Generalized Haldane map from the matrix product state path integral to the critical theory of the J1-J2 chain

Fariha Azad; Adam McRoberts; Chris Hooley; Andrew Green

doi:10.1103/PhysRevResearch.7.L012037

Generalized Haldane map from the matrix product state path integral to the critical theory of the J1-J2 chain

Fariha Azad, Adam McRoberts, Chris Hooley, Andrew Green

Research output: Contribution to journal › Article › peer-review

Abstract

We study the J1-J2 spin-1/2 chain using a path integral constructed over matrix product states (MPS). By virtue of its nontrivial entanglement structure, the MPS ansatz captures the key phases of the model even at a semiclassical, saddle-point level, and, as a variational state, is in good agreement with the field theory obtained by Abelian bosonization. Going beyond the semiclassical 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 generalization 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 dimerization transition is particularly clear in the MPS formulation—an explicit dimerization potential becomes relevant, gapping out the magnetic fluctuations.

Original language	English
Article number	L012037
Number of pages	6
Journal	Physical Review Research
Volume	7
Issue number	1
DOIs	https://doi.org/10.1103/PhysRevResearch.7.L012037
Publication status	Published - 19 Feb 2025
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2025 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Funding

This work was in part supported by the Deutsche Forschungsgemeinschaft under Grant No. SFB 1143 (Project Id. 247310070) and the cluster of excellence ct.qmat (EXC 2147, Project Id. 390858490), by the EPSRC under Grants No. EP/S021582/1 and No. EP/I031014, and by the ERU under 'Perspectives of a Quantum Digital Transformation'. We acknowledge fruitful discussions in workshops funded by EP/W026872/1. Acknowledgments. This work was in part supported by the Deutsche Forschungsgemeinschaft under Grant No. SFB 1143 (Project Id. 247310070) and the cluster of excellence ct.qmat (EXC 2147, Project Id. 390858490), by the EPSRC under Grants No. EP/S021582/1 and No. EP/I031014, and by the ERU under \u2018Perspectives of a Quantum Digital Transformation\u2019. We acknowledge fruitful discussions in workshops funded by EP/W026872/1.

Funders	Funder number
Deutsche Forschungsgemeinschaft	EXC 2147, 247310070, 390858490, SFB 1143
Deutsche Forschungsgemeinschaft
Einstein Research Unit	EP/W026872/1
Engineering and Physical Sciences Research Council	EP/I031014, EP/S021582/1
Engineering and Physical Sciences Research Council

Keywords

Path integrals
Quantum criticality
Quantum field theory
Gauge theory techniques
Methods in magnetism
Spin chains
Spin ladders
Tensor network methods
Variational wave functional methods

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abstract = "We study the J1-J2 spin-1/2 chain using a path integral constructed over matrix product states (MPS). By virtue of its nontrivial entanglement structure, the MPS ansatz captures the key phases of the model even at a semiclassical, saddle-point level, and, as a variational state, is in good agreement with the field theory obtained by Abelian bosonization. Going beyond the semiclassical 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 generalization 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 dimerization transition is particularly clear in the MPS formulation—an explicit dimerization potential becomes relevant, gapping out the magnetic fluctuations.",

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Generalized Haldane map from the matrix product state path integral to the critical theory of the J1-J2 chain

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