Finite-temperature transport in one-dimensional quantum lattice models

Over the last decade impressive progress has been made in the theoretical understanding of transport properties of clean, one-dimensional quantum lattice systems. Many physically relevant models in one dimension are Bethe-ansatz integrable, including the anisotropic spin-$1/2$ Heisenberg (also called the spin-$1/2$ $\mathit{XXZ}$ chain) and the Fermi-Hubbard model. Nevertheless, practical computations of correlation functions and transport coefficients pose hard problems from both the conceptual and technical points of view. Only because of recent progress in the theory of integrable systems, on the one hand, and the development of numerical methods, on the other hand, has it become possible to compute their finite-temperature and nonequilibrium transport properties quantitatively. Owing to the discovery of a novel class of quasilocal conserved quantities, there is now a qualitative understanding of the origin of ballistic finite-temperature transport, and even diffusive or superdiffusive subleading corrections, in integrable lattice models. The current understanding of transport in one-dimensional lattice models, in particular, in the paradigmatic example of the spin-$1/2$ $\mathit{XXZ}$ and Fermi-Hubbard models, is reviewed, as well as state-of-the-art theoretical methods, including both analytical and computational approaches. Among other novel techniques, matrix-product-state-based simulation methods, dynamical typicality, and, in particular, generalized hydrodynamics are covered. The close and fruitful connection between theoretical models and recent experiments is discussed, with examples given from the realms of both quantum magnets and ultracold quantum gases in optical lattices.

Figure 1

The three different scenarios that one can envision for the behavior of the regular part of the optical conductivity ${\sigma }^{\prime }(\omega )$ (solid lines) at finite temperature. The point at $\omega =0$ indicates the Drude weight, which may coexist with a nonzero regular part.

Figure 2

Different scenarios for the time-dependent diffusion constant $D^{(\mathrm{S})}(t)$: ballistic, superdiffusive, and diffusive (top to bottom). The behavior in the short-time limit is always ballistic, and the typical exponents in the longtime dynamics are indicated as $t^1$, $t^{\alpha }$ ($0<\alpha <1$), and $t^0$, respectively.

Figure 3

Diffusive spreading of a spin-density perturbation as a function of (a) time $t$ and real space $r$, (b) time $t$ and momentum space $q$, and (c) frequency $\omega$ and momentum space $q$.

Figure 4

Lower bound for the spin Drude weight ${\mathcal{D}}_{\mathrm{w}}^{(\mathrm{S})}$ of the spin-$1/2$ $XXZ$ chain according to Eq. (78), as obtained by [483], and another lower bound for ${\mathcal{D}}_{\mathrm{w}}^{(\mathrm{S})}$, as previously obtained by [478]. Both bounds exhibit a pronounced fractal-like (i.e., nowhere continuous) dependence on the anisotropy parameter $\mathrm{\Delta }$.

Figure 5

Exact results for the energy Drude weight of the spin-$1/2$ $XXZ$ chain given in Eq. (1) at zero magnetization. The data were taken from [335].

Figure 6

TBA results for the spin Drude weight ${\mathcal{D}}_{\mathrm{w}}^{(\mathrm{S})}$ of the spin-$1/2$ $XXZ$ chain vs $\mathrm{\Delta }$ for different temperatures $T$ (measured in units of $J$). From [678].

Figure 7

Finite-size scaling of the spin Drude weight ${\mathcal{D}}_{\mathrm{w}}^{(\mathrm{S})}$ in the high-temperature limit $\beta =0$, as presented by [585], [586]. Crosses indicate dynamical quantum typicality (DQT) data while other symbols indicate ED data. Similar ED data were given by [684], [243], [494], [249], and [306].

Figure 8

Frequency dependence of the charge conductivity in the Fermi-Hubbard chain at $U/t_{\mathrm{h}}=16$ and $\beta =0$ and at half filling, as obtained from Fourier transforming real-time data that uses (a) short times that are $L$ independent, $t_{\max }{t}_{\mathrm{h}}=4$, and (b) long times of $t_{\max }{t}_{\mathrm{h}}=100$ ([287]). See Sec. 7 and Eq. (178) for the definition of the Hamiltonian.

Figure 9

Accuracy of the DQT approximation, illustrated for the spin-current autocorrelation function in the spin-$1/2$ $XXZ$ chain at the isotropic point $\mathrm{\Delta }=1$ and infinite temperatures $\beta =0$ ([586]). (a) ED vs DQT for a chain (with a total Hilbert-space dimension of $\dim =2^L$) and an uncoupled ladder ($\dim =4^L\gg 2^L$) with $L=10$. (b) DQT for $L=33$ and two randomly drawn pure states. For the behavior of spin-spin correlations see [26].

Figure 10

Benchmark of the improved finite-$T$ tDMRG algorithm for the spin-$1/2$ $XXZ$ chain [Eq. (1)]. (a) Spin autocorrelation function for $\mathrm{\Delta }=0$ computed using both the standard algorithm [Eq. (149)] and the improved version ($U$ replaced by $\tilde{U}$) with a fixed bond dimension of $\chi =60$. The exact solution is shown as a reference. The data were taken from [304]. (b) Spin-current autocorrelation function at $\mathrm{\Delta }=0.5$ calculated using Eq. (150) with a fixed discarded weight. The data were taken from [308]. Both the analytical result given by [483] (horizontal dashed line) and the timescale reached in the tDMRG calculation given by [563] (vertical dotted line) are shown for comparison. (c) Same as (b) but for discarded weights that each differ by 1 order of magnitude [the two values denote the discarded weight during the two different time evolutions in Eq. (150); see 318]. Data obtained using a fixed bond dimension of $\chi =200$ are shown for comparison.

Figure 11

Top panel: NESS boundary Lindblad driving. Bottom panel: typical NESS density profiles for diffusive (blue line), superdiffusive (red line), and ballistic transport (black line).

Figure 12

Comparison of results for the spin-diffusion constant $D^{(\mathrm{S})}$ obtained from NESS simulations (empty symbols) and unitary domain-wall spreading (solid symbols) in the spin-$1/2$ $XXZ$ chain with a quasiperiodic magnetic field of amplitude $\lambda$. Details were given by [674]. Unitary domain-wall spreading is a particular case of a bipartitioning protocol; see Sec. 9b for details.

Figure 13

Schmidt spectrum for the NESS of a boundary-driven spin-$1/2$ Heisenberg chain ($\mathrm{\Delta }=1$) with $L=64$ sites, $\mu =0.005$, and MPO bond dimensions $\chi =60–500$. The dashed line is the best-fitting asymptotic decay. Inset: convergence of the NESS current $j_{\chi }$ obtained from a fixed-$\chi$ calculation, with the dashed line the predicted error from the main plot $(j_{\infty }-j_{\chi })/j_{\infty }\approx L/90\chi$ (no fitting parameters).

Figure 14

(a) Overview of all known exact results [free fermions, Bethe ansatz (BA) at $T=0$, and Mazur bounds] as well as predictions obtained using GHD and numerics for the spin Drude weight of the spin-$1/2$ $XXZ$ chain at zero magnetization. (b) Overview of the high-temperature results for the leading contribution at low but finite frequencies. In the regime $|\mathrm{\Delta }|<1$, this low-frequency contribution is either superdiffusive or diffusive.

Figure 15

Comparison of different results for the spin Drude weight ${\mathcal{D}}_{\mathrm{w}}^{(\mathrm{S})}$ in the high-temperature limit $\beta =0$ at zero magnetization: the solid line shows the lower bound ([483]), which is given by Eq. (78) and which at infinite temperature coincides with the TBA result ([678]; [626]; [457]) and the GHD prediction ([274]; [85]). Moreover, we show ED ([249]), DQT ([585], [586]), and tDMRG data taken from [310] and [308]; see also [304], [306], and [302]. For the tDMRG data, the Drude weight is taken as the value of $⟨{\mathcal{J}}^{(\mathrm{S})}(t){\mathcal{J}}^{(\mathrm{S})}⟩/L$ at the largest accessible time without any further extrapolation. Note that while the lower bound was computed only for commensurate $\mathrm{\Delta }=\cos (\ell \pi /m)$, numerical data are also shown away from these points.

Figure 16

(a) Comparison of different results for the time-dependent diffusion constant $D^{(\mathrm{S})}(t)$ of the spin-$1/2$ $XXZ$ chain at $\mathrm{\Delta }=1.5$ in the infinite-temperature limit $\beta =0$: ED ([584]), DQT ([586]), and tDMRG ([311]). (b) tDMRG data for $D^{(\mathrm{S})}(t)$ at various $\mathrm{\Delta }$ [obtained from integrating the current autocorrelation function given by [310] and 308].

Figure 17

Diffusion constant of the spin-$1/2$ $XXZ$ chain at infinite temperature as a function of $1/\mathrm{\Delta }$ for $\mathrm{\Delta }>1$. tDMRG results ([311]) are compared to the GHD prediction for $\mathrm{\Delta }\to \infty$ [cf. Eq. (173)] and $\mathrm{\Delta }<\infty$ ([132]). The tDMRG results close to $\mathrm{\Delta }=1$ give only a lower bound to the true diffusion constant.

Figure 18

Real part of the optical conductivity $\sigma (\omega )$ of the spin-$1/2$ $XXZ$ chain, as obtained by [473] from ED and the MCLM for $\mathrm{\Delta }=2$ and infinite temperature $\beta =0$. The anomalous scaling with system size is still consistent with a well-behaved low-frequency part and a finite dc conductivity in the thermodynamic limit. The solid line results from computing frequency moments (FM).

Figure 19

Results from the Lindblad quantum master equation for simulating spin transport in the spin-$1/2$ $XXZ$ chain, as obtained by [668] for $\mathrm{\Delta }=1$. (a) Superdiffusive scaling of the spin current as $j^{(S)}\propto 1/\sqrt{L}$. (b) $L$-independent magnetization profile.

Figure 20

Energy Drude weight of the Fermi-Hubbard chain as a function of temperature computed from the finite-$T$ tDMRG ([309]; [302]) and GHD ([273]).

Figure 21

Charge Drude weight of the Fermi-Hubbard chain at half filling and infinite temperature versus system size obtained from dynamical typicality ([287]), plotted using a logarithmic scale on both axes.

Figure 22

(a) Spin-$1/2$ ladder with a coupling strength $J$ and $J_{\perp }$ along the legs and rungs, respectively, and (b) a frustrated chain with a next-nearest-neighbor coupling of strength $J^{\prime }$.

Figure 23

Infinite-temperature energy Drude weight of frustrated spin-$1/2$ Heisenberg chains computed with exact diagonalization ([243], [245]).

Figure 24

Spin conductivity of a spin-$1/2$ $XX$ ladder for various rung couplings $J_{\perp }$ ([308]).

Figure 25

Real part of the conductivity of a nonintegrable model vs $\omega$ (in units of $t$), namely, spinless fermions with a nearest-neighbor repulsion of strength $V=2t$ and an additional next-to-nearest-neighbor hopping $t^{\prime }=t$, where $t$ is the nearest-neighbor hopping matrix element. The exact-diagonalization data indicate that the low-frequency behavior is incompatible with a simple Lorentzian; see [430] for a discussion. Other examples of a similar shape were reported by [680] and [246]. From [430].

Figure 26

Spin-diffusion constant of the spin-$1/2$ $XX$ ladder as a function of $r=J_{\perp }/J$ at infinite temperature, as obtained from dynamical typicality for $L=12,14,17$ rungs (i.e., $2L$ sites in the ladder) ([588]) and perturbation theory (PT) ([506]), with $D^{(\mathrm{S})}/J=1/(2\gamma r^2)$ and $\gamma \approx 0.63$ in the limit of small $r$ with no free parameter.

Figure 27

Temperature dependence of the thermal conductivity $\kappa$ in a Heisenberg chain with a staggered field, as given by [586]; see also [266]. $\tau$ is the relaxation time, defined as the time at which the current correlation has decayed to a fraction of $1/e$ ([586]).

Figure 28

Dependence of the NESS spin current on system size $L$ for a spin-$1/2$ $XXZ$ chain ($\mathrm{\Delta }=0.5$) with a single impurity of strength $h$. Adapted from [73].

Figure 29

Densities as a function of time $t$ and position $r$ for a local quench inducing (a) spin dynamics and (b) energy dynamics in the spin-$1/2$ $XXZ$ chain at $\mathrm{\Delta }=1.5$ at $T=\infty$. The dynamics is induced by introducing a local perturbation in the initial state. Adapted from [311].

Figure 30

Spatial dependence of magnetization profiles at different times as obtained by [594] for a spin-$1/2$ $XXZ$ chain at $\mathrm{\Delta }=1.5$, the full Hilbert space of $L=36$ sites, and a randomly chosen initial pure state with a $\delta$ peak on top of a many-particle background at high temperatures. These profiles are well described by Gaussian fits over several orders of magnitude. Similar Gaussian profiles were found by [374].

Figure 31

Pictorial representation of a generic bipartitioning protocol. The two halves of the chain are prepared in two different homogeneous states at time $t=0$. A nonequilibrium region emerges from the junction at the middle and expands at the maximal allowed speed.

Figure 32

Profiles of the local currents in the spin-$1/2$ $XXZ$ chain for different values of $\mathrm{\Delta }=\cos (\eta )$ as a function of rescaled position $x/t$. Symbols denote time-evolving block-decimation data for a chain of length $L=60$ (top panel), $L=120$ (bottom panel), and different times; full black lines are the GHD predictions. Top panel: energy current after the two halves of the system were initially prepared at inverse temperatures ${\beta }_{\mathrm{L}}=1$ and ${\beta }_{\mathrm{R}}=2$. Bottom panel: spin current after the two halves were prepared in two ferromagnetic states with opposite magnetization. Inset: time-dependent approach of $j_x^{(S)}$ (solid colored lines) to the prediction (dashed lines). Adapted from [49].

Figure 33

Top panel: experimental data for the thermal conductivity of ${\mathrm{SrCuO}}_2$ for two sample purities [2N (black curves) and 4N (red curves)]. The high-quality samples (4N) show a high thermal conductivity ${\kappa }_c$ in the crystal direction that is parallel to the spin chains, which is attributed to spin excitations. In the transverse directions, presumably only phonons contribute. From [259]. Bottom panel: extracted magnetic mean free paths of spinons. From [260].

Figure 34

Expansion velocity of a cloud of bosons that are released from a trap into an empty optical lattice. Main panel: experimental and DMRG data for the expansion velocity as a function of interaction strength $U/J_x$ extracted from the half width at half maximum. $J_x$ is the hopping-matrix element along the $x$ direction of the two-dimensional lattice and $J_x=J$ for the one-dimensional case. $U$ is the on-site interaction strength in the Bose-Hubbard model. Inset: DMRG data for the radial velocity as a function of $U/J$. Both noninteracting and strongly interacting bosons expand ballistically with the same expansion velocity ([513]).