Nonthermal Melting of Néel Order in the Hubbard Model

We study the unitary time evolution of antiferromagnetic order in the Hubbard model after a quench starting from the perfect Néel state. In this setup, which is well suited for experiments with cold atoms, one can distinguish fundamentally different pathways for melting of long-range order at weak and strong interaction. In the Mott insulating regime, melting of long-range order occurs due to the ultrafast transfer of energy from charge excitations to the spin background, while local magnetic moments and their exchange coupling persist during the process. The latter can be demonstrated by a local spin-precession experiment. At weak interaction, local moments decay along with the long-range order. The dynamics is governed by residual quasiparticles, which are reflected in oscillations of the off-diagonal components of the momentum distribution. Such oscillations provide an alternative route to study the prethermalization phenomenon and its influence on the dynamics away from the integrable (noninteracting) limit. The Hubbard model is solved within nonequilibrium dynamical mean-field theory, using the density-matrix renormalization group as an impurity solver.

Popular Summary

Symmetry-broken states such as magnetic order and superconductivity are the hallmark of complex properties in solids. An intriguing new direction in condensed-matter physics is to manipulate these properties on the fastest possible time scales using ultrashort laser pulses. Theoretically, however, the collective many-particle dynamics that is responsible for the formation and melting of long-range order is associated with many open questions.

Here, we combine two state-of-the-art numerical techniques—time-dependent density matrix renormalization group and nonequilibrium dynamical mean-field theory—to create a model system that represents interacting electrons on a bipartite lattice in which electrons can tunnel between sites. We prepare this model such that particles on neighboring sites initially align their magnetic moments in an antiparallel manner (i.e., representing antiferromagnetic order). The particles can then move between lattice sites, which leads to the melting of the magnetic order. We theoretically show that the precise movement mechanism depends strongly on the interaction between the particles: For strong interactions, the system behaves like a collection of localized magnetic moments. For weak interactions, on the other hand, the dynamics reflects the existence of coherent quasiparticles, which are typically restricted to excitations close to the ground state. In our case, these quasiparticles prevail on short times even though the system is strongly excited.

Our setup, which is well suited for experiments using cold atoms, has the ability to reveal the crossover between localized and itinerant behavior. In the future, similar studies of systems with several active orbitals may make it possible to better understand how complex solids can relax into entirely new—and possibly thermodynamically hidden—phases.

Figure 1

(a) Time evolution of the order parameter $M(t)=M_{\text{stagg}}/[1-2d(t)]$ for different values of the Coulomb repulsion in the range $U=0$ to $U=10$. (b) The double occupation $d(t)$ at the same values of $U$. Arrows indicate the double occupation in a thermalized state at the same total energy as the quenched state (as obtained from equilibrium DMFT, using a quantum Monte Carlo impurity solver [54]). All thermalized states are in the paramagnetic phase; the corresponding inverse temperature $1/T_{\mathrm{eff}}$ is plotted in (c).

Figure 2

Diagonal component $n({\epsilon }_k,t)=⟨c_k^{†}{c}_k⟩$ of the momentum occupation [(a), (c), and (e)] and off-diagonal component $\mathrm{Re}\pi ({\epsilon }_k)=⟨c_k^{†}{c}_{\overline{k}}⟩$ [(b), (d), and (f)], where $k$ and $\overline{k}$ are pairs of single-particle states coupled by a staggered potential, plotted for quenches to three different values of $U$ as a function of the energy ${\epsilon }_k$ ranging from $-2$ to 2 in the band of the Bethe lattice. The bold lines indicate momentum distributions obtained in the (paramagnetic) equilibrium state at the same energy. Note that the symmetry of the curves for $\epsilon \to -\epsilon$ is a consequence of particle-hole symmetry.

Figure 3

(a) Line cut of $\pi (\epsilon ,t)$ for $\epsilon =-1.5$ and various values of $U$. Solid lines are fits with the sum of a decaying exponential background and decaying oscillations, $f(t)=b\exp [-c(t-t_0)]+a\exp [-\mathrm{\Gamma }(t-t_0)]\cos (2{\epsilon }^{\prime }t+\varphi )$ for $t>t_0=1.5$. (b) Amplitudes of the background $b$, the oscillations $a$, the quasiparticle energy ${\epsilon }^{\prime }$, and the quasiparticle decay rate $\mathrm{\Gamma }$ as a function of $U$. (Note that $\mathrm{\Gamma }$ and ${\epsilon }^{\prime }$ are no longer well defined if $a$ becomes small.)

Figure 4

Dynamics at a probe site $o$, where the spin is initially flipped to the $x$ direction. (a)–(c) Expectation values $⟨S_z(t)⟩$, $⟨S_y(t)⟩$, and $⟨S_x(t)⟩$ for various values of $U$ [see legend in (d)]. (d) Trajectory of the spin $⟨\mathit{S}⟩$ in the $S_x\text{−}{S}_y$ plane. The inset illustrates the initial state, with one spin flipped to the $x$ direction, and the effective exchange field. (e) Effective exchange interaction, $d\varphi (t)/dt/|M_{\text{stagg}}|$, where $\varphi (t)=\arctan (S_y/S_x)$ is the angle in the $S_x\text{−}{S}_y$ plane, for $U=3,4,5,7,10$. The dotted lines correspond to the perturbative value of the exchange interaction, $J_{\mathrm{ex}}=4J_{*}^2/U$. (f) Temperature-dependent local moment in equilibrium, defined by $(1/\beta ){\int }_0^{\beta }d\tau ⟨S_z(\tau )S_z(0)⟩$, obtained using a continuous-time quantum Monte Carlo impurity solver.

Figure 5

Data for the following quench protocol: $t<0$: $J_{*}=0$ (Néel state); $0\le t<0.5$: $J_{*}=1$, $U=U_i$; $t\ge 0.5$: $J_{*}=1$, $U=8$. The intermediate step controls the excitation density in the final state. (a) Time evolution of the double occupancy or various values of $U_i$. (b) Time evolution of the order parameter $M(t)$. (c) Number of spin flips per charge-carrier density $n_{\delta }$, $[1-M(t)]/n_{\delta }$, compared to the mean displacement $R(t)$ of the initially localized particle in the $t\text{−}{J}_z$ model for $J_{\mathrm{ex}}^{*}=0$ and $J_{\mathrm{ex}}^{*}=0.5$ (dotted and dashed black lines; see text). The inset shows the average of $d(t)$ for $2\le t\le 5$ (open circles), and the density of mobile carriers $n_{\delta }$, obtained from the integrated weight in the upper Hubbard band (red crosses).

Figure 6

(a) Occupied density of states $N_{\sigma }(\omega ,t)$ after the quench: $t<0$: $v=0$ (Néel state); $0\le t<0.5$: $v=1$, $U=U_i$; $t\ge 0.5$: $v=1$, $U=8$, for various values of $U_i$; $N_{\uparrow }(\omega ,t)$ and $N_{\downarrow }(\omega ,t)$ refer to majority and minority spin, respectively, and the upper Hubbard band is scaled by a factor of 5. Dashed and solid lines denote time $t=4$ and the largest simulation time $t=t_{\max }$, respectively. (b) Integrated weight in the upper Hubbard band (red crosses and blue stars).