Exploiting translational invariance in Matrix Product State simulations of spin chains with periodic boundary conditions

Abstract

We present a matrix product state (MPS) algorithm to approximate ground states of translationally invariant systems with periodic boundary conditions. For a fixed value of the bond dimension D of the MPS, we discuss how to minimize the computational cost to obtain a seemingly optimal MPS approximation to the ground state. In a chain of N sites and correlation length ξ, the computational cost formally scales as g(D,ξ/N)D^3, where g(D,ξ/N) is a nontrivial function. For ξ<< N, this scaling reduces to D^3, independent of the system size N, making our algorithm N times faster than previous proposals. We apply the method to obtain MPS approximations for the ground states of the critical quantum Ising and Heisenberg spin-1/2 models as well as for the noncritical Heisenberg spin-1 model. In the critical case, for any chain length N, we find a model-dependent bond dimension D(N) above which the polynomial decay of correlations is faithfully reproduced throughout the entire system.

17 pages, 15 figures

References in corpus (17)

• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Efficient simulation of one-dimensional quantum many-body systems
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• DMRG and periodic boundary conditions: a quantum information perspective
• The iTEBD algorithm beyond unitary evolution
• A class of ansatz wave functions for 1D spin systems and their relation to DMRG
• Matrix product operator representations
• Theory of finite-entanglement scaling at one-dimensional quantum critical points
• Scaling of entanglement support for Matrix Product States
• Variational quantum Monte Carlo simulations with tensor-network states
• Product Wave Function Renormalization Group Method
• Numerical Renormalization Group at Criticality
• Efficient Matrix Product State Method for periodic boundary conditions
• Magnetization Process of One-Dimensional Quantum Antiferromagnet: The Product Wavefunction Renormalization Group Approach
• Universal Asymptotic Eigenvalue Distribution of Density Matrices and the Corner Transfer Matrices in the Thermodynamic Limit
• Density Matrix Renormalization Group of Gapless Systems
• Product Wave Function Renormalization Group: construction from the matrix product point of view

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