Phase diagram and entanglement of two interacting topological Kitaev chains

A superconducting wire described by a $p$-wave pairing and a Kitaev Hamiltonian exhibits Majorana fermions at its edges and is topologically protected by symmetry. We consider two Kitaev wires (chains) coupled by a Coulomb-type interaction and study the complete phase diagram using analytical and numerical techniques. A topological superconducting phase with four Majorana fermions occurs until moderate interactions between chains. For large interactions, both repulsive and attractive, by analogy with the Hubbard model, we identify Mott phases with Ising-type magnetic order. For repulsive interactions, the Ising antiferromagnetic order favors the occurrence of orbital currents spontaneously breaking time-reversal symmetry. By strongly varying the chemical potentials of the two chains, quantum phase transitions towards fully polarized (empty or full) fermionic chains occur. In the Kitaev model, the quantum critical point separating the topological superconducting phase and the polarized phase belongs to the universality class of the critical Ising model in two dimensions. When increasing the Coulomb interaction between chains, then we identify an additional phase corresponding to two critical Ising theories (or two chains of Majorana fermions). We confirm the existence of such a phase from exact mappings and from the concept of bipartite fluctuations. We show the existence of negative logarithmic corrections in the bipartite fluctuations, as a reminiscence of the quantum critical point in the Kitaev model. Other entanglement probes such as bipartite entropy and entanglement spectrum are also used to characterize the phase diagram. The limit of large interactions can be reached in an equivalent setup of ultracold atoms and Josephson junctions.

Figure 1

Sketch of the phase diagram of the interacting ladder at $\mathrm{\Delta }=t$ obtained with analytical (bosonization, exact mappings) and numerical methods [exact diagonalization (ED), density matrix renormalization group (DMRG)]. The chemical potentials of the two chains are taken to be equal. $4\text{MF}$ is the $\text{SPT}$ gapped phase presenting two Majorana fermions at each of the ladder extremities. $\text{MI-AF}$ and $\text{MI-F}$ are two gapped Mott phases, either antiferromagnetic or ferromagnetic. Polarized corresponds to a trivial phase with a quasiempty or quasifull ladder. In red, the gapless $\text{DCI}$ phase embodies an extension of the critical point at $g=0$. It acquires an extension of order $t$ as $g$ goes to $+\infty$.

Figure 2

Exact phase diagram of the Hubbard model at zero magnetic field, obtained from Bethe ansatz [52]. Charge M-H corresponds to the celebrated Mott-Heisenberg phase, opening at arbitrarily low $g>0$. The charge mode is gapped and the electronic density is fixed at half-filling, while the spin mode is free [and its effective model is a $\text{SU}\left(2\right)$-invariant Heisenberg model]. Spin M-H is its equivalent for $g<0$, inversing the role of the two sectors. Both of them are characterized by a central charge $c=1$. Luttinger liquid (LL phases) corresponds to two phases of free diluted electrons (with a central charge $c=2$). Polarized phases are trivial phases with totally empty or full wires.

Figure 3

Linear and logarithmic contributions to the bipartite charge fluctuations as a function of $\mathrm{\Delta }/t$ for a single Kitaev wire at the transition point, from theoretical computations and DMRG simulations [60, 61] with a 90-site wire. Linear contributions are easily extracted from the simulations and match very well the theoretical values. On the other hand, logarithmic contributions are more complex to extract and convergence is slower. We observe yet a good agreement for $\mathrm{\Delta }/t>0.3$. Below this value, finite-size effects cannot be neglected. Fits have been realized using the infinite wire form of the fluctuations, as, at long range, there is a precision loss due to the dominant linear term.

Figure 4

Linear and logarithmic contributions to the bipartite charge fluctuations as a function of $\mu /t$ for a single Kitaev wire across the transition point, at $\mathrm{\Delta }=t$, from theoretical computations and DMRG simulations with a 90-site wire. Linear contributions are easily extracted from the simulations and match very well the theoretical values. This coefficient becomes constant at $\mu =-2t$. Logarithmic contributions increase as we reach the transition, and are maximum at this point.

Figure 5

The first four levels of the energy spectrum on the line $\mu =0$ for $t=\mathrm{\Delta }$ and with open boundaries for a 60-site ladder. We observe a fourfold degeneracy in the topological phase, and a twofold degeneracy in the Ising phases.

Figure 6

The first five levels of the entanglement spectrum on the line $\mu =0$ for $t=\mathrm{\Delta }$ and periodic boundary conditions for a 60-site ladder. We observe a fourfold degeneracy in the topological phase, and a twofold degeneracy in the Ising phase due to the degeneracy of the ground state with PBC.

Figure 7

Effective pairing (in red) and its derivative (in blue) as a function of the interaction strength on the line $\mu =0$ for $\mathrm{\Delta }=t$ and periodic boundary conditions for a 60-site ladder.

Figure 8

Unraveling scheme to obtain the model far from half-filling. The initial fermions are split into two Majoranas. Each fermionic wire is then reorganized into a Majorana wire with an alternating hopping term. Finally, recombination of the Majorana wires into new fermions composed of a Majorana of each wire.

Figure 9

Linear regression of the width of the $\text{DCI}$ phase for $g=5t$ and $\mathrm{\Delta }=t$ based on the analysis of the energy spectrum. This width converges towards a finite nonzero value of order $0.06t$.

Figure 10

First four levels of the energy spectrum at $g=+\infty$ for a 68-site ladder at $\mathrm{\Delta }=t$ from DMRG with PBC. The visible double degeneracy of the central phase reveals the survival of the $\text{DCI}$ phase. Scaling analysis and entanglement entropy confirms its gaplessness.

Figure 11

Central charge obtained from entanglement entropy for a 68-site ladder at $\mathrm{\Delta }=t$ and $g=+\infty$ from DMRG with PBC, using the model given in Eq. (47). The results are in qualitative agreement with our theoretical prediction and the results at finite $g$ in Fig. 14.

Figure 12

First four levels of the energy spectrum of the periodic ladder for $g=5t$ and $\mathrm{\Delta }=t$, as a function of the chemical potential for a 76-site wire. On the left, the system is in the polarized phase, on the right it is in the $4\text{FM}$ phase. The double degeneracy of the ground state clearly reveals the $\text{DCI}$ phase. Symmetry analysis reveals that the parity of the ground states follow the stated rules [even-even in polarized, odd-odd in $4\text{FM}$ and (even-odd, odd-even) in $\text{DCI}$].

Figure 13

Entanglement spectrum of the periodic ladder for $g=5t$ and $\mathrm{\Delta }=t$, as a function of the chemical potential for a 90-site wire. On the left, the system is in the polarized phase, on the right it is in the $4\text{FM}$ phase. We observe a progressive lifting of the degeneracy from 4 in the $4\text{FM}$ to 2 in the $\text{DCI}$ and finally no degeneracy in the Polarized phase.

Figure 14

Central charge extracted from a fit of the entanglement entropy of the periodic ladder for $g=5t$ and $\mathrm{\Delta }=t=1$, as a function of the chemical potential for a 76-site wire.

Figure 15

Logarithmic contribution to the bipartite charge fluctuations in the first wire in the $\text{DCI}$ phase for a 90-site wire at $g=5t$ and $\mu =-3.85t$. The pic corresponds to the middle of this phase.

Figure 16

Fermionic density as a function of $\mu$ (in units of $t$) for the noninteracting Kitaev wire for different values of $\mathrm{\Delta }$ (in red, $\mathrm{\Delta }=2t$, in green $\mathrm{\Delta }=t$, in blue $\mathrm{\Delta }=0.5t$).

Figure 17

Effective pairing as a function of $\mu$ (in units of $t$) for noninteracting Kitaev wire for different values of $\mathrm{\Delta }$ (in red, $\mathrm{\Delta }=2t$, in green $\mathrm{\Delta }=t$, in blue $\mathrm{\Delta }=2t$). It serves as a good order parameter in the thermodynamic limit as it is constant in the topological phase.