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Anisotropic higher rank $\mathbb{Z}_N$ topological phases on graphs
by Hiromi Ebisu, Bo Han
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Submission summary
| Authors (as registered SciPost users): | Hiromi Ebisu |
| Submission information | |
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| Preprint Link: | scipost_202209_00064v2 (pdf) |
| Date accepted: | 2023-02-13 |
| Date submitted: | 2022-12-18 19:24 |
| Submitted by: | Ebisu, Hiromi |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study unusual gapped topological phases where they admit $\mathbb{Z}_N$ fractional excitations in the same manner as topologically ordered phases, yet their ground state degeneracy depends on the local geometry of the system. Placing such phases on 2D lattice, composed of an arbitrary connected graph and 1D line, we find that the fusion rules of quasiparticle excitations are described by the Laplacian of the graph and that the number of superselection sectors is related to the kernel of the Laplacian. Based on this analysis, we further show that the ground state degeneracy is given by $\bigl[N\times \prod_{i}\text{gcd}(N, p_i)\bigr]^2$, where $p_i$'s are invariant factors of the Laplacian that are greater than one and gcd stands for the greatest common divisor. We also discuss braiding statistics between quasiparticle excitations.
List of changes
Layout of the figures has been changed. Minor modification has been made in Fig. 1.
For clearer illustration on the generic lattice that we introduce in Sec. III E, a new figure has been added (fig. 5).
A footnote [21] has been added to address why we regard our lattice as 2D. (i.e., why we regard the graph as 1D)
Published as SciPost Phys. 14, 106 (2023)