Probing Quantum Criticality and Symmetry Breaking at the Microscopic Level

We report on an experimental study of the Lipkin-Meshkov-Glick model of quantum spins interacting at infinite range in a transverse magnetic field, which exhibits a ferromagnetic phase transition in the thermodynamic limit. We use dysprosium atoms of electronic spin $J=8$, subjected to a quadratic Zeeman light shift, to simulate $2J=16$ interacting spins $1/2$. We probe the system microscopically using single magnetic sublevel resolution, giving access to the spin projection parity, which is the collective observable characterizing the underlying ${\mathbb{Z}}_2$ symmetry. We measure the thermodynamic properties and dynamical response of the system, and we study the quantum critical behavior around the transition point. In the ferromagnetic phase, we achieve coherent tunneling between symmetry-broken states, and we test the link between symmetry breaking and the appearance of a finite order parameter.

Figure 1

(a) Scheme of experiment, based on laser-induced dynamics of the electronic spin of dysprosium atoms (quadratic light shift of $\propto -\lambda J_x^2$) in the presence of a magnetic field (Zeeman coupling ${\omega }_z{J}_z$). (b), (c), and (d) Classical energy landscapes on the southern hemisphere of the generalized Bloch sphere for $\lambda =0$, ${\omega }_z$, and $1.5{\omega }_z$, respectively. (e) Finite-size phase diagram, showing the spin pair correlator $⟨{\sigma }_{1x}{\sigma }_{2x}⟩$, with a ferromagnetic phase in the thermodynamic limit for $\lambda >{\omega }_z$ (green line). For a finite $N$, the phase transition is smoothened over a quantum critical region (dashed red area).

Figure 2

(a), (c), and (e) Measured projection probabilities ${\mathrm{\Pi }}_m(\widehat{\mathbf{n}})$ for $\widehat{\mathbf{n}}=\widehat{\mathbf{x}}$, $\widehat{\mathbf{y}}$, and $\widehat{\mathbf{z}}$ [Figs. 2, 2, and 2, respectively] as a function of the interaction strength $\lambda$. (b), (d), and (f) Evolution of the spin pair correlator $⟨{\sigma }_{1x}{\sigma }_{2x}⟩$ [Fig. 2], its variance [inset of Fig. 2], the correlator $⟨{\sigma }_{1y}{\sigma }_{2y}⟩$ [Fig. 2], and the mean parity $p_z$ [Fig. 2]. Solid blue, dotted black, and dashed red lines correspond to the LMGM, the classical mean-field model, and the critical Hamiltonian, respectively. No averaging is performed in Fig. 2. In other panels, all data are the averages of about five independent measurements. In all figures, error bars represent the $1\text{−}\sigma$ statistical uncertainty.

Figure 3

(a) Parity gap $\delta$ between even- and odd-parity sectors, and dynamical gap $\mathrm{\Delta }$ between the ground and first even-parity states as a function of the coupling $\lambda$. Solid (dashed) lines are LMGM (mean-field) predictions. [(a) inset] Energy level scheme of the six lowest eigenstates for $\lambda =0.5{\omega }_z$. (b) Breathing mode oscillation performed for $\lambda =1.04(2){\omega }_z$. The solid line is a sine fit of frequency $\mathrm{\Delta }$. (c) Dipole mode oscillation performed for $\lambda =0.79(2){\omega }_z$. The solid line is a sine fit of frequency $\delta$.

Figure 4

(a) Parity gap $\delta$ as a function of $\lambda$ (blue dots) compared with LMGM (blue line), mean-field theory (black dotted line), semiclassical tunneling (red dashed line), and perturbation theories (green dash-dotted line). The solid black line is the mean value of $\delta$ expected from the LMGM and averaged over magnetic field fluctuations. (b) and (c) Time evolutions of projection probabilities ${\mathrm{\Pi }}_m(\widehat{\mathbf{x}})$ during dipole mode oscillation for $\lambda =0.79(2){\omega }_z$ [Fig. 4] and $\lambda =1.36(2){\omega }_z$ [Fig. 4]. The most probable projection $m^{*}$ is plotted as a blue line.

Figure 5

(a) and (b) Projection probabilities ${\mathrm{\Pi }}_m(\widehat{\mathbf{x}})$ [Fig. 5] and ${\mathrm{\Pi }}_m(\widehat{\mathbf{z}})$ [Fig. 5] in the ground state as a function of ${\omega }_x$ for $\lambda =1.40(3){\omega }_z$. (c) and (d) Order parameter $⟨{\sigma }_{1x}⟩$ and mean parity $p_z$ computed from Figs. 5 and 5 and compared to the LMGM (solid lines) and the mean-field order parameter values (dotted lines).