Vortices with Magnetic Field Inversion in Noncentrosymmetric Superconductors Julien Garaud, Maxim N

Vortices with magnetic field inversion in noncentrosymmetric superconductors Julien Garaud, Maxim N. Chernodub, Dmitri E. Kharzeev

To cite this version:

Julien Garaud, Maxim N. Chernodub, Dmitri E. Kharzeev. Vortices with magnetic field inversion in noncentrosymmetric superconductors. Physical Review B, American Physical Society, 2020, 102 (18), pp.184516. 10.1103/PhysRevB.102.184516. hal-02542879

HAL Id: hal-02542879 https://hal.archives-ouvertes.fr/hal-02542879 Submitted on 9 Nov 2020

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J. Garaud,1, ∗ M. N. Chernodub,1, 2, † and D. 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: March 25, 2020) Superconducting materials with non-centrosymmetric lattices lacking the space inversion symmetry exhibit a variety of interesting parity-breaking phenomena, including magneto-electric effect, spin-polarized currents, helical states, and unusual Josephson effect. We demonstrate, within a Ginzburg-Landau framework describing non-centrosymmetric 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 a 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 appearance of metastable vortex/anti-vortex bound states.

I. INTRODUCTION

Non-centrosymmetric superconductors are superconducting materials whose crystal structure is not symmet- ric under the spatial inversion. These parity-breaking materials have attracted much theoretical [1,2] and experimental [3–5] interest, as they open the possibility to investigate spontaneous breaking of a continuous symmetry in a parity-violating medium (for recent reviews, see [6–8]). The parity-breaking nature of the superconducting order parameter [4,5] in the non-centrosymmetric superconductors leads to various unusual magnetoelectric phenomena due to the mixing of singlet and triplet 0.015 0.05 0.1 0.2 0.5 1 3.2 components of the superconducting condensate, correla- tions between supercurrents and spin polarization, to the |B| existence of helical states, and unusual structure of vortex lattices. Figure 1. Inversion patterns of the magnetic field B of a vor- Moreover, parity breaking in the non-centrosymmetric tex in a non-centrosymmetric superconductor. The magnetic superconductors also results in an unconventional field forms helicoidal patterns around a straight static vortex. Josephson effect, where the junction features a phase- As the distance from the vortex core increases, the longitudi- shifted relation for the Josephson current [9, 10]. Un- nal (parallel to the vortex core) component of the magnetic conventional Josephson junctions consisting of two non- field may change its sign. The magnetic field may exhibit sev- centrosymmetric superconductors linked by a uniaxial eral sign reversals in the normal plane. In the picture, which ferromagnet were recently proposed as the element of is a result of a numerical simulation of the Ginzburg-Landau a qubit that avoids the use of an offset magnetic flux, theory, the colors encode the amplitude of the magnetic field enabling a simpler and more robust architecture [11]. B, in a normal plane with respect to the vortex line while the In the macroscopic description of such superconducting arrows demonstrate the orientation of the field. states, the lack of inversion symmetry yields new terms

arXiv:2003.10917v1 [cond-mat.supr-con] 24 Mar 2020 in the Ginzburg-Landau free energy represented by the so-called the Lifshitz invariants. These terms directly and, at the same time, are invariant under spatial ro- couple the magnetic field B to the supercurrent j and tations. The corresponding Lifshitz invariant featur- thus lead to a variety of new effects that are absent in ing these symmetries is described by a simple, parity- conventional superconductors. The explicit form of the violating isotropic term, γj · B, where the coupling γ de- allowed Lifshitz invariant depends on the point symmetry termines the strength of the parity breaking. This partic- group of the underlying crystal structure. ular structure describes non-centrosymmetric supercon- In this paper, we consider a particular class of non- ductors with O point group symmetry such as Li2Pt3B centrosymmetric superconductors whose macroscopic in- [5, 12], Mo3Al2C[13, 14], and PtSbS [15]. teractions break the discrete group of parity reversals Vortex states in cubic non-centrosymmetric supercon- 2 ductors feature a transverse magnetic field, in addition to II. THEORETICAL FRAMEWORK the ordinary longitudinal field. Consequently, they also carry a longitudinal current on top of the usual trans- We consider non-centrosymmetric superconductors verse screening currents [16–18]. Therefore, as illustrated with the crystal structure possessing the O point group in Fig.1, both the superconducting current and the mag- symmetry. Such materials are described, in the vicin- netic field form a helical-like structure that winds around ity of the superconducting critical temperature, 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. [6, 20]): AppendixB, and animations [19] ). The previous theoretical papers studied vortices in the perturbative regime B2 β F = + k|Dψ|2 + γj · B + (|ψ|2 − ψ2)2 , (1) where the coupling to the Lifshitz invariant γ is small, ei- 8π 2 0 ther in the London limit (with a large Ginzburg-Landau where j = 2e Im (ψ∗Dψ); we use =c=1. Here, the sin- parameter) [16, 17], or beyond it [18]. For currently ~ gle component order parameter ψ = |ψ|eiϕ is a complex known non-centrosymmetric materials, these approxima- scalar field that is coupled to the vector potential A of tions are valid since the magnitude of the Lifshitz invari- the magnetic field B = ∇×A through the gauge deriva- ants, which can be estimated in a weak-coupling approx- tive D ≡ ∇ − ieA, where e is a gauge coupling. The imation, is typically small. We propose here a general explicit breaking of the inversion symmetry is accounted study of vortices, for all possible values of the Lifshitz in- by the Lifshitz invariant term with the prefactor γ, that variant coupling, both in the London limit and beyond. directly couples the magnetic field B and the supercurrent j = 2e|ψ|2(∇ϕ − eA). The current j is the usual We demonstrate that vortices may feature an inver- superconducting current at γ = 0, i.e. in the absence of sion of the magnetic field at distance of about 4λL from parity breaking. The parameter γ can be chosen to be the vortex center. Moreover, for rather high values of the positive without loss of generality. At a nonzero parity- coupling γ, alternating reversals may occur several times, breaking coupling γ, the current gets an additional con- at different distances from the vortex core. Such an in- tribution from the Lifshitz term [20]. The other coupling version of the magnetic field is illustrated, in Fig.1. The constants k and β describe, respectively, the magnitude reversal of the magnetic field, which is in stark contrast to of the kinetic and potential terms in the free energy (1). conventional superconducting vortices, becomes increas- The variation of the free energy (1) with respect to the ingly important for larger couplings of the Lifshitz invari- scalar field ψ∗ yields the Ginzburg-Landau equation for ant term. This property of field inversion is responsible the superconducting condensate, for other unusual behaviours, also absent in conventional 2 2 type-2 superconductors. Indeed, we show that it leads kD + 2ieγB · Dψ = β(|ψ| − ψ0)ψ , (2) to the formation of vortex bound states, vortex clusters, while the variation of the free energy with respect to the and meta-stable pairs of vortex and anti-vortex. These gauge potential A gives the Ampère-Maxwell equation: phenomena should have numerous physical consequences B on the response of non-centrosymmetric superconductors ∇× + γj = kj + 2γe2|ψ|2B . (3) to an external magnetic field. 4π The physical length scales of the theory are the coherence The paper is organized as follows. In Sec.II, we length ξ and the London penetration depth λL, introduce the phenomenological Ginzburg-Landau the- k 1 ory that describes the superconducting state of a non- 2 2 ξ = , and λL = 2 2 , (4) centrosymmetric material with the O point group sym- 2βψ0 8πke ψ0 metry. Next, in Sec.III we investigate the properties respectively. The Ginzburg-Landau parameter, κ = of single vortices both in the London limit and beyond λL/ξ, is given by the ratio of these characteristic length it. We also demonstrate that the parity-breaking super- scales. conductors can feature an inversion of the longitudinal Note that since the parity-violating term in the magnetic field. This observation suggests that the inter- Ginzburg-Landau model (1) is not positively defined, vortex interaction in parity-odd superconductors might the strength of the parity violation cannot be arbitrar- be much richer than that for a conventional supercon- ily large. For the free energy to be bounded from below ductor. Hence we derive analytically the intervortex in- in the ground state, the parity-odd parameter γ cannot teraction energy in the London limit in Sec.IV. We show exceed a critical value, that the interaction potential depends non-monotonically s on the intervortex distance, which leads to the existence k 0 γ < γ , where γ = = kλ . (5) of vortex bound states. Using numerical minimization 6 ? ? 8πe2ψ2 L of the Ginzburg-Landau free energy, we further observe 0 that such bound states persist beyond the London limit. A detailed discussion of the positive definiteness, and Our conclusions and discussion of further prospects are the derivation of the range of validity are given in Ap- given in the last section. pendixA. The bound (5) implies that the parity breaking 3 should not be too strong in order to ensure the validity of current the minimalistic Ginzburg-Landau model (1). Note however that the upper bound on the parity-violating cou- 1 1 B = ∇×∇ϕ − 2 ∇×j . (7) pling applies only to the form of the free energy functional e 2eψ0 (1). If the parity violating coupling γ exceeds the critical The constant density approximation, together with value (5), the model has to be supplemented with higher- Eq. (7), is then used to rewrite the the Ampère-Maxwell order gradient terms, for the energy to be bounded. equation (3) as the London equation for the current:

2 γ λL∇×∇×j + j − 2 ∇×j = S, (8) III. VORTICES IN NON-CENTROSYMMETRIC k SUPERCONDUCTORS where the source term on the right hand side,

Vortices are the elementary topological excitations in 1 γ S = ∇×∇×∇ϕ − 2 ∇×∇ϕ superconductors. Below, in the London limit, we derive 4πke kλL vortex solutions for any values of the coupling γ < γ?. Φ γ 1 = 0 ∇×v − v , with v = ∇×∇ϕ . (9) As previously stated, the new solutions described here 4πk kλ2 2π exhibit very unusual properties like the inversion of the L magnetic field, which allows for the vortex bound states Here Φ0 = 2π/e is the elementary flux quantum, and v formation. While this property is interesting by itself, is the density of vortex field that accounts for the phase it is also important to verify that the overall physical singularities. x picture advocated here is not merely an artifact of the In the dimensionless units, x˜ = , ∇˜ = λL∇, the λL London limit. Consequently, we check that the results London equation is obtained in the London limit are consistent with the nu- Φ merical solutions of the full nonlinear problem, by using ∇˜ ×∇˜ ×j + j − 2Γ∇˜ ×j = 0 ∇˜ ×v − Γv the following procedure. 4πkλL

The Lifshitz invariant is a scalar under rotations, thus and B(x˜) = Φ0v − 4πkλL∇˜ ×j . (10) solutions are the same for any orientation of the surface normal. It thus makes sense to consider the case of field For the energy to be bounded, the criterion (5) implies configurations that are translationally invariant along the that the dimensionless coupling Γ = γ/kλL introduced z-axis. Thus, the fields should respect symmetries gener- here, satisfies 0 6 Γ < 1. Defining the amplitude A = ated by the Killing vector K = ∂/∂z. Since all internal Φ0 , the momentum space London equation reads as (z) 4πkλL symmetries of the theory are gauged, there exist a gauge where the fields do not depend on z [21]. A reasonable − p × p × jp + jp − 2iΓp × jp = A ip × vp − Γvp , field ansatz is thus and Bp = Φ0v − 4πikλLp × jp . (11) A = (Ax(x, y),Ay(x, y),Az(x, y)) and ψ = ψ(x, y) . (6) where jp is the Fourier component of the current j in the space of the dimensionless momenta p: To investigate the properties of the vortex solutions, the physical degrees of freedom ψ and A are dis- Z d3p cretized within a finite-element formulation [22], and the j(x˜) = eip·x˜ j . (12) (2π)3 p Ginzburg-Landau free energy (1) is subsequently mini- mized using a non-linear conjugate gradient algorithm. Similarly, the quantities vp and Bp are, respectively, the Given a starting configuration where the condensate has Fourier components of v(x˜) and B(x˜). The solution of iθ a specified phase winding (at large distances ψ ∝ e the algebraic equation (11) in the momentum space is and θ is the polar angle relative to the vortex center), the minimization procedure leads, after convergence of An jm = − Γ(1 − p2)δ + (Ω + 2)pmpn the algorithm, to the vortex solution of the full nonlinear p Σ mn theory [23]. 2 lo n mn n + i(Ω + 2Γ )mlnp vp := Φ0Λp vp , (13)

m Φ0 n 2 2 2 m n Bp = [1 + (1 − 2Γ )p ]δmn + (Ω + 2Γ )p p A. London limit solutions Σ 2 lo n mn n + iΓ(1 − p )mlnp vp := Φ0Υp vp , (14) In the London limit, κ → ∞, the superconducting condensate is approximated to have a constant den- with the polynomials Σ ≡ Σ(p2) = (1 + p2)2 − 4Γ2p2 2 2 2 sity, |ψ| = ψ0. Hence the supercurrent now reads as and Ω ≡ Ω(p ) = 1 + p − 4Γ . Here δmn and mln are, 2 j = 2eψ0 (∇ϕ − eA). It leads to the second London respectively, the Kronecker and the Levi-Civita symbols, equation that relates the magnetic ﬁeld and the super- and the silent indices are summed over. 4 London limit Ginzburg-Landau 0.8 0.8 Parameters: 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 0.2 πλ πλ 2 2 0 /

/ -0.4 0 0 0.4 0.1 B Bθ (Φ

(Φ θ

0 /

/ 0 -0.1 B

B γ/kλL = 0.00 0.20 γ/kλL = 0.07 0.20 -0.4 0.45 0.70 0.85 0.30 0.47 0.67 -0.3 -0.8 0.90 0.95 0.97 0.73 0.80 0.87 -0.5 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 ρ/λL ρ/λL 0 0.2 0.6 1 0 0.5 1 1.5 2 2.5 0 0.2 0.6 1 -2 -1 0 1 0 0.2 0.6 1 8

8 . B B j j 4 |ψ|/ψ0 z ⊥ z ⊥ L 0 = 0 L y/λ -4 −2 −1 -8 ×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 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 supercurrents , 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 at 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.

Thus, the vortex field v completely determines, via its B. Single vortex Fourier image vp, the momentum-space representations of the supercurrent (13) and of the magnetic field (14). The analysis becomes particularly simple for a single The corresponding real space solutions are obtained by elementary vortex with a unit winding number (n1 = the Fourier transformation (12). Assuming the transla- 1) located at the origin (x1 = 0). The corresponding tion invariance along z-axis, a set of N vortices located magnetic field reads as follows: at the positions xã, and characterized by the individual winding numbers n (with a = 1, ··· ,N), is described by  2  a iΓ(1 − p )py the Fourier components 2πΦ0δ(pz) 2 Bp = 2  −iΓ(1 − p )px  . (16) λLΣ (1 − 2Γ2)p2 + 1

δ(p )e N Next, we express the position, x˜ = (˜ρ cos θ, ρ˜sin θ, z˜), z z X −ip·xã vp = 2π nae , (15) and momentum, p = (q cos ϑ, q sin ϑ, p ), in cylindrical λ2 z L a=1 coordinates. An integration over the angular degrees of 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: comes with the inversion of the supercurrents. Note that the distance from the vortex center ρ ' 4λ where the Z ∞ 2 2 L ρ Φ0Γ q (1 − q )dq qρ longitudinal component of the magnetic field first van- Bθ = 2 2 2 2 2 J1 λL 2πλL 0 (1 + q ) − 4Γ q λL ishes, corresponds to the radius where the in-plane cur- ∞ ρ Φ Z q[(1 − 2Γ2)q2 + 1]dq qρ rent jθ reverses its sign. Similarly, the longitudinal cur- B = 0 J . (17) z 2 2 2 2 2 0 rent jz vanishes for the first time at the shorter distance λL 2πλL 0 (1 + q ) − 4Γ q λL to the vortex core, ρ ' 2λL, where the circular magnetic Similarly, the nonzero components of the current are: field cancels, Bθ = 0. These observations are consistent with the results from the perturbative regime γ γ Z ∞ 2 2 2 ? ρ Φ0 q (q + 1 − 2Γ )dq qρ [18]. Interestingly, these specific radii are pretty much jθ = 2 3 2 2 2 2 J1 λL 8π kλL 0 (1 + q ) − 4Γ q λL unaffected by the value of the parity-breaking coupling. ρ −Φ Γ Z ∞ q(1 − q2)dq qρ j = 0 J . (18) z λ 8π2kλ3 (1 + q2)2 − 4Γ2q2 0 λ L L 0 L IV. VORTEX INTERACTIONS In the absence of parity breaking, Γ = 0, these integrals can be solved analytically, in terms of the modified Bessel The possibility of having an inversion of the magnetic function of the second kind Km. This expectedly gives field suggests that the interaction between two vortices the textbook expressions for the nonvanishing compo- might be much more involved than the pure repulsion Φ0 nents of the magnetic field, Bz(ρ/λL) = 2πλ2 K0(ρ/λL), that occurs in conventional type-2 superconductors. In- L deed, since the conventional long-range intervortex re- and of the superconducting current, 4πkjθ(ρ/λL) = Φ0 pulsion is due to the magnetic field, it is quite likely that 2πλ3 K1(ρ/λL). The general case with a nonzero parity- L the interaction here might be not only quantitatively, but breaking coupling, Γ 6= 0, requires a numerical evaluation also qualitatively altered. To investigate these, we con- of the integrals (17) and (18). sider the London limit free energy F written in the previ- Figure2 shows the magnetic field of a single vortex ously used dimensionless coordinates. Using Eq. (12) to both in the London limit and for the full Ginzburg- express the quantities j and B in terms of their Fourier Landau problem. First, although the solutions are ex- components, yields the expression of the free energy in pected to differ at the vortex core, the overall behaviour the momentum space: remains qualitatively similar in both cases. Indeed, the London solutions are divergent at the vortex core, and 3 Z 3 λL d p n thus they require a sharp cut-off at the coherence length F = Bp · B−p + (4πkλLj ) · (4πkλLj ) 8π (2π)3 p −p ξ. Solutions beyond the London limit, on the other hand, o are regular everywhere. The bottom row of Fig.2 shows + 2Γ(4πkλLjp) · B−p . (19) a typical vortex solution obtained numerically beyond the London limit. This is a close-up view of the vor- Replacing the Fourier components of the magnetic field tex core structure, while the actual numerical domain Bp and of the current jp with the corresponding expres- is much larger in order to prevent any finite size effect. sions in terms of the vortex field vp, Eqs. (13) and (14), While the density profile is similar to that of common respectively, yields the free energy: vortices, the magnetic field shows a pretty unusual pro- 3 Z 3 file featuring a slight inversion, away from the center. For λL d p m mn n F = 3 vp G v−p (20) the current parameter set where γ = 0.8γ?, the amplitude 8π (2π) of the reversed field compared to the maximal amplitude where Gmn = ΥlmΥln + ΛlmΛln + 2ΓΛlmΥln . is rather small. Yet, as illustrated on the top-right pan- p −p p −p p −p els of Fig.2, the amplitude of inversion of the magnetic The interaction matrix G has a rather involved structure. field, typically increase with the parity-breaking coupling Yet, given that only the axial Fourier components of the zz γ. Thus when γ is close to the critical coupling γ?, the vortex field (15) are nonzero, only the component G magnitude of the responses and field inversions become will contribute to the energy. Up to terms that are pro- rather important. portional to pz, and thus will be suppressed by the Dirac zz When the parity-breaking coupling γ is small com- delta δ(pz), G takes the simple form pared to the upper bound γ?, the longitudinal compo- (1 − Γ2)(1 + p2) nent Bz of the magnetic field is monotonic and exponen- Gzz = + terms ∝ p . (21) tially localized, as for conventional vortices. The vor- Σ z tex configurations start to deviate from the conventional Finally, using the vortex field ansatz (15), together with case when the parity breaking strengthens. When γ ap- the expressions (21) and (20) determines the free energy proaches the critical value γ , the magnetic field B do ? z associated with a set of translationally invariant vortices not vary monotonically any longer. It can be reversed, and even feature several local minima as can be seen in N Φ2(1−Γ2) X Z d2p 1 + p2 the top-right panel of Fig.2. Note that, the complicated 0 ip·(xã−x˜b) F = nanb 2 e . (22) 8πλL 2π Σ(p ) spatial structure and inversion of the magnetic field also a,b=1 6

1.5 25 γ/kλL = 0.00 0.20 1 0.45 0.70 0.85 20 0.90 0.95 0.97 Repuls.

) 0.5

L 15 L 0 d/λ

( 10 d/λ

U -0.5 Attract. 5 Repuls. -1 0 2 4 6 8 10 12 14 0 0.1 0.3 0.5 0.7 0.9 d/λL γ/kλL

0 0.2 0.6 1 0 0.5 1 1.5 2 2.5 0 0.2 0.6 1 -2 -1 0 1 0 0.2 0.6 1 6

8 . B B j j 4 |ψ|/ψ0 z ⊥ z ⊥ L 0 = 0 L

x/λ -4 −2 −1 -8 ×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 and repulsive regions depending on the parity breaking coupling γ and the intervortex distance d. The panels in the bottom row show various physical quantities in the transverse plane of a vortex bound state, for the parity-breaking coupling γ = 0.6γ?. 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 two dimensional integration in (22) can further be (Γ = 0), the integral in (23) can be calculated analyt- simplified and finally, the free energy reads as: ically, providing again the textbook expression, for the 2 interaction energy Vint(d/ΛL) = Φ0K0(d/ΛL)/4πλL. 2 2 N Φ0(1 − Γ ) X |xa − xb| Figure3 displays the function U(d/λL) which controls F = nanbU (23) the interacting potential between vortices, calculated in 8πλL λL a,b=1 the London limit. For vanishing γ, the interaction is Z q(1 + q2) purely repulsive, and it is altered by a nonzero coupling. where U(x) = J0(qx) . (1 + q2)2 − 4Γ2q2dq As shown on the left panel, when increasing γ/kλL the interaction can become non-monotonic with a minimum Hence the free energy of a set of vortices reads as follows: at a finite distance of about 4λL. Upon further increase of the coupling γ toward the critical coupling γ?, the inter- 2 2 N acting potential can even develop several local minima. Φ0(1 − Γ ) X |xa − xb| F = F0 + nanbU , (24) The phase diagram on the right panel of Fig.3 shows the 4πλL λL a,b>a different attractive and repulsive regions as functions of γ and of the vortex separation. 2 2 Φ0(1−Γ ) P 2 where the term F0 = n U(ξ) accounts for 8πλL a a The fact that the interaction energy features a mini- the self-energy of individual vortices. Since U(x) diverges mum at a finite distance implies that a pair of vortices at small separations x, the self energy has to be regular- tends to form a bound state. As can be seen in the bot- ized at the coherence length ξ λL which determines tom row of Fig.3, the tendency of vortices to form bound- the size of the vortex core. The interaction energy of the states persists beyond the London limit approximation. vortices separated by a distance d is thus determined by This configuration is obtained numerically by minimiz- the function U(d/λL). In the absence of parity-breaking ing the Ginzburg-Landau free energy (1). Notice that 7 these bottom panels show close-up view of the vortex all presently known non-centrosymmetric materials with pair, while the actual numerical domain is much larger O point group symmetry, weak coupling estimates sug- [24]. The fact that vortices can form a bound state can gest that the parity-breaking coupling is small γ γ?. heuristically be understood as a compromise between the However, there are a priori no known theoretical restric- axial magnetic repulsion of Bz which competes with in- tions on the existence of superconducting systems with a plane attraction mediated by B⊥. The bound state for- strong breaking of the inversion parity. If such materials mation can alternatively be understood to originate from exist, then according to our analysis they should exhibit the competition between the in-plane and axial contribu- the exotic vortex properties investigated here. tions of the currents. First of all, the in-plane screening currents mediate, as usual, repulsion between vortices. The interaction between axial component of the currents, V. CONCLUSIONS on the other hand, mediates an attraction, just like the force between parallel wires carrying co-directed currents. In this paper we have demonstrated that the vortices in The non-monotonic behavior of the magnetic field and non-centrosymmetric cubic superconductors feature un- currents thus leads to non-monotonic intervortex inter- usual properties induced by the possible reversal of the actions, and therefore allows for bound state of vortices magnetic field around them. Indeed, the longitudinal or cluster to form. Such a situation is known to exist (i.e., parallel to the vortex line) component of the mag- in multicomponent superconductors due to the competi- netic field changes sign at a certain distance away from tion between various length scales (see e.g. [25–29]). In the vortex core. Contrary to the vortices in a conven- an applied external field, the existence of non-monotonic tional superconductor, the magnetic-field reversal in the interactions allows for a macroscopic phase separation parity-broken superconductor leads to non-monotonic in- into domains of vortex clusters and vortex-less Meiss- tervortex forces which can act both attractively and re- ner domains. The situation here contrasts with the pulsively depending on the distance separating individual multicomponent case, as it occurs only due to the ex- vortices. istence of Lifshitz invariants. In two-dimensional sys- We have demonstrated these properties using mostly tems of interacting particles, multi-scale potentials and analytical calculations in the London limit. Full non-monotonic interactions are known to be responsible nonlinear numerical analysis within the Ginzburg- for the formation of rich hierarchical structures. These Landau description proves that these properties of non- structures include clusters of clusters, concentric rings, centrosymmetric superconductors survive beyond the clusters inside a ring, or stripes [30, 31]. It thus can London approximation. be expected that very rich structures would appear in Due to the nonmonotonic intervortex interactions, the non-centrosymmetric superconductors as well. However, vortices in the parity-breaking superconductors may form a verification of this conjecture is beyond the scope of the unusual states of vortex matter, such as bound states current work, as it deserves a separate detailed investi- and clusters of vortices. The structure of the interaction gation. potential strongly suggests that very rich vortex matter As shown in Fig.3, the interaction energy Vv/v(x) ∝ structures can emerge. For example, hierarchically struc- U(x) between two vortices with unit winding n1 = n2 = 1 tured quasi-regular vortex clusters, stripes and more, are can thus lead to the formation of a vortex bound state. typical features of the interacting multi-scale and non- A very interesting property is that it also opens the pos- monotonic interaction potentials [30, 31]. sibility of vortex/anti-vortex bound states. Indeed, ac- Moreover, given the possibility to form vortex/anti- cording to Eq. (24) the interaction of a vortex n1 = 1 vortex bound states, we can anticipate important con- and an anti-vortex n2 = −1 corresponds to a reversal sequences for the statistical properties and phase transi- of the interacting potential: Vv/av(x) ∝ −U(x). Thus tions in such models. from Fig.3 it is clear that if a vortex/anti-vortex pair is Note added: In the process of completion of this small enough, it will collapse to zero size and thus lead work, we were informed about an independent work to the vortex/anti-vortex annihilation. Now, if the size by Samoilenka and Babaev [32] showing similar results of the vortex/anti-vortex pair is larger than 4λL, there about vortices and their interactions. The submission of exists an energy barrier that prevents the pair from fur- this work was coordinated with that of [32]. ther collapse. Hence the vortex/anti-vortex pair should relax to a local minimum of the interaction energy. The resulting vortex/anti-vortex bound state has thus a size ACKNOWLEDGMENTS of approximately 7λL. We find that the most interesting physical effects ap- We acknowledge fruitful discussions with D. F. Agter- pear when the parity-breaking coupling γ of the Lif- berg, E. Babaev and F. N. Rybakov. The work of M.C. shitz invariant becomes non-negligible with respect to was partially supported by Grant No. 0657-2020-0015 the critical coupling γ?. The actual values of the coeffi- of the Ministry of Science and Higher Education of Rus- cients in front of the Lifschitz invariant in the Ginzburg- sia. The work of D.K. was supported by the U.S. De- Landau theory are hard to specify. Unfortunately, for partment of Energy, Office of Nuclear Physics, under 8 contracts DE-FG-88ER40388 and DE-AC02-98CH10886, resources provided by the Swedish National Infrastruc- and by the Office of Basic Energy Science under contract ture for Computing (SNIC) at National Supercomputer DE-SC-0017662. The computations were performed on Center at Linköping,Sweden.

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[28] E. Babaev, J. Carlström,M. Silaev, and J.M. Speight, where λL is the London penetration depth (4). “Type-1.5 superconductivity in multicomponent sys- In the London limit, the superconducting density |ψ|2 tems,” Physica C: Superconductivity and its Applica- is a fixed constant quantity regardless of the external tions 533, 20 – 35 (2017). conditions. Therefore the Ginzburg-Landau theory for [29] Mihail Silaev, Thomas Winyard, and Egor Babaev, the NCS superconductor in the London limit is always “Non-london electrodynamics in a multiband london bounded from below provided the Lifshitz–invariant cou- model: Anisotropy-induced nonlocalities and multiple magnetic field penetration lengths,” Phys. Rev. B 97, pling γ satisfies Eq. (A4). 174504 (2018). [30] C. J. Olson Reichhardt, C. Reichhardt, and A. R. Bishop, “Structural transitions, melting, and intermedi- Positive definiteness beyond London limit ate phases for stripe- and clump-forming systems,” Phys. Rev. E 82, 041502 (2010). [31] Christopher N Varney, Karl A H Sellin, Qing-Ze Wang, The issue of the positive definiteness is less obvious Hans Fangohr, and Egor Babaev, “Hierarchical struc- beyond the London limit. Indeed, let us first assume ture formation in layered superconducting systems with that the values of the parameters (e, k, γ) are chosen in multi-scale inter-vortex interactions,” Journal of Physics: such as way the formal criterium (A3) is satisfied. If we Condensed Matter 25, 415702 (2013). neglect the fluctuations of the condensate ψ (this require- [32] A. Samoilenka and E. Babaev, private communication – ment is always satisfied in the London regime) then we arXiv preprint. (2020). indeed find that the ground state resides in a locally stable regime so that all terms in the free energy (A2c) are positive defined. However, the density |ψ| is, in principle, Appendix A: Positive definiteness of the energy allowed to take any value and large enough fluctuations of |ψ| might trigger an instability. A possible signature The free energy (1) should be bounded from below in of the instability can indeed be spotted in the property order to be able to describe the ground state of the NCS that a variation of the absolute value of the condensate superconductor. To demonstrate the boundedness, we about the ground state, |ψ| = ψ0 + δ|ψ| gives a negative 2 3 use the relations contribution to the free energy, δF = −kδ|ψ|/(2e ψ0) in the linear order, provided all other parameters are fixed. j = 2e|ψ|2 (∇ϕ − eA) , (A1a) In order to illustrate a possible mechanism of the devel- |Dψ|2 = (∇|ψ|)2 + |ψ|2 (∇ϕ − eA)2 (A1b) opment of the instability inside the non-centrosymmetric 2 superconductor, let us consider a large enough local re- 2 |j| = (∇|ψ|) + , (A1c) gion characterized by a uniform, coordinate-independent 4e2|ψ|2 condensate ψ. For this configuration, the third (gradi- to rewrite the energy density in the following form: ent) term in the free energy density (A2c) is identically 2 2 zero. Gradually increasing the value of the condensate B 2 k|j| F = + k (∇|ψ|) + + γj · B + V [ψ] (A2a) beyond the ground state value ψ0, we increase the fourth 8π 4e2|ψ|2 (potential) term in Eq. (A2c) which make this change 1 k|j|2 energetically unfavorable. On the other hand, as the = B2 + 8πγj · B + 8π 4e2|ψ|2 condensate crosses the threshold of the applicability of β Eq. (A3), then the second term in the free energy (A2c) + k (∇|ψ|)2 + (|ψ|2 − ψ2)2 (A2b) becomes negatively defined, and the development of the 2 0 current j leads to the unbounded decrease this term. The 1 k 2 2 2 rise of in the current j will, in turn, affect the first (mag- = B + 4πγj + 2 2 − 2πγ |j| 8π 4e |ψ| netic) term, what may be compensated by a rearranging 2 β of the magnetic field B with the local environment in + k (∇|ψ|) + (|ψ|2 − ψ2)2 . (A2c) 2 0 such a way that the combination B + 4πγj keeps a small Leaving aside all terms with the perfect squares in value in the discussed region. Eq. (A2c), we find that the only criterion for the free Notice that the presence of isolated vortices make the energy to be bounded from below is to require the pref- system stable as in a vortex core the condensate vanishes, actor in front of the |j|2 term to be positive. We arrive ψ → 0, and the second, potentially-unbounded term in to the following condition of the stability of the system (A2c) becomes positively defined. In our numerical sim- (1): ulations we were also spotting certain unstable patterns especially in the regimes when the Lifshitz-invariant cou- k γ2 < . (A3) pling γ was chosen to close to the critical value γ? in 8πe2|ψ|2 the ground state (A4). For example, a system of ran- domly placed multiple elementary vortices relax their free In the ground state with |ψ| = ψ0, the stability condition (A3) reduces to the simple inequality: energy via mutual attraction and formation of a common bound state. Since the vortex bound state hosts a γ < γ? = kλL . (A4) stronger circular electric current, it becomes possible to 10

Figure 4. Helical structure of the magnetic field streamlines around vortices in non-centrosymmetric 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 magnetic field between both planes. The left panel shows the helical structure of the streamlines for 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. overcome the stability by ‘compressing’ the vortex clus- Appendix B: Vortex helicity ter, and then destabilizing the whole system. As emphasized in the main body of the paper, the magnetic field of vortex states in cubic non-centrosymmetric superconductors feature helicoidal structure around the core. This is illustrated in Fig.4, that displays typical magnetic field structure around vortex cores. Fig.4 shows two qualitatively different situation of moderate (left panel) and important (right panel) parity-breaking coupling γ. For moderate parity-breaking coupling, the We conclude that the processes that permit fluctua- magnetic field streamlines have helical structure with a tions of the condensate |ψ| towards the large values (as pitch that varies with the distance from the vortex core. compared to the ground state value ψ0) could activate Note that all streamlines have the same chirality, which the destabilization of the whole model. Theoretically, the is specified by the sign of the parity-breaking coupling unboundedness of the free energy from below may appear γ. On the other hand, as discussed in the main body, to be an unwanted feature of the model. However, one vortices features inversion of the magnetic field B for should always keep in mind that the Ginzburg-Landau important parity-breaking coupling γ. As a result, the functional is a leading part of the gradient expansion of chirality of the streamline depends on whether the longi- an effective model, and there always exist higher power tudinal component of the magnetic field is inverted. More gradients that will play a stabilizing role preventing the details about the helical structure of the magnetic field unboundedness to be actually realized in a physically rel- can be seen from animations in Supplemental material evant model. [19].