Abstract
Spontaneous symmetry breaking underlies much of our classification of phases of matter and their associated transitions^{1,2,3}. The nature of the underlying symmetry being broken determines many of the qualitative properties of the phase; this is illustrated by the case of discrete versus continuous symmetry breaking. Indeed, in contrast to the discrete case, the breaking of a continuous symmetry leads to the emergence of gapless Goldstone modes controlling, for instance, the thermodynamic stability of the ordered phase^{4,5}. Here, we realize a twodimensional dipolar XY model that shows a continuous spinrotational symmetry using a programmable Rydberg quantum simulator. We demonstrate the adiabatic preparation of correlated lowtemperature states of both the XY ferromagnet and the XY antiferromagnet. In the ferromagnetic case, we characterize the presence of a longrange XY order, a feature prohibited in the absence of longrange dipolar interaction. Our exploration of the manybody physics of XY interactions complements recent works using the Rydbergblockade mechanism to realize Isingtype interactions showing discrete spin rotation symmetry^{6,7,8,9}.
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Acknowledgements
We acknowledge the insights of and discussions with M. Aidelsburger, L. Henriet, V. Lienhard, J. Moore, C. Laumann, B. Halperin, E. Altman, B. Ye, E. Davis and M. Block. We are especially indebted to H.P. Büchler for insightful comments and discussions about the role of dipolar interactions in the XY model. The computational results presented were performed in part using the FASRC Cannon cluster supported by the FAS Division of Science Research Computing Group at Harvard University, the Savio computational cluster resource provided by the Berkeley Research Computing programme at the University of California, Berkeley and the Vienna Scientific Cluster. This work is supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 817482 (PASQuanS), the Agence Nationale de la Recherche (ANR, project nos. RYBOTIN and ANR22PETQ0004, project QuBitAF) and the European Research Council (advanced grant no. 101018511ATARAXIA). J.H. acknowledges support from the NSF OIA Convergence Accelerator programme under award number 2040549, and the Munich Quantum Valley, which is supported by the Bavarian state government with funds from the Hightech Agenda Bayern Plus. M.S. and A.M.L. acknowledge support by the Austrian Science Fund (FWF) through grant no. I 4548. D.B. acknowledges support from grant no. MCIN/AEI/10.13039/501100011033 (grant nos. RYC2018 025348I, PID2020119667GAI00 and European Union NextGenerationEU PRTRC17.I1). M.P.Z. acknowledges support from the Department of the Environment (DOE) Early Career programme and the Alfred P. Sloan foundation. N.Y.Y. acknowledges support from the Army Research Office (ARO) (grant no. W911NF2110262), the AFOSR MURI programme (grant no. W911NF2010136), the David and Lucile Packard foundation, and the Alfred P. Sloan foundation. M.B. and V.L. acknowledge support from NSF QLCI programme (grant no. OMA2016245). S.C. acknowledges support from the ARO through the MURI programme (grant no. W911NF1710323) and from the US DOE, Office of Science, Office of Advanced Scientific Computing Research, under the Accelerated Research in Quantum Computing programme.
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C.C., G.B., M.B. and G.E. contributed equally to this work. C.C., G.B., G.E., P.S. and D.B. carried out the experiments. M.B., L.L., V.S.L., J.H., S.C. and M.S. conducted the theoretical analysis and simulations. A.M.L., M.P.Z., T.L., N.Y.Y. and A.B. supervised the work. All authors contributed to the data analysis, progression of the project and on both the experimental and theoretical side. All authors contributed to the writing of the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Experimental procedures and sequence.
a, Fluorescence image of the atoms in a fully assembled 6 × 7 array. b, Scheme for the preparation of the initial staggered state. c, Detected staggered state, corresponding to the situation for which all the atoms in sublattice A are in \(\left\uparrow \right\rangle \), and all the atoms in sublattice B are in \(\left\downarrow \right\rangle \). d, Schematics of the atomic level diagram. e, Experimental sequence.
Extended Data Fig. 2 Simplified error tree associated to the preparation of the initial Néel state, for a the atoms in sublattice A (nonaddressed), and b in sublattice B (addressed).
For simplicity, the events with a probability of order 2 or higher in the η_{i}, ϵ, \({{\epsilon }}^{{\prime} }\) are disregarded.
Extended Data Fig. 3 Time dependence of the correlations along x in the FM case for a 10 × 10 lattice.
a, Time evolution of the nearestneighbour correlations along x (different colors correspond to different times). b, Spatial correlations as a function of distance, measured at different times t = {0.0, 0.5, 1.0, 2.0, 8.0} μs indicated by dashed lines in a.
Extended Data Fig. 4 DMRG ground state calculations.
a, Realspace correlation profile ∣C^{x}(d)∣ on L × L square clusters with open boundary conditions. The ground state of \({H}_{{\rm{XY}}}^{{\rm{FM}}}\) clearly exhibits XY LRO at all system sizes. For \({H}_{{\rm{XY}}}^{{\rm{AFM}}}\) and H_{nn}, the correlations decrease at long distances, but this decay is reduced as L increases. b, Finitesize scaling of the magnetization \({m}_{{\rm{FM}}/{\rm{AFM}}}^{2}\). All three models are consistent with \({m}_{{\rm{FM}}/{\rm{AFM}}}^{2} > 0\) as L → ∞. c, Dependence of \({m}_{{\rm{FM}}/{\rm{AFM}}}^{2}\) on the interaction distance cutoff \({R}_{\max }\). At each system size, the ground state correlations are wellconverged by \({R}_{\max }\approx 4\). d–f, Ground state properties of H_{XY} + H_{Z} as a function of δ. There is a smooth crossover from the XY ordered state at δ = 0 to the staggered paramagnet as δ → ∞. The − dm^{2}/dδ peaks (f) are finitesize incarnations of the quantum phase transition expected in the thermodynamic limit; we use their centers to define the crossover point ħδ_{c}/J.
Extended Data Fig. 5 Excitation gap for two adiabatic preparation protocols.
a, Minimal energy gaps of \({H}_{{\rm{XY}}}^{{\rm{AFM(FM)}}}+{H}_{{\rm{Z}}}\) to the lowest excited state in the M^{z} = 0 sector as a function of ħδ/J. We here only consider gaps among states with momentum \({\bf{k}}=0\) and fully symmetric under the lattice pointgroup, which reflects the setup in the (ideal) experiment. Blue (red) curves show the results for the AFM (FM) model. Darker colors correspond to larger system sizes. The inset shows a sketch of the expected phase diagram. b, Same as a, but for the protocol with Hamiltonian \({H}_{{\rm{XY}}}^{{\rm{AFM(FM)}}}+{H}_{{\rm{X}}}\). Here we cannot restrict the analysis to a single M^{z} sector since it is not conserved. c, Cumulatively integrated \(1/{\Delta }_{\min }^{2}\) [starting from the largest value ħδ/J = 24] for the gaps shown in a. The values at ħδ/J = 0 measure how difficult it is to prepare the ground state of \({H}_{{\rm{XY}}}^{{\rm{AFM(FM)}}}\) by sweeping δ. d, Same as c but for the gaps along Ω, as shown in b. The inset shows a sketch of the expected phase diagrams for \({H}_{{\rm{XY}}}^{{\rm{AFM(FM)}}}+{H}_{{\rm{X}}}\).
Extended Data Fig. 6 Numerical simulation of the adiabatic preparation for the 6 × 7 lattice.
We compare the predictions from the tMPS simulations (disorder ensemble average in dark teal, standard deviation in light teal) to the experimental data (grey), as measured at lightshift δ(t) = δ. We also show the groundstate expectation value from DMRG (purple) a, The staggered polarization P_{z} of the FM. Theory and experiment agree remarkably well, except for an offset at small δ, due to the lightshiftinduced depumping. Inset: ramp δ(t) used for the FM simulation. b, The ferromagnetic magnetization \({m}_{{\rm{FM}}}^{2}(\delta )\). We find excellent agreement between experiment and numerics for δ > 2, including near the phase transition at \({\delta }_{c}^{{\rm{FM}}}=5.5\) (red dashed line). The two diverge somewhat at smaller δ (later times), likely due to decoherence and unmodeled systematic measurement errors. c, d, Corresponding results for the AFM. For P_{z}, the tMPS simulation accurately reproduces the experimental data across the whole δ(t) sweep. For δ far above \({\delta }_{c}^{{\rm{AFM}}}\,=\,0.6\) (blue dashed line), there are manybody Rabi oscillations characteristic of the paramagnetic phase. c, Inset: ramp δ(t) used for the AFM simulation. d, Inset: zoomin of lower left corner. At small δ (late times), the magnetization \({m}_{{\rm{AFM}}}^{2}\) measured in experiment is below that predicted from the simulations.
Extended Data Fig. 7 Energetics of the simulated adiabatic preparation.
a, b, Interaction energy density \({E}_{{\rm{XY}}}/\bar{N}(\delta )\) in the tMPS simulations of the 6 × 7 lattice. The teal line and envelope are the disorder ensemble average and standard deviation, respectively. Following a single state with minimal initialization errors (pink line), we see that E_{XY} tightly follows the DMRG ground state value (purple), confirming that diabatic errors are negligible. c, d, Energy gaps Δ_{0} between the ground state and the first excited state in the S^{z} = 0 sector, obtained from DMRG. For the nearideal initial state, the final energy density (pink dotted line) falls below the gap in both the FM and AFM case.
Extended Data Fig. 8 Finitetemperature properties of H_{XY} + H_{Z}.
a, Phase diagram of \({H}_{{\rm{XY}}}^{{\rm{FM}}}+{H}_{{\rm{Z}}}\) at finite temperature T and lightshift δ, computed from METTS on a 6 × 7 array in the M^{z} = 0 sector. We also include T = 0 points calculated from DMRG. The region with large magnetization \({m}_{{\rm{FM}}}^{2}\) at small δ and small T should correspond to the LRO phase in the thermodynamic limit. The colorbar is chosen so that dark red corresponds to the final \({m}_{{\rm{FM}}}^{2}\) calculated in the tMPS simulation, absent measurement errors. Thin black lines are equalmagnitude contours to guide the eye. b,c, Estimated temperature of a quench experiment with final lightshift δ_{f} and quench magnitude δ_{q}, taking the prequench configuration to be either the DMRG ground state (b) or the tMPS ramp simulation ensemble (c). The oscillatory behavior seen in c stems from the paramagnetic Rabi oscillations discussed in Sec. D3. d, e Corresponding magnetization \({m}_{{\rm{FM}}}^{2}\) of the system at temperature T_{eff}(δ_{f}, δ_{q}). fj Analogous results for the antiferromagnet. The region with finite magnetization \({m}_{{\rm{AFM}}}^{2}\) is expected to become an algebraicordered (BKT) phase in the thermodynamic limit.
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Chen, C., Bornet, G., Bintz, M. et al. Continuous symmetry breaking in a twodimensional Rydberg array. Nature 616, 691–695 (2023). https://doi.org/10.1038/s41586023058592
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DOI: https://doi.org/10.1038/s41586023058592
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