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arXiv:cond-mat/0401115

Tensor Product Variational Formulation for Quantum Systems

Yukinobu Nishio, Nobuya Maeshima, Andrej Gendiar, Tomotoshi Nishino

Published Jan 8, 2004• 42 citations
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Abstract

We consider a variational problem for the two-dimensional (2D) Heisenberg and XY models, using a trial state which is constructed as a 2D product of local weights. Variational energy is calculated by use of the the corner transfer matrix renormalization group (CTMRG) method, and its upper bound is surveyed. The variational approach is a way of applying the density matrix renormalization group method (DMRG) to infinite size 2D quantum systems.

DOI
10.48550/arxiv.cond-mat/0401115

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•Jan 1, 2011•0 citations
We consider a variational problem for the two-dimensional (2D) Heisenberg and XY models, using a trial state which is constructed as a 2D product of local weights. Variational energy is calculated by use of the the corner transfer matrix renormalization group (CTMRG) method, and its upper bound is surveyed. The variational approach is a way of applying the density matrix renormalization group method (DMRG) to infinite size 2D quantum systems.
We propose a numerical self-consistent method for 3D classical lattice models, which optimizes the variational state written as two-dimensional product of tensors. The variational partition function is calculated by the corner transfer matrix renormalization group (CTMRG), which is a variant of the density matrix renormalization group (DMRG). Numerical efficiency of the method is observed via its application to the 3D Ising model. Variational states represented as products of local tensols have been used widely for calculating the ground-state energy in one-dimensional (1D) quantum systems and the free energy in two-dimensional (2D) classical statistical systems. In 1941 Kramers and Wannier calculated a lower bound of the partition function of the 2D
In order to extend the density-matrix renormalization-group (DMRG) method to two-dimensional systems, we formulate two alternative methods to prepare the initial states. We find that the number of states that is needed for accurate energy calculations grows exponentially with the linear system size. We also analyze how the states kept in the DMRG method manage to preserve both the intrablock and interblock Hamiltonians, which is the key to the high accuracy of the method. We also prove that the energy calculated on a finite cluster is always a variational upper bound.
In the context of tensor network states, we for the first time reformulate the corner transfer matrix renormalization group (CTMRG) method into a variational bilevel optimization algorithm. The solution of the optimization problem corresponds to the fixed-point environment pursued in the conventional CTMRG method, from which the partition function of a classical statistical model, represented by an infinite tensor network, can be efficiently evaluated. The validity of this variational idea is demonstrated by the high-precision calculation of the residual entropy of the dimer model, and is further verified by investigating several typical phase transitions in classical spin models, where the obtained critical points and critical exponents all agree with the best known results in literature. Its extension to three-dimensional tensor networks or quantum lattice models is straightforward, as also discussed briefly.
Tensor networks (TNs) have become one of the most essential building blocks for various fields of theoretical physics such as condensed matter theory, statistical mechanics, quantum information, and quantum gravity. This review provides a unified description of a series of developments in the TN from the statistical mechanics side. In particular, we begin with the variational principle for the transfer matrix of the 2D Ising model, which naturally leads us to the matrix product state (MPS) and the corner transfer matrix (CTM). We then explain how the CTM can be evolved to such MPS-based approaches as density matrix renormalization group (DMRG) and infinite time-evolved block decimation. We also elucidate that the finite-size DMRG played an intrinsic role for incorporating various quantum information concepts in subsequent developments in the TN. After surveying higher-dimensional generalizations like tensor product states or projected entangled pair states, we describe tensor renormalization groups (TRGs), which are a fusion of TNs and Kadanoff-Wilson type real-space renormalization groups, focusing on their fixed point structures. We then discuss how the difficulty in TRGs for critical systems can be overcome in the tensor network renormalization and the multi-scale entanglement renormalization ansatz.
The concept of the corner transfer matrix (CTM) was first discovered by Baxter in 1968, when he derived a set of variational matrix equations, which for finite matrix sizes, could be numerically solved to obtain a sequence of approximations for the statistical mechanical properties of a system of monomers and dimers on a rectangular lattice [1]. It was not until 1978, however, in a seminal paper entitled "Variational Approximations for Square Lattice Models in Statistical Mechanics" [4], that Baxter outlined his CTM variational method, which brought to light the potential power of the former objects to obtain numerics and series expansions for unsolved models and to calculate the order parameter of solved ones. Subsequent numerical work led to the realisation that the method, though general, was not very efficient; and increasing efficiency required making model specific modifications, which restricted the transferability of the resulting algorithm to other models. The CTM variational method was thus not widely adopted, despite holding much promise. More recently, however, Nishino and Okunishi discovered that White's density matrix renormalisation group (DMRG) algorithm [55, 56] could be efficiently extended to study two-dimensional classical lattice models, if the density matrix was approximated by Baxters CTMs. Numerical tests of their algorithm, the corner transfer matrix renormalisation group (CTMRG) method, were met with much success. Notable among these is its implementation within the infinite projected entangled-pair states (iPEPS) algorithm of Orus and Vidal to simulate the ground state of infinite two-dimensional quantum systems. In this thesis we review CTM derived variational methods: Baxter's original CTM iterative method, and developments of the CTM concept within the DMRG algorithm by Nishino and Okunishi, which led to the CTMRG method to calculate critical phenomena of classical and quantum systems. This will begin with elucidating the theoretical formalisms of both methods in two dimensions and their generalisations to three dimensions; followed by review of important application, namely, within the iPEPS algorithm of Orus and Vidal to numerically study infinite planar quantum systems.