A Generalised Haldane Map from the Matrix Product State Path Integral to the Critical Theory of the $J_1$-$J_2$ Chain
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| Format: | Preprint |
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2024
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| author | Azad, F. McRoberts, Adam J. Hooley, Chris Green, A. G. |
| author_facet | Azad, F. McRoberts, Adam J. Hooley, Chris Green, A. G. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_16088 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Generalised Haldane Map from the Matrix Product State Path Integral to the Critical Theory of the $J_1$-$J_2$ Chain Azad, F. McRoberts, Adam J. Hooley, Chris Green, A. G. Strongly Correlated Electrons Quantum Physics 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. |
| title | A Generalised Haldane Map from the Matrix Product State Path Integral to the Critical Theory of the $J_1$-$J_2$ Chain |
| topic | Strongly Correlated Electrons Quantum Physics |
| url | https://arxiv.org/abs/2404.16088 |