This chapter discusses the properties of matrix product state (MPS). It starts with a simple proof that the wave function generated by DMRG is an MPS. Then three different but equivalent canonical forms or representations of MPS are introduced. An MPS generally has redundant gauge degrees of freedom on each bond linking two neighboring local tensors. One can convert it into a canonical form by taking a canonical transformation to remove the gauge redundancy in the local tensors. Finally, the implementation of symmetries, including both the U(1) and SU(2) symmetries, is discussed.

This chapter constructs the MPS representation of a quantum state in the continuous limit. It starts with an MPS representation for the corresponding state in a discretized lattice system. Then the limit of the discretized lattice constant going to zero is taken to obtain its continuous presentation. The formulas for determining the expectation values are also derived. Finally, we discuss the scheme of canonicalization and the method for optimizing the local tensors of the continuous MPS.

This chapter discusses the properties of infinite MPS and their associated transfer matrices. The formulas for determining the expectation values of physical observables are derived and expressed using the leading eigenvectors of the transfer matrix. The concept of the string order parameter is introduced and exemplified with the AKLT state, followed by a statement on the condition for the existence of string order. Furthermore, the procedure of canonicalizing an infinite MPS with one or more than one site in a unit cell is discussed.

This chapter introduces two commonly used methods of determining the local tensors of an MPS. The first is the variational optimization method, which determines an MPS by minimizing the energy expectation value. This method is equivalent to solving a generalized eigenequation around the extreme point of the ground-state energy. The second is an update method based on an imaginary time evolution, which cools down a quantum state from finite to zero temperature. We discuss three update approaches: update via canonicalization, full update, and simple update. For an MPS, the canonicalization approach is accurate and easy to implement. However, the full and simple update can be generalized to higher dimensions and applied to, for example, PEPS. The full update is a global minimization approach. It is accurate but has a higher computational cost than the simple update. The simple update is a local optimization approach based on an entanglement mean-field approximation and is easy to implement. Finally, we discuss the purification technique and apply it to evaluate the thermal density matrix or solve a quenched disorder problem in the framework of MPS.

Coarse-graining renormalization aims to reformulate a tensor network model with a coarse-grained one at a larger scale. It has attracted particular attention in recent years because it opens a new avenue to unveil the entanglement structure of a tensor network model under the scaling transformation. This chapter reviews and compares the tensor renormalization group (TRG) and other coarse-graining methods developed in the past two decades. The methods can be divided into two groups according to whether or not the renormalization effect of the environment tensors is incorporated in the optimization of local tensors. The local optimization methods include TRG, HOTRG (a variant of TRG based on the higher-order singular value decomposition), tensor network renormalization (TNR), and loop-TNR. The global optimization methods include the second renormalized TRG and HOTRG, referred to as SRG and HOSRG, respectively. Among all these coarse-graining methods, HOTRG and HOSRG are the only two that can be readily extended and efficiently applied to three-dimensional classical or two-dimensional quantum lattice models.

The purpose of this chapter is to introduce the diagrammatic representation of tensors and tensor network states and discuss some basic properties or formulas of matrices or tensors used in various renormalization group methods. It includes, for example, the singular value decomposition and the polar decomposition of matrices, the higher-order singular value decomposition of tensors, and the automatic differentiation and its backpropagation scheme. Several Trotter-Suzuki decomposition formulas of non-commutate operators are also introduced.

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.

This chapter discusses the truncation criteria in the RG treatment of a non-Hermitian matrix, starting with a modified definition of the reduced density matrix using the leading left and right eigenvectors. As the reduced density matrix so defined is not Hermitian, there is no theorem to protect or guarantee that its eigenvalues are semi-positive definite. This non-Hermitian problem causes trouble in the determination of an optimized truncation scheme. Three truncation schemes for determining the RG transformation matrices are introduced, relying on the canonical diagonalization of the reduced density matrix, biorthonormalization, and lower-rank approximation of the environment density matrix, respectively. The canonical diagonalization scheme is optimal if the reduced density matrix is semi-positive definite. The scheme of biorthonormalization may not be optimal, but it is mathematically more stable.

Truncation of basis states is a vital step in the tensor network renormalization. This chapter introduces the concept of reduced density matrices and discusses the criterion of judging which state should be retained and which not in the basis truncation. In a Hermitian system, the reduced density matrix of a quantum state is semi-positive definite, and its eigenvalues measure the probabilities of the corresponding eigenvectors. Therefore, we should do the truncation according to the eigenvalues of the reduced density matrix. This criterion is equivalent to taking a Schmidt decomposition for the wave function of the quantum state and truncating the basis states according to their singular values. It is also equivalent to maximizing the fidelity of the targeted state before and after truncation. We also introduce the edge and bond density matrices and show that they have the same eigen-spectra as the reduced density matrix.