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Physica Scripta

A new general approach for obtaining matrix product operators for spin Hamiltonians with periodic boundary conditions

Payal D. Solanki, M. S. Ramkarthik

Published Jan 24, 2023• 0 citations
Publisher

Abstract

Abstract This article concerns the matrix product operators for one-dimensional spin Hamiltonians under open and periodic boundary conditions. A novel approach called ‘the dynamics of blocks’ is proposed to obtain the matrix product operators in the case of Hamiltonians with periodic boundary conditions. This approach works universally for any Hamiltonian with two-particle as well as single-particle interactions. The main significance of the proposed approach is that, the dimensions of the matrix product operators obtained are the same for both open and periodic boundary conditions. This method gives us a generic way of constructing MPOs for any Hamiltonian with arbitrary neighboring interactions, immaterial of how the interaction strength falls. A general numerical implementation of the ‘dynamics of blocks’ method is also carried out. The DMRG algorithm was implemented, and the ground state eigenvalue was obtained using these MPOs to numerically validate the method.

DOI
10.1088/1402-4896/acb5cf

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Efficient MPS Algorithm for Periodic Boundary Conditions and Applications Section: Solid matter Original Author's Text: English Abstract: We present the implementation of an efficient algorithm for the calculation of the spectrum of one-dimensional quantum systems with periodic boundary conditions. This algorithm is based on a matrix product representation for quantum states (MPS) and a similar representation for Hamiltonians and other operators (MPO). It is significantly more efficient for systems of about 100 sites and more than for small quantum systems. We apply the formalism to calculate the ground state and the first excited state of a spin-1 Heisenberg ring and deduce the size of a Haldane gap. The results are compared to previous high-precision DMRG calculations. Furthermore, we study the spin-1 systems with a biquadratic nearest-neighbor interaction and show the first results of an application to a mesoscopic Hubbard ring of spinless fermions, which carries a persistent current. Key words: matrix product representation for quantum states (MPS) and Hamiltonians (MPO), spin-1 Heisenberg ring, density matrix renormalization group (DMRG). List Issues Configure
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We introduce an efficient method to calculate the ground state of one-dimensional lattice models with periodic boundary conditions. The method works in the representation of matrix product states (MPS), related to the density matrix renormalization group (DMRG) method. It improves on a previous approach by Verstraete et al. We introduce a factorization procedure for long products of MPS matrices, which reduces the computational effort from m5{m}^{5}m5 to m3{m}^{3}m3, where mmm is the matrix dimension, and $m\ensuremath{\simeq}100--1000$ in typical cases. We test the method on the S=12S=\frac{1}{2}S=21​ and S=1S=1S=1 Heisenberg chains. It is also applicable to nontranslationally invariant cases. The method makes ground-state calculations with periodic boundary conditions about as efficient as traditional DMRG calculations for systems with open boundaries.
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.
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We investigate the application of matrix product states to the Hubbard model in one spatial dimension with both of open and periodic boundary conditions. We develop the variatinal method that the optimization of the variational parameters is carried out locally and sequentially in the framework of matrix product operators (MPO) by including the sign, due to the anti-commutation relation of fermion operators, in the matrix element of MPO. The numerical accuracy of the ground state energy is examined.