Relation between chiral central charge and ground-state degeneracy in ( 2 + 1 ) -dimensional topological orders
A bosonic topological order on $d$-dimensional closed space $\Sigma^d$ may have degenerate ground states. The space $\Sigma^d$ with different shapes (different metrics) form a moduli space ${\cal M}_{\Sigma^d}$. Thus the degenerate ground states on every point in the moduli space ${\cal M}_{\Sigma^d...
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| Format: | Article |
| Language: | English |
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American Physical Society (APS)
2021
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| Online Access: | https://hdl.handle.net/1721.1/135326 |
| _version_ | 1826208639603441664 |
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| author | Kong, Liang Wen, Xiao-Gang |
| author2 | Massachusetts Institute of Technology. Department of Physics |
| author_facet | Massachusetts Institute of Technology. Department of Physics Kong, Liang Wen, Xiao-Gang |
| author_sort | Kong, Liang |
| collection | MIT |
| description | A bosonic topological order on $d$-dimensional closed space $\Sigma^d$ may
have degenerate ground states. The space $\Sigma^d$ with different shapes
(different metrics) form a moduli space ${\cal M}_{\Sigma^d}$. Thus the
degenerate ground states on every point in the moduli space ${\cal
M}_{\Sigma^d}$ form a complex vector bundle over ${\cal M}_{\Sigma^d}$. It was
suggested that the collection of such vector bundles for $d$-dimensional closed
spaces of all topologies completely characterizes the topological order. Using
such a point of view, we propose a direct relation between two seemingly
unrelated properties of 2+1-dimensional topological orders: (1) the chiral
central charge $c$ that describes the many-body density of states for edge
excitations (or more precisely the thermal Hall conductance of the edge), (2)
the ground state degeneracy $D_g$ on closed genus $g$ surface. We show that $c
D_g/2 \in \mathbb{Z},\ g\geq 3$ for bosonic topological orders. We explicitly
checked the validity of this relation for over 140 simple topological orders.
For fermionic topological orders, let $D_{g,\sigma}^{e}$ ($D_{g,\sigma}^{o}$)
be the degeneracy with even (odd) number of fermions for genus-$g$ surface with
spin structure $\sigma$. Then we have $2c D_{g,\sigma}^{e} \in \mathbb{Z}$ and
$2c D_{g,\sigma}^{o} \in \mathbb{Z}$ for $g\geq 3$. |
| first_indexed | 2024-09-23T14:08:48Z |
| format | Article |
| id | mit-1721.1/135326 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T14:08:48Z |
| publishDate | 2021 |
| publisher | American Physical Society (APS) |
| record_format | dspace |
| spelling | mit-1721.1/1353262023-03-15T19:46:01Z Relation between chiral central charge and ground-state degeneracy in ( 2 + 1 ) -dimensional topological orders Kong, Liang Wen, Xiao-Gang Massachusetts Institute of Technology. Department of Physics A bosonic topological order on $d$-dimensional closed space $\Sigma^d$ may have degenerate ground states. The space $\Sigma^d$ with different shapes (different metrics) form a moduli space ${\cal M}_{\Sigma^d}$. Thus the degenerate ground states on every point in the moduli space ${\cal M}_{\Sigma^d}$ form a complex vector bundle over ${\cal M}_{\Sigma^d}$. It was suggested that the collection of such vector bundles for $d$-dimensional closed spaces of all topologies completely characterizes the topological order. Using such a point of view, we propose a direct relation between two seemingly unrelated properties of 2+1-dimensional topological orders: (1) the chiral central charge $c$ that describes the many-body density of states for edge excitations (or more precisely the thermal Hall conductance of the edge), (2) the ground state degeneracy $D_g$ on closed genus $g$ surface. We show that $c D_g/2 \in \mathbb{Z},\ g\geq 3$ for bosonic topological orders. We explicitly checked the validity of this relation for over 140 simple topological orders. For fermionic topological orders, let $D_{g,\sigma}^{e}$ ($D_{g,\sigma}^{o}$) be the degeneracy with even (odd) number of fermions for genus-$g$ surface with spin structure $\sigma$. Then we have $2c D_{g,\sigma}^{e} \in \mathbb{Z}$ and $2c D_{g,\sigma}^{o} \in \mathbb{Z}$ for $g\geq 3$. 2021-10-27T20:22:58Z 2021-10-27T20:22:58Z 2020 2021-07-09T15:25:50Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/135326 en 10.1103/PHYSREVRESEARCH.2.033344 Physical Review Research Creative Commons Attribution 4.0 International license https://creativecommons.org/licenses/by/4.0/ application/pdf American Physical Society (APS) APS |
| spellingShingle | Kong, Liang Wen, Xiao-Gang Relation between chiral central charge and ground-state degeneracy in ( 2 + 1 ) -dimensional topological orders |
| title | Relation between chiral central charge and ground-state degeneracy in ( 2 + 1 ) -dimensional topological orders |
| title_full | Relation between chiral central charge and ground-state degeneracy in ( 2 + 1 ) -dimensional topological orders |
| title_fullStr | Relation between chiral central charge and ground-state degeneracy in ( 2 + 1 ) -dimensional topological orders |
| title_full_unstemmed | Relation between chiral central charge and ground-state degeneracy in ( 2 + 1 ) -dimensional topological orders |
| title_short | Relation between chiral central charge and ground-state degeneracy in ( 2 + 1 ) -dimensional topological orders |
| title_sort | relation between chiral central charge and ground state degeneracy in 2 1 dimensional topological orders |
| url | https://hdl.handle.net/1721.1/135326 |
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