(2014); Kim et al. As the system size increases, this is a statement about eigenstates whose energies are that of a thermal ensemble at infinite temperature, which constitute the overwhelming majority of states in the spectrum of large systems. We denote the total number of sites in the system by N. First, it is important to mention some symmetries of this model in the square lattice, which is a bipartite lattice. Because of this, its Tc(g) is not known to us, and we are not able to report results for Ec(g) as we do for the ferromagnetic case. This is a clear signature of quantum chaos. following a quantum quench in a spin chain,”, D. Rossini, S. Suzuki, (2015), and indeed the onset of eigenstate thermalization has been seen to coincide with the onset of quantum chaos in some one-dimensional systems Santos and Rigol (2010a, b). We report on the realization of long-range Ising interactions in a cold gas of cesium atoms by Rydberg dressing. Using the results for Tc(g) from the latter study, we have calculate Ec(g) in all clusters (for g<3.044, which is the critical value for the ground-state phase transition). The quantization axis is set by a 1 G magnetic field B. We compute. Within the GOE: SstrGOE≈0.3646; i.e., it is, to leading order, independent of D. Hence, this quantity allows one to compare eigenvectors in different symmetry sectors without the need of extra manipulations Santos and Rigol (2010a). the user has read and agrees to our Terms and Also, in addition to the ferromagnetic 2D-TFIM considered in Ref. Mod. To address this, we have been improving access via several different mechanisms. Figure 2 shows the numerical results obtained for P(r) averaged over all momentum sectors excluding k=(0,0) and k=(π,π). Mech. So as long as xthr≆1, our statements about the eigenstate to eigenstate fluctuations are not restricted to eigenstates whose energy is that of a thermal ensemble at infinite temperature (for which Eα≅0 and xthr≅1). observables in small Hubbard lattices,”, E. Khatami, G. Pupillo, To make this point even clearer, in Fig. and thermalization in one-dimensional fermionic systems,”, L. F. Santos and M. Rigol, “Localization and the effects (10) and (11). Ising ferromagnet,” Eur. 2 display the average value of r as a function of the strength of the fields in the sectors with k≠(0,0) and k≠(π,π), in which all symmetries are resolved. 4 when ε=g≈2. We note that, the highly symmetric clusters with 16 and 18 sites [P(r) is not shown for those clusters] exhibit space symmetries (not necessarily inversion) in all momentum sectors. thermalization in systems with spontaneously broken symmetry,”, J. M. Deutsch, “Quantum The 1D transverse field Ising model can be solved exactly by mapping it to free fermions. Solid lines show Floquet mean-field model for the measured values χτR and hτX with no contrast loss, while edge of shaded region accounts for contrast C. Sign up to receive regular email alerts from Physical Review Letters. eigenstates obey the eigenstate thermalization hypothesis,”, S. Sorg, L. Vidmar, phases and transitions in transverse Ising models, R. M. Stratt, “Path-integral The first application of HZZ is split into two, with the second Rydberg pulse after the last microwave rotation, to keep the fixed points along the ϕ=0 meridian. of symmetries in the thermalization properties of one-dimensional quantum (1987); Suzuki et al. We should add that PGOE(r) in Eq. in the transverse field Ising chain: I. In order to do that, we need to identify which eigenstates fall in the part of the spectrum that exhibits long-range order. We note that this model has a Z2 symmetry associated with its invariance under the transformation ^σzi→−^σzi. -Provided two independent frameworks on how to think about the Ising Model, and ordering transitions, and how to obtain the observable thermodynamic quantities. electric fields,” Phys. -Showed that using a macorscopic or a microscopic mean MEAN-FIELD TREATMENTS OF THE ISING MODEL 89 Figure 5. In integrable regimes, on the other hand, the Poisson distribution results in. We study both models in various clusters with periodic boundary conditions, which allows us to present a finite size scaling analysis of the quantities of interest. (2013). one-dimensional bosonic and fermionic systems and its relation to 1) does not accommodate the Néel state; besides, it displays larger finite size effects in comparison to cluster 20A (see Appendix). An understanding of how quantum chaos onsets in different parts of the spectrum can be gained by studying the delocalization of the energy eigenstates in the basis used to diagonalize the Hamiltonian Santos and Rigol (2010b, a). The one-dimensional TFIM has been extensively studied theoretically in recent years in the context of quantum quenches in integrable systems Rossini et al. systems,”, C. Neuenhahn and F. Marquardt, “Thermalization of interacting fermions and delocalization in Fock properties,” J. -The Ising Model can be solved approximately by mean-field methods equivalent to those applied to obtain regular solution theory. In quantum chaotic systems, the presence of unresolved symmetries results in a distribution P(r) that is between PGOE(r) and PP(r). In general, the results for the antiferromagnetic model are slightly better than for the ferromagnetic one. Using a rate function approach we compute the leading order corrections to the mean field behavior analytically. All the clusters considered in this work are shown in Fig. lattice,” Nature, S. Sachdev, K. Sengupta, and S. M. Girvin, “Mott insulators in strong J. Flores, M. Horoi, The one-dimensional Ising model with a transverse field is solved exactly by transforming the set of Pauli operators to a new set of Fermi operators. One can see that, as expected, the distributions become increasingly peaked about (ΔSF)α=(ΔSAF)α=0 as the system size increases, and their support decreases significantly (consistent with decreasing exponentially fast) as the system size is increases. On the other hand, as per Berry-Tabor’s conjecture Berry and Tabor (1977), one expects a Poisson distribution when the system is integrable.

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