Heating-Induced Long-Range $\eta$ Pairing in the Hubbard Model

We show how, upon heating the spin degrees of freedom of the Hubbard model to infinite temperature, the symmetries of the system allow the creation of steady states with long-range correlations between $\eta$ pairs. We induce this heating with either dissipation or periodic driving and evolve the system towards a nonequilibrium steady state, a process which melts all spin order in the system. The steady state is identical in both cases and displays distance-invariant off-diagonal $\eta$ correlations. These correlations were first recognized in the superconducting eigenstates described in Yang’s seminal Letter [Phys. Rev. Lett. 63, 2144 (1989)], which are a subset of our steady states. We show that our results are a consequence of symmetry properties and entirely independent of the microscopic details of the model and the heating mechanism.

Figure 1

(a),(b) Evolution of the correlator $⟨{\eta }_i^{+}{\eta }_{i+j}^{-}⟩$, averaged over all sites separated by a distance $j$, for the 10 site Hubbard model at half filling and $⟨S^z⟩=0$. The system is initialized in the ground state of $H$ for $U=4.0\tau$ and then evolved with a system-environment coupling of $\gamma =2.0\tau$ for (a) and $\gamma =0.0$ for (b). In both quenches the interaction strength is changed to $U=\tau$. (c),(d) Matrix of correlations of $|⟨{\eta }_i^{+}{\eta }_j^{-}⟩|$ associated with the density matrix at $t\tau =20.0$ for the simulations in (a) and (b), respectively.

Figure 2

Projections, ${\rho }^{\prime }={\mathcal{P}}_{s,d}\rho {\mathcal{P}}_{s,d}$, of the density matrix in the spin ($s$) and $\eta$ ($d$) sectors of the $M=6$ Hubbard model. The system is at symmetric half filling, $⟨N_{\uparrow }⟩=⟨N_{\downarrow }⟩=M/2$, and the projections have been renormalized [$\mathrm{Tr}({\rho }^{\prime })=1$]. The color indicates the magnitude of the matrix elements.. Following the projection, the indices $i$ and $j$ run over the remaining basis states in lexicographic order when they are converted to binary strings, where $\uparrow =1$, $\downarrow =0$ and $\uparrow \downarrow =1$, $\mathrm{vac}=0$ for the projections in $s$ and $d$, respectively. As an example, for plots (a) and (c), when $i=1$, this corresponds to the basis vector $|\uparrow \uparrow \uparrow \downarrow \downarrow \downarrow ⟩=|111000⟩$, and when $i=20$, $|\downarrow \downarrow \downarrow \uparrow \uparrow \uparrow ⟩=|000111⟩$. (a),(b) Projections for the thermal state of the $U=2.5\tau$ Hubbard model: $\rho \propto \exp (-\beta H)$ with $\beta =0.8/\tau$. (c),(d) Projections following spin dephasing of this initial state by the map in Eq. (4) in the long-time limit with quench parameters $U=\tau$ and $\gamma =2.0\tau$.

Figure 3

Dynamics of the averaged $\eta$ correlations $|⟨{\eta }_i^{+}{\eta }_{i+j}^{-}⟩|$ following a quench from the $U=\tau$ ground state of the $M=8$ half-filled Hubbard model with $⟨S^z⟩=0$. The blue curves reflect a quench with the periodically driven Hamiltonian in Eq. (7), where $U=6.0\tau$ and $\mathrm{\Omega }=\tau$. The green markers are for a quench under the dephasing map in Eq. (4) with $U=\tau$ and $\gamma =2.0\tau$. The dashed lines indicate the prediction from the grand-canonical ensemble in Eq. (6). Insets show the dependence of the correlations on distance, extracted for the blue curve at times $t\tau =0.0$ and $t\tau =60.0$. (a) $V=2.0U$ and $f(i)=i$. (b) $V=4.0U$ and $f(i)\in \mathrm{rand}[0,2.0]$, a uniform random number on the specified interval.