Comments on abelian Higgs models and persistent order

Abstract

A natural question about Quantum Field Theory is whether there is a deformation to a trivial gapped phase. If the underlying theory has an anomaly, then symmetric deformations can never lead to a trivial phase. We discuss such discrete anomalies in Abelian Higgs models in 1+1 and 2+1 dimensions. We emphasize the role of charge conjugation symmetry in these anomalies; for example, we obtain nontrivial constraints on the degrees of freedom that live on a domain wall in the VBS phase of the Abelian Higgs model in 2+1 dimensions. In addition, as a byproduct of our analysis, we show that in 1+1 dimensions the Abelian Higgs model is dual to the Ising model. We also study variations of the Abelian Higgs model in 1+1 and 2+1 dimensions where there is no dynamical particle of unit charge. These models have a center symmetry and additional discrete anomalies. In the absence of a dynamical unit charge particle, the Ising transition in the 1+1 dimensional Abelian Higgs model is removed. These models without a unit charge particle exhibit a remarkably persistent order: we prove that the system cannot be disordered by either quantum or thermal fluctuations. Equivalently, when these theories are studied on a circle, no matter how small or large the circle is, the ground state is non-trivial.

Global anomaly matching in the higher-dimensional CPN−1 model

“…(68) is the two-dimensional CP N −1 model with a θ term at θ = π. At this θ angle, this 2d model has the charge-conjugation symmetry C and also possesses the PSU(N ) F × C [16,20] and R × C anomalies [18]. In Ref.…”

“…In general, T Mag can be different from T Néel , but the anomalies yield the important 7 Note that T Néel and T Mag are analogous to the temperatures for CP symmetry breaking and deconfinement in the pure Yang-Mills theory at θ = π, respectively [23]. See also similar discussions for other gauge theories [20,25,27,32]. constraint for the transition temperatures:…”

“…Quantum field theoretical approach to low-dimensional spin systems provides one representative ground where 't Hooft anomalies play a pivotal role in understanding their possible low-energy behaviors such as properties of their ground states and energy spectra [11,[16][17][18][19][20][21]. One can see this because the 't Hooft anomaly can be regarded as an avatar of the Lieb-Shultz-Mattis (LSM) theorem for the lattice model [43][44][45][46] in the continuum field theory [15,17] (see also Refs.…”

“…[23] for an analogous restriction in the pure Yang-Mills theory at θ = π and Refs. [20,25,27,32] for the discussion on other gauge theories). We also mention its consistency with the large-N analysis on the CP N −1 nonlinear sigma model.…”

“…to keep the constraint(20). This 1-form gauge invariance removes the redundant diagonal part in U(N ), which acts on no gauge-invariant operators as discussed in Sec.…”