The density-matrix renormalization group applied to transfer matrices: Static and dynamical properties of one-dimensional quantum systems at finite temperature

TL;DR

This paper extends the density-matrix renormalization group (DMRG) method to transfer matrices, enabling the calculation of static and dynamical properties of one-dimensional quantum systems at finite temperature, including real-time dynamics.

Contribution

It introduces a new approach using transfer matrices and two different mappings to compute thermodynamic and dynamical properties of quantum chains at finite temperature.

Findings

Successfully calculated thermodynamic properties of a spin-1/2 Heisenberg chain.

Demonstrated real-time dynamics via classical systems with complex weights.

Analyzed boundary contributions for open spin chains.

Abstract

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.