Renormalization and tensor product states in spin chains and lattices

TL;DR

This paper surveys tensor-product-based descriptions of many-body quantum states, arguing that a small-set of structured ansatzes (MPS, TTN, MERA, PEPS) efficiently capture the physics of systems with local interactions due to area-law entanglement. It connects these states to renormalization group ideas, showing how DMRG corresponds to MPS, how TTN and MERA implement hierarchical RG with varying entanglement properties, and how PEPS extends these concepts to higher dimensions. The work discusses practical methods for computing expectation values, ground states, and dynamics within these frameworks, and elaborates on the computational challenges and approximate contraction schemes, especially for PEPS in 2D. It concludes with perspectives on open questions, such as convergence guarantees for PEPS, time evolution limitations, and the development of more efficient higher-dimensional algorithms, possibly aided by Monte Carlo hybrids. Overall, the paper highlights the central role of TPS in understanding and simulating the corner of Hilbert space relevant to realistic quantum many-body systems.

Abstract

We review different descriptions of many--body quantum systems in terms of tensor product states. We introduce several families of such states in terms of known renormalization procedures, and show that they naturally arise in that context. We concentrate on Matrix Product States, Tree Tensor States, Multiscale Entanglement Renormalization Ansatz, and Projected Entangled Pair States. We highlight some of their properties, and show how they can be used to describe a variety of systems.

Figures (13)

• Figure 1: Construction of a MPS in terms of entangled auxiliary particles. (a) Original spin system; (b) We replace each spin by two auxiliary particles (except at the ends of the chain), which are in a maximally entangled state with their neighbors; (c) The final state is obtained after mapping the state of each pair of auxiliary particles locally onto the original spins.
• Figure 2: Graphical representation of an MPS in terms of contracted tensors (tensor network). (a) The set of matrices $A^n$ are represented in terms of a rank--3 tensor where the index $n$ is pointing vertically; (b) We consider the set of tensors corresponding to each spins and (c) contract them according to the horizontal indices; (d) the same can be done with periodic boundary conditions by adding an extra bond on the end spins; (e) Tensor representation of an operator acting on a spin; (f.1) In order to calculate $\langle \Psi|\Psi\rangle$ we contract the tensor corresponding to $\Psi$ with that of $\bar{\Psi}$, giving rise to (f.2) a row of tensors which are contracted to give a number. The tensors can be viewed as matrices (one double-index to the left and another to the right). (g.1) and (g.2) the same but with an expectation value.
• Figure 3: Sequential generation of MPS. (a) Using an ancilla with Hilbert space of $D$ dimensions, we act sequentially on the first, second, etc spins with unitary operators; (b) This process can be understood with the graphical language introduced before. After each interaction, the spins get entangled in a MPS with the ancilla itself. (c) We can replace the ancilla by the $log D$ spin which are to the right of the spin we are acting on.
• Figure 4: Comparison of the two renormalization procedures. (a) At each step, we add a new spin (ball) to the previous system (square) obtaining a new Hilbert space, which we truncate to obtain the one of a smaller dimension (new square). (b) At each step, we take two neighboring systems (squares) and truncate the Hilbert space to obtain the new one of the new systems.
• Figure 5: Tensor network representation of the TPS. (a) Each tensor $T$ is represented by a square with three indices. (b) the fact that $T$ is an isometry can be represented as a line when we contract $T$ and $\bar{T}$. (c) A TTS.