Multi-Field Relativistic Continuous Matrix Product States

TL;DR

This paper introduces a Riemannian optimization framework for relativistic continuous matrix product states (RCMPS), enabling their application to multi-field quantum field theories and capturing complex phenomena like phase transitions.

Contribution

It develops a novel Riemannian optimization method for multi-field RCMPS, overcoming divergence issues and broadening their applicability to complex quantum models.

Findings

Successfully applied to a two-field scalar model in 1+1 dimensions

Captured symmetry-breaking phases and BKT transition signatures

Demonstrated improved variational results for multi-field quantum theories

Abstract

Relativistic continuous matrix product states (RCMPS) are a powerful variational ansatz for quantum field theories of a single field. However, they inherit a property of their non-relativistic counterpart that makes them divergent for models with multiple fields, unless a regularity condition is satisfied. This has so far restricted the use of RCMPS to toy models with a single self-interacting field. We address this long standing problem by introducing a Riemannian optimization framework, that allows to minimize the energy density over the regular submanifold of multi-field RCMPS, and thus to retain purely variational results. We demonstrate its power on a model of two interacting scalar fields in $1+1$ dimensions. The method captures distinct symmetry-breaking phases, and the signature of a Berezinskii-Kosterlitz-Thouless (BKT) transition along an $O(2)$-symmetric parameter line. This makes RCMPS usable for a far larger class of problems than before.