Quenches and dynamical phase transitions in a nonintegrable quantum Ising model

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.

Figure 1

The schematic phase diagram of the model given in Eq. (3); the solid line extending from $(\mathrm{\Gamma }=0,h=2)$ to $(\mathrm{\Gamma }=0,h=-2)$ through $(\mathrm{\Gamma }=1,h=0)$ separates the antiferromagnetic (AFM) phase from the paramagnetic (PM) phase. The points $(\mathrm{\Gamma }=0,h=\pm 2)$ denote the first order transition while $(\mathrm{\Gamma }=1,h=0)$ corresponds to the integrable quantum critical point. Throughout this paper, $\mathrm{\Gamma }$ is set equal to 1 and $h$ is quenched along the dashed line shown with an arrow.

Figure 2

Our numerical results show prominent periodic occurrences of DPTs when the longitudinal field is slowly ramped from a large positive value to a large negative value as $h\sim -t/\tau$. This periodic pattern can be qualitatively explained by studying $I\left(t\right)$ of the equivalent integrable Hamiltonian (4) as demonstrated in Appendix pp2.

Figure 3

Numerically obtained $I\left(t\right)$ showing the absence and occurrence of the nonanalyticities (DPTs) in different situations when $h$ is suddenly quenched. (a) No DPTs are observed for a sudden quench of small amplitude of $h$ even if the system crosses the QCP in the process; (b) DPTs occur when $h$ is quenched within the same phase; (c) DPTs also appear when $h$ is quenched from a large positive value to the QCP at $h=0$; (d) a regular (but not periodic) occurrence of DPTs is observed when $h$ is suddenly quenched from a large positive to a large negative value. The inset shows that the DPTs are rounded off when the quench amplitude is even larger, leading to Rabi osscillations.

Figure 4

Scaling of residual energy for slow and sudden quenching by changing the longitudinal field $h$ with the transverse field $\mathrm{\Gamma }={\mathrm{\Gamma }}_c=1$, as obtained from DMRG studies. When $h$ is changed linearly as $-t/\tau$ to the QCP ($h=0$), ${\epsilon }_{\mathrm{res}}\sim {\tau }^{-4/3}$ which is in perfect agreement with the KZ scaling. In the inset, we verify the scaling of ${\epsilon }_{\mathrm{res}}\sim h^2$ for a sudden quench starting from the QCP.

Figure 5

The rate function for quenching $h=3$ to $h=-3$ shows sharp nonanalyticities at periodic intervals in time with system size $N=400$ and several $\tau$'s.