Lattice Abelian Higgs Model Research Articles

Lattice gauge theories in the presence of a linear gauge-symmetry breaking.

We study the effects of gauge-symmetry breaking (GSB) perturbations in three-dimensional lattice gauge theories with scalar fields. We study this issue at transitions in which gauge correlations are not critical and the gauge symmetry only selects the gauge-invariant scalar degrees of freedom that become critical. A paradigmatic model in which this behavior is realized is the lattice CP^{1} model or, more generally, the lattice Abelian-Higgs model with two-component complex scalar fields and compact gauge fields. We consider this model in the presence of a linear GSB perturbation. The gauge symmetry turns out to be quite robust with respect to the GSB perturbation: the continuum limit is gauge invariant also in the presence of a finite small GSB term. We also determine the phase diagram of the model. It has one disordered phase and two phases that are tensor and vector ordered, respectively. They are separated by continuous transition lines, which belong to the O(3), O(4), and O(2) vector universality classes, and which meet at a multicritical point. We remark that the behavior at the CP^{1} gauge-symmetric critical point substantially differs from that at transitions in which gauge correlations become critical, for instance at transitions in the noncompact lattice Abelian-Higgs model that are controlled by the charged fixed point: in this case, the behavior is extremely sensitive to GSB perturbations.

Lattice Abelian-Higgs model with noncompact gauge fields

We consider a noncompact lattice formulation of the three-dimensional electrodynamics with $N$-component complex scalar fields, i.e., the lattice Abelian-Higgs model with noncompact gauge fields. For any $N\ensuremath{\ge}2$, the phase diagram shows three phases differing for the behavior of the scalar-field and gauge-field correlations: The Coulomb phase (short-ranged scalar and long-ranged gauge correlations), the Higgs phase (condensed scalar-field and gapped gauge correlations), and the molecular phase (condensed scalar-field and long-ranged gauge correlations). They are separated by three transition lines meeting at a multicritical point. Their nature depends on the phases they separate, and on the number $N$ of components of the scalar field. In particular, the Coulomb-to-molecular transition line (where gauge correlations are irrelevant) is associated with the Landau-Ginzburg-Wilson ${\mathrm{\ensuremath{\Phi}}}^{4}$ theory sharing the same SU($N$) global symmetry but without explicit gauge fields. On the other hand, the Coulomb-to-Higgs transition line (where gauge correlations are relevant) turns out to be described by the continuum Abelian-Higgs field theory with explicit gauge fields. Our numerical study is based on finite-size scaling analyses of Monte Carlo simulations with ${C}^{*}$ boundary conditions (appropriate for lattice systems with noncompact gauge variables, unlike periodic boundary conditions), for several values of $N$, i.e., $N=2$, 4, 10, 15, and 25. The numerical results agree with the renormalization-group predictions of the continuum field theories. In particular, the Coulomb-to-Higgs transitions are continuous for $N\ensuremath{\gtrsim}10$, in agreement with the predictions of the Abelian-Higgs field theory.

’t Hooft surface in lattice gauge theory

We discuss the lattice formulation of the 't Hooft surface, that is, the two-dimensional surface operator of a dual variable. The 't Hooft surface describes the world sheets of topological vortices. We derive the formulas to calculate the expectation value of the 't Hooft surface in the multiple-charge lattice Abelian Higgs model and in the lattice non-Abelian Higgs model. As the first demonstration of the formula, we compute the intervortex potential in the charge-2 lattice Abelian Higgs model.

Large-N behavior of three-dimensional lattice CPN−1 models

We consider a three-dimensional lattice CPN−1 model, which corresponds to the lattice Abelian–Higgs model in the infinite gauge-coupling limit. We investigate its phase diagram and critical behavior in the large-N limit. We obtain numerical evidence that the model undergoes a first-order transition for sufficiently large values of N, i.e. for any N > 2 up to N = 100. The transition becomes stronger—both the latent heat and the surface tension increase—as N increases. Moreover, on the high-temperature side, gauge fields decorrelate on distances of the order of one lattice spacing for all values of N considered. Our results are consistent with a simple scenario, in which the transition is of first order for any N, including . We critically discuss the analytic large-N calculations that predicted a large-N continuous transition, showing that one crucial assumption made in these computations fails for the model we consider.

Universal features of the Abelian Polyakov loop in 1+1 dimensions

We show that the Polyakov loop of the two-dimensional lattice Abelian Higgs model can be calculated using the tensor renormalization group approach. We check the accuracy of the results using standard Monte Carlo simulations. We show that the energy gap produced by the insertion of the Polyakov loop obeys universal finite-size scaling which persists in the time continuum limit. We briefly discuss the relevance of these results for quantum simulations.

The role of vortex strings in the non-compact lattice Abelian Higgs model

The non-compact lattice version of the Abelian Higgs model is studied in terms of its topological excitations. The Villain form of the partition function is represented as a sum over world-sheets of gauge-invariant “vortex” strings. The phase transition of the system is then related to the density of these excitations. Through Monte Carlo simulations the density of the vortex sheets is shown to be a good order parameter for the system. The vortex density essentially vanishes in the Higgs phase, and the Coulomb phase consists of a single vortex condensate.

Logarithmic triviality of scalar quantum electrodynamics.

Using finite size scaling and histogram methods we obtain numerical results from lattice simulations indicating the logarithmic triviality of scalar quantum electrodynamics, even when the bare gauge coupling is chosen large. Simulations of the non-compact formulation of the lattice abelian Higgs model with fixed length scalar fields on $L^{4}$ lattices with $L$ ranging from $6$ through $20$ indicate a line of second order critical points. Fluctuation-induced first order transitions are ruled out. Runs of over ten million sweeps for each $L$ produce specific heat peaks which grow logarithmically with $L$ and whose critical couplings shift with $L$ picking out a correlation length exponent of $0.50(5)$ consistent with mean field theory. This behavior is qualitatively similar to that found in pure $\lambda\phi^{4}$.

Z2Abelian Higgs: a cluster expansion approach

The Z2 lattice Abelian Higgs model in D=2+1 is studied. New cluster expansion techniques which involve consideration of two perturbative terms are developed. These are used to obtain strong coupling expansions, for the vacuum energy density and the mass gaps of both 'meson' and 'plaquette' excitations. The expansions are used, via Pade approximant techniques and duality considerations, to explore the phase diagram of the theory. The location of the second-order phase boundaries are in good agreement with those obtained by other methods. The behaviour of the line of first-order transitions is examined and is estimated to terminate at lambda >or approximately=4.5.

Mean field analysis of the lattice Abelian Higgs model

We study an Abelian Higgs model in three, four and five dimensions using the mean field perturbation expansion with covariant gauge fixing. In four and five dimensions the mean field analysis shows three phases; a confined phase, a Coulomb phase with massless spin waves, and a Higgs phase in which the spin waves aquire a mass. The Higgs and confined phases are shown to be connected. In three dimensions we find a single phase.

Rigorous Analyses on Order Parameters and Charged States in the Lattice Abelian-Higgs Model

Rigorous Analyses on Order Parameters and Charged States in the Lattice Abelian-Higgs Model

Symmetry breaking in the lattice Abelian Higgs model.

A new gauge-invariant order parameter is introduced for the Abelian Higgs model and used to prove the existence of a phase transition for the lattice theory in three or more dimensions. In Landau gauge this order parameter is the limit of as Vertical Barx-yVertical Bar..-->..infinity.

One-loop mean-field theory for lattice Abelian Higgs model

The author considers the lattice Abelian Higgs model with frozen radial degrees of freedom using the mean-field approximation with one-loop corrections. In the weak-coupling region the behaviour of the frequencies arising in the expansion of the action allows the Higgs and Coulomb phases to be distinguished. Analytical results are presented for all phase transition lines. The phase structure obtained is in qualitative agreement with Monte Carlo calculations for Higgs charges q=1, 2 and 6.

On spontaneous symmetry breaking in the lattice abelian Higgs model

We investigate the order and temperature dependence of the Coulomb-Higgs phase transition in lattice scalar electrodynamics. Numerical results from Monte Carlo simulations are compared with analytical techniques such as mean field theory and the one-loop effective potential.

Phase structure, magnetic monopoles, and vortices in the lattice Abelian Higgs model

We present Monte Carlo calculations of lattice Abelian Higgs models in four dimensions and with charges of the Higgs particles equal to $q=1, 2, \mathrm{and} 6$. The phase transitions are studied in the plane of the two coupling constants considering separately average plaquette and average link expectation values. The density of topological excitations is studied. In the confinement phase we find finite densities of magnetic-monopole currents, electric currents, and vortex currents. The magnetic-monopole currents vanish exponentially in the Coulomb phase. The density of electric currents and vortex currents is finite in the Coulomb phase and vanishes exponentially in the Higgs phase.

Topological charge in the lattice abelian Higgs model

A lattice version of the two-dimensional abelian Higgs model is derived with emphasis on preserving the topological charge aspect of the continuum theory. The modulus of the Higgs field is not frozen, and a θ angle exists in the lattice quantum theory. It allows us to observe topological charge in Monte Carlo simulation and thus verify the semiclassical dilute vortex gas picture.

Finite-temperature behavior of the lattice abelian Higgs model

We study phase transitions in the lattice version of the abelian Higgs model, a model which can exhibit both spontaneous symmetry breaking and confinement. When the Higgs charge is the basic U(1) unit, we find that the Higgs and confinement regions are not separated by a phase transition and form a single homogenous phase which we call the total screening phase. The model does not undergo a symmetry restoring phase transition at finite temperature. If the Higgs charge is some multiple of the basic unit the model follows the conventional wisdom: there are 3 phases (normal, Higgs and confinement) at zero temperature, two of which disappear above some critical point. We apply the lessons learned from the lattice Higgs model to understand the behavior of the weak interactions at high temperature. In a long appendix we give an intuitive physical picture for the Polyakov-Susskind quark liberating phase transition and show that it is related to the Hagedorn spectrum of a confining model. We end with a collection of effective field theory approximations to various lattice theories.