Higgs Phases and Boundary Criticality

TL;DR

This work demonstrates that global symmetries attached to Higgs fields become physically manifest at boundaries of lattice gauge theories when electric flux can escape, leading to boundary spontaneous symmetry breaking in the Higgs regime. Through analytical arguments and extensive Monte Carlo simulations, the authors show that the boundary undergoes second-order phase transitions in distinct universality classes: 3D XY for Abelian $U(1)$, and 3D $O(2N)$ (or related chiral realizations) for non-Abelian SU(N) cases, with higher-form Higgs models yielding boundary confinement–deconfinement transitions. They develop boundary theories in the infinite-$ ext{kappa}$ limit, relate them to gauged nonlinear sigma models and principal chiral models, and illustrate how boundary criticality persists under bulk fluctuations and via tunable boundary couplings. The study also frames these boundary phenomena within the HiggsSPT paradigm, discussing dual pictures and higher-form generalizations, and highlighting implications for the relation between Higgs and SPT phases in both Abelian and non-Abelian settings. Overall, the results reveal a universal boundary mechanism by which Higgs phases reveal edge modes and criticality, with broad implications for understanding bulk-boundary correspondence in gauge theories.

Abstract

Motivated by recent work connecting Higgs phases to symmetry protected topological (SPT) phases, we investigate the interplay of gauge redundancy and global symmetry in lattice gauge theories with Higgs fields in the presence of a boundary. The core conceptual point is that a global symmetry associated to a Higgs field, which is pure-gauge in a closed system, acts physically at the boundary under boundary conditions which allow electric flux to escape the system. We demonstrate in both Abelian and non-Abelian models that this symmetry is spontaneously broken in the Higgs regime, implying the presence of gapless edge modes. Starting with the U(1) Abelian Higgs model in 4D, we demonstrate a boundary phase transition in the 3D XY universality class separating the bulk Higgs and confining regimes. Varying the boundary coupling while preserving the symmetries shifts the location of the boundary phase transition. We then consider non-Abelian gauge theories with fundamental and group-valued Higgs matter, and identify the analogous non-Abelian global symmetry acting on the boundary generated by the total color charge. For SU($N$) gauge theory with fundamental Higgs matter we argue for a boundary phase transition in the O($2N$) universality class, verified numerically for $N=2,3$. For group-valued Higgs matter, the boundary theory is a principal chiral model exhibiting chiral symmetry breaking. We further demonstrate this mechanism in theories with higher-form Higgs fields. We show how the higher-form matter symmetry acts at the boundary and can spontaneously break, exhibiting a boundary confinement-deconfinement transition. We also study the electric-magnetic dual theory, demonstrating a dual magnetic defect condensation transition at the boundary. We discuss some implications and extensions of these findings and what they may imply for the relation between Higgs and SPT phases.

Figures (13)

• Figure 1: A sketch of the phase diagram of the 4D U(1) lattice Abelian-Higgs phase diagram discussed in \ref{['sec:abelian']}. The confined and Higgs regimes belong to the same thermodynamic phase, though we demonstrate that in presence of symmetry-preserving boundary they are sharply separated by a second order boundary phase transition in the 3D XY universality class. Along the line $\kappa = 0$ the model has an exact electric 1-form symmetry, while along the line with $\beta = \infty$ the model has an exact magnetic 1-form symmetry. The confined regime is smoothly connected to the $\beta = 0$ limit where the gauge field is maximally disordered. Along the $\kappa = \infty$ line, the bulk is completely frozen, and the boundary reduces to a 3D XY model. The boundary $U(1)$ symmetry is spontaneously broken on the Higgs side of the transition line. A review of the structure of this phase diagram is provided in \ref{['apx:u1_phase_diagram']}.
• Figure 2: A demonstration of the boundary conditions considered in this work. The gauge field takes values on the bulk links (black) as well as a set of links extending out of the bulk (green). The holonomy of the gauge field is defined on all bulk plaquettes (gray), as well as on the set of plaquettes extending out of the bulk (green). The matter field takes values only in the bulk of the system (white spheres), with the Gauss law satisfied at all bulk vertices. The outside vacuum (blue sites, links, plaquettes) has no dynamical fields. With these boundary conditions electric flux is capable of passing through the boundary, allowing for non-trivial charge sectors in the bulk. We denote the bulk cells (white, black, gray) by $X$ and the boundary layer cells (green) by $\partial X$.
• Figure 3: Boundary criticality for the 4D $U(1)$ Higgs phase: (a) Average boundary plaquette for $\kappa=$0.55, 0.64, 0.85, 0.94, 1.0 for $L=16$. The transition shifts towards smaller $\beta$ for larger values of $\kappa$. (b) The Binder ratio $U_4$ for the magnetization at the boundary as a function of $\beta$ for $\kappa=2.0$ and $L=16,20,24,28,32$. (c) Rescaled Binder ratio for $\kappa=2.0$ showing collapse for $\nu=0.67$ corresponding to the 3D XY universality class.
• Figure 4: Boundary phase transition lines for the U(1) Abelian Higgs model, for different $\alpha=\beta_{\text{bdry}}/\beta_{\text{bulk}}$, overlaid on the bulk plaquette susceptibility. From right to left, $\alpha$= 0.4, 0.5, 0.6, 1.0, 2.0, 3.0, 10.0 for system size $L=16$.
• Figure 5: Bulk phase diagram of the SU(2) Higgs model, showing (a) the link expectation value $\langle \Lambda_{\ell}\rangle$, (b) the link variance $\sigma^2(\mathrm{Re}\, \Lambda_{\ell})$, (c) the Wilson plaquette average $\langle \mathrm{Re}\, W_p \rangle$, and the plaquette variance $\sigma^2(\mathrm{Re}\, W_p)$. The small cyan-colored point is the location of the critical endpoint identified in Ref. bonatiPhaseDiagramLattice2010, $(\beta_c,\kappa_c)\approx (2.73,0.70)$. From the critical endpoint there is a first-order transition line extending to larger $\beta$ with nearly-constant $\kappa$, which is clearly seen in (b). A rapid-crossover region extends from the critical point to smaller $\beta$ and larger $\kappa$, signaled in both (b) and (d) by strong bulk fluctuations, indicating the "supercritical" region which roughly delineates the Higgs and confined regimes.