Emanant and emergent symmetry-topological-order from low-energy spectrum
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| Format: | Preprint |
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2025
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| _version_ | 1866915877845204992 |
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| author | Chen, Zixin Jessie Aksoy, Ömer M. Xu, Cenke Wen, Xiao-Gang |
| author_facet | Chen, Zixin Jessie Aksoy, Ömer M. Xu, Cenke Wen, Xiao-Gang |
| contents | Low-energy emanant and emergent symmetries can be anomalous, higher-group, or non-invertible. A way to systematically capture the properties of such symmetries is through the topological orders in one-higher dimension, known as symmetry topological orders (symTOs). Consequently, identifying the emergent or emanant symmetry of a system is not simply a matter of determining its group structure, but rather of computing the corresponding symTO. In this work, we develop a method to compute the symTO of 1+1D systems by analyzing their low-energy spectra under closed boundary conditions with all possible symmetry twists. Following this approach, we show that the gapless antiferromagnetic (AF) spin-$1/2$ Heisenberg model possesses an exact emanant symTO corresponding to the $D_8$ quantum double, when the global symmetry is restricted to the $\mathbb{Z}_2^x \times \mathbb{Z}_2^z$ subgroup of the $SO(3)$ spin-rotation symmetry and lattice translations. Moreover, this model exhibits an emergent $SO(4)$ symmetry, whose exact components are described jointly by automorphisms of the $D_8$ quantum double and the $SO(3)$ spin-rotations. Using the condensable algebras of the emanant symTO, we further identify several other phases that may be accessible by modifying interactions among low-energy excitations: (1) a gapped dimer phase, connected to the AF phase via an $SO(4)$ rotation, (2) a commensurate collinear ferromagnetic phase that breaks translation by one site with a $ω\sim k^2$ mode, (3) an incommensurate, translation-symmetric ferromagnetic phase featuring both $ω\sim k^2$ and $ω\sim k$ modes, (4) and an incommensurate ferromagnetic phase that breaks translation by one site with both $ω\sim k^2$ and $ω\sim k$ modes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_08879 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Emanant and emergent symmetry-topological-order from low-energy spectrum Chen, Zixin Jessie Aksoy, Ömer M. Xu, Cenke Wen, Xiao-Gang Strongly Correlated Electrons High Energy Physics - Theory Low-energy emanant and emergent symmetries can be anomalous, higher-group, or non-invertible. A way to systematically capture the properties of such symmetries is through the topological orders in one-higher dimension, known as symmetry topological orders (symTOs). Consequently, identifying the emergent or emanant symmetry of a system is not simply a matter of determining its group structure, but rather of computing the corresponding symTO. In this work, we develop a method to compute the symTO of 1+1D systems by analyzing their low-energy spectra under closed boundary conditions with all possible symmetry twists. Following this approach, we show that the gapless antiferromagnetic (AF) spin-$1/2$ Heisenberg model possesses an exact emanant symTO corresponding to the $D_8$ quantum double, when the global symmetry is restricted to the $\mathbb{Z}_2^x \times \mathbb{Z}_2^z$ subgroup of the $SO(3)$ spin-rotation symmetry and lattice translations. Moreover, this model exhibits an emergent $SO(4)$ symmetry, whose exact components are described jointly by automorphisms of the $D_8$ quantum double and the $SO(3)$ spin-rotations. Using the condensable algebras of the emanant symTO, we further identify several other phases that may be accessible by modifying interactions among low-energy excitations: (1) a gapped dimer phase, connected to the AF phase via an $SO(4)$ rotation, (2) a commensurate collinear ferromagnetic phase that breaks translation by one site with a $ω\sim k^2$ mode, (3) an incommensurate, translation-symmetric ferromagnetic phase featuring both $ω\sim k^2$ and $ω\sim k$ modes, (4) and an incommensurate ferromagnetic phase that breaks translation by one site with both $ω\sim k^2$ and $ω\sim k$ modes. |
| title | Emanant and emergent symmetry-topological-order from low-energy spectrum |
| topic | Strongly Correlated Electrons High Energy Physics - Theory |
| url | https://arxiv.org/abs/2509.08879 |