Degeneracy lifting of Majorana bound states due to electron-phonon interactions

We study theoretically how electron-phonon interaction affects the energies and level broadening (inverse lifetime) of Majorana bound states (MBSs) in a clean topological nanowire at low temperatures. At zero temperature, the energy splitting between the right and left MBSs remains exponentially small with increasing nanowire length $L$. At finite temperatures, however, the absorption of thermal phonons leads to the broadening of energy levels of the MBSs that does not decay with system length, and the coherent absorption/emission of phonons at opposite ends of the nanowire results in MBSs energy splitting that decays only as an inverse power law in $L$. Both effects remain exponential in temperature. In the case of quantized transverse motion of phonons, the presence of Van Hove singularities in the phonon density of states causes additional resonant enhancement of both the energy splitting and the level broadening of the MBSs. This is the most favorable case to observe the phonon-induced energy splitting of MBSs as it becomes much larger than the broadening even if the topological nanowire is much longer than the coherence length. We also calculate the charge and spin associated with the energy splitting of the MBSs induced by phonons. We consider both a spinless low-energy continuum model, which we evaluate analytically, as well as a spinful lattice model for a Rashba nanowire, which we evaluate numerically.

Figure 1

Feynman diagram for the self-energy ${\mathrm{\Sigma }}_{\pm \pm }$ corresponding to the process where a phonon (curly black line) promotes the system from a bound state ${\mathrm{\Phi }}_{\pm }$ to a bulk state $n$ (straight red line).

Figure 2

Plot of the dimensionless function $|F_0\left(L\right)|$ defined in Eq. (25) that describes the $L$ dependence of the energy splitting $\delta {\varepsilon }_{+}$ of the MBSs [see Eq. (24)]. $|F_0\left(L\right)|$ shows oscillatory behavior with a period determined by $k_F$ and an amplitude exponentially decaying as $exp(-L/\xi )$. Here, we took $k_F=10/\xi$.

Figure 3

Energy splitting $2\delta {\varepsilon }_{+}$ (blue solid line) and broadening $\gamma$ (green line) of MBSs caused by interaction with 1D acoustic phonons [see Eqs. (26) and (27)]. Dashed blue line shows analytical estimation for $2\delta {\varepsilon }_{+}$ given by Eq. (29). The broadening $\gamma$ saturates at a finite value as the length $L$ of the nanowire grows, while the energy splitting $2\delta {\varepsilon }_{+}$ oscillates with a characteristic oscillation scale governed by both Fermi wavelength $2\pi /k_F$ and phonon wavelength $c_s/\mathrm{\Delta }$. The amplitude of $2\delta {\varepsilon }_{+}$ decays as a power law $\propto L^{-1/2}$ with increasing length. We took $(\mathrm{\Delta },v_F,c_s,C_{\mathrm{ac}},a)=(0.2\mathrm{meV},4.6\times {10}^4\mathrm{m}/\mathrm{s},4.4\times {10}^3\mathrm{m}/\mathrm{s},5\mathrm{eV},5\AA$), $T=0.1\mathrm{\Delta }$, $k_F=5/\xi$ for numerical estimations.

Figure 4

Energy splitting $2\delta {\varepsilon }_{+}$ (blue line) and broadening $\gamma$ (green line) [see Eqs. (33, 34, 35)] of the MBSs due to the second branch of confined phonons in case when the band bottom of this branch, ${\mathrm{\Omega }}_1$, is less than the gap $\mathrm{\Delta }$, ${\mathrm{\Omega }}_1=0.9\mathrm{\Delta }$. In this case, neither the broadening $\gamma$, nor the energy splitting $2\delta {\varepsilon }_{+}$ exhibit resonance peaks. Comparing with Fig. 3 one can see that the dependence of $\gamma$ and $2\delta {\varepsilon }_{+}$ on $L$ is qualitatively the same as for purely 1D acoustic phonons without VHS. We took $g_1=g$, and the phenomenological relaxation rate $\mathrm{\Gamma }=0.01\mu \mathrm{eV}$. The other parameters are the same as in Fig. 3.

Figure 5

The same as Fig. 4 but for the case when ${\mathrm{\Omega }}_1$ slightly exceeds the gap $\mathrm{\Delta }$, ${\mathrm{\Omega }}_1=1.1\mathrm{\Delta }$. If the nanowire length $L$ is varied, each time ${\mathrm{\Omega }}_1$ coincides with energies of the bulk quasiparticle states, both the broadening $\gamma$ and splitting $2\delta {\varepsilon }_{+}$ are enhanced, manifesting resonance peaks. The splitting $2\delta {\varepsilon }_{+}$ remains significant and of order of $\gamma$ even for $L\gg \xi$. The other parameters are the same as in Fig. 4.

Figure 6

The same as Fig. 5 but as a function of the energy ${\mathrm{\Omega }}_1$ of the band bottom of the phonon branch. If ${\mathrm{\Omega }}_1<\mathrm{\Delta }$ the broadening $\gamma$ and the amplitude of the energy splitting $\delta {\varepsilon }_{+}$ depend weakly on ${\mathrm{\Omega }}_1$. The absorption of phonons and, thus, $\delta {\varepsilon }_{+}$ and $\gamma$, are enhanced as ${\mathrm{\Omega }}_1$ gets close to $\mathrm{\Delta }$ due to the VHS in the phonon branch, and the resonance peaks correspond to the case when ${\mathrm{\Omega }}_1$ coincides with an energy of a bulk quasiparticle mode ${\varepsilon }_n$. At increasing ${\mathrm{\Omega }}_1$, both $\gamma$ and $\delta {\varepsilon }_{+}$ decay exponentially in temperature, $\delta {\varepsilon }_{+},\gamma \propto exp\left\{-{\mathrm{\Omega }}_1/T\right\}$. We took $L=20\xi$. The other parameters are the same as in Fig. 5.

Figure 7

(a) The energy splitting $2\delta {\varepsilon }_{+}$ (blue line) and broadening $\gamma$ (green line) of the MBSs and (b) the ratio $\delta {\varepsilon }_{+}/\gamma$ as a function of $k_F$ for $\mathrm{\Gamma }=0.01\mu \mathrm{eV}$ (blue solid line) and $\mathrm{\Gamma }=1\mu \mathrm{eV}$ (red dashed line). By tuning $k_F$ (for example, by gates) one can achieve a resonance when the band bottom of the phonon branch ${\mathrm{\Omega }}_1$ coincides with an energy of one of the quasiparticle bulk modes. The resonant absorption of phonons boosts both $\delta {\varepsilon }_{+}$ and $\gamma$. Around the resonance, the energy splitting $\delta {\varepsilon }_{+}$ can largely exceed the broadening $\gamma$. Thus a fine-tuning of $k_F$ makes it possible to observe the phonon-induced energy splitting of the MBSs. The height of the resonance peaks is determined by the value of $\mathrm{\Gamma }$, however, this dependence is logarithmically weak. We took $L=20\xi$. The other parameters are the same as in Fig. 5.

Figure 8

Energy splitting $2\delta {\varepsilon }_{+}$ (blue line) and broadening $\gamma$ (green line) of the MBSs as functions of (a) number of sites $N$ and (b) bottom of phonon branch ${\mathrm{\Omega }}_1$ for the spinful lattice model [see Eq. (41)] of a topological Rashba nanowire. A higher mode of confined phonons interacts with electrons causing the splitting and broadening of the MBSs. The behavior is qualitatively similar to the case of the low-energy continuum model (see Figs. 5 and 6). If the length $L=N{a}_0$ of the 1D TSC is fine-tuned so that ${\mathrm{\Omega }}_1$ coincides with one of the energy levels of the bulk quasiparticle states, both the broadening $\gamma$ and the energy splitting $2\delta {\varepsilon }_{+}$ show pronounced resonance peaks. Importantly, the energy shift ${\varepsilon }_{+}$ remains significant and of order of $\gamma$ even in long samples, $L\gg \xi$. The parameters chosen are $(E_{so},{\mathrm{\Delta }}_{sc},{\mathrm{\Delta }}_Z,{\mathrm{\Omega }}_1,\mu )=(0.6,0.2,0.4,0.22,0)\mathrm{meV}$, corresponding to $t=10\mathrm{meV}$ and coherence length $\xi =50a_0$. The rest of the parameters are the same as for Fig. 5.

Figure 9

Energy splitting $2\delta {\varepsilon }_{+}$ (blue line) and broadening $\gamma$ (green line) of the MBSs as functions of chemical potential $\mu$ for VHS energies (a) ${\mathrm{\Omega }}_1=1.1{\mathrm{\Delta }}_{sc}$ and (b) ${\mathrm{\Omega }}_1=0.9{\mathrm{\Delta }}_{sc}$ calculated numerically for the spinful lattice model of a topological Rashba nanowire, see Eq. (41). By tuning $\mu$, one can achieve the resonance as the bottom of the phonon band coincides with an energy of one of the quasiparticle bulk eigenstates. The resonant absorption of phonons results in the growth of both $\delta {\varepsilon }_{+}$ and $\gamma$ by several orders of magnitude. (b) If ${\mathrm{\Omega }}_1<|{\mathrm{\Delta }}_Z-{\mathrm{\Delta }}_{sc}|$, the quasiparticle gap is larger than ${\mathrm{\Omega }}_1$ for small values of chemical potential $\left|\mu \right|$ (shaded yellow region). Thus, the broadening $\gamma$ does not depend on $\mu$, while the splitting $|2\delta {\varepsilon }_{+}|$ remains smaller than the broadening $\gamma$. As $\left|\mu \right|$ increases, the topological gap $\mathrm{\Delta }=|{\mathrm{\Delta }}_Z-\sqrt{{\mathrm{\Delta }}_{sc}^2-{\mu }^2}|$ decreases. At some point, ${\mathrm{\Omega }}_1$ becomes greater than $\mathrm{\Delta }$, and both broadening $\gamma$ and energy splitting $|2\delta {\varepsilon }_{+}|$ show resonant behavior, such that, depending on $\mu$, the energy splitting can become comparable to the broadening, and, thus, can be observed experimentally. We took $L=20\xi$, the rest of parameters are the same as for Fig. 8.

Figure 10

(a) Charge $\delta Q_{+}$ as well as components of the spin (b) $\delta S_{+}^x$ along the $x$ and (c) $\delta S_{+}^y$ along $y$ axis of the MBSs induced by electron-phonon interaction as a function of number of sites $N$ found numerically in the spinful Rashba lattice model. The $z$ component of the spin, $S_{\pm }^z$, remains zero due to a symmetry of the system [58]. A finite energy splitting $|2\delta {\varepsilon }_{+}|$ (blue solid line [see also Fig. 8]) gives rise to a finite charge and spin of the MBSs. Both charge and spin oscillate with increasing the system length $N$, showing resonant enhancement similar to the energy splitting [cf. Figs. 5 and 8]. The parameters chosen are $(E_{so},{\mathrm{\Delta }}_{sc},{\mathrm{\Delta }}_Z,{\mathrm{\Omega }}_1)=(0.6,0.2,0.4,0.22)\mathrm{meV}$, corresponding to $t=10\mathrm{meV}$, coherence length $\xi =50a_0$. The rest of the parameters are the same as for Fig. 5.

Figure 11

Charge density $\delta {\rho }_{+}\left(x\right)$ of the MBSs induced by the electron-phonon interaction as a function of lattice site $j$ (a) close to the resonance, $N=640$ [see Fig. 10], and (b) away from the resonance, $N=500$. While in the nonresonant case (b), the charge density changes its sign and oscillates on the characteristic scale $c_s/\mathrm{\Delta }$ corresponding to the phonon wavelength, in case of the resonance (a), the total charge on the negatively charged sites (for the given values of parameters) is larger than the one on the positively charged site such that the overall charge carried by the MBSs is nonzero. The parameters chosen are the same as for Fig. 10.

Figure 12

Phonon density of states (DOS) corresponding to the phonon dispersion in Eq. (D1) with $c_t/c_l=0.5$ at fixed energy $\mathrm{\Delta }=1\mathrm{meV}$ (a) as a function of the nanowire radius $R$ and (b) as a function the phonon energy $\hslash \mathrm{\Omega }$ and $R$. The pronounced peaks correspond to VHSs, which have energies inversely proportional to $R$. The bottom of phonon branch $\hslash \mathrm{\Omega }=0.2\mathrm{meV}$ (dashed horizontal line) corresponds to $R=100a=50\mathrm{nm}$.