In order to check whether eigenstate thermalization occurs in the models studied in Sec. 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. What is the best way to remove 100% of a software that is not yet installed? systems,”, C. Neuenhahn and F. Marquardt, “Thermalization of interacting fermions and delocalization in Fock High temperature expansion,”, P. Pfeuty and R. J. Elliott, “The Ising 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. We have shown how quantum chaos and eigenstate thermalization onset in those systems as one departs from integrable limits and increases the system size. (2011); Foini et al. Rev. for the antiferromagnetic 2D-TFIM with a longitudinal field.  and B. K. Chakrabarti, Quantum Ising thermalization in systems with spontaneously broken symmetry,”, J. M. Deutsch, “Quantum (2012a, b). We report results for the largest, the fifth largest, and the tenth largest values of those quantities in the windows selected, as well as the average value (which is dominated by the aforementioned “infinite-temperature” states). How to calculate the exchange constant of the Ising model? (1987); Suzuki et al. observables in small Hubbard lattices,”, E. Khatami, G. Pupillo, 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. The narrowest support for Sstrα within a small energy window in the middle of the spectrum is seen in Fig. • In the ferromagnetic case there is a phase transition. B, D. Rossini, A. Silva, H &= -\Gamma\sum_i S_i^z -J\sum_i S_i^x S_{i+1}^x \tag{1} \\ ground for many-body quantum chaos,” Phys. Should we leave technical astronomy questions to Astronomy SE? Section IV is devoted to the analysis of eigenstate thermalization indicators and their scaling. (2013); Beugeling et al. (1981); Flores et al. A. Polkovnikov,  and M. Rigol, “From quantum chaos and eigenstate 8(a) and 8(b) we plot the results obtained for (ΔSF)α and (ΔSAF)α, respectively, as a function of the number of lattice sites for two values of xthr. This transformation maps the uniform magnetization per site M=⟨∑iσzi⟩/N, which is the order parameter in the ferromagnetic case, onto the staggered magnetization per site Mstag=⟨∑i(−1)ix+iyσzi⟩/N, which is the order parameter in the antiferromagnetic case, and vice versa. You can find more about that e.g. P. Calabrese, F. H. L. Essler,  and M. Fagotti, “Quantum (2013); Beugeling et al. Evaluating the quality of Monte Carlo simulations for 3D Ising model, Magnetization in Quantum Transverse Ising Model: Mean Field Theory vs Reality, OOP implementation of Rock Paper Scissors game logic in Java, What modern innovations have been/are being made for the piano. Correlation functions, Thermodynamic properties of a Heisenberg antiferromagnet. 5 we compare the structural entropy of the ferromagnetic 2D-TFIM (g=1) for systems with N=12 and 16 sites when: (a) one accounts for translational, Z2, and inversion symmetry (when present), and (b) one does not resolve any symmetry (in which case we can fully diagonalize the Hamiltonian only up to N=16). M. Srednicki,  and M. Rigol, “Fluctuation-dissipation theorem in an 9(a) and 9(b), respectively. quantum thermalization,” Phys. Another way to state this is that the eigenstate to eigenstate fluctuations of the expectation values of the observables is very small, more precisely, exponentially small in the system size D’Alessio et al. 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. (2012); Khatami et al. It is worth stressing that, given the fact that our Hamiltonian contains only short-range interactions, the GOE prediction is valid only away from the edges of the spectrum Brody et al. Can it be justified that an economic contraction of 11.3% is "the largest fall for more than 300 years"? K. R. Fratus and M. Srednicki, “Eigenstate Quantum phase diagram of two-dimensional transverse field Ising model: unconstrained tree tensor network and mapping analysis. In the momentum sectors with k=(0,0) and k=(π,π), Fig. We should stress, however, that clusters with the same number of sites but different geometries can display differences in the energy-eigenstate expectation values. in the transverse-field Ising chain,”, L. Foini, L. F. Cugliandolo,  and A. Gambassi, “Fluctuation-dissipation relations and critical quenches in the transverse Quick link too easy to remove after installation, is this a problem? How to measure the spin-spin correlation in a Monte Carlo simulation of the Ising model? 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. 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 Transverse field quantum Ising chain. 10-year back file (where available). Fratus and Srednicki (2015), here we study the antiferromagnetic 2D-TFIM in the presence of a longitudinal field. 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). II, we introduce the model and discuss the numerical approach used to study it. You will only need to do this once. The results in Fig. What is the cost of health care in the US? We study the onset of eigenstate thermalization in the two-dimensional transverse field Ising model (2D-TFIM) in the square lattice. in the transverse field Ising chain: I. eigenstate thermalization,”, H. Kim, T. N. Ikeda,  and D. A. Huse, “Testing whether all The perturbation series is classified with respect to the small parameter 1/z, where z is the number of psins interacting with a given spin; this generates the molecular-field approximation and the random-phase approximation as the lowest-order description of the thermodynamic functions and correlation functions respectively. (2013). Figure 10 shows the energy-eigenstate expectation values of ^SF for the clusters 20A and 20B (see Fig. B, E. Dagotto, “Correlated States at the edges of the spectrum need to be removed because, as mentioned before, they neither exhibit quantum chaos nor eigenstate thermalization. These estimates are significantly lower than those made in Ref. The one-dimensional TFIM has been extensively studied theoretically in recent years in the context of quantum quenches in integrable systems Rossini et al. Find out more about journal subscriptions at your site. integrable quantum spin chain: Local versus nonlocal operators and effective J. Rev. F. M. Izrailev, “Simple models Rev. In Table 1, we show the breakup of the Hilbert space for all the clusters studied. It features a lattice with nearest neighbour interactions determined by the alignment or anti-alignment of spin projections along the $${\displaystyle z}$$ axis, as well as an external magnetic field perpendicular to the $${\displaystyle z}$$ axis (without loss of generality, along the $${\displaystyle x}$$ axis) which creates an energetic bias for one x-axis spin direction over the other. While numerical degeneracies lead to obvious quantitative differences between panels (a) and (b), the results are qualitatively similar and, with increasing system size, one could potentially identify that there is quantum chaos in the system even if one does not resolve any of the symmetries of the model. one-dimensional bosonic and fermionic systems and its relation to This was possible via a mapping of the site occupation of the bosonic atoms onto pseudo-spins Sachdev et al. a transverse field: A density-matrix renormalization calculation,”, O. Nagai, Y. Yamada,  and Y. Miyatake, “Monte Carlo Simulations The decrease of the average value is consistent with an exponential for the systems with N≥12, independent of the value of xthr. (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). It was examined by two of us (KRF and MS) Fratus and Srednicki (2015) to understand whether eigenstate thermalization Deutsch (1991); Srednicki (1994); Rigol et al. We restrict our analysis to the case ε=g. of a continuous time cluster algorithm to the two-dimensional random quantum Why did MacOS Classic choose the colon as a path separator? O. Giraud,  and G. Roux, “Distribution of the ratio of consecutive level In this work, we present an in depth study of quantum chaos and eigenstate thermalization indicators in the 2D-TFIM in the square lattice. MathJax reference. 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.

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