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arXiv:cond-mat/0510124 · Physical Review B

Finite-temperature density matrix renormalization using an enlarged Hilbert space

Adrian E. Feiguin, Steven R. White

Published Dec 2, 2005• 327 citations
arXivHTMLPDF

Abstract

We apply a generalization of the time-dependent density matrix renormalization group (DMRG) to study finite-temperature properties of several quantum spin chains, including the frustrated ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model. We discuss several practical issues with the method, including use of quantum numbers and finite-size effects. We compare with transfer-matrix DMRG, finding that both methods produce excellent results.

DOI
10.1103/physrevb.72.220401

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We apply a generalization of the time-dependent density matrix renormalization group (DMRG) to study finite-temperature properties of several quantum spin chains, including the frustrated J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ model. We discuss several practical issues with the method, including use of quantum numbers and finite-size effects. We compare with transfer-matrix DMRG, finding that both methods produce excellent results.
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.
The density matrix renormalization group (DMRG) method is applied to the anisotropic Heisenberg chain at finite temperatures. The free energy of the system is obtained using the quantum transfer matrix which is iteratively enlarged in the imaginary time direction. The magnetic susceptibility and the specific heat are calculated down to T=0.01J and compared with the Bethe ansatz results. The agreement including the logarithmic correction in the magnetic susceptibility at the isotropic point is fairly good.
The density matrix renormalization group (DMRG) is a numerical method for studying low dimensional strongly correlated models. The study of these models is one of the most active areas of condensed matter physics today and has been spurred not only by the discovery of high temperature superconductivity but also by experiments on new compounds. In this thesis we study fundamentals of the density matrix renormalization group method as well as applications of the method to Heisenberg spin chains. The introductory text is meant as a brief overview of the DMRG as well as Heisenberg spin systems and provides an introduction to the papers. In papers I and II we investigate the density matrix renormalization group with the aim to better understand the underlying structure of the method. We find that if the renormalization converges to a fixed point in the thermodynamic limit, the resulting basis states can be written as a matrix product form. We also show an explicit relation of the DMRG to variational calculations using the fact that the matrix product states can be rederived through a variational ansatz without any reference to the DMRG calculation. It is then shown how the matrix product states can be used to define a set of Bloch waves for describing elementary excitations of the Heisenberg spin-1 chain. The low lying energy spectrum is calculated. In papers III and IV we investigate the thermodynamics of impurities in a spin-1/2 chain. For this purpose we have shown that a recently developed transfer matrix DMRG method can be used to calculate impurity properties. Both impurity corrections to thermodynamic properties and local properties at finite temperature can be obtained. The method gives quicker and more accurate results than quantum Monte Carlo. The numerical results show good agreement with field theory calculations for a number of impurity configurations and the cross-over function for the magnetic susceptibility from high- to low-temperature behavior shows the expected data collapse. Our results also give quantitative predictions for experiments on doped spin-1/2 chains.
We apply the transfer-matrix density matrix renormalization group (DMRG) approach to frustrated quantum spin chains, down to T=0.025KT\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.025\mathrm{K}T=0.025K in the thermodynamic limit. The incommensurability problem of exact diagonalization and the negative sign problem of quantum Monte Carlo vanish completely. For illustration, we give results for chains with next-nearest-neighbor frustration and for the delta chain, which has been a testbed for many thermodynamic methods. The DMRG approach proves to be an extremely powerful method for the old problem of the thermodynamics of frustrated systems.
The density matrix renormalization group (DMRG) method and its applications to finite temperatures and two-dimensional systems are reviewed. The basic idea of the original DMRG method, which allows precise study of the ground-state properties and low-energy excitations, is presented for models which include long-range interactions. The DMRG scheme is then applied to the diagonalization of the quantum transfer matrix for one-dimensional systems, and a reliable algorithm at finite temperatures is formulated. Dynamic correlation functions at finite temperatures are calculated from the eigenvectors of the quantum transfer matrix with analytical continuation to the real frequency axis. An application of the DMRG method to two-dimensional quantum systems in a magnetic field is demonstrated and reliable results for quantum Hall systems are presented.
Based on transfer-matrix density matrix renormalization group (TMRG) method, a general procedure to calculate the finite-temperature pairwise entanglement of low-dimensional quantum chains is proposed. The reduced pairwise density matrix is reconstructed with TMRG, and measures of quantum entanglement can be calculated from the pairwise density matrix. The finite-temperature entanglement of the diamond chain model and the spin trimerized model, which are two typical models revealing 1∕31∕31∕3 plateaus in the magnetization curves, is calculated. For the diamond chain model, the anisotropy coefficient $\ensuremath{\Delta}$ is found to have a great effect on the appearance of the magnetization plateau, and the plateau disappears when $\ensuremath{\Delta}=0.5$. Moreover, our results show that the pairwise entanglement can provide information complementary to that obtained from bulk properties. For the trimerized model, the temperature dependence of the pairwise entanglement is calculated, and the threshold temperature Tc{T}_{c}Tc​, above which the thermal entanglement vanishes, is found to be independent of the external magnetic field BBB. In addition, the scaling behavior of the thermal entanglement is calculated in the Trotter space. With the augmentation of the system in the Trotter direction, we find that the low-temperature entanglement shows obvious variation in the vicinity of quantum phase transition (QPT) point Bc{B}_{c}Bc​ and converges fast in noncritical regions, which provides another way to identify QPT of one-dimensional quantum systems.