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arXiv:1205.5636 · Physical Review B

Optimized decimation of tensor networks with super-orthogonalization for two-dimensional quantum lattice models

Shi-Ju Ran, Wei Li, Bin Xi, Zhe Zhang, Gang Su

Published Oct 26, 2012• 67 citations
arXivHTMLPDF

Abstract

An algorithm based on the optimized decimation of tensor networks with super-orthogonalization (ODTNS) that can be applied to simulate efficiently and accurately not only the ground state but also the thermodynamic properties of two-dimensional (2D) quantum lattice models is proposed. By transforming the 2D quantum model into a three-dimensional (3D) closed tensor network (TN) comprised of the tensor product density operator and a 3D brick-wall TN, the free energy of the system can be calculated with the imaginary time evolution, in which the network Tucker decomposition is suggested for the first time to obtain the optimal lower-dimensional approximation on the bond space by transforming the TN into a super-orthogonal form. The efficiency and accuracy of this algorithm are testified, which are fairly comparable with the quantum Monte Carlo calculations. Besides, the present ODTNS scheme can also be applicable to the 2D frustrated quantum spin models with nice efficiency.

DOI
10.1103/physrevb.86.134429

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An algorithm based on the optimized decimation of tensor networks with super-orthogonalization (ODTNS) that can be applied to simulate efficiently and accurately not only the ground state but also the thermodynamic properties of two-dimensional (2D) quantum lattice models is proposed. By transforming the 2D quantum model into a three-dimensional (3D) closed tensor network (TN) comprised of the tensor product density operator and a 3D brick-wall TN, the free energy of the system can be calculated with the imaginary time evolution, in which the network Tucker decomposition is suggested for the first time to obtain the optimal lower-dimensional approximation on the bond space by transforming the TN into a super-orthogonal form. The efficiency and accuracy of this algorithm are testified, which are fairly comparable with the quantum Monte Carlo calculations. Besides, the present ODTNS scheme can also be applicable to the 2D frustrated quantum spin models with nice efficiency.
We propose a novel algorithm with a modified Tucker decomposition for tensor network that allows for efficiently and precisely calculating the ground state and thermodynamic properties of two-dimensional (2D) quantum spin lattice systems, and is coined as the fast linearized tensor renormalization group (fLTRG). Its amazing efficiency and precision are examined by studying the spin-1/2 anisotropic Heisenberg antiferromagnet on a honeycomb lattice, and the results are found to be fairly in agreement with the quantum Monte Carlo calculations. It is also successfully applied to tackle a quasi-2D spin-1/2 frustrated bilayer honeycomb Heisenberg model, where a quantum phase transition from an ordered antiferromagnetic state to a gapless quantum spin liquid phase is found. The thermodynamic behaviors of this frustrated spin system are also explored. The present fLTRG algorithm could be readily extended to other quantum lattice systems.
Tensor-network states (TNS) are a promising but numerically challenging tool for simulating two-dimensional (2D) quantum many-body problems. We introduce an isometric restriction of the TNS ansatz that allows for highly efficient contraction of the network. We consider two concrete applications using this ansatz. First, we show that a matrix-product state representation of a 2D quantum state can be iteratively transformed into an isometric 2D TNS. Second, we introduce a 2D version of the time-evolving block decimation algorithm for approximating of the ground state of a Hamiltonian as an isometric TNS-which we demonstrate for the 2D transverse field Ising model.
We have proposed a novel numerical method to calculate accurately physical quantities of the ground state using the tensor network wave function in two dimensions. The tensor network wave function is determined by an iterative projection approach which uses the Trotter-Suzuki decomposition formula of quantum operators and the singular value decomposition of matrix. The norm of the wave function and the expectation value of a physical observable are evaluated by a coarse-grain tensor renormalization group approach. Our method allows a tensor network wave function with a high bond degree of freedom (such as D=8) to be handled accurately and efficiently in the thermodynamic limit. For the Heisenberg model on a honeycomb lattice, our results for the ground state energy and the staggered magnetization agree well with those obtained by the quantum Monte Carlo and other approaches.
The exact contraction of a generic two-dimensional (2D) tensor network state (TNS) is known to be exponentially hard, making simulation of 2D systems difficult. The recently introduced class of isometric TNS (isoTNS) represents a subset of TNS that allows for efficient simulation of such systems on finite square lattices. The isoTNS ansatz requires the identification of an ``orthogonality column'' of tensors, within which one-dimensional matrix product state (MPS) methods can be used for calculation of observables and optimization of tensors. Here we extend isoTNS to infinitely long strip geometries and introduce an infinite version of the Moses Move algorithm for moving the orthogonality column around the network. Using this algorithm, we iteratively transform an infinite MPS representation of a 2D quantum state into a strip isoTNS and investigate the entanglement properties of the resulting state. In addition, we demonstrate that the local observables can be evaluated efficiently. Finally, we introduce an infinite time-evolving block decimation algorithm (iTEBD2{\mathrm{iTEBD}}^{2}iTEBD2) and use it to approximate the ground state of the 2D transverse field Ising model on lattices of infinite strip geometry.
Strongly correlated layered 2D systems are of central importance in condensed matter physics, but their numerical study is very challenging. Motivated by the enormous successes of tensor networks for 1D and 2D systems, we develop an efficient tensor network approach based on infinite projected entangled-pair states for layered 2D systems. Starting from an anisotropic 3D infinite projected entangled-pair state ansatz, we propose a contraction scheme in which the weakly interacting layers are effectively decoupled away from the center of the layers, such that they can be efficiently contracted using 2D contraction methods while keeping the center of the layers connected in order to capture the most relevant interlayer correlations. We present benchmark data for the anisotropic 3D Heisenberg model on a cubic lattice, which shows close agreement with quantum Monte Carlo and full 3D contraction results. Finally, we study the dimer to Néel phase transition in the Shastry-Sutherland model with interlayer coupling, a frustrated spin model that is out of reach of quantum Monte Carlo due to the negative sign problem.
Isometric tensor network states (isoTNS) generalize the isometric form of the one-dimensional matrix product states to tensor networks in two and higher dimensions. Here, we introduce an alternative isometric form for isoTNS by incorporating auxiliary tensors to represent the orthogonality hypersurface. We implement the time evolving block decimation algorithm on this new isometric form and benchmark the method by computing ground states and the real time evolution of the transverse field Ising model in two dimensions on large square lattices of up to 1250 sites. Our results demonstrate that isoTNS can efficiently capture the entanglement structure of two-dimensional area law states. The short-time dynamics is also accurately reproduced even at the critical point. Our isoTNS formulation further allows for a natural extension to different lattice geometries, such as the honeycomb or kagome lattice.