Optimal Matrix Product States for the Heisenberg Spin Chain

TL;DR

This paper derives exact results for the optimal Matrix Product State approximation of the ground state of the infinite Heisenberg spin-1/2 chain, revealing the relationship between different simulation methods and symmetry properties.

Contribution

It introduces an analytical minimization approach for optimal MPS approximation of the Heisenberg chain ground state, clarifying the properties of standard simulation results.

Findings

Standard methods yield the same optimal energy but different properties.

Translational and rotational symmetries cannot be simultaneously preserved in minimal energy MPS.

Explicit constructions for symmetry-preserving MPS are provided.

Abstract

We present some exact results for the optimal Matrix Product State (MPS) approximation to the ground state of the infinite isotropic Heisenberg spin-1/2 chain. Our approach is based on the systematic use of Schmidt decompositions to reduce the problem of approximating for the ground state of a spin chain to an analytical minimization. This allows to show that results of standard simulations, e.g. density matrix renormalization group and infinite time evolving block decimation, do correspond to the result obtained by this minimization strategy and, thus, both methods deliver optimal MPS with the same energy but, otherwise, different properties. We also find that translational and rotational symmetries cannot be maintained simultaneously by the MPS ansatz of minimum energy and present explicit constructions for each case. Furthermore, we analyze symmetry restoration and quantify it to uncover new scaling relations. The method we propose can be extended to any translational invariant Hamiltonian.