Phase diagram of the lattice superconductor

In a mean-field approximation the phase transition of a lattice superconductor is understood in terms of an orientation of its topological excitations (vortex loops and magnetic field loops). The loops are obtained through a duality transformation of the original variables. The phase boundary Tc(e2,b−1) is found as a function of the charge e and a Ginzburg-Landau quartic term coefficient b. Second-order behavior is found for a region b > bc(e2) and a tricritical point with first-order behavior appears for b < bc(e2), consistent with the Monte Carlo results of Dasgupta and Halperin and of Bartholomew. A crossover between XY and inverted XY behavior, in the second-order regime, is conjectured. An outer first-order phase boundary THLM(e2,b−1)≳Tc(e2, b−1) is also found, consistent with the early continuum results of Halperin, Lubensky, and Ma (HLM). THLM(e2,b−1) is associated with ordering within a superconducting grain, while Tc(e2,b−1) involves phase locking between grains