A classification of pure states on quantum spin chains satisfying the split property with on-site finite group symmetries
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- by Yoshiko Ogata;
- Trans. Amer. Math. Soc. Ser. B 8 (2021), 39-65
- DOI: https://doi.org/10.1090/btran/51
- Published electronically: February 2, 2021
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Abstract:
We consider a set $SPG(\mathcal {A})$ of pure split states on a quantum spin chain $\mathcal {A}$ which are invariant under the on-site action $\tau$ of a finite group $G$. For each element $\omega$ in $SPG(\mathcal {A})$ we can associate a second cohomology class $c_{\omega ,R}$ of $G$. We consider a classification of $SPG(\mathcal {A})$ whose criterion is given as follows: $\omega _{0}$ and $\omega _{1}$ in $SPG(\mathcal {A})$ are equivalent if there are automorphisms $\Xi _{R}$, $\Xi _L$ on $\mathcal {A}_{R}$, $\mathcal {A}_{L}$ (right and left half infinite chains) preserving the symmetry $\tau$, such that $\omega _{1}$ and $\omega _{0}\circ \left ( \Xi _{L}\otimes \Xi _{R}\right )$ are quasi-equivalent. It means that we can move $\omega _{0}$ close to $\omega _{1}$ without changing the entanglement nor breaking the symmetry. We show that the second cohomology class $c_{\omega ,R}$ is the complete invariant of this classification.References
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Bibliographic Information
- Yoshiko Ogata
- Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo, 153-8914, Japan
- MR Author ID: 719505
- Received by editor(s): November 11, 2019
- Received by editor(s) in revised form: June 17, 2020
- Published electronically: February 2, 2021
- Additional Notes: The author was supported in part by the Grants-in-Aid for Scientific Research, JSPS. This work was supported by JSPS KAKENHI Grant Number 16K05171 and 19K03534.
- © Copyright 2021 by the author under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 8 (2021), 39-65
- MSC (2020): Primary 46L30
- DOI: https://doi.org/10.1090/btran/51
- MathSciNet review: 4207892