Dynamics of the transverse Ising model with next-nearest-neighbor interactions

We study the effects of next-nearest-neighbor (NNN) interactions on the dynamics of the one-dimensional spin-$1/2$ transverse Ising model in the high-temperature limit. We use exact diagonalization to obtain the time-dependent transverse correlation function and the corresponding spectral density for a tagged spin. Our results for chains of 13 spins with periodic boundary conditions produce results which are valid in the infinite-size limit. In general we find that the NNN coupling produces slower dynamics accompanied by an enhancement of the central mode behavior. Even in the case of a strong transverse field, if the NNN coupling is sufficiently large, then there is a crossover from collective mode to central mode behavior. We also obtain several recurrants for the continued fraction representation of the relaxation function.

Figure 1

Time-dependent transverse correlation function of a tagged spin in the TI model when $B=J=1.0$ for chain sizes $L=9,11,$ and 13, together with the known Gaussian result of the infinite chain. Here and in the next figures we use $J=1$ as the unit of energy.

Figure 2

Spectral density for the TI model for $B=1$ and several chain sizes, (a) $L=9$, (b) $L=11$, and (c) $L=13$. The solid line is the exact Gaussian result for the infinite system.

Figure 3

Time-dependent correlation function for $B=0.5$ and several values of the NNN coupling $J_2$. Each curve is the overlap of two curves: one obtained for $L=12$ and other for $L=13$.

Figure 4

Spectral density for $B=0.5$ and several values of $J_2$. All the curves were obtained for chains with $L=13$.

Figure 5

Time-dependent correlation function for $B=1$ and several values of $J_2$. The curves were obtained with $L=12$ and 13.

Figure 6

Spectral density for $B=1$ and several values of $J_2$, obtained for chains of length $L=13$. The vertical scales of (a) and (b) were chosen to show more clearly the details of each of the curves.

Figure 7

Time-dependent correlation function for $B=2.0$ and various values of $J_2$. The curves were obtained for chain sizes $L=12$ and 13.

Figure 8

Spectral density for the case $B=2$ and several values of $J_2$. The plots were obtained for chains of size $L=13$.

Figure 9

Recurrants for $B=J_2=1.0$ and chain sizes $L=12$ and 13.

Figure 10

Recurrants of the infinite transverse ANNNI model with $B=1.0$ and several values of $J_2$ (data taken from Table 3).