Tensor network methods for quantum lattice systems

Xinliang Lyu, Naoki Kawashima

Tensor-network renormalization group (TNRG) is an efficient real-space renormalization group method for studying the criticality in both classical and quantum lattice systems. Exploiting symmetries of a system in a TNRG algorithm can simplify the implementation of the algorithm and can help produce correct tensor RG flows. Although a general framework for considering a global on-site symmetry has been established, it is still unclear how to incorporate a lattice symmetry in TNRG. As a first step for lattice symmetries, we propose a method to incorporate the lattice-reflection symmetry in the context of a TNRG with entanglement filtering in both two and three dimensions (2D and 3D). To achieve this, we write down a general definition of lattice-reflection symmetry in tensor-network language. Then, we introduce a transposition trick for exploiting and imposing the lattice-reflection symmetry in two basic TNRG operations: projective truncations and entanglement filtering. Using the transposition trick, the detailed algorithms of the TNRG map in both 2D and 3D are laid out, where the lattice-reflection symmetry is preserved and imposed. Finally, we demonstrate how to construct the linearization of the TNRG maps in a given lattice-reflection sector, with the help of which it becomes possible to extract scaling dimensions in each sector separately. Our work paves the way for understanding the lattice-rotation symmetry in TNRG.

Nir Gutman

Simulating many-body quantum systems on a classical computer is difficult due to the large number of degrees of freedom, causing the computational complexity to grow exponentially with system size. Tensor Networks (TN) is a framework that breaks down large tensors into a network of smaller tensors, enabling efficient simulation of certain many-body quantum systems. To calculate expectation values of local observables or simulate nearest-neighbor interactions, a contraction of the entire network is needed. This is a known hard problem, which cannot be done exactly for systems with spatial dimension D>1 and is the major bottleneck in all tensor-network based algorithms. Various approximate-contraction algorithms have been suggested, all with their strengths and weaknesses. Nevertheless, contracting a 2D TN remains a major numerical challenge, limiting the use of TN techniques for many interesting systems. Recently, a close connection between TN and Probabilistic Graphical Models (PGM) has been shown. In the PGM framework, marginals of complicated probability distributions can be approximated using iterative message passing algorithms such as Belief Propagation (BP). The BP algorithm can be adapted to the TN framework as an efficient contraction algorithm. While BP is extremely efficient and easy to parallelize, it often yields inaccurate results for highly correlated quantum states or frustrated systems. To overcome this, we suggest the BlockBP algorithm, which coarse-grains the system into blocks and performs BP between them. This thesis focuses on: (i) development and implementation of the BlockBP algorithm for infinite lattices; (ii) using this algorithm to study the anti-ferromagnetic Heisenberg model on the Kagome lattice in the thermodynamic limit - a frustrated 2D model that is difficult to simulate using existing numerical methods.

William J. Huggins, Edwin Miles Stoudenmire, Norm M. Tubman +2 more

Gang Su

Shenghao Li, Yuanyuan Wu, Yuancheng Luo

Tensor network (TN) algorithms for strongly correlated electron lattice systems have become one of the most important numerical methods for studying such systems. Based on the tensor network algorithm for spin ladder systems, the study of quantum phase transitions (QPTs) in spin ladder systems through string order parameters has shown that quantum information theory provides a universal method for studying quantum phase transitions: without knowing the local or non-local order parameters of the system, or whether local or non-local order parameters exist, the ground-state wave function of the system can be obtained. If the ground-state lacks the traditional Landau long-range order and thus has no local order parameters, or if several non-local order parameters are found through other means, some topological quantum phases can be completely described, thereby greatly enriching the traditional Landau symmetry-breaking phase transition theory.

Wei Li, Bin-Bin Chen, yun-jing liu +1 more