As one departs from the integrable limits g=0 and g=∞, and as one increases the system size, the structural entropy away from the edges of the spectrum becomes a smoother function of the energy of the eigenstates. 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. J. (2013); Beugeling et al. (2013). Leeuwen (1998)). (2002). The insets in Fig. systems,”, C. Neuenhahn and F. Marquardt, “Thermalization of interacting fermions and delocalization in Fock (c) Twisting strength Q vs k measured with (τR,τX)=(10,0)  μs in the four regions of the atomic cloud (i)-(iv) used in (a). The 1D transverse field Ising model can be solved exactly by mapping it to free fermions. observables in small Hubbard lattices,”, E. Khatami, G. Pupillo, (a) Trajectories S(k) for initial states |θ,ϕ⟩ (square data points) and up to k=4 Floquet cycles, obtained with dressing parameters (Ω,Δ)=2π×(2.8,25)  MHz. From the eigenstate thermalization hypothesis (ETH) D’Alessio et al. While many systematic studies of these topics have been undertaken in one-dimensional lattices Rigol (2009a, b); Santos and Rigol (2010b, a); Neuenhahn and Marquardt (2012); Genway et al. electrons in high-temperature superconductors,”, O. Bohigas, M. J. Giannoni,  and C. Schmit, “Characterization In the latter two sectors inversion is not the only space symmetry. properties,” J. of Three-Dimensional Heisenberg and Transverse-Ising Magnets,” in, Quantum Monte Carlo -Provided two independent frameworks on how to think about the Ising Model, and ordering transitions, and how to obtain the observable thermodynamic quantities. 3 and 4 are qualitatively similar. O. Giraud,  and G. Roux, “Distribution of the ratio of consecutive level The interactions are enhanced by coupling to Rydberg states in the vicinity of a Förster resonance. This highlights the importance of resolving all symmetries for one to be able to identify the presence of quantum chaos in the distribution of level spacings. (2013). 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. We used exact diagonalization to obtain the ground-state energies and corresponding eigenvectors for lattice sizes up to 24 spins. II, we introduce the model and discuss the numerical approach used to study it. Kim et al. By continuing you agree to the use of cookies. There are some momentum sectors that exhibit space symmetries. (2012); Khatami et al. We compute. Fratus and Srednicki (2015) using fluctuation-corrected mean-field theory Stratt (1986), indicating that much of the branch structure for the magnetization seen in Ref. For k=(0,0) and k=(π,π), ⟨r⟩ is close to ⟨r⟩P for all values of g studied. E, M. Rigol, V. Dunjko,  and M. Olshanii, “Thermalization and its A, M. Srednicki, “Chaos and 9(a) and 9(b), respectively. We make use of translation symmetry to break up the Hamiltonian in momentum sectors. 2 were obtained using the central half of the spectrum in each subspace analyzed. isolated system of quantum dipolar bosons after a quench,”, W. Beugeling, R. Moessner, L. Pollet,  and F. Heidrich-Meisner, “Relaxation and thermalization in the -The Ising Model can be solved approximately by mean-field methods equivalent to those applied to obtain regular solution theory. Ground state properties,”, M. S. L. du Croo de Jongh and J. M. J. van Leeuwen, “Critical behavior of the two-dimensional Ising model in (c) Alternating between interactions (HZZ) and microwave rotations (HX) produces an effective transverse-field Ising model. High temperature expansion,”, P. Pfeuty and R. J. Elliott, “The Ising You can find more about that e.g. correlators,” J. Your comment should inspire ideas to flow and help the author improves the paper. field Ising chain,”. B, E. Dagotto, “Correlated Y. Y. Atas, E. Bogomolny, Section IV is devoted to the analysis of eigenstate thermalization indicators and their scaling. All the clusters considered in this work are shown in Fig. (1987); Suzuki et al. Bloch spheres show average spin ⟨S⟩ at select times for two different initial states |θ⟩ (blue and red). Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made. Keep your question short and to the point. effects of fluctuations,”, Journal of Physics C: Solid State Physics. and the inverse participation ratio (IPR). An analysis of finite size effects reveals that quantum chaos and eigenstate thermalization occur in those systems whenever the fields are nonvanishing and not too large. The ferromagnetic 2D-TFIM has been intensively studied in the past (Refs. (b) Energy level diagrams for a single atom (left) and for a pair of atoms (right)., Physical Review Physics Education Research, Log in with individual APS Journal Account », Log in with a username/password provided by your institution », Get access through a U.S. public or high school library ». Obtaining R. (May 27, 2013) Leonard Susskind develops the Ising model of ferromagnetism to explain the mathematics of. In order to study quantum chaos indicators and calculate the expectation values of observables in eigenstates of the Hamiltonian, we use full exact diagonalization of clusters with different sizes and periodic boundary conditions. This can be done using the critical temperature for the phase transition Tc. A. Polkovnikov,  and M. Rigol, “From quantum chaos and eigenstate In the transverse field Ising model, the order parameter is not conserved, and increasing the transverse field enhances quantum fluctuations until they become strong enough to restore the symmetry of the ground state. model with a transverse field. The three-dimensional TFIM was used by DeGennes to characterize the ferroelectric phase of KH2PO2 De Gennes (1963), and the one-dimensional TFIM was recently realized in experiments with ultracold bosons in tilted optical lattices Simon et al. of symmetries in the thermalization properties of one-dimensional quantum 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. On the other hand, as per Berry-Tabor’s conjecture Berry and Tabor (1977), one expects a Poisson distribution when the system is integrable. Physical and Engineering Sciences, V. Oganesyan and D. A. Huse, “Localization of This is a clear indication of the occurrence of eigenstate thermalization. The results in Fig. We characterize the interactions by measuring the mean-field shift of the clock transition via Ramsey spectroscopy, observing one-axis twisting dynamics. (2008). In general, the results for the antiferromagnetic model are slightly better than for the ferromagnetic one. The decrease of the average value is consistent with an exponential for the systems with N≥12, independent of the value of xthr. In order to make a stronger statement about the eigenstate to eigenstate fluctuations, we compute their largest values, as well as their average, after removing all states with energy Eα such that (Eα−E0)/|E0|.

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