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arXiv:1812.03657 · Physical review. B./Physical review. B

Gradient optimization of fermionic projected entangled pair states on directed lattices

Shaojun Dong, Chao Wang, Yong‐Jian Han, Guang‐Can Guo, Lixin He

Published May 30, 2019• 19 citations
arXivHTMLPDF

Abstract

The recently developed stochastic gradient method combined with Monte Carlo sampling techniques [W. Y. Liu et al., Phys. Rev. B 95, 195154 (2017)] offers a low scaling and accurate method to optimize the projected entangled pair states (PEPS). We extended this method to the fermionic PEPS (fPEPS). To simplify the implementation, we introduce a Fermi arrow notation to specify the order of the fermion operators in the virtual entangled EPR pairs. By defining some local operation rules associated with the Fermi arrows, one can implement fPEPS algorithms very similar to that of standard PEPS. We benchmark the method for the interacting spinless fermion models and the $t\text{\ensuremath{-}}J$ models. The numerical calculations show that the gradient optimization greatly improves the results of simple update method. Furthermore, very large virtual bond dimensions ($D$) and truncation dimensions (${D}_{c}$) are necessary to converge the results of these models. The method therefore offers a powerful tool to simulate fermion systems because it has much lower scaling than the direct contraction methods.

DOI
10.1103/physrevb.99.195153

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The recently developed stochastic gradient method combined with Monte Carlo sampling techniques [W. Y. Liu et al., Phys. Rev. B 95, 195154 (2017)] offers a low scaling and accurate method to optimize the projected entangled pair states (PEPS). We extended this method to the fermionic PEPS (fPEPS). To simplify the implementation, we introduce a Fermi arrow notation to specify the order of the fermion operators in the virtual entangled EPR pairs. By defining some local operation rules associated with the Fermi arrows, one can implement fPEPS algorithms very similar to that of standard PEPS. We benchmark the method for the interacting spinless fermion models and the t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J models. The numerical calculations show that the gradient optimization greatly improves the results of simple update method. Furthermore, very large virtual bond dimensions (DDD) and truncation dimensions (Dc{D}_{c}Dc​) are necessary to converge the results of these models. The method therefore offers a powerful tool to simulate fermion systems because it has much lower scaling than the direct contraction methods.
Projected entangled pair states (PEPS) methods have been proven to be powerful tools to solve strongly correlated quantum many-body problems in two dimensions. However, due to the high computational scaling with the virtual bond dimension DDD, in a practical application, PEPS are often limited to rather small bond dimensions, which may not be large enough for some highly entangled systems, for instance, frustrated systems. Optimization of the ground state using the imaginary time evolution method with a simple update scheme may go to a larger bond dimension. However, the accuracy of the rough approximation to the environment of the local tensors is questionable. Here, we demonstrate that by combining the imaginary time evolution method with a simple update, Monte Carlo sampling techniques and gradient optimization will offer an efficient method to calculate the PEPS ground state. By taking advantage of massive parallel computing, we can study quantum systems with larger bond dimensions up to D=10D=10D=10 without resorting to any symmetry. Benchmark tests of the method on the ${J}_{1}\ensuremath{-}{J}_{2}$ model give impressive accuracy compared with exact results.
In a recent contribution [P. Corboz, R. Or\'us, B. Bauer, and G. Vidal, Phys. Rev. B 81, 165104 (2010)] fermionic projected entangled-pair states (PEPSs) were used to approximate the ground state of free and interacting spinless fermion models, as well as the $t\text{\ensuremath{-}}J$ model. This paper revisits these three models in the presence of an additional next-nearest hopping amplitude in the Hamiltonian. First we explain how to account for next-nearest neighbor Hamiltonian terms in the context of fermionic PEPS algorithms based on simulating time evolution. Then we present benchmark calculations for the three models of fermions and compare our results against analytical, mean-field, and variational Monte Carlo results, respectively. Consistent with previous computations restricted to nearest neighbor Hamiltonians, we systematically obtain more accurate (or better converged) results for gapped phases than for gapless ones.
We adapt and optimize the projected-pair-entangled-state (PEPS) algorithm on finite lattices (fPEPS) for two-dimensional Hubbard models and apply the algorithm to the Hubbard model with nearest-neighbor hopping on a square lattice. In particular, we formulate the PEPS algorithm using projected entangled pair operators, incorporate SU(2) symmetry in all tensor indices, and optimize the PEPS using both iterative-diagonalization-based local bond optimization and gradient-based optimization of the PEPS. We discuss the performance and convergence of the algorithm for the Hubbard model on lattice sizes of up to $8\ifmmode\times\else\texttimes\fi{}8$ for PEPS states with U(1) symmetric bond dimensions of up to D=8D=8D=8 and SU(2) symmetric bond dimensions of up to D=6D=6D=6. Finally, we comment on the relative and overall efficiency of schemes for optimizing fPEPS.
We explain how to implement, in the context of projected entangled-pair states (PEPSs), the general procedure of fermionization of a tensor network introduced in P. Corboz and G. Vidal, Phys. Rev. B 80, 165129 (2009). The resulting fermionic PEPS, similar to previous proposals, can be used to study the ground state of interacting fermions on a two-dimensional lattice. As in the bosonic case, the cost of simulations depends on the amount of entanglement in the ground state and not directly on the strength of interactions. The present formulation of fermionic PEPS leads to a straightforward numerical implementation that allowed us to recycle much of the code for bosonic PEPS. We demonstrate that fermionic PEPS are a useful variational ansatz for interacting fermion systems by computing approximations to the ground state of several models on an infinite lattice. For a model of interacting spinless fermions, ground state energies lower than Hartree-Fock results are obtained, shifting the boundary between the metal and charge-density wave phases. For the $t\text{\ensuremath{-}}J$ model, energies comparable with those of a specialized Gutzwiller-projected ansatz are also obtained.
We present an improved version of the algorithm contracting and optimizing finite projected entangled pair states (fPEPS) in conjunction with projected entangled pair operators (PEPOs). Our work has two components to it. First, we explain in detail the characteristic contraction patterns that occur in fPEPS calculations and how to slice them such that peak memory occupation remains minimal while ensuring efficient parallel computation. Second, we combine controlled bond expansion [A. Gleis, J.-W. Li, and J. von Delft, ] with randomized singular value decomposition [V. Rokhlin, A. Szlam, and M. Tygert, ] and apply it throughout the fPEPS algorithm. We present benchmark results for the Hubbard model for system sizes up to 8 × 8 and SU(2) symmetric bond dimension of up to D = 6 for PEPS bonds and χ = 500 for the environment bonds. Finally, we comment on the state and future of the fPEPS-PEPO framework.
The current leading approach to the variational optimization of projected entangled-pair states (PEPS) is based on automatic differentiation, which allows for a convenient evaluation of the energy gradient with respect to the local variational degrees of freedom. However, evaluating the energy gradient not only remains a major computational bottleneck of the optimization procedure, but also suffers from frequent numerical instabilities. In this work, we adopt recent advances in implicit differentiation techniques to address these challenges in PEPS optimization. By reformulating the core step of the gradient computation in terms of a single characteristic equation for the contraction environment, we reduce the cost of the gradient computation and improve its scaling with the problem size. By choosing a suitable parametrization of this characteristic equation based on the intrinsic symmetries of the contraction environment, we can directly remove instabilities from the global gradient computation that would otherwise arise from the derivatives of subroutines of the contraction algorithm. Finally, we demonstrate how this approach drastically simplifies the practical implementation of stable gradient-based PEPS optimization.