Abstract
We study the competition between two different topological orders in three dimensions by considering the X-cube model and the three-dimensional toric code. The corresponding Hamiltonian can be decomposed into two commuting parts, one of which displays a self-dual spectrum. To determine the phase diagram, we compute the high-order series expansions of the ground-state energy in all limiting cases. Apart from the topological order related to the toric code and the fractonic order related to the X-cube model, we found two new phases which are adiabatically connected to classical limits with nontrivial sub-extensive degeneracies. All phase transitions are found to be first order.
Highlights
Quantum systems with topological order are an important research field due to their intriguing physical properties as well as their potential relevance for quantum technological applications
Paradigmatic examples of these two categories of topological order are the 3D toric code (TC) model [19, 20], which is a direct extension of the celebrated model introduced by Kitaev in two dimensions [4] for fault-tolerant quantum computation and has a finite ground-state degeneracy, and the X-cube model (XC) proposed by Vijay, Haah, and Fu [28], which has a sub-extensive ground-state degeneracy
An exact decomposition of the system allows for a quantitative determination of the ground-state phase diagram in the full parameter space
Summary
Quantum systems with topological order are an important research field due to their intriguing physical properties as well as their potential relevance for quantum technological applications. One defining characteristic of fracton phases is that their elementary excitations have a restricted mobility under the action of local operators so that they are considered as attractive candidates for 3D quantum memories [26] Paradigmatic examples of these two categories of topological order are the 3D toric code (TC) model [19, 20], which is a direct extension of the celebrated model introduced by Kitaev in two dimensions [4] for fault-tolerant quantum computation and has a finite ground-state degeneracy, and the X-cube model (XC) proposed by Vijay, Haah, and Fu [28], which has a sub-extensive ground-state degeneracy. The Hamiltonian (1) is not exactly solvable for arbitrary couplings but there are some limiting cases where H can be solved analytically
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