We investigate the dynamics following sudden quenches across quantum critical points belonging to different universality classes. Specifically, we use matrix product state methods to study the quantum Ising chain in the presence of two additional terms which break integrability. We find that in all models the rate function for the return probability to the initial state becomes a nonanalytic function of time in the thermodynamic limit. This so-called dynamical phase transition'' was first observed in a recent work by Heyl, Polkovnikov, and Kehrein [Phys. Rev. Lett. 110, 135704 (2013)] for the exactly-solvable quantum Ising chain, which can be mapped to free fermions. Our results for interacting theories'' indicate that nonanalytic dynamics is a generic feature of sudden quenches across quantum critical points. We discuss potential connections to the dynamics of the order parameter.
We investigate sudden quenches across the critical point in the transverse field Ising chain with a perturbing nonintegrable next-nearest-neighbor interaction. Expressions for the return (Loschmidt) amplitude and associated rate function are derived to linear order in the next-nearest-neighbor coupling. In the thermodynamic limit these quantities exhibit nonanalytic behavior at a set of critical times, a phenomenon referred to as a dynamical quantum phase transition. We quantify the effect of the integrability breaking perturbation on the location and shape of these nonanalyticities. Our results agree with those of earlier numerical studies and offer further support for the assertion that the dynamical quantum phase transitions exhibited by this model are a generic feature of its postquench dynamics and are robust with respect to the inclusion of nonintegrable perturbations.
We study quenching dynamics of a one-dimensional transverse Ising chain with nearest neighbor antiferromagnetic interactions in the presence of a longitudinal field which renders the model nonintegrable. The dynamics of the spin chain is studied following a slow (characterized by a rate) or sudden quenches of the longitudinal field. Analyzing the temporal evolution of the Loschmidt overlap, we find different possibilities of the presence (or absence) of dynamical phase transitions (DPTs) manifested in the nonanalyticities of the rate function of the return probability. Even though the model is nonintegrable, there are periodic occurrences of DPTs when the system is slowly ramped across the quantum critical point (QCP) as opposed to the ferromagnetic version of the model; this numerical finding is qualitatively explained by mapping the original model to an effective integrable spin model which is appropriate for describing such slow quenches. Furthermore, concerning the sudden quenches, our numerical results show that in some cases, DPTs can be present even when the spin chain is quenched within the same phase or even to the QCP, while in some other situations they completely disappear even after quenching across the QCP. These observations lead us to the conclusion that it is the change in the nature of the ground state that determines the presence of DPTs following a sudden quench.
Dynamical phase transitions (DPTs) occur after quenching some global parameters in quantum systems, and are signalled by the nonanalytical time evolution of the dynamical free energy, which is calculated from the Loschmidt overlap between the initial and time evolved states. In a recent Letter [M. Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)], it was suggested that DPTs are closely related to equilibrium phase transitions (EPTs) for the transverse field Ising model. By studying a minimal model, the XY chain in a transverse magnetic field, we show analytically that this connection does not hold generally. We present examples where DPT occurs without crossing any equilibrium critical lines by the quench, and a nontrivial example with no DPT but crossing a critical line by the quench. Although the nonanalyticities of the dynamical free energy on the real time axis do not indicate the presence or absence of an EPT, the structure of Fisher lines for complex times reveals a qualitative difference.
In this work, we investigate quenches in a free-fermion chain with long-range hopping, which decay with the distance with an exponent $\ensuremath{\nu}$ and has range
D. By exploring the exact solution of the model, we found that the dynamic free energy is nonanalytical in the thermodynamic limit, whenever the sudden quench crosses the equilibrium quantum critical point. We were able to determine the nonanalyticities of dynamic free energy
f(t) at some critical times
tc by solving nonlinear equations. We also show that the Yang-Lee-Fisher (YLF) zeros cross the real-time axis at those critical times. We found that the number of nontrivial critical times,
Ns, depends on $\ensuremath{\nu}$ and
D. In particular, we show that for small $\ensuremath{\nu}$ and large
D the dynamic free energy presents nonanalyticities in any time interval $\mathrm{\ensuremath{\Delta}}t\ensuremath{\sim}1/D\ensuremath{\ll}1$, i.e., there are nonanalyticities at almost all times. For the spacial case $\ensuremath{\nu}=0$, we obtain the critical times in terms of a simple expression of the model parameters and also show that
f(t) is nonanalytical even for finite system under antiperiodic boundary condition, when we consider some special values of quench parameters. We also show that, generically, the first derivative of the dynamic free energy is discontinuous at the critical time instant when the YLF zeros are nondegenerate. On the other hand, when they become degenerate, all derivatives of
f(t) exist at the associated critical instant.
We consider sudden quenches across quantum phase transitions in the S=1 XXZ model starting from the Haldane phase. We demonstrate that dynamical phase transitions may occur during these quenches that are identified by nonanalyticities in the rate function for the return probability. In addition, we show that the temporal behavior of the string order parameter is intimately related to the subsequent dynamical phase transitions. We furthermore find that the dynamical quantum phase transitions can be accompanied by enhanced two-site entanglement.
We study the dynamics arising from a double quantum quench where the parameters of a given Hamiltonian are abruptly changed from being in an equilibrium phase A to a different phase B and back ($\mathrm{A}\ensuremath{\rightarrow}\mathrm{B}\ensuremath{\rightarrow}\mathrm{A}$). As prototype models, we consider the (integrable) transverse Ising field as well as the (nonintegrable) ANNNI model. The return amplitude features nonanalyticities after the first quench through the equilibrium quantum critical point ($\mathrm{A}\ensuremath{\rightarrow}\mathrm{B}$), which is routinely taken as a signature of passing through a so-called dynamical quantum phase transition. We demonstrate that nonanalyticities after the second quench ($\mathrm{B}\ensuremath{\rightarrow}\mathrm{A}$) can be avoided and reestablished in a recurring manner upon increasing the time
T spent in phase B. The system retains an infinite memory of its past state, and one has the intriguing opportunity to control at will whether or not dynamical quantum phase transitions appear after the second quench.