Vortices with Magnetic Field Inversion in Noncentrosymmetric

Vortices with magnetic ﬁeld inversion in noncentrosymmetric superconductors

Julien Garaud,1, ∗ Maxim N. Chernodub,1, 2, † and Dmitri E. Kharzeev3, 4, 5, ‡ 1Institut Denis Poisson CNRS/UMR 7013, Universitéde 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 Orléans,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 superconducting materials whose crystal structure is not symmet- ric under the spatial inversion. These parity-breaking materials have attracted much theoretical [1–6] and experimental [7–11] interest, as they make it possible to investigate spontaneous breaking of a continuous symmetry in a parity-violating medium (for recent reviews, see [12–14]). The parity-breaking nature of the superconducting order parameter [8,9] in noncentrosymmetric superconductors 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- |B| 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 invariants’. 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(ψ∗Dψ) · 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 transverse 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 = + k|Dψ|2 + γj · B + (|ψ|2 − ψ2)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 (ψ∗Dψ); we use ~=c=1. Here, the sin- (with a large Ginzburg-Landau parameter) [22, 23], or iϕ beyond it [24]. For currently known noncentrosymmetric gle component order parameter ψ = |ψ|e 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 = ∇×A through the gauge in a weak-coupling approximation, is typically small. We derivative D ≡ ∇ − 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 = 2e|ψ|2(∇ϕ − 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 ψ∗ 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 · Dψ = β(|ψ| − ψ0)ψ , (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 Ampère-Maxwell equation: and meta-stable pairs of vortex and anti-vortex. These B ∇× + γj = kj + 2γe2|ψ|2B ≡ 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 2βψ0 8πke ψ0 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 ψ0 3

2 A detailed discussion of the positive definiteness, and ψ0. Hence the current now reads as j = 2eψ0 (∇ϕ − eA). the derivation of the range of validity are given in Ap- It leads to the second London equation that relates the pendixA. The bound (5) implies that the parity breaking magnetic field and j should not be too strong in order to ensure the validity of the minimalistic Ginzburg-Landau model (1). Note, 1 1 B = ∇×∇ϕ − 2 ∇×j . (7) however, that the upper bound on the parity-violating e 2eψ0 coupling applies only to the form of the free-energy functional (1). If the parity-violating coupling γ exceeds the The constant density approximation, together with critical value (5), the model has to be supplemented with Eq. (7), is then used to rewrite the Ampère-Maxwell higher-order terms, for the energy to be bounded. The equation (3) as the London equation for the current: Lifshitz invariant in the free energy (1) is given by a γ λ2 ∇×∇×j + j − 2 ∇×j = S, (8) higher-order term that becomes gradually irrelevant as L k the system approaches a phase transition to the normal phase. In our work, we stay away from the criticality where the source term on the right hand side reads to highlight the importance of the Lifshitz term for the 1 γ dynamics of the vortices. S = ∇×∇×∇ϕ − 2 ∇×∇ϕ 4πke kλL Φ γ 1 = 0 ∇×v − v , with v = ∇×∇ϕ . (9) III. VORTICES IN NONCENTROSYMMETRIC 2 4πk kλL 2π SUPERCONDUCTORS Here Φ0 = 2π/e is the elementary flux quantum, and v Vortices are the elementary topological excitations in is the density of vortex field that accounts for the phase superconductors. Below, in the London limit, we derive singularities. x In the dimensionless units, x˜ = , ∇˜ = λL∇, the vortex solutions for any values of the coupling γ < γ?. λL While the London limit is itself an interesting regime, it London equation is is important to verify that the overall physical picture ad- Φ vocated here is not merely an artifact of that particular ∇˜ ×∇˜ ×j + j − 2Γ∇˜ ×j = 0 ∇˜ ×v − Γv approximation. Consequently, we check that the results 4πkλL obtained in the London limit are consistent with the nu- and B(x˜) = Φ0v − 4πkλL∇˜ ×j . (10) merical solutions of the full nonlinear problem, by using the following procedure. For the energy to be bounded, the criterion (5) implies First of all, since the Lifshitz invariant behaves as a that the dimensionless coupling Γ = γ/kλL introduced scalar under rotations, the solutions should not depend here, satisfies 0 6 Γ < 1. Defining the amplitude A = Φ0 , the momentum space London equation reads on a particular orientation of the surface normal. Hence, 4πkλL there is no loss of generality to consider straight vortices along the z-axis. Such translationally invariant − p × p × jp + jp − 2iΓp × jp = A ip × vp − Γvp , (straight) vortices are described, with all generality, by the two-dimensional field ansatz in the xy-plane (see re- and Bp = Φ0v − 4πikλLp × jp . (11) mark [28]): where jp is the Fourier component of the current j in the space of the dimensionless momenta p: A = (Ax(x, y),Ay(x, y),Az(x, y)) and ψ = ψ(x, y) . (6) Z d3p Next, in order to numerically investigate the properties j(x˜) = eip·x˜ j . (12) of the vortex solutions, the physical degrees of freedom ψ (2π)3 p and A are discretized within a finite-element formulation [29], and the Ginzburg-Landau free energy (1) is subse- Similarly, the quantities vp and Bp are, respectively, the quently minimized using a non-linear conjugate gradient Fourier components of v(x˜) and B(x˜). The solution of algorithm. Given a starting configuration where the con- the algebraic equation (11) in the momentum space is densate has a specified phase winding (at large distances An ψ ∝ eiθ and θ is the polar angle relative to the vortex jm = − Γ(1 − p2)δ + (Ω + 2)pmpn p Σ mn center), the minimization procedure leads, after conver- 2 lo n mn n gence of the algorithm, to the vortex solution of the full + i(Ω + 2Γ )mlnp vp := Φ0Λp vp , (13) nonlinear theory [30]. Φ n Bm = 0 [1 + (1 − 2Γ2)p2]δ + (Ω + 2Γ2)pmpn p Σ mn 2 lo n mn n A. London limit solutions + iΓ(1 − p )mlnp vp := Φ0Υp vp , (14)

In the London limit, κ → ∞, the superconducting con- with the polynomials Σ ≡ Σ(p2) = (1 + p2)2 − 4Γ2p2 2 2 2 densate is approximated to have a constant density, |ψ| = and Ω ≡ Ω(p ) = 1 + p − 4Γ . Here δmn and mln are, 4 London limit Ginzburg-Landau 0.8 0.8 Parameters: 0.6 0.6 Bz Bz ψ =β =k =1; e=0.133 0 √ ) ) 0.4 λL =1.5; ξ =1/ 2 0.4 2 L 2 L κ=λL/ξ =2.12

0.2 0.2 πλ πλ 2 2 0 0 / / 0 0

Bθ Bθ (Φ (Φ

0 0 / /

-0.2 -0.2 B

B γ/kλL = 0.05 0.15 γ/kλL = 0.05 0.15 0.25 0.45 0.75 0.25 0.45 0.75 -0.4 -0.4 0.80 0.85 0.90 0.80 0.85 0.90 -0.6 -0.6 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 ρ/λL ρ/λL 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 -2 -1 0 1 0 2 4 6 8 10 8

8 . B B j j 4 |ψ|/ψ0 z ⊥ z ⊥ L 0 = 0 L y/λ -4 −1 −1 −2 −2 -8 ×10 ×10 ×10 ×10 -8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8 γ/kλ x/λL x/λL x/λL x/λL x/λL

Figure 2. The upper row displays the longitudinal (Bz) and circular (Bθ) components of the magnetic field of a single vortex, as functions of the radial distance ρ from the vortex center, and for various values of the parity-odd coupling γ. The left and right panels show the magnetic field in the London limit and beyond the London approximation, respectively. The panels in the bottom row, result from the minimization of the Ginzburg-Landau free energy at the parity-breaking coupling γ = 0.8γ?. They show the superconducting condensate |ψ|, the longitudinal and transverse components of the magnetic field, Bz and B⊥, and currents , jz and j⊥, in the transverse plane of the vortex. In the case of a weak parity violation, γ γ?, the longitudinal component of the magnetic field is similar to that of conventional Abrikosov vortices for which Bz(ρ) is monotonic and exponentially localized around the vortex core at ρ = 0. When the parity-breaking term becomes large, with γ approaching the critical value γ?, the longitudinal component Bz becomes a non-monotonic function as the distance ρ from the vortex core increases. respectively, the Kronecker and the Levi-Civita symbols, B. Single vortex and the silent indices are summed over. Thus, the vortex field v completely determines, via its The analysis becomes particularly simple for a single Fourier image vp, the momentum-space representations elementary vortex with a unit winding number (n1=1) of the current (13) and of the magnetic field (14). The located at the origin (x1 = 0). The corresponding mag- corresponding real-space solutions are obtained by the netic field reads as follows: Fourier transformation (12). Assuming the translation  2  invariance along the z-axis, a set of N vortices located iΓ(1 − p )py 2πΦ0δ(pz) 2 at the positions xã, and characterized by the individual Bp = 2  −iΓ(1 − p )px  . (16) λ Σ 2 2 winding numbers na (with a = 1, ··· ,N), is described by L (1 − 2Γ )p + 1 the Fourier components Next, we express the position, x˜ = (˜ρ cos θ, ρ˜sin θ, z˜), δ(p )e N and momentum, p = (q cos ϑ, q sin ϑ, pz), in cylindrical z z X −ip·xã vp = 2π 2 nae , (15) coordinates. An integration over the angular degrees of λL a=1 freedom ϑ nullifies the radial part Bρ of the magnetic field and generates the Bessel functions of the first kind, J0 where the Dirac delta for the momentum pz specifies the and J1. Hence, the nonzero components of the magnetic translation invariance of the configuration. field can be expressed as one-dimensional integrals over 5 the radial momentum q: top-right panels of Fig.2, the amplitude of inversion of the magnetic field, typically increases with the parity- Z ∞ 2 2 ρ Φ0Γ q (1 − q )dq qρ breaking coupling γ. Thus when γ is close to the critical Bθ = 2 2 2 2 2 J1 , λL 2πλL 0 (1 + q ) − 4Γ q λL coupling γ?, the magnitude of the responses and field in- ρ Φ Z ∞q[(1 − 2Γ2)q2 + 1]dq qρ versions become more important. B = 0 J . (17) z 2 2 2 2 2 0 When the parity-breaking coupling γ is small com- λL 2πλL 0 (1 + q ) − 4Γ q λL pared to the upper bound γ?, the longitudinal component Similarly, the nonzero components of the current are: Bz of the magnetic field is monotonic and exponentially localized, as for conventional vortices. The vortex con- Z ∞ 2 2 2 ρ Φ0 q (q + 1 − 2Γ )dq qρ figurations start to deviate from the conventional case jθ = 2 3 2 2 2 2 J1 , λL 8π kλL 0 (1 + q ) − 4Γ q λL when the parity breaking strengthens. Indeed, when γ Z ∞ 2 increases, the magnetic field Bz does not vary monotoni- ρ −Φ0Γ q(1 − q )dq qρ jz = J0 . (18) cally any longer. As can be seen in the top-right panel of λ 8π2kλ3 (1 + q2)2 − 4Γ2q2 λ L L 0 L Fig.2, it can be reversed, and even features several local Using the Hankel transform [31], as demonstrated in de- minima as γ approaches its critical value γ?. Note that, tail in AppendixD, these integrals can be solved an- the complicated spatial structure and inversion of the alytically in terms of the modified Bessel functions of magnetic field also comes with the inversion of the supercurrents. The distance from the vortex center ρ ' 4λL, the second√ kind Kν . Introducing the complex number η = Γ − i 1 − Γ2, the nonzero components of the mag- where the longitudinal component of the magnetic field netic field read first vanishes, corresponds to the radius where the in- plane current jθ reverses its sign. Similarly, the longitu- ρ Φ iηρ dinal current j vanishes for the first time at the shorter B = 0 Re iη2K , z θ 2 1 distance to the vortex core, ρ ' 2λ , where the circular λL 2πλL λL L magnetic field cancels, Bθ = 0. These observations are ρ −Φ0 2 iηρ Bz = 2 Re η K0 . (19) consistent with the results from the perturbative regime, λL 2πλ λL L γ γ? [24]. Interestingly, these specific radii are pretty Similarly, the nonzero components of j are: much unaffected by the value of the parity-breaking coupling. ρ −Φ0 iηρ Note that the structure of the zeros of the modified jθ = 2 3 Re iηK1 , λL 8π kλL λL Bessel functions with complex arguments shows that any non-zero value of the parity-breaking coupling γ exhibits ρ Φ0 iηρ j = Re ηK . (20) zeros at some distance away from the singularity. In prac- z λ 8π2kλ3 0 λ L L L tice, for small values of γ, the first zero is pushed very In the absence of parity breaking (Γ = 0 and η = −i), the far from the vortex core, and the amplitude of the field above integrals expectedly give the textbook expressions inversion is vanishingly small. Hence while the field inver- for the nonvanishing components of the magnetic field, sion formally occurs at all finite γ, it becomes noticeable Φ0 when γ is not too small. The structure of magnetic field Bz(ρ/λL) = 2πλ2 K0(ρ/λL), and of the superconducting L inversion as a function of γ qualitatively resembles the Φ0 current, 4πkjθ(ρ/λL)= 3 K1(ρ/λL). 2πλL alternating attractive/positive regions displayed in the Figure2 shows the magnetic field of a single vortex right panel of Fig.3. both in the London limit and for the full Ginzburg- Landau problem. First, although the solutions are expected to differ at the vortex core, the overall behavior remains qualitatively similar in both cases. Indeed, IV. VORTEX INTERACTIONS the London solutions are divergent at the vortex core, and thus they require a sharp cut-off at the coherence length ξ. Solutions beyond the London limit, on the The possibility of having an inversion of the magnetic other hand, are regular everywhere. The bottom row field suggests that the interaction between two vortices of Fig.2 shows a typical vortex solution obtained numer- might be much more involved than the pure repulsion ically beyond the London limit. This is a close-up view that occurs in conventional type-2 superconductors. In- of the vortex core structure, while the actual numerical deed, since the conventional long-range intervortex re- domain is much larger in order to prevent any finite-size pulsion is due to the magnetic field, it is quite likely that effect. While the density profile is similar to that of com- the interaction here might be not only quantitatively, but mon vortices, the magnetic field shows a pretty unusual also qualitatively altered. To investigate these, we con- profile featuring a slight inversion, away from the cen- sider the London limit free energy F written in the previ- ter. For the current parameter set, where γ = 0.8γ?, the ously used dimensionless coordinates. Using Eq. (12) to amplitude of the reversed field compared to the maxi- express the quantities j and B in terms of their Fourier mal amplitude is rather small. Yet, as illustrated on the components, we find the expression of the free energy in 6

0.4 25 γ/kλL = 0.05 0.15 0.3 0.25 0.45 0.75 20 0.2 0.80 0.85 0.90 )

L 0.1 15 L

d/λ 0

( 10 d/λ

U -0.1 Attract. 5 -0.2 Repuls. -0.3 0 2 4 6 8 10 12 14 0 0.1 0.3 0.5 0.7 0.9 d/λL γ/kλL

0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 -2 -1 0 1 0 2 4 6 8 10 8

8 . B B j j 4 |ψ|/ψ0 z ⊥ z ⊥ L 0 = 0 L y/λ -4 −1 −1 −2 −2 -8 ×10 ×10 ×10 ×10 -8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8 γ/kλ x/λL x/λL x/λL x/λL x/λL

Figure 3. The top-left panel shows the function U(d/λL) that controls the intervortex interactions, as a function of the distance d between the vortices, for various values of the parity-odd coupling γ. The top-right panel displays a phase diagram showing the attractive (blue) and repulsive (yellow) regions depending on the parity breaking coupling γ and the intervortex distance d. The panels in the top row corresponds to the London limit. The panels on the bottom row, obtained within the Ginzburg-Landau calculation, show various physical quantities in the transverse plane of a vortex bound state, for the parity-breaking coupling γ = 0.8γ? (the other parameters are the same as in Fig.2). This vortex pair, obtained after convergence of minimization of the Ginzburg-Landau free energy, demonstrates that the property of the non-monotonic interactions can survive beyond the London limit. The London limit estimates the intervortex separation dLL = 3.6λL, while the Ginzburg-Landau calculation finds dGL = 4.6λL. Despite the fact that the GL simulation is far from the London limit (here κ = 2.12), both values are in qualitative agreement. the momentum space: will contribute to the energy. Up to terms that are proportional to pz, and thus will be suppressed by the Dirac λ3 Z d3p n zz F = L B · B + (4πkλ j ) · (4πkλ j ) delta δ(pz), G takes the simple form 8π (2π)3 p −p L p L −p o + 2Γ(4πkλLjp) · B−p . (21) (1 − Γ2)(1 + p2) Gzz = + (terms ∝ p ) . (23) Replacing the Fourier components of the magnetic field Σ z Bp and of the current jp with the corresponding expressions in terms of the vortex field vp, Eqs. (13) and (14), respectively, yields the free energy: Finally, using the vortex field ansatz (15), together with λ3 Z d3p the expressions (22) and (23) determines the free energy F = L vmGmnvn (22) 8π (2π)3 p −p associated with a set of translationally invariant vortices mn lm ln lm ln lm ln where G = Υp Υ−p + Λp Λ−p + 2ΓΛp Υ−p .

The interaction matrix G has a rather involved structure. N Φ2(1−Γ2) X Z d2p 1 + p2 Yet, given that only the axial Fourier components of the 0 ip·(x˜a−x˜b) F = nanb 2 e . (24) zz 8πλL 2π Σ(p ) vortex ﬁeld (15) are nonzero, only the component G a,b=1 7

The two dimensional integration in (24) can further be between parallel wires carrying co-directed electric cur- simplified and finally, the free energy reads as: rents. The non-monotonic behavior of the magnetic field and 2 2 N Φ0(1 − Γ ) X |xa − xb| currents thus leads to non-monotonic intervortex inter- F = nanbU (25) actions, and therefore allows for bound-states of vortices 8πλL λL a,b=1 or clusters to form. Such a situation is known to exist in Z ∞ q(1 + q2) dq multicomponent superconductors due to the competition where U(x) = J (qx) 2 2 2 2 0 between various length scales (see e.g. [33–37] as well as 0 (1 + q ) − 4Γ q iη [38, 39] for superconducting/superfluid systems). In an = Re √ K0 (iηx) . (26) applied external field, the existence of non-monotonic in- 2 1 − Γ teractions allows for a macroscopic phase separation into Hence the free energy of a set of vortices reads as: domains of vortex clusters and vortex-less Meissner domains. The situation here contrasts with the multicom- 2 2 N ponent case, as it occurs only due to the existence of Lif- Φ0(1 − Γ ) X |xa − xb| F = F0 + nanbU , (27) shitz invariants. In two-dimensional systems of interact- 4πλL λL a,b>a ing particles, multi-scale potentials and non-monotonic interactions are known to be responsible for the forma- 2 2 Φ0(1−Γ ) P 2 tion of rich hierarchical structures. These structures in- where the term F0 = naU(ξ) accounts for 8πλL a clude clusters of clusters, concentric rings, clusters inside the self-energy of individual vortices. Since U(x) di- a ring, or stripes [40, 41]. It can thus be expected that verges at small separations x, the self-energy has to be very rich structures would appear in noncentrosymmetric regularized at the coherence length ξ λ , which de- L superconductors as well. However, a verification of this termines the size of the vortex core. The interaction en- conjecture is beyond the scope of the current work, as it ergy of the vortices separated by a distance d is thus deserves a separate detailed investigation. determined by the function U(d/λ ). In the absence L As shown in Fig.3, the interaction energy V (x) ∝ of parity breaking (Γ = 0 and η = −i), Eq. (27) leads v/v U(x) between two vortices with unit winding n = n = 1 again to the textbook expression for the interaction en- 1 2 can thus lead to the formation of a vortex bound state. ergy V (d/λ ) = Φ2K (d/λ )/4πλ . int L 0 0 L L A very interesting property is that it also opens the pos- Figure3 displays the function U(d/λ ), which controls L sibility of vortex/anti-vortex bound-states. Indeed, ac- the interacting potential between vortices, calculated in cording to Eq. (27) the interaction of a vortex n = 1 the London limit. For vanishing γ, the interaction is 1 and an anti-vortex n = −1 corresponds to a reversal purely repulsive, and it is altered by a nonzero coupling. 2 of the interacting potential: V (x) ∝ −U(x). Thus As shown in the left panel, when increasing γ/kλ the in- v/av L from Fig.3 it is clear that if a vortex/anti-vortex pair teraction can become non-monotonic with a minimum at is small enough, it will collapse to zero and thus lead a finite distance of about 4λ . Upon further increase of L to the vortex/anti-vortex annihilation. Now, considering the coupling γ toward the critical coupling γ , the inter- ? for example the curve γ/kλ = 0.9 in Fig.3, if the size of acting potential can even develop several local minima. L the vortex/anti-vortex pair is larger than 4λ , there ex- The phase diagram on the right panel of Fig.3 shows L ists an energy barrier that prevents the pair from further the different attractive and repulsive regions as functions collapse. Hence the vortex/anti-vortex pair should relax of γ and of the vortex separation. to a local minimum of the interaction energy. The re- The fact that the interaction energy features a mini- sulting vortex/anti-vortex bound state has thus a size of mum at a finite distance implies that a pair of vortices approximately 7λ . Note that the above analysis demon- tends to form a bound state. As can be seen in the bot- L strates that vortex/anti-vortex bound-states do exist as tom row of Fig.3, the tendency of vortices to form bound- meta-stable states in the London limit. It is quite likely states persists beyond the London approximation. This that these results are still qualitatively valid beyond the configuration is obtained numerically by minimizing the London approximation at least for strong type-2 super- Ginzburg-Landau free energy (1). Notice that these bot- conductors. The possibility to realize vortex/anti-vortex tom panels show a close-up view of the vortex pair, while pairs for weakly type-2 superconductors requires careful the actual numerical domain is much larger [32]. The analysis and is beyond the scope of the current work. formation of a vortex bound state can heuristically be understood as a compromise between the axial magnetic repulsion of Bz which competes with in-plane attraction V. CONCLUSIONS mediated by B⊥. The bound-state formation can alter- natively be understood to originate from the competition between the in-plane and axial contributions of the cur- In this paper we have demonstrated that the vortices in rents. First of all, the in-plane screening currents me- noncentrosymmetric cubic superconductors feature un- diate, as usual, repulsion between vortices. The inter- usual properties induced by the possible reversal of the action between the axial components of the currents, on magnetic field around them. Indeed, the longitudinal the other hand, mediates an attraction, just like the force (i.e., parallel to the vortex line) component of the mag- 8 netic field changes sign at a certain distance away from statistical properties and phase transitions in such mod- the vortex core. Contrary to the vortices in a conven- els. tional superconductor, the magnetic-field reversal in the Note added: In the process of completion of this parity-broken superconductor leads to non-monotonic in- work, we were informed about an independent work tervortex forces which can act both attractively and re- by Samoilenka and Babaev [42] showing similar results pulsively depending on the distance separating individual about vortices and their interactions. The submission of vortices. this work was coordinated with that of [42]. These properties have been demonstrated analytically within the London limit. Our numerical analysis of ACKNOWLEDGMENTS the nonlinear Ginzburg-Landau theory proves that the magnetic-field reversal and the non-monotonic intervor- We acknowledge fruitful discussions with D. F. Agter- tex forces survive beyond the London approximation in berg, E. Babaev and F. N. Rybakov and A. Samoilenka. noncentrosymmetric superconductors. We especially thank A. Samoilenka for suggesting to use Due to the nonmonotonic intervortex interactions, the the Hankel transform in our analytical calculations. The vortices in the parity-breaking superconductors may form work of M.C. was partially supported by Grant No. 0657- unusual states of vortex matter, such as bound states 2020-0015 of the Ministry of Science and Higher Ed- and clusters of vortices. The structure of the interaction ucation of Russia. The work of D.K. was supported potential strongly suggests that very rich vortex matter by the U.S. Department of Energy, Office of Nuclear structures can emerge. For example, hierarchically struc- Physics, under contracts DE-FG-88ER40388 and DE- tured quasi-regular vortex clusters, stripes and more, AC02-98CH10886, and by the Office of Basic Energy Sci- are typical features of the interacting multi-scale and ence under contract DE-SC-0017662. The computations non-monotonic interaction potentials [40, 41]. Moreover, were performed on resources provided by the Swedish Na- given the possibility to form vortex/anti-vortex bound tional Infrastructure for Computing (SNIC) at National states, we can anticipate important consequences for the Supercomputer Center at Linköping,Sweden.

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Appendix A: Positive definiteness of the energy regime so that all terms in the free energy (A2c) are positively defined. However, the density |ψ| is, in principle, The free energy (1) should be bounded from below in allowed to take any value, and large enough fluctuations order to be able to describe the ground state of the NCS of |ψ| might trigger an instability. A possible signature superconductor. To demonstrate the boundedness, we of the instability can indeed be spotted in the property use the relations that a variation of the absolute value of the condensate about the ground state, |ψ| = ψ0 + δ|ψ|, gives a negative 2 2 3 j = 2e|ψ| (∇ϕ − eA) , (A1a) contribution to the free energy, δF = −kδ|ψ|/(2e ψ0), in the linear order, provided all other parameters are fixed. |Dψ|2 = (∇|ψ|)2 + |ψ|2 (∇ϕ − eA)2 (A1b) |j|2 2 To illustrate a possible mechanism of the development = (∇|ψ|) + 2 2 , (A1c) 4e |ψ| of the instability inside the noncentrosymmetric super- to rewrite the energy density in the following form: conductor, let us consider a large enough local region characterized by a uniform, coordinate-independent con- 2 2 densate ψ. For this configuration, the third (gradient) B 2 k|j| F = + k (∇|ψ|) + + γj · B + V [ψ] (A2a) term in the free-energy density (A2c) is identically zero. 8π 4e2|ψ|2 Gradually increasing the value of the condensate beyond 1 k|j|2 = B2 + 8πγj · B + the ground state value ψ0, we increase the fourth (poten- 8π 4e2|ψ|2 tial) term in Eq. (A2c), which make this change energet- β ically unfavorable. On the other hand, as the condensate + k (∇|ψ|)2 + (|ψ|2 − ψ2)2 (A2b) 2 0 crosses the threshold of applicability of Eq. (A3), then the second term in the free energy (A2c) becomes nega- 1 2 k 2 2 = B + 4πγj + − 2πγ |j| 8π 4e2|ψ|2 tively defined, and the development of the current j leads to the unbounded decrease of this term. The rise in the 2 β 2 2 2 current j will, in turn, affect the first (magnetic) term, + k (∇|ψ|) + (|ψ| − ψ0) . (A2c) 2 which may be compensated by a rearranging of the mag- Leaving aside all terms with the perfect squares in netic field B with the local environment in such a way Eq. (A2c), we find that the only criterion for the free that the combination B + 4πγj keeps a small value in energy to be bounded from below is to require the pref- the discussed region. actor in front of the |j|2 term to be positive. We arrive at the following condition of the stability of the system (1): Notice that the presence of isolated vortices makes the system stable as in a vortex core the condensate vanishes, k γ2 < . (A3) ψ → 0, and the second, potentially unbounded term in 8πe2|ψ|2 (A2c) becomes positively defined. In our numerical simulations, we were also spotting certain unstable patterns In the ground state with |ψ| = ψ0, the stability condi- especially in the regimes when the Lifshitz-invariant cou- tion (A3) reduces to the simple inequality: pling γ was chosen to close to the critical value γ? in the ground state (A4). For example, a system of ran- γ < γ = kλ . (A4) ? L domly placed multiple elementary vortices relaxes their free energy via mutual attraction and the formation of a where λL is the London penetration depth (4). In the London limit, the superconducting density |ψ|2 common bound state. Since the vortex bound state hosts is a fixed constant quantity regardless of the external a stronger circular electric current, it becomes possible to conditions. Therefore, the Ginzburg-Landau theory for overcome the stability by ‘compressing’ the vortex clus- the NCS superconductor in the London limit is always ter, and then destabilizing the whole system. bounded from below provided the Lifshitz–invariant coupling γ satisfies Eq. (A4). We conclude that the processes that permit fluctuations of the condensate |ψ| towards the large values (as compared to the ground-state value ψ0) could activate Positive definiteness beyond the London limit the destabilization of the whole model. Theoretically, the unboundedness of the free energy from below may appear The issue of positive definiteness is less obvious be- to be an unwanted feature of the model. However, one yond the London limit. Indeed, let us first assume that should always keep in mind that the Ginzburg-Landau the values of the parameters (e, k, γ) are chosen in such functional is a leading part of the gradient expansion of a way that the formal criterium (A3) is satisfied. If we an effective model, and there always exist higher power neglect the fluctuations of the condensate ψ (this require- gradients that will play a stabilizing role preventing the ment is always satisfied in the London regime) then we in- unboundedness from actually being realized in a physi- deed find that the ground state resides in a locally stable cally relevant model. 11

Figure 4. Helical structure of the magnetic field streamlines around vortices in noncentrosymmetric superconductors. The magnetic field is displayed on the two planes normal with respect to the vortex line. The colors encode the amplitude |B|, while the arrows demonstrate the orientation of the field. The tubes represent streamlines of the magnetic field between both planes. The left panel shows the helical structure of the streamlines for a moderate value of the parity-breaking coupling, γ/kλ = 0.2. The streamlines here feature all the same chirality. The right panel corresponds to rather important parity-breaking coupling γ/kλ = 0.8, for which the longitudinal component of the magnetic field is inverted at some distance from the core. The chirality of the streamline depends on whether the longitudinal component of the magnetic field is inverted.

Appendix B: Vortex helicity body of the paper, vortices feature inversion of the magnetic field B for important parity-breaking coupling γ. As a result, the chirality of the streamline depends on As emphasized in the main body of the paper, the mag- whether the longitudinal component of the magnetic field netic field of vortex states in cubic noncentrosymmetric is inverted. More details about the helical structure of superconductors features a helicoidal structure around the magnetic field can be seen from animations in the the core. This is illustrated in Fig.4, which displays supplemental material [25]. a typical magnetic field structure around vortex cores. The magnetic field there is determined within the Lon- don approximation (19). Each helical tube represents a Appendix C: Description of the supplementary streamline of the magnetic field which is tangent to the animations (see ancillary files) magnetic field in every point. Fig.4 clearly shows that every streamline forms an helix along the axis z. Each There are three animations that illustrate the results helix has a period that depends on the distance from the of presented in the manuscript. The magnetic field forms helix to the center of the vortex. The latter property helical patterns around a straight static vortex, in a non- indicates that, contrary to the magnetic field itself, the centrosymmetric superconductor. For important values streamlines of the magnetic field are not invariant for the of the parity-breaking coupling γ, the magnetic field B translations along the z axis. can further show inversion patterns around the vortex. Fig.4 shows two qualitatively different situation of That is, as the distance from the vortex core increases, moderate (left panel) and important (right panel) parity- the longitudinal component of the magnetic field may breaking coupling γ. For moderate parity-breaking cou- change it sign. pling, the magnetic field streamlines have a helical struc- The supplementary animations display the following: ture with a pitch that varies with the distance from the On the two static planes, normal with respect to the vor- vortex core. Note that all streamlines have the same chi- tex line, the colors encode the amplitude of the magnetic rality, which is specified by the sign of the parity-breaking field B, while the arrows demonstrate the orientation of coupling γ. On the other hand, as discussed in the main the field. The tubes represent streamlines of the mag- 12 netic magnetic field between both planes. The integral in (D4) has a form of an Hankel transform [31] which is an integral transformation whose kernel is a • movie-1.avi and movie-2.avi: Helical structure Bessel function. In short, the ν-th Hankel transform Fν of the magnetic field streamlines around vortices of a given function f(q) with q > 0 is defined as in noncentrosymmetric superconductors. The fisrt movie (movie-1.avi) shows the helical structure of Z ∞ the streamlines for moderate value of the parity- Fν (x) := f(q)Jν (qx)qdq . (D5) breaking coupling γ/kλ = 0.2. The streamlines 0 here feature all the same chirality. The second The inverse of the Hankel transform is also a Hankel animation (movie-2.avi) corresponds to rather im- transform. The functions portant parity-breaking coupling γ/kλ = 0.8, for ν q ν which the longitudinal component of the magnetic f(q) = ←→ Fν (x) = a Kν (ax) , (D6) field is inverted at some distance from the core. q2 + a2 The chirality of the streamline depends on whether are related to each other via the Hankel transforma- the longitudinal component of the magnetic field is tion (D5). Identifying a = iη, the integral in (D4) thus inverted. That is depending on the chirality, some read as of the streamlines propagate forward (along positive z-direction), while other propagate backward Z ∞ qν J (qx)qdq = (iη)ν K (iηx) . (D7) (along negative z- direction). 2 2 ν ν 0 q − η • movie-3.avi: Magnetic streamlines, emphasizing This is a well-defined expression since the constant η is forward propagating lines (solid tubes), in the case a complex number and the integral does not cross any of field inversion due to important parity-breaking pole. As a result, the integral (D1) reads as follows: coupling γ/kλ = 0.8. The transparent tubes propagate backward (along the negative z-direction). ν Gν (x) = 2Re [C(iη) Kν (iηx)] . (D8) movie-4.avi: Magnetic streamlines, emphasizing • This generic relation determines both j, the magnetic backward propagating lines (solid tubes), in the field B and the interaction U(x). Notice that the mod- case of field inversion due to important parity- ified Bessel function of the second kind in Eq. (D8) can breaking coupling γ/kλ = 0.8. The transpar- be related to the Hankel function of first kind using the ent tubes propagate forward (along the positive z- relation direction). i K (z) = eiνπ/2H(1)(iz) , if − π < arg z ≤ π/2 . (D9) ν 2 ν Appendix D: Calculation of the integrals In our case, z = iηx and therefore arg z ∈ [0, π/2]. Below The intervortex interaction (26), the components of the we calculate the potential U(x) as an example. magnetic field (17), and the components of the current (18) are expressed in terms of integrals of the generic form Example: calculation of the interaction Z ∞ P (q) ν+1 The interaction U(x), defined in the main body in Gν (x)= 2 2 2 2 q Jν (qx)dq , (D1) 0 (1 + q ) − 4Γ q equation (25), reads as where ν = 0, 1. Introducing the complex number η = √ Z ∞ Γ − i 1 − Γ2, the quotient of the polynomials P (q) and P (q) U(x) = J0(qx)qdq 2 2 2 2 2 2 2 2 (1 + q ) − 4Γ q can be written as 0 (1 + q ) − 4Γ q 2 P (q) C C∗ where P (q) = 1 + q . (D10) = + , (D2) (1 + q2)2 − 4Γ2q2 q2 − η2 q2 − η∗2 Solving (D2) for the√ given polynomial P (q), gives the where ∗ stands for the complex conjugation (since q is coefficient C = iη/(2 1 − Γ2). The solution (D8) for real). The coefficient C is the solution of the equation this particular problem is thus P (q) = C(q2 − η∗2) + C∗(q2 − η2) . (D3) iη U(x) = Re √ K0 (iηx) . (D11) The integral (D1) becomes as follows: 1 − Γ2 Z ∞ C In the absence of parity-breaking Γ = 0, η = −i. This G (x) = 2 Re qν+1J (qx)dq ν q2 − η2 ν provides the textbook expression for the interaction of 0 vortices in the London limit: U(x) = K (x). Similar Z ∞ qν 0 calculations determine the solutions (19) and (20) for the = 2Re C 2 2 Jν (qx)qdq . (D4) 0 q − η components of B and j.