Finding the ground states of symmetric infinite-dimensional Hamiltonians: explicit constrained optimizations of tensor networks

TL;DR

This paper introduces a tensor network method for efficiently finding ground states of symmetric infinite-dimensional Hamiltonians, leveraging constrained optimizations of matrix product states without truncation, applicable to highly symmetric infinite-range models.

Contribution

The authors develop a simplified, mathematically elegant tensor network algorithm for ground state optimization in symmetric infinite-dimensional systems, with potential for efficient brute-force solutions.

Findings

Successfully computed the ground state of the U(1)-symmetric infinite-dimensional antiferromagnetic XX Heisenberg model.

Demonstrated the method's effectiveness and explained its features through a toy-model example.

Provided a comparison showing slightly higher polynomial complexity but simpler structure than existing methods.

Abstract

Understanding extreme non-locality in many-body quantum systems can help resolve questions in thermostatistics and laser physics. The existence of symmetry selection rules for Hamiltonians with non-decaying terms on infinite-size lattices can lead to finite energies per site, which deserves attention. Here, we present a tensor network approach to construct the ground states of nontrivial symmetric infinite-dimensional spin Hamiltonians based on constrained optimizations of their infinite matrix product states description, which contains no truncation step, offers a very simple mathematical structure, and other minor advantages at the cost of slightly higher polynomial complexity in comparison to an existing method. More precisely speaking, our proposed algorithm is in part equivalent to the more generic and well-established solvers of infinite density-matrix renormalization-group and variational uniform matrix product states, which are, in principle, capable of accurately representing the ground states of such infinite-range-interacting many-body systems. However, we employ some mathematical simplifications that would allow for efficient brute-force optimizations of tensor-network matrices for the specific cases of highly-symmetric infinite-size infinite-range models. As a toy-model example, we showcase the effectiveness and explain some features of our method by finding the ground state of the U(1)-symmetric infinite-dimensional antiferromagnetic $XX$ Heisenberg model.