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arXiv:1105.2596 · Journal of Statistical Mechanics Theory and Experiment

Biorthonormal matrix-product-state analysis for the non-Hermitian transfer-matrix renormalization group in the thermodynamic limit

Yukun Huang

Published Jul 6, 2011• 6 citations
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Abstract

We give a thorough biorthonormal matrix-product-state (BMPS) analysis of the transfer-matrix renormalization group (TMRG) for non-Hermitian matrices in the thermodynamic limit. The BMPS is built on dual series of reduced biorthonormal bases for the left and right Perron states of a non-Hermitian matrix. We propose two alternative infinite-size biorthonormal TMRG (iBTMRG) algorithms and compare their numerical performances for both finite and infinite systems. We show that both iBTMRGs produce dual infinite-BMPS (iBMPS) which are translationally invariant in the thermodynamic limit. We also develop an efficient wavefunction transformation of the iBTMRG, in analogy with that of McCulloch for the infinite-DMRG (McCulloch, 2008 arXiv: 0804.2509 ), for predicting the wavefunction as the lattice size is increased. The resulting iBMPS allows for probing bulk properties of the system in the thermodynamic limit without boundary effects and allows for reducing the computational cost to be independent of the lattice size; these findings are illustrated by calculating the magnetization as a function of the temperature and the critical spin–spin correlation in the thermodynamic limit for a 2D classical Ising model.

DOI
10.1088/1742-5468/2011/07/p07003

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We give a thorough biorthonormal matrix-product-state (BMPS) analysis of the transfer-matrix renormalization group (TMRG) for non-Hermitian matrices in the thermodynamic limit. The BMPS is built on dual series of reduced biorthonormal bases for the left and right Perron states of a non-Hermitian matrix. We propose two alternative infinite-size biorthonormal TMRG (iBTMRG) algorithms and compare their numerical performances for both finite and infinite systems. We show that both iBTMRGs produce dual infinite-BMPS (iBMPS) which are translationally invariant in the thermodynamic limit. We also develop an efficient wavefunction transformation of the iBTMRG, in analogy with that of McCulloch for the infinite-DMRG (McCulloch, 2008 arXiv: 0804.2509 ), for predicting the wavefunction as the lattice size is increased. The resulting iBMPS allows for probing bulk properties of the system in the thermodynamic limit without boundary effects and allows for reducing the computational cost to be independent of the lattice size; these findings are illustrated by calculating the magnetization as a function of the temperature and the critical spin–spin correlation in the thermodynamic limit for a 2D classical Ising model.
A biorthonormal transfer-matrix renormalization-group (BTMRG) method for non-Hermitian matrices is presented. This BTMRG produces a dual set of biorthonormal bases to construct the renormalized transfer matrix with only half the dimensions of the matrix of a conventional transfer-matrix renormalization group (TMRG). We show that under generic conditions, such biorthonormal bases always exist. Based on a special E·S·E scheme (where S and E represent the system and environment blocks, respectively, and the two dots in between represent two additional physical sites), the BTMRG method can achieve zero truncation of any reduced state in describing both current left and right Perron states so as to reach a high degree of efficiency and accuracy. We believe that the BTMRG constitutes a more powerful and robust tool than conventional TMRG for non-Hermitian matrices and that it would allow us to better understand the collective behaviors and emerging phenomena of strongly correlated many-body systems. We also show that this scheme is particularly adapted to the calculation of the two-site correlation function of a one-dimensional quantum or two-dimensional classical lattice model.
We seek numerical calculations of partition functions of general Markov random fields (MRFs) by using the biorthonormal transfer-matrix renormalization-group (BTMRG) method. The BTMRG is a modification of the conventional TMRG, a variant of the density-matrix renormalization-group (DMRG), which automatically truncates the Hilbert space so that the properties of large systems can be precisely calculated while the dimensions of the renormalized transfer matrix remain constant. We apply the BTMRG to the decimation of the fundamental transfer matrix for general MRFs. Four binary 2nd-order MRFs are selected for numerical simulations. Results of simulations show that all four models exhibit the phase transition phenomenon. This work shows that our BTMRG method is superior to the conventional TMRG in accuracy, computational speed, and in the possibility of treating a much larger system.
A biorthonormal-block density-matrix renormalization group algorithm is proposed to accurately compute properties of large-scale non-Hermitian many-body systems, in which a renormalized-space partition of the non-Hermitian reduced density matrix is implemented to fulfill the prerequisite for the biorthonormality of the renormalization group (RG) transformation and to optimize the construction of saved Hilbert spaces. A redundancy in saved spaces of the reduced density matrix is exploited to reduce a condition number resulting from the nonunitarity of the left and right transformation matrices, in order to ensure the numerical stability of the RG procedure. The algorithm is successfully applied to an interacting fermionic Su-Schrieffer-Heeger model with nonreciprocal hoppings and staggered complex chemical potential, exhibiting novel many-body phenomena.
I revisit the infinite-size variant of the Density Matrix Renormalization Group (iDMRG) algorithm for obtaining a fixed-point translationally invariant matrix product wavefunction in the context of one-dimensional quantum systems. A crucial ingredient of this algorithm is an efficient transformation for obtaining the matrix elements of the wavefunction as the lattice size is increased, and I introduce here a versatile transformation that is demonstrated to be much more effective than previous versions. The resulting algorithm has a surprisingly close relationship to Vidal's Time Evolving Block Decimation for infinite systems, but allows much faster convergence. Access to a translationally invariant matrix product state allows the calculation of correlation functions based on the transfer matrix, which directly gives the spectrum of all correlation lengths. I also show some advantages of the Matrix Product Operator (MPO) technique for constructing expectation values of higher moments, such as the exact variance $$.
We propose a formalism to study dynamical properties of a quantum many-body system in the thermodynamic limit by studying a finite system with ``infinite boundary conditions'' where both finite-size effects and boundary effects have been eliminated. For one-dimensional systems, infinite boundary conditions are obtained by attaching two boundary sites to a finite system, where each of these two sites effectively represents a semi-infinite extension of the system. One can then use standard finite-size matrix product state techniques to study a region of the system while avoiding many of the complications normally associated with finite-size calculations such as boundary Friedel oscillations. We illustrate the technique with an example of time evolution of a local perturbation applied to an infinite (translationally invariant) ground state, and use this to calculate the spectral function of the S=1S=1S=1 Heisenberg spin chain. This approach is more efficient and more accurate than conventional simulations based on finite-size matrix product state and density-matrix renormalization-group approaches.
We apply the biorthonormal transfer-matrix renormalization group (BTMRG) [Huang, Phys. Rev. E 83, 036702 (2011)] to study low-temperature properties of quantum spin chains. Simulations on anisotropic Heisenberg spin-1/21/21/2 chains demonstrate that the BTMRG outperforms the conventional transfer-matrix renormalization group by successfully accessing far lower temperature than previously reported, while retaining the same level of accuracy. The power of the method is further illustrated by the calculation of the low-temperature specific heat for a frustrated spin chain.