Vortices with magnetic field inversion in noncentrosymmetric superconductors Julien Garaud,1, ∗ Maxim N. Chernodub,1, 2, y and Dmitri E. Kharzeev3, 4, 5, z 1Institut Denis Poisson CNRS/UMR 7013, Universit´ede Tours, 37200 France 2Pacific Quantum Center, Far Eastern Federal University, Sukhanova 8, Vladivostok, 690950, Russia 3Department of Physics and Astronomy, Stony Brook University, New York 11794-3800, USA 4Department of Physics and RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA 5Le Studium, Loire Valley Institute for Advanced Studies, Tours and Orl´eans,France (Dated: November 30, 2020) Superconducting materials with noncentrosymmetric lattices lacking space inversion symmetry exhibit a variety of interesting parity-breaking phenomena, including the magneto-electric effect, spin-polarized currents, helical states, and the unusual Josephson effect. We demonstrate, within a Ginzburg-Landau framework describing noncentrosymmetric superconductors with O point group symmetry, that vortices can exhibit an inversion of the magnetic field at a certain distance from the vortex core. In stark contrast to conventional superconducting vortices, the magnetic-field reversal in the parity-broken superconductor leads to non-monotonic intervortex forces, and, as a consequence, to the exotic properties of the vortex matter such as the formation of vortex bound states, vortex clusters, and the appearance of metastable vortex/anti-vortex bound states. I. INTRODUCTION noncentrosymmetric superconductors are supercon- ducting materials whose crystal structure is not symmet- ric under the spatial inversion. These parity-breaking materials have attracted much theoretical [1{6] and ex- perimental [7{11] interest, as they make it possible to investigate spontaneous breaking of a continuous sym- metry in a parity-violating medium (for recent reviews, see [12{14]). The parity-breaking nature of the supercon- ducting order parameter [8,9] in noncentrosymmetric su- perconductors leads to various unusual magneto-electric phenomena due to the mixing of singlet and triplet com- 0.015 0.05 0.1 0.2 0.5 1 3.2 ponents of the superconducting condensate, correlations between supercurrents and spin polarization, to the exis- jBj tence of helical states, and to unusual structure of vortex lattices. Figure 1. Inversion patterns of the magnetic field B of a vor- Moreover, parity breaking in noncentrosymmetric su- tex in a noncentrosymmetric superconductor. The magnetic perconductors also results in an unconventional Joseph- field forms helicoidal patterns around a straight static vortex. son effect, where the junction features a phase-shifted re- As the distance from the vortex core increases, the longitudi- lation for the Josephson current [15, 16]. Unconventional nal (parallel to the vortex core) component of the magnetic Josephson junctions consisting of two noncentrosymmet- field may change its sign. The magnetic field may exhibit sev- ric superconductors linked by a uniaxial ferromagnet eral sign reversals in the normal plane. In the picture, which were recently proposed as the element of a qubit that is a result of a numerical simulation of the Ginzburg-Landau avoids the use of an offset magnetic flux, enabling a sim- theory, the colors encode the amplitude of the magnetic field pler and more robust architecture [17]. B, in a normal plane with respect to the vortex line while the In the macroscopic description of such superconducting arrows show the orientation of the field. states, the lack of inversion symmetry yields new terms arXiv:2003.10917v2 [cond-mat.supr-con] 26 Nov 2020 in the Ginzburg-Landau free energy termed `Lifshitz in- variants'. These terms directly couple the magnetic field are invariant under spatial rotations. The correspond- with an electric current and thus lead to a variety of new ing Lifshitz invariant featuring these symmetries is de- effects that are absent in conventional superconductors. scribed by the simple, parity-violating isotropic term, The explicit form of the allowed Lifshitz invariant de- / Im( ∗D ) · B. This particular structure describes pends on the point symmetry group of the underlying noncentrosymmetric superconductors with O point group crystal structure. symmetry such as Li2Pt3B[9, 18], Mo3Al2C[19, 20], and In this paper, we consider a particular class of non- PtSbS [21]. centrosymmetric superconductors that break the dis- Vortex states in cubic noncentrosymmetric supercon- crete group of parity reversals and, at the same time, ductors feature a transverse magnetic field, in addition to 2 the ordinary longitudinal field. Consequently, they also II. THEORETICAL FRAMEWORK carry a longitudinal current on top of the usual trans- verse screening currents [22{24]. Therefore, as illustrated We consider noncentrosymmetric superconductors in Fig.1, both the superconducting current and the mag- with the crystal structure possessing the O point netic field form a helical-like structure that winds around group symmetry. Such materials are described by the the vortex core (for additional material illustrating the Ginzburg-Landau free energy F = R d3x F with the free- helical spatial structure of the magnetic streamlines, see energy density given by (see e.g. [12, 26]): AppendixB, and supplemental animations [25] described in AppendixC). Previous theoretical papers studied vor- B2 β F = + kjD j2 + γj · B + (j j2 − 2)2 ; (1) tices in the perturbative regime where the coupling to 8π 2 0 the Lifshitz invariant is small, either in the London limit where j = 2e Im ( ∗D ); we use ~=c=1. Here, the sin- (with a large Ginzburg-Landau parameter) [22, 23], or i' beyond it [24]. For currently known noncentrosymmetric gle component order parameter = j je is a com- materials, these approximations are valid since the mag- plex scalar field that is coupled to the vector potential nitude of the Lifshitz invariants, which can be estimated A of the magnetic field B = r×A through the gauge in a weak-coupling approximation, is typically small. We derivative D ≡ r − ieA, where e is a gauge coupling. propose here a general study of vortices, for all possi- The explicit breaking of the inversion symmetry is ac- ble values of the Lifshitz invariant coupling, both in the counted for by the Lifshitz invariant term with the pref- London limit and beyond. actor γ, which directly couples the magnetic field B and j = 2ej j2(r' − eA). In the absence of parity breaking, when γ = 0, the vector j matches the usual supercon- We demonstrate that vortices may feature an inversion ducting current. The parameter γ, which controls the of the magnetic field at a distance of about 4λL from the strength of the parity breaking, can be chosen to be pos- vortex center. Moreover, for rather high values of the itive without loss of generality. The other coupling con- coupling γ, alternating reversals may occur several times, stants k and β describe, respectively, the magnitude of at different distances from the vortex core. Such an in- the kinetic and potential terms in the free energy (1). version of the magnetic field is illustrated, in Fig.1. The The variation of the free energy (1) with respect to the reversal of the magnetic field, which is in stark contrast to scalar field ∗ yields the Ginzburg-Landau equation for conventional superconducting vortices, becomes increas- the superconducting condensate, ingly important for larger couplings of the Lifshitz invari- 2 2 ant term. This property of field inversion is responsible kD + 2ieγB · D = β(j j − 0) ; (2) for other unusual behaviors, also absent in conventional type-2 superconductors. Indeed, we show that it leads while the variation of the free energy with respect to the to the formation of vortex bound states, vortex clusters, gauge potential A gives the Amp`ere-Maxwell equation: and meta-stable pairs of vortex and anti-vortex. These B r× + γj = kj + 2γe2j j2B ≡ J : (3) phenomena should have numerous physical consequences 4π on the response of noncentrosymmetric superconductors to an external magnetic field. The supercurrent J, is defined via the variation of the free energy (1) with respect to the vector potential: J = δF/δA. Nonzero parity-breaking coupling γ gives The paper is organized as follows. In Sec.II, we intro- an additional contribution from the Lifshitz term, which duce the phenomenological Ginzburg-Landau theory that is proportional to B [26] (see also remark [27]). The describes the superconducting state of a noncentrosym- physical length scales of the theory are, respectively, the metric material with O point group symmetry. Next, in coherence length ξ and the London penetration depth Sec. III we investigate the properties of single vortices λ , both in the London limit and beyond it. We also demon- L strate that the parity-breaking superconductors can fea- k 1 ξ2 = ; and λ2 = : (4) ture an inversion of the longitudinal magnetic field. This L 2 2 2β 0 8πke 0 observation suggests that the intervortex interaction in parity-odd superconductors might be much richer than The Ginzburg-Landau parameter, κ = λL/ξ, is given by that for a conventional superconductor. Hence we de- the ratio of these characteristic length scales. rive analytically the intervortex interaction energy in the Note that since the parity-violating term in the London limit in Sec.IV. We show that the interaction Ginzburg-Landau model (1) is not positively defined, potential depends non-monotonically on the intervortex the strength of the parity violation cannot be arbitrar- distance, which leads to the existence of vortex bound ily large. For the free energy to be bounded from below states. Using numerical minimization of the Ginzburg- in the ground state, the parity-odd parameter γ cannot Landau free energy, we further observe that such bound exceed a critical value, states persist beyond the London limit. Our conclusions s and discussion of further prospects are given in the last k 0 6 γ < γ? ; where γ? = 2 2 = kλL : (5) section. 8πe 0 3 2 A detailed discussion of the positive definiteness, and 0.