research28 min studyLesson 569 in Physics

Projected entangled-pair states

Extend virtual-bond representations to two dimensions while separating state expressiveness from environment-contraction cost.

Before you begin

On this page

The question for today

What you’ll understand

Count PEPS tensor entries with open-boundary coordination.
Derive a cut-rank and entropy bound from severed virtual bonds.
Identify state-bond and environment-bond convergence as separate requirements.

A matrix product state follows a chain, so representing a wide two-dimensional system through a one-dimensional ordering can require large bonds across long cuts. Projected entangled-pair states instead place virtual bonds directly on the spatial lattice. Their geometry naturally supports boundary-proportional entanglement, but contracting their expectation values introduces a new obstacle. A compact local tensor description does not imply that its normalization or observables can always be evaluated efficiently or with a trustworthy error bound.

Place virtual indices on the lattice edges

For a square lattice, each bulk tensor has one physical index and four virtual indices, one for each neighboring site. Contract matching virtual labels along every edge to obtain a physical-state amplitude. At open boundaries, edges and corners have fewer virtual legs. With physical dimension d and uniform bond dimension D, a bulk tensor has $dD^4$ entries. This is a raw representation count; gauge freedom and normalization mean these entries are not all independent physical parameters.

The virtual construction can be viewed as entangled pairs on edges followed by a linear map at each vertex to its physical space. The map need not be unitary or a physical operation applied in an experiment. Increasing D enlarges the family of representable states, but no fixed small D represents every two-dimensional many-body wavefunction. Symmetries can reduce storage through block structure, while fermionic states require consistent graded signs or another appropriate fermionic tensor construction.

\psi(\{s_i\})=\sum_{\{\alpha_e\}}\prod_i A_i^{s_i}(\{\alpha_{e\ni i}\}),\qquad A_{\mathrm{bulk}}:\ dD^4\ \text{entries}

s_i labels a d-dimensional physical space and alpha_e runs from 1 to D on each virtual edge. The equation describes an open square-lattice PEPS with scalar coefficients after all virtual indices are contracted.↔ Scroll wide equations horizontally.

A boundary cut bounds entanglement capacity

If a bipartition severs n virtual bonds, its connection space has dimension at most $D^n$ . The physical Schmidt rank is bounded by that number and by the dimensions of the two physical blocks. Entropy therefore grows no faster than n times the logarithm of D. This supplies an area-law capacity at fixed D on a regular two-dimensional lattice. It is an upper bound: many PEPS have substantially smaller rank or entropy because their tensors do not use every virtual direction independently.

An area law is useful motivation rather than a universal approximation theorem for all ground states. Critical correlations, topological structure, fermionic surfaces, and desired global fidelity can impose additional demands. A state may require a larger D than local energies initially suggest. Mapping a two-dimensional lattice onto a narrow MPS cylinder also remains useful, but its one-dimensional bond must carry the entanglement crossing the cylinder width. Comparing methods requires the same geometry, boundaries, and target observable.

\operatorname{rank}_{\mathrm{Schmidt}}\leq\min(d^{N_A},d^{N_B},D^{n_{\mathrm{cut}}}),\qquad S_A\leq n_{\mathrm{cut}}\ln D

N_A and N_B count physical sites on the two sides, with uniform local dimension d. The entropy bound uses natural logarithms and may be tightened by either physical-block dimension.↔ Scroll wide equations horizontally.

Contracting a network is a second approximation problem

The squared norm and expectation numerator form a double-layer tensor network. Contracting rows can grow intermediate boundary spaces exponentially with width, so practical methods approximate the environment using a separate truncation dimension, often denoted chi. Increasing physical-state D and increasing environment chi solve different problems. A better optimized state may still be evaluated inaccurately if its environment is too small. Conversely, exact contraction of a small-D state cannot cure the state-family restriction.

General PEPS contraction is computationally hard; special structures and practical approximation schemes do not establish a universal efficient algorithm. A variational energy is an upper bound when evaluated exactly for the normalized state, but approximate contraction can spoil that numerical bound. Check environment convergence, normalization consistency, and stability across update schemes before extrapolating in D. The spectral lab below illustrates how a larger retained representation can change a local value nonmonotonically. It is a finite sine sum, not a PEPS contractor, quantum wavefunction, or area-law test.

\langle O\rangle=\frac{\operatorname{Contract}(A,\overline A;O)}{\operatorname{Contract}(A,\overline A;I)}

The numerator inserts O into the physical legs of the double-layer network; the denominator is the state norm squared. Approximate contraction must control both quantities consistently, including their ratio and environment cutoff.↔ Scroll wide equations horizontally.

Work it through

Count a small open PEPS

Consider a $4\times4$ open square lattice with $d=2$ and $D=2$ . Count raw tensor entries using actual corner, edge, and interior coordination. Bound Schmidt rank for a vertical cut into equal halves.

Four corners have two virtual legs, eight other boundary sites have three, and four interior sites have four. Their entry counts per tensor are $2(2^2)=8$ , $2(2^3)=16$ , and $2(2^4)=32$ .
The total is $4(8)+8(16)+4(32)=288$ raw entries, compared with $2^{16}=65536$ full product-basis amplitudes. This smaller tensor family is restricted and gauge redundant, not an arbitrary reparameterization of every full state at D equal to two.
The vertical cut crosses four virtual edges, so rank is at most $2^4=16$ . Each physical half has dimension $2^8=256$ , so the virtual bound is tighter; its entropy bound is $4\ln2\simeq2.773$ nats.

The result

The open PEPS uses 288 raw tensor entries and has cut rank at most sixteen, with entropy at most 2.773 nats across that cut.

The living notebook

Explore the model

Interactive lab

Interactive lab · spectral

A retained representation need not change pointwise monotonically

Reset and set mode count $N=1$ , decay $p=0$ at time zero. Predict the sum at $x=\pi/2$ is one. Repeat from Reset with both controls $N=3$ , $p=0$ : the sum becomes zero because the third sine mode cancels the first there. This fixed finite sum $\sum\sin(nx-t)/n^p$ has no PEPS tensors, environment contraction, or many-body dynamics.

Finite sumHighest included mode

Sum at x = π/2: 0.867
Smallest wavelength: 2π / 5
Highest-mode coefficient: 0.2

Mode count N5Coefficient decay p1

t = 0 s

A finite sum of sin(nx − t)/nᵖ. Playback changes t; increasing the cutoff includes finer spatial scales. This is not a proof about a continuum limit or renormalization. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forIncreasing a representation cutoff can change a local readout in either direction, so convergence needs explicit comparisons.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

The state bond D controls the variational family, while an environment cutoff controls its approximate contraction. Both need convergence checks.

Your turn

Try it yourself

A PEPS cut crosses three virtual bonds of dimension four, and both physical block dimensions are at least 64. Bound its Schmidt rank and entropy. Would those bounds certify accurate contraction of a correlation function?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Multiply virtual dimensions across the cut and take the natural logarithm for the entropy bound.

Compare with the solution

The virtual connection dimension is $4^3=64$ , so Schmidt rank is at most 64 and entropy is at most $\ln64=3\ln4\simeq4.159$ nats. The physical blocks do not tighten the stated bound. These are capacity constraints on the represented state. They neither estimate a contraction algorithm’s error nor guarantee that the state approximates a desired Hamiltonian ground state; environment and state convergence remain separate.

Keep the idea

Take it with you

PEPS assigns local physical tensors to a spatial network of auxiliary bonds.
Cut virtual dimensions bound Schmidt rank and area-law entropy capacity.
Compact storage, accurate state optimization, and controlled contraction are distinct computational achievements.

Read further

Original sources and deeper explanations to follow at your own pace.

Optional progress, saved in this browser when available. Revisit any lesson at any time.

Lesson not marked complete.