Entanglement Spectrum as a diagnostic of chirality of Topological Spin Liquids: Analysis of an $\mathrm{SU}(3)$ PEPS
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| Format: | Preprint |
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2023
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| _version_ | 1866916540710912000 |
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| author | Arildsen, Mark J. Chen, Ji-Yao Schuch, Norbert Ludwig, Andreas W. W. |
| author_facet | Arildsen, Mark J. Chen, Ji-Yao Schuch, Norbert Ludwig, Andreas W. W. |
| contents | We address the key question of representation of chiral topological quantum states in (2+1) dimensions (i.e., with non-zero chiral central charge) by Projected Entangled Pair States (PEPS). A noted result (due to Wahl, Tu, Schuch, and Cirac [Phys. Rev. Lett. 111, 236805 (2013)], and Dubail and Read [Phys. Rev. B 92, 205307 (2015)]) says that this is possible for non-interacting fermions, but the answer is as yet unknown for interacting systems. Characteristic counting of degeneracies of low-lying states in the entanglement spectrum (ES) at fixed transverse momentum of bipartitioned long cylinders ("Li-Haldane counting") provides often-used supporting evidence for chirality. However, non-chiral PEPS (with zero chiral central charge), yet with strong breaking of time-reversal and reflection symmetries, with invariance under the product of these two operations (i.e., "apparently" chiral states), are known whose low-lying ES exhibits the same Li-Haldane counting as a chiral state in certain topological sectors [Kurečić, Vanderstraeten, and Schuch, Phys. Rev. B 99, 045116 (2019); Arildsen, Schuch, and Ludwig, Phys. Rev. B 108, 245150 (2023)]. In the present work, we identify a distinct indicator and hallmark of chirality in the ES of PEPS with global $\mathrm{SU}(3)$ symmetry: the splittings of conjugate irreps. We prove that in the ES of the chiral states conjugate irreps are exactly degenerate, because the operators that would split them [related to the cubic Casimir invariant of $\mathrm{SU}(3)$] are forbidden. By contrast, in the ES of non-chiral states, conjugate splittings are demonstrably non-vanishing. Such a diagnostic provides an unambiguous and powerful tool to distinguish chiral and non-chiral topological states in (2+1) dimensions via their entanglement spectra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_13240 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Entanglement Spectrum as a diagnostic of chirality of Topological Spin Liquids: Analysis of an $\mathrm{SU}(3)$ PEPS Arildsen, Mark J. Chen, Ji-Yao Schuch, Norbert Ludwig, Andreas W. W. Strongly Correlated Electrons Statistical Mechanics Quantum Physics We address the key question of representation of chiral topological quantum states in (2+1) dimensions (i.e., with non-zero chiral central charge) by Projected Entangled Pair States (PEPS). A noted result (due to Wahl, Tu, Schuch, and Cirac [Phys. Rev. Lett. 111, 236805 (2013)], and Dubail and Read [Phys. Rev. B 92, 205307 (2015)]) says that this is possible for non-interacting fermions, but the answer is as yet unknown for interacting systems. Characteristic counting of degeneracies of low-lying states in the entanglement spectrum (ES) at fixed transverse momentum of bipartitioned long cylinders ("Li-Haldane counting") provides often-used supporting evidence for chirality. However, non-chiral PEPS (with zero chiral central charge), yet with strong breaking of time-reversal and reflection symmetries, with invariance under the product of these two operations (i.e., "apparently" chiral states), are known whose low-lying ES exhibits the same Li-Haldane counting as a chiral state in certain topological sectors [Kurečić, Vanderstraeten, and Schuch, Phys. Rev. B 99, 045116 (2019); Arildsen, Schuch, and Ludwig, Phys. Rev. B 108, 245150 (2023)]. In the present work, we identify a distinct indicator and hallmark of chirality in the ES of PEPS with global $\mathrm{SU}(3)$ symmetry: the splittings of conjugate irreps. We prove that in the ES of the chiral states conjugate irreps are exactly degenerate, because the operators that would split them [related to the cubic Casimir invariant of $\mathrm{SU}(3)$] are forbidden. By contrast, in the ES of non-chiral states, conjugate splittings are demonstrably non-vanishing. Such a diagnostic provides an unambiguous and powerful tool to distinguish chiral and non-chiral topological states in (2+1) dimensions via their entanglement spectra. |
| title | Entanglement Spectrum as a diagnostic of chirality of Topological Spin Liquids: Analysis of an $\mathrm{SU}(3)$ PEPS |
| topic | Strongly Correlated Electrons Statistical Mechanics Quantum Physics |
| url | https://arxiv.org/abs/2305.13240 |