Applying matrix product operators to model systems with long-range interactions

An algorithm is presented which computes a translationally invariant matrix product state approximation of the ground state of an infinite one-dimensional (1D) system. It does this by embedding sites into an approximation of the infinite “environment” of the chain, allowing the sites to relax and then merging them with the environment in order to refine the approximation. By making use of matrix product operators, our approach is able to directly model any long-range interaction that can be systematically approximated by a series of decaying exponentials. We apply these techniques to compute the ground state of the Haldane-Shastry model [Phys. Rev. Lett. 60, 635 (1988) and Phys. Rev. Lett. 60, 639 (1988)] and present the results.

Figure 1

(Color online) Finite state automata representation of a sum of exponentially decaying $XX$ interactions.

Figure 2

(Color online) (Top) Decaying three-, six-, and nine-term exponential approximations to $1/r^2$ potential. (Bottom) Two-point correlator for selected values of $\chi$ using the nine-term exponential approximation; a residual of the correlator for each value of $\chi$ is plotted simultaneously and labeled by ${\epsilon }_{\chi }$.

Figure 3

(Color online) Difference between the energy of the computed state and the energy of the exact ground state plotted for each of the three exponential expansions that were employed.