Topological anomalous skin effect in Weyl superconductors

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Topological anomalous skin effect in Weyl superconductors

Tsz Chun Wu, Hridis K. Pal, and Matthew S. Foster

Phys. Rev. B 103, 104517 – Published 30 March 2021

Abstract

We show that a Weyl superconductor can absorb light via a surface-to-bulk mechanism, which we dub the topological anomalous skin effect. This occurs even in the absence of disorder for a single-band superconductor, and is facilitated by the topological splitting of the Hilbert space into bulk and chiral surface Majorana states. In the clean limit, the effect manifests as a characteristic absorption peak due to surface-bulk transitions. We also consider the effects of bulk disorder, using the Keldysh response theory. For weak disorder, the bulk response is reminiscent of the Mattis-Bardeen result for $s$ -wave superconductors, with strongly suppressed spectral weight below twice the pairing energy, despite the presence of gapless Weyl points. For stronger disorder, the bulk response becomes more Drude-like and the $p$ -wave features disappear. We show that the surface-bulk signal survives when combined with the bulk in the presence of weak disorder. The topological anomalous skin effect can therefore serve as a fingerprint for Weyl superconductivity. We also compute the Meissner response in the slab geometry, incorporating the effect of the surface states.

Received 23 November 2020
Revised 9 March 2021
Accepted 9 March 2021

DOI:https://doi.org/10.1103/PhysRevB.103.104517

Physics Subject Headings (PhySH)

Impurities in superconductors Meissner effect Optical conductivity Skin effect Surface states

Topological superconductors

Classical electromagnetism Linear response theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Tsz Chun Wu ¹, Hridis K. Pal², and Matthew S. Foster^1,3

¹Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
²Department of Physics, IIT Bombay, Powai, Mumbai 400076, India
³Rice Center for Quantum Materials, Rice University, Houston, Texas 77005, USA

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Vol. 103, Iss. 10 — 1 March 2021

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Images

Figure 1
Geometry for the topological anomalous skin effect. (a) We consider a Weyl superconductor described by Eq. (3a). Due to bulk $k_{x} + i k_{y}$ pairing, a single pair of Weyl points arises at $k_{z} = \pm k_{F}$ . In the figure, the green plane depicts the dispersion of the chiral surface state $E_{k_{y}}^{s} = - Δ k_{y}$ , for a superconductor occupying the half space $x \geq 0$ . Since we consider the surface only at $x = 0$ , there is only one branch for the surface states. The topological anomalous skin effect arises via absorption due to optical transitions between surface and bulk states, as indicated by the red vertical arrow in the sketch. (b) We consider a plane electromagnetic wave impacting the Weyl superconductor occupying the $x \geq 0$ half space at normal incidence. The labels $I, R$ , and $T$ denote, respectively, the incident, reflected, and transmitted component of the radiation. $(E, B, S)$ stands for (E field, B field, and Poynting vector). A polarization along $\hat{y}$ (i.e., perpendicular to the plane of incidence) is assumed for the electric field $E$ in this sketch. The topological anomalous skin effect mainly takes place in the yellow region, extending up to the scale of $min (λ_{L}, l_{coh})$ . Here $λ_{L}$ denotes the London depth, and $l_{coh}$ is the coherence length (which is the minimal confinement depth for the chiral Majorana surface fluid).
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Figure 2
The topological anomalous skin effect in the clean limit. The panels in this figure plot the frequency ( $Ω$ ) dependence of the absorbance $A^{y y} (Ω)$ (blue) and $A^{z z} (Ω)$ (orange) due to the surface-bulk transition, Eq. (1.3). Panels show results at zero temperature for different combinations of the coherence length $l_{coh}$ and the diamagnetic (London) penetration depth $λ_{L}$ . The Weyl nodes lie along the $k_{z}$ axis in our model, so that $A^{y y} (Ω) [A^{z z} (Ω)]$ encodes absorption for electric-field polarization perpendicular (parallel) to the surface Majorana Fermi arc. Parameters used in panels (a)–(c) correspond to the type-I superconductor regime, while those used in panels (d)–(f) correspond to the type-II regime. Here, the frequency is normalized by the pairing gap energy $Δ_{0} = Δ k_{F}$ ; $Δ$ is the $p + i p$ -wave pairing amplitude in the model [Eq. (3a)]. We set $k_{F} = 1$ for all the plots.
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Figure 3
The bulk optical conductivity of the dirty Weyl superconductor. The panels in this figure plot the normalized optical conductivity $Re σ_{bb, R}^{μ ν} / Re σ_{Drude}^{μ ν}$ as a function of the reduced frequency $Ω / Δ_{0}$ at zero temperature, with disorder strength (a) $Υ_{el} = 0.01 Δ_{0}$ , (b) $Υ_{el} = 0.5 Δ_{0}$ , (c) $Υ_{el} = Δ_{0}$ , and (d) $Υ_{el} = 10 Δ_{0}$ . Here $Υ_{el} = 1 / (2 τ_{el})$ , where $τ_{el}$ is the elastic lifetime due to impurity scattering in the normal state, and $σ_{Drude}^{μ ν} = δ^{μ ν} σ_{dc} / [1 + {(ω τ_{el})}^{2}]$ . In the weak disorder regime, the frequency dependence of the optical conductivity is reminiscent of the Mattis-Bardeen result for a dirty $s$ -wave superconductor. The gapless excitations around the Weyl points cause a finite zero-temperature response for $Ω < 2 Δ_{0}$ . As the disorder strength increases, the optical conductivity gradually approaches the Drude result. This is consistent with the formation of a thermal metallic phase in the dirty limit [40, 48, 94].
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Figure 4
Topological anomalous skin effect in the presence of weak disorder. The panels in this figure plot the total absorbance ( $A$ ) as a function of the reduced frequency $\hat{Ω} = Ω / Δ_{0}$ , in the weak disorder limit with different coherence lengths $l_{coh}$ , London penetration depths $λ_{L}$ , and disorder strengths $Υ_{el}$ in the (a)–(c) type-I and (d)–(f) type-II superconductor regimes. In all the panels, orange and blue curves represent the $y y$ and $z z$ components of the absorbance, respectively (compare to Fig. 2). Long dashed lines correspond to the surface-bulk contribution, while short dashed lines correspond to the disordered bulk contribution. The total absorbance is depicted by the solid lines. We use $k_{F} = 1$ for all the plots. In the type-I case, the features from the surface-bulk absorption survive up to a relatively large amount of disorder. In the type-II case, those features are gradually suppressed by the bulk absorption as the disorder strength increases.
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Figure 5
Schematic illustration of the degeneracy in the bulk band, for bulk eigenenergies $E_{q, k}^{b}$ [Eq. (3.6)] plotted vs the standing-wave momentum $q \geq 0$ . Eigenstates are semi-infinite standing waves in the $x$ direction [due to the slab geometry, Fig. 1], plane waves transverse to this with momenta $k = (k_{y}, k_{z})$ . (a) For $k_{F}^{2} - k_{y}^{2} - k_{z}^{2} - 2 m^{2} Δ^{2} > 0$ , there exists a $q_{-} < q_{\min}$ such that $E_{q, k}^{b} = E_{q_{-}, k}^{b}$ for each $q \in (q_{\min}, q_{0}]$ . The minimum of $E_{q, k}^{b}$ is located at $q_{\min} = \sqrt{{k_{F}^{2} - k_{y}^{2} - k_{z}^{2} - 2 m^{2} Δ^{2}}^{}}$ . In this region, the two degenerate bulk states are $Ψ_{λ}^{b (1)} (q, k; x)$ and $Ψ_{λ}^{b (2)} (q, k; x)$ . Beyond $q_{0}$ , the bulk states are given by $ψ_{λ}^{b >} (q, k; x)$ . (b) For $k_{F}^{2} - k_{y}^{2} - k_{z}^{2} - 2 m^{2} Δ^{2} \leq 0, q_{\min} = 0$ and there is no degeneracy for all $q$ . The bulk states are just $ψ_{λ}^{b >} (q, k; x)$ .
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Figure 6
Plot of the frequency dependence of $Γ (ω)$ based on self-consistently solving Eq. (4.29) at different disorder strengths $Υ_{el} = 1 / 2 τ_{el}$ .
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