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arXiv:2409.05285

Scalable tensor network algorithm for thermal quantum many-body systems in two dimension

Meng Zhang, Hao Zhang, Chao Wang, Lixin He

Published Sep 9, 2024• 0 citations
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Abstract

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 $J_1$-$J_2$ 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).

DOI
10.48550/arxiv.2409.05285

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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).
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 to represent the vectorization of the quantum thermal state, and we utilize a stochastic reconfiguration method to cool down the quantum states from infinite temperature. We validate our method by benchmarking it against the two-dimensional antiferromagnetic Heisenberg model, the ${J}_{1}\ensuremath{-}{J}_{2}$ 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, exact diagonalization, and determinant quantum Monte Carlo.
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.
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.
Based on the scheme of variational Monte Carlo sampling, we develop an accurate and efficient two-dimensional tensor-network algorithm to simulate quantum lattice models. We find that Monte Carlo sampling shows huge advantages in dealing with finite projected entangled pair states, which allows significantly enlarged system size and improves the accuracy of tensor network simulation. We demonstrate our method on the square-lattice antiferromagnetic Heisenberg model up to $32\ifmmode\times\else\texttimes\fi{}32$ sites, as well as a highly frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ model up to $24\ifmmode\times\else\texttimes\fi{}24$ sites. The results, including ground state energy and spin correlations, are in excellent agreement with those of the available quantum Monte Carlo or density matrix renormalization group methods. Therefore, our method substantially advances the calculation of 2D tensor networks for finite systems, and potentially opens a new door towards resolving many challenging strongly correlated quantum many-body problems.
The two-dimensional (2D) Hubbard model, widely believed to capture the essential physics of high-T_c cuprate compounds, has attracted immense research interest ever since the discovery of superconductivity in these materials. Despite its superficial simplicity, the Hubbard model incubates abundant fascinating phenomena owing to the strong interactions between charge carriers, encompassing antiferromagnetism, the pseudogap, and unconventional superconductivity. Yet these very interactions induce strong correlations, rendering canonical perturbative approaches unreliable. With advances in modern computational hardware, numerical techniques — notably tensor network methods and Artificial Intelligence (AI) algorithms — have become indispensable to our expedition. Tensor networks encode the amplitude information of many-body quantum states in a network of interconnected tensors. Leveraging the entanglement area law, tensor networks provide efficient, systematic, and controllable representations of quantum states that would otherwise require an exponentially large number of parameters. Past decades have witnessed the development of various tensor network ansatzes, such as Matrix Product States/Operators (MPS/MPO) for one-dimensional systems and Projected Entangled-Pair States (PEPS) for two-dimensional systems, together with diverse algorithms for optimizing these ansatzes for both zero- and finite temperature scenarios. Over the past years, AI technologies have revolutionized the way we analyze scientific data. The transformer architecture, in particular, has demonstrated exceptional capabilities in the domain of natural language processing. Its attention mechanism excels at capturing long-range correlations in sequential data, making it a promising tool for studying strongly correlated many-body systems. The interpretable nature of the attention mechanism further offers valuable insights into the AI's perception of the underlying physics. In this thesis, we assemble an all-inclusive numerical toolchain for analyzing strongly correlated lattice systems. As a demonstration, we investigate the zero- and finite-temperature properties of the 2D Hubbard model on a square lattice via the infinite PEPS (iPEPS) and the eXponential Tensor Renormalization Group (XTRG), respectively. The native two-dimensional iPEPS tensor network faithfully embodies the entanglement area law and helps settle long-standing debates regarding superconducting order in the ground state of the Hubbard model. The XTRG algorithm efficiently constructs thermal density matrices across a broad temperature range via a highly effective exponential-cooling protocol, thereby enabling detailed probes of pairing and pseudogap phenomena. Afterwards, we generate a comprehensive snapshot dataset which furnishes subsequent training of an optimized encoder-only transformer model. Our novel AI architecture features a Markovian interpretation of the attention design and affords improved parallelism. The collective technological stack opens a brand new avenue for exploring the rich physics of strongly correlated many-body systems.
Aimed at a more realistic classical description of natural quantum systems, we present a two-dimensional tensor network algorithm to study finite temperature properties of frustrated model quantum systems and real quantum materials. For this purpose, we introduce the infinite projected entangled simplex operator ansatz to study thermodynamic properties. To obtain state-of-the-art benchmarking results, we explore the highly challenging spin-1/2 Heisenberg antiferromagnet on the Kagome lattice, a system for which we investigate the melting of the magnetization plateaus at finite magnetic field and temperature. Making a close connection to actual experimental data of real quantum materials, we go on to studying the finite temperature properties of Ca10Cr7O28{\mathrm{Ca}}_{10}{\mathrm{Cr}}_{7}{\mathrm{O}}_{28}Ca10​Cr7​O28​. We compare the magnetization curve of this material in the presence of an external magnetic field at finite temperature with classically simulated data. As the first theoretical tool that incorporates both thermal fluctuations as well as quantum correlations in the study of this material, our work contributes to settling the existing controversy between the experimental data and previous theoretical works on the magnetization process.