Universal chiral magnetic effect in the vortex lattice of a Weyl superconductor

It was shown recently that Weyl fermions in a superconducting vortex lattice can condense into Landau levels. Here we study the chiral magnetic effect in the lowest Landau level: The appearance of an equilibrium current $I$ along the lines of magnetic flux $\Phi$, due to an imbalance between Weyl fermions of opposite chirality. A universal contribution $dI/d\Phi=(e/h)^2\mu$ (at equilibrium chemical potential $\mu$ relative to the Weyl point) appears when quasiparticles of one of the two chiralities are confined in vortex cores. The confined states are charge-neutral Majorana fermions.

It was shown recently that Weyl fermions in a superconducting vortex lattice can condense into Landau levels. Here we study the chiral magnetic effect in the lowest Landau level: The appearance of an equilibrium current I along the lines of magnetic flux Φ, due to an imbalance between Weyl fermions of opposite chirality. A universal contribution dI/dΦ = (e/h) 2 µ (at equilibrium chemical potential µ relative to the Weyl point) appears when quasiparticles of one of the two chiralities are confined in vortex cores. The confined states are charge-neutral Majorana fermions.

I. INTRODUCTION
This paper combines two topics of recent research on Weyl fermions in condensed matter. The first topic is the search for the chiral magnetic effect in equilibrium [1][2][3][4][5][6][7][8][9]. The second topic is the search for Landau levels in a superconducting vortex lattice [10][11][12][13]. What we will show is that the lowest Landau level in the Abrikosov vortex lattice of a Weyl superconductor supports the equilibrium chiral magnetic effect at the universal limit of (e/h) 2 , unaffected by any renormalization of the quasiparticle charge by the superconducting order parameter. Let us introduce these two topics separately and show how they come together.
The first topic, the chiral magnetic effect (CME) in a Weyl semimetal, is the appearance of an electrical current I along lines of magnetic flux Φ, in response to a chemical potential difference µ + − µ − between Weyl fermions of opposite chirality. The universal value [14][15][16] dI dΦ = e 2 h 2 (µ + − µ − ) (1.1) follows directly from the product of the degeneracy (e/h)Φ of the lowest Landau level and the current per mode of (e/h)(µ + − µ − ). A Weyl semimetal in equilibrium must have µ + = µ − , hence a vanishing chiral magnetic effect -in accord with a classic result of Levitov, Nazarov, and Eliashberg [17,18] that the combination of Onsager symmetry and gauge invariance forbids a linear relation between electrical current and magnetic field in equilibrium.
Because superconductivity breaks gauge invariance, a Weyl superconductor is not so constrained: As demonstrated in Ref. 8, one of the two chiralities can be gapped out by the superconducting order parameter. When a magnetic flux Φ penetrates uniformly through a thin film (no vortices), an equilibrium current appears along the flux lines, of a magnitude set by the equilibrium chemical potential µ ± of the ungapped chirality. The renormalized charge e * < e determines the degeneracy (e * /h)Φ of the lowest Landau level in the superconducting thin film. The second topic, the search for Landau levels in an Abrikosov vortex lattice, goes back to the discovery of massless Dirac fermions in d -wave superconductors [19,20]. In that context scattering by the vortex lattice obscures the Landau level quantization [21][22][23], however, as discovered recently [13], the chirality of Weyl fermions protects the zeroth Landau level by means of a topological index theorem. The same index theorem enforces the (e/h)Φ degeneracy of the Landau level, even though the charge of the quasiparticles is renormalized to e * < e. Does this topological protection extend to the equilibrium chiral magnetic effect, so that we can realize Eq. (1.2) with e * replaced by e? That is the question we set out to answer in this work.
The outline of the paper is as follows. In the next section we formulate the problem of a Weyl superconductor in a vortex lattice. We then show in Sec. III that a flux bias of the superconductor can drive the quasiparticles into a topologically distinct phase where one chirality is exponentially confined to the vortex cores. The unconfined Landau bands contain electron-like or hole-like Weyl fermions, while the vortex-core bands are chargeneutral Majorana fermions. The consequences of this topological phase transition for the chiral magnetic effect are presented in Sec. IV. We support our analytical calculations with numerical simulations and conclude in Sec. V. Figure 1 shows the system we are considering, a Weyl superconductor in a magnetic field, in either a flux-biased or a current-biased circuit. For the Weyl superconductor we take the heterostructure configuration of Meng and Balents [24]: a stack in the z-direction of layers of Weyl semimetal alternating with an s-wave superconductor. A magnetization β perpendicular to the layers separates the Weyl cones along k z in opposite chiralities. Each Weyl cone is twofold degenerate in the electron-hole degree of freedom, mixed by the superconducting pair potential ∆ 0 . arXiv:1911.00312v1 [cond-mat.mes-hall] 1 Nov 2019 < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > The Bogoliubov-De Gennes Hamiltonian is [1,24,25]

II. FORMULATION OF THE PROBLEM
The Pauli matrices σ i and τ i (i = x, y, z, with i = 0 for the unit matrix) act on the spin and orbital degrees of freedom, respectively. The wave vector k = (k x , k y , k z ) is measured in units of the inverse of the lattice constant a 0 of a cubic atomic lattice. Energies are measured in units of the nearest-neighbor hopping energy t 0 (taken isotropic for simplicity). The chemical potential is µ, the vector potential is A, and the pair potential has amplitude ∆ 0 and phase φ. We set ≡ 1 and take the electron charge e > 0. For definiteness we also fix the sign β > 0. The Fermi velocity v F = a 0 t 0 / is unity for our chosen units. The superconductor has length L parallel to the applied magnetic field B = ∇ × A. The dimensions in the perpendicular direction are W × W , large compared to the London penetration length λ. This is the key difference with Ref. 8, where W < λ was assumed in order to prevent the formation of Abrikosov vortices. For W λ l m ξ 0 (with l m = /eB the magnetic length and ξ 0 = v F /∆ 0 the superconducting coherence length) we are in the vortex phase of a strong-type-II superconductor, where the magnetic field penetrates in the form of vortices of magnetic flux Φ 0 = h/2e. The vortex lattice has two vortices per unit cell, we take the square array (lattice constant d 0 ) indicated in Fig. 1.
In the gauge with ∇·A = 0 the superconducting phase is determined by The first equation specifies a 2π winding of the phase around each vortex core at R n , and the second equation ensures that the superconducting velocity has vanishing divergence. Since the vortex cores occupy only a small fraction (ξ 0 /l m ) 2 of the volume, we may take a uniform pair potential amplitude |∆| = ∆ 0 and a uniform magnetic field strength |B| = B 0 . The dominant effect of the vortex lattice is the purely quantum mechanical scattering of quasiparticles by the superconducting phase [22]. The vector potential contains a constant contribution A z = Λ/e in the z-direction controlled by either the flux bias or the current bias [26]: (2.4)

III. CHIRALITY CONFINEMENT IN A VORTEX LATTICE
In the absence of a vortex lattice, for W < λ, it was shown in Ref. 8 that a flux bias or current bias confines Weyl fermions of one definite chirality to the surfaces parallel to the magnetic field, gapping them out in the bulk. Here we consider the opposite regime W λ in which a vortex lattice forms in the Weyl superconductor. We will show that effect of the Λ bias is qualitatively different: both chiralities remain gapless in the bulk, but one of the two chiralities is confined to the vortex cores.
The analytics is greatly simplified if the magnetic field is along the same z-axis as the separation of the Weyl cones. The corresponding vector potential is where for definiteness we take Λ ≥ 0. This is the fluxbiased geometry of Fig. 1b. Numerical simulations indicate that the current-biased geometry of Fig. 1c, with B along the y-axis, is qualitatively similar -but we have not succeeded in obtaining a complete analytical treatment in that geometry.

A. Landau bands
We have calculated the eigenvalues and eigenfunctions of the tight-binding Hamiltonian (2.1) using the Kwant code [28] as described in Ref. 13. We take parameters β = t 0 , ∆ 0 = 0.5 t 0 , µ = 0. We arrange h/2e vortices on the square lattice shown in Fig. 1a. The lattice constant d 0 = N a 0 of the vortex lattice determines the magnetic field B 0 = (h/e)d −2 0 . In the numerics the full nonlinear k-dependence of H(k) is used, while for the analytical expressions we expand near k = 0.
The zero-field spectra in Figs. 2a and 2b reproduce the findings of Ref. 8: For small Λ and provided that ∆ 0 < β one sees two pairs of oppositely charged gapless Weyl cones, symmetrically arranged around k z = 0 at momenta K ± and −K ± given by The pair at |k z | = K − is displaced relative to the other pair at |k z | = K + by the flux bias Λ, becoming gapped when Λ is in the critical range Application of a magnetic field in Figs. 2c and 2d shows the formation of chiral zeroth-order Landau bands: a pair of electron-like Landau levels of opposite chirality and a similar pair of hole-like Landau levels. The Landau bands have a linear dispersion in the z-direction, along the magnetic field, while they are dispersionless flat bands in the x-y plane.
For k z near K ± the electron-like and hole-like dispersions are given by [13] and similarly near −K ± the dispersions are The k z -dependent factor cos θ renormalizes the charge and velocity of the quasiparticles, according to [8,27] cos θ(k) = |k z | The degeneracy of a Landau band is not affected by charge renormalization [13], each electron-like or holelike Landau band contains chiral modes, determined by the ratio of the enclosed flux Φ = B 0 W 2 and the bare single-electron flux quantum h/e. While the dispersion of a Landau band in the Brillouin zone changes only quantitatively with the flux bias, it does have a pronounced qualitative effect on the spatial extension in the x-y plane. As shown in Fig. 3, the intensity profile |ψ ± (x, y)| 2 of a zeroth-order Landau level at |k z | = K ± peaks when r = (x, y) approaches a vortex core at R n . The dependence on the separation δr = |r − R n | is a power law [13], (3.7) When Λ enters the critical range (3.3) this power law decay applies only to one of the two chiralities: the two Landau bands at k z = K + and k z = −K + with dE/dk z < 0 still have the power law decay (3.7), but the other two bands with dE/dk z > 0 merge at k z = 0 and become exponentially confined to a vortex core. As we shall derive in the next subsection,

B. Vortex core bands
To demonstrate the exponential confinement in a vortex core of the τ z = +1 chirality we expand the Hamiltonian (2.1) to first order in k x , k y at k z = 0, µ = 0, The applied magnetic field does not contribute on length scales below l m , so we only need to include the constant eA z = Λ term in the vector potential. The winding of the superconducting phase is accounted for by the factor e iϕ , in polar coordinates (x, y, z) = (r cos ϕ, r sin ϕ, z) centered on the vortex core.

A. Charge renormalization
We summarize the formulas from Ref. 8 that show how charge renormalization by the superconductor affects the CME.
The equilibrium expectation value I z of the electrical current in the z-direction is given by The sum over n is over transverse modes with energy In the insets in panel c the same data is presented using a loglog scale (for the zeroth Landau level) and log-linear scale (for the vortex-core band). The Landau band is spread over the magnetic unit cell, with an algebraic divergence at the vortex cores, whereas the vortex-core band is exponentially localized at the vortices. The profiles were calculated for the same set of parameters as the spectra in Fig. 2, with the Landau band corresponding to the state marked with a square, and the vortex-core band corresponding to the state marked with a circle. To improve the spatial resolution, we used a larger ratio d0/a0 = 102.
expectation values that we need are those of the velocity operator v z = ∂H/∂k z and the charge operator Q = −e∂H/∂µ, given by Following Ref. 8 we also define the "vector charge" which may be different from the average (scalar) charge Q 0 ≡ Q E because the average of the current as the product of charge and velocity may differ from the product of the averages.
The CME is a contribution to I z that is linear in the equilibrium chemical potential µ, measured relative to the Weyl points. We extract this contribution by taking the derivative ∂ µ I z in the limit µ → 0. Two terms appear, an on-shell term from the Fermi level and an off-shell term from energies below the Fermi level, At low temperatures, when −f (E) → δ(E) becomes a delta function, the on-shell contribution J on-shell involves only Fermi surface properties. It is helpful to rewrite it as a sum over modes at E = 0. For that purpose we replace the integration over k z by an energy integration weighted with the density of states: (4.6) In the T → 0 limit a sum over modes remains, where we have restored the units of = h/2π.

B. On-shell contributions
We apply Eq. (4.7) to the vortex lattice of the fluxbiased Weyl superconductor. Derivatives with respect to A z are then derivatives with respect to the flux bias Λ. According to the dispersion relation (3.4a), the electronlike Landau band near K + has renormalized charges in the limit k z → K + , µ → 0. The charge renormalization factors cancel, so this Landau band with sign v z < 0 contributes to J on-shell an amount − 1 2 e/h times the degeneracy N 0 = (e/h)Φ, totalling − 1 2 (e/h) 2 Φ. Similarly, for the hole-like Landau band near −K − Eq. (3.4a) gives for the same contribution of − 1 2 (e/h) 2 Φ. The total onshell contribution for this chirality is (4.10) We can repeat the calculation for the electron-like band near K − and the hole-like band near −K − , the only change is the sign v z > 0, resulting in J on-shell (|k z | = K − ) = (e/h) 2 Φ. (4.11) We conclude that the Dirac fermions in the Landau bands of opposite chirality give identical opposite on-shell contributions ±(e/h) 2 Φ to ∂ µ I z . The net result vanishes when Λ is outside of the critical region (Λ c1 , Λ c2 ). When Λ c1 < Λ < Λ c2 one of the two chiralities is transformed into unpaired Majorana fermions confined to the vortex cores. The vortex-core bands have Q 0 = 0 at E = 0, so they have no on-shell contribution, resulting in (4.12) The coefficient (e/h) 2 contains the bare charge, unaffected by the charge renormalization.

C. Off-shell contributions
Turning now to the off-shell contributions (4.5c), we note that the Landau bands do not contribute in view of Eq. (3.4): For the vortex-core bands, off-shell contributions cancel because of particle-hole symmetry. This does not exclude off-shell contributions from states far below the Fermi level, where our entire lowenergy analysis no longer applies. In fact, as we show in Figs. 4 and 5, we do find a substantial off-shell contribution to ∂ µ I z in our numerical calculations (see App. A for details). Unlike the on-shell contribution (4.12), which has a discontinuity at Λ = Λ c1 , Λ c2 , the off-shell contribution depends smoothly on the flux bias and can therefore be extracted from the data.

V. CONCLUSION
In summary, we have demonstrated that a flux bias in a Weyl superconductor drives a confinement/deconfinement transition in the vortex phase: For weak flux bias the subgap excitations are all delocalized in the plane perpendicular to the vortices. With increasing flux bias a transition occurs at which half of the states become exponentially localized inside the vortex cores. The localized states have a definite chirality, meaning that they all propagate in the same direction along the vortices. (The sign of the velocity is set by the sign of the external magnetic field B 0 .) As a physical consequence of this topological phase transition we have studied the chiral magnetic effect. The states confined to the vortex cores are charge-neutral Majorana fermions, so they carry no electrical current. The states of opposite chirality, which remain delocalized, are charged, and because they all move in the same direction they can carry a nonzero current density j parallel to the vortices. This is an equilibrium supercurrent, proportional to the magnetic field B 0 and to the chemical potential µ (measured relative to the Weyl point).
We have calculated that the supercurrent along the vortices jumps at the topological phase transition by an amount which for a large system size tends to the universal limit Remarkably enough, the proportionality constant contains the bare electron charge e, even though the quasiparticles have a renormalized charge e * < e.  Fig. 4, but now for a fixed flux bias eAz = 1.05/a0 in the two-cone regime, showing the contributions to ∂µIz from different momenta kz along the magnetic field. We distinguish between the total current and the off-shell contribution. The difference between the two is the on-shell contribution, which peaks at the momenta where the Fermi level crosses the chiral Landau bands. The vortexcore bands at kz = 0 have vanishing on-shell contribution.
The chiral fermions confined in the vortex cores are a superconducting realization of the "topological coaxial cable" of Schuster et al. [29], where the fermions are confined to vortex lines in a Higgs field. There is one difference: the chiral fermions in the Higgs field are charge-e Dirac fermions, while in our case they are charge-neutral Majorana fermions. The difference manifests itself in the physical observable that serves as a signature of the confinement: for Schuster et al. this is a quantized current dI/dV = e 2 /h per vortex out of equilibrium, in our case it is a quantized current dI/dµ = 1 2 e/h per vortex in equilibrium. Top: low-energy dispersion relation for the corresponding system. The on-shell contribution to the current response, which is the difference between the total and off-shell contributions, only appears at momenta for which a band crosses the Fermi energy. In the four-cone regime four peaks are present, the contributions of which cancel out. In the two-cone regime the vortex-core band at kz = 0 has a vanishing on-shell contribution, whereas the contribution of the other two Landau levels remains unchanged. The plots were obtained for a system size N = 18. and (A3), in the two-cone regime at eAz = 1.05/a0 for a finite chemical potential µ. The colored data points give the total response, as well as the off-shell and on-shell contributions. The dotted line µe 2 Φ/h 2 is the theoretical prediction (4.12) for the on-shell contribution to first order in µ, which is a good approximation to the numerical result for small µ. The plots were obtained for a system size N = 18.