Dynamics of the spin-1/2 Ising two-leg ladder with four-spin plaquette interaction and transverse field

Multiple-spin coupling is a key tool for investigating magnetic quantum systems. In particular, models with four-spin interactions became of great interest since they have been applied to describe material properties such as superconductivity. In this framework, the dynamics of the spin-1/2 Ising two-leg ladder with four-spin plaquette interaction in a transverse field is investigated. By means of the recurrence relations method, the first four recurrants are determined exactly for the general model, while five exact recurrants are calculated for two particular models. In all cases, higher-order recurrants are computed through extrapolation to obtain time-dependent autocorrelation functions and spectral densities in the long-time regime and infinite-temperature limit. It is found that the relaxation functions decay slower than the expected behavior of the one-dimensional transverse Ising model. The spectral lines, characterized by a Lorentzian behavior in the central body and a Gaussian shape in the tails, display exchange narrowing as the coupling intensities increase.

Figure 1

Representation of the spin-$1/2$ Ising two-leg ladder with plaquette exchange and transverse field. $J_1$ ($J_2$) is the Ising interaction along chain 1 (2), $I$ is the rung interaction, $J_r$ is the four-spin plaquette exchange, and $B_1$ ($B_2$) is the transverse field in the chain 1 (2). The $z$ direction is perpendicular to the plane.

Figure 2

(a) Exact recurrants of the standard two-leg ladder Ising model ($J_r=0$), for $J_1=J_2=B_1=B_2=1.0$, and different values of the rung coupling $I$, together with the 1D TIM from Ref. [46]. (The lines connecting the symbols are just an aid to the eye.) (b) Time-dependent spin autocorrelation functions $C_z\left(t\right)$ evaluated from the exact recurrants. The expected Gaussian decay (1D TIM) is also shown for the sake of comparison.

Figure 3

(a) Exact recurrants of the Ising ladder with $I=0$, $J_1=J_2=B_1=B_2=1.0$, and some values of the plaquette interaction $J_r$. (The lines connecting the symbols are just an aid to the eye.) (b) Time-dependent spin autocorrelation functions $C_z\left(t\right)$ obtained from the first five recurrants.

Figure 4

(a) Time-dependent correlation functions for $J_r=0$, $J_1=J_2=B_1=B_2=1.0$, and some values of $I$, using 100 recurrants ${\mathrm{\Delta }}_{\nu }$. The exact Gaussian decay of the 1D TIM is also shown on account of comparison (black line). (b) Spectral densities obtained from Eq. (16). They also correspond to the Fourier transforms of the correlation functions shown in (a).

Figure 5

(a) Best fits for the autocorrelation functions. At short times, the Gaussian decay is observed (blue squares). For clarity, only the fit corresponding to $I=0.0$ and $J_r=1.0$ is shown in this region. For longer times, the best fits are exponential (open circles). A transition between both forms occurs in the region $0.5\lesssim t\lesssim 1.3$. (b) Spectral density line shape. The results are very well adjusted by a combination of a Lorentzian peak (for $\left|w\right|\lesssim 2.0$) and Gaussian wings ($\left|w\right|\gtrsim 3.3$). A transition between both regimes is observed in the region $2.0\lesssim \left|w\right|\lesssim 3.3$. Similar behavior is detected for other values of $I$ and $J_r$.

Figure 6

(a) Time-dependent correlation functions for $I=0$, $J_1=J_2=B_1=B_2=1.0$, and different values of $J_r$, using 100 recurrants ${\mathrm{\Delta }}_{\nu }$. (b) Corresponding spectral densities.

Figure 7

(a) Correlation functions $C_z\left(t\right)$ for $J_1=J_2=B_1=B_2=1.0$ and some values of $I$ and $J_r$, using 100 recurrants ${\mathrm{\Delta }}_{\nu }$. (b) Corresponding spectral densities.

Figure 8

(a) Time-dependent correlation functions for particular cases and the general model. We used the first four (general model) and five (cases $J_r=0$ and $I=0$) exact recurrants besides the extrapolated ones up to $\nu =100$. (b) Corresponding spectral densities.