  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.
