Dynamical quantum phase transitions in systems with continuous symmetry breaking

Interacting many-body systems that are driven far away from equilibrium can exhibit phase transitions between dynamically emerging quantum phases, which manifest as singularities in the Loschmidt echo. Whether and under which conditions such dynamical transitions occur in higher-dimensional systems with spontaneously broken continuous symmetries is largely elusive thus far. Here, we study the dynamics of the Loschmidt echo in the three-dimensional O($N$) model following a quantum quench from a symmetry-breaking initial state. The O($N$) model exhibits a dynamical transition in the asymptotic steady state, separating two phases with a finite and vanishing order parameter, that is associated with the broken symmetry. We analytically calculate the rate function of the Loschmidt echo and find that it exhibits periodic kink singularities when this dynamical steady-state transition is crossed. The singularities arise exactly at the zero crossings of the oscillating order parameter. As a consequence, the appearance of the kink singularities in the transient dynamics is directly linked to a dynamical transition in the order parameter. Furthermore, we argue, that our results for dynamical quantum phase transitions in the O($N$) model are general and apply to generic systems with continuous symmetry breaking.

Figure 1

Dynamical criticality in the Loschmidt echo for systems with spontaneous symmetry breaking. (a) We study dynamical quantum phase transitions of the O($N$) model following quantum quenches from an initial bare mass $r_i^0$ to a final bare mass $r_f^0$. The initial state is chosen to break the continuous symmetry of the O($N$) model and hence is described by a finite order parameter. Our system exhibits a steady-state dynamical phase transition at $r_c^{\text{dyn}}$, which separates the dynamically ordered phase in which the long-time average of the order parameter $\overline{\varphi }$ remains finite from the disordered phase in which $\overline{\varphi }$ vanishes. We calculate analytically the Loschmidt echo, and we find that the associated rate function remains smooth for quenches within the dynamically ordered phase (b) but exhibits nonanalytic kink singularities when crossing the dynamical critical point $r_c^{\text{dyn}}$ (c).

Figure 2

Dynamical phase diagram of the O($N$) model in three spatial dimensions. The system is prepared in the equilibrium symmetry-broken phase at zero temperature. For quenches to a point inside of the dynamically symmetry-broken phase, $r_{\mathrm{f}}^0<r_{\mathrm{c}}^{\text{dyn}}$, the order parameter relaxes to a finite value $\overline{\varphi }$ (red line) and the effective mass $r_{\mathrm{f}}$ remains zero, indicating the presence of gapless excitations in the steady state. For quenches into the symmetric phase, $r_{\mathrm{f}}^0>r_{\mathrm{c}}^{\text{dyn}}$, the long-time average of the order parameter is zero, $\overline{\varphi }=0$, and the effective mass $r_{\mathrm{f}}$ becomes finite (blue dashed line). Close to the critical point $r_{\mathrm{c}}^{\text{dyn}}$ the long-time average $\overline{\varphi }$ vanishes as ${(r_{\mathrm{c}}^{\text{dyn}}-r_{\mathrm{f}}^0)}^{1/4}$ and the effective mass vanishes as $r_{\mathrm{f}}\sim (r_{\mathrm{f}}^0-r_{\mathrm{c}}^{\text{dyn}})$ [17].

Figure 3

Definition of the angle $\theta$ and surface of element of the $N$-dimensional sphere. The ground-state manifold of the $\text{O}\left(N\right)$ model can be pictured as a sphere with radius ${\varphi }_0$ in an $N$-dimensional space. The return probability $\mathcal{L}\left(t\right)$ to the ground-state manifold is obtained from the integral of the overlap $|\langle \chi |\mathrm{\Psi }\left(t\right)\rangle {|}^2$ over this sphere. Defining $\theta$ as the angle between the vector $\chi$ and the initial order parameter $({\varphi }_0,0,\cdots ,0)$ and making use of the rotational symmetry around the $(a=1)$ axis, one can write the integration element $d^N\chi \delta \left(\right|\chi |-{\varphi }_0)$ as the product of the arc length ${\varphi }_{0}d\theta$ and the surface area of the sphere in $N-1$ dimensions $S^{(N-1)}({\varphi }_{0}sin\theta )$ generated by rotating $\chi$ around the $a=1$ axis with $\theta$ fixed. To obtain a probability measure, we finally divide the integration element by the total available surface area $S^N\left({\varphi }_0\right)$.

Figure 4

Order-parameter landscape for the angle $\theta$: The return probability, Eq. (13), can be interpreted as a classical partition function for the variable $\theta \in [0,\pi ]$ moving in an effective free-energy landscape $\mathcal{F}(\theta ,\varphi )$. The landscape has the form of a double-well potential, where the order parameter $\varphi \left(t\right)$ is acting as an external field, shifting the two wells against each other. For (a) $\varphi \left(t\right)>0$, the left minimum is energetically more favorable, while for (c) $\varphi \left(t\right)<0$, the situation is reversed. As the system size $L$ is increased ($L=1,5,\infty$, darker lines correspond to larger $L$), the left (right) potential well is shifted toward $\theta =0$ ($\theta =\pi$), as indicated by the arrows in panel (b). The angles $\theta =0$ ($\theta =\pi$) correspond to the states having an order parameter parallel (antiparallel) to the initial state. (b) When $\varphi \left(t\right)$ changes sign, the most relevant value of $\theta$ jumps from one well to the other, which gives rise to the kinks in the Loschmidt rate function $\mathcal{R}\left(t\right)$. All plots are for $N=10$.

Figure 5

Loschmidt rate function. The Loschmidt rate function $\mathcal{R}\left(t\right)$ of the return probability to the ground-state manifold contains a squeezed state contribution ${\mathcal{R}}_{\mathrm{sq}}\left(t\right)$ that scales extensively with system size, and a coherent state contribution ${\mathcal{R}}_{\mathrm{coh}}\left(t\right)$ that scales subextensively. Whereas ${\mathcal{R}}_{\mathrm{sq}}\left(t\right)$ is a smooth function of time, ${\mathcal{R}}_{\mathrm{coh}}\left(t\right)$ shows kinks when the system is quenched across the dynamical quantum phase transition. Due to the subextensive scaling of the coherent state contribution ${\mathcal{R}}_{\mathrm{coh}}$, the nonanalytic behavior is not visible in the full rate-function $\mathcal{R}\left(t\right)$ of the return probability, (a). Nevertheless the nonanalytic behavior is clearly observable in the second derivative $\ddot{\mathcal{R}}\left(t\right)$ as $\delta$ peaks, since the squeezed state contribution relaxes on a much shorter time scale than the one of the coherent state, (b). The system parameters are $L=2.5\times {10}^4$, $\lambda =1.0$, $r_{\mathrm{i}}^0=-1.0$, and $r_{\mathrm{f}}^0=0.0$.

Figure 6

Coherent state contribution to the rate function and order-parameter dynamics. The coherent state contribution ${\mathcal{R}}_{\mathrm{coh}}$ to the rate function exhibits kinks at the zero crossings of the order parameter $\varphi \left(t\right)$. The kinks appear periodically, and the time between them, $\mathrm{\Delta }{T}_{\mathrm{kink}}$, is determined by final effective mass $r_{\mathrm{f}}$: $\mathrm{\Delta }{T}_{\mathrm{kink}}\sim {(r_{\mathrm{f}}^0-r_{\mathrm{c}}^{\mathrm{dyn}})}^{-\frac{1}{2}}$. The data are evaluated for the same parameters as in Fig. 5