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arXiv:2409.18206 · Physical review. B./Physical review. B

Entanglement renormalization of fractonic anisotropic Z N Laplacian models

Yuan Xue, Pranay Gorantla, Zhu-Xi Luo

Published May 5, 2025• 1 citation
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Abstract

Gapped fracton phases constitute a new class of quantum states of matter, which connects to topological orders but does not fit easily into existing paradigms. They host unconventional features such as sub-extensive and robust ground-state degeneracies as well as sensitivity to lattice geometry. We investigate the anisotropic ${\mathbb{Z}}_{N}$ Laplacian model [Phys. Rev. B 107, 125121 (2023)], which can describe a family of fracton phases defined on arbitrary graphs. Focusing on representative geometries where the 3D lattices are extensions of 2D square, triangular, honeycomb, and the Kagome lattices into the third dimension, we study their ground-state degeneracies and mobility of excitations, and examine their entanglement renormalization group (ERG) flows. All models show bifurcating behaviors under ERG but have distinct ERG flows sensitive to both $N$ and lattice geometry. In particular, we show that the anisotropic ${\mathbb{Z}}_{N}$ Laplacian models defined on the extensions of triangular and honeycomb lattices are equivalent when $N$ is coprime to 3. We also point out that, in contrast to previous expectations, the model defined on the extension of Kagome lattice is robust against local perturbations if and only if $N$ is coprime to 6.

DOI
10.1103/physrevb.111.205105

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We introduce lattice gauge theories which describe three-dimensional, gapped quantum phases exhibiting the phenomenology of both conventional three-dimensional topological orders and fracton orders, starting from a finite group GGG, a choice of an Abelian normal subgroup NNN, and a choice of foliation structure. These hybrid fracton orders, examples of which were introduced in a previous paper [Tantivasadakarn et al., Phys. Rev. B 103, 245136 (2021)], can also host immobile, pointlike excitations that are non-Abelian, and therefore give rise to a protected degeneracy. We construct solvable lattice models for these orders which interpolate between a conventional, three-dimensional GGG gauge theory and a pure fracton order, by varying the choice of normal subgroup NNN. We demonstrate that certain universal data of the topological excitations and their mobilities are directly related to the choice of GGG and NNN, and also present complementary perspectives on these orders: certain orders may be obtained by gauging a global symmetry which enriches a particular fracton order, by either fractionalizing on or permuting the excitations with restricted mobility, while certain hybrid orders can be obtained by condensing excitations in a stack of initially decoupled, two-dimensional topological orders.