Vinod Mamale

We study the evolution of Abelian $U(1)$ electromagnetic as well as non-Abelian $SU(2)$ gauge fields, in the presence of spacetime oscillations. Analysis of the time evolution of Abelian gauge fields shows the presence of parametric resonance in spatial modes. A similar analysis in the case of non-Abelian gauge fields, in the linear approximation, shows the presence of the same resonant spatial modes. The resonant modes induce large fluctuations in physical observables including those that break the $CP$ symmetry. We also carry out time evolution of small random fluctuations of the gauge fields, using numerical simulations in $2+1$ and $3+1$ dimensions. These simulations help to study nonlinear effects in the case of non-Abelian gauge theories. Our results show that there is an increase in energy density with the coupling, at late times. These results suggest that gravitational waves may excite non-Abelian gauge fields more efficiently than electromagnetic fields. Also, gravitational waves in the early Universe and from the merger of neutron stars, black holes, etc., may enhance $CP$ violation and generate an imbalance in chiral charge distributions, magnetic fields, etc.

We study the evolution of abelian $U(1)$ electromagnetic as well as non-abelian $SU(2)$ gauge fields, in the presence of space-time oscillations. Analysis of the time evolution of abelian gauge fields shows the presence of parametric resonance in spatial modes. A similar analysis in the case of non-abelian gauge fields, in the linear approximation, shows the presence of the same resonant spatial modes. The resonant modes induce large fluctuations in physical observables including those that break the $CP-$symmetry. We also carry out time evolution of small random fluctuations of the gauge fields, using numerical simulations in $2+1$ and $3+1$ dimensions. These simulations help to study non-linear effects in the case of non-abelian gauge theories. Our results show that there is an increase in energy density with the coupling, at late times. These results suggest that gravitational waves may excite non-abelian gauge fields more efficiently than electromagnetic fields. Also, gravitational waves in the early Universe and from the merger of neutron stars, black holes etc. may enhance $CP-$violation and generate an imbalance in chiral charge distributions, magnetic fields etc.

We study lattice cutoff effects on the confinement-deconfinement transition and the $Z_3$ symmetry in $SU(3)$-Higgs theory in $3+1$ dimensions. The Higgs in this study is a complex triplet with vanishing bare mass and quartic coupling. The lattice cutoff is regulated by varying the number of temporal lattice sites, $N_{\tau }$. Our results show that the nature of the confinement-deconfinement transition depends on $N_{\tau }$. For $N_{\tau }=2$ the transition is found to be the end point of a first-order transition and is first order for $N_{\tau }\ge 3$. The distributions of the Polyakov loop and other observables, sensitive to the $Z_3$ symmetry, show that the strength of $Z_3$ explicit breaking decreases with $N_{\tau }$. Up to $T\simeq 2T_c$, the free energy difference between $Z_3$ states decreases with $N_{\tau }$, suggesting the realization of $Z_3$ symmetry in the continuum limit.

We study lattice cutoff effects on the confinement-deconfinement transition and the $Z_3$ symmetry in $SU(3)$-Higgs theory in $3+1$ dimensions. The Higgs in this study is a complex triplet with vanishing bare mass and quartic coupling. The lattice cutoff is regulated by varying the number of temporal lattice sites, $N_\tau$. Our results show that the nature of the confinement-deconfinement transition depends on $N_\tau$. For $N_\tau=2$ the transition is found to be the end point of a first-order transition and is first order for $N_\tau \ge 3$. The distributions of the Polyakov loop and other observables, sensitive to the $Z_3$ symmetry, show that the strength of $Z_3$ explicit breaking decreases with $N_\tau$. Up to $T\simeq 2T_c$, the free energy difference between $Z_3$ states decreases with $N_\tau$, suggesting the realization of $Z_3$ symmetry in the continuum limit.

At finite temperature, the field along a linear stretch of correlation length size is supposed to trace the shortest path in the field space given the two end point values, known as the Geodesic rule. In this study, we compute the probability that, the field variations over distances of correlation length follow this rule in theories with $O(2)$ global symmetry. We consider a simple ferromagnetic $O(2)$ spin-model and a complex $\phi^4$ theory. The computations are carried out on an ensemble of equilibrium configurations, generated using Monte Carlo simulations. The numerical results suggest significant deviation to the Geodesic rule, relevant for formation of topological defects during quench in 2nd order phase transition. Also for the case of $O(2)-$spins in two dimensions, distribution and density of vortices, have been studied. It is found that, for quench temperatures close to the transition point, the Kibble-Zurek Mechanism underestimates equilibrium density of defects. The exponents corresponding to width of the distributions, are found to be smaller than Kibble Mechanism estimates and match only when there is no deviation from the geodesic rule.

Using Monte Carlo methods, we study $Z_2$ symmetry in $Z_2+$Higgs theory. In $3+1$ space-time dimensions, our simulation results suggest that the $Z_2$ symmetry is realized at large number of temporal lattice points ($N_\tau$) in the Higgs symmetric phase. In order to see the dependence of $Z_2$ symmetry on the number of temporal lattice points we have also studied a simple temporal one dimensional model for a given spatial site. We show that the $Z_2$ symmetry is observed at the level of free energy at large $N_\tau$ limit. For this model, we also compute the density of states (DoS) for various $N_\tau$ values, where we show that the realization of the $Z_2$ symmetry happens at higher $N_\tau$. Therefore, the realization of $Z_2$ symmetry may be due to dominance of the DoS.

We study $Z_N$ symmetry in $SU(N)$ gauge theories in the presence of matter fields in the fundamental representation, by restricting the lattice partition function integration to matter fields which are uniform in spatial directions and gauge fields with vanishing spatial components. In this approximation the gauge matter field interaction effectively reduces to a 1-dimensional gauged chain. This makes analytical calculations of the matter field contribution to the Polyakov loop free energy possible. We show that in the limit of large number of temporal sites the explicit breaking of $Z_N$ symmetry in this free energy vanishes, driven by dominance of the density of states. We argue that the spatial links as well as the spatial modes of the matter fields determine the boundaries separating regions where $Z_N$ symmetry is realised from rest of the phase diagram.