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Variational optimization with infinite projected entangled-pair states

Philippe Corboz

arXiv:1605.03006v2cond-mat.str-elquant-ph

TL;DR

The paper addresses the challenge of optimizing iPEPS tensors accurately and efficiently for two-dimensional ground states. It introduces iterative variational energy minimization with CTM-based summation of Hamiltonian contributions. Across challenging spin and electron models, the scheme is more accurate than full-update imaginary time evolution, converges faster, and has similar computational cost.

  • Problem

    Optimizing iPEPS tensors requires balancing accuracy and efficiency because imaginary time evolution incurs truncation and Trotter-Suzuki errors and may require many small time steps.

  • Method

    The method sweeps over iPEPS tensors, minimizing the energy of each tensor while using CTM to systematically sum the infinite Hamiltonian contributions.

  • Results

    The variational scheme yields more accurate energies and order parameters than full-update imaginary time evolution, with faster convergence and similar computational cost.

  • Takeaways & Limitations

    Variational optimization provides a broadly applicable route to improved iPEPS simulations of challenging frustrated spin and strongly correlated electron systems.

Abstract

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We present a scheme to perform an iterative variational optimization with infinite projected entangled-pair states (iPEPS), a tensor network ansatz for a two-dimensional wave function in the thermodynamic limit, to compute the ground state of a local Hamiltonian. The method is based on a systematic summation of Hamiltonian contributions using the corner transfer-matrix method. Benchmark results for challenging problems are presented, including the 2D Heisenberg model, the Shastry-Sutherland model, and the t-J model, which show that the variational scheme yields considerably more accurate results than the previously best imaginary time evolution algorithm, with a similar computational cost and with a faster convergence towards the ground state.

I. INTRODUCTION

iPEPS provides an efficient variational ansatz for two-dimensional ground states, but optimizing its tensors remains challenging. The paper introduces variational energy minimization as an alternative to imaginary time evolution, using CTM to handle infinite Hamiltonian contributions.

  • Motivation: Tensor optimization is the central challenge because iPEPS algorithms must find variational parameters that best approximate a Hamiltonian ground state.The ansatz can be powerful, but further improvements are needed for higher accuracy.
  • Optimization challenge: Imaginary time evolution introduces truncation and Trotter-Suzuki errors, while small time steps require many computationally inefficient iterations.The full update accounts for the entire two-dimensional wave function during bond truncation and is more accurate than the simple update.
  • Variational optimization: The proposed variational scheme sweeps over tensors and minimizes the energy of one tensor while keeping the others fixed.It is reported to converge faster and produce considerably more accurate results than full-update imaginary time evolution, even as τ →0.
  • Variational optimization: For iPEPS, CTM systematically sums infinitely many Hamiltonian contributions, while a practical scheme addresses the resulting highly nonlinear tensor optimization.These are additional complications beyond applying the related strategy to finite PEPS.
  • A. iPEPS ansatz: iPEPS represents two-dimensional ground states in the thermodynamic limit with a tensor unit cell whose accuracy is controlled by bond dimension D.Larger unit cells can represent spontaneously broken translational symmetry.

B. Contraction of an iPEPS

The CTM method approximates infinite iPEPS contractions by representing the surrounding environment with corner and edge tensors. Iterative growth and renormalization absorb unit-cell columns or rows while retaining a boundary dimension χ.

  • Environment construction: CTM contracts the infinite reduced-tensor network by constructing four corner tensors for quadrants and four edge tensors for half-rows or half-columns.The same environment construction supports expectation-value calculations for arbitrary unit-cell sizes.
  • CTM iterations: The environment is initialized from boundary tensors and iteratively grown in different directions through left, right, top, and bottom moves.Iterations continue until quantities such as the energy converge.
  • Growth and renormalization: During a growth move, a new tensor column is attached to the boundary and then renormalized using projectors.The renormalization updates the corner and edge tensors after the enlarged boundary is compressed.
  • Growth and renormalization: The boundary bond dimension χ controls the accuracy of the approximate contraction.For a full left move, the entire Lx × Ly unit cell is absorbed into the left boundary.

C. Optimization based on imaginary time evolution

Imaginary time evolution approximates ground-state projection by decomposing the evolution operator into two-site bond operators and repeatedly truncating the resulting state. Simple update is cheaper, whereas full update is more accurate because it uses the entire two-dimensional wave function.

  • ITE procedure: Imaginary time evolution starts from an initial iPEPS and applies a Trotter-Suzuki decomposition into two-site operators on nearest-neighbor bonds.The time step is τ = β/n, and the decomposition error decreases as τ becomes smaller.
  • ITE procedure: After each two-site operation, the resulting state is truncated back to the same iPEPS bond dimension until convergence.The truncation is required because the exact evolution generally increases bond dimensions.
  • Truncation schemes: Simple update performs local singular-value-decomposition truncations, whereas full update incorporates the entire two-dimensional wave function.Simple update is computationally cheaper but less accurate than full update.

A. Basic idea

The method optimizes each iPEPS tensor by solving a generalized eigenvalue problem, while CTM-derived H-environments systematically incorporate the infinite Hamiltonian sum.

  • A. Basic idea: Each tensor update minimizes the energy with the other tensors fixed by solving a generalized eigenvalue problem built from H and N.H and N are reshaped into matrices, while tensor A is reshaped into a vector; the lowest-eigenvalue vector is reshaped into the updated tensor.
  • A. Basic idea: The matrix H is difficult because it contains an infinite sum of Hamiltonian expectation values and depends on the repeatedly occurring iPEPS tensor.This makes the optimization highly nonlinear rather than quadratic.
  • Systematic summation with CTM: CTM H-environment tensors encode Hamiltonian contributions from infinite corners, half-rows, connecting regions, and the center site's neighboring terms.The construction includes corner, edge, horizontal-corner, vertical-corner, and local center-neighbor contributions.
  • Systematic summation with CTM: The CTM construction tracks nearest-neighbor terms systematically and updates H-environments using the same projectors as norm environments.Matching renormalized indices allows the resulting diagrams to be added consistently.
  • Systematic summation with CTM: Additional edge-related tensors retain Hamiltonian contributions between edges so they can later enter the horizontal and vertical corner environments.These tensors may not appear directly in the central expectation-value diagram but are needed to propagate their contributions.

C. Practical schemes

Practical optimization combines generalized-eigenvector updates with one-parameter interpolation and repeated tensor sweeps until the energy converges.

  • C. Practical schemes: Because iPEPS tensors occur infinitely many times, directly replacing a tensor by the lowest-eigenvalue solution is not generally optimal.The dependence of H and N on that tensor makes the local problem highly nonlinear.
  • C. Practical schemes: The line search first evaluates E(1) and E(0.5), accepting the eigenvector solution immediately when E(0.5) is lower.Otherwise, it selects a direction using a small probe h and an initial step size Δ0.
  • C. Practical schemes: The search typically uses only a few energy evaluations and stops once it finds a lower-energy solution rather than optimizing λ exactly at every iteration.The method prioritizes reaching the global minimum after many sweeps over each iteration's local optimum.
  • C. Practical schemes: The algorithm repeats local minimization for every tensor in the unit cell until the desired energy convergence is reached.Environment tensors can be initialized from the previous iteration, requiring only a few additional CTM iterations per step.

IV. BENCHMARK RESULTS

Benchmarks across the Heisenberg, Shastry-Sutherland, and t-J models show that variational optimization improves iPEPS energies and order parameters at each bond dimension.

  • IV. BENCHMARK RESULTS: Variational optimization considerably improves iPEPS results for the Heisenberg, Shastry-Sutherland, and t-J models, including both energies and order parameters.Larger-bond-dimension calculations exploit U(1) symmetry to increase computational efficiency.
  • IV. BENCHMARK RESULTS: The Heisenberg benchmark compares simple update, full update, and variational optimization using energy error, staggered magnetization, Trotter-step dependence, and runtime.The runtime comparison uses D = 4 and χ = 50, while the Trotter comparison uses D = 4.

A. Heisenberg model

For the 2D S=1/2 Heisenberg model, variational iPEPS optimization improves energy and order-parameter accuracy over simple and full updates, while converging faster and approaching reference energies efficiently.

  • A. Heisenberg model: A relative energy error roughly 2× lower than full-update results is obtained with variational optimization, while the staggered magnetization is also considerably improved.The comparison varies the bond dimension D across simple, full, and variational optimization methods.
  • A. Heisenberg model: Even as τ →0, the full update remains less accurate because single-bond truncations are only locally optimal and need not be globally optimal across the ansatz.Applying and truncating all bonds simultaneously could improve global optimality but would be computationally much more expensive.
  • A. Heisenberg model: Variational optimization typically converges faster: for D = 4 and χ = 50, one variational step taking 1 minute outperforms the full update in the long-runtime limit.The comparison starts from the simple-update result on a Macbook Pro laptop.
  • A. Heisenberg model: At D = 6 and χ = 250, the variational energy per site is −0.669408J, close to the Monte Carlo value −0.6694421(4)J with relative error 5e-5.This precision is comparable to 2D DMRG on a width-10 cylinder with m = 3000 states.
  • A. Heisenberg model: A width-12 DMRG cylinder with the same number of states is an order of magnitude worse, while D = 6 iPEPS uses roughly 2.6e3 parameters per tensor versus about 1.8e7 for m = 3000.The iPEPS accuracy applies directly to the infinite system, illustrating a more efficient 2D wave-function representation than MPS.
  • A. Heisenberg model: Overall, full update fails to reach the most optimal fixed-D result, whereas variational optimization gives more accurate results and typically converges faster toward the ground state.These conclusions summarize the Heisenberg benchmark comparison.

B. Shastry-Sutherland model

The variational optimization recovers accurate Shastry–Sutherland states even when initialized incorrectly, outperforming simpler optimization schemes in energy and symmetry preservation.

  • Zero-field plaquette phase: The full update can remain trapped in an antiferromagnetic state near the plaquette phase, whereas variational optimization converges to the correct plaquette state from the wrong initial state.The plaquette state has lower plaquette bond energies and vanishing local spins.
  • Finite-field benchmark: Variational optimization lowers both the symmetry error and the variational energy relative to the simple- and full-update methods.The symmetry error is the mean sitewise standard deviation among symmetry-equivalent sites and vanishes for perfect rotational invariance.

C. t-J model

The variational scheme is also tested on the fermionic t-J model, where it substantially improves energies over full-update results for a representative doping and coupling.

  • Setup: The fermionic calculation uses a fermionic iPEPS ansatz and a formalism that accounts for fermionic exchange statistics.The t-J model is treated as a challenging fermionic system and an effective strongly interacting-limit model of the Hubbard model.
  • t-J benchmark: For J/t = 0.4 and hole density δ = 0.12, variational optimization substantially improves the energy per hole at every tested bond dimension D.The improvement is obtained by adding only 3–5 variational update steps to full-update results.
  • Scope: The t-J benchmark extends the accuracy improvements beyond frustrated spin models to strongly correlated electron systems.The authors identify similar improvements for the 2D Hubbard model, although those results are not shown.

V. SUMMARY AND OUTLOOK

The paper concludes that CTM-based variational optimization improves iPEPS energies and order parameters over full-update imaginary-time evolution while often converging faster.

  • Summary: For fixed bond dimension D, the variational scheme yields more accurate energies and order parameters than the previously best full-update imaginary-time evolution.The conclusion is supported across the paper’s benchmark systems, including Heisenberg, Shastry–Sutherland, and t-J models.
  • Summary: The variational approach can also speed convergence toward the ground state.A key benchmark is the Shastry–Sutherland case where full update fails to recover the correct ground state but variational optimization reaches the plaquette state.
  • Outlook: Next-nearest-neighbor Hamiltonians require only minor modifications, while longer-range interactions require additional H-environment tensors.This marks the supported extension boundary of the presented scheme.
  • Outlook: Alternative implementations can use channel environments or different energy-minimization methods such as conjugate gradients.These options provide routes beyond the specific CTM and generalized-eigenvalue implementation discussed here.

Appendix A: Two-site variational optimization

The appendix extends the variational optimization from one-site updates to two-site updates, allowing two tensors to be optimized together and the bond dimension to change dynamically.

  • Two-site update: A two-site update optimizes two tensors at once, analogously to two-site MPS optimization.The main text discusses one-site updates, where tensors in the unit cell are optimized iteratively one after another.
  • Two-site update: The two-site approach can dynamically adjust the bond dimension during optimization.This is identified as an advantage over the one-site update formulation.
  • Update procedure: The update constructs new tensors by splitting the involved tensors with an SVD, solving a generalized eigenvalue problem, and combining the solution with the initial tensor.The construction includes an interpolation parameter λ before producing the updated tensors.
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