Non-Equilibrium Phase Transition in a Boundary-Driven Dissipative Fermionic Chain

TL;DR

The paper investigates a boundary-driven open quantum system of two coupled Kitaev chains subjected to a boundary Floquet drive, showing that localized driving can induce non-equilibrium long-range order in a bulk that is gapped in the absence of drive. A rotating-frame transformation yields a time-independent effective Hamiltonian, enabling the use of static methods (third-quantization) to obtain the Floquet-Lindblad steady state and its correlation matrix. The key mechanism is a resonance where drive frequency bridges bulk energy gaps, allowing boundary-injected particles and holes to propagate and generate bulk correlations, with the long-range order scaling as $\chi \propto \gamma^2$ when the drive bridges a particle-hole gap. These results demonstrate that localized coherent boundary control can generate macroscopic order in open quantum systems, with potential applications to quantum information transfer and state preparation in driven-dissipative platforms.

Abstract

We demonstrate that a boundary-localized periodic (Floquet) drive can induce nontrivial long-range correlations in a non-interacting fermionic chain which is additionally subject to boundary dissipation. Surprisingly, we find that this phenomenon occurs even when the corresponding isolated bulk is in a trivial gapped phase with exponentially decaying correlations. We argue that this boundary-drive induced non-equilibrium transition (as witnessed through the correlation matrix) is driven by a resonance mechanism whereby the drive frequency bridges bulk energy gaps, allowing boundary-injected particles and holes to propagate and mediate long-range correlations into the bulk. We also numerically establish that when the drive bridges a particle-hole gap, the induced long-range order scales as a power law with the bulk pairing potential ($χ\sim γ^2$). Our results highlight the potential of localized coherent driving for generating macroscopic order in open quantum systems.

Figures (8)

• Figure 1: Schematic representation of the model: two fermionic chains (labeled by $\sigma = \uparrow, \downarrow$), each with hopping, pairing, and uniform on-site energy ($t_\sigma, \gamma_\sigma, h_\sigma$) are coupled at the first site by a monochromatic drive $H_D(t)$ with frequency $\omega$. The system is connected to Markovian baths at its boundaries (sites $j=1$ and $j=L$) with gain and loss rates given by $\Gamma_{j,\sigma,g}$ and $\Gamma_{j,\sigma,l}$, respectively.
• Figure 2: Spatial structure of the steady-state correlation matrix elements for varying system sizes and driving frequencies. The axes correspond to indices $j$ (rows, top-down) and $k$ (columns, left-right). The colors are proportional to $\log |C_{jk}|$, which range from $\log 10^{-16}$ (dark blue) to $\log 1$ (bright yellow). The grid consists of three rows corresponding to system sizes $L = 64, 256, 512$ (top to bottom) and nine columns corresponding to driving frequencies $\omega = 2, 4, 6, 8, 10, 12, 14, 16, 18$ (left to right). The system is driven with amplitude $F=3$, and the static parameters are fixed at $\gamma=0.5$ and $h=3$ (deep in the trivial regime). Resonant driving frequencies (e.g., specific columns) exhibit non-trivial long-range correlations connecting the boundaries, distinct from the short-range behavior observed at off-resonant frequencies.
• Figure 3: Phase diagram of the long-range correlation index $\chi$ (log scale) as a function of driving strength $F$ and frequency $\omega$. Parameters: $\gamma=0.5, h=3, L=512$. Red regions indicate phases with drive-induced long-range correlations. The distinct domains correspond to specific resonance conditions: $\omega \in [0,4]$ (intra-band) and $\omega \in [8,16]$ (inter-band).
• Figure 4: Detailed analysis of the critical resonance transition at $\omega_c = 4$ ($F=3, \gamma=0.5, h=3$). (a) Spatial profile of correlations $|C(r)|$ for a fixed system size $L=512$ while scanning the driving frequency $\omega$ across the intra-band resonance cutoff. The critical point at $\omega=4$ (black line) exhibits algebraic decay, while slight detuning (red and blue gradients indicate $\omega < 4$ and $\omega > 4$ with $|\Delta\omega| = 2^{-n}$) results in the correlated and trivial phases, respectively. (b) Finite-size scaling analysis at the critical frequency $\omega=4$. The main plot shows $|C(r)|$ (with $r=j-k$ and $j+k=L$) for system sizes $L=64, 128, 256, 512$, following a power-law decay $\sim r^{-2}$ (dashed line). The inset demonstrates the data collapse of the scaled correlations $\log(C L^\nu)$ versus $r/L$ (with $\nu=2.0$), confirming the scaling hypothesis.
• Figure 5: Scaling of the correlation index $\chi$ with the pairing potential $\gamma$ on a log-log scale. The parameters are fixed at $h=2$, $F=5$, and $\omega=6$, placing the system in the inter-band resonance regime. (The system size is $L=512$.) The linear fit indicates a power-law dependence $\chi \propto \gamma^2$, confirming that finite pairing is essential for the drive-induced resonance mechanism.