First-order topological phase transitions and disorder-induced Majorana modes in interacting fermion chains

Using a combination of the mean-field Bogoliubov–de Gennes approach and the density matrix renormalization group method, we discover first-order topological transitions between topological superconducting and trivial insulating phases in a sawtooth lattice of intersite attractive fermions. The topological characterization of different phases is achieved in terms of winding numbers, Majorana edge modes, and entanglement spectra. By studying the effect of disorder on first-order topological phase transitions, we establish disorder-induced topological phase coexistence as a mechanism for generating a finite density of Majorana particles.

Figure 1

Schematic representation of the sawtooth lattice model. $t,t^{\prime }$ indicate the hopping amplitudes and $V,V^{\prime }$ the attractive interactions between electrons on neighboring sites. Inequivalent sites are labeled $A$ and $B$.

Figure 2

(a)–(d) Zero-temperature variations of ${\mathrm{\Delta }}_1, {\mathrm{\Delta }}_2, \delta n$, and $n$ with chemical potential for $V=V^{\prime }=1$ and for (a) $t=0.2$, (b) $t=0.4$, (c) $t=1.0$, (d) $t=2.0$. (e) Ground state phase diagram in the $t\text{−}\mu$ plane constructed from the data similar to that shown in (a)–(d). (f) Same as (e), for $V=3$ and $V^{\prime }=1$. Horizontal dashed lines in (e) indicate the scans for which the data in (a)–(d) are displayed. Dashed lines at $t=2$ in (e) and at $t=1$ in (f) also represent the scans for which the DMRG results of Fig. 3 are obtained. (g) Spectral gap, charge modulation, and winding numbers as a function of $\mu$ for $t^{\prime }=V=V^{\prime }=1$ and $t=0.2$.

Figure 3

(a) Inverse exponent of the density-density and pair-pair correlation functions as well as the Tomonaga-Luttinger parameter as a function of density per site for $t=2.0$ and $t^{\prime }=V=V^{\prime }=1.0$. The corresponding phase diagram is also shown on the top. (b) Plot similar to (a) for $t=t^{\prime }=V^{\prime }=1.0$ and $V=3.0$. (c) Edge component of the single-particle correlation function for $t=2.0, t^{\prime }=V=V^{\prime }=1.0$ (red) and $t=t^{\prime }=V^{\prime }=1.0, V=3.0$ (blue) with $N=201$ open cluster. The open and solid circles denote the correlations between edged $A$ sites and between edged $B$ sites, respectively. (d) Entanglement spectra as a function of density per site for $t=2.0, t^{\prime }=V=V^{\prime }=1.0$ with $N=60$ periodic cluster.

Figure 4

${\mathrm{\Delta }}_{i,1}$ and zero-energy LDOS as a function of site index $i$ for (a) $D=0.4, n=0.46$ and (b) $D=0.8, n=0.435$. The other parameter values are $V=3, V^{\prime }=1, t=1, \mu =-2.93$. Note that the ${\mathrm{\Delta }}_{i,1}$ values are scaled up in order to emphasize correlations with the LDOS data.