We describe boundary effects in superconducting systems with Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting instability, using Bogoliubov-de-Gennes and Ginzburg-Landau formalisms. Firstly, we show that in dimensions larger than one the standard Ginzburg-Landau (GL) functional formalism for FFLO superconductors is unbounded from below. This is demonstrated by finding solutions with zero Laplacian terms near boundaries. We generalize the GL formalism for these systems by retaining higher order terms. Next, we demonstrate that a cubic superconductor with imbalanced fermions, at a mean-field level has a sequence of the phase transitions. At low temperatures it forms Larkin-Ovchinnikov state in the bulk but has a different modulation pattern close to the boundaries. When temperature is increased the first phase transition occurs when the bulk of the material becomes normal while the surfaces remain superconducting. The second transition occurs at higher temperature where the system retains superconductivity on the edges. The third transition is associated with the loss of edge-superconductivity while retaining superconducting gap in the vertices. We obtain the same sequence of phase transition by numerically solving the Bogoliubov-de Gennes model.

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Fulde, Ferrell, Larkin, and Ovchinnikov (FFLO) predicted inhomogeneous superconducting and superfluid ground states, spontaneously breaking translation symmetries. In this Letter, we demonstrate that the transition from the FFLO to the normal state as a function of temperature or increased Fermi surface splitting is not a direct one. Instead the system has an additional phase transition to a different state where pair-density-wave superconductivity (or superfluidity) exists only on the boundaries of the system, while the bulk of the system is normal. The surface pairdensity-wave state is very robust and exists for much larger fields and temperatures than the FFLO state.

Larkin-Ovchinnikov superconducting state has spontaneous modulation of Cooper pair density, while Fulde-Ferrell state has a spontaneous modulation in the phase of the order parameter. We report that a quasi-two-dimensional Dirac metal, under certain conditions has principally different inhomogeneous superconducting states that by contrast have spontaneous modulation in a submanifold of a multiple-symmetries-breaking order parameter. The first state we find can be viewed as a nematic superconductor where the nematicity vector spontaneously breaks rotational and translational symmetries due to spatial modulation. The other demonstrated state is a chiral superconductor with spontaneously broken time-reversal and translational symmetries. It is characterized by an order parameter, which forms a lattice pattern of alternating chiralities.For most superconductors, the ground state represents a configuration where the superconducting fields are homogeneous and can be classified according to the pairing symmetries. A generalization was theoretically proposed by Larkin and Ovchinnikov [1] and independently by Fulde and Ferrell [2]. It was demonstrated that not only U (1) symmetry can be broken in such a superconducting state but also the translational symmetry due to formation of Cooper pairs occurring with finite momentum. That state is called Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) state. In the simplest case, it can be caused by the pair-breaking effect of the Zeeman field in conventional superconductors. There are other mechanisms for the formation of inhomogeneous states in different systems such as cold atoms [3], or dense quark matter in neutron star interiors [4]. This made periodically modulated superconducting and superfluid states a subject of wide interest (for reviews see Refs. [5,6]).In this Rapid Communication we discuss a class of materials that supports inhomogeneous states which are principally different from the LOFF solutions. Namely, we find an inhomogeneous counterpart of the chiral superconducting state, where the system spontaneously forms a pattern of alternating chiralities, thereby breaking both translational and time reversal symmetries. Since the time reversal shares Z 2 symmetry with Ising magnets, we term this state as "antichiral" state in analogy to the antiferromagnetism. Another state we find is an inhomogeneous counterpart of the nematic superconducting state where the nematic vector is modulated, forming a nematicity-wave.We show that these states occur for the type of microscopic physics like that found in the recently discovered doped topological insulators [7][8][9]. Experimental studies of these materials suggest the presence of nematicity in the superconducting state with two components and odd-parity symmetric order parameter. However the chiral state might be more energetically favorable in the quasi-two-dimensional limit of these Dirac materials, in which the Fermi surface is cylindrical. The type of the superconducting pairing and the Majorana surface states are subjects of in...

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