Quantum phases of a one-dimensional Majorana-Bose-Hubbard model

Majorana zero modes (MZM-s) occurring at the edges of a one-dimensional (1D), $p$-wave, spinless superconductor, in the absence of fluctuations of the phase of the superconducting order parameter, are quintessential examples of topologically protected zero-energy modes occurring at the edges of 1D symmetry-protected topological phases. In this work, we numerically investigate the fate of the topological phase in the presence of phase fluctuations using the density matrix renormalization group (DMRG) technique. To that end, we consider a one-dimensional array of MZM-s on mesoscopic superconducting islands at zero temperature. Cooper-pair and MZM-assisted single-electron tunneling, together with finite charging energy of the mesoscopic islands, give rise to a rich phase diagram of this model. We show that the system can be in either a Mott-insulating phase, a Luttinger liquid (LL) phase of Cooper pairs, or a second gapless phase. In contrast to the LL of Cooper pairs, this second phase is characterized by nonlocal string correlation functions which decay algebraically due to gapless charge-$e$ excitations. The three phases are separated from each other by phase transitions of either Kosterlitz-Thouless or Ising type. Using a Jordan-Wigner transformation, we map the system to a generalized Bose-Hubbard model with two types of hopping and use DMRG to analyze the different phases and the phase transitions.

Figure 1

Schematic of a 1D array of mesoscopic superconducting islands (denoted by yellow boxes) with MZM-s (denoted by red dots). In addition to the rotor degrees of freedom, ${\varphi }_i,n_i$, each island has two MZM-s, denoted by ${\gamma }_i^a,{\gamma }_i^b$.

Figure 2

Phase diagram of the MBH model for $E_M=0$ (left panel) and $E_J=0$ (right panel) obtained using DMRG. The system is in a MI phase within the lobes, while outside the system is in a charge-$2e$ (charge-$e$) LL phase, as shown in white (gray) in the left (right) panel. In the MI phase, the average occupation on each island is pinned to integer (half-integer) value for $E_M\left(E_J\right)=0$ as shown in the left (right) panel. The phase transition at constant density is of KT type, where the LL-parameter $K_{2\left(1\right)}=1/2$ for $E_{M\left(J\right)}=0$, as shown by the blue dot (square). The phase transition along the sides of the lobe is accompanied by a change in the density. The LL-parameter for this phase transition is given by $K_{2\left(1\right)}=1$ for $E_{M\left(J\right)}=0$ [the maroon (green) dots (squares)]. The characteristics of the different lobes on each of the panels are identical. We have only annotated the bottom-most lobes in the figure for brevity.

Figure 3

The phase diagram of the MBH model for nonzero $E_J/E_C,E_M/E_C$. We chose $E_g=\rho =0$. The charge-$2e$ LL phase and the MI phase are separated by a KT transition (blue squares), where $K_2=1/2$. This KT transition line meets an Ising transition line (violet-red stars), which separates the two LL phases. This Ising line continues beyond the KT line and separates part of the MI and the charge-$e$ LL phases. Finally, this line turns into a KT line (blue dots). At the latter KT transition, $K_1=1/2$. Inset: Zoom in on the region where the three phases meet.

Figure 4

Qualitatively different phase transitions the system undergoes to enter the charge-$e$ LL phase from the MI and the charge-$2e$ LL phase. We chose $\rho =0$. [(a), (c), (e)] Transition from the MI to the charge-$e$ LL. We chose $E_J/E_C\simeq 0.48$. (a) The system is a MI inside the lobe and a charge-$e$ LL everywhere else. Both charge-$e$ (dark red dots) and charge-$2e$ (green squares) excitation gaps are computed to obtain the gate voltages at the boundary of the lobe. For any nonzero $E_M/E_C$, both the charge-$e$ and charge-$2e$ spectra are gapless simultaneously at the sides and the tip of the lobe. (c) Zoom in at the tip of the lobe (blue dot). The nature of the phase transition is of KT type, where $K_1$ (dark red diamonds) $=1/2$ (blue line). (e) Scaling of entanglement entropy ($S$) vs. log of correlation length ($\xi$) in the charge-$e$ LL phase used to extract the central charge ($c$). [(b), (d), and (f)] Transition from the charge-$2e$ LL to the charge-$e$ LL. We chose $E_J/E_C=2$. (b) The system is a charge-$2e$ LL inside the white lobe, while being a charge-$e$ LL everywhere else. (d) Zoom in around the tip of the lobe. The Luttinger parameter $K_2$ is below 1/2, while $K_1$ abruptly drops from $>10$ in the charge-$2e$ LL phase (not shown) to below 1/2. (f) Scaling of $S$ vs. $ln\xi$ used to confirm the nature of the transition at the lobe-tip (in violet-red) contrasted with the same outside the lobe (in purple). The extra 1/2 in the central charge at the tip of lobe indicates the additional gapless Ising degree of freedom on top of the gapless $U\left(1\right)$.

Figure 5

[(a)–(c)] Characteristic decay of $S_b$, $S_{\mathrm{triv}}$, and $S_{\mathrm{top}}$ in the MI, charge-$2e$ LL, and the charge-$e$ LL phases (see Table 1) using iDMRG technique. We chose $E_J/E_C=0,E_M/E_C=0.02$ for the MI phase, $E_J/E_C=0$, $E_M/E_C=0.6$ for the charge-$e$ LL phase, and $E_J/E_C=1$, $E_M/E_C=2\times {10}^{-4}$ in the charge-$2e$ LL phase. (a) In the MI phase, $S_{\mathrm{triv}}$ is constant and nonzero, while both $S_{\mathrm{top}},S_b$ are exponentially decaying. (b) In the charge-$2e$ LL phase, $S_{\mathrm{triv}}$ is constant and nonzero, while $S_{\mathrm{top}}$ is exponentially decaying. On the other hand, $S_b$ decays algebraically. The extracted Luttinger parameter is $K_2\simeq 0.41$. (c) In the charge-$e$ LL phase, all three correlators decay algebraically. The extracted Luttinger parameters are $K_1\simeq 0.36$, $K_2\simeq 1.43$, and $K^{\prime }\simeq 2.82$. (d) Variation of the entanglement entropy ($S$) with the log of the correlation length ($\xi$) in the charge-$e,2e$ LL phases. Both phases show the characteristic linear dependence with the extracted central charge $c=1$ (with precision to the second decimal place).

Figure 6

Verification of the relative magnitude of the Luttinger parameters deep in the charge-$e$ LL phase for $E_J=\rho =0$ using iDMRG technique. The values of the $E_M/E_C$ are shown on the top $x$ axis. The green diamonds (red triangles) are the evaluated values of $K_2$ ($1/K^{\prime }$) as a function of the evaluated values of $K_1$. The dashed lines are linear fits, which confirm the relations between the different Luttinger parameters: $K_2/K_1=4$, $K^{\prime }=1/K_1$.

Figure 7

Schematic of a 1D array of mesoscopic superconducting islands (denoted by yellow boxes) with MZM-s (denoted by red dots). In addition to the rotor degrees of freedom, ${\varphi }_i,n_i$, each island has four MZM-s, denoted by ${\gamma }_i^a$, ${\gamma }_i^b$, ${\gamma }_i^c$, and ${\gamma }_i^d$.

Figure 8

iDMRG results for obtaining the location of the tip of the Mott lobe for $E_J=\rho =0$. (a) Variation of the Luttinger parameter, $K_1$ (dark red diamonds), with the $E_M/E_C$. The phase transition point is when $K_1$ crosses $1/2$ (blue horizontal line). The location of the transition is obtained by linear interpolation between the two points closest to $1/2$. (b) Variation of $S_{\mathrm{top}}(i,j)$ with $|i-j|$ in the charge-$e$ LL phase. To accentuate the difference between the curves, we have added (subtracted) 0.1 to the $y$ axis values for the indigo (maroon) curves. The slope of the linear fit gives the Luttinger parameter $K_1$. (c) Variation of entanglement entropy ($S$) with the log of the correlation length ($\xi$). As expected for a critical system, $S$ scales linearly with $ln\xi$, with the slope providing the central charge $c$. We have again added (subtracted) 0.1 to the $y$ axis values for the indigo (maroon) curves to make them distinct on this scale.