Quantum impurity coupled to Majorana edge fermions

We study a quantum impurity coupled to the edge states of a two-dimensional helical topological superconductor, that is, to a pair of counterpropagating Majorana fermion edge channels with opposite spin polarizations. For an impurity described by the Anderson impurity model, we show that the problem maps onto a variant of the interacting resonant two-level model which, in turn, maps onto the ferromagnetic Kondo model. Both magnetic and nonmagnetic impurities are considered. For magnetic impurities, an analysis relying on bosonization and the numerical renormalization group shows that the system flows to a fixed point characterized by a residual $\ln 2$ entropy and anisotropic static and dynamical impurity magnetic susceptibilities. For nonmagnetic impurities, the system flows instead to a fixed point with no residual entropy and we find diamagnetic impurity response at low temperatures. We comment on the Schrieffer-Wolff transformation for problems with nonstandard conduction band continua and on the differences which arise when we describe the impurities by either Anderson or Kondo impurity models.

Figure 1

Schematic representation of the single-impurity problem studied in this work. A single-impurity level (circle) is hybridized with two continua of counterpropagating Majorana fermions with opposite spin polarizations (dashed lines).

Figure 2

Thermodynamic properties (impurity entropy and impurity susceptibilities) at the particle-hole symmetric point $\delta =0$ for a range of values of the interaction strength $U$. The arrows indicate the direction of increasing $U$. In the absence of the external magnetic field, one has ${\chi }_{xx,\mathrm{imp}}={\chi }_{zz,\mathrm{imp}}$ due to symmetry. The out-of-diagonal magnetic susceptibilities are all zero. On the curves for $U=0.02$, we label the positions of the temperatures $T_1$ and $T_2$ (see the main text for their definitions).

Figure 3

The temperature $T_2$ where the effective transverse magnetic moment ${\mu }_x^2=T{\chi }_{xx,\mathrm{imp}}\left(T\right)$ is reduced to $1/8$. The full curve is a fit which corresponds to Eq. (34).

Figure 4

Renormalization-group flow diagram. The NRG iteration step numbers corresponding to the temperature scales $T_1$ and $T_2$ are indicated.

Figure 5

Thermodynamic behavior away from the particle-hole symmetric point at $\delta =0$. The arrows indicate the direction of increasing $\delta$. The transition point corresponds to $\left|\delta \right|\sim U/2-c\Gamma$ with $c$ of order 1.

Figure 6

Phase diagram in the ($\Gamma ,\delta$) plane for fixed interaction parameter $U$.

Figure 7

An impurity in an external magnetic field along $x$, $y$, and $z$ directions, respectively, with constant strength $\left|\mathbf{B}\right|$. For each direction of the field, the parameter $U$ is swept from the weak-interaction to the strong-interaction (Kondo) regime. The arrows indicate the direction of increasing $U$.

Figure 8

Dynamical magnetic susceptibility functions for an external magnetic field applied along the transverse ($x$) direction (top panels) and along the longitudinal ($y$) direction (bottom panels).

Figure 9

Spectral properties of the impurity. The nonzero spectral functions are shown. We introduce the notation $d_0=d_{\downarrow }^{†}$, $d_1=d_{\uparrow }^{†}$, $d_2=d_{\downarrow }$, $d_3=d_{\uparrow }$ and define the spectral functions as $A_{ij}\left(\omega \right)=-(1/\pi )\mathrm{Im}{\langle \langle d_i;d_j^{†}\rangle \rangle }_{\omega +i\delta }$. Nonzero anomalous spectral functions $A_{02}$ and $A_{13}$ thus indicate the presence of pairing correlations. The inset in the bottom panel represents the power-law exponent as a function of the interaction strength.