Pairing-induced phase transition in the non-reciprocal Kitaev chain

Abstract

Investigating the robustness of non-reciprocity in the presence of competing interactions is central to understanding non-reciprocal quantum matter. In this work, we use reservoir engineering to induce non-reciprocal hopping and pairing in the fermionic Kitaev chain, and reveal the emergence of a pairing-induced phase transition. The two phases appear in the spectrum of the non-Hermitian Kitaev Hamiltonian describing the dynamics of correlations, separated by an exceptional point. In the non-reciprocal phase, dynamics are characterized by directionality and slow relaxation, and the steady state supports non-reciprocal density and spatial correlations. At strong pairing, we uncover an unexpected density wave phase, featuring short relaxation times, a modulation in particle occupation and strikingly different correlation spreading depending on pairing non-reciprocity. Our work highlights the non-trivial breakdown of non-reciprocity due to superconducting pairing and invites experimental investigation of non-reciprocal fermionic systems.

Figures (10)

• Figure 1: Schematic depiction of the model. Each site can host at most one spinless fermion, with an energy cost corresponding to the chemical potential $\mu$. Neighboring sites are coupled coherently through a hopping term $w$ and a pairing interaction $\Delta$ which breaks $U(1)$ symmetry [Eq. (\ref{['Eq:H']})]. The dissipative part of the model gives rise to incoherent hopping and pairing terms with rates $\Gamma_h$ and $\Gamma_p$, and phases $\theta_h$ and $\theta_p$, respectively [Eq. (\ref{['Eq:L']})]. These interfere with the analogous coherent processes, generating non-reciprocity.
• Figure 2: Real and imaginary part of the OBC spectrum, red and blue curves, for coherent (a) and non-reciprocal pairing (b) as a function of $\Delta$ for $\Gamma_h = 2w$, $\theta_h = \frac{\pi}{2}$ and $\mu = 0$; system size is $N = 100$. (a) For coherent pairing the spectrum shows a transition, marked by a gap opening in the real part. The gapped phase hosts two eigenvalues with $\Re[E_n]=0$, reminiscent of the Majorana zero modes. (b) For non-reciprocal pairing the complex spectrum is gapless. However, an exceptional point appears at $\Delta = w$.
• Figure 3: (a)-(d) Density dynamics in the non-reciprocal Kitaev chain. As $\Delta <\Delta_c = w$ ($\Delta = 0.1w$ is shown here) both the purely coherent ($\Gamma_p = 0$) and non-reciprocal ($\Gamma_p = 2\Delta$) pairing cases show a clear unidirectional lightcone. Non-reciprocity is further highlighted by the relaxation to the steady state shown in the insets. At $\Delta>\Delta_c$ ($\Delta = 10w$ is shown here) a density wave pattern emerges. Crucially, when $\Gamma_p = 2\Delta$ the density wave is completely non-reciprocal and spreads to the right only. (e)-(f) Relaxation time $\tau$ as a function of $\Delta$ for different system sizes $N$. Both for $\Gamma_p = 0$ (e) and for non-reciprocal pairing (f) the relaxation time decays as a fast power-law in the non-reciprocal phase $\Delta<w$ (light blue dashed lines). In the density wave phase, the power-law drastically changes exponent, showing a very slow decay (red dashed lines).
• Figure 4: (a)-(b) Steady state density as a function of pairing amplitude $\Delta$. In the non-reciprocal phase $\langle\hat{n}_j\rangle_{ss}$ is inhomogeneous up to a characteristic lengthscale $\xi_{NR}$, while in the density wave phase it acquires a modulation penetrating from the boundaries up to a lengthscale $\xi_{DW}$. (c)-(d) Central site correlation functions decay asymmetrically in the non-reciprocal phase and depend strongly on $\Gamma_p$ in the density wave phase. (e) Characteristic lenghtscales $\xi_{NR}$ and $\xi_{DW}$ have opposite behavior across the transition: $\xi_{NR}$ decays as a power-law and vanishes at the critical point, while $\xi_{DW}$ increases linearly in the density wave phase. (f) Correlation functions decay length $\zeta$. Deep in the non-reciprocal phase, decay is asymmetric, $\zeta_R\neq \zeta_L$, while in the density wave phase $\zeta$ increases for $\Gamma_p =0$ and it tends to saturate for non-reciprocal pairing.
• Figure 5: Density dynamics $\langle\hat{n}_j(t)\rangle$ under PBC show a transition similar to the one presented in the main text. (a)-(b) At weak pairing, particles spread only towards the right in a unidirectional lightcone. (c)-(d) Above the critical point $\langle\hat{n}_j(t)\rangle$ shows the typical density wave pattern, which becomes non-reciprocal upon making pairing non-reciprocal. Notice that the steady state density is completely homogeneous, hence these features concern the transient dynamics.