Sharp Phase Transitions in a Small Frustrated Network of Trapped Ion Spins

Sharp quantum phase transitions typically require a large system with many particles. Here we show that, for a frustrated fully connected Ising spin network represented by trapped atomic ions, the competition between different spin orders leads to rich phase transitions whose sharpness scales exponentially with the number of spins. This unusual finite-size scaling behavior opens up the possibility of observing sharp quantum phase transitions in a system of just a few trapped ion spins.

Figure 1

Illustration of spin orders in two frustrated Ising networks with competing long-range interaction for $N=7$ ions. (a) Ferromagnetic order with detuning $\mu /{\omega }_{\perp }=0.9886$. (b) Kink order with $\mu /{\omega }_{\perp }=0.9900$. The thickness of the edge in the graph represents the strength of each coupling. Positive (antiferromagnetic) spin couplings are indicated in red, while negative (ferromagnetic) couplings are indicated in black. [Throughout this paper, we set the trap aspect ratio (transverse:axial) ${\omega }_{\perp }/{\omega }_{\parallel }=10$ [4, 5, 6].]

Figure 2

(a) Ground-state phases at $B=0$ characterized by the corresponding spin orders for $N=3,5,7,9$ ions. The transition points are positioned by the values of $\mu$, whereas the phonon mode frequencies are presented in parentheses. (b) Average polarization $⟨{\Psi }_G|\sum _n{\sigma }_n^x|{\Psi }_G⟩/N$ for $N=7$ ions at finite fields $B$ with $|{\Psi }_G⟩$ denoting the ground state. The right figure is a close-up near the critical point.

Figure 3

(a) The schematic phase diagrams for odd numbers of ions in the region between the 2nd and 3rd highest modes. For a small number of ions, the solid line represents a sharp transition, whereas the dashed lines represent a continuous crossover to the polarized state. (b),(c) The calculated theoretical phase diagrams for (b) $N=5$ and (c) $N=9$ ions. Color shows the order parameter defined by $P_{\mathrm{FM}}-P_K$, where $P_{\mathrm{FM}}\equiv \sum _{s=\uparrow ,\downarrow }|⟨s,s,\dots ,s|{\Psi }_G⟩{|}^2$ and $P_K\equiv \sum _{s=\uparrow ,\downarrow }|⟨s^{(N+1/2)},{\overline{s}}^{(N-1/2)}|{\Psi }_G⟩{|}^2+|⟨s^{(N-1/2)},{\overline{s}}^{(N+1/2)}|{\Psi }_G⟩{|}^2$ are the projection probabilities of the ground state $|{\Psi }_G⟩$ of the system to the Hilbert subspace with the ferromagnetic and the kink orders, respectively. ($|s^{(m)},{\overline{s}}^{(N-m)}⟩\equiv {\Pi }_{i=1}^m|s_i⟩{\Pi }_{i=m+1}^N|{\overline{s}}_i⟩$, where $s$ denotes the spin orientation with $\overline{\uparrow }\equiv \downarrow$ and vice versa.)

Figure 4

(a) The structure of the lowest four energy levels around the sharp ferromagnetic-kink phase transition. The two lowest states have a ferromagnetic (kink) order on the left (right) side. The transition width $W$ is defined as the distance between two points $A_1$ and $A_2$ located in the ferromagnetic (kink) phase, respectively, with the order parameter $P_{\mathrm{FM}}-P_K$ defined in the Fig. 3 caption changing from 0.71 (at $A_1$) to $-0.68$ (at $A_2$) for $N=9$ and $B/\overline{J}=0.05$. (b) Data points $\Delta E$ (in log scale) as a function of the ion number $N$ and magnetic field $B$, on top of a solid line representing the relation $\kappa \equiv \log (\Delta E/\overline{J})/[\frac{N-1}{2}\log (B/\overline{J})]=1$. For each $N$, those dots correspond to $B/\overline{J}$ increasing from 0.005 to 0.05 with a step size of 0.005. The largest deviation in this figure occurs at $N=11$, $B/\overline{J}=0.05$, where $\kappa -1=3.4\%$.