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Proximity effect in SSH -superconductor junction

I. A. Belkovich, A. A. Radkevich

TL;DR

The paper addresses proximity-induced superconductivity in a 1D SSH chain in contact with a (massive) superconductor. It develops a microscopic, analytically tractable framework based on functional integration to derive an effective, nonlocal action $S_{\text{eff}} = S_{SSH} + \int d\tau d\tau' \sum_{m,m'} \Phi_m^{\dagger}(\tau) \hat{V}_{m,m'}(\tau-\tau') \Phi_{m'}(\tau')$, where $\hat{V}$ encodes the nonlocal proximity interaction. The results show real energy corrections for bulk states inside the superconducting gap, and imaginary parts (finite lifetimes) outside the gap; edge-state energies are shifted by the induced interaction, with additional dissipation arising from phase fluctuations of the low-dimensional superconductor. A naive, phenomenological proximity model misses dissipation and nonlocal effects and can introduce unphysical symmetries. Overall, the work provides a rigorous microscopic basis for proximity effects in SSH-like topological systems and clarifies when coherence is preserved versus when dissipation dominates.

Abstract

A model of microscopic interaction between a superconductor and a one-dimensional topological insulator, an SSH chain, is considered. Using the functional integration method, the effective action of the interaction between a superconductor and a topological insulator is obtained. We obtain corrections to the quasiparticle excitation spectrum of the SSH chain due to tunneling in various limits and discuss the influence of phase fluctuations. We find that for bulk superconductors, the states of the chain are stable for energies lying inside the superconducting gap while in lower-dimensional superconductors phase fluctuations yield finite temperature-dependent lifetimes even inside the gap. We also discuss whether these results can be reproduced within a simple phenomenological approach.

Proximity effect in SSH -superconductor junction

TL;DR

The paper addresses proximity-induced superconductivity in a 1D SSH chain in contact with a (massive) superconductor. It develops a microscopic, analytically tractable framework based on functional integration to derive an effective, nonlocal action , where encodes the nonlocal proximity interaction. The results show real energy corrections for bulk states inside the superconducting gap, and imaginary parts (finite lifetimes) outside the gap; edge-state energies are shifted by the induced interaction, with additional dissipation arising from phase fluctuations of the low-dimensional superconductor. A naive, phenomenological proximity model misses dissipation and nonlocal effects and can introduce unphysical symmetries. Overall, the work provides a rigorous microscopic basis for proximity effects in SSH-like topological systems and clarifies when coherence is preserved versus when dissipation dominates.

Abstract

A model of microscopic interaction between a superconductor and a one-dimensional topological insulator, an SSH chain, is considered. Using the functional integration method, the effective action of the interaction between a superconductor and a topological insulator is obtained. We obtain corrections to the quasiparticle excitation spectrum of the SSH chain due to tunneling in various limits and discuss the influence of phase fluctuations. We find that for bulk superconductors, the states of the chain are stable for energies lying inside the superconducting gap while in lower-dimensional superconductors phase fluctuations yield finite temperature-dependent lifetimes even inside the gap. We also discuss whether these results can be reproduced within a simple phenomenological approach.
Paper Structure (9 sections, 88 equations, 2 figures)

This paper contains 9 sections, 88 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Set up of the problem
  3. Elimination of superconductor's degrees of freedom
  4. Bulk states
  5. Edge states
  6. Effect of the collective modes
  7. Naive proximity effect
  8. Discussion
  9. Calculation of the corrections to the bulk energy spectrum

Figures (2)

  • Figure 1: The considered model of junction between a TI and a superconductor
  • Figure 2: Perturbative numerical calculation of the spectrum. For the superconductor we used the model spectrum of a cubic lattice $\xi_k = -2t[\cos{(k_x a)}+\cos{(k_y a)}+\cos{(k_z a)}]-\mu$ with the following parameters: (a) $\varepsilon_{g} = 2$, $t_{1} = 1$, $t_{2} = 0.5$, $t_{0} = 0.8$, $t = 1$, $\Delta = 4$, $\mu = 0$; (b) $\varepsilon_{g} = 0$, $t_{1} = 1$, $t_{2} = 0.5$, $t_{0} = 0.8$, $t = 1$, $\Delta = 4$, $\mu = 1$. The branches of the unperturbed spectrum are shown in blue, while the spectrum with corrections is shown in red. The black horizontal lines mark the boundaries of the superconductor gap region.