Nonthermal symmetry-broken states in the strongly interacting Hubbard model

We study the time evolution of the antiferromagnetic order parameter after interaction quenches in the Hubbard model. Using the nonequilibrium dynamical mean-field formalism, we show that the system, after a quench from intermediate to strong interaction, is trapped in a nonthermal state which is reminiscent of a photodoped state and protected by the slow decay of doublons. If the effective doping of this state is low enough, it exhibits robust antiferromagnetic order, even if the system is highly excited and the thermal state is thus expected to be paramagnetic. We comment on the implication of our findings for the stability of nonthermal superconducting states.

Figure 1

Left panel: Antiferromagnetic phase diagram for the half-filled Hubbard model. The blue line with circles shows the exact DMFT result (from Ref. 26), the red line with open circles the NCA approximation, and the green line with stars the OCA approximation. Diamonds indicate $T_{\mathrm{eff}}\left(U\right)$ for interaction quenches from $U=4$, $T=0.1$. Right panel: NCA phase boundary as a function of filling at $U=8$. The star indicates the effective temperature of the doped Hubbard model with the same magnetization as in the trapped state (see text).

Figure 2

Top panel: Time evolution of the magnetization for quenches from $T=0.1$ and $U=4$ to $U=6$, 7, and 8. The effective temperatures after these quenches are $T_{\mathrm{eff}}=0.113$, $0.124$ and $T=0.732$, respectively, and the arrows indicate the corresponding thermal values of the magnetization. Inset: Same data on a different scale. Bottom panel: Time evolution of $A(\omega ,t)$ for the minority spin after a quench from $T=0.1$ and $U=4$ to $U=8$, and comparison to the thermal result (dashed). The pink curve ($t=18$) corresponds to the spectral function of the long-lived nonthermal state.

Figure 3

Time evolution of the double occupancy $\langle n_{\uparrow }{n}_{\downarrow }\rangle \left(t\right)$ for the quench from $U=4$, $T=0.1$ to $U=8$, 3, $2.5$, 2 (clockwise from the top left) and comparison to the thermal result (black arrow). The right panels show a narrow window around the thermal value and (as dashed line) an exponential fit to the long-time behavior. The left-pointing arrows in the left panels show the initial value of the double occupancy.

Figure 4

Comparison of the spectral function $A(\omega ,t=18)$ of the trapped state after the quench from $U=4$, $T=0.1$ to $U=8$ (bold red line) to the minority-spin spectral function of a half-filled Hubbard model with $T=0.0986$ and to an infinitesimally doped $t$-$J$ model with $J=4t^2/U=0.5$ (top panel). The blue curve in the bottom panel corresponds to a doped Hubbard model with $T=0.0917$ and chemical potential chosen such that the magnetization and double occupancy of the trapped state are reproduced. (The $t$-$J$ and doped Hubbard spectra are shifted by $+4.2$ and $+2.4$, respectively.)

Figure 5

Time evolution of $A^{<}(\omega ,t)$ for a quench from $U=4$, $T=0.1$ to $U=8$. While the occupied states rapidly accumulate at the lower band edge in the AFM calculation, the occupation is nearly time independent in the PM case. The curves are slightly broadened by introducing a Gaussian factor $e^{-\alpha {(t-t^{\prime })}^2}$ in the integrand of Eq. (5), which ensures a smooth cutoff of the upper integration limit.

Figure 6

Time evolution of the magnetization (top left panel) and double occupancy (top right panel) for quenches from indicated initial $U$, $T=0.1$ to $U=8$. The bottom panel shows the inverse of the relaxation time $\tau$ as a function of initial $U$.