Efficient matrix-product state method for periodic boundary conditions

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 to m³, where m is the matrix dimension, and m100--1000 in typical cases. We test the method on the S=1/2 and $S=1$ 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.