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arXiv:2106.05451 · Physical review. B./Physical review. B

Model for fractons, fluxons, and free vertex excitations

Jintae Kim, Jung Hoon Han

Published Sep 16, 2021• 4 citations
arXivHTMLPDF

Abstract

We propose a lattice spin model on a cubic lattice that shares many of the properties of the 3D toric code and the X-cube fracton model. The model, made of ${\mathbb{Z}}_{3}$ degrees of freedom at the links, has the vertex, the cube, and the plaquette terms. Being a stabilizer code the ground states are exactly solved. With only the vertex and the cube terms present, we show that the ground state degeneracy (GSD) is ${3}^{{L}^{3}+3L\ensuremath{-}1}$, where $L$ is the linear dimension of the cubic lattice. In addition to fractons, there are free vertex excitations we call the freeons. With the addition of the plaquette terms, GSD is vastly reduced to ${3}^{3}$, with fracton, fluxon, and freeon excitations, among which only the freeons are deconfined. The model is called the AB model if only the vertex $({A}_{v})$ and the cube $({B}_{c})$ terms are present and the ABC model if in addition the plaquette terms $({C}_{p})$ are included. The AC model consisting of vertex and plaquette terms is the ${\mathbb{Z}}_{3}$ 3D toric code. The extensive GSD of the AB model derives from the existence of both local and nonlocal logical operators that connect different ground states. The latter operators are identical to the logical operators of the ${\mathbb{Z}}_{3}$ X-cube model. Fracton excitations are immobile and accompanied by the creation of fluxons---plaquettes having nonzero flux. In the ABC model, such fluxon creation costs energy and ends up confining the fractons. Unlike past models of fractons, vertex excitations are free to move in any direction and pick up a nontrivial statistical phase when passing through a fluxon or a fracton cluster.

DOI
10.1103/physrevb.104.115128

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We propose a lattice spin model on a cubic lattice that shares many of the properties of the 3D toric code and the X-cube fracton model. The model, made of Z3{\mathbb{Z}}_{3}Z3​ degrees of freedom at the links, has the vertex, the cube, and the plaquette terms. Being a stabilizer code the ground states are exactly solved. With only the vertex and the cube terms present, we show that the ground state degeneracy (GSD) is 3L3+3L\ensuremath−1{3}^{{L}^{3}+3L\ensuremath{-}1}3L3+3L\ensuremath−1, where LLL is the linear dimension of the cubic lattice. In addition to fractons, there are free vertex excitations we call the freeons. With the addition of the plaquette terms, GSD is vastly reduced to 33{3}^{3}33, with fracton, fluxon, and freeon excitations, among which only the freeons are deconfined. The model is called the AB model if only the vertex (Av)({A}_{v})(Av​) and the cube (Bc)({B}_{c})(Bc​) terms are present and the ABC model if in addition the plaquette terms (Cp)({C}_{p})(Cp​) are included. The AC model consisting of vertex and plaquette terms is the Z3{\mathbb{Z}}_{3}Z3​ 3D toric code. The extensive GSD of the AB model derives from the existence of both local and nonlocal logical operators that connect different ground states. The latter operators are identical to the logical operators of the Z3{\mathbb{Z}}_{3}Z3​ X-cube model. Fracton excitations are immobile and accompanied by the creation of fluxons---plaquettes having nonzero flux. In the ABC model, such fluxon creation costs energy and ends up confining the fractons. Unlike past models of fractons, vertex excitations are free to move in any direction and pick up a nontrivial statistical phase when passing through a fluxon or a fracton cluster.
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