Relationship between the transverse-field Ising model and the $XY$ model via the rotating-wave approximation

In a large transverse field, there is an energy cost associated with flipping spins along the axis of the field. This penalty can be employed to relate the transverse-field Ising model in a large field to the $XY$ model in no field (when measurements are performed at the proper stroboscopic times). We describe the details for how this relationship works and, in particular, we also show under what circumstances it fails. We examine wave-function overlap between the two models and observables, such as spin-spin Green's functions. In general, the mapping is quite robust at short times, but will ultimately fail if the run time becomes too long. There is also a tradeoff between the length of time one can run a simulation out to and the time jitter of the stroboscopic measurements that must be balanced when planning to employ this mapping.

Figure 1

Energy levels of the transverse-field Ising Hamiltonian in a field of $B/J_0=10$ and of the $XY$ Hamiltonian for a chain of six ions and a longitudinal trapping frequency of 950 kHz and $\alpha \approx 0.63$. The organization of the transverse-field Ising levels into approximate $S_{\text{tot}}^z$ blocks is evident. The $S_{\text{tot}}^z\approx 0$ Ising levels and the $S_{\text{tot}}^z=0$ $XY$ levels are colored in red.

Figure 2

(a) Transverse-field Ising model energy levels for $B/J_0=10$ and $XY$ model energy levels for a chain of six ions in a longitudinal trapping frequency of 950 kHz ($\alpha \approx 0.63$). Levels in the $S_{\text{tot}}^z\approx 0$ block of the Ising model and the $S_{\text{tot}}^z=0$ block of the $XY$ model are colored in red. (b) Difference between the corresponding $XY$ and transverse-field Ising model energy levels is plotted as a function of field strength for $B/J_0=7$, 10, 20, 30, 50, 70, and 100. The particular levels used to measure the difference are shown in the inset with the rows.

Figure 3

(a) $XY$ and transverse-field Ising model energy levels for $S_{\text{tot}}^z=-\frac{1}{2}$ in an external field of $B/J_0=10$ on a chain of seven ions with a longitudinal trapping frequency of 650 kHz ($\alpha \approx 1$). The $S_{\text{tot}}^z=\frac{1}{2}$ energy levels are plotted in panel (b). Panels (c) and (d) plot the field dependence of the difference between $S_{\text{tot}}^z=-\frac{1}{2}$ and $\frac{1}{2}$ energy levels, respectively. The levels used to calculate the differences are identified in the inset by the arrows.

Figure 4

Plot of the modulus squared of the overlap of the time-evolved $XY$ and transverse-field Ising state vectors, $\langle {\mathrm{\Psi }}_{\text{Ising}}\left(t\right)|{\mathrm{\Psi }}_{XY}\left(t\right)\rangle$, for a five-ion chain with a longitudinal trapping frequency of $950\mathrm{kHz}$ ($\alpha \approx 0.63$) and various transverse-field strengths. Panels (a)–(d) plot the squared overlap between $t{J}_0=0$ and 2 for $B/J_0=5$, 10, 15, and 20, respectively. The black dots are plotted at the Larmor frequency, ${\omega }_L=4\pi$. The red dots are plotted at a numerically optimized frequency, given by ${\omega }_{\text{opt}}=4\pi \sqrt{B^2+{\left(1.67J_0\right)}^2}$.

Figure 5

Plot of the pure-wave-function retarded Green's function of the transverse-field Ising Hamiltonian, in blue, and of the $XY$ Hamiltonian, in green, for a chain of seven ions in a longitudinal trapping frequency of $650\mathrm{kHz}$ ($\alpha \approx 1$) and various transverse-field strengths. Panels (a)–(d) plot the Green's functions between $t{J}_0=0$ and 2 for $B/J_0=5$, 10, 15, and 20, respectively. The black dots are plotted at the Larmor frequency, ${\omega }_L=4\pi$, and the red dots are plotted at ${\omega }_{\text{opt}}=4\pi \sqrt{B^2+{\left(0.84J_0\right)}^2}$.

Figure 6

Color map for the spatiotemporal evolution of $C_{0,j^{\prime }}\left(t\right)$ on an 11-ion chain with a longitudinal trapping frequency of $560\mathrm{kHz}$ ($\alpha \approx 1.19$) and a field strength of $B/J_0=10$. Panel (a) plots the evolution of the $XY$ model correlation function between $t{J}_0=0$ and 0.3. The white curve is a power-law fit for the light cone of the correlations, reproduced from Ref. [6]. Panels (b) and (c) plot the evolution of the transverse-field Ising model, but panel (b) plots only the stroboscopic points with a sampling rate of twice the Larmor frequency, $2{\omega }_L=8\pi B$. Panel (c) samples the system too frequently, so it does not produce the $XY$ model accurately.