Influence of topological degeneracy on the boundary Berezinskii-Kosterlitz-Thouless quantum phase transition of a dissipative resonant level

The interplay between a topological degeneracy and the residue degeneracy (also known as the residue entropy) of quantum criticality remains an important but not thoroughly understood topic. We find that this topological degeneracy, provided by a Majorana zero mode pair, relaxes the otherwise strictly requested symmetry requirement, to observe the boundary Berezinskii-Kosterlitz-Thouless (BKT) quantum phase transition (QPT) of a dissipative resonant level. Our work indicates that the topological degeneracy can be potentially viewed as an auxiliary symmetry that realizes a robust boundary QPT. The relaxation of the symmetry requirement extends the transition from a point to a finite area, thus greatly reducing the difficulty to experimentally observe the QPT. This topology-involved exotic BKT phase diagram, on the other hand, provides another piece of evidence that can further confirm the existence of a Majorana zero mode.

Figure 1

(a) The model of the system. The MZM (red dot) couples to the dot of the DRLM, with the amplitude $t_{\text{M}}$. The dot couples to two leads with corresponding amplitudes $V_L$ and $V_R$. The leads are dissipative with the total impedance $\mathcal{R}$. When $t_{\text{M}}=0$, the system reduces to the DRLM, with the zero-temperature equilibrium conductance in (b) (when $r=2.5$), as a function of dimensionless quantities $V_L{\tau }_c$ and $V_R{\tau }_c$. Here, ${\tau }_c$ is the cutoff in imaginary time. In real systems, ${\tau }_c\sim 1/D$, with $D$ the relevant bandwidth [34]. In (b), the conductance is determined by the fixed points [with four candidates in (c)] to which the system approaches [figured out by RG equations (2) and (3)] at zero temperature. The marked-out phases A–D are illustrated in (c), with red crosses indicating vanishing lead-dot communications. Of phase B, both dot-lead couplings are transparent, leading to a perfect equilibrium conductance $e^2/h$ at zero temperature. Otherwise (phases A, C, and D), the dot decouples with one or both leads. The corresponding zero-temperature equilibrium conductance vanishes.

Figure 2

Phase diagrams (indicated by the corresponding zero-temperature equilibrium conductance) and RG flow curves when $r=2.5$. (a and b) Phase diagrams with larger ($t_{\text{M}}=0.14T_0$) and smaller ($t_{\text{M}}=0.05T_0$) values of $t_{\text{M}}$, respectively. Here, $T_0$ is the initial temperature of RG flow (i.e., the temperature at which parameters in Hamiltonian are obtained) that requires experimental calibration. Similar to that in Fig. 1, the conductance is obtained after identifying the fixed point [A or B, cf. Fig. 1] to which the system flows at zero temperature. (c) High-temperature RG flow curves of $K_{\text{charge}}$ and $K_{\text{asy}}$, for the symmetric (red curves) and asymmetric (blue curves) situations. The red and blue diamonds mark points that belong to the ballistic-transmission ($K_{\text{charge}}<K_C$) and isolated-impurity ($K_{\text{charge}}>K_C$) phases, respectively, if high-temperature RG flow terminates at their corresponding $l$ ($l=0$ when RG initially begins). (d) Low-temperature RG flow of $K_{\text{charge}}$ and $K_{\text{asy}}$, with their initial values given by the corresponding diamond.

Figure 3

Lead-impurity tunneling events. (a) In a regular DRLM, the dot state alternates (between two degenerate dot states) in time. (b) In great contrast, with tunneling into the MZM degeneracy (the gray arrow), two neighboring dot-lead tunneling events (shown by the blue and red arrows) can both contain a creation operator in the dot.

Figure 4

The effect of left-right asymmetry (assuming $V_L<V_R$) in a regular (a and b) and Majorana-involved DRLM (c and d). The red and blue dots refer to the MZM and the residue degeneracy ($g=\sqrt{1+r}$) of the ballistic-transmission phase, respectively. The gray dot instead indicates the hybridized dot degree of freedom. RG flow prefers the fixed point with a smaller degeneracy.