Anomalous behavior of the energy gap in the one-dimensional quantum $XY$ model

We reexamine the well-studied one-dimensional spin-$1/2 XY$ model to reveal its nontrivial energy spectrum, in particular the energy gap between the ground state and the first excited state. In the case of the isotropic $XY$ model, the $XX$ model, the gap behaves very irregularly as a function of the system size at a second order transition point. This is in stark contrast to the usual power-law decay of the gap and is reminiscent of the similar behavior at the first order phase transition in the infinite-range quantum $XY$ model. The gap also shows nontrivial oscillatory behavior for the phase transitions in the anisotropic model in the incommensurate phase. We observe a close relation between this anomalous behavior of the gap and the correlation functions. These results, those for the isotropic case in particular, are important from the viewpoint of quantum annealing where the efficiency of computation is strongly affected by the size dependence of the energy gap.

Figure 1

The phase diagram of the $XY$ model in Eq. (1), which undergoes a first order phase transition across the surface indicated in gray between the phases with positive and negative expectation values of ${\sigma }_j^x$. Along the isotropic line of $\gamma =0$ indicated in black, for $0\le \mathrm{\Gamma }<1$, the transition is of second order.

Figure 2

(a): Wave numbers for the presumed ground state in the case of $k_1$ on the $\mathrm{complex}\text{−}k$ plane. The black points mark the occupied modes, and the white ones are for the unoccupied ones. The number of occupied modes is odd. (b): Likewise, the number of occupied states for $k_2$ is even.

Figure 3

System-size dependence of the energy gap for $\gamma =0$. Panels (a), (b), (c), and (d) show the energy gap for $\mathrm{\Gamma }=0.1,\mathrm{\Gamma }=0.3,\mathrm{\Gamma }=0.5$, and $\mathrm{\Gamma }=0.999$, respectively.

Figure 4

System-size dependence of the energy gap for the infinite-range $XY$ model in Eq. (35) for the longitudinal field $h=0$ and various, fixed values of the transverse field $\mathrm{\Gamma }$. Panels (a), (b), and (c) show the data for $\mathrm{\Gamma }=\pi /10,\mathrm{\Gamma }=1/3$, and $\mathrm{\Gamma }=1-\pi /300$, respectively.

Figure 5

System-size dependence of the gap multiplied by the size for $\gamma =0$ and $\mathrm{\Gamma }=0.1$.

Figure 6

System-size dependence of the energy gap for an anisotropic $XY$ model in Eq. (1) for $h=0$ and in the ferromagnetic phase. Dots and circles show the energy gap for $(\mathrm{\Gamma },\gamma )=(0.85,0.95)$ and $(\mathrm{\Gamma },\gamma )=(0.2,0.15)$, respectively. In the second case the gap falls toward zero very rapidly at periodically observed dips.