Nontopological nature of the edge current in a chiral $p$-wave superconductor

Nontopological nature of the edge current in a chiral $p$ -wave superconductor

Wen Huang, Samuel Lederer, Edward Taylor, and Catherine Kallin

Phys. Rev. B 91, 094507 – Published 12 March 2015

Abstract

The edges of time-reversal symmetry breaking topological superconductors support chiral Majorana bound states as well as spontaneous charge currents. The Majorana modes are a robust, topological property, but the charge currents are nontopological—and therefore sensitive to microscopic details—even if we neglect Meissner screening. We give insight into the nontopological nature of edge currents in chiral $p$ -wave superconductors using a variety of theoretical techniques, including lattice Bogoliubov–de Gennes equations, the quasiclassical approximation, and the gradient expansion, and we describe those special cases in which edge currents do have a topological character. While edge currents are not quantized, they are generically large, but they can be substantially reduced for a sufficiently anisotropic gap function, a scenario of possible relevance for the putative chiral $p$ -wave superconductor ${Sr}_{2} {RuO}_{4}$ .

Received 29 December 2014

DOI:https://doi.org/10.1103/PhysRevB.91.094507

Authors & Affiliations

Wen Huang¹, Samuel Lederer², Edward Taylor¹, and Catherine Kallin^1,3

¹Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1
²Department of Physics, Stanford University, Stanford, California 94305, USA
³Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8

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Vol. 91, Iss. 9 — 1 March 2015

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Images

Figure 1
The integrated edge current calculated from $T = 0$ BdG (solid curves) as a function of the chemical potential over the entire bandwidth for (a) $t^{'} = (3 / 8) t$ (inset, $t^{'} = 0$ ) and (b) $t^{'} = t$ . $Δ_{0} = 0.2 t$ for all plots; this requires varying the interaction $g$ as the chemical potential is varied. In the inset of (a), for instance, $g / t$ is varied from 11.2 at $μ = - 4 t$ to 3.25 at $μ = 0$ . Calculations are carried out for $N_{x} = N_{y} = 300$ lattice sites. The “topological current” obtained from (1), with details given in Appendix pp1, is also shown (dashed lines) and coincides with the BdG result for $t^{'} = 0$ . Regions of $μ$ with different Chern numbers are separated by a dotted vertical line.
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Figure 2
The effect of order parameter anisotropy on the edge current. The integrated edge current for two different order parameters is shown for $t^{'} = 3 t / 8$ : $Δ_{0} (k) = Δ_{0} (sin k_{x} + i sin k_{y})$ [solid curve; same as in Fig. 1] and $Δ_{0} (k) = Δ_{0} (sin k_{x} cos k_{y} + i sin k_{y} cos k_{x})$ (red dot-dashed curve). The Chern number in the latter case is equal to 1 for $- 5.5 t \leq μ < 0, - 3$ for $0 < μ < 1.5 t$ , and $- 1$ for $1.5 t < μ < 2.5 t$ . The topological current value is shown by the blue dashed curves and coincides for the two order parameters.
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Figure 3
Plots of the integrated edge current from BdG for $t^{'} = 3 t / 8$ [see also Fig. 1] with an edge at $x = 0$ and an edge potential $A_{0} (x) = (μ + 5.5 t) [1 - tanh (x / λ)]$ for $μ < 1.5 t$ and $A_{0} (x) = (μ - 2.5 t) [1 - tanh (x / λ)]$ for $μ > 1.5 t$ . As the edge becomes progressively softer ( $λ / ξ_{0}$ increasing), the BdG results approach the topological value (A3) obtained from (1). All results should coincide near the bottom ( $μ = - 5.5 t$ ) and top ( $μ = 2.5 t$ ) of the band. The van Hove singularity at $μ = 1.5 t$ pushes the region of agreement near the top of the band to values of $μ$ very close to $2.5 t$ . For $λ = 6.5 ξ_{0}$ , the current does not vanish at the top of the band since we had to use a large value of the order parameter, $Δ_{0} = 0.4 t$ , to keep the coherence length small.
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Figure 4
Spectral flow plots showing the evolution of BdG eigenvalues for $E_{k_{y} = - 0.29 π}^{(j)}, Δ (k) = Δ_{0} (sin k_{x} + i sin k_{y})$ (left) and $E_{k_{y} = - 0.26 π}^{(j)}, Δ (k) = Δ_{0} (sin k_{x} cos k_{y} + i sin k_{y} cos k_{x})$ (right) for NN-only hopping and $μ = - t$ as the edge width $λ$ is evolved.
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Figure 5
Dispersion (left) and spectral asymmetry (right) for the NNN-pairing model $Δ (k) = Δ_{0} (sin k_{x} cos k_{y} + i sin k_{y} cos k_{x})$ for NN-only hopping at $μ = - t$ where one edge is sharp and the other is soft. Left: arrows point to the Majorana branch at the sharp edge. Unlike the soft-edge branch, which only crosses zero at $k_{y} = 0$ , the sharp-edge branch has an additional zero crossing away from $k_{y} = 0$ , as expected from the spectral flow shown in Fig. 4. This extra zero crossing gives rise to the nonzero spectral asymmetry shown in the right panel.
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