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arXiv:2602.22113

Lowering the temperature of two-dimensional fermionic tensor networks with cluster expansions

Sander De Meyer, Atsushi Ueda, Yuchi He, Nick Bultinck, Jutho Haegeman

cond-mat.str-elquant-phSubmitted Feb 25, 2026• Announced Feb 26, 2026• 1 citation
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Abstract

Representing the time-evolution operator as a tensor network constitutes a key ingredient in several algorithms for studying quantum lattice systems at finite temperature or in a non-equilibrium setting. For a Hamiltonian composed of strictly short-ranged interactions, the Suzuki-Trotter decomposition is the main technique for obtaining such a representation. In [B.~Vanhecke, L.~Vanderstraeten and F.~Verstraete, Physical Review A, L020402 (2021)], an alternative strategy, the cluster expansion, was introduced. This approach naturally preserves internal and lattice symmetries and can more easily be extended to higher-order representations or longer-ranged interactions. We extend the cluster expansion to two-dimensional fermionic systems, and employ it to construct projected entangled-pair operator (PEPO) approximations of Gibbs states. We also discuss and benchmark different truncation schemes for multiplying layers of PEPOs together. Applying the resulting framework to a two-dimensional spinless fermion model with attractive interactions, we resolve a clear phase boundary at finite temperature.

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Representing the time-evolution operator as a tensor network constitutes a key ingredient in several algorithms for studying quantum lattice systems at finite temperature or in a non-equilibrium setting. For a Hamiltonian composed of strictly short-ranged interactions, the Suzuki-Trotter decomposition is the main technique for obtaining such a representation. In [B.~Vanhecke, L.~Vanderstraeten and F.~Verstraete, Physical Review A, L020402 (2021)], an alternative strategy, the cluster expansion, was introduced. This approach naturally preserves internal and lattice symmetries and can more easily be extended to higher-order representations or longer-ranged interactions. We extend the cluster expansion to two-dimensional fermionic systems, and employ it to construct projected entangled-pair operator (PEPO) approximations of Gibbs states. We also discuss and benchmark different truncation schemes for multiplying layers of PEPOs together. Applying the resulting framework to a two-dimensional spinless fermion model with attractive interactions, we resolve a clear phase boundary at finite temperature.
Representing the time-evolution operator as a tensor network constitutes a key ingredient in several algorithms for studying quantum lattice systems at finite temperature or in a nonequilibrium setting. For a Hamiltonian composed of strictly short-range interactions, the Suzuki-Trotter decomposition is the main technique for strategy-the cluster expansion. This approach naturally preserves internal and lattice symmetries and can more easily be extended to higher-order representations or longer-range interactions. We extend the cluster expansion to two-dimensional fermionic systems and employ it to construct projected entangled-pair operator (PEPO) approximations of Gibbs states. We also discuss and benchmark different truncation schemes for multiplying layers of PEPOs together. Applying the resulting framework to a two-dimensional spinless fermion model with attractive interactions, we resolve a clear phase boundary at finite temperature.
A Gibbs operator ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ for a two-dimensional (2D) lattice system with a Hamiltonian HHH can be represented by a 3D tensor network, with the third dimension being the imaginary time (inverse temperature) $\ensuremath{\beta}$. Coarse graining the network along $\ensuremath{\beta}$ results in a 2D projected entangled-pair operator (PEPO) with a finite bond dimension. The coarse graining is performed by a tree tensor network of isometries. They are optimized variationally to maximize the accuracy of the PEPO as a representation of the 2D thermal state ${e}^{\ensuremath{-}\ensuremath{\beta}H}$. The algorithm is applied to the two-dimensional Hubbard model on an infinite square lattice. Benchmark results at finite temperature 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.
Within the Projected Entangled Pair State (PEPS) tensor network formalism, a simple update (SU) method has been used to investigate the time evolution of a two-dimensional U(1) critical spin-1/2 spin liquid under Hamiltonian quench [Phys. Rev. B 106, 195132 (2022)]. Here we introduce two different variational frameworks to describe the time dynamics of SU(2)-symmetric translationally-invariant PEPS, aiming to improve the accuracy. In one approach, after using a Trotter-Suzuki decomposition of the time evolution operator in term of two-site elementary gates, one considers a single bond embedded in an environment approximated by a Corner Transfer Matrix Renormalization Group (CTMRG). A variational update of the two tensors on the bond is performed under the application of the elementary gate and then, after symmetrization of the site tensors, the environment is updated. In the second approach, a cluster optimization is performed on a finite (periodic) cluster, maximizing the overlap of the exact time-evolved state with a symmetric finite-size PEPS ansatz. Observables are then computed on the infinite lattice contracting the infinite-PEPS (iPEPS) by CTMRG. We show that the variational schemes outperform the SU method and remain accurate over a significant time interval before hitting the entanglement barrier. Studying the spectrum of the transfer matrix, we find that the asymptotic correlations are very well preserved under time evolution, including the critical nature of the singlet correlations, as expected from the Lieb-Robinson (LR) bound theorem. Consistently, the system (asymptotic) boundary is found to be described by the same Conformal Field Theory of central charge c = 1 c = 1 during time evolution. We also compute the time-evolution of the short distance spin-spin correlations and estimate the LR velocity.
Progress in describing thermodynamic phase transitions in quantum systems is obtained by noticing that the Gibbs operator ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ for a two-dimensional (2D) lattice system with a Hamiltonian HHH can be represented by a three-dimensional tensor network, the third dimension being the imaginary time (inverse temperature) $\ensuremath{\beta}$. Coarse graining the network along $\ensuremath{\beta}$ results in a 2D projected entangled-pair operator (PEPO) with a finite bond dimension DDD. The coarse graining is performed by a tree tensor network of isometries. The isometries are optimized variationally, taking into account full tensor environment, to maximize the accuracy of the PEPO. The algorithm is applied to the isotropic quantum compass model on an infinite square lattice near a symmetry-breaking phase transition at finite temperature. From the linear susceptibility in the symmetric phase and the order parameter in the symmetry-broken phase, the critical temperature is estimated at Tc=0.0606(4)J{\mathcal{T}}_{c}=0.0606(4)JTc​=0.0606(4)J, where JJJ is the isotropic coupling constant between S=12S=\frac{1}{2}S=21​ pseudospins.
Simulating strongly-correlated quantum many-body systems at finite temperatures is a significant challenge in computational physics. In this work, we present a scalable finite-temperature tensor network algorithm for two-dimensional quantum many-body systems. We employ the (fermionic) projected entangled pair state (PEPS) to represent the vectorization of the quantum thermal state and utilize a stochastic reconfiguration method to cool down the quantum states from infinite temperature. We validate our method by benchmarking it against the 2D antiferromagnetic Heisenberg model, the J1J_1J1​-J2J_2J2​ model, and the Fermi-Hubbard model, comparing physical properties such as internal energy, specific heat, and magnetic susceptibility with results obtained from stochastic series expansion (SSE), exact diagonalization, and determinant quantum Monte Carlo (DQMC).
We describe a practical and efficient approach to represent physically realistic long-range interactions in two-dimensional tensor network algorithms via projected entangled-pair operators (PEPOs). We express the long-range interaction as a linear combination of correlation functions of an auxiliary system with only nearest-neighbor interactions. To obtain a smooth and radially isotropic interaction across all length scales, we map the physical lattice to an auxiliary lattice of expanded size. Our construction yields a long-range PEPO as a sum of ancillary PEPOs, each of small, constant bond dimension. This representation enables efficient numerical simulations with long-range interactions using projected entangled pair states.