Odd-frequency pairs in chiral symmetric systems: Spectral bulk-boundary correspondence and topological criticality

Odd-frequency Cooper pairs with chiral symmetry emerging at the edges of topological superconductors are a useful physical quantity for characterizing the topological properties of these materials. In this work, we show that the odd-frequency Cooper pair amplitudes can be expressed by a winding number extended to a nonzero frequency, which is called a “spectral bulk-boundary correspondence,” and can be evaluated from the spectral features of the bulk. The odd-frequency Cooper pair amplitudes are classified into two categories: the amplitudes in the first category have the singular functional form $\sim 1/z$ (where $z$ is a complex frequency) that reflects the presence of a topological surface Andreev bound state, whereas the amplitudes in the second category have the regular form $\sim z$ and are regarded as nontopological. We discuss the topological phase transition by using the coefficient in the latter category, which undergoes a power-law divergence at the topological phase transition point and is used to indicate the distance to the critical point. These concepts are established based on several concrete models, including a Rashba nanowire system that is promising for realizing Majorana fermions.

Figure 1

(a) Schematic of the correspondence between odd-frequency Cooper pair correlation $F_{\mathrm{edge}}^{\mathrm{odd}}\left(z\right)$ and the extended winding number $w_{\mathrm{bulk}}\left(z\right)$ with complex frequency $z$. (b) Graphs of singular and regular frequency dependence of $F_{\mathrm{edge}}^{\mathrm{odd}}$. $z=\mathrm{i}{\omega }_n$ is a Matsubara frequency and if $z$ is purely imaginary, $F_{\mathrm{edge}}^{\mathrm{odd}}$ is also purely imaginary.

Figure 2

(a) Schematic of one-dimensional $p$-wave pairing Kitaev chain. (b) The imaginary part of total amount of the odd frequency Cooper pair amplitude $\mathrm{Im}\left[F_{\mathrm{edge}}^{\mathrm{odd}}\left(z\right)\right]$ is plotted as a function of ${\omega }_n=z/\mathrm{i}$ at $\mathrm{\Delta }/t=0.05$ with several $\mu$. In the inset, $\varepsilon$ is plotted as a function of ${\omega }_n$ for $\mu /t=0.5$ and 2.01.

Figure 3

(a) The winding number $W$ (left vertical axis) and $\chi$ (right vertical axis) are plotted as a function of $\mu$ for $\mathrm{\Delta }/t=0.05$. (b) $\left|\chi \right|$ is plotted near ${\mu }_{\mathrm{c}}$ as a function of $|\mu -{\mu }_{\mathrm{c}}|/t$ for $\mu >{\mu }_{\mathrm{c}}$ and $\mu <{\mu }_{\mathrm{c}}$ with ${\mu }_{\mathrm{c}}=2t$ and $\delta \mu =\mu -{\mu }_{\mathrm{c}}$.

Figure 4

Critical regimes for the Kitaev chain near $\mu ={\mu }_{\mathrm{c}}$ and $\mathrm{\Delta }=0$, illustrated in the plane of the mass $m(\propto \mu -{\mu }_c)$ and velocity $v(\propto \mathrm{\Delta })$. Colored regions (QCR1 to QCR3) are characterized by different critical exponents $\chi \sim v^{\alpha }{m}^{\beta }$ with fixed $\alpha +2\beta =-4$. The orange line shows the critical points at which the energy gap closes.

Figure 5

(a) Schematic of Rashba nanowire. (b) Energy dispersion of nanowire for $\mathrm{\Delta }/t=0.01,V_{\mathrm{ex}}/t=1,\lambda /t=0.5$, and $\mu =0$. The red and green shaded areas are the topological regime. (c) $W$ (red, left vertical axis) and $\chi$ (blue, right vertical axis) are plotted as a function of $\mu /t$. ${\mu }_{\mathrm{c}1}=-\sqrt{V_{\mathrm{ex}}^2-{\mathrm{\Delta }}^2}, {\mu }_{\mathrm{c}2}=\sqrt{V_{\mathrm{ex}}^2-{\mathrm{\Delta }}^2}, {\mu }_{\mathrm{c}3}=4t-\sqrt{V_{\mathrm{ex}}^2-{\mathrm{\Delta }}^2}$, and ${\mu }_{\mathrm{c}4}=4t+\sqrt{V_{\mathrm{ex}}^2-{\mathrm{\Delta }}^2}$. (d) Schematic of the QCRs.

Figure 6

$\varepsilon$ for the Kitaev chain with $\mathrm{\Delta }/t=1$ is plotted as functions of (a) $\mu /t$ and ${\omega }_n/\mathrm{\Delta }$ ($z=\mathrm{i}{\omega }_n$) and (b) $\mu /t$ and $\omega /\mathrm{\Delta }$ ($z=\omega +\mathrm{i}\eta$ with $\eta ={10}^{-2}$).

Figure 7

(a) $\mathrm{Im}\left[F_{\mathrm{edge}}^{\mathrm{odd}}\left(z\right)\right]$, (b) $\mathrm{Re}[w_{\mathrm{bulk}}\left(z\right)/z]$, and (c) $\varepsilon$ for Kitaev chain with $\mathrm{\Delta }/t=0.05$ are plotted as functions of $\mu /t$ and ${\omega }_n/\mathrm{\Delta }$ ($z=\mathrm{i}{\omega }_n$). (d) $\mathrm{Re}\left[F_{\mathrm{edge}}^{\mathrm{odd}}\left(z\right)\right]$, (e) $\mathrm{Im}\left[F_{\mathrm{edge}}^{\mathrm{odd}}\left(z\right)\right]$, (f) $\mathrm{Re}[w_{\mathrm{bulk}}\left(z\right)/z]$, (g) $\mathrm{Im}[w_{\mathrm{bulk}}\left(z\right)/z]$, and (h) $\varepsilon$ for Kitaev chain with $\mathrm{\Delta }/t=0.05$ are plotted as functions of $\mu /t$ and $\omega /\mathrm{\Delta }$ ($z=\omega +\mathrm{i}\eta$ with $\eta ={10}^{-2}$).

Figure 8

(a) $\varepsilon$ with surface modulation of the gap function $[tanh(j\mathrm{\Delta }/t\left)\right]$ is plotted as functions of $\mu /t$ and ${\omega }_n/\mathrm{\Delta }$ for $\mathrm{\Delta }/t=0.05$. (b) randomly chosen $f_{\mathrm{\Delta }}\left(j\right)$ and $f_{\mu }\left(j\right)$ are plotted as a function of $j$. (c) $\varepsilon$ with surface modulation shown in (b) is plotted as functions of $\mu /t$ and ${\omega }_n/\mathrm{\Delta }$ for $\mathrm{\Delta }/t=0.05$.

Figure 9

(a) The real part of $z{F}_{\mathrm{edge}}^{\mathrm{odd}}\left(z\right)$, (b) the real part of $w_{\mathrm{bulk}}\left(z\right)$, and (c) $\varepsilon$ are plotted as functions of $\mu /t$ and $\omega /\mathrm{\Delta }$ for $\mathrm{\Delta }/t=0.01,V_{\mathrm{ex}}/t=1$, and $\lambda /t=0.5$. (d) The real part of $z{F}_{\mathrm{edge}}^{\mathrm{odd}}\left(z\right)$, (e) the real part of $w_{\mathrm{bulk}}\left(z\right)$, and (f) $\varepsilon$ are plotted as functions of $V_{\mathrm{ex}}/t$ and $\omega /\mathrm{\Delta }$ for $\mathrm{\Delta }/t=0.01,\mu /t=1$, and $\lambda /t=0.5$. Note that $z{F}_{\mathrm{edge}}^{\mathrm{odd}}\left(z\right)$ and $w_{\mathrm{bulk}}\left(z\right)$ are real valued function for purely imaginary $z$.

Figure 10

(a) $W$ (left vertical axis) and $\chi t^2$ (right vertical axis) are plotted as a function of $\mu /t$ for $\mathrm{\Delta }/t=0.9,V_{\mathrm{ex}}/t=1$ and $\lambda /t=0.5$. (b) $\chi t^2$ is plotted as a function of $\mu /t$ for $\mathrm{\Delta }/t=0.01,V_{\mathrm{ex}}/t=1$ and $\lambda /t=0.5$ with ${\mu }_{\mathrm{c}1}=-\sqrt{V_{\mathrm{ex}}^2-{\mathrm{\Delta }}^2}, {\mu }_{\mathrm{c}2}=\sqrt{V_{\mathrm{ex}}^2-{\mathrm{\Delta }}^2}, {\mu }_{\mathrm{c}3}=4t-\sqrt{V_{\mathrm{ex}}^2-{\mathrm{\Delta }}^2}$, and ${\mu }_{\mathrm{c}4}=4t+\sqrt{V_{\mathrm{ex}}^2-{\mathrm{\Delta }}^2}$. (c) $\left|\chi \right|t^2$ is plotted as a function of $|\mu -{\mu }_{\mathrm{c}}|/t$ close to $\mu ={\mu }_{\mathrm{c}1},{\mu }_{\mathrm{c}2},{\mu }_{\mathrm{c}3}$, and ${\mu }_{\mathrm{c}4}$. (d) Energy dispersion is plotted as a function of $k$ with $V_{\mathrm{ex}}=0.5,2$,and 3.5 for $\mathrm{\Delta }/t=0.01,\mu /t=1$, and $\lambda /t=0.5$. Green shaded area and red shaded one is a topological regime with $W=-1$ for green and $W=1$ for red, respectively. (e) $W$ (left vertical axis) and $\chi t^2$ (right vertical axis) are plotted as a function of $V_{\mathrm{ex}}/t$ for $\mathrm{\Delta }/t=0.01,\mu /t=1,$ and $\lambda /t=0.5$ with $V_{\mathrm{c}1}=\sqrt{{\mu }^2-{\mathrm{\Delta }}^2},V_{\mathrm{c}2}=\sqrt{{(4t-\mu )}^2-{\mathrm{\Delta }}^2}$. $W$ and $\chi$ are even function of $V_{\mathrm{ex}}$. (f) $\left|\chi \right|t^2$ is plotted as a function of $|V_{\mathrm{ex}}-V_{\mathrm{c}}|/t$ close to $V_{\mathrm{ex}}=V_{\mathrm{c}1}$ and $V_{\mathrm{c}2}$.

Figure 11

(a) Schematic illustration of the unit cell. A and B indicate sublattices. (b) Imaginary part of $F_{\mathrm{edge}}^{\mathrm{odd}}\left(z\right)$ is plotted as functions of $k_y$ and ${\omega }_n/\mathrm{\Delta }$ with $\mathrm{\Delta }/t=0.01$ and $\mu /t=-1$. (c) $\varepsilon$ is plotted as functions of $k_y$ and ${\omega }_n/\mathrm{\Delta }$ with $\mathrm{\Delta }/t=0.01$ and $\mu /t=-1$. At $k_y=0$, there is no value because $w_{\mathrm{bulk}}\left(z\right)$ and $F_{\mathrm{edge}}^{\mathrm{odd}}\left(z\right)$ are exactly zero. (d) $\chi t^2$ is plotted as a function of $k_y$. $\left|\chi \right|t^2$ is plotted as a function of $|k_y-k_{\mathrm{c}}|$ near (e) $k_{\mathrm{c}}=0$ and (f) $2arccos(-\mu /4t)$.