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Multifield relativistic continuous matrix product states

Laboratoire de Physique de l’École Normale Supérieure, Mines Paris—PSL, Inria, CNRS, ENS-PSL, Sorbonne Université, PSL Research University, Paris, France

^*Contact author: karantiwana21693@gmail.com
^†Contact author: antoine.tilloy@minesparis.psl.eu

Phys. Rev. D 113, L071901 – Published 7 April, 2026

DOI: https://doi.org/10.1103/154w-vkm8

Abstract

Relativistic continuous matrix product states (RCMPSs) are a powerful variational ansatz for quantum field theories of a single field. However, they inherit a property of their nonrelativistic counterpart that makes them divergent for models with multiple fields, unless a regularity condition is satisfied. This has so far restricted the use of RCMPSs to toy models with a single self-interacting field. We address this long-standing problem by introducing a Riemannian optimization framework, that allows us to minimize the energy density over the regular submanifold of multifield RCMPSs, 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 transition along an $O (2)$ -symmetric parameter line. This makes RCMPSs usable for a far larger class of problems than before.

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Multifield relativistic continuous matrix product states

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