Odd-frequency pairing and proximity effect in Kitaev chain systems including a topological critical point

In this paper, we investigate the relation between odd-frequency pairing and proximity effect in nonuniform Kitaev chain systems with a particular interest in the topological critical point. First, we correlate the odd-frequency pairing and Majorana fermion in a semi-infinite Kitaev chain, where we find that the spatial dependence of the odd-frequency pair amplitude coincides with that of the local density of states at low frequencies. Second, we demonstrate that, contrary to the standard view, the odd-frequency pair amplitude spreads into the bulk of a semi-infinite Kitaev chain at the topological critical point. Finally, we show that odd-frequency Cooper pairs cause the proximity effect in a normal metal/diffusive normal metal/Kitaev chain junction even at the topological critical point. Our results hold relevance to the investigation of odd-frequency pairing and topological superconductivity in more complicated systems that involve Rashba nanowires with magnetic fields.

Figure 1

The energy dispersion of the Kitaev chain in the normal state with $\mathrm{\Delta }=0$. When a chemical potential is in the energy band $-2t<\mu <2t$, a topological phase with Majorana fermion is realized. At the topological critical point, $\mu$ is located at the edge of the energy band.

Figure 2

(a) Semi-infinite system of Kitaev chain. (b) Normal metal/diffusive normal metal/Kitaev chain junction. $j$: The number of sites in the chain. $j\le 0$: A ballistic metal without impurities. $1\le j\le L$: Diffusive normal metal containing impurities. $L+1\le j$: Kitaev chain. $L$: The number of sites in the diffusive normal metal region.

Figure 3

(a) Imaginary part of normal Green's function $\mathrm{Im}\left[G_{1,1}\right]$ and anomalous Green's function $\mathrm{Im}\left[{\tilde{F}}_{1,1}\right]$ are plotted as a function of $\mu$ in the semi-infinite Kitaev chain. ${\omega }_n$: Matsubara frequency. $\mathrm{Im}\left[{\tilde{F}}_{1,1}\right]$: The odd-frequency spin-triplet even-parity pair amplitude. In the low ${\omega }_n$ limit, $-\mathrm{Im}\left[G_{1,1}\right]$: the local density of states of Majorana fermion in topological phase with $\left|\mu \right|<2t$. ${\omega }_n/t={10}^{-5}$. (b), (c) The site dependence of probability density of Majorana wave function $|a_j{|}^2$, odd-frequency spin-triplet even-parity pair amplitude $\mathrm{Im}\left[f^{\mathrm{OTE}}\left(j\right)\right]$ with ${\omega }_n/t={10}^{-5}$, and the local density of states $\rho (E=0,j)$ with ${\delta }_{\epsilon }/t={10}^{-5}$. All these quantities are calculated in the semi-infinite Kitaev chain and are normalized by their values at $j=1$. (b) $\mu /t=1.9$, (c) $\mu /t=2, \mathrm{\Delta }/t=0.1$.

Figure 4

(a)–(c) Odd-frequency spin-triplet even-parity (OTE) pair amplitude in the normal metal/diffusive normal metal/Kitaev chain (N/DN/KC) junction; (d)–(f) even-frequency spin-triplet odd-parity (ETO) pair amplitude in the normal metal/diffusive normal metal/Kitaev chain junction; (g)–(i) even-frequency spin-singlet even-parity (ESE) pair amplitude in the normal metal/diffusive normal metal/$s$-wave superconductor (N/DN/S) junction; ${\mu }_{\mathrm{N}}/t=0.5, \mathrm{\Delta }/t=0.1, {\omega }_n/t={10}^{-5}, L=50$, and $W/t=3$. (a), (d) Topological regime: $\mu /t=0$, 0.5, 1; (b), (e) topological regime near topological critical point: $\mu /t=1.95$, 1.99, 1.999; (c), (f) topological critical point: $\mu /t=2$; nontopological regime: $\mu =2.1$, 2.5. (g) metallic regime: $\mu /t=0$, 0.5, 1; (h) metallic regime near metal-insulator transition point: $\mu /t=1.95$, 1.99, 1.999; (i) metal-insulator transition point: $\mu /t=2$; insulating regime: $\mu =2.1$, 2.5.

Figure 5

(a)–(c) The local density of states in the diffusive normal metal for energy $E$ in the normal metal/diffusive normal metal/Kitaev chain junction: $j=20$; (d)–(f) differential conductance for bias voltage eV in the normal metal/diffusive normal metal/Kitaev chain junction: $j=-10$; ${\mu }_{\mathrm{N}}/t=0.5, \mathrm{\Delta }/t=0.1, {\omega }_n/t={10}^{-8}, L=20$, and $W/t=1$. (a), (d) Topological regime: $\mu /t=0$, 0.5, 1; (b), (e) topological regime near topological critical point: $\mu /t=1.95$, 1.99, 1.999; (c), (f) topological critical point: $\mu /t=2$; nontopological regime: $\mu =2.1$.

Figure 6

Half width at half maximum of the zero-bias conductance peak is plotted as a function of (a) $L$ and (b) $W$ in the diffusive normal metal in the normal metal/diffusive normal metal/Kitaev chain junction. $L$: The length of the diffisive normal metal, $W$: the intensity of random potential. ${\mu }_{\mathrm{N}}/t=0.5, \mathrm{\Delta }/t=0.1, {\mu }_{\mathrm{N}}/t=\mu =0, j=-10$, and ${\delta }_{\epsilon }/t={10}^{-8}$. (a) $W/t=1$, (b) $L=20$.

Figure 7

The definition of the peak width. In this paper, the peak width is defined as the half width at half maximum.

Figure 8

Schematics for obtaining the half width at half maximum of the peak using the false position method.

Figure 9

Half width at half maximum of the zero-energy peak of the local density of states is plotted as a function of (a) $L$, (b) $W$, of the diffusive normal metal in the normal metal/diffusive normal metal/Kitaev chain junction. $L$: The length of the diffusive normal metal, $W$: the magnitude of random potential. $\mathrm{\Delta }/t=0.1, {\mu }_{\mathrm{N}}=\mu =0, j=L/2$, and ${\delta }_{\epsilon }/t={10}^{-8}$. (a) $W=1$, (b) $L=20$.

Figure 10

Standard deviation $\sigma$ of differential conductance in the normal metal/diffusive normal metal/Kitaev chain junction is plotted as a function of bias voltage eV. It is independent of $j$ in the normal metal region. ${\mu }_{\mathrm{N}}/t=0.5, \mathrm{\Delta }/t=0.1, {\delta }_{\epsilon }/t={10}^{-8}, j=-10$, and $W/t=1$. Topological regime near topological critical point: $\mu =1.99$, 1.95, 1.999.