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Acta Physica Sinica
Vol. 69, Issue 11, 117401-1 (2020)

Theory of topological superconductivity based on Yu-Shiba-Rusinov states

Jian Li^1、2、*

Author Affiliations

¹School of Science, Westlake University, Hangzhou 310024, China

²Institute of Natural Sciences, Westlake Institute for Advanced Study, Hangzhou 310024, China

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DOI: 10.7498/aps.69.20200831 Cite this Article

Jian Li. Theory of topological superconductivity based on Yu-Shiba-Rusinov states[J]. Acta Physica Sinica, 2020, 69(11): 117401-1 Copy Citation Text

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Abstract

Yu-Shiba-Rusinov states are subgap bound states induced by magnetic impurity atoms in a superconductor. These states can be used as building blocks in constructing an effective topological superconductor. Here we formulate a unified theory of topological superconductivity in different dimensions based on Yu-Shiba-Rusinov states, and demonstrate its application with simple but illustrative examples. Such a theory underlies a number of recent experiments on the related platform.

Keywords

magnetic impurity Majorana fermion topological phase topological superconductivity Yu-Shiba-Rusinov state

$ \hat{H}_{\rm{YSR}} = \hat{H}_{\rm{SC}} + \hat{H}_{\rm{M}}+ \hat{H}_{\rm{T}}, $

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$ \begin{split} \hat{H}_{\rm{SC}} =\; & \int{{\rm{d}}{{k}}}\Bigg[ {\sum\limits_{\alpha\alpha' \atop ss'} \xi_{\alpha s, \alpha' s'}({{k}}) c_{\alpha s}^{\dagger}({{k}})c_{\alpha' s'}({{k}})} \Bigg. \\ &\Bigg. {+ \sum\limits_{\alpha} \varDelta c_{\alpha {\uparrow}}^{\dagger}({{k}})c_{\bar{\alpha} {\downarrow}}^{\dagger}(-{{k}})+ {\rm{h.c.}}} \Bigg], \end{split} $

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$\begin{split} \hat{H}_{\rm{M}} = \sum\limits_{{{r}}_{\rm{M}},{{r}}'_{\rm{M}}}\sum\limits_{\beta\beta' \atop ss'} h_{\beta s, \beta' s'}({{r}}_{\rm{M}}-{{r}}'_{\rm{M}}) d_{\beta s}^{\dagger}({{r}}_{\rm{M}})d_{\beta' s'}({{r}}'_{\rm{M}})\\ = \int_{ {{BZ}}}{{\rm{d}}{{k}}_{\rm{M}}}\sum\limits_{\beta\beta' \atop ss'} h_{\beta s, \beta' s'}({{k}}_{\rm{M}}) d_{\beta s}^{\dagger}({{k}}_{\rm{M}})d_{\beta' s'}({{k}}_{\rm{M}}), \end{split} $

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$ \hat{H}_{\rm{T}} \!=\! \sum\limits_{{{r}}_{\rm{M}}}\sum\limits_{\alpha\beta \atop ss'}\! \int\! {{\rm{d}} {{r}}}\, v_{\alpha s, \beta s'}({{r}}-{{r}}_{\rm{M}}) c_{\alpha s}^{\dagger}({{r}})d_{\beta s'}({{r}}_{\rm{M}}) + {\rm{h.c.}}, $

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$ H_{\rm{SC}}({{k}}) = \left( {\begin{array}{*{20}{c}} {{\xi}}({{k}}) & \varDelta \\ \varDelta & -{{\xi}}({{k}}) \end{array}} \right), $

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$\begin{split} &{\rm G}_{\rm{SC}}^{(0)}(|E| < \varDelta,{{k}}) \equiv \left[E-H_{\rm{SC}}({{k}})\right]^{-1} =\\ {}& [E^2-\varDelta ^2-{{\xi}}({{k}})^2]^{-1} \left( {\begin{array}{*{20}{c}} E+{{\xi}}({{k}}) & \varDelta \\ \varDelta & E-{{\xi}}({{k}}) \end{array}} \right).\end{split} $

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$\begin{split} {}& \hat{H}_{\rm{T}} = \sum\limits_{{{r}}_{\rm{M}}}\sum\limits_{\alpha\beta {\rm{s}}} v_{\alpha\beta} c_{\alpha {\rm{s}}}^{\dagger}({{r}}_{\rm{M}})d_{\beta {\rm{s}}}({{r}}_{\rm{M}}) + {\rm{h.c.}} \\ =\;&\int_{{\rm{BZ}}}{{{\rm{d}}}{{k}}_{{\rm{M}}}}\sum\limits_{\alpha\beta {\rm{s}}} v_{\alpha\beta} \bigg[\sum\limits_{{{n}}}c_{\alpha {\rm{s}}}^{\dagger}({{k}}_{{\rm{M}}}+{{n}}\cdot{{K}}_{{\rm{M}}}, {{r}}_\perp=0)\bigg]\\ &\times d_{\beta {\rm{s}}}({{k}}_{{\rm{M}}}) + {\rm{h.c}}.,\end{split} $

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$ \varSigma_{\rm{M}}(|E| < \varDelta,{{k}}_{\rm{M}}) = {V}^{\dagger}\tilde{{\rm G}}_{\rm{SC}}^{(0)}(E,{{k}}_{\rm{M}}){V}, $

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$ \begin{split} & \tilde{{\rm G}}_{\rm{SC}}^{(0)}(E,{{k}}_{\rm{M}}) \equiv \sum\limits_{{{n}}} {\rm G}_{\rm{SC}}^{(0)}(E,{{k}}_{\rm{M}}+{{n}}\cdot{{K}}_{\rm{M}},{{r}}_{\perp} = 0) \\ {}& = \sum\limits_{{{n}}}\int{{\rm d}{{k}}_{\perp}} {\rm G}_{\rm{SC}}^{(0)}(E,{{k}}_{\rm{M}}+{{n}}\cdot{{K}}_{\rm{M}},{{k}}_{\perp}), \\[-22pt] \end{split} $

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$ G_{\rm{M}}(E^+,{{k}}_{\rm{M}}) = \left[E^+ - {H}_{\rm{M}}({{k}}_{\rm{M}}) - \varSigma_{\rm{M}}(E^+,{{k}}_{\rm{M}})\right]^{-1}, $

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$ {\rm G}_{\rm{SC}}^{({\rm{M}})}(E^+,{{k}}_{\rm{M}}) = \left[\tilde{{\rm G}}_{\rm{SC}}^{(0)}(E^+,{{k}}_{\rm{M}})^{-1} - J{{S}}\cdot{\hat{ s}}\right]^{-1}. $

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$ \begin{split} H({{k}}_{\rm{M}}) \equiv{}& -G_{\rm{M}}(E = 0,{{k}}_{\rm{M}})^{-1} \\ ={}& {H}_{\rm{M}}({{k}}_{\rm{M}}) + \varSigma_{\rm{M}}(E = 0,{{k}}_{\rm{M}}). \end{split} $

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$ {{P}} \equiv ( {{\rm{i}}}\tau_2)\otimes{{T}} = \begin{pmatrix} & {{T}} \\ -{{T}} & \end{pmatrix} , $

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$ {{P}}{\rm G}_{\rm{SC}}^{(0)}(E = 0,{{k}}){ {P}}^{-1} = - {\rm G}_{\rm{SC}}^{(0)}(E = 0,-{{k}}), $

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$ {{P}}\varSigma_{\rm{M}}(E = 0,{{k}}_{\rm{M}}){{P}}^{-1} = -\varSigma_{\rm{M}}(E = 0,-{{k}}_{\rm{M}}). $

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$ (-1)^{\nu_0} = {\rm{sgn}}\left[ {\rm{Pf}}(A)\right], $

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$A \equiv - {{\rm{i}}} U^{\dagger} H U, $

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$ (-1)^{\nu_1} = {\rm{sgn}}\left[ {\rm{Pf}}(A_0) {\rm{Pf}}(A_{{\text{π}}})\right], $

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$ A_{0,{\text{π}}} \equiv - {{\rm{i}}} U^{\dagger} H({{k}}_{\rm{M}} = 0,{\text{π}}) U. $

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$ \nu_2 = \frac{1}{2{\text{π}}}\sum\limits_{E_b < 0}\int_{ {\rm{BZ}}}{{\rm{d}}{{S}}_{{{k}}_{\rm{M}}}}\cdot{{\bf{\nabla}}_{{{k}}_{\rm{M}}}}\times{\left\langle {u_b({{{k}}_{\rm{M}}})}|{ {{\rm{i}}}{{\bf{\nabla}}_{{{k}}_{\rm{M}}}}u_b({{{k}}_{\rm{M}}})}\right\rangle}, $

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$U = \frac{1}{2} \left( {\begin{array}{*{20}{c}} \sigma_0 & {{\rm{i}}}\sigma_1 \\ - {{\rm{i}}}\sigma_2 & {{\rm{i}}}\sigma_3 \end{array}} \right). $

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$ {{\xi}}({{k}}) = \left(\frac{\hbar^2 k^2}{2m}-\mu\right)\sigma_0, $

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$ \begin{split} &\tilde{\rm G}_{\rm{SC}}^{(0)}(|E| < \varDelta, {{r}} = 0) \simeq -\frac{{\text{π}}\rho}{\sqrt{\varDelta ^2-E^2}}(E\tau_0+\varDelta \tau_1)\sigma_0, \end{split} $

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$ \begin{split} {{\rm G}_{\rm{M}}}({E^ + }) =\; & \left[ {\left( {1 + \frac{\varGamma }{{\sqrt {{\varDelta ^2} - {E^2}} }}} \right)} \right.{E^ + } - {\varepsilon _{\rm{M}}}{\sigma _3} \\ & + {\mu _{\rm{M}}}{ \tau _3} + {{\left. {\frac{{\varGamma \varDelta }}{{\sqrt {{\varDelta ^2} - {E^2}} }}{\tau _1}} \right]}^{ - 1}}, \end{split} $

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$ H = {\varepsilon _{\rm{M}}}{\sigma _3} - {\mu _{\rm{M}}}{\tau _3} - \varGamma {\tau _1}. $

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$ E_{\rm YSR} \simeq \pm\varDelta \frac{\varGamma^2+\mu_{\rm{M}}^2-\varepsilon_{\rm{M}}^2}{\sqrt{(\varGamma^2+\mu_{\rm{M}}^2-\varepsilon_{\rm{M}}^2)^2+(2\varGamma\varepsilon_{\rm{M}})^2}}. $

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$ E_{\rm{YSR}}(\mu_{\rm{M}} = 0) = \pm\varDelta \frac{1-(\varepsilon_{\rm{M}}/\varGamma)^2}{1+(\varepsilon_{\rm{M}}/\varGamma)^2}. $

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$ (-1)^{\nu_0} = {\rm{sgn}}(\varGamma^2+\mu_{\rm{M}}^2-\varepsilon_{\rm{M}}^2). $

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$ \begin{split}& {G}_{\rm{SC}}^{(0)}(|E| < \varDelta, k_{//}, {{r}}_{\perp} = 0) \\ \simeq{}& -\dfrac{{\text{π}}\rho(k_{//})}{\sqrt{\varDelta ^2-E^2}}(E+\varDelta \tau_1). \end{split} $

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$ \varSigma_{\rm{M}}(|E| < \varDelta, k_{\rm{M}}) \simeq -\frac{\varGamma(k_{\rm{M}})}{\sqrt{\varDelta ^2-E^2}}(E+\varDelta \tau_1), $

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$ {{h}}(k_{\rm{M}}) = \varepsilon_0(k_{\rm{M}})-\mu_{\rm{M}} + \varepsilon_{\rm{SO}}(k_{\rm{M}})\sigma_2 + \varepsilon_{\rm{M}}\sigma_3, $

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$ \begin{split} & {\rm G}_{\rm{M}}(E^+, k_{\rm{M}}) \\ =\; & \bigg\{ {\bigg( {1+\frac{\varGamma(k_{\rm{M}})}{\sqrt{\varDelta ^2-E^2}}} \bigg)} E^+ +\frac{\varGamma(k_{\rm{M}})\varDelta }{\sqrt{\varDelta ^2-E^2}}\tau_1 \\ & -[\varepsilon_0(k_{\rm{M}}) \!-\! \mu_{\rm{M}} \!+\! \varepsilon_{\rm{SO}}(k_{\rm{M}})\sigma_2]\tau_3 \! -\! \varepsilon_{\rm{M}}\sigma_3\bigg\}^{-1}, \end{split} $

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$ \begin{split} H(k_{\rm{M}}) =\; & [\varepsilon_0(k_{\rm{M}})-\mu_{\rm{M}} + \varepsilon_{\rm{SO}}(k_{\rm{M}})\sigma_2]\tau_3 \\ & + \varepsilon_{\rm{M}}\sigma_3 - \varGamma(k_{\rm{M}})\tau_1. \end{split} $

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$ E_{\rm{YSR}}(k_{\rm{M}}) \simeq \pm\varDelta \sqrt{\frac{(P-\varepsilon_{\rm{SO}}^2)^2+4\varepsilon_{\rm{SO}}^2 \varGamma^2}{(P-\varepsilon_{\rm{SO}}^2)^2+4(\varepsilon_{\rm{M}}^2+\varepsilon_{\rm{SO}}^2)\varGamma^2}}, $

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$ P(k_{\rm{M}}) \equiv \varGamma(k_{\rm{M}})^2+[\varepsilon_0(k_{\rm{M}})-\mu_{\rm{M}}]^2-\varepsilon_{\rm{M}}^2. $

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$ (-1)^{\nu_1} = {\rm{sgn}}[P(k_{\rm{M}} = 0)P(k_{\rm{M}} = {\text{π}})]. $

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$ \begin{split}& {{G}}_{\rm{SC}}^{(0)}(|E| < \varDelta, {{k}}_{//}, r_{\perp} = 0) \\ \simeq & -\frac{{\text{π}}\rho({{k}}_{//})}{\sqrt{\varDelta ^2-E^2}}(E+\varDelta \tau_1), \end{split} $

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$ \varSigma_{\rm{M}}(|E| < \varDelta, {{k}}_{\rm{M}}) \simeq -\frac{\varGamma({{k}}_{\rm{M}})}{\sqrt{\varDelta ^2-E^2}}(E+\varDelta \tau_1), $

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$ {{h}}({{k}}_{\rm{M}}) = \varepsilon_0({{k}}_{\rm{M}})-\mu_{\rm{M}} + {{\epsilon}}_{\rm{SO}}({{k}}_{\rm{M}})\cdot{{\sigma}}_{//} + \varepsilon_{\rm{M}}\sigma_3, $

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$ \begin{split} &G_{\rm{M}}(E^+, {{k}}_{\rm{M}}) \\=\; & \bigg\{ {\left( {1+\frac{\varGamma({{k}}_{\rm{M}})}{\sqrt{\varDelta ^2-E^2}}} \right)} E^+ +\frac{\varGamma({{k}}_{\rm{M}})\varDelta }{\sqrt{\varDelta ^2-E^2}}\tau_1 \\ &-[\varepsilon_0({{k}}_{\rm{M}})-\mu_{\rm{M}} + {{\epsilon}}_{\rm{SO}}({{k}}_{\rm{M}})\cdot{{\sigma}}_{//}]\tau_3 -\varepsilon_{\rm{M}}\sigma_3\bigg\}^{-1}, \end{split} $

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$ \begin{split} H({{k}}_{\rm{M}}) =\; & [\varepsilon_0({{k}}_{\rm{M}})-\mu_{\rm{M}} + {{\epsilon}}_{\rm{SO}}({{k}}_{\rm{M}})\cdot{{\sigma}}_{//}]\tau_3 \\ & + \varepsilon_{\rm{M}}\sigma_3 - \varGamma({{k}}_{\rm{M}})\tau_1. \end{split} $

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$ {( - 1)^{{\nu _2}}} = {\rm{sgn}}\left[ {\prod\limits_{{{{k}}_{\rm{M}}} \in {\rm{TRIM}}} P ({{{k}}_{\rm{M}}})} \right]. $

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Jian Li. Theory of topological superconductivity based on Yu-Shiba-Rusinov states[J]. Acta Physica Sinica, 2020, 69(11): 117401-1

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