Probing Ground-State Phase Transitions through Quench Dynamics

The study of quantum phase transitions requires the preparation of a many-body system near its ground state, a challenging task for many experimental systems. The measurement of quench dynamics, on the other hand, is now a routine practice in most cold atom platforms. Here we show that quintessential ingredients of quantum phase transitions can be probed directly with quench dynamics in integrable and nearly integrable systems. As a paradigmatic example, we study global quench dynamics in a transverse-field Ising model with either short-range or long-range interactions. When the model is integrable, we discover a new dynamical critical point with a nonanalytic signature in the short-range correlators. The location of the dynamical critical point matches that of the quantum critical point and can be identified using a finite-time scaling method. We extend this scaling picture to systems near integrability and demonstrate the continued existence of a dynamical critical point detectable at prethermal timescales. We quantify the difference in the locations of the dynamical and quantum critical points away from (but near) integrability. Thus, we demonstrate that this method can be used to approximately locate the quantum critical point near integrability. The scaling method is also relevant to experiments with finite time and system size, and our predictions are testable in near-term experiments with trapped ions and Rydberg atoms.

Figure 1

(a) Schematic picture of the quench. We evolve the initial state $|{\psi }_{\mathrm{in}}⟩$, which consists of all spins polarized in the Ising direction, for a time $t$ under the Hamiltonian defined in Eq. (1) (shown) or Eq. (2). Then we measure the nearest-neighbor correlator $G(t)$ defined in Eq. (3). (b) The dependence of $G_{\mathrm{av}}^{\infty }$ on $B$ as given by Eq. (5).

Figure 2

Finite-time scaling for the integrable case. The first derivative of the correlator, ${\partial }_B{G}_{\mathrm{av}}(t)$, exhibits a sharper jump across the transition ($B=B_c^{\mathrm{dy}}=J$) at later times. The curves are obtained by differentiating Eq. (4) with a system size $L=100$ [32]. The curves at different times cross at the same point, thus revealing a DCP. The dotted line in black indicates the expected discontinuity in the derivative in the thermodynamic limit ($L\to \infty$) and at long times ($Jt\to \infty$). Inset: We extract the time dependence by performing a scaling collapse after rescaling the magnetic field $B$ by $t$. The scaling function near the critical point is linear, and is shown as a dashed red line.

Figure 3

Comparison between the finite-time scaling of the time-averaged nearest-neighbor correlator $G_{\mathrm{av}}(t)$ and the ground-state Binder cumulant ${\mathcal{U}}_4^{\mathrm{gs}}$. The first column [(a),(c)] corresponds to the TFIM with next-nearest-neighbor interaction $\mathrm{\Delta }=0.1$ [Eq. (1)], while the second column [(b),(d)] corresponds to the TFIM with long-range interaction $\alpha =6$ [Eq. (2)]. The finite-time scaling of the quench dynamics is shown in the first row, and the finite-size scaling for the ground state is shown in the second row. The DCPs ($B_c^{\mathrm{dy}}$) identified in (a) and (b) using finite-time scaling are close but different from the locations of the quantum critical points ($B_c^{\mathrm{gs}}$) identified in (c) and (d). The ground-state simulations were done using a density matrix renormalization group algorithm with bond dimension $\chi =32$.