Dynamical Quantum Phase Transitions in Spin Chains with Long-Range Interactions: Merging Different Concepts of Nonequilibrium Criticality

We theoretically study the dynamics of a transverse-field Ising chain with power-law decaying interactions characterized by an exponent $\alpha$, which can be experimentally realized in ion traps. We focus on two classes of emergent dynamical critical phenomena following a quantum quench from a ferromagnetic initial state: The first one manifests in the time-averaged order parameter, which vanishes at a critical transverse field. We argue that such a transition occurs only for long-range interactions $\alpha \le 2$. The second class corresponds to the emergence of time-periodic singularities in the return probability to the ground-state manifold which is obtained for all values of $\alpha$ and agrees with the order parameter transition for $\alpha \le 2$. We characterize how the two classes of nonequilibrium criticality correspond to each other and give a physical interpretation based on the symmetry of the time-evolved quantum states.

Figure 1

wDynamical phase diagram. We study the quantum dynamics of an Ising chain with power-law decaying interactions by preparing the system in a fully polarized state and abruptly switching on a finite transverse field $h_f$. We identify the DQPT through two mechanisms: One introduces an order parameter (DQPT-OP) that is finite only in the dynamical ferromagnetic phase, whereas the other is based on nonanalytic kinks in the Loschmidt rate function (DQPT-LO) that only arise for quenches across the transition. When the interaction exponent $\alpha <2$, the DQPT occurs simultaneously for both cases along the red line, separating the dynamical ferromagnetic (I) and the dynamical paramagnetic (III) phase. We argue that for $\alpha >2$ the system does not establish a finite order parameter and thus the DQPT-OP ends at $\alpha =2$. Yet, the DQPT-LO persists for arbitrarily large $\alpha$ (dashed line) separating two dynamical phases characterized by a monotonic decay (II) and an oscillating decay (III) of the magnetization.

Figure 2

Time evolving the order parameter. (a) For $\alpha =1.5$ and $h_f=0.7J$ the order parameter ${\sigma }^x(t)$ approaches a finite value, describing a dynamical symmetry-broken state with ferromagnetic order, whereas it decays to 0 with strong oscillations for quenches to $h_f=1.5J$. (b) Even though for shorter-ranged interactions $\alpha =3$ the order parameter reaches 0 for all values of $h_f$, the nature of the decay is very different: For small fields $h_f=0.7J$ it decays with a timescale much longer than the microscopic scales, whereas for large fields $h_f=1.5J$ it oscillates around 0 and decays rapidly.

Figure 3

Dynamical phase diagram of the order parameter. We estimate the asymptotic value of the order parameter $\overline{{\sigma }^x}$ as a function of the quenched transverse field $h_f$ for different system sizes $N$ and interaction exponents (a) $\alpha =0.1$, (b) $\alpha =1.5$, and (c) $\alpha =3$. For both values of $\alpha <2$ we find that the finite-size flow of the order parameter indicates a DQPT-OP with the critical point $h_c\sim J$. For very long-ranged interactions, (a), the order parameter $\overline{{\sigma }^x}$ approaches the mean-field predictions, $\alpha =0$ (dashed lines), with increasing system size. By contrast, for relatively short-ranged interactions (c) $\alpha =3$, the finite-size flow suggests that the order parameter $\overline{{\sigma }^x}$ flows toward 0 in the thermodynamic limit for all values of the transverse field.

Figure 4

Dynamical quantum phase transitions in the Loschmidt echo. We compute an extension of the Loschmidt echo, which is the return probability to the degenerate ground-state manifold, Eq. (3), for different values of the transverse field $h_f$ and interaction exponent (a) $\alpha =1.8$ and (b) $\alpha =2.5$. We observe nonanalyticities in the associated rate function $\lambda (t)$ for arbitrary values of the interaction exponent $\alpha$, provided the final transverse field $h_f$ is sufficiently large. The insets compare the typical rate of kinks in $\lambda (t)$, solid line, with the zero crossings of the order parameter ${\sigma }^x(t)$. The right panels show the evolution of the magnetization $\overrightarrow{\sigma }(t)$ projected onto the $xy$ plane of the Bloch sphere. When quenching across the dynamical transition, the magnetization spreads over both hemispheres (black curves) whereas it remains located on one hemisphere for quenches within the same dynamical phase (blue and red curves), indicating a bifurcation of the dynamics.