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 $m^5$ to $m^3$, where $m$ is the matrix dimension, and $m\simeq 100–1000$ in typical cases. We test the method on the $S=\frac{1}{2}$ 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.

Figure 1

(Color online) SVD of a product $E_{\mathbb{1}}^{\left[1\right]}\cdots E_{\mathbb{1}}^{\left[l\right]}$ with $m=10$. The logarithm of the singular values ${\sigma }_k$ is shown for different $l$ in the case of a spin 1 Heisenberg chain of length 100 with periodic boundary conditions. The inset shows data for a spin $\frac{1}{2}$ Heisenberg chain.

Figure 2

(Color online) Scaling of the circular version of the algorithm. CPU time per sweep (one update of each site) is measured on a 100 site spin $\frac{1}{2}$ Heisenberg chain for different system sizes. The time is fitted to a function $m^k$.

Figure 3

(Color online) Relative error of the ground-state energy of the spin 1 Heisenberg model versus the dimension $m$ of the reduced Hilbert space. DMRG results with obc and pbc are shown, as well as Matrix Product State results with pbc. The inset shows MPS results with pbc for a spin $\frac{1}{2}$ Heisenberg chain of 100 sites.