Deconfinement transitions in three-dimensional compact lattice Abelian Higgs models with multiple-charge scalar fields

We investigate the nature of the deconfinement transitions in three-dimensional lattice Abelian Higgs models, in which a complex scalar field of integer charge $Q\ge 2$ is minimally coupled with a compact $U\left(1\right)$ gauge field. Their phase diagram presents two phases separated by a transition line where static charges $q$, with $q<Q$, deconfine. We argue that these deconfinement transitions belong to the same universality class as transitions in generic three-dimensional ${\mathbb{Z}}_Q$ gauge models. In particular, they are Ising-like for $Q=2$, of first order for $Q=3$, and belong to the three-dimensional gauge $XY$ universality class for $Q\ge 4$. This general scenario is supported by numerical finite-size scaling analyses of the energy cumulants for $Q=2, Q=4$, and $Q=6$.

Figure 1

Sketch of the $J\text{−}K$ phase diagram of the 3D one-component CLAH model, in which a compact $U\left(1\right)$ gauge field is coupled with a (unit-length) complex scalar field with charge $Q\ge 2$. A confined phase (for small $J$ or small $K$) and a deconfined phase (for large $J$ and $K$) are present, separated by a deconfinement transition line. For $K\to \infty$ and $J\to \infty$ the model reduces to the $XY$ vector model and to the lattice ${\mathbb{Z}}_Q$ gauge model with Wilson action, respectively. See Sec. 2 for more details.

Figure 2

Sketch of the $J\text{−}K$ phase diagram of the 3D multicomponent CLAH model, in which a compact $U$(1) gauge field is coupled with an $N$-component unit-length complex scalar field of charge $Q\ge 2$, for generic $N\ge 2$. There are three phases, the disordered-confined (DC), the ordered-deconfined (OD), and the ordered-confined (OC) phases. The AH model is equivalent to the $C{P}^{N-1}$ model for $K=0$, to the $O\left(2N\right)$ vector model for $K\to \infty$. For $J\to \infty$ and $Q\ge 2$, we obtain the lattice ${\mathbb{Z}}_Q$ gauge model as in the one-component case.

Figure 3

Rescaled $C_3$ cumulant along the $J=0.7$ line (top) and the $K=1$ line (bottom) for $Q=2$. Data approach an asymptotic FSS curve, supporting the Ising nature of the transition, although significant scaling corrections are visible for $K=1$. The continuous curves are computed using Eq. (37), with $a=0.67, p{\left(ac\right)}^{-3}=2.15$ for $J=0.7$, and with $a=0.78, p{\left(ac\right)}^{-3}=0.46$ for $K=1$; the scaling curve ${\mathcal{U}}_3^{\left(I\right)}\left(X\right)$ is reported in Appendix pp3. In the insets we plot $L^{{\omega }_I}[L^{3-3/{\nu }_I}{C}_3(J,K,L)-{\mathcal{C}}_3\left(X\right)]$ versus $X$ with ${\omega }_I=0.83$. The observed behavior is consistent with Eq. (39), as expected for a transition in the Ising universality class.

Figure 4

Rescaled $C_4$ cumulant along the $J=0.7$ line (top) and the $K=1$ line (bottom) for $Q=2$. The continuous curves are computed using Eq. (37) with $a=0.67, p{\left(ac\right)}^{-4}=2.55$ for $J=0.7$, and with $a=0.78, p{\left(ac\right)}^{-4}=0.35$ for $K=1$. We use the relation ${\mathcal{U}}_4^{\left(I\right)}\left(X\right)={\partial }_X{\mathcal{U}}_3^{\left(I\right)}\left(X\right)$; the scaling function ${\mathcal{U}}_3^{\left(I\right)}\left(X\right)$ is reported in Appendix pp3. We do not report MC data for $K=1$ and $L=56$ because they are too noisy. In the insets we plot $L^{{\omega }_I}[L^{3-4/{\nu }_I}{C}_4(J,K,L)-{\mathcal{C}}_4\left(X\right)]$ versus $X$ with ${\omega }_I=0.83$. The observed behavior is consistent with Eq. (39), as expected for a transition in the Ising universality class.

Figure 5

Results for $Q=2$ along the line $K=1$. Rescaled cumulants $C_{J,3}$ (top) and $C_{J,4}$ (bottom), for $Q=2$ along the line $K=1$, using the Ising critical exponent ${\nu }_I=0.629971$. The continuous curves are computed using Eq. (33), with $a=0.78, p=1.04, c_J=0.97$; the scaling curve ${\mathcal{U}}_3^{\left(I\right)}\left(X\right)$ is reported in Appendix pp3. We do not report estimates of $C_{J,4}$ for $K=1$ and $L=56$ because data are too noisy. In the insets we plot the combination defined as in Eq. (39) versus $X$ with ${\omega }_I=0.83$.

Figure 6

FSS behavior of the third cumulant $C_3$ for $Q=6$ along the line $J=1.0$, using the $XY$ critical exponent ${\nu }_{XY}=0.6717$. We also report (solid curve) the scaling curve (40), obtained by using the parametrization of ${\mathcal{U}}_3^{\left(XY\right)}\left(X\right)$ reported in Appendix pp4. In the inset we plot $L^{{\omega }_{XY}}[L^{3-3/{\nu }_{XY}}{C}_3(J,L)-{\mathcal{C}}_3\left(X\right)]$, with ${\nu }_{XY}=0.6717$ and ${\omega }_{XY}=0.789$.

Figure 7

FSS behavior of the third cumulant $C_3$ in the $Q=6$ model along the $K=4$ line, with ${\nu }_{XY}=0.6717$ and $J_c=0.4684$.

Figure 8

FSS behavior of the third cumulant $C_3$ in the $XY$ vector model, with ${\nu }_{XY}=0.6717$ and $J_c=0.22708234$.

Figure 9

FSS behavior of the third cumulant $C_3$ for $Q=4$ along the line $J=1.0$, with ${\nu }_{XY}=0.6717, J_c=1.0205$. Data approach the FSS scaling curve computed using Eq. (40); we use the parametrization of ${\mathcal{U}}_3^{\left(XY\right)}\left(X\right)$ reported in Appendix pp4, $a=16, p{\left(ac\right)}^{-3}=9\times {10}^{-5}$.

Figure 10

The phase diagram of the $N$-component NCLAH model (D1), in the Hamiltonian parameter space $\kappa \text{−}J$, for $N=1$ (top) and generic $N\ge 2$ (bottom). For $N=1$, there are two phases, the Coulomb (C) and Higgs (H) phases, characterized by the confinement and deconfinement of charged gauge-invariant excitations, respectively. For $N\ge 2$, the scalar field is disordered and gauge correlations are long ranged in the small-$J$ Coulomb (C) phase. For large $J$ two phases occur, the molecular (M) and Higgs (H) ordered phase, in which the global $SU\left(N\right)$ symmetry is spontaneously broken. The two phases are distinguished by the behavior of the gauge modes: The gauge field is long ranged in the M phase (small $\kappa$), while it is gapped in the H phase (large $\kappa$). Moreover, while the C and M phases are confined phases, the H phase shows the deconfinement of charged gauge-invariant excitations. See, e.g., Ref. [98] for more details.

Figure 11

FSS behavior of the energy cumulants ${\tilde{C}}_3$ (top) and ${\tilde{C}}_4$ (bottom), for the 3D ${\mathbb{Z}}_2$ gauge model. The solid line in the upper panel is the interpolation ${\mathcal{U}}_3^{\left(I\right)}\left(X\right)$ reported in Eq. (C3), while the solid line in the lower panel corresponds to $\frac{d}{dX}{\mathcal{U}}_3^{\left(I\right)}\left(X\right)$.

Figure 12

Plot of ${\tilde{B}}_3(J,L)L^{-3/{\nu }_I+3}$ versus $X$ in the extended range $-8\le X\le 9$ and interpolated scaling curve ${\mathcal{U}}_3^{\left(I\right)}\left(X\right)$.

Figure 13

Top: Plot of the scaling curves ${\mathcal{U}}_{2a}^{\left(I\right)}\left(X\right)$ and ${\mathcal{U}}_{2b}^{\left(I\right)}\left(X\right)$ (thin lines) and of their average, which is what we consider our best estimate (thick central line). Bottom: Plot of $({\tilde{C}}_2(J,L)-b_c)L^{-2/{\nu }_I+3}$ with $b_c=-4.10$. The thick line is the estimate of ${\mathcal{U}}_2^{\left(I\right)}\left(X\right)$.

Figure 14

Scaling plot of ${\tilde{C}}_3$ as a function of $X$ for the inverted $XY$ model. We also report the interpolating curve given in Eq. (D4).