Single-site dissipation stabilizes a superconducting nonequilibrium steady state in a strongly correlated lattice

TL;DR

The paper addresses stabilizing superconducting order in a strongly correlated lattice under open-system dynamics by introducing a minimal, strictly local dissipative seed. It develops a rotated local jump protocol for the particle–hole symmetric Hubbard model, yielding a nonequilibrium η-paired steady state with off-diagonal long-range order through a projected Liouvillian anchored in the η-multiplet, with a dissipative gap Δ that governs relaxation. The authors demonstrate, both analytically and numerically, that a single-site dissipative seed can phase-lock η pseudospins across the lattice, producing ⟨η_i^+⟩_ss=1/2 and ⟨η_i^+ η_j^-⟩_ss=1/4 in the steady state, and they map out the regime of disorder robustness against various static perturbations. They identify perturbations that destroy ODLRO (e.g., onsite potential disorder, transverse fields, angle-disorder, and pair-breaking loss) and those that mostly renormalize dynamics without erasing the order, showing a disorder-tolerant route to stabilizing superconductivity as a nonequilibrium attractor via minimal local quantum-jump control. The results imply a scalable, experimentally accessible path to long-range quantum coherence in bipartite lattices and motivate extensions to higher dimensions and time-dependent pumping schemes.

Abstract

Can superconducting order be made a robust attractor of open-system dynamics in strongly correlated lattices? We demonstrate that it can by proposing a minimal dissipation-engineering protocol for the particle--hole symmetric Hubbard model. By applying a rotated quantum jump operator, a locally transformed $η$-pair lowering operator, on as little as a single lattice site, we show that the Lindblad evolution autonomously pumps the system from the vacuum into a nonequilibrium steady state (NESS) with macroscopic $η$-pair off-diagonal long-range order (ODLRO). Crucially, this local-to-global synchronization contrasts with schemes requiring spatially extensive reservoirs: here, a strictly local dissipative seed suffices to establish coherence across the interacting system. We elucidate the mechanism via local dark-state selection, controlled elimination of off-manifold excursions induced by hopping, and a Liouvillian invariant-subspace structure that yields an attractive fixed point with a finite dissipative gap. Furthermore, we classify the stability of this NESS against static disorder, identifying a broad regime where the superconducting attractor is resilient to Hamiltonian perturbations that leave the effective subspace structure intact, while pinpointing specific perturbations that directly dephase the $η$-pseudospin coherence and suppress ODLRO. Our results establish a disorder-tolerant route to stabilizing superconducting order as a non-thermal attractor via minimal local quantum-jump control.

Figures (11)

• Figure 1: Schematic of the nonequilibrium phase-locking protocol that stabilizes an $\eta$-paired superconducting NESS. Starting from a strongly correlated Hubbard system (with optional disorder in the interaction $U_i$, onsite potential $\mu_i$, and/or bond hopping $t_{ij}$), we apply local rotated dissipation on selected sites with jump operators $L_j=\eta_j^- e^{i\frac{\pi}{2}\eta_j^x}$. Unlike the unrotated channel $\tilde{L}_j=\eta_j^-$, which simply annihilates pairs and leaves the vacuum as the local dark state, the rotation tilts the dissipative measurement axis so that the locally protected state is the $+y$$\eta$-pseudospin eigenstate, $|\eta_j^{y}=+1/2\rangle \propto e^{-i\frac{\pi}{2}\eta_j^x}|0\rangle_j$, i.e., a coherent holon--doublon superposition. Each quantum jump then acts like a small compass correction: it damps misaligned components while repumping transverse $\eta$ coherence. In the presence of strong correlations, this local axis selection propagates through the many-body dynamics, effectively phase-locking the $\eta$ pseudospins across the lattice. At long times the system relaxes to a unique NESS with $\eta$ pseudospins aligned along the $+y$ direction, yielding a finite $\eta$-pair amplitude and ODLRO.
• Figure 2: Real-time buildup of $\eta$-pair coherence under open boundary conditions (OBC) for $N=4$ at $t=\gamma$, $U=8/\gamma$, and the rotation angle $\theta=\pi/2$. Top panels: site-resolved one-point amplitude $|\Phi_i(\tau)|=|\langle\eta_i^+(t)\rangle|$ for (a1) a single driven site $\mathcal{J}=\{1\}$ and (a2) two driven sites $\mathcal{J}=\{1,4\}$; the dashed line indicates the projected benchmark $|\Phi_i|=1/2$. Bottom panels: real-space profile of the two-point correlator magnitude $|C(r,\tau)|=|\langle \eta_i^+(t)\eta_{i+r}^-(t)\rangle|$ for multiple separations $r$ (here $r=1,2,3$), shown for (b1) $\mathcal{J}=\{1\}$ and (b2) $\mathcal{J}=\{1,4\}$; the dashed line marks the predicted ODLRO plateau $|C|=1/4$. At intermediate times short-range correlations develop first, while at late times the correlator becomes nearly independent of $r$ and converges to an $r$-independent plateau $|C(r,t\!\to\!\infty)|\simeq 1/4$ (up to finite-size effects), providing direct evidence for ODLRO according to Eq. \ref{['eq:ODLRO_condition']}.
• Figure 3: Finite-size scaling of global $\eta$-pair order parameters for dissipation applied only on the first site $\mathcal{J}=\{1\}$. (a) Global $\eta$-pair amplitude $|\Phi(\tau)|$. (b) $\eta$-pair structure factor $|S_{\eta}(\tau)|$. Curves are shown for different system sizes (as labeled). Both observables rapidly approach steady plateaus that are essentially insensitive to $N$, indicating that the observed macroscopic order is not a finite-size artifact. In particular, the saturation of $|S_{\eta}(\tau)|$ to a finite value, consistent with the real-space long-distance ODLRO limit, provides a complementary, bulk-sensitive confirmation of the dissipatively stabilized $\eta$-paired NESS. Parameters: 1D chain, $t=\gamma$, $U=4/\gamma$, and $\theta=\pi/2$.
• Figure 4: Robustness against interaction disorder $U_i=U+\delta U_i$ as a function of $W_U/\gamma$. (a) $|\Phi_m|_{\mathrm{ss}}$ and (b) $|C_m(r\!\to\!N/2)|_{\mathrm{ss}}$. Points show disorder averages over $100$ realizations; dashed lines mark the projected benchmarks $1/2$ and $1/4$. Parameters: 1D chain with PBC, $t=\gamma$, $U=8/\gamma$, $\theta=\pi/2$, $N=5$, and dissipation applied to site $1$. Both correlators remain close to their projected values over the explored range of $W_U/\gamma$, indicating that the $\eta$-paired NESS is weakly affected by interaction inhomogeneity.
• Figure 5: Robustness against dissipation-strength disorder $\gamma_i=\gamma+\delta\gamma_i$ as a function of $W_{\gamma}/\gamma$. (a) $|\Phi_m|_{\mathrm{ss}}$, and (b) $|C_m(r\!\to\!N/2)|_{\mathrm{ss}}$. Points show disorder averages over $100$ realizations; dashed lines mark the projected benchmarks $1/2$ and $1/4$. Parameters: 1D chain with PBC, $t=\gamma$, $U=8/\gamma$, $\theta=\pi/2$, $N=5$, and dissipation applied to site $1$. Both correlators remain close to their projected values over a broad range of $W_{\gamma}/\gamma$, indicating that dissipative inhomogeneity weakly renormalizes the projected generator $\mathcal{L}_{\mathrm{eff}}^{(0)}$. Error bars denote the standard error of the mean.