We propose an easily implemented approach to study time-dependent correlation functions of one-dimensional systems at finite-temperature T using the density matrix renormalization group. The entanglement growth inherent to any time-dependent calculation is significantly reduced if the auxiliary degrees of freedom which purify the statistical operator are time evolved with the physical Hamiltonian but reversed time. We exploit this to investigate the long-time behavior of current correlation functions of the XXZ spin-1/2 Heisenberg chain. This allows a direct extraction of the Drude weight D at intermediate to large T. We find that D is nonzero--and thus transport is dissipationless--everywhere in the gapless phase. At low temperatures we establish an upper bound to D by comparing with bosonization.
We revisit the numerical problem of computing the high temperature spin stiffness, or Drude weight,
D of the spin-
1/2 XXZ chain using exact diagonalization to systematically analyze its dependence on system symmetries and ensemble. Within the canonical ensemble and for states with zero total magnetization, we find
D vanishes exactly due to spin-inversion symmetry for all but the anisotropies ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}}_{MN}=cos(\ensuremath{\pi}M/N)$ with $N,M\ensuremath{\in}{\mathbb{Z}}^{+}$ coprimes and
N>M, provided system sizes $L\ensuremath{\ge}2N$, for which states with different spin-inversion signature become degenerate due to the underlying
sl2 loop algebra symmetry. All these loop-algebra degenerate states carry finite currents which we conjecture [based on data from the system sizes and anisotropies ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}}_{MN}$ (with
N<L/2) available to us] to dominate the grand-canonical ensemble evaluation of
D in the thermodynamic limit. Including a magnetic flux not only breaks spin-inversion in the zero magnetization sector but also lifts the loop-algebra degeneracies in all symmetry sectors---this effect is more pertinent at smaller $\mathrm{\ensuremath{\Delta}}$ due to the larger contributions to
D coming from the low-magnetization sectors which are more sensitive to the system's symmetries. Thus we generically find a finite
D for fluxed rings and arbitrary $0<\mathrm{\ensuremath{\Delta}}<1$ in both ensembles. In contrast, at the isotropic point and in the gapped phase ($\mathrm{\ensuremath{\Delta}}\ensuremath{\ge}1$)
D is found to vanish in the thermodynamic limit, independent of symmetry or ensemble. Our analysis demonstrates how convergence to the thermodynamic limit within the gapless phase ($\mathrm{\ensuremath{\Delta}}<1$) may be accelerated and the finite-size anomalies overcome:
D extrapolates nicely in the thermodynamic limit to either the recently computed lower bound or the thermodynamic Bethe ansatz result provided both spin inversion is broken and the additional degeneracies at the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}}_{MN}$ anisotropies are lifted.
We report a detailed analysis of the Drude weights for both thermal and spin transport in one-dimensional spin-
21 systems by means of exact diagonalization and analytic approaches at finite temperatures. Transport properties are studied first for the integrable
XXZ model and second for various nonintegrable systems such as the dimerized chain, the frustrated chain, and the spin ladder. We compare our results obtained by exact diagonalization and mean-field theory with those of the Bethe ansatz, bosonization, and other numerical studies in the case of the anisotropic Heisenberg model both in the gapless and gapped regime. In particular, we find indications that the Drude weight for spin transport is finite in the thermodynamic limit for the isotropic chain. For the nonintegrable models, a finite-size analysis of the numerical data for the Drude weights is presented, covering the entire parameter space of the dimerized and frustrated chain. We also discuss which conclusions can be drawn from bosonization regarding the question of whether the Drude weights are finite or not. One of our main results is that the Drude weights vanish in the thermodynamic limit for nonintegrable models.
We calculate the finite temperature thermal conductivity of a time-reversal invariant chiral
Z3 clock model along an integrable line in the parameter space using the time-dependent density matrix renormalization group. The thermal current itself is not a conserved charge, unlike in the case of XXZ model, but has a finite overlap with a local conserved charge
Q(2) obtained from the transfer matrix. We find that the Drude weight is finite at nonzero temperature, and the Mazur bound from
Q(2) saturates the Drude weight, allowing us to obtain an asymptotic expression for the Drude weight at high temperatures. The numerical estimates of the conductivity are validated using a sum rule for thermal conductivity. On the computational side, we also explore the effectiveness of the ancilla disentangler in the integrable and nonintegrable regimes of the model. We find that at high temperature, the disentangler helps by localizing the entanglement growth around the quench location allowing us to obtain the Drude weight in the integrable model. The improvement is insufficient to extract the asymptotic value of the correlator in the nonintegrable regime. The ancilla disentangler also provides no additional advantage at low temperatures.
We use the concept of typicality to study the real-time dynamics of spin and energy currents in spin-
1/2 models in one dimension and at nonzero temperatures. These chains are the integrable XXZ chain and a nonintegrable modification due to the presence of a staggered magnetic field oriented in the
z direction. In the framework of linear response theory, we numerically calculate autocorrelation functions by propagating a single pure state, drawn at random as a typical representative of the full statistical ensemble. By comparing to small-system data from exact diagonalization (ED) and existing short-time data from time-dependent density matrix renormalization group, we show that typicality is satisfied in finite systems over a wide range of temperature and is fulfilled in both integrable and nonintegrable systems. For the integrable case, we calculate the long-time dynamics of the spin current and extract the spin Drude weight for large systems outside the range of ED. We particularly provide strong evidence that the high-temperature Drude weight vanishes at the isotropic point. For the nonintegrable case, we obtain the full relaxation curve of the energy current and determine the heat conductivity as a function of magnetic field, exchange anisotropy, and temperature.
Finite-temperature Drude weight (spin stiffness)
D(T) is evaluated within the anisotropic spin-
1/2 Heisenberg model on a chain using the exact diagonalization for small systems. It is shown that odd-side chains allow for more reliable scaling and results, in particular, if one takes into account corrections due to low-frequency finite-size anomalies. At high
T and zero magnetization
D is shown to scale to zero approaching the isotropic point $\ensuremath{\Delta}=1$. On the other hand, for $\ensuremath{\Delta}>2$ at all magnetizations
D is nearly exhausted with the overlap with the conserved energy current. Results for the
T variation
D(T) are also presented.