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Projected Entangled Pair States at Finite Temperature: Iterative Self-Consistent Bond Renormalization for Exact Imaginary Time Evolution
Piotr Czarnik, Jacek Dziarmaga
TL;DR
After a Suzuki-Trotter gate, PEPS bond growth requires renormalization using a full environment whose tensors have enlarged bond dimension. The paper iteratively optimizes renormalized tensors with a D-dimensional environment, then benchmarks thermal Ising states on finite and infinite square lattices across a finite-temperature transition.
Problem
After each Suzuki-Trotter gate, the bond dimension grows from D to k×D, and preserving decomposition accuracy requires renormalization using the full enlarged-tensor environment.
Source: §I · p. 1Method
The scheme repeatedly optimizes the renormalized tensors using a full environment of D-dimensional tensors, updating that environment until convergence.
Source: §I · p. 2, §VI · p. 4Results
The Ising benchmarks evolve both infinite and finite square lattices across a finite-temperature second-order phase transition.
Source: §I · p. 2Takeaways & Limitations
The protocol brings imaginary-time evolution closer to numerical exactness for obtaining thermal states of strongly correlated systems.
Source: §IX · p. 6
Abstract
from arXiv · showhide
A projected entangled pair state (PEPS) with ancillas can be evolved in imaginary time to obtain thermal states of a strongly correlated quantum system on a 2D lattice. Every application of a Suzuki-Trotter gate multiplies the PEPS bond dimension $D$ by a factor $k$. It has to be renormalized back to the original $D$. In order to preserve the accuracy of the Suzuki-Trotter (S-T) decomposition, the renormalization has in principle to take into account full environment made of the new tensors with the bond dimension $k\times D$. Here we propose a self-consistent renormalization procedure operating with the original bond dimension $D$, but without compromising the accuracy of the S-T decomposition. The iterative procedure renormalizes the bond using full environment made of renormalized tensors with the bond dimension $D$. After every renormalization, the new renormalized tensors are used to update the environment, and then the renormalization is repeated again and again until convergence. As a benchmark application, we obtain thermal states of the transverse field quantum Ising model on a square lattice - both infinite and finite - evolving the system across a second-order phase transition at finite temperature.
I. INTRODUCTION
PEPS extends matrix product states to two dimensions, and ancilla-based imaginary-time evolution provides a route to finite-temperature PEPS. This paper addresses the enlarged-environment cost of bond renormalization with a self-consistent scheme and benchmarks it on the square-lattice quantum Ising model.
I. INTRODUCTION
- PEPS generalizes matrix product states to two dimensions, while finite-temperature states had been explored mostly with MPS and also with PEPS and alternative approaches.The cited alternatives include direct partition-function contraction and linear optimization of local density matrices.Source: §I · p. 1
- The paper’s stated aim is to improve the numerical efficiency of the earlier ancilla-based PEPS approach to finite-temperature evolution.Source: §I · p. 1
- After each Suzuki-Trotter gate, the PEPS bond dimension grows by a factor k and must be reduced from k×D to D using the full environment to preserve decomposition accuracy.That environment contains tensors with the enlarged bond dimension, and its calculation is the most time-consuming part of the algorithm.Source: §I · pp. 1–2
- The proposed scheme repeatedly renormalizes with a full environment of D-dimensional tensors, updating that environment after each step until convergence.The converged tensors become the PEPS tensors after the Suzuki-Trotter gate.Source: §I · p. 2
- The benchmark evolves the quantum Ising model on both infinite and finite square lattices across a finite-temperature second-order phase transition.Source: §I · p. 2
II. PURIFICATION OF THERMAL STATES AS PEPS
Thermal states are represented by purifications in an enlarged spin-and-ancilla Hilbert space. The state is initialized at infinite temperature and represented during evolution as a translationally invariant PEPS.
II. PURIFICATION OF THERMAL STATES AS PEPS
- The infinite-temperature state is represented by a purification in the enlarged Hilbert space, with each spin paired with an ancilla.The initial state can be represented with bond dimension D = 1.Source: §II · p. 2
- The infinite-temperature purification is a product of maximally entangled states between each spin and its ancilla.Source: §II · p. 2
- At finite β, the thermal state ρ(β) ∝ e−βH is obtained from its purification, represented by a translationally invariant PEPS with one tensor per site.The tensor has spin and ancilla indices and four bond indices contracted with nearest-neighbor tensors; the resulting contraction gives the amplitude Ψ_A.Source: §II · p. 2
III. TRANSVERSE FIELD QUANTUM ISING MODEL ON A SQUARE LATTICE
The square-lattice transverse-field quantum Ising model has a ferromagnetic phase and reported thermal and zero-temperature critical points.
III. TRANSVERSE FIELD QUANTUM ISING MODEL ON A SQUARE LATTICE
- The model has nonzero spontaneous magnetization in its ferromagnetic phase at small h and large β.Source: §III · p. 2
- At h = 0 the thermal critical point is β0 = 0.441, while the zero-temperature quantum critical point is h0 = 3.044.The passage cites Ref.21 for the zero-temperature value.Source: §III · p. 2
IV. SUZUKI-TROTTER DECOMPOSITION
The Suzuki-Trotter construction expresses a small imaginary-time step through bond gates and tensor products. Its exact map increases the PEPS bond dimension, motivating subsequent truncation.
IV. SUZUKI-TROTTER DECOMPOSITION
- The second-order Suzuki-Trotter decomposition represents a small time step as a product of operators.Source: §IV · p. 3
- The operator UZZ(dβ) is a product over nearest-neighbor bonds and can be rearranged as a tensor product operator using bond indices s⟨m,m′⟩.The bond indices take values in {0, 1}.Source: §IV · p. 3
- Applying the tensor product operator maps A exactly to B, whose bond dimension is 2D rather than the original D.Source: §IV · p. 3
V. TENSOR RENORMALIZATION
A Suzuki–Trotter gate creates a PEPS tensor with bond dimension 2D, which must be truncated to D. The isometry W performs this renormalization while the method seeks to avoid the doubled-bond overhead without compromising second-order Suzuki–Trotter accuracy.
V. TENSOR RENORMALIZATION
- The isometry W is a variational parameter that determines A′ and is optimized to maximize fidelity between the exact and renormalized PEPS.The fidelity objective involves tensor B, whose bond dimension is doubled.Source: §V · p. 3
- The method seeks to avoid the doubled-bond overhead without compromising the accuracy of the second-order Suzuki–Trotter decomposition.The proposed optimization targets A′ rather than requiring the doubled-bond tensor B in the fidelity calculation.Source: §V · p. 3
- W maps the exact new tensor B from bond dimension 2D back to D, producing the renormalized PEPS tensor A′.The renormalization is chosen to least distort the exact new PEPS.Source: Figure 3 · p. 3, §V · p. 3
VI. SELF-CONSISTENT OPTIMIZATION
The isometry is optimized iteratively against the PEPS environment, updating the other bonds and repeating until convergence. The procedure uses a bond environment matrix to optimize one bond at a time and accepts the converged tensor as the new PEPS tensor.
VI. SELF-CONSISTENT OPTIMIZATION
- The algorithm optimizes one bond’s isometry with other isometries fixed, applies that choice to the other bonds, and repeats until convergence.The converged tensor A′ is accepted after the infinitesimal time step.Source: §VI · pp. 3–4
- Instead of evaluating fidelity with the exact tensor B, the method maximizes the norm of the PEPS with tensor A′ using the same iterative update steps.The replacement is justified when W is close to optimal; for large enough D and optimal W, the fidelity is not distorted.Source: §VI · p. 4
- The bond environment matrix E is formed from the norm network with the chosen bond in the bra left uncontracted and unrenormalized.Its indices are the uncontracted bond indices along the chosen bond.Source: §VI · p. 4, Figure 5 · p. 4
- At fixed E, the isometry is optimized by diagonalizing the symmetric environment matrix and selecting its leading D eigenvectors.Only the symmetric part of E contributes to the norm because the projector is symmetric.Source: §VI · p. 4
VII. BENCHMARK RESULTS
With a small longitudinal bias, the infinite-lattice Ising benchmark crosses its transition smoothly, while local observables converge at smaller environmental bond dimension than long-range correlations. The reported correlator has a finite correlation length of 224.
VII. BENCHMARK RESULTS
- A tiny longitudinal bias is added to smooth the transition at finite βc(h), with results shown for h = 2 3hc.The selected transverse field introduces substantial quantum fluctuations and raises βc above Onsager’s β0.Source: §VII · p. 5
- βc = 0.589 = 1.33β0 marks the steepest order-parameter slope, and ⟨Z⟩ converges for D ≥6 and M ≥12.The calculation uses δ = 10−6.Source: §VII · p. 5, Figure 6 · p. 5
- M ≥20 is needed to converge the finite correlation length ξ ≈224 for the biased long-range correlator.The finite value is attributed to the longitudinal bias.Source: §VII · p. 5, Figure 7 · p. 5
- At δ = 0, ⟨Z⟩ jumps discontinuously near the transition but overlaps the smooth biased curve deeper in the ferromagnetic phase.The passage attributes the jump to lack of convergence in M and says the bias influence becomes negligible deeper in the phase.Source: §VII · p. 5, Figure 6 · p. 5
- ξ = 224 and η = 0.28 for the reported correlator, which is converged at D ≥6 and M ≥24.The correlation function has exponential tails at large R and power-law decay at intermediate R.Source: §VII · p. 5, Figure 7 · p. 5
VIII. FINITE LATTICE
On a finite lattice, the algorithm optimizes bond environments separately and repeats lattice sweeps until convergence. The 11×11 benchmark uses finite size to smooth the transition, without a longitudinal bias.
VIII. FINITE LATTICE
- Because finite-lattice bonds are inequivalent, each bond is optimized separately and sweeps repeat until all bonds converge.The finite-network contraction can use matrix product states instead of corner matrix renormalization.Source: §VIII · p. 5
- The 11×11 diagonal correlator is calculated at δ = 0 because finite size itself smooths the transition.The reported correlation is between sites (3, 3) and (9, 9).Source: §VIII · p. 5, Figure 9 · p. 6
- The correlator uses M_MPS = 16 and is reported as converged for D ≥5 and M_MPS ≥16.The caption identifies the transverse field as g = 2 3gc.Source: Figure 9 · p. 6
IX. CONCLUSION
The paper presents self-consistent PEPS bond renormalization after a Suzuki-Trotter gate, using an environment built from renormalized tensors. Its corner-matrix environment approximates infinite-network sectors with finite tensors.
IX. CONCLUSION
- Self-consistent bond renormalization uses the full environment of renormalized tensors to preserve Suzuki-Trotter accuracy.The paper reports a one-day MATLAB run on a laptop; formal cost is dominated by corner matrix renormalization, scaling as M^3D^6 or O(M^3D^2) in more efficient versions.Source: §IX · p. 6
- The protocol brings imaginary-time evolution closer to numerical exactness for thermal states of strongly correlated systems.This is the conclusion stated for the proposed protocol.Source: §IX · p. 6
Appendix A: Corner matrix renormalization
- Finite corner matrices C and top tensors T approximate infinite-network sectors as an effective environment for the central tensor.Increasing environmental bond dimension M makes the effective environment more accurate; finite correlation length is expected to permit convergence at finite M.Source: Appendix A · p. 7
- Corner renormalization repeatedly enlarges the corner with one tensor a and two T tensors, then converges C and T for observable or bond-environment calculations.The Ising-model tensors are assumed isotropic in their four bond indices, making C and the environmental indices of T symmetric.Source: Appendix A · p. 7
C T C
The corner and top tensors provide a finite effective environment for a central transfer tensor. The gauge accelerator improves the starting environment when the renormalization is iterated after updating the isometry.
C T C
- C and T approximate the infinite sectors so the central transfer tensor sees an environment as close as possible to its exact one.Their environmental bonds have dimension M, and the tensors are iterated to convergence.Source: Figure 10 · p. 7
- Each iteration enlarges C, uses its M leading eigenvectors to form an isometry, and truncates the corner and top tensor back to dimension M.The procedure repeats until the M leading eigenvalues converge; its leading cost in this variant scales as M^3D^6.Source: Figure 11 · p. 7
Appendix B: Gauge accelerator for corner matrix renormalization
- Updating the isometry requires reconverging the environment with the new renormalized tensor A, and repeating until A converges.The iterative environment update is the most time-consuming part of the algorithm.Source: Appendix B · p. 7
- A gauge transformation derived from the overlap of old and new isometries gives T a better starting point for the next environment iteration.The transformation approximates replacing the old isometry with the new one, which cannot be done exactly because the old isometry is not invertible.Source: Appendix B · p. 8, Figure 13 · p. 8
Appendix C: Onsager’s critical point at finite M
A finite PEPS bond dimension can represent the Onsager critical point exactly, but a finite environmental bond dimension M limits the effective environment. At nonzero transverse field, a tiny longitudinal bias keeps the correlation length and required M finite.
Appendix C: Onsager’s critical point at finite M
- At criticality, the correlation length grows roughly as M^2 while spontaneous magnetization decays as M^-1/4, showing that finite M limits the environment rather than finite D.The figure fits are ξ = 1.05M^1.93 and ⟨Z⟩ = 0.784M^-0.246; the text relates the magnetization scaling to ξ^-1/8.Source: Appendix C · p. 8, Figure 14 · p. 9
- Although a finite-D PEPS can represent the Onsager transition exactly, the effective environment converges toward the exact one non-exponentially and finite M cannot accurately provide infinite correlation length.Source: Appendix C · p. 8
- For nonzero transverse field, adding a tiny longitudinal bias makes both the correlation length and the necessary environmental bond dimension M finite.Source: Appendix C · p. 8