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Variational tensor network renormalization in imaginary time: benchmark results in the Hubbard model at finite temperature

Piotr Czarnik, Marek M. Rams, Jacek Dziarmaga

arXiv:1607.04016v2cond-mat.str-elquant-ph

TL;DR

Thermal states have been explored less extensively with tensor networks than ground states, and existing numerical methods for the Hubbard model have limited parameter ranges. This paper applies variational imaginary-time coarse-graining to the fermionic Hubbard model and reports results meeting benchmarks in the regime where established methods agree.

  • Problem

    Thermal states have received less tensor-network study than ground states, while DQMC and DCA face sign problems and NLCE breaks down at larger βt.

    Source: §I · p. 1, §II · p. 2

  • Method

    The method coarse-grains a 3D imaginary-time network into a 2D PEPO using a tree of isometries optimized variationally with full tensor environments.

    Source: §I · p. 1, §V · p. 10

  • Results

    At βt = 1 and βt = 2 for densities n = 1, 0.85, and 0.875, results are consistent with DCA, NLCE, and DQMC benchmarks.

    Source: §V · p. 10

  • Takeaways & Limitations

    The method meets benchmark standards in the parameter regime where the other numerical methods give mutually consistent results.

    Source: §V · p. 10

    The current dimensional-reduction scheme is limited to orthogonal projections, and the authors note room for algorithmic improvement.

    Source: §V · p. 10

Abstract

from arXiv · show

A Gibbs operator $e^{-\beta H}$ for a 2D lattice system with a Hamiltonian $H$ can be represented by a 3D tensor network, the third dimension being the imaginary time (inverse temperature) $\beta$. Coarse-graining the network along $\beta$ results in an accurate 2D projected entangled-pair operator (PEPO) with a finite bond dimension. The coarse-graining is performed by a tree tensor network of isometries that are optimized variationally to maximize the accuracy of the PEPO. The algorithm is applied to the two-dimensional Hubbard model on an infinite square lattice. Benchmark results are obtained that are consistent with the best cluster dynamical mean-field theory and power series expansion in the regime of parameters where they yield mutually consistent results.

I. INTRODUCTION

The paper applies variational imaginary-time coarse-graining to the fermionic Hubbard model, testing it where established numerical methods agree. The reported results meet that benchmark test.

I. INTRODUCTION

  • Thermal-state tensor-network work was less developed than ground-state work; earlier PEPS methods were less accurate, while direct local updates were expected to converge slowly with refinement.Source: §I · p. 1

III. ALGORITHM

  • The Gibbs operator is built from Suzuki-Trotter time slices, then coarse-grained along imaginary time by full-environment variationally optimized isometries into a 2D PEPO.Source: §I · p. 1

I. INTRODUCTION

  • In the mutually consistent regime for linked-cluster expansion and cluster dynamical mean-field theory, the Hubbard results meet the benchmark test.Source: §I · pp. 1–2

II. HUBBARD MODEL

  • The benchmark scope is restricted to parameters where DQMC, DCA, and NLCE give mutually consistent results; sign problems and convergence limits constrain their reach.Source: §II · p. 2

A. Suzuki-Trotter decomposition

B. Fermionic tensor networks

Fermionic tensor networks preserve the signs required by anticommutation through parity-aware indices and swap gates. Parity-preserving tensors also allow indices to be rearranged without changing those signs.

B. Fermionic tensor networks

  • Fock-state indices encode the parity of their fermion number, bond indices encode transfer parity, and the parity of an index set is the product of its members' parities.Source: B. Fermionic tensor networks · p. 3
  • Only parity-preserving tensors are used, so a tensor vanishes when the product of its index parities is −1.Source: B. Fermionic tensor networks · p. 3
  • Thanks to fermionic anticommutators, the amplitude acquires a factor (−1)n1(−1)n2, which the tensor-network formalism can account for by introducing rank-4 swap-gate tensors.Source: B. Fermionic tensor networks · p. 3
  • A swap gate contributes −1 when both crossing indices are odd and +1 otherwise, with bond-index parity marking odd-fermion transfer.The crossing indices are odd when they label odd-fermion Fock states.Source: B. Fermionic tensor networks · p. 3
  • For the hopping example, paired swap factors cancel, allowing the corresponding gates to be removed from the final tensor network.Source: B. Fermionic tensor networks · p. 4
  • An index can be dragged over a parity-preserving tensor without changing swap factors, enabling lower-cost tensor contractions.Source: B. Fermionic tensor networks · p. 4

C. Hopping gates

The hopping gate is assembled from local fermionic operators into a three-dimensional Trotter network. Contracting over Fock indices produces elementary tensors whose layers represent the infinitesimal gate and, when stacked, the Gibbs operator.

C. Hopping gates

  • Four local operators are grouped into a rank-6 tensor Fσ(dβ), whose internal structure retains the lattice's Fock ordering.The twisted bond lines are hidden in the compact tensor representation.Source: C. Hopping gates · p. 4
  • A chessboard layer of Fσ(dβ) and F†σ(dβ) contracts shared nearest-neighbor bond indices, with line crossings encoding swap factors.Source: C. Hopping gates · p. 4
  • Three chessboard layers form the hopping gate; two such sandwiches, dressed with on-site gates, make U(dβ), which is contracted over Fock indices into elementary sublattice tensors.The layered network cannot be contracted exactly in practice.Source: C. Hopping gates · p. 4
  • Fusing the bond indices gives an elementary-tensor layer for U(dβ), up to Trotter error controlled by dβ; stacking N layers yields U(β).Source: C. Hopping gates · p. 4, Figure 3 · p. 5

D. Coarse graining and renormalization in imaginary time

Imaginary-time coarse-graining combines tensors step by step while isometries keep bond dimensions tractable. The resulting transfer tensors form a PEPO representation of the Gibbs operator and its partition function.

D. Coarse graining and renormalization in imaginary time

E. Figure of merit

The figure of merit is constructed from a cut-bond environment and maximized over isometries to limit distortion from the bond projections. Translational invariance and iterative optimization enforce self-consistency across equivalent bonds and directions.

E. Figure of merit

  • Contracting transfer tensors inserts a projection at every bond, so the isometries are optimized to limit distortion of the partition function.The projection is onto a D′-dimensional subspace; sufficiently large D′ is expected to make it effectively an identity.Source: E. Figure of merit · p. 7
  • The orthogonal projection ansatz is considered sufficient at half-filling, where particle-hole symmetry makes its tensor environment real-symmetric.More general non-orthogonal projections are deferred to future work.Source: E. Figure of merit · p. 7
  • The bond environment is defined by cutting a bond, and its projected form Pl El Pl is optimized to approximate the original environment El.The figure of merit fl is maximized with respect to the bond isometry.Source: E. Figure of merit · pp. 7–8
  • The same isometries are applied at equivalent bonds and optimized iteratively until self-consistency holds across bonds and directions.Convergence is monitored using two-site reduced density matrices, required here to converge to six decimal points.Source: E. Figure of merit · p. 8

F. Variational optimization

The variational procedure approximates bond environments, then uses tensor environments to update isometries through down- and up-sweeps of the tree. The loop continues until convergence; its dominant numerical cost scales as D^8 and is typically subleading to CTM.

F. Variational optimization

  • The corner transfer matrix method (CTM) supplies a systematically improvable approximation to the otherwise intractable infinite bond environment.Source: F. Variational optimization · p. 8
  • Derivatives of the figure of merit with respect to adjacent transfer tensors are obtained from their environments, which are then converted to PEPO-tensor environments for isometry updates.The procedure forms separate contributions from the A and B sublattices.Source: F. Variational optimization · p. 8
  • Down- and up-sweeps propagate environments through the tree tensor network and update its isometries until the optimization loop converges.An isometry update uses a singular-value decomposition of its environment.Source: F. Variational optimization · p. 8, Figure 10 · p. 8
  • The dominant numerical cost scales as D^8 and is typically subleading to the cost of CTM.Source: F. Variational optimization · p. 9

A. Non-interacting case

The non-interacting, spinless-fermion test shows improving accuracy with increasing bond dimension at βt = 2.56. At D = 16, the relative errors in energy and density are below 0.1%.

A. Non-interacting case

  • The test uses the exactly solvable U = 0 Hamiltonian with one species of spinless fermions, whose finite-temperature Fermi sea is strongly entangled in position space at large β.Source: A. Non-interacting case · p. 9
  • Below 0.1% relative error is obtained for energy E and fermion density n at D = 16 and βt = 2.56.The plotted results compare μ = 0 and μ = 0.5.Source: Figure 13 · p. 9, A. Non-interacting case · p. 9
  • Relative errors in energy and density improve systematically as bond dimension D increases.Source: A. Non-interacting case · p. 9

B. Strongly correlated case

For U/t = 8, the simulations benchmark the Hubbard model at βt = 1 and 2 and densities n = 1, 0.875, and 0.85 against DCA. At D = 20, energies and double occupancies agree well with DCA in the parameter cases where independent methods also support it.

B. Strongly correlated case

C. Numerical details

The numerical results were produced with MATLAB and ncon using real-valued tensors. The reported U/t = 8 simulations used three desktops over three weeks, and the results were checked for convergence in environmental bond dimension and Trotter step.

C. Numerical details

V. CONCLUSION

The paper applies a fermionic variational tensor-network algorithm to the Hubbard model on an infinite square lattice. Its results meet the benchmark standards in the reported regime, while the current dimensional-reduction scheme remains limited to orthogonal projections.

V. CONCLUSION

  • The Gibbs operator is represented as a 3D tensor network with imaginary time as its third dimension, then coarse-grained into a finite-bond-dimension PEPO.Source: §V · p. 10
  • A tree tensor network of isometries is optimized variationally to maximize the accuracy of the final PEPO.Source: §V · p. 10
  • At βt = 1 and 2 and densities n = 1, 0.85, and 0.875, the Hubbard-model results meet the benchmark standards set by mutually consistent DCA, NLCE, and DQMC results.Source: §V · p. 10
  • The fermionic algorithm generalizes the spin-model method from Refs. 27, 28, and its current dimensional-reduction scheme is limited to orthogonal projections.The authors identify projection-scheme design as an area for further algorithmic improvement.Source: §V · p. 10

Appendix A: Corner transfer matrix renormalization

The appendices describe an efficient CTM update, compare variational renormalization with direct imaginary-time evolution, and examine convergence of the renormalized bond-environment spectrum. The reported comparisons show faster convergence with bond dimension for the variational approach and rapid spectral decay as the renormalized dimension grows.

Appendix A: Corner transfer matrix renormalization

  • A CTM up move splits the enlarged west and east half-plane environments, computes projections, and uses them to truncate enlarged edge and corner tensors.Fig. 14 illustrates the steps, including the symmetric split of singular values.Source: Appendix A · p. 11, Figure 14 · p. 11
  • A QR middle step reduces the projection SVD from a D′M × D′M matrix to an M × M matrix, and the modified update converges to the same local observables within the main algorithm’s precision.The authors report that the procedure is almost as stable as the original; for the stated D′ range, edge-tensor renormalization becomes the most expensive part.Source: Appendix A · p. 11
  • The faster CTM move cannot systematically increase M in one step, so the authors combine it with a more expensive move every 10–20 faster moves.The combined strategy is reported to reduce CTM convergence time while allowing systematic increases of M.Source: Appendix A · p. 11

Appendix B: Comparison with direct imaginary time evolution.

  • The variational approach achieves larger accessible bond dimensions and faster convergence of observables in D than direct imaginary-time evolution, even for non-interacting fermions at relatively high temperature.The comparison is shown for 4 ≤ D ≤ 16 for the variational approach and 4 ≤ D ≤ 8 for imaginary-time evolution.Source: Appendix B · p. 12, Figure 15 · p. 12

Appendix C: Spectrum of the bond environment

  • The singular-value spectrum of the renormalized D′ × D′ bond environment decays rapidly and converges with increasing D′, supporting its use to approximate the much larger unrenormalized environment.Source: Appendix C · p. 12, Figure 16 · p. 12
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