Abstract
Superconductivity owes its properties to the phase of the electron pair condensate that breaks the $U(1)$ symmetry. In the most traditional ground state, the phase is uniform and rigid. The normal state can be unstable towards special inhomogeneous superconducting states: the Abrikosov vortex state, and the Fulde-Ferrell-Larkin-Ovchinnikov state. Here we show that the phase-uniform superconducting state can go into a fundamentally different and more ordered non-uniform ground state, that we denote as a phase crystal. The new state breaks translational invariance through formation of a spatially periodic modulation of the phase, manifested by unusual superflow patterns and circulating currents, that also break time-reversal symmetry. We list the general conditions needed for realization of phase crystals. Using microscopic theory we then derive an analytic expression for the superfluid density tensor for the case of a non-uniform environment in a semi-infinite superconductor. We demonstrate how the surface quasiparticle states enter the superfluid density and identify phase crystallization as the main player in several previous numerical observations in unconventional superconductors, and predict existence of a similar phenomenon in superconductor-ferromagnetic structures. This analytic approach provides a new unifying aspect for the exploration of boundary-induced quasiparticles and collective excitations in superconductors. More generally, we trace the origin of phase crystallization to non-local properties of the gradient energy, which implies existence of similar pattern-forming instabilities in many other contexts.
Highlights
The defining characteristic of superfluidity and superconductivity is spontaneous symmetry breaking of the global U(1) phase χ, associated with the order parameter = | | exp(iχ )
We have described a superconducting state where the global U(1) phase spontaneously forms a modulation in space, breaking continuous translational invariance
The phase modulation results in a pattern of loop currents and breaking of time-reversal symmetry
Summary
The phase crystal, on the other hand, is associated with a modification of the symmetry variable χ describing the degeneracy manifold of the superconducting state and can occur even when the order parameter amplitude | | is large, i.e., deep inside the superconducting state far from the normal to superconductor transition; the phase crystal does maintain nontrivial particle currents It is different from the textures appearing in systems with multicomponent order parameters and a more complex degeneracy space, such as 3He and liquid crystals [20,21,22]. The patterns are formed on the much shorter coherence length scale ξ0 = hvF/2π kBTc, where vF is the Fermi velocity, Tc is the superconducting transition temperature, and kB is the Boltzmann constant (h = kB = 1 in the following) To describe this physics we ignore the amplitude gradient terms in the free energy and generalize the kinetic superflow energy in the limit of small ps as.
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