A Generalised Haldane Map from the Matrix Product State Path Integral to the Critical Theory of the $J_1$-$J_2$ Chain

TL;DR

This paper develops a generalized Haldane map from matrix product state path integrals to the critical field theory of the $J_1$-$J_2$ spin chain, revealing topological and symmetry properties of the phase transitions.

Contribution

It introduces a novel continuum limit of MPS path integrals that reproduces the critical field theory with topological terms and emergent symmetries, linking microscopic states to field theory.

Findings

Recovered the correct topological term in the field theory

Linked MPS states to emergent $SO(4)$ symmetry

Clarified the dimerisation transition in the MPS framework

Abstract

We study the $J_1$-$J_2$ spin-$1/2$ chain using a path integral constructed over matrix product states (MPS). By virtue of its non-trivial entanglement structure, the MPS ansatz captures the key phases of the model even at a semi-classical, saddle-point level, and, as a variational state, is in good agreement with the field theory obtained by abelian bosonisation. Going beyond the semi-classical level, we show that the MPS ansatz facilitates a physically-motivated derivation of the field theory of the critical phase: by carefully taking the continuum limit -- a generalisation of the Haldane map -- we recover from the MPS path integral a field theory with the correct topological term and emergent $SO(4)$ symmetry, constructively linking the microscopic states and topological field-theoretic structures. Moreover, the dimerisation transition is particularly clear in the MPS formulation -- an explicit dimerisation potential becomes relevant, gapping out the magnetic fluctuations.