Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems

TL;DR

The paper surveys variational renormalization group methods built on matrix product states (MPS) and their higher-dimensional generalization, projected entangled pair states (PEPS). It explains why ground states of local gapped Hamiltonians admit compact MPS/PEPS representations under area laws, enabling scalable algorithms for 1D and 2D quantum spin systems, including DMRG/VMPS, TEBD, and imaginary-time evolution. It also discusses MPOs for finite-temperature and spectral computations, time evolution, and extensions to 2D via PEPS, with iPEPS for infinite lattices; Hastings’ results and MERA are discussed in the context of entanglement scaling. The work provides practical algorithmic frameworks, trade-offs for boundary conditions, and example applications to spin chains, impurity problems, and 2D quantum magnets, illustrating a unifying, information-theoretic perspective on simulating quantum many-body systems.

Abstract

This article reviews recent developments in the theoretical understanding and the numerical implementation of variational renormalization group methods using matrix product states and projected entangled pair states.

Figures (25)

• Figure 1: The entropy of a block of spins scales like the perimeter of the block.
• Figure 2: A one dimensional spin chain with finite correlation length $\xi_{corr}$; $l_{AB}$ denotes the distance between the block $A$ (left) and $B$ (right). Because $\l_{AB}$ is much larger than the correlation length $\xi_{corr}$, the state $\rho_{AB}$ is essentially a product state.
• Figure 4: A general Matrix Product State
• Figure 5: a) An MPS with open boundary conditions as a tensor network. The open bonds correspond to the uncontracted physical indices and the closed bonds to contracted indices arising from taking the products; b) calculating the expectation value of an nearest neighbour operator over a MPS with open boundary conditions by contracting the tensor network completely; c) an extra bond has to be contracted in the case of periodic boundary conditions.
• Figure 6: A MPS in the form of a Cayley tree with coordination number 3: just as in the case of spin chains, the big open circles represent projections of three virtual spins to one physical one.

Theorems & Definitions (2)

• Lemma 1
• Lemma 2