One-to-one quantum simulation of the low-dimensional frustrated quantum magnet TmMgGaO$_4$ with 256 qubits

Abstract

Low-dimensional materials exhibit exotic properties due to enhanced quantum fluctuations, making the understanding of their microscopic origin central in condensed matter physics. Analogue quantum simulators offer a powerful approach for investigating these systems at the microscopic level, particularly in large-scale regimes where quantum entanglement limits classical numerical methods. To date, analogue simulators have largely focused on universal Hamiltonians rather than material-specific quantitative comparisons. Here we use a Rydberg-based quantum simulator to study the bulk-layered frustrated quantum magnet TmMgGaO$_4$. Magnetisation measurements obtained from the quantum simulator show excellent agreement with independent measurements performed in a magnetic laboratory facility, validating the proposed effective two-dimensional microscopic Hamiltonian. Building on this quantitative correspondence, we investigate on both platforms the antiferromagnetic phase transition. We further probe the role of quantum fluctuations via snapshot analysis, connecting our results to integrated inelastic neutron scattering data. Finally, we access, with the simulator, non-equilibrium dynamics on picosecond material timescales, including frequency response and thermalisation of observables.

Figures (15)

• Figure 1: Bridging macroscopic measurements and quantum simulation of a frustrated magnet. a, Sketch of experimental setup for thermodynamic measurements of AC susceptibility $\chi^z_{\rm AC}(\mu_0H,T)$ on a TmMgGaO$_4$ single-crystal sample in longitudinal magnetic fields up to $18\,\mathrm{T}$ and at temperatures down to $20\,\mathrm{mK}$. Zoom on the crystal internal structure. b, Microscopic description of TmMgGaO$_4$ and mapping to a programmable quantum simulator. (left) Two-dimensional triangular lattice of Tm$^{3+}$ ions; inset shows the effective pseudospin-$1/2$ doublet $\ket{\Phi_\pm}$ with crystalline-field induced splitting $\Delta_x/2$ and separated from higher energy levels. (centre) Microscopic model with easy-axis Ising interactions $J_1$ and $J_2$, and transverse and longitudinal fields $(\Delta_x,\Delta_z)$. (right) Effective Rydberg Hamiltonian with qubit states $\ket{g}$, $\ket{r}$ coupled by Rabi frequency $\Omega/2\pi$ and detuning $\delta/2\pi$, interacting via a distance-dependent potential $U(r_{ij})$. c, Neutral-atom QPU with $N=256$ atoms, time-dependent adiabatic protocols $\Omega(t),\delta(t)$ and read out by site-resolved projective measurements $z_i$. d, Zero-temperature phase diagram $(\Delta_z/J_1,\Delta_x/J_1)$ from density-matrix-renormalisation group algorithm (Methods), showing paramagnetic (all spins up; hollow dots) and $1/3$-filling ordered (one sublattice with spins down; filled dots) phases; $\Delta_x/J_1$ of TmMgGaO$_4$li_kosterlitz-thouless_2020 used for transition scan (solid black). Classical and quantum critical points estimates (gray). e, (left axis) AC magnetisation $M^z_{\rm AC}(\Delta_z/J_1)$ at $T=20~\mathrm{mK}$ (blue) compared to QPU magnetisation $M^z_{\rm QPU}(\Delta_z/J_1)$ at $N=256$ (green dots); error bars reflect the finite number of QPU measurements. Magnetisation obtained from cut of DMRG phase diagram of $\textbf{d}$ in the quasi-classical regime $\Delta_x/J_1=0.2$ (dotted line). Horizontal axes map experimental ($\mu_0H$), microscopic-model ($\Delta_z/J_1$), and quantum simulator ($\delta/2\pi$) parameters.
• Figure 1: Macroscopic photograph of the TmMgGaO$_4$ single-crystal sample synthesised. The exact colours of the sample in this photograph are influenced by lighting conditions and should not be relied upon for precise colour representation. The samples appear white and slightly opaque under normal viewing conditions.
• Figure 2: Probing the paramagnet to $1/3$-order quantum phase transition.a, AC magnetisation $M^z_{\mathrm{AC}}(\Delta_z/J_1)$ of TmMgGaO$_4$ at decreasing $T$; inset: AC susceptibility $\chi^z_{\rm AC}$ highlighting emerging features (arrows). b, AC susceptibility $\chi^z_{\mathrm{AC}}(\Delta_z/J_1,T)$. Transition points extracted from field (squares) and temperature (diamonds) sweeps; previous results from hu_evidence_2020qin_field-tuned_2022 shown for comparison. c, QPU magnetisation $M^z_{\mathrm{QPU}}(\Delta_z/J_1)$ for $N=100$, measured on three devices : FC1 (red), FM2 (blue) and FM1 (green dots); error bars reflect statistical noise from the finite number of measurements taken. DMRG data obtained at smaller $\Delta_x/J_1$ (black line). d, Structure factor $S_{\rm QPU}^z(\mathbf{q}_{1/3})$ from FM1 data at $N=49,100,169,256$ (dots); a cubic fit (solid line) for $N=256$ smooths the noise fluctuations to help estimating the location of the quantum phase transition (dashed) with $68\%$ confidence interval (shaded) obtained from varying fit windows. Insets: momentum-space maps of $S^{zz}_{\rm QPU}(\mathbf{q})$ across the transition for $|\mathbf{q}|\leq2\pi$.
• Figure 2: Estimation of quantum and thermal critical points through AC susceptibility measurements of TmMgGaO$_4$.a, AC susceptibility as a function of the applied magnetic field along the c axis at different temperatures and for excitation frequency of $200$ Hz Inset: Frequency dependence of $\chi^z_{\mathrm{AC}}$ at $0.02$ K. b, Comparison of $\chi^z_\text{AC}(20\text{mK})$ and its second field derivative. The minima in the second derivative are used to locate the $\mu_0 H$ fields corresponding to the quantum phase transition and the thermal bump in Fig. \ref{['fig:fig2']}b. The derivatives were smoothed using a Savitzky–Golay filter with a 50-point window. c, Temperature dependence of $\chi^z_{\mathrm{AC}}$ for different magnetic fields applied along the $c$ axis. The curves are offset in the y-axis for better visualisation. The white points serve as guides to the eye, indicating the positions extracted for the thermal dome (T sweeps) in Fig. \ref{['fig:fig2']}b.
• Figure 3: Quantum fluctuations and emergence of $1/3$ order.a, (left axis) Low-temperature integrated inelastic neutron scattering signal from qin_field-tuned_2022 (blue squares), given by the difference between total scattering at $T=0.13\,\mathrm{K}$ and elastic scattering at $T=40\,\mathrm{mK}$. (right axis) Variance of the QPU magnetisation $(\Delta M^z_{\mathrm{QPU}})^2$ as a function of applied field $\Delta_z/J_1$ for $N=256$. QPU measurements (green dots) with error bars reflecting statistical noise from the finite number of shots. b, Representative single-shot spin configurations from the QPU, with plaquettes coloured by triangle configuration. c, Phase diagram of the mean $1/3$-cluster size versus temperature $T/J_1$ (we set $k_\text{B}/\hbar=1$) and field $\Delta_z/J_1$ from QMC at $N=256$. d, Vertical cuts from c at $\Delta_z/J_1=3,~5.2,~7.7$ (light to dark blue).