Magnetism in the two-dimensional dipolar XY model

Motivated by a recent experiment on a square-lattice Rydberg atom array realizing a long-range dipolar XY model [C. Chen et al., Nature (London) 616, 691 (2023)], we numerically study the model's equilibrium properties. We obtain the phase diagram, critical properties, entropies, variance of the magnetization, and site-resolved correlation functions. We consider both ferromagnetic and antiferromagnetic interactions and apply quantum Monte Carlo and pseudo-Majorana functional renormalization group techniques, generalizing the latter to a $U\left(1\right)$ symmetric setting. Our simulations perform extensive thermometry in dipolar Rydberg atom arrays and establish conditions for adiabaticity and thermodynamic equilibrium. On the ferromagnetic side of the experiment, we determine the entropy per particle $S/N\approx 0.5$, close to the one at the critical temperature, $S_c/N=0.585\left(15\right)$. The simulations suggest the presence of an out-of-equilibrium plateau at large distances in the correlation function, thus motivating future studies on the nonequilibrium dynamics of the system.

Figure 1

FM case: (a) Phase diagram for the XY model with FM dipolar interactions and a staggered $z$-field $\delta$; see Eq. (1). The dotted line is a guide to the eye separating the paramagnetic phase (PM) and in-plane XY FM order. (b) Finite-$T$ transition: In-plane magnetization squared $m_{\perp }^2$ multiplied by $L^{\eta }$ with $\eta =1$ as a function of inverse temperature $\beta$ for linear system sizes $L=17,33,65,127,257$ (with increasing slope) for the model without staggered field, $\delta =0$. (c) Quantum critical point (QCP): Finite-size scaling of the spin stiffness ${\chi }_{\mathrm{s}}$ multiplied by the inverse temperature $\beta$ as a function of staggered field $\delta$ for system sizes as in (b) and scaling the inverse temperature as $\beta \sim \sqrt{L}$ initiated by $\beta (L=33)=16$. Error bars are smaller than marker size in all subplots.

Figure 2

Upper: Entropy as a function of temperature for the dipolar system with $\delta =0$. The green curve for the AFM case is obtained from the free energy computed with pm-fRG; the blue curve is for the FM case where the entropy is determined from integration of energy (from QMC), Eq. (3). This is done separately in the disordered and ordered phases, each initiated by their analytically known asymptotic behavior (solid purple for spin waves and dashed purple for second-order high-temperature expansions; see Appendix pp2). The entropy per site at the FM critical temperature $T_c/J=1.923\left(1\right)$ (vertical dashed black line) is $S_c/N=0.585\left(15\right)$ (horizontal dashed black line shows the uncertainty window). Lower: Variance of the $z$ magnetization as a function of temperature for the AFM case ($\delta =0$) obtained by pm-fRG and for the FM case with $\delta =0$ and $\delta =2$ computed by QMC for a system with linear system size $L=33$. The dashed lines depict linear-in-temperature behavior as predicted by spin-wave theory (see text).

Figure 3

Experimental correlators for a $10\times 10$ system with open boundary conditions obtained for various durations $t=1\mu \mathrm{s}$ (green dots) and $t=8\mu \mathrm{s}$ (magenta dots), taken from Ref. [41] (and corrected for experimental uncertainties; see Ref. [41] and Appendix pp1). QMC curves at temperatures corresponding to the entropy of the initial system (dashed cyan line where error bars take the uncertainty on the initial state preparation into account) and the final system (dashed magenta line), obtained as a best possible fit. The green dots and green solid line (constant), seen in experiment for short durations $t\sim 1–2\mu \mathrm{s}$, are not compatible with the QMC, where the correlator decays linearly (cf. magenta solid line).

Figure 4

Equilibrium equal-time correlator $C_{ij}^{xx}$ versus distance. (a) FM case: The data are computed with QMC (dots) for linear system size $L=65$ and for various temperatures above and below the critical temperature $T_c/J=1.923\left(1\right)$. Statistical error bars are smaller than the symbol size. The magenta curve corresponds to an entropy comparable to the initial state entropy in the (much smaller) $N=42$ experimental system; the green curve is close to the one reported in Fig. 3(a) of Ref. [41] after readout errors have been taken into account (this agreement is coincidental; see text). The pm-fRG data (diamonds) for $T/J=2.222$ are shown for comparison. (b) AFM case: The data are computed with pm-fRG for an essentially infinite system. Lines are guides to the eye.

Figure 5

Entropy per site (left axis) and $m_{\perp }^2$ (right axis) for systems of size $L=33$ and $L=65$ with periodic boundary conditions (PBC) and for a system of size $L=10$ with open boundary conditions (OBC).

Figure 6

Dots denote spin correlation functions obtained for various experimental durations $t=1\mu \mathrm{s}$ (green), $t=2\mu \mathrm{s}$ (blue), and $t=8\mu \mathrm{s}$, taken from Ref. [41] (and multiplied by 1.208; see text and Ref. [41]). Note that the values of the experimental data at short distances are underestimated because of the presence of holes and the fact that $H_{\mathrm{XY}}$ is active during the $X$ to $Z$ rotation. Also shown are the spin correlation functions obtained in quantum Monte Carlo simulations (dashed lines) for a $10\times 10$ system with open boundary conditions, zero magnetic field, and zero staggered field $\delta$. The radial averaging is performed over all sites of the lattice. We argue that an equilibrium approach is justifiable for times $t\gtrsim 2\mu \mathrm{s}$. The cyan data indicate the range of values compatible with the error bars on the initial entropy, $S/N=0.30\left(3\right)$. The dashed yellow curve is incompatible with the experimental data, and therefore provides a clear upper bound to the temperature in experiment.

Figure 7

Variance of the $z$ magnetization per site as a function of temperature for a system of linear size $L=10$ with open boundary conditions inspired by the experimental setting [41]. The linear temperature dependence, predicted by linear spin-wave theory, is respected at low temperatures but with a different slope because of the finite system size. The linear temperature dependence holds however far beyond the regime of linear spin-wave theory (almost everywhere in the ordered phase), and thus enables experimental thermometry independent of numerical simulations, provided that the data are in equilibrium.

Figure 8

AFM case: Experimental spin correlation functions measured after $t=2\mu \mathrm{s}$ (blue dots), $t=4\mu \mathrm{s}$ (orange dots), and $t=8\mu \mathrm{s}$ (green dots), obtained from the authors of Ref. [41] (private communication). As in the FM case, we have multiplied the measured data with a factor of 1.208 to take into account measurement errors. The data for small distances are well reproduced by thermal pm-fRG simulations (dashed line, infinite system) at temperatures $T/\left|J\right|=0.56$, 0.78, and 0.96, respectively. The corresponding entropies are $S/N=0.550$, 0.596, and 0.618.

Figure 9

Energy per site as a function of inverse temperature $\beta J$ in the high-temperature regime. Results are shown for the AFM case obtained by pm-fRG (blue dots) and for the FM case obtained by QMC (red and cyan dots for $L=33$ and $L=65$, respectively, with periodic boundary conditions and using the bare potential). The energies are compared with the predictions of Eq. (B4) with truncations to second order (dashed black line) and third order (dashed green line for the FM case and dashed purple line for the AFM case).

Figure 10

Upper: Energy per site as a function of temperature $T/J$ for a system with periodic boundary conditions, showing strong finite-size effects. The energy is fitted against the predictions of Eq. (D4) (dashed lines). Lower: At low temperatures, the energy difference with the ground state energy is given by a form predicted by linear spin wave theory (dashed line, $\sim T^5$). It remains valid up to $T/J\approx 0.8$, although the fits work best up to $T/J\approx 0.4$.

Figure 11

Lower: Data for the rescaled in-plane magnetization squared can be collapsed onto a single curve in the critical region for a system with $\delta =0$ in the vicinity of the critical temperature. Critical exponents are $\nu =1$ and $\eta =1$, and the applied inverse critical temperature is ${\beta }_c=0.5198$. Upper: Collapse of the spin stiffness for the same system.

Figure 12

Lower: Data for the rescaled in-plane magnetization squared can be collapsed onto a single curve in the critical region in the vicinity of the quantum critical point ${\delta }_c=4.03$. Critical exponents are $z=1/2, \nu =1$, and $\eta =1$. Upper: Collapse of the spin stiffness for the same system.

Figure 13

The spin exchange correlation function decays asymptotically with a power law with power 3 in the paramagnetic phase, power 1 at the critical temperature, and approaches a constant in the ordered phase. The usage of the cord function on the $x$ axis is convenient to remove the leading order effects of the periodic boundary conditions. Simulations were performed for $\delta =0$ and $L=129$ with the inverse temperature indicated in the figure. The results are plotted along the $x$ axis only.

Figure 14

Extracting the decay of the $C^{+-}\left(x\right)-\langle m_{\perp }^2\rangle \sim x^{-p}$ and $|C^{zz}\left(x\right)|\sim x^{-q}$ correlation functions in the ordered phase on approach to the ground state for a system of linear system size $L=33$, with periodic boundary conditions and making use of the bare potential.

Figure 15

Benchmarking the pm-fRG on the AFM side ($J=1$): (a) Energy per spin over temperature as calculated from pm-fRG, both via $\langle H_{\mathrm{XY}}\rangle$ from equal-time correlators $C_{ij}^{xx}$ (dots) and from $E/N={\partial }_{\beta }\left(f\beta \right)$ (crosses) where $f$ is the free energy per spin. The horizontal line denotes the DMRG ground state energy reproduced from Ref. [41]. (b) Ratio of local spin susceptibilities (at $\nu =0$) calculated via the 4-point pseudo-Majorana vertex $\mathrm{\Gamma }$ and directly from the Majorana propagator $g$. We show the susceptibility ratio both for in-plane ($x$) and perpendicular ($z$) direction.