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arXiv:1607.04016 · Physical review. B./Physical review. B

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

Piotr Czarnik, Marek M. Rams, Jacek Dziarmaga

Published Dec 16, 2016• 69 citations
arXivHTMLPDF

Abstract

A Gibbs operator ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ for a two-dimensional (2D) lattice system with a Hamiltonian $H$ 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.

DOI
10.1103/physrevb.94.235142

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A Gibbs operator e\ensuremath−\ensuremathβH{e}^{\ensuremath{-}\ensuremath{\beta}H}e\ensuremath−\ensuremathβH 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) \ensuremathβ\ensuremath{\beta}\ensuremathβ. Coarse graining the network along \ensuremathβ\ensuremath{\beta}\ensuremathβ 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−\ensuremathβH{e}^{\ensuremath{-}\ensuremath{\beta}H}e\ensuremath−\ensuremathβH. 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.
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.
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.
The variational tensor network renormalization approach to two-dimensional (2D) quantum systems at finite temperature is applied to a model suffering the notorious quantum Monte Carlo sign problem---the orbital eg{e}_{g}eg​ model with spatially highly anisotropic orbital interactions. Coarse graining of the tensor network along the inverse temperature $\ensuremath{\beta}$ yields a numerically tractable 2D tensor network representing the Gibbs state. Its bond dimension DDD---limiting the amount of entanglement---is a natural refinement parameter. Increasing DDD we obtain a converged order parameter and its linear susceptibility close to the critical point. They confirm the existence of finite order parameter below the critical temperature Tc{T}_{c}Tc​, provide a numerically exact estimate of Tc{T}_{c}Tc​, and give the critical exponents within 11%1 of the 2D Ising universality class.
Tensor network methods have become a powerful class of tools to capture strongly correlated matter, but methods to capture the experimentally ubiquitous family of models at finite temperature beyond one spatial dimension are largely lacking. We introduce a tensor network algorithm able to simulate thermal states of two-dimensional quantum lattice systems in the thermodynamic limit. The method develops instances of projected entangled pair states and projected entangled pair operators for this purpose. It is the key feature of this algorithm to resemble the cooling down of the system from an infinite temperature state until it reaches the desired finite-temperature regime. As a benchmark, we study the finite-temperature phase transition of the Ising model on an infinite square lattice, for which we obtain remarkable agreement with the exact solution. We then turn to study the finite-temperature Bose-Hubbard model in the limits of two (hard-core) and three bosonic modes per site. Our technique can be used to support the experimental study of actual effectively two-dimensional materials in the laboratory, as well as to benchmark optical lattice quantum simulators with ultracold atoms.
The projected entangled pair state (PEPS) ansatz can represent a thermal state in a strongly correlated system. We introduce a variational algorithm to optimize this tensor network whose essential ingredient is an auxiliary tree tensor network (TTN). Since the full tensor environment is taken into account, with increasing bond dimension the PEPS-TTN ansatz provides the exact Gibbs state. Our presentation opens with a 1D version for a matrix product state (MPS-TTN) and then generalizes to PEPS-TTN in 2D. Benchmark results in the quantum Ising model are presented.
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).