From fractionally charged solitons to Majorana bound states in a one-dimensional interacting model

We consider one-dimensional topological insulators hosting fractionally charged midgap states in the presence and absence of induced superconductivity pairing. Under the protection of a discrete symmetry, relating positive and negative energy states, the solitonic midgap states remain pinned at zero energy when superconducting correlations are induced by proximity effect. When the superconducting pairing dominates the initial insulating gap, Majorana fermion phases develop for a class of insulators. As a concrete example, we study the Creutz model with induced $s$-wave superconductivity and repulsive Hubbard-type interactions. For a finite wire, without interactions, the solitonic modes originating from the nonsuperconducting model survive at zero energy, revealing a fourfold-degenerate ground state. However, interactions break the aforementioned discrete symmetry and completely remove this degeneracy, thereby producing a unique ground state which is characterized by a topological bulk invariant with respect to the product of fermion parity and bond inversion. In contrast, the Majorana edge modes are globally robust to interactions. Moreover, the parameter range for which a topological Majorana phase is stabilized expands when increasing the repulsive Hubbard interaction. The topological phase diagram of the interacting model is obtained using a combination of mean-field theory and density matrix renormalization group techniques.

Figure 1

Left: Creutz lattice model. The filled (empty) bullets represent spin-up (-down) states of ${\sigma }_3$. Each site encompasses a vertical bond, containing two spin states. The spin degeneracy is lifted by an onsite coupling $w$. There is a spin-conserving hopping term $t$, and a “spin-orbit” term $g$ that encodes the probability of electrons to spin flip during hopping to a different site. Right: Energy dispersion of the lattice Creutz model. One-dimensional Dirac cones can be realized at $k=0$ and $\pi$ for a parameter choice $w=\pm g$. Otherwise, the system is insulating with gapped Dirac cones as shown in the figure.

Figure 2

Phase diagram of the Creutz-Majorana model. The blue region represents the topological superconductor with zero-energy Majorana bound states ($\mathcal{M}=-1$). The gray and yellow regions represent phases with $\mathcal{M}=1$. The yellow region hosts chiral bound states at zero energy, even if it is trivial with respect to Majorana number $\mathcal{M}$.

Figure 3

Phase diagram of the noninteracting superconducting model. The black continuous lines represent the bulk-gap closing. The dashed line on the $x$ axis represents the Creutz model, with CBS at $|g/w|>1$ (in red) and a trivial gapped phase for $|g/w|<1$ (in black). The yellow phases contiguous to the red line represent the domain where the solitonic modes (CBS) survive in the superconducting region. In the bulk, the yellow region is an inversion symmetry-protected topological phase (SPT). The blue region represents the topologically nontrivial superconductor where Majorana bound states can develop. On the dashed green line, the superconducting pairing increases and produces a transition from the nontrivial insulating Creutz model, to a superconducting phase with CBS, and, subsequently, to the demise of CBS, and the formation of Majorana modes. This transition is investigated in a continuum model in Sec. 3b1.

Figure 4

Kink in the Dirac fermion mass at the interface. The topologically nontrivial “inverted” gap regime is realized for $x>0$.

Figure 5

Representation of the energy eigenstates of the BdG Hamiltonian near zero energy. Top: phase transitions at $\Delta /w=1,3$. Majorana modes are present when $\Delta /w\in (1,3)$. Bottom: removal of the fourfold degeneracy when an onsite scalar potential is added on the first and last sites. The wire length is $L=500a$.

Figure 6

Local expectation value of the chiral operator $\overline{\mathcal{S}}$ in the zero-energy eigenstates at the first 10 and the last 10 sites of a $L=500a$ wire. Two CBS, positive chiral eigenstates are localized at the right edge (red circles and green squares). The remaining two CBS, negative chiral eigenstates are localized at the left edge (blue triangles and magenta diamonds). The total “chiral density” obeys Eq. (26).

Figure 7

Different phases in the system under increasing pairing $\Delta$. The images stand for the possible ground states forming in the CM wires by studying the zero-energy states' occupation. The filled (empty) circles represent filled (empty) electronic states. (a) At zero pairing, i.e., in the Creutz model, there are fractionally charged bound states forming at the two ends of the wire. An electron filling a state at the left (right) edge has ${\sigma }_2$ projection spin $\frac{1}{2}$ ($-\frac{1}{2}$). (b) For finite pairing $\Delta$, but below the topological transition $\Delta <{\Delta }_{\mathrm{crit}}$, there are CBS zero modes. In the BdG there is a doubling of states (represented in gray) near zero energy. However, this redundant doubling does not modify the number of physical states, which remain four. The charge is no longer a good quantum number. (c) After crossing the transition to the Majorana phase ($\Delta >{\Delta }_{\mathrm{crit}}$), there are again just two states in the spectrum, two Majorana fermions forming at the two system edges. These states can be either filled or empty with a single electron. This is represented by having a fractionalized empty or filled electronic state at the two edges of the system.

Figure 8

Topological phase diagram of the Creutz-Majorana-Hubbard model at different interaction strengths. The dotted line represents transitions between topological phases obtained at the mean-field level, while the dots represent critical points from the DMRG calculations. The topological superconductor phase which can host Majorana modes (MBS), represented in the central region, increases under the effects of interactions. The region at large “spin orbit” $\left|g\right|$ and weak pairing $\left|\Delta \right|$ is a symmetry-protected topological phase. The rest of the outer regions are trivial superconductors with a unique ground state. The agreement between mean field and DMRG breaks down at very large $g$, for strong interactions.

Figure 9

Energy eigenvalues of a finite Creutz-Majorana model at three different interaction strengths. The interaction pushes the CBS located in the trivial superconducting region at finite energy, rendering the ground state unique. Model parameters are ${\Delta }^2/w^2=0.5$, $g^2/w^2=4$ and the wire length in $L=500a$.

Figure 10

DMRG results for the energy spectrum measured from the ground-state energy $E_0$ for $L=64a$ open wires. Top row: ${\Delta }^2=0.0625$, $g^2=1$. The double degeneracy in the topological phase with MBS survives the addition of interactions $U$. Bottom row: ${\Delta }^2=0.0625$, $g^2=20$. The fourfold energy degeneracy of the phase hosting CBS is removed by interactions $U$.

Figure 11

Phases found in the Creutz-Majorana-Hubbard model at fixed ${\Delta }^2=0.25$, from iDMRG results. By increasing $g$, we sweep through the trivial phase, the TSC, and an inversion symmetry-protected topological (SPT) phase. The bulk entanglement spectrum is obtained from two different cuts of the chain, shown in the diagrams in the right corners. The topological phase with Majorana edge states (blue) displays a twofold entanglement degeneracy for both kinds of cuts, unlike the phases surrounding it.

Figure 12

Top: Topological transitions are tracked by peaks in the correlation length $\xi$ at different interaction strengths. Bottom: Finite-entanglement scaling of iDMRG results of the phase transition at $U$=1, ${\Delta }^2=0.25$, and $g_c=1.80840\left(4\right)$. Left: Scaling of the entanglement entropy with the correlation length at $g_c$. Right: Scaling of the correlation length with the distance to the critical coupling $g_c$. Results are representative of all transitions observed and consistent with the Ising universality class.

Figure 13

Phases in the Creutz-Majorana-Hubbard model at ${\Delta }^2=0.25$ from iDMRG simulations. Top: $U$ = 0, the transition into the TSC phase is marked by a rise in particle-number fluctuations for simulations initialized at half filling $\langle n_i\rangle =1$. Bottom: $U$=$0.5$, repulsive interactions lower the average number of fermions in the system, thereby inducing an effective chemical potential, which lift the chiral bound states at large $g$.

Figure 14

Robustness of the topological phase to large interactions from iDMRG results. $g^2={\Delta }^2=1$. Top: The entanglement spectrum remains twofold degenerate. Bottom: The correlation length increases but remains finite.

Figure 15

Bulk spectral function $A(k,\omega )$ for $U=0$ (left column) and $U=1$ (right column). (a), (b) Trivial phase for $g^2=0.5$, ${\Delta }^2=5$. (c), (d) TSC phase with Majorana edge modes for $g^2={\Delta }^2=1$. (e) Symmetry-protected topological phase with CBS edge modes and (f) without them for $g^2=5,{\Delta }^2=0.5$. Color map is a TEBD calculation and dashed lines come from the mean-field theory.

Figure 16

Local spectral function $A_1\left(\omega \right)$ at the edge of an open wire for $U=0$ (left column) and $U=1$ (right column). (a), (b) The trivial phase at $g^2=0.5$ and ${\Delta }^2=5$ has no quasiparticle peak at zero energy. (c), (d) A Majorana edge fermion in the topological phase at $g^2={\Delta }^2=1$ is seen as a sharp peak at zero energy, which survives finite interactions. (e), (f) The superconducting phase at $g^2=5$ and ${\Delta }^2=0.5$. (e) The presence of zero-energy states is reflected in the spectral function peak at zero energy. (f) The CBS are not robust to interactions $U$, which removes them from zero energy, leading to a unique ground state. Each subfigure combines TEBD (red open points) with mean-field results (lines), scaled by an overall factor.

Figure 17

Energy gap for the interacting Creutz model at $U/t=1$ in mean-field theory (blue dots). Ferromagnetic order parameter $\eta$ in the $x$ direction (red crosses; in dimensionless units). Phase transitions are denoted by dashed green lines. The closing of the gap in the intermediary phase marks the metallic phase. The wire length is $L=500a$.

Figure 18

Finite-entanglement scaling of DMRG results for the interacting Creutz model at $g^2/w^2=2$ and $U/t=1$. The central charge $c\simeq 1$ matches that of a normal metallic phase.