Phase structure of (2+1)-dimensional compact lattice gauge theories and the transition from Mott insulator to fractionalized insulator

Large-scale Monte Carlo simulations are employed to study phase transitions in the three-dimensional compact Abelian Higgs model in adjoint representations of the matter field, labeled by an integer q, for $q=2,3,4,5.\mathrm{}$ We also study various limiting cases of the model, such as the $Z_q$ lattice gauge theory, dual to the three-dimensional (3D) spin model, and the 3D $\mathrm{XY}$ spin model which is dual to the $Z_q$ lattice gauge theory in the limit $\overrightarrow{q}\infty .$ In addition, for benchmark purposes, we study the square lattice eight-vertex model, which is exactly solvable and features nonuniversal critical exponents. We have computed the first, second, and third moments of the action to locate the phase transition of the compact Abelian Higgs model in the parameter space $(\beta ,\kappa ),$ where $\beta$ is the coupling constant of the matter term and $\kappa$ is the coupling constant of the gauge term. We have found that for $q=3,$ the three-dimensional compact Abelian Higgs model has a phase-transition line ${\beta }_{\mathrm{c}}\left(\kappa \right)$ which is first order for $\kappa$ below a finite tricritical value ${\kappa }_{\mathrm{tri}}$ and second order above. The $\beta =\infty$ first order phase transition persists for finite $\beta$ and joins the second order phase transition at a tricritical point $({\beta }_{\mathrm{tri}},{\kappa }_{\mathrm{tri}})=(1.23\mathrm{}\pm 0.03,1.73\pm 0.03).$ For all other integer $q>~2$ we have considered, the entire phase-transition line ${\beta }_c\left(\kappa \right)$ is critical. We have used finite-size scaling of the second and third moments of the action to extract critical exponents $\alpha$ and $\nu$ without invoking hyperscaling, for the $\mathrm{XY}$ model, the $Z_2$ spin and lattice gauge models, as well as the compact Abelian Higgs model for $q=2\mathrm{}$ and $q=3.\mathrm{}$ In all cases, we have found that for practical system sizes, the third moment gives scaling of superior quality compared to the second moment. We have also computed the exponent ratio for the $q=2\mathrm{}$ compact $U\left(1\right)\mathrm{}$ Higgs model along the critical line, finding a continuously varying ratio $(1+\alpha )/\nu ,$ as well as continuously varying $\alpha$ and $\nu$ as $\kappa$ is increased from $0.76$ to $\infty ,$ with the Ising universality class $(1+\alpha )/\nu =1.763\mathrm{}$ as a limiting case for $\overrightarrow{\beta }\infty ,\overrightarrow{\kappa }0.761,$ and the $\mathrm{XY}$ universality class $(1+\alpha )/\nu =1.467\mathrm{}$ as a limiting case for $\overrightarrow{\beta }0.454,\overrightarrow{\kappa }\infty .$ However, the critical line exhibits a remarkable resilience of $Z_2$ criticality as $\beta$ is reduced along the critical line. Thus, the three-dimensional compact Abelian Higgs model for $q=2\mathrm{}$ appears to represent a fixed-line theory defining a new universality class. We relate these results to a recent microscopic description of zero-temperature quantum phase transitions within insulating phases of strongly correlated systems in two spatial dimensions, proposing the above to be the universality class of the zero-temperature quantum phase transition from a Mott-Hubbard insulator to a charge-fractionalized insulator in two spatial dimensions, which thus is that of the 3D Ising model for a considerable range of parameters.