It is shown that permanent confinement in three-dimensional compact U(1) gauge theory can be destroyed by matter fields in a deconfinement transition. This follows from a nontrivial infrared fixed point caused by matter, and an anomalous scaling dimension of the gauge field. This leads to a logarithmic interaction between the defects of the gauge fields, which form a gas of magnetic monopoles. For logarithmic interactions, the original electric charges are unconfined. The confined phase, which is permanent in the absence of matter fields, is reached at a critical electric charge, where the interaction between magnetic charges is screened by a pair-unbinding in a Kosterlitz-Thouless–like phase transition.

Quantum phase transitions in Mott insulators do not fit easily into the Landau-Ginzburg-Wilson paradigm. A recently proposed alternative to it is the so-called deconfined quantum criticality scenario, providing a new paradigm for quantum phase transitions. In this context it has recently been proposed that a second-order phase transition would occur in a two-dimensional spin $1/2$ quantum antiferromagnet in the deep easy-plane limit. A check of this conjecture is important for understanding the phase structure of Mott insulators. To this end we have performed large-scale Monte Carlo simulations on an effective gauge theory for this system, including a Berry-phase term that projects out the $S=1/2$ sector. The result is a first-order phase transition, thus contradicting the conjecture.

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.

We study the effective interactions between Dirac fermions on the surface of a three-dimensional topological insulator due to the proximity coupling to the magnetic fluctuations in a ferromagnetic or antiferromagnetic insulator. Our results show that the magnetic fluctuations can mediate attractive interactions between Dirac fermions of both Amperean and BCS types. In the ferromagnetic case, we find pairing between fermions with parallel momenta, so-called Amperean pairing, whenever the effective Lagrangian for the magnetic fluctuations does not contain a quadratic term. The pairing interaction also increases with increasing Fermi momentum and is in agreement with previous studies in the limit of high chemical potential. If a quadratic term is present, the pairing is instead of BCS type above a certain chemical potential. In the antiferromagnetic case, BCS pairing occurs when the ferromagnetic coupling between magnons on the same sublattice exceeds the antiferromagnetic coupling between magnons on different sublattices. Outside this region in parameter space, we again find that Amperean pairing is realized.