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arXiv:2302.04192 · Physical review. B./Physical review. B

Finite projected entangled pair states for the Hubbard model

Markus Scheb, R. M. Noack

Published Apr 7, 2023• 14 citations
arXivHTMLPDF

Abstract

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 $D=8$ and SU(2) symmetric bond dimensions of up to $D=6$. Finally, we comment on the relative and overall efficiency of schemes for optimizing fPEPS.

DOI
10.1103/physrevb.107.165112

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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×\else\texttimes\fi88\ifmmode\times\else\texttimes\fi{}88\ifmmode×\else\texttimes\fi8 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 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.
We demonstrate the use of finite-size fermionic projected entangled pair states, in conjunction with variational Monte Carlo, to perform accurate simulations of the ground state of the 2D Hubbard model. Using bond dimensions of up to D=28, we show that we can surpass state-of-the-art density matrix renormalization group energies that use up to m=32 000 SU(2) multiplets on eight-leg ladders. We further apply our methodology to 10×16, 12×16, and 16×16 lattices at 1/8 hole doping and observe the dimensional crossover between stripe orientations. Our Letter shows the power of finite-size fermionic tensor networks to resolve the physics of the 2D Hubbard model and related problems.
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 (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.
We present a general graph-based projected entangled-pair state (gPEPS) algorithm to approximate ground states of nearest-neighbor local Hamiltonians on any lattice or graph of infinite size. By introducing the structural matrix, which codifies the details of tensor networks on any graphs in any dimension ddd, we are able to produce a code that can be essentially launched to simulate any lattice. We further introduce an optimized algorithm to compute simple tensor updates as well as expectation values and correlators with a mean-field-like effective environments. Though not being variational, this strategy allows to cope with PEPS of very large bond dimension (e.g., D=100D=100D=100) and produces remarkably accurate results in the thermodynamic limit in many situations, and specially when the correlation length is small and the connectivity of the lattice is large. We prove the validity of our approach by benchmarking the algorithm against known results for several models, i.e., the antiferromagnetic Heisenberg model on a chain, star and cubic lattices, the hardcore Bose-Hubbard model on square lattice, the ferromagnetic Heisenberg model in a field on the pyrochlore lattice, as well as the three-state quantum Potts model in field on the kagome lattice and the spin-1 bilinear-biquadratic Heisenberg model on the triangular lattice. We further demonstrate the performance of gPEPS by studying the quantum phase transition of the 2d2d2d quantum Ising model in transverse magnetic field on the square lattice, and the phase diagram of the Kitaev-Heisenberg model on the hyperhoneycomb lattice. Our results are in excellent agreement with previous studies.
Infinite projected entangled pair states (iPEPS) have emerged as a powerful tool for studying interacting two-dimensional fermionic systems. In this review, we discuss the iPEPS construction and some basic properties of this tensor network (TN) ansatz. Special focus is put on (i) a gentle introduction of the diagrammatic TN representations forming the basis for deriving the complex numerical algorithm, and (ii) the technical advance of fully exploiting non-abelian symmetries for fermionic iPEPS treatments of multi-band lattice models. The exploitation of non-abelian symmetries substantially increases the performance of the algorithm, enabling the treatment of fermionic systems up to a bond dimension D=24 D = 24 on a square lattice. A variety of complex two-dimensional (2D) models thus become numerically accessible. Here, we present first promising results for two types of multi-band Hubbard models, one with 2 2 bands of spinful fermions of \mathrm{SU}(2)_\mathrm{spin} \otimes \mathrm{SU}(2)_\mathrm{orb} S U ( 2 ) s p i n ⊗ S U ( 2 ) o r b symmetry, the other with 3 3 flavors of spinless fermions of \mathrm{SU}(3)_\mathrm{flavor} S U ( 3 ) f l a v o r symmetry.