This chapter extends DMRG from real space to an arbitrary basis space in which each basis state, such as a momentum eigenstate or a molecular orbital, serves as an effective lattice site. Unlike in real space, the interaction potentials become nonlocal and off-diagonal in an arbitrary basis representation. To solve this nonlocal problem, one should optimize the order of basis states and introduce the so-called complementary operators to minimize the number of operators whose matrix elements must be computed and stored. We illustrate the momentum-space DMRG using the Hubbard model and discuss its application in other interacting fermion models. Finally, we introduce a DMRG scheme for optimizing the single-particle basis states and their order simultaneously in a more general basis space without momentum conservation.

Implementation of symmetries can significantly enhance the expression power of DMRG and effectively allow us to retain more basis states in the DMRG calculation. This chapter discusses the skills of imposing symmetries, including spin reflection, spatial reflection, continuous U(1) or other symmetries with additive quantum numbers, and non-Abelian SU(2) symmetry, in a DMRG calculation.

This chapter introduces the quantum transfer matrix renormalization group (QTMRG). It is a method of studying the thermodynamic and correlation functions of one-dimensional quantum lattice models. The RG transformation matrices are determined using the criteria presented in the preceding chapter and used to update the transfer matrix and other physical quantities. The spin-1/2 and spin-1 antiferromagnetic Heisenberg models are used to demonstrate the accuracy and efficiency of the method.

This chapter introduces the density matrix renormalization group (DMRG) in real space. The infinite and finite lattice algorithms of DMRG, and the approaches for targeting more than one eigenstate and for implementing DMRG in two dimensions by mapping a two-dimensional lattice onto a one-dimensional one, are discussed. The one-dimensional antiferromagnetic Heisenberg model of both integer and half-integer spins is used to demonstrate the method.

This chapter introduces two kinds of RG methods for solving the leading eigenvalue and eigenvectors of a transfer matrix: TMRG (transfer matrix renormalization group) and CTMRG (corner transfer matrix renormalization group). These methods are developed to study the thermodynamic properties of two-dimensional classical statistical models. Furthermore, in the framework of MPS, the fixed-point equations of these methods are derived, and the steps for efficiently solving these equations are outlined.

This chapter reformulates QTMRG using the language of MPS and introduces the concept of bicanonical MPS and the method of biorthogonalization. The fixed-point equations for determining the local tensors of MPS in a translation-invariant system of one or more than one site in a unit cell are derived. The steps for solving these equations in the scheme of biorthonormalization are discussed.

This chapter starts with an introductory survey on the physical background and historical events that lead to the emergence of the density matrix renormalization group (DMRG) and its tensor network generalization. We then briefly overview the major progress on the renormalization group methods of tensor networks and their applications in the past three decades. The tensor network renormalization was initially developed to solve quantum many-body problems, but its application field has grown constantly. It has now become an irreplaceable tool for investigating strongly correlated problems, statistical physics, quantum information, quantum chemistry, and artificial intelligence.

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.

This chapter introduces the tensor network ansatz for a quantum state whose entanglement entropy obeys the so-called area law with or without logarithmic corrections. This ansatz represents a quantum many-body wave function by a network product of local tensors defined on the lattice sites and treats all tensor elements as variational parameters. It includes, for example, one-dimensional matrix product states (MPS) and two-dimensional projected entangled pair states (PEPS) or projected entangled simplex states (PESS). A typical example is the spin-1 AKLT chain, whose ground state can be exactly represented as an MPS. If a logarithmic correction to the entanglement area law emerges, a tensor network state termed the multi-scale entanglement renormalization ansatz (MERA) describes the entanglement structure of the ground state more accurately in one dimension.

This chapter presents the methods of calculating dynamical correlation functions, including the continued-fraction expansion, Lanczos-DMRG, Lanczos-MPS, Chebyshev-MPS, correction vector, conjugate gradient, and dynamical DMRG methods. In the practical application of the Lanczos-MPS or Chebyshev-MPS method, a reorthogonalization scheme is introduced to optimize all the MPS generated with these methods. As an example of application, we investigate the dynamical spectra of the spin-1/2 Heisenberg model using the Chebyshev-MPS method.

This chapter discusses the methods of solving PEPS or other two-dimensional tensor network states, including variational optimization and the annealing simulation. The variation optimization determines the local terms by minimizing the ground-state energy. The annealing simulation takes the full or simple update strategy to filter out the ground state through the imaginary time evolution. The nonlinear effect arises in evaluating the derivative of uniform PEPS and is avoided by utilizing automatic differentiation. Both variational optimization and the annealing simulation involve a contraction of double-layer tensor network states. This contraction is the primary technical barrier in the study of PEPS. A nested tensor network approach is introduced to combat this difficulty.

This chapter introduces the method for solving time-dependent problems of quantum many-body systems. It includes the pace-keeping DMRG, time-evolving block decimation (TEBD), adaptive time-dependent DMRG, and folded transfer matrix methods. The pace-keeping DMRG, which solves the time-dependent Schrodinger equation, works independently of the dimensionality, nor the model Hamiltonian, with or without impurities. The time-evolving block decimation (TEBD) is more efficient than the pace-keeping DMRG if a one-dimensional Hamiltonian with the nearest-neighboring interactions is studied. The adaptive time-dependent DMRG provides an efficient scheme to implement TEBD with the skill of DMRG. On the other hand, the folded transfer matrix method handles the transfer matrix like TMRG by folding the transfer matrix so that the entanglement entropy along the positive and negative time evolution directions can partially cancel each other. This folding scheme significantly extends the time scale within which a time-dependent problem can be reliably investigated.

Several numerical methods used in the study of tensor network renormalization are introduced, including the power, Lanczos, conjugate gradient, Arnoldi methods, and quantum Monte Carlo simulation.