Nonequilibrium Criticality in Quench Dynamics of Long-Range Spin Models

Long-range interacting spin systems are ubiquitous in physics and exhibit a variety of ground-state disorder-to-order phase transitions. We consider a prototype of infinite-range interacting models known as the Lipkin-Meshkov-Glick model describing the collective interaction of $N$ spins and investigate the dynamical properties of fluctuations and correlations after a sudden quench of the Hamiltonian. Specifically, we focus on critical quenches, where the initial state and/or the postquench Hamiltonian are critical. Depending on the type of quench, we identify three distinct behaviors where both the short-time dynamics and the stationary state at long times are effectively thermal, quantum, and genuinely nonequilibrium, characterized by distinct universality classes and static and dynamical critical exponents. These behaviors can be identified by an infrared effective temperature that is finite, zero, and infinite (the latter scaling with the system size as $N^{1/3}$), respectively. The quench dynamics is studied through a combination of exact numerics and analytical calculations utilizing the nonequilibrium Keldysh field theory. Our results are amenable to realization in experiments with trapped-ion experiments where long-range interactions naturally arise.

Figure 1

(a) Schematic of the LMG model [Eq. (1)]. Spins interact via anisotropic $XY$ Hamiltonian parametrized by $J({\gamma }_x,{\gamma }_y)$ with a transverse magnetic field $\mathrm{\Delta }$. (b) Ground-state phase diagram of the LMG model. The critical (dashed) line defines the disorder-to-order transition. Three different initial states, corresponding to type-I, type-II, and type-III quenches, are shown as open circles, and the postquench Hamiltonian is denoted by a solid circle.

Figure 2

Evolution of fluctuations, $(1/N)⟨S_x^2⟩$, for quench types I, II and III (see Fig. 1) reported in the first [(a)–(b)], second [(c)–(d)] and third [(e)–(f)] rows, respectively. The right column shows the rescaled fluctuations and dynamics with the critical exponents $(\alpha ,\zeta )$ [see Eq. (5)] given by $(1/2,1/4)$ [panel (b)] $(1/3,1/3)$ [panel (d)] and $(2/3,1/6)$ [panel (f)] corresponding to types I, II and III, respectively.

Figure 3

(a) The correlation and response functions within the stationary state for the type-III quench for different system sizes. $t_2=20$ is chosen to ensure that the stationary state is reached. Scaling collapse is consistent with the exponents $\zeta =1/6$ and $\alpha =2/3$. (b) Time-dependent effective temperature $T(t)$ extracted from the two-time correlators for $N=3000$ shown in (a). (c) Scaling of $T(t)$ with system size in the limit $t\to 0$. It is consistent with $T(t\to 0)\sim N^{1/3}$.