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arXiv:1111.4508 · Physical Review Letters

Finite-Temperature Dynamical Density Matrix Renormalization Group and the Drude Weight of Spin- 1/2 Chains

Christoph Karrasch, Jens Hjorleifur Bardarson, Joel E. Moore

Published May 31, 2012• 211 citations
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Abstract

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.

DOI
10.1103/physrevlett.108.227206

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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.
Finite-temperature transport properties of one-dimensional systems can be studied using the time dependent density matrix renormalization group via the introduction of auxiliary degrees of freedom which purify the thermal statistical operator. We demonstrate how the numerical effort of such calculations is reduced when the physical time evolution is augmented by an additional time evolution within the auxiliary Hilbert space. Specifically, we explore a variety of integrable and non-integrable, gapless and gapped models at temperatures ranging from T = ∞ down to T /bandwidth = 0.05 and study both (i) linear response where (heat and charge) transport coefficients are determined by the current–current correlation function and (ii) non-equilibrium driven by arbitrary large temperature gradients. The modified density matrix renormalization algorithm removes an 'artificial' build-up of entanglement between the auxiliary and physical degrees of freedom. Thus, longer time scales can be reached.
We revisit the problem of the spin Drude weight DDD of the integrable spin-1/21/21/2 XXZ chain using two complementary approaches, exact diagonalization (ED) and the time-dependent density-matrix renormalization group (tDMRG). We pursue two main goals. First, we present extensive results for the temperature dependence of DDD. By exploiting time translation invariance within tDMRG, one can extract DDD for significantly lower temperatures than in previous tDMRG studies. Second, we discuss the numerical quality of the tDMRG data and elaborate on details of the finite-size scaling of the ED results, comparing calculations carried out in the canonical and grand-canonical ensembles. Furthermore, we analyze the behavior of the Drude weight as the point with SU(2)-symmetric exchange is approached and discuss the relative contribution of the Drude weight to the sum rule as a function of temperature.
We illustrate how finite-temperature charge and thermal Drude weights of one-dimensional systems can be obtained from the relaxation of initial states featuring global (left–right) gradients in the chemical potential or temperature. The approach is tested for spinless interacting fermions as well as for the Fermi-Hubbard model, and the behavior in the vicinity of special points (such as half filling or isotropic chains) is discussed. We present technical details on how to implement the calculation in practice using the density matrix renormalization group and show that the non-equilibrium dynamics is often less demanding to simulate numerically and features simpler finite-time transients than the corresponding linear response current correlators; thus, new parameter regimes can become accessible. As an application, we determine the thermal Drude weight of the Hubbard model for temperatures T which are an order of magnitude smaller than those reached in the equilibrium approach. This allows us to demonstrate that at low T and half filling, thermal transport is successively governed by spin excitations and described quantitatively by the Bethe ansatz Drude weight of the Heisenberg chain.
We present time-dependent density matrix renormalization group simulations (t-DMRG)(t\text{-DMRG})(t-DMRG) at finite temperatures. It is demonstrated how a combination of finite-temperature t-DMRGt\text{-DMRG}t-DMRG and time-series prediction allows for an easy and very accurate calculation of spectral functions in one-dimensional quantum systems, irrespective of their statistics for arbitrary temperatures. This is illustrated with spin structure factors of XXXXXX and XXXXXXXXX spin-12\frac{1}{2}21​ chains. For the XXXXXX model we can compare against an exact solution, and for the XXXXXXXXX model (Heisenberg antiferromagnet) against a Bethe ansatz solution and quantum Monte Carlo data.
The density-matrix renormalization group (DMRG) applied to transfer matrices allows it to calculate static as well as dynamical properties of one-dimensional quantum systems at finite temperature in the thermodynamic limit. To this end the quantum system is mapped onto a two-dimensional classical system by a Trotter-Suzuki decomposition. Here we discuss two different mappings: The standard mapping onto a two-dimensional lattice with checkerboard structure as well as an alternative mapping introduced by two of us. For the classical system an appropriate quantum transfer matrix is defined which is then treated using a DMRG scheme. As applications, the calculation of thermodynamic properties for a spin-1/2 Heisenberg chain in a staggered magnetic field and the calculation of boundary contributions for open spin chains are discussed. Finally, we show how to obtain real time dynamics from a classical system with complex Boltzmann weights and present results for the autocorrelation function of the XXZ-chain.
This paper provides a study and discussion of earlier as well as novel more efficient schemes for the precise evaluation of finite-temperature response functions of strongly correlated quantum systems in the framework of the time-dependent density matrix renormalization group (tDMRG). The computational costs and bond dimensions as functions of time and temperature are examined for the example of the spin-1/2 XXZ Heisenberg chain in the critical XY phase and the gapped Néel phase. The matrix product state purifications occurring in the algorithms are in a one-to-one relation with the corresponding matrix product operators. This notational simplification elucidates implications of quasi-locality on the computational costs. Based on the observation that there is considerable freedom in designing efficient tDMRG schemes for the calculation of dynamical correlators at finite temperatures, a new class of optimizable schemes, as recently suggested in Barthel, Schollwöck and Sachdev (2012 arXiv:1212.3570), is explained and analyzed numerically. A specific novel near-optimal scheme that requires no additional optimization reaches maximum times that are typically increased by a factor of 2, when compared against earlier approaches. These increased reachable times make many more physical applications accessible. For each of the described tDMRG schemes, one can devise a corresponding transfer matrix renormalization group variant.