0and in a high-frequency peak located around ω/J & .
95, which corresponds to the spin gap.(ii) While the low frequency peak (including the Drude weight) contains a large fraction of thetotal weight in the case of the thermal conductivity, the spectral weight of the spin conductivityis mainly concentrated in the high-frequency peak. The latter is illustrated in Figs. 3(b) and3(d), showing the integrated spectral weight I th[s] ( ω ) /I , where I is the full spectral weight of κ [ σ ]( ω ). The low-frequency peak is present in σ ( ω ) as well. The inset and the main panel ofFig. 3(c) show that the low-frequency peak extends up to ω/J ∼ .
1, which corresponds to thewidth of the one-triplet band.Furthermore, by integrating the low-frequency peak in κ reg ( ω ) over ω up to ω/J ≈ . I reg ( ω/J = 0 . I th ( ω/J = 0 .
5) is plotted in the inset of Fig. 3(a) for N = 16 and N = 18sites. Hence, a significant redistribution of spectral weight as the system size increases is notexpected. One further observes a maximum in I th ( ω/J = 0 .
5) at roughly
T /J ∼ .
35 and a1 /T -dependence at high temperatures.In summary, both models exhibit an intriguing behavior of the frequency dependence ofboth the spin and thermal conductivity that deserves further investigations. As mentioned in the introduction, and as is evident form the phenomenological analysis ofexperimental data for spin ladder [9,10,11,83,82] as well as spin chain materials [13,14,84,85],it is important to include external scattering processes to arrive at a realistic theory of thermaltransport in quasi 1D magnetic materials. For instance, doping with nonmagnetic impurities in(Sr,Ca,La) Cu O [82] – substitution of Zn for Cu – has been found to result in a suppressionof the thermal conductivity linear in the Zn content. Mobile charge carriers effectively suppressthe magnon thermal transport in spin ladder systems [83]. As heat transport via magneticsystems in a material requires the heat to be transferred from the lattice to the spin system,inevitably, spin-phonon scattering needs to be modeled by theory.First studies have addressed the thermal conductivity of spin-phonon coupled spin chains[77,86,87,88] as well as spin ladders [89]. Some of these works [77,89] start from effective fieldtheories and describe transport within the Memory-matrix formalism [90,91] by first identi-fying the slowest decaying modes, following the spirit of Ref. [71]. These then determine thelong-time behavior of current-current correlation functions. For the case of spin chains, an ex-ponentially large thermal conductivity κ total ∝ exp( a Θ/ T ) is predicted [77], where Θ is theDebye temperature. A peculiar result of the Boltzmann theory of Refs. [86,87] is the constantspin thermal conductivity at high temperatures. For spin ladders, Ref. [89] highlights the rel-evance of spin-phonon drag terms contributing to the total thermal conductivity, with a richinterplay of energy scales influencing the low-temperature behavior. A direct comparison ofthese results with experiments, however, needs to be done in future, in particular, as disordermay be of relevance in the structurally disordered spin ladder compounds (Sr,Ca,La) Cu O Will be inserted by the editor [9,10,89]. Finally, note that spin phonon coupling has also been studied in the context of spintransport in the spin-1/2 chain by means of QMC [92].Note that via the Jordan-Wigner transformation Heisenberg type of models can be mappedonto spinless fermions [8], the transport properties of which have extensively been studied inthe context of localization [93]. We just mention an incomplete list of recent, closely relatedworks addressing Heisenberg chains [94,95,96,97,98,99], spin ladders [73], or effective low-energymodels [86,87,100]. Interestingly, some works seem to indicate that the dc spin conductivity maybe finite for interacting systems in the case of off-diagonal disorder [95,96]. Also, even if the dcspin conductivity vanishes, the same is not necessarily true for thermal transport as energy canstill be transfered over a weak link [100].Finally, only results from a mean-field theory are available for the thermal conductivity ofdoped spin ladders in the literature [101]. Transport properties of 1D t - J and Hubbard modelshave widely been investigated (see, e.g., Ref. [33] for an overview), and it is beyond the scopeof this work to discuss the charge and spin transport of these systems. Their thermal transportproperties have, however, not been studied sufficiently [102]. Note that the Hubbard model,being integrable, is expected to exhibit ballistic thermal transport, which also holds for thesupersymmetric point of the t - J model [35]. We may conclude that the intrinsic thermal transport properties of the spin-1/2
XXZ chainin zero and finite longitudinal fields are well understood. The spin transport of this model stillposes some challenges to theorists, such as an analytical calculation of the spin Drude weightof the spin-1/2 Heisenberg chain. As for nonintegrable systems and within linear response the-ory, it seems that generically, ballistic transport in the sense of finite Drude weights is notrealized. Rather, the relevant information is encoded in the frequency dependence of the con-ductivities. The challenge to computational scientists is to devise algorithms that can simulatelow temperature regimes. Analytical approaches face the problem that effective field-theoriesof nonintegrable models are typically integrable, with diverging transport coefficients. Hence,the definition of a low-energy theory that describes transport accurately is a nontrivial task.Promising results with respect to the interpretation of experiments have been obtained fromfirst studies incorporating phonons or disorder, but a consistent picture has not emerged yet.Highly interesting and potentially new physics is expected from both experiments and noveltheoretical methods such as the time-dependent density matrix renormalization group method[103] that investigate transport and relaxation of strongly-correlated electron systems awayfrom equilibrium.
Acknowledgments
This work has been possible only through collaborations and discus-sions with B. B¨uchner, D.C. Cabra, and C. Hess, which we gratefully acknowledge. We wouldalso like to thank N. Andrei, J. Gemmer, C. Gros, P. Jung, A. Kl¨umper, T. Lorenz, K. Louis,M. Michel, A. Rosch, A. Sologubenko, and X. Zotos for fruitful discussions.
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