PEPS Overview

A projected entangled-pair state (PEPS) assigns one tensor to each lattice site and contracts virtual indices along lattice bonds. This preserves the spatial connectivity of a two-dimensional problem, but expectation values require a difficult two-dimensional contraction. A small state bond dimension does not make its norm exactly cheap to evaluate.

Required background. Tensor-network contraction supplies index and cost conventions. MPS provides the boundary-state representation used in many contraction algorithms.

The square-lattice tensor and the double layer

At an interior square-lattice site use $A^s_{lrud}$, with physical dimension $d$ and four virtual dimensions bounded by $D$. Storage per site is $dD^4$. Boundary legs may have dimension one for a finite patch; periodic or infinite systems need a specified cell.

Contracting the physical index between a ket and bra gives a double tensor,

$$E_{(ll')(rr')(uu')(dd')} =\sum_s A^s_{lrud}\overline{A^s_{l'r'u'd'}}.$$

Each paired virtual leg has dimension $D^2$. Norms and local expectations are contractions of this double-layer network, with operator insertions replacing selected $E$ tensors. Unlike an open MPS chain, a two-dimensional loop network cannot generally be reduced to a sequence of small exact environments.

Approximate environments introduce a second dimension

Boundary-MPS methods absorb successive rows into a one-dimensional boundary state, then compress that state to dimension $\chi$. Corner-transfer approaches approximate the environment with corner and edge tensors, also controlled by a boundary dimension. The state parameter $D$ and contraction parameter $\chi$ play distinct roles:

• Increasing $D$ enriches the variational state.
• Increasing $\chi$ improves the evaluation and optimization environment for that state.
• Increasing system size or unit cell addresses an additional geometric approximation.

For every fixed $D$, converge observables in $\chi$ before interpreting a change with $D$. A calculation at larger $D$ but insufficient $\chi$ can be less reliable.

Local updates and the environment metric

An imaginary-time gate changes a pair of physical tensors and increases their shared effective bond. A simple update truncates using a local bond-weight approximation to the environment. This is inexpensive and resembles a tree environment; loops are only approximately represented. A full update optimizes the changed tensors using a more complete approximate environment.

The norm to be minimized is physically determined by the environment, not merely by the Frobenius norm of isolated tensors. Environment approximations may yield a noisy or nearly singular local metric. Gauge conditioning, regularization, and residual checks are then necessary. A successful linear solve does not establish that the environment itself is accurate.

A benchmark hierarchy

At $D=1$, PEPS reduces to a product state and all contractions factorize exactly. This is the first check for arbitrary complex local spinors and local operator insertions. Next, on a $2\times2$ open patch, contract the finite network exactly and compare with every approximate environment routine at fixed tensors. This separates contraction errors from optimization errors.

Finally optimize a small Hamiltonian for which exact diagonalization is feasible. With exact contractions, a normalized variational energy cannot lie below the true ground energy. Approximate contractions can violate that bound; an apparently excellent low energy may therefore expose an inaccurate environment rather than a better state.

What a PEPS result must report

Give geometry, boundary conditions or infinite unit cell, $D$, environment dimension, environment stopping tolerance, and update method. Report norm consistency and changes in energy and representative correlations under increasing $\chi$. Near phase competition, use alternative initial tensors and unit cells. A low energy alone does not settle long-distance order or a phase boundary.

This page concerns established numerical controls. It does not claim that a fixed contraction scheme or finite sequence of dimensions guarantees the thermodynamic answer.

Exercises

Why does a PEPS virtual cut crossing $b$ bonds imply an entropy bound $S\leq b\log D$?

Solution

The virtual indices crossing the cut span at most $D^b$ states. Factoring the physical wavefunction through that space bounds its Schmidt rank by $D^b$, so $S\leq\log D^b$. The bound permits an area-law representation but says nothing by itself about how accurately a particular state or its environment is approximated.

References

• Orús, Román, “A practical introduction to tensor networks: Matrix product states and projected entangled pair states,” Annals of Physics 349, 117–158 (2014), arXiv:1306.2164 — representations and approximate contractions.