Chiral Meissner effect in time-reversal invariant Weyl superconductors

Weyl semimetals have nodes in their electronic structure at which electrons attain a definite chirality. Due to the chiral anomaly, the non-conservation of charges with given chirality, the axion term appears in their effective electromagnetic action. We determine how this affects the properties of time-reversal invariant Weyl {\it superconductors} (SCs) in the London regime. For type II SCs the axion coupling generates magnetic $B$-fields transverse to vortices, which become unstable at a critical coupling so that a transition into type I SC ensues. In this regime an applied $B$-field not only decays inside the SC within the London penetration depth, but the axion coupling generates an additional perpendicular field. Consequently, when penetrating into the bulk the $B$-field starts to steadily rotate away from the applied field. At a critical coupling the screening of the magnetic field breaks down. The novel chiral superconducting state that emerges has a periodically divergent susceptibility that separates onsets of chiral Meissner regimes. The chiral anomaly thus leaves very crisp experimental signatures in structurally chiral Weyl SCs with an axion response.

Since the presence of Weyl nodes modifies electromagnetic properties of a superconducting system, we will explore here the Meissner and magnetic vortex states of Weyl superconductors (SCs) in the London limit.The low-energy effective theory of Weyl semimetals is governed by the axion action [9][10][11][12][13][14][15], S a = α 4π 2 dt d 3 rϑ(t, r)E • B, where α is the fine-structure constant and the axion field is assumed to have the explicit form ϑ(t, r) = b • r − b 0 t.Here b and b 0 represent the separation between Weyl nodes in momentum and energy, respectively [14,15].Specifically, we will be interested in the case where time-reversal invariance holds, which leads to a net b = 0 due to the presence of time reversed Weyl node pairs.In the normal state at equilibrium Weyl nodes of opposite chirality cancel out the axion action term proportional to b 0 [19][20][21].This is not necessarily so in Weyl SCs, in particular when Weyl cones of one particular chirality are gapped out due to the superconducting pairing [22,23], as in this case the remaining gapless Weyl nodes of opposite chirality induce axion electrodynamics.This scenario is relevant in cases where the normal groundstate Weyl nodes of opposite chirality have different energies.In a TRI crystal this requires the absence of any mirror symmetry and, as inversion must be broken to produce Weyl nodes in the first place, thus a chiral crystal structure [24,25].Superconductivity in such chiral crystals permits gapping out of Weyl nodes of a single chirality in equilibrium so that the axion action term proportional to b 0 appears in the effective electromagnetic response.
In such cases the London electrodynamics of TRI Weyl SCs is captured by the following Lagrangian, where ρ s is the superconducting stiffness, q = 2e the charge, θ the phase of the order parameter, ϕ and A are electric and vector potentials, respectively.We use units such that ℏ = c = 1.As we are interested in the static regime, we obtain the following generalized London equation from the Lagrangian above, where from Eq. ( 1) we identify the axion coupling constant as given by a = q 2 b 0 /(2π).
Within the London theory the superconducting current is given by j SC = qρ s (∇θ − qA), while a typical physical consequence of the axion response is the chiral magnetic effect (CME) [26][27][28], which implies that the current density contains a contribution j CME = −aB/(4π).Therefore, in a Weyl superconductor with time-reversal invariance the total current density is given by j = j SC +j CME , which is reflected in Eq. ( 2).Here we uncover a number of novel electrodynamic features that follow from the interplay between the axion CME and superconductivity in chiral Weyl semimetallic materials.
As will be shown, while the magnetic field expulsion from a superconductor is ensured by its current having a term proportional to A, the CME contribution, which is linear in B, leads to a rotation of the screened magnetic field.The latter behavior may be understood by first considering the non-superconducting phase.In this case ∇ × B = −aB and we see that ∇ 2 B + a 2 B = 0, which yields spatially rotating magnetic field profiles.When the system becomes superconducting, the screening of the magnetic field twists as a response to the rotation induced by the CME, and the chiral Meissner state ensues.
We find that the electromagnetic response of Weyl SCs differs drastically depending on whether the axion coupling a is above or bellow a critical value a c , which is related to the superconducting stiffness.Investigating the vortex properties of type II Weyl SCs, one observes that for a < a c a component of the magnetic induction transverse to the vortex line is generated in addition to the one directed along the vortex line.The rotation of the magnetic induction around the vortex results in a screening current directed not only around but also along the vortex line.However, due to the competition between the axion term and superconductivity with the increase of the ratio a/a c , a transition occurs at a c from the chiral vortex state to a Meissner state without vortices.
For type I Weyl SCs we show that the chiral Meissner state present for a < a c breaks at a = a c and the system enters a new state with no magnetic field screening.Instead of decaying, the magnetic field rotates inside the sample for a ≥ a c .Nevertheless, in this regime the magnetic susceptibility χ = −1/(4π) at quantized values of a implying diamagnetic behavior.

II. VORTEX IN TRI WEYL SC
To understand the electromagnetic response of a Weyl SC, we first focus on the fate of a single magnetic vortex.The analysis here differs significantly from previous discussions of axion electrodynamics based on the Witten effect [29], where the field of the vortex induces a fractional charge at the interface between a SC and a topological insulator [30], as well as a fractional angular momentum [31,32].The vortex axion physics discussed below does not involve the electric field and is intrinsic to TRI Weyl SCs, so proximity to a topological material needs not be assumed.Taking the curl of Eq. ( 2) we obtain, where M 2 = 4πq 2 ρ s represents the inverse square of the London penetration depth λ, Φ 0 = 2π/q is the elementary flux quantum, and Ω = ∇ × ∇θ is the vorticity (recall that the curl of a gradient vanishes everywhere, except where topological defects like vortices exist [33]).
We consider an infinite system with a single vortex line directed along the z-axis and with the origin located at r = 0.This vortex configuration leads to the vorticity taking a simple form, Ω = 2πδ 2 (r) ẑ.The exact solution to Eq. ( 3) is obtained by performing a Fourier transform, which leads to and yields in real space B(r) = B φ (r) φ + B z (r) ẑ, with where K α (x) are modified Bessel functions of the second kind, and where a c = 2M .Note that despite the complex values of M ± , the components of the magnetic induction in Eqs. ( 5) and ( 6) are real.These solutions are only valid for a < a c .When a ≥ a c , M ± become purely imaginary resulting in poles in Eq. ( 4) and the integrals leading to B(r) are no longer defined [34].Consequently, there is no vortex solution for a ≥ a c and one has to consider Eq.
(3) with a trivial vorticity (Ω = 0).Therefore, the only solution for the infinite system in this case is B = 0, a perfect diamagnetic response.Interestingly, expressions similar to Eqs. (3)(4)(5)(6) have arisen in the context of noncentrosymmetric superconductors [35][36][37].Due to the axion, a φ-component of the magnetic field is generated and, as a consequence, a component of the current parallel to the vortex is induced.The total current screening the vortex is thus encircling it in a helical manner, with a handedness/chirality determined by the sign of a. Figure 1 displays the magnetic induction components corresponding to the vortex solution of Eqs. ( 5) and ( 6) for different values of a.Note that the fields start to develop more spatial structure with increasing a, so the Meissner screening is spatially oscillating.For a closer to a c the oscillations are progressively enhanced, see Fig. 1(c-d).
Since at a ≥ a c the vortex solution breaks down, the system transitions into a superconducting phase without vortices.As at the SC phase transition the penetration depth λ diverges and subsequently M tends to zero, for any finite (and possibly small) intrinsic a, the regime a ≥ a c is always realized close to the SC phase transition.Moreover, if the case a ≥ a c is reached at any temperature, the Weyl SC only features a regime where no vortices are present -a behavior characteristic of a type I SC.Such a perfect diamagnetic character is also observed for a vortex solution in a finite slab of thickness L, with the vortex line perpendicular to the surface, see Appendix A. There too the vortex solution does not exist in the case of a ≥ a c leading to a type I London response.

III. MEISSNER STATE
Having discussed vortex solutions, we consider now a superconducting state without vortices.Here we find a crucial difference relative to the usual London electrodynamics: the Meissner screening works differently, since application of an external magnetic field generates an additional component of the magnetic induction due to the axion term.As a first example, let us consider a semiinfinite superconductor located in the region x > 0 in the presence of an applied magnetic field B ap = B ap ŷ parallel to the surface.For this simple geometry one obtains from Eq. (3), with the boundary conditions, tor, û(x) = cos (ax/2) ŷ + sin (ax/2) ẑ, (10) and one observes that the field inside the SC rotates with respect to the applied one; a chiral oscillatory feature we found in the vortex as well, for instance, illustrated in Fig. 1.Thus, applying a magnetic field in the y-direction does not only lead to a Meissner state with an exponentially decaying y-component of the field, but also generates a similarly decaying field along the z-direction as a consequence of the axion coupling.The corresponding field profile is depicted in panel (a) of Fig. 2 for an exemplary value of a below a c .Note that the axion leads to a larger penetration depth, λ = 1/ M 2 − a 2 /4, compared to the one in the non-Weyl SC, λ = 1/M .For a ≥ a c there is no solution that fulfills the boundary conditions B y (x → ∞) = 0 and B z (x → ∞) = 0. Hence, instead of demanding the magnetic field to vanish inside the sample, we only enforce the boundary conditions at the surface B y (x = 0) = B ap , B z (x = 0) = 0, and require the solutions to be real, which yields Remarkably, the magnetic field inside of the Weyl SC exhibits a purely oscillatory behavior illustrated in Fig. 2-(b) for a = 2.5a c .There is no magnetic field screening inside the bulk of the SC for a ≥ a c .Instead, it rotates around the surface normal while penetrating all the way into the bulk.While the solution above corresponds to the instance where the external magnetic field is applied parallel to the surface, the Weyl SC response to the field perpendicular to the surface is rather simple.For B ap = B ap x continuity of the normal component implies that the CME current vanishes, in which case the usual London equation follows.Therefore, the magnetic induction inside the semi-infinite slab has the conventional London form, B = B ap e −M x x.This angle-dependence of the chiral magnetic field screening is quantifiable through the magnetic helicity, H = dxA • B, which is a measure for the twistedness of the B field.For an arbitrary directed external field, B ap = B ap cos θê ⊥ + sin θê ∥ (where ê⊥ and ê∥ are unit vectors perpendicular and parallel to the surface, respectively) the helicity per area has a sinusoidal dependence on θ, where θ is the angle between the applied field and the surface normal of a Weyl SC.Hence, the conventional exponential decay for B ap perpendicular to the surface corresponds to vanishing helicity, while for the semi-infinite slab in a parallel field and a < a c the helicity per unit area is finite, , and diverges at a c .This brings to the fore once more the important role of the critical axion coupling a c in modifying the nature of the Meissner state.
To explore further regimes bellow and above a c , we consider a finite slab geometry, where the TRI Weyl SC is between two surfaces such that |x| ≤ L/2 = L.This geometry additionally allows to compute the magnetic susceptibility, as this requires us to determine the average magnetic induction over the system, which cannot be done easily in a convergent manner in a semi-infinite system.Having applied the boundary conditions that require B(x = ± L) = B ap , we obtain, The expression above holds for any value of a. Figure 2-(c-f) shows the magnetic induction profiles corresponding to Eq. for increasing values of a up to slightly above a c .Similarly to the findings above, the slab solution features two distinct regimes for a < a c and a ≥ a c .The magnetic induction inside the SC rotates for any a, but for a < a c it is screened, while for a ≥ a c the field penetrates into the whole slab.Remarkably, in the latter regime the oscillation amplitude can get larger than the applied field.This exotic behavior in a system with a nonzero superconducting order parameter may raise a question whether the energy of such a state is bounded below.To check that the solutions for a ≥ a c are physically valid, one must go beyond the linearized London theory and consider the Ginzburg-Landau energy in the London approximation (see Appendix B).We determine the diamagnetic susceptibility χ via the spatial average of the magnetic induction.It is clear that axion-induced field component B z averages to zero for any a.Hence, the susceptibility is determined by the field component parallel to the applied field.Considering the case of a large slab thickness compared to the London penetration depth, M L ≫ 1, one finds that χ = −1/(4π) for all a ≤ a c , as can be seen from the form of χ = 2 M 2 L 2 [1 − cos(M L)] − 1/(4π) for a → a c specifically.This implies a perfect diamagnetic response of the bulk superconductor.
For a > a c , on the other hand, the situation is more delicate and interesting.From the numerator of the first term in Eq. ( 13), one observes that χ = −1/(4π) is reached for values of a given by Interestingly, for any a min (n) the magnetic induction inside the slab satisfies a force-free equation characteristic of a TRI Weyl semimetal (see Appendix C).Thus, in this case the Weyl superconductor with M ̸ = 0 has exactly the same magnetic induction as a normal M = 0 Weyl semimetal.On the other hand, the susceptibility diverges for Consequently, at quantized values of the axion coupling the system becomes unstable.Exactly at this progression of critical couplings, the magnetic helicity of the slab diverges as well; see Appendix D. The two sets of relations characterize two distinct behaviors of the electromagnetic response to the external magnetic field.When a takes the values a min (n), the susceptibility χ → −1/(4π) irrespective of the slab thickness L in contrast to the conventional Meissner effect, where χ → −1/(4π) only for sufficiently thick samples.The values a max (m) separate transitions between these different onsets of chiral Meissner regimes, and occur due to poles present in the Green function of the operator (−∇ 2 + M 2 + a∇×) in the slab geometry.Hence the regime for a > a c exhibits a helimagnetic behavior, since the Green function in this case features poles at nonzero momenta.

IV. CONCLUSIONS AND OUTLOOK
As the axion term affects the properties of timereversal invariant Weyl superconductors in a quite nontrivial manner, several distinct experimentally testable predictions follow from our results.In the vortex state the currents parallel to magnetic vortices are induced by the axion coupling.In future work it will be interesting to establish how this affects the vortex lattice and its stability.Additionally, the rotating B-field causes vortices close to the surface to cant with respect to the applied field, which results in magnetic stray fields outside the Weyl SC [38].This is a rather intricate consequence of the emergent field components transverse to the flux line, as simple addition of a mirror vortex cannot fulfill the boundary conditions on the surface.
We find that the the vortex becomes unstable at the critical axion coupling implying a transition from a type II to a type I superconducting state.Such a transition between type I and II SC in the same material is known as type 1.5 superconductivity for multiband systems [39,40].As close to the SC phase transition the London penetration depth diverges, the critical axion coupling vanishes there.Thus for any given axion coupling intrinsic to the TRI Weyl SC material, close enough to the SC transition the system automatically enters the strong coupling regime where the vortex state becomes unstable and a type 1.5 regime may ensue.Interestingly, studies of chiral effects in an astrophysical setting have established that for vanishing critical axion coupling a chiral plasma instability (CPI) emerges, which is strongly affected by field fluctuations [41].For Weyl SCs close to T c the CPI must then be considered on the same footing as superconducting order parameter fluctuations.The fate of this fluctuating chiral SC is an open question for future work.
In type I SCs the axion induced magnetic field component perpendicular to the applied field and parallel to the surface may be explored by surface sensitive probes, e.g. the magneto-optic Kerr effect.For strong coupling the axion renormalization of the London penetration depth may be probed experimentally.Also the periodically divergent susceptibility associated with the chiral Meissner state is a marked experimental signature.
Note added.Recently, we became aware of the work of Stålhammar et al. [42], who report semi-infinite slab and cylinder solutions for a < a c similar to our results.(D2) From the expressions for magnetic field components and the vector potential we can calculate total helicity per area by integrating over the sample's thickness, Similarly to magnetic susceptibility mentioned in the main text, the helicity is divergent for values of a that satisfy the relation in Eq. ( 14) of the main text.

2 FIG. 1 .
FIG. 1. Magnetic induction profiles for different values of the axion coupling a [panels (a) and (b) for the field components Bφ and Bz, respectively].Fields are plotted in units of M 2 Φ0/(2π) against the radial coordinate r in units of M .Panels (c) and (d) correspond to a situation where a is very close to the critical value ac = 2M for which the vortex solution ceases to exist.We note that as a approaches ac from bellow the field profiles start to become more oscillatory.The onset of these spatial oscillations is illustrated by the threedimensional plots for Bz in panels (e) and (f).

2 cFIG. 2 .
FIG.2.Panels (a) and (b): Magnetic field components of a semi-infinite superconductor located at x > 0 and in the presence of an applied magnetic field, Bap = Bap ŷ.The chiral magnetic screening occurs for a < ac (a) and is absent for a ≥ ac (b).Panels (c)-(f): Magnetic field profiles in a finite TRI Weyl SC.At panels (c-e) we have a < ac, corresponding to a = 0.4ac, a = 0.75ac, and a = 0.95ac, respectively.We can see once more the onset of spatial oscillations in the magnetic induction as a increases.In panel (f) a = 1.1ac leading to the case where the axion coupling completely dominates over Meissner screening.

2 c − a 2 sinh 2 L a 2 c − a 2 × 2 c − a 2 +
(D3) × −2a L a 2 c − a 2 cos a L cosh L a 2 c − a 2 + 2a L a 2 c − a 2 − a sinh 2 L a 2a cos a L + 2 a 2 − a 2 c L sin a L sinh L a 2 c − a 2 .