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 with $N$ sites and correlation length $\ensuremath{\xi}$, the computational cost formally scales as $g(D,\ensuremath{\xi}/N){D}^{3}$, where $g(D,\ensuremath{\xi}/N)$ is a nontrivial function. For $\ensuremath{\xi}\ensuremath{\ll}N$, this scaling reduces to ${D}^{3}$, independent of the system size $N$, making our method $N$ times faster than previous proposals. We apply the algorithm 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.