Effective spin quantum phases in systems of trapped ions

A system of trapped ions under the action of off-resonant standing waves can be used to simulate a variety of quantum spin models. In this work, we describe theoretically quantum phases that can be observed in the simplest realization of this idea: quantum Ising and $XY$ models. Our numerical calculations with the density matrix renormalization group method show that experiments with ion traps should allow one to access general properties of quantum critical systems. On the other hand, ion trap quantum spin models show a few unusual features due to the peculiarities of induced effective spin-spin interactions which lead to interesting effects like long-range quantum correlations and the coexistence of different spin phases.

Figure 1

(a) Evolution of the averaged effective transverse magnetization in $1∕r^3$ and linear trap Ising models ($N=100$ ions). We also plot $⟨{\sigma }_j^x⟩$ in the case of the central ion $(j=50)$ in a linear trap. (b) Local strength of the Ising interaction in a linear ion trap, and local value of the critical field.

Figure 2

(a) Second derivative of $m^x$ as a function of $B^x$ in ion trap Ising models ($N=100$ ions). Dashed line, $1∕r^3$ Ising model; solid line, average over the 20 central ions in a linear trap Ising model; dotted line, central $(j=50)$ ion in a linear trap Ising model (solid and dotted lines are almost on top of each other). (b) Contour plot of $⟨{\sigma }_j^x⟩$ on the plane of the coordinate $j$ and the magnetic field $B^x$.

Figure 3

(a) Fluctuation of the longitudinal effective magnetization, and (b) Néel order parameter. In both plots we consider the $1∕r^3$ Ising model and the central region (20 ions) of the linear trap Ising model.

Figure 4

(a) Absolute value of correlations $\mid C_{j_0,j_0+j}^{\alpha \alpha }\mid$ between the central ion $(j_0=50)$ and the rest of a chain with $N=100$ ions, in the case of the $1∕r^3$ Ising model. (i) $C_{j_0,j_0+j}^{xx}$, $B^x=0.72<B_c^x$; (ii) $C_{j_0,j_0+j}^{zz}$, $B^x=1.32>B_c^x$. (b) Zoom of the long-range tail of $C^{zz}$, which follows an algebraic decay with an exponent $\alpha =-3$.

Figure 5

(a) Absolute value of correlations $\mid C_{j_0,j_0+j}^{\alpha \alpha }\mid$ between the central ion $(j_0=50)$ and the rest of a chain with $N=100$ ions, in the case of the linear trap Ising model. (i) $C^{xx}$, $B^x=0.89<B_c^x$; (ii) $C^{zz}$, $B^x=1.72>B_c^x$. (b) Plot of $C_{j_0,j_0+j}^{xx}$ for both $1∕r^3$ and linear ion trap models, exactly at the critical point $B^x=B_c^x$. At intermediate distances, both curves are well fitted by a power-law decay $C_{j_0,j_0+j}^{xx}\propto {\mid j\mid }^p$, with $p=-2$, the same critical exponent that corresponds to the NN Ising model.

Figure 6

Correlation lengths ${\left({\xi }^{\alpha \alpha }\right)}^{-1}$ as a function of $B^x$ in both $1∕r^3$ and linear ion trap models. $\left(\mathrm{i}\right),\left({\mathrm{i}}^{\prime }\right)$ curves correspond to ${\left({\xi }^{xx}\right)}^{-1}$, whereas $\left(\mathrm{ii}\right),\left({\mathrm{ii}}^{\prime }\right)$, (ii) correspond to ${\left({\xi }^{zz}\right)}^{-1}$. ${\xi }^{zz}$ is not shown in the antiferromagnetic phase because $C_{i,j}^{zz}$ tends to a constant value in the absence of symmetry breaking.

Figure 7

(a) Absolute value of correlations $\mid C_{j_0,j_0+j}^{\alpha \alpha }\mid$ ($j_0=50$, $N=100$ ions) in the case of the $1∕r^3$ Ising model. Points, numerical results from the DMRG calculations; solid line, calculation with the HP transformation.

Figure 8

Phase diagram of $1∕r^3$ and linear trap $XY$ models. We plot the averaged global magnetization, as well as the evolution of $⟨{\sigma }_j^z⟩$ at the center of a linear trap ($N=50$ ions). The dotted line corresponds to the thermodynamical limit of the NN $XY$ model.

Figure 9

(a) Local critical field calculated from the evolution of $⟨{\sigma }_j^z⟩$ in the linear trap $XY$ model. (b) Spatial dependence of $⟨{\sigma }_j^z⟩$ in a linear trap $XY$ model with different values of $B^z$.

Figure 10

Correlation $C_{j_0,j_0+j}^{xx}$ between the central ion $j_0=26$ and the rest of the chain in the critical region as a function of ion separation $j$. $C_{j_0,j_0+j}^{xx}$ is well fitted by a power-law decay with the exponents $\alpha$ shown in the figure.