The Higgs Mechanism in Higher-Rank Symmetric $U(1)$ Gauge Theories

TL;DR

This work analyzes how the Higgs mechanism operates in a broad family of rank-2 symmetric U(1) lattice gauge theories, linking gapless higher-rank phases to gapped fracton and conventional topological orders. By classifying the $(m,n)$ scalar and vector theories on square/cubic lattices and constructing local rotor models, the authors show that Higgsing to a discrete subgroup often yields conventional $ ext{Z}_2$ topological order, while specific $(2r,2s{+}1)$ scalar-charge theories in $d=3$ realize the X-Cube fracton phase, sometimes via transitions that break continuous rotational symmetry down to a cubic lattice symmetry. They provide explicit solvable models (commuting projector constructions) and analyze phase diagrams, revealing rich structures including multiple toric-code copies and possible direct transitions between $ ext{Z}_2^k$ topological orders and fracton order. Collectively, the results illuminate how fracton order can emerge within a lattice-Higgs framework, and they propose a continuum-field perspective connecting gapless higher-rank gauge theories to gapped fracton phases. The findings suggest a broader landscape in which gapped fracton phases may be obtained as Higgs phases of stable gapless theories, with implications for quantum information and novel topological orders.

Abstract

We use the Higgs mechanism to investigate connections between higher-rank symmetric $U(1)$ gauge theories and gapped fracton phases. We define two classes of rank-2 symmetric $U(1)$ gauge theories: the $(m,n)$ scalar and vector charge theories, for integer $m$ and $n$, which respect the symmetry of the square (cubic) lattice in two (three) spatial dimensions. We further provide local lattice rotor models whose low energy dynamics are described by these theories. We then describe in detail the Higgs phases obtained when the $U(1)$ gauge symmetry is spontaneously broken to a discrete subgroup. A subset of the scalar charge theories indeed have X-cube fracton order as their Higgs phase, although we find that this can only occur if the continuum higher rank gauge theory breaks continuous spatial rotational symmetry. However, not all higher rank gauge theories have fractonic Higgs phases; other Higgs phases possess conventional topological order. Nevertheless, they yield interesting novel exactly solvable models of conventional topological order, somewhat reminiscent of the color code models in both two and three spatial dimensions. We also investigate phase transitions in these models and find a possible direct phase transition between four copies of $\mathbb{Z}_2$ gauge theory in three spatial dimensions and X-cube fracton order.

Figures (30)

• Figure 1: Arrangement of the gauge field degrees of freedom for a $U(1)$ symmetric lattice gauge theory in (a) $d=2$ and (b) $d=3$. Each dot is a rotor variable, and the circles indicate that $d$ spins live on each site.
• Figure 2: Charge configurations created in the $(m,n)$ scalar charge theory by (a) $e^{iA_{xx}}$, the raising operator for $E_{xx}$, acting on the center site (b) $e^{iA_{xy}}$, the raising operator for $E_{xy}$, acting on the black plaquette.
• Figure 3: Transverse hopping operator for a unit dipole in the $(1,2)$ scalar charge theory. The $e^{iA_{ij}}$ operators act on the black rotors and create the charge configuration shown. Because $e^{iA_{xy}}$ only creates charge-2 objects in the $(1,2)$ theory, it cannot cause a unit dipole to hop, unlike the $(1,1)$ theory.
• Figure 4: Charge configurations created in the $(m,n)$ vector charge theory by (a) $e^{iA_{xx}}$, the raising operator for $E_{xx}$, acting on the black site (b) $e^{iA_{xy}}$, the raising operator for $E_{xy}$, acting on the black plaquette.
• Figure 5: Effect of the Higgs mechanism on the local operator $e^{iA_{xx}}$ in the $(1,1)$ scalar charge theory. The original charge configuration (top) is modified when charge-2 particles are condensed; the charge-2 particle is absorbed into the condensate, and charges $+1$ and $-1$ become equivalent.