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arXiv:2307.08083 · PRX Quantum

Generating Function for Projected Entangled-Pair States

Wei-Lin Tu, Laurens Vanderstraeten, Norbert Schuch, Hyun‐Yong Lee, Naoki Kawashima, Ji-Yao Chen

Published Mar 1, 2024• 16 citations
arXivHTMLPDF

Abstract

Diagrammatic summation is a common bottleneck in modern applications of projected entangled-pair states, especially in computing low-energy excitations of a two-dimensional quantum many-body system. To solve this problem, here we extend the generating-function approach for tensor-network diagrammatic summation, a scheme previously proposed in the context of matrix product states. Taking the form of a one-particle excitation, we show that the excited state can be computed efficiently in the generating-function formalism, which can further be used in evaluating the dynamical structure factor of the system. Our benchmark results for the spin- 1 / 2 transverse-field Ising model and Heisenberg model on the square lattice provide a desirable accuracy, showing good agreement with known results. We then study the spin- 1 / 2 J 1 - J 2 model on the same lattice and investigate the dynamical properties of the putative gapless spin liquid phase. We conclude with a discussion on generalizations to multiparticle excitations. Published by the American Physical Society 2024

DOI
10.1103/prxquantum.5.010335

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Diagrammatic summation is a common bottleneck in modern applications of projected entangled-pair states, especially in computing low-energy excitations of a two-dimensional quantum many-body system. To solve this problem, here we extend the generating-function approach for tensor-network diagrammatic summation, a scheme previously proposed in the context of matrix product states. Taking the form of a one-particle excitation, we show that the excited state can be computed efficiently in the generating-function formalism, which can further be used in evaluating the dynamical structure factor of the system. Our benchmark results for the spin- 1 / 2 transverse-field Ising model and Heisenberg model on the square lattice provide a desirable accuracy, showing good agreement with known results. We then study the spin- 1 / 2 J 1 - J 2 model on the same lattice and investigate the dynamical properties of the putative gapless spin liquid phase. We conclude with a discussion on generalizations to multiparticle excitations. Published by the American Physical Society 2024
The understanding of complex quantum many-body systems has been vastly boosted by tensor network (TN) methods. Among others, excitation spectrum and long-range interacting systems can be studied using TNs, where one however confronts the intricate summation over an extensive number of tensor diagrams. Here, we introduce a set of generating functions, which encode the diagrammatic summations as leading-order series expansion coefficients. Combined with automatic differentiation, the generating function allows us to solve the problem of TN diagrammatic summation. We illustrate this scheme by computing variational excited states and the dynamical structure factor of a quantum spin chain, and further investigating entanglement properties of excited states. Extensions to infinite-size systems and higher dimension are outlined.
Calculation of observables with three-dimensional projected entangled pair states is generally hard, as it requires a contraction of complex multi-layer tensor networks. We utilize the multi-layer structure of these tensor networks to largely simplify the contraction. The proposed approach involves the usage of the layer structure both to simplify the search for the boundary projected entangled pair states and the single-layer mapping of the final corner transfer matrix renormalization group contraction. We benchmark our results on the cubic lattice Heisenberg model, reaching the bond dimension D = 7 , and find a good agreement with the previous results. Published by the American Physical Society 2024
Numerical treatment of two-dimensional strongly-correlated systems is both extremely challenging and of fundamental importance. Infinite projected entangled-pair states (PEPS), a class of tensor networks, have demonstrated cutting-edge performance for ground-state calculations, working directly in the thermodynamic limit. Furthermore, in recent years the application of PEPS has been extended to also low-lying excited states, using an ansatz that targets quasiparticle states above the ground state with high accuracy. A major technical challenge for those simulations is the accurate evaluation of summations of two- and three-point correlation functions with reasonable computational cost. In this paper, we show how a reformulation of nnn-point functions in the context of PEPS leads to extra contributions to the results that prove to play an important role. Benchmarks for the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model illustrate the improved precision, efficiency, and stability of the simulations compared to previous approaches. Leveraging automatic differentiation to generate the most tedious and error-prone parts of the computation, the straightforward implementation presented here is a step towards broader adoption of the PEPS excitation ansatz in future applications.
We present an extension of a framework for simulating single quasiparticle or collective excitations on top of strongly correlated quantum many-body ground states using infinite projected entangled pair states, a tensor network ansatz for two-dimensional wave functions in the thermodynamic limit. Our approach performs a systematic summation of locally perturbed states in order to obtain excited eigenstates localized in momentum space, using the corner transfer matrix method, and generalizes the framework to arbitrary unit cell sizes, the implementation of global Abelian symmetries and fermionic systems. Results for several test cases are presented, including the transverse Ising model, the spin-1/21/21/2 Heisenberg model, and a free fermionic model, to demonstrate the capability of the method to accurately capture dispersions. We also provide insight into the nature of excitations at the $k=(\ensuremath{\pi},0)$ point of the Heisenberg model.
Numerical treatment of two dimensional strongly-correlated systems is both extremely challenging and of fundamental importance. Infinite projected entangled-pair states (PEPS), a class of tensor networks, have demonstrated cutting-edge performance for ground state calculations, working directly in the thermodynamic limit. Furthermore, in recent years the application of PEPS has been extended to also low-lying excited states, using an ansatz that targets quasiparticle states above the ground state with high accuracy. A major technical challenge for those simulations is the accurate evaluation of summations of two- and three-point correlation functions with reasonable computational cost. In this work, we show how a reformulation of nnn-point functions in the context of PEPS leads to extra contributions to the results that prove to play an important role. Benchmarks for the frustrated J1−J2J_1-J_2J1​−J2​ Heisenberg model illustrate the improved precision, efficiency and stability of the simulations compared to previous approaches. Leveraging automatic differentiation to generate the most tedious and error-prone parts of the computation, the straightforward implementation presented here is a step towards broader adoption of the PEPS excitation ansatz in future applications.
The tensor network algorithm, a family of prevalent numerical methods for quantum many-body problems, aptly captures the entanglement properties intrinsic to quantum systems, enabling precise representation of quantum states. However, its computational cost is notably high, particularly in calculating physical observables like correlation functions. To surmount the computational challenge and enhance efficiency, we propose integrating the Green's function Monte Carlo (GFMC) method with the projected entangled pair state (PEPS) ansatz. This approach combines the high-efficiency characteristics of Monte Carlo with the sign-free nature of tensor network states and proves effective in addressing the computational bottleneck. To showcase its prowess, we apply this hybrid approach to investigate the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice, a model notorious for its sign problem in quantum Monte Carlo simulations. Our results reveal a substantial improvement in the accuracy of ground-state energy when utilizing a preliminary PEPS as the guiding wave function for GFMC. By calculating the structure factor and spin-spin correlation functions, we further characterize the phase diagram, identifying a possible columnar valence-bond state phase within the intermediate parameter range of 0.52<J2/J1<0.580.52<{J}_{2}/{J}_{1}<0.580.52<J2​/J1​<0.58. This comprehensive study underscores the efficacy of our combined approach, demonstrating its ability to accurately simulate frustrated quantum spin systems while ensuring computational efficiency.