Lattice Abelian Higgs Model

• The lattice Abelian Higgs model is a discretized U(1) gauge theory coupled with charged scalar matter, enabling nonperturbative studies of scalar electrodynamics.
• It distinguishes multiple regimes—confinement, Coulomb, molecular, and Higgs phases—with transitions influenced by gauge compactness, matter charge, and scalar multiplicity.
• Critical phenomena and universality classes are analyzed via Monte Carlo simulations and renormalization-group techniques, highlighting thresholds for continuous and first-order transitions.

The lattice Abelian Higgs model is a lattice gauge theory in which a $U(1)$ gauge field on links is coupled to charged scalar matter on sites, providing a nonperturbative discretization of scalar electrodynamics or the Abelian-Higgs field theory. In the continuum, the reference field theory is commonly written as

$${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$

with an $N$-component complex scalar field ${\bm \Phi}$ and electromagnetic field $A_\mu$ (Bonati et al., 2022). Lattice formulations are used to determine when this continuum theory actually controls critical behavior, to distinguish confinement, Coulomb, molecular, and Higgs regimes, and to analyze how compactness of the gauge field, matter charge, dimension, and the number of scalar components reorganize the phase diagram (Bonati et al., 28 May 2026).

1. Continuum correspondence and lattice formulations

A standard compact three-dimensional Wilson formulation employs an $N$-component complex matter field ${\bm z}_{\bm x}$ of unit length on lattice sites and compact link variables $\lambda_{\bm x,\mu}\in U(1)$, with Hamiltonian

$$H = - \beta N \sum_{\bm x,\mu} \left( \bar{\bm z}_{\bm x}\cdot \lambda_{\bm x,\mu}\, {\bm z}_{\bm x+\hat\mu} + {\rm c.c.}\right) -\beta_g \sum_{\bm x,\mu\neq\nu} \left( \lambda_{\bm x,\mu} \,\lambda_{\bm x+\hat{\mu},\nu} \,\bar{\lambda}_{\bm x+\hat{\nu},\mu} \,\bar{\lambda}_{\bm x,\nu} + {\rm c.c.}\right)$$

(Pelissetto et al., 2019). In higher-charge compact models the matter term is replaced by $\lambda^q$, for example

$${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$0

(Bonati et al., 2022). Because the gauge field is compact, the charge $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$1 is not removable by field redefinitions, so $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$2 is a genuine model parameter (Bonati et al., 28 May 2026).

The noncompact formulation replaces link phases by real variables $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$3, with $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$4 and a Maxwell term

$${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$5

(Bonati et al., 2020). In that noncompact formulation the charge $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$6 can be scaled away, so effectively $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$7 (Bonati et al., 2022).

Several singular limits organize the model. In compact unit-charge models, $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$8 yields a lattice $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$9 model, while $N$0 yields an $N$1 vector model (Pelissetto et al., 2019). In noncompact models, $N$2 maps to an inverted $N$3 model, and in compact higher-charge models the $N$4 limit reduces to a $N$5 gauge theory (Bonati et al., 2020, Bonati et al., 2020).

2. Symmetries, gauge-invariant observables, and gauge fixing

The defining symmetry structure combines local $N$6 gauge invariance with a global flavor symmetry, typically $N$7 or $N$8, acting on the scalar multiplet (Bonati et al., 2020, Bonati et al., 2023). The canonical gauge-invariant local order parameter is the traceless Hermitian bilinear

$N$9

used throughout the numerical literature to build two-point functions, susceptibilities, second-moment correlation lengths, Binder parameters, and the RG-invariant ratio ${\bm \Phi}$0 (Bonati et al., 28 May 2026).

This choice is not merely conventional. In compact unit-charge three-dimensional multicomponent models, gauge correlations are never critical: gauge excitations are massive for any finite coupling, and the transition is governed by the gauge-invariant composite ${\bm \Phi}$1 rather than by critical gauge photons (Pelissetto et al., 2019). By contrast, along the noncompact Coulomb–Higgs line, gauge and matter fields may both be critical for sufficiently large ${\bm \Phi}$2, and gauge-dependent observables become meaningful only after explicit gauge fixing (Bonati et al., 2023).

Gauge fixing is structurally important in the noncompact theory. Because noncompact gauge variables generate gauge-invariant zero modes, the finite-volume partition function is ill-defined with periodic boundary conditions; ${\bm \Phi}$3 boundary conditions are therefore used in Monte Carlo studies (Bonati et al., 2020). In the hard Lorenz gauge,

${\bm \Phi}$4

the scalar field coincides with a gauge-invariant Dirac-dressed operator

${\bm \Phi}$5

and scalar correlations become critical; in the axial gauge they do not (Bonati et al., 2023). A related construction with charge conjugate boundary conditions yields a locally gauge-invariant charged scalar operator and permits spectroscopy of charged states without fixing the gauge (Woloshyn, 2017).

3. Phase structure across compact, noncompact, and higher-charge variants

A common simplification is that the lattice Abelian Higgs model has only “confinement” and “Higgs” phases. That is accurate only for restricted parameter choices. The broader phase structure depends strongly on compactness and matter charge.

For compact unit-charge multicomponent models in three dimensions, numerical work for ${\bm \Phi}$6 and ${\bm \Phi}$7 identifies two phases: a disordered confined phase and an ordered Higgs phase, separated by a single transition line (Pelissetto et al., 2019). In this setting the nature of the transition is independent of the gauge coupling for any finite positive ${\bm \Phi}$8: it is continuous in the Heisenberg universality class for ${\bm \Phi}$9 and first order for $A_\mu$0 (Pelissetto et al., 2019).

For compact higher-charge models with $A_\mu$1, the phase diagram is richer. The literature identifies three phases: disordered-confined (DC), ordered-confined (OC), and ordered-deconfined (OD), separated by DC–OC, OC–OD, and DC–OD transition lines meeting at a multicritical point (Bonati et al., 2020, Bonati et al., 2022). The OC–OD line is a deconfinement transition, while the DC–OD line is the line on which scalar ordering and gauge deconfinement emerge together (Bonati et al., 2022).

For noncompact three-dimensional models with $A_\mu$2, the phase diagram contains Coulomb (C), molecular (M), and Higgs (H) phases, separated by Coulomb–molecular (CM), molecular–Higgs (MH), and Coulomb–Higgs (CH) lines (Bonati et al., 2020). The molecular phase is ordered in the scalar sector but retains long-ranged gauge correlations, and is therefore absent from the corresponding compact formulation (Bonati et al., 2020).

Formulation	Phases	Characteristic transition lines
Compact, unit charge	Disordered confined; ordered Higgs	Single ordering line
Compact, $A_\mu$3	DC; OC; OD	DC–OC, OC–OD, DC–OD
Noncompact, $A_\mu$4	Coulomb; molecular; Higgs	CM, MH, CH

This classification suggests that compactness controls whether long-ranged gauge correlations survive as an independent thermodynamic sector, while matter charge controls whether confinement and scalar ordering can decouple. That implication is explicit in the higher-charge literature but absent in the unit-charge compact case (Bonati et al., 2020).

4. Universality classes and renormalization-group structure

The central renormalization-group question is whether the Abelian-Higgs field theory possesses a stable charged fixed point (CFP). In the field theory this fixed point exists only for sufficiently large $A_\mu$5: near four dimensions one has $A_\mu$6, whereas in three dimensions resummed perturbation theory gives $A_\mu$7, and perturbative, functional-RG, and large-$A_\mu$8 analyses in the noncompact literature give $A_\mu$9 (Bonati et al., 28 May 2026, Bonati et al., 2023).

The lattice evidence shows that different transition lines access different continuum descriptions. In the noncompact model, the CM line is governed by a gauge-invariant Landau-Ginzburg-Wilson $N$0 theory; for $N$1 it is $N$2, while for $N$3 the cubic invariant typically drives first-order behavior (Bonati et al., 2020). The CH line is instead described by the Abelian-Higgs field theory with explicit gauge fields; Monte Carlo data support weak first-order behavior for small $N$4 and continuous transitions for $N$5 (Bonati et al., 2020). Representative estimates are

$N$6

(Bonati et al., 2020).

The MH line belongs to a different charged universality class. It is $N$7-independent, coincides with the one-component noncompact model, and numerically matches inverted $N$8 criticality, with gauge anomalous dimension $N$9 and charged-scalar exponent ${\bm z}_{\bm x}$0 (Bonati et al., 2023). This is not captured by the standard perturbative AHFT flow near four dimensions (Bonati et al., 2023).

In compact higher-charge deconfinement problems with a single complex scalar of charge ${\bm z}_{\bm x}$1, the transition line is argued to belong to the same universality class as three-dimensional ${\bm z}_{\bm x}$2 gauge models: Ising-like for ${\bm z}_{\bm x}$3, first order for ${\bm z}_{\bm x}$4, and ${\bm z}_{\bm x}$5 for ${\bm z}_{\bm x}$6, with a special ${\bm z}_{\bm x}$7 limit at ${\bm z}_{\bm x}$8 (Bonati et al., 2024).

5. Higher charge, multicomponent matter, and threshold phenomena

Higher matter charge reorganizes the compact model most sharply in the multicomponent setting. For doubly charged matter, the DC–OD line is the primary candidate for realizing the three-dimensional Abelian-Higgs universality class (Bonati et al., 2022). Monte Carlo studies for ${\bm z}_{\bm x}$9 and $\lambda_{\bm x,\mu}\in U(1)$0 show that compact $\lambda_{\bm x,\mu}\in U(1)$1 transitions on the DC–OD line fall on the same universal $\lambda_{\bm x,\mu}\in U(1)$2 versus $\lambda_{\bm x,\mu}\in U(1)$3 curve as noncompact CH transitions, and thus belong to the same universality class for any $\lambda_{\bm x,\mu}\in U(1)$4 examined (Bonati et al., 2022).

The dependence on $\lambda_{\bm x,\mu}\in U(1)$5 is nontrivial. For the doubly charged model, earlier work had already established $\lambda_{\bm x,\mu}\in U(1)$6 first order and $\lambda_{\bm x,\mu}\in U(1)$7 continuous, implying

$\lambda_{\bm x,\mu}\in U(1)$8

(Bonati et al., 28 May 2026). The more recent finite-size scaling analysis for $\lambda_{\bm x,\mu}\in U(1)$9, with lattices up to $$H = - \beta N \sum_{\bm x,\mu} \left( \bar{\bm z}_{\bm x}\cdot \lambda_{\bm x,\mu}\, {\bm z}_{\bm x+\hat\mu} + {\rm c.c.}\right) -\beta_g \sum_{\bm x,\mu\neq\nu} \left( \lambda_{\bm x,\mu} \,\lambda_{\bm x+\hat{\mu},\nu} \,\bar{\lambda}_{\bm x+\hat{\nu},\mu} \,\bar{\lambda}_{\bm x,\nu} + {\rm c.c.}\right)$$0, sharpens this threshold. It finds strong evidence for a continuous DC–OD transition at $$H = - \beta N \sum_{\bm x,\mu} \left( \bar{\bm z}_{\bm x}\cdot \lambda_{\bm x,\mu}\, {\bm z}_{\bm x+\hat\mu} + {\rm c.c.}\right) -\beta_g \sum_{\bm x,\mu\neq\nu} \left( \lambda_{\bm x,\mu} \,\lambda_{\bm x+\hat{\mu},\nu} \,\bar{\lambda}_{\bm x+\hat{\nu},\mu} \,\bar{\lambda}_{\bm x,\nu} + {\rm c.c.}\right)$$1, with

$$H = - \beta N \sum_{\bm x,\mu} \left( \bar{\bm z}_{\bm x}\cdot \lambda_{\bm x,\mu}\, {\bm z}_{\bm x+\hat\mu} + {\rm c.c.}\right) -\beta_g \sum_{\bm x,\mu\neq\nu} \left( \lambda_{\bm x,\mu} \,\lambda_{\bm x+\hat{\mu},\nu} \,\bar{\lambda}_{\bm x+\hat{\nu},\mu} \,\bar{\lambda}_{\bm x,\nu} + {\rm c.c.}\right)$$2

weak first-order transitions for $$H = - \beta N \sum_{\bm x,\mu} \left( \bar{\bm z}_{\bm x}\cdot \lambda_{\bm x,\mu}\, {\bm z}_{\bm x+\hat\mu} + {\rm c.c.}\right) -\beta_g \sum_{\bm x,\mu\neq\nu} \left( \lambda_{\bm x,\mu} \,\lambda_{\bm x+\hat{\mu},\nu} \,\bar{\lambda}_{\bm x+\hat{\nu},\mu} \,\bar{\lambda}_{\bm x,\nu} + {\rm c.c.}\right)$$3, and inconclusive behavior for $$H = - \beta N \sum_{\bm x,\mu} \left( \bar{\bm z}_{\bm x}\cdot \lambda_{\bm x,\mu}\, {\bm z}_{\bm x+\hat\mu} + {\rm c.c.}\right) -\beta_g \sum_{\bm x,\mu\neq\nu} \left( \lambda_{\bm x,\mu} \,\lambda_{\bm x+\hat{\mu},\nu} \,\bar{\lambda}_{\bm x+\hat{\nu},\mu} \,\bar{\lambda}_{\bm x,\nu} + {\rm c.c.}\right)$$4, leading to the estimate

$$H = - \beta N \sum_{\bm x,\mu} \left( \bar{\bm z}_{\bm x}\cdot \lambda_{\bm x,\mu}\, {\bm z}_{\bm x+\hat\mu} + {\rm c.c.}\right) -\beta_g \sum_{\bm x,\mu\neq\nu} \left( \lambda_{\bm x,\mu} \,\lambda_{\bm x+\hat{\mu},\nu} \,\bar{\lambda}_{\bm x+\hat{\nu},\mu} \,\bar{\lambda}_{\bm x,\nu} + {\rm c.c.}\right)$$5

(Bonati et al., 28 May 2026). If one assumes that the lattice threshold $$H = - \beta N \sum_{\bm x,\mu} \left( \bar{\bm z}_{\bm x}\cdot \lambda_{\bm x,\mu}\, {\bm z}_{\bm x+\hat\mu} + {\rm c.c.}\right) -\beta_g \sum_{\bm x,\mu\neq\nu} \left( \lambda_{\bm x,\mu} \,\lambda_{\bm x+\hat{\mu},\nu} \,\bar{\lambda}_{\bm x+\hat{\nu},\mu} \,\bar{\lambda}_{\bm x,\nu} + {\rm c.c.}\right)$$6 coincides with the field-theory threshold $$H = - \beta N \sum_{\bm x,\mu} \left( \bar{\bm z}_{\bm x}\cdot \lambda_{\bm x,\mu}\, {\bm z}_{\bm x+\hat\mu} + {\rm c.c.}\right) -\beta_g \sum_{\bm x,\mu\neq\nu} \left( \lambda_{\bm x,\mu} \,\lambda_{\bm x+\hat{\mu},\nu} \,\bar{\lambda}_{\bm x+\hat{\nu},\mu} \,\bar{\lambda}_{\bm x,\nu} + {\rm c.c.}\right)$$7, this implies $$H = - \beta N \sum_{\bm x,\mu} \left( \bar{\bm z}_{\bm x}\cdot \lambda_{\bm x,\mu}\, {\bm z}_{\bm x+\hat\mu} + {\rm c.c.}\right) -\beta_g \sum_{\bm x,\mu\neq\nu} \left( \lambda_{\bm x,\mu} \,\lambda_{\bm x+\hat{\mu},\nu} \,\bar{\lambda}_{\bm x+\hat{\nu},\mu} \,\bar{\lambda}_{\bm x,\nu} + {\rm c.c.}\right)$$8, lower than the earlier perturbative estimate $$H = - \beta N \sum_{\bm x,\mu} \left( \bar{\bm z}_{\bm x}\cdot \lambda_{\bm x,\mu}\, {\bm z}_{\bm x+\hat\mu} + {\rm c.c.}\right) -\beta_g \sum_{\bm x,\mu\neq\nu} \left( \lambda_{\bm x,\mu} \,\lambda_{\bm x+\hat{\mu},\nu} \,\bar{\lambda}_{\bm x+\hat{\nu},\mu} \,\bar{\lambda}_{\bm x,\nu} + {\rm c.c.}\right)$$9 (Bonati et al., 28 May 2026).

At smaller $\lambda^q$0, the higher-charge compact model exhibits distinct line-by-line criticality. For $\lambda^q$1, $\lambda^q$2 has a continuous O(3) DC–OC line, a continuous Ising OC–OD line, and a first-order DC–OD line; for $\lambda^q$3, the DC–OC line is first order, the OC–OD line remains Ising-like, and the DC–OD line is continuous with

$\lambda^q$4

(Bonati et al., 2020).

6. Rigorous constructions, Wilson observables, and implementations

Beyond Monte Carlo criticality, the lattice Abelian Higgs model has a substantial rigorous and constructive literature. In a two-dimensional Villain formulation, integrating out the Higgs field yields a gauge-field marginal with a positive loop expansion,

$\lambda^q$5

which implies quantitative diamagnetic inequalities and ultraviolet stability after gauge fixing (Chandra et al., 2022). This provides a probabilistic control of the gauge sector that is largely absent from conventional numerical formulations.

Wilson loops and Wilson lines play different roles once matter is dynamical. In the $\lambda^q$6 fixed-length Abelian lattice Higgs model on $\lambda^q$7, low-temperature asymptotics show that Wilson loops are governed by rare localized defects and by an auxiliary $\lambda^q$8 gradient model; for $\lambda^q$9, this reduces to the Ising model (Forsström et al., 2021). Open Wilson lines are more relevant in the presence of Higgs matter, and their expectations factor asymptotically into a gauge-dressing term and a matter-field correlation term (Forsström, 2021). The Marcu–Fredenhagen ratio exists in all predicted phases of the $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$00 model, is strictly positive in nontrivial subsets of the Higgs and confinement phases, and vanishes in a nontrivial subset of the free phase, so it undergoes a phase transition and functions as an order parameter (Forsström, 2024). For compact charge-$${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$01 models, charged Wilson loops and charged Marcu–Fredenhagen ratios rigorously distinguish three phases when $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$02 (Forsström, 25 Feb 2026).

Hamiltonian and quantum-simulation formulations make the same structure accessible in real time. A Kogut–Susskind Hamiltonian with electric fields $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$03, link operators $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$04, vertex charges $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$05, and Gauss-law generators

$${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$06

has been mapped to ultracold-atom architectures in which exact local gauge symmetry is enforced by hyperfine angular-momentum conservation, and plaquette terms are generated perturbatively by auxiliary bosons (González-Cuadra et al., 2017). In $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$07 dimensions, a gauge-invariant tensor formulation with chemical potential has no sign problem, admits exact blocking formulas in the $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$08 limit, and yields spin-1 Hamiltonians related to two-species Bose–Hubbard models (Bazavov et al., 2015). A separate $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$09D study identifies a line of first-order transitions separating Higgs and confined regions, ending at a quantum critical point with central charge $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$10, interpreted as a product of a massless free fermion and a massless free boson, though with unresolved anomalies in the scaling data (Chanda et al., 2021).

Taken together, these results show that the lattice Abelian Higgs model is not a single universality class or a single phase diagram, but a family of gauge theories whose infrared behavior depends decisively on compactness, matter charge, dimension, and the flavor multiplicity $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$11. The recent higher-charge compact studies place this dependence in especially sharp form: the three-dimensional Abelian-Higgs charged fixed point is realized on the lattice only beyond a finite threshold in $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$12, and the best current compact-lattice estimate for the doubly charged case is $${\cal L}=|D_\mu{\bm\Phi}|^2+r\,{\bm \Phi}^*{\bm \Phi}+\frac{1}{6}u\,({\bm \Phi}^*{\bm \Phi})^2+\frac{1}{4g^2}F_{\mu\nu}^2, \qquad D_\mu=\partial_\mu+iA_\mu,$$13 (Bonati et al., 28 May 2026).