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arXiv:cond-mat/0504091 · Physical Review B

Real-time dynamics at finite temperature by the density-matrix renormalization group: A path-integral approach

Jesko Sirker, Andreas Klümper

Published Jun 2, 2005• 66 citations
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Abstract

We propose a path-integral variant of the density-matrix renormalization group method to calculate real-time correlation functions at arbitrary finite temperatures. To illustrate the method we study the longitudinal autocorrelation function of the $XXZ$ chain. By comparison with exact results at the free fermion point, we show that our method yields accurate results up to a limiting time which is determined by the spectrum of the reduced density matrix.

DOI
10.1103/physrevb.71.241101

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We propose a path-integral variant of the density-matrix renormalization group method to calculate real-time correlation functions at arbitrary finite temperatures. To illustrate the method we study the longitudinal autocorrelation function of the XXZXXZXXZ chain. By comparison with exact results at the free fermion point, we show that our method yields accurate results up to a limiting time which is determined by the spectrum of the reduced density matrix.
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.
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.
We present a numerical method for the evaluation of dynamical response functions at finite temperatures in one-dimensional strongly correlated systems. The approach is based on the density-matrix renormalization group method, combined with the finite-temperature Lanczos diagonalization. The feasibility of the method is tested on the example of dynamical spin correlations in the anisotropic Heisenberg chain, in particular, it yields nontrivial results for the critical behavior in the isotropic case.
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 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 present a real-space renormalization group approach for the corner Hamiltonian, which is relevant to the reduced density matrix in the density matrix renormalization group. A set of self-consistent equations that the renormalized Hamiltonian should satisfy in the thermodynamic limit is also derived from the fixed point of the recursion relation for the corner Hamiltonian. We demonstrate the renormalization group algorithm for the S=1/2S=1/2S=1/2 XXZ spin chain and show that the results are consistent with the exact solution. We further examine the renormalization group for the S=1 Heisenberg spin chain and then discuss the nature of the eigenvalue spectrum of the corner Hamiltonian for the non-integrable model.