Quasiparticle dynamics in a quasiperiodic Ising model with temporally fluctuating transverse fields

We study quasiparticle dynamics in a quasiperiodic Ising model with temporally fluctuating transverse fields. Specifically, we calculate the dynamical exponents of the standard deviation of a quasiparticle spreading under a field chosen randomly from binary values $\pm h$ at every time interval. We find that the short-time behavior of the dynamical exponents depends on the interval of the temporally fluctuating fields. We also reveal how the quasiparticle dynamics affects the relaxation of spin-spin correlation functions. The dynamics can be explained via the overlap between the eigenvectors of a Hamiltonian with $\pm h$.

Figure 1

Phase diagram of the QP-TFIM. The complexity is due to the various symmetries in individual parameter regimes. The abbreviation PM and FM indicate paramagnetic and ferromagnetic, respectively. The diagram was obtained by Chandran et al. [41].

Figure 2

Dynamics of inverse dynamical exponents, where all calculations were performed with $L=2048,N_{\mathrm{samp}}={10}^4$. (a) Dependence of the fluctuation times $\tau$ on the dynamics of $1/z\left(t_n\right)$ at $A_J=0.5$. The inverse dynamical exponent $1/z\left(t_n\right)$ approaches to the diffusive value 0.5 for most fluctuation times $\tau$, except for $\tau /2\pi =1.2,2.4$. (b) Dependence of the quasiperiodic spin couplings $A_J$ at $\tau /2\pi =0.25$. The dynamical exponent at $A_J=0$ (TFIM) does not drop to the diffusive value 0.5. In contrast, the other dynamical exponents reach the diffusive value in the long-time scale, with the speed of the decay tending to be faster with increasing $A_J$. (c) Log-log plot of the relaxation curves in (b). The dynamical exponents exhibit power-law decay, and the slopes of the curves become the same in the long-time scale.

Figure 3

Scaling of the shapes of the probability distributions for $A_J=0,0.1,0.2$ [(a)–(c), respectively]. For $A_J=0$ (TFIM), two outside peaks can be seen, and the distributions are not on the quadratic curve of the diffusive scaling. The peaks are suppressed as $A_J$ increasing. For $A_J=0.2,t_n/\tau =2^8$, the distribution can be fit by the quadratic curve.

Figure 4

Time dependence of the spin-spin correlation function with $L=1024,l=510,J=1,A_J=0.5,h=1,N_{\mathrm{samp}}=1000$, and $\tau /2\pi =1.0–1.4$. The inset depicts the time dependence of the dynamical exponents for the same setting as depicted in Fig. 2.

Figure 5

Density of states (DOS) of the matrix $\mathrm{H}$ with $L=2048,h=1,J=1$, and $A_J=0,0.2,0.5,0.7,1$. The introduction of $A_J$ to the TFIM ($A_J=0$) causes gaps in the continuous density of states. The red arrows in the DOS for $A_J=0.5$ depict peaks corresponding to ${\epsilon }_{\mu }=\pm 0.833$.

Figure 6

Overlap between the eigenvectors of $H_{+}$ and $H_{-}$. The vertical axis $\mu$ and horizontal axis $\nu$ represent the numbers of eigenvectors of $H_{+}$ and $H_{-}$, respectively.