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arXiv:1003.2743 · Physical Review B

Fermionic implementation of projected entangled pair states algorithm

Iztok Pižorn, Frank Verstraete

Published Jun 9, 2010• 53 citations
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Abstract

We present and implement an efficient variational method to simulate two-dimensional finite-size fermionic quantum systems by fermionic projected entangled pair states. The approach differs from the original one due to the fact that there is no need for an extra string bond for contracting the tensor network. The method is tested on a bilinear fermionic model on a square lattice for sizes up to ten by ten where good relative accuracy is achieved. Qualitatively good results are also obtained for an interacting fermionic system.

DOI
10.1103/physrevb.81.245110

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We present and implement an efficient variational method to simulate two-dimensional finite-size fermionic quantum systems by fermionic projected entangled pair states. The approach differs from the original one due to the fact that there is no need for an extra string bond for contracting the tensor network. The method is tested on a bilinear fermionic model on a square lattice for sizes up to ten by ten where good relative accuracy is achieved. Qualitatively good results are also obtained for an interacting fermionic system.
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.
Projected entangled-pair states (PEPS) provide excellent variational wave functions for ground states of strongly correlated quantum systems in two dimensions. Determining the expectation value of such a PEPS wave function requires the contraction of a two-dimensional tensor network. Here, the authors show that this contraction can itself be formulated as a variational problem, allowing one to compare the accuracies of two commonly used contraction schemes. The authors find good agreement between the different schemes, demonstrating the robustness of PEPS simulations.
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.
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 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.
Tensor network algorithms have proven to be very powerful tools for studying one- and two-dimensional quantum many-body systems. However, their application to three-dimensional (3D) quantum systems has so far been limited, mostly because the efficient contraction of a 3D tensor network is very challenging. In this paper, we develop and benchmark two contraction approaches for infinite projected entangled-pair states (iPEPS) in 3D. The first approach is based on a contraction of a finite cluster of tensors including an effective environment to approximate the full 3D network. The second approach performs a full contraction of the network by first iteratively contracting layers of the network with a boundary iPEPS, followed by a contraction of the resulting quasi-2D network using the corner transfer matrix renormalization group. Benchmark data for the Heisenberg and Bose-Hubbard models on the cubic lattice show that the algorithms provide competitive results compared to other approaches, making iPEPS a promising tool to study challenging open problems in 3D.