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Cambridge University Press eBooks

Tensor Network Representation of Classical Statistical Models

Tao Xiang

Published Aug 31, 2023• 0 citations
Publisher

Abstract

This chapter introduces the tensor network representation of the partition functions of classical statistical models. A classical statistical model may have different tensor network representations. Using simple examples, we show how to represent a classical statistical model as a tensor network model, for example, on the lattice where the Hamiltonian is defined or on its dual lattice. A tensor network representation of a model defined on a vertex-sharing lattice is also introduced. Finally, we discuss the duality properties of the q-state clock and q-state Potts models and determine their self-dual points using their tensor network models on the square lattice.

DOI
10.1017/9781009398671.004

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Cambridge University Press eBooks•Aug 31, 2023•0 citations
This chapter introduces the tensor network representation of the partition functions of classical statistical models. A classical statistical model may have different tensor network representations. Using simple examples, we show how to represent a classical statistical model as a tensor network model, for example, on the lattice where the Hamiltonian is defined or on its dual lattice. A tensor network representation of a model defined on a vertex-sharing lattice is also introduced. Finally, we discuss the duality properties of the q-state clock and q-state Potts models and determine their self-dual points using their tensor network models on the square lattice.
We have discussed the tensor-network representation of classical statistical or interacting quantum lattice models, and given a comprehensive introduction to the numerical methods we recently proposed for studying the tensor-network states/models in two dimensions. A second renormalization scheme is introduced to take into account the environment contribution in the calculation of the partition function of classical tensor-network models or the expectation values of quantum tensor-network states. It improves significantly the accuracy of the coarse-grained tensor renormalization-group method. In the study of the quantum tensor-network states, we point out that the renormalization effect of the environment can be efficiently and accurately described by the bond vector. This, combined with the imaginary-time evolution of the wave function, provides an accurate projection method to determine the tensor-network wave function. It reduces significantly the truncation error and enables a tensor-network state with a large bond dimension, which is difficult to be accessed by other methods, to be accurately determined.
This chapter introduces the tensor network representation of physical operators, especially the matrix product representation of model Hamiltonians, called the matrix product operators (MPO), and the quantum transfer matrix representation of partition functions with different boundary conditions or with an impurity. The leading eigenvalue and eigenvectors of the quantum transfer matrix determine all thermodynamic quantities. It allows us to investigate thermodynamics without solving the full energy spectra of the Hamiltonian.
For any D-dimensional quantum lattice system, the fidelity between two ground state many-body wave functions is mapped onto the partition function of a D-dimensional classical statistical vertex lattice model with the same lattice geometry. The fidelity per lattice site, analogous to the free energy per site, is well defined in the thermodynamic limit and can be used to characterize the phase diagram of the model. We explain how to compute the fidelity per site in the context of tensor network algorithms, and demonstrate the approach by analyzing the two-dimensional quantum Ising model with transverse and parallel magnetic fields.
This work proposes a minimal model extending the duality between classical statistical spin systems and fermionic systems beyond the case of free fermions. A Jordan-Wigner transformation applied to a two-dimensional tensor network maps the partition sum of a classical statistical mechanics model to a Grassmann variable integral, structurally similar to the path integral for interacting fermions in two dimensions. The resulting model is simple, featuring only two parameters: one governing spin-spin interaction (dual to effective hopping strength in the fermionic picture), the other measuring the deviation from the free fermion limit. Nevertheless, it exhibits a rich phase diagram, partially stabilized by elements of topology, and featuring three phases meeting at a multicritical point. Besides the interpretation as a spin and fermionic system, the model is closely related to loop gas and vertex models and can be interpreted as a parity-preserving (non-unitary) circuit. Its minimal construction makes it an ideal reference system for studying non-linearities in tensor networks and deriving results by means of duality.
We discuss in detail algorithms for implementing tensor network renormalization (TNR) for the study of classical statistical and quantum many-body systems. First, we recall established techniques for how the partition function of a 2D2D2D classical many-body system or the Euclidean path integral of a 1D1D1D quantum system can be represented as a network of tensors, before describing how TNR can be implemented to efficiently contract the network via a sequence of coarse-graining transformations. The efficacy of the TNR approach is then benchmarked for the 2D2D2D classical statistical and 1D1D1D quantum Ising models; in particular the ability of TNR to maintain a high level of accuracy over sustained coarse-graining transformations, even at a critical point, is demonstrated.
This chapter discusses the properties of tree tensor network states and the methods for evaluating the ground state and thermodynamic properties of quantum lattice models on a Bethe lattice or, more generally, a Husimi lattice. It starts with a brief discussion of the canonical form of a tree tensor network state. Then, a canonicalization scheme is proposed. To calculate the ground state through the imaginary time evolution, the full and simple update methods are introduced to renormalize the local tensors. Finally, as the correlation length of a quantum system is finite even at a critical point, an accurate and efficient method is described to compute the thermodynamic quantities of quantum lattice models on the Bethe lattice.