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arXiv:1201.6121 · Physical Review B

Matrix-product-based projected wave functions ansatz for quantum many-body ground states

Chung-Pin Chou, Frank Pollmann, Ting-Kuo Lee

Published Jul 19, 2012• 14 citations
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Abstract

We introduce a projected wave function approach based on projection operators in the form of matrix-product operators (MPOs). Our approach allows us to variationally improve the short-range entanglement of a given trial wave function by optimizing the matrix elements of the MPOs while the long-range entanglement is contained in the initial guess of the wave function. The optimization is performed using standard variational Monte Carlo techniques. We demonstrate the efficiency of our approach by considering a one-dimensional model of interacting spinless fermions. In addition, we indicate how to generalize this approach to higher dimensions using projection operators which are based on tensor products.

DOI
10.1103/physrevb.86.041105

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We introduce a projected wave function approach based on projection operators in the form of matrix-product operators (MPOs). Our approach allows us to variationally improve the short-range entanglement of a given trial wave function by optimizing the matrix elements of the MPOs while the long-range entanglement is contained in the initial guess of the wave function. The optimization is performed using standard variational Monte Carlo techniques. We demonstrate the efficiency of our approach by considering a one-dimensional model of interacting spinless fermions. In addition, we indicate how to generalize this approach to higher dimensions using projection operators which are based on tensor products.
We propose an efficient numerical method, which combines the advantages of recently developed tensor-network based methods and standard trial wave functions, to study the ground-state properties of quantum many-body systems. In this approach, we apply a projector in the form of a tensor-product operator to an input wave function, such as a Jastrow-type or Hartree-Fock wave function, and optimize the tensor elements via variational Monte Carlo. The entanglement already contained in the input wave function can considerably reduce the bond dimensions compared to the regular tensor-product state representation. In particular, this allows us to also represent states that do not obey the area law of entanglement entropy. In addition, for fermionic systems, the fermion sign structure can be encoded in the input wave function. We show that the optimized states provide good approximations of the ground-state energy and correlation functions in the cases of two-dimensional bosonic and fermonic systems.
Matrix product states and matrix product operators (MPOs) are one-dimensional tensor networks that underlie the modern density matrix renormalization group (DMRG) algorithm. The use of MPOs accounts for the high level of generality and wide range of applicability of DMRG. However, current algorithms for two-dimensional (2D) tensor network states, known as projected entangled-pair states, rarely employ the associated 2D tensor network operators, projected entangled-pair operators (PEPOs), due to their computational cost and conceptual complexity. To lower these two barriers, we describe how to reformulate a PEPO into a set of tensor network operators that resemble MPOs by considering the different sets of local operators that are generated from sequential bipartitions of the 2D system. The expectation value of a PEPO can then be evaluated on the fly using only the action of MPOs and generalized MPOs at each step of the approximate contraction of the 2D tensor network. This technique allows for the simpler construction and more efficient energy evaluation of 2D Hamiltonians that contain finite-range interactions, and provides an improved strategy to encode long-range interactions that is orders of magnitude more accurate and efficient than existing schemes.
Recent work by Wu et al. [arXiv:1910.11011] proposed a numerical method, so-called matrix product operator--matrix product state (MPO-MPS) method, by which several types of quantum many-body wave functions, in particular, the projected Fermi sea state, can be efficiently represented as a tensor network. In this paper, we generalize the MPO-MPS method to study Gutzwiller projected paired states of fermions, where the maximally localized Wannier orbitals for Bogoliubov quasiparticles/quasiholes have been adapted to improve the computational performance. The study of SO(3)SO(3)SO(3)-symmetric spin-1 chains reveals that this new method has better performance than variational Monte Carlo for gapped states and similar performance for gapless states. Moreover, we demonstrate that dynamic correlation functions can be easily evaluated by this method cooperating with other MPS-based accurate approaches, such as the Chebyshev MPS method.
By taking inspiration from the backflow transformation for correlated systems, we introduce a tensor network Ansatz which extends the well-established matrix product state representation of a quantum many-body wave function. This structure provides enough resources to ensure that states in dimensions larger than or equal to one obey an area law for entanglement. It can be efficiently manipulated to address the ground-state search problem by means of an optimization scheme which mixes tensor-network and variational Monte Carlo algorithms. We benchmark the Ansatz against spin models both in one and two dimensions, demonstrating high accuracy and precision. We finally employ our approach to study the challenging S=1/2S=1/2S=1/2 two-dimensional (2D) ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, demonstrating that it is competitive with the state-of-the-art methods in 2D.
We show how to construct relevant families of matrix product operators (MPOs) in one and higher dimensions. These form the building blocks for the numerical simulation methods based on matrix product states and projected entangled pair states. In particular, we construct translationally invariant MPOs suitable for time evolution, and show how such descriptions are possible for Hamiltonians with long-range interactions. We show how these tools can be exploited for constructing new algorithms for simulating quantum spin systems.
We apply a series of projection techniques on top of tensor networks to compute energies of ground state wave functions with higher accuracy than tensor networks alone with minimal additional cost. We consider both matrix product states as well as tree tensor networks in this work. Building on top of these approaches, we apply fixed-node quantum Monte Carlo, Lanczos steps, and exact projection. We demonstrate these improvements for the triangular lattice Heisenberg model, where we capture up to 57 percent of the remaining energy not captured by the tensor network alone. We conclude by discussing further ways to improve our approach.