Dynamical quantum phase transitions in spin-$S U\left(1\right)$ quantum link models

Dynamical quantum phase transitions (DQPTs) are a powerful concept of probing far-from-equilibrium criticality in quantum many-body systems. With the strong ongoing experimental drive to quantum simulate lattice gauge theories, it becomes important to investigate DQPTs in these models in order to better understand their far-from-equilibrium properties. In this work, we use infinite matrix product state techniques to study DQPTs in spin-$S U\left(1\right)$ quantum link models. Although we are able to reproduce literature results directly connecting DQPTs to a sign change in the dynamical order parameter in the case of $S=1/2$ for quenches starting in a vacuum initial state, we find that for different quench protocols or different values of the link spin length $S>1/2$ this direct connection is no longer present. In particular, we find that there is an abundance of different types of DQPTs not directly associated with any sign change of the order parameter. Our findings indicate that DQPTs are fundamentally different between the Wilson-Kogut-Susskind limit and its representation through the quantum link formalism.

Figure 1

Illustration of DQPT behavior in the quench dynamics of spin-$S U\left(1\right)$ QLMs. The ${\mathbb{Z}}_2$ symmetry-broken phase of the latter hosts two doubly degenerate ground states, $|{\psi }_0^{+}\rangle$ and $|{\psi }_0^{-}\rangle$. Starting in $|{\psi }_0^{+}\rangle$ and quenching across the equilibrium quantum critical point, rich dynamical criticality arises in the total return rate $r\left(t\right)$ (1a) as the system explores the Hilbert space in the wake of the quench. When $r\left(t\right)$ switches from the primary return rate ${\lambda }_1^{+}\left(t\right)$ (1b) onto $|{\psi }_0^{+}\rangle$ to the secondary return rate ${\lambda }_1^{-}\left(t\right)$ (1c) onto $|{\psi }_0^{-}\rangle$ or vice versa, a manifold DQPT appears at a critical time $t_c$ when $r\left(t_c\right)={\lambda }_1^{+}\left(t_c\right)={\lambda }_1^{-}\left(t_c\right)$. Furthermore, DQPTs can also appear in the form of nonanalyticities in $r\left(t\right)$ that are not related to a switch between ${\lambda }_1^{+}\left(t\right)$ and ${\lambda }_1^{-}\left(t\right)$. Such a branch DQPT occurs at critical times $t_c$ when $r\left(t_c\right)={\lambda }_1^{\alpha }\left(t_c\right)={\lambda }_2^{\alpha }\left(t_c\right)$, where ${\lambda }_1^{\alpha }\left(t_c\right)=min\{{\lambda }_1^{+}\left(t_c\right),{\lambda }_1^{-}\left(t_c\right)\}$ and ${\lambda }_2^{\alpha }\left(t\right)$ is the second rate-function branch; see Sec. 2a for details.

Figure 2

Dynamics of the total return rate (1a) (blue), which is the minimum of the primary (1b) and secondary (1c) return rates (solid and dotted gray curves, respectively) and the order parameter (5) (red) for quenches starting in a ${\mathbb{Z}}_2$ symmetry-broken product state, which is one of two doubly degenerate ground states of the spin-$S U\left(1\right)$ QLM (3) at $\kappa =0.1\sqrt{J}$ and $\mu \to \pm \infty$ (positive sign for half-integer $S$, negative for integer $S$). The solid (dotted) green curves designate the second primary (secondary) rate-function branch; see Sec. 2b. (a) For the case of $S=1/2$, periodic manifold DQPTs correspond directly to order-parameter zeros occurring at roughly the same period, in full agreement with the conclusions of Ref. [47]. (b,c) For $S>1/2$, this direct connection is no longer present, and many aperiodic manifold and branch DQPTs (see Sec. 2b for definition) appear in $r\left(t\right)$ even when there is only a single order-parameter zero in the accessible time-evolution window.

Figure 3

Quench dynamics of the return rate (1a) and order parameter (5) for the $U\left(1\right)$ QLM with half-integer (a) $S=1/2$ and (b) $S=3/2$ in the wake of a quench from $\mu =J$ to $\mu =-J$ at $\kappa =0.1\sqrt{J}$. As before, this quench is from the ${\mathbb{Z}}_2$ symmetry-broken phase to the ${\mathbb{Z}}_2$-symmetric phase of Eq. (3) for the case of half-integer $S$. In both cases, there is no direct connection between manifold DQPTs and order-parameter zeros, where we get a plethora of aperiodic manifold DQPTs and several branch DQPTs, but only a few order-parameter zeros within the accessible evolution times.

Figure 4

Quench dynamics of the return rate (1a) and order parameter (5) for the $U\left(1\right)$ QLM with integer (a) $S=1$ and (b) $S=2$ in the wake of a quench from $\mu =-J$ to $\mu =J$ at $\kappa =0.1\sqrt{J}$. As before, this quench is from the ${\mathbb{Z}}_2$ symmetry-broken phase to the ${\mathbb{Z}}_2$-symmetric phase of Eq. (3) for the case of integer $S$. In both cases, there is no direct connection between manifold DQPTs and order-parameter zeros, where we get a plethora of aperiodic manifold DQPTs and several branch DQPTs, but only a few (or no) order-parameter zeros within the accessible evolution times.

Figure 5

Convergence analysis of the (a) return rate and (b) electric flux of Fig. 2, which involves a quench of the initial state $|{\psi }_0^{+}\rangle$ (physical vacuum) with the Hamiltonian $\widehat{H}$ of Eq. (3) at $\mu =0.655J/\left[6\sqrt{S(S+1)}\right]$, with $S=3/2$, and $\kappa =0.1\sqrt{J}$. We see excellent convergence at a maximal bond dimension of ${\mathcal{D}}_{\text{max}}=550$.