We study the onset of eigenstate thermalization in the two-dimensional transverse field Ising model (2D-TFIM) in the square lattice. We consider two nonequivalent Hamiltonians: the ferromagnetic 2D-TFIM and the antiferromagnetic 2D-TFIM in the presence of a uniform longitudinal field. We use full exact diagonalization to examine the behavior of quantum chaos indicators and of the diagonal matrix elements of operators of interest in the eigenstates of the Hamiltonian. 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.

A strongly non-integrable system is expected to satisfy the eigenstate thermalization hypothesis, which states that the expectation value of an observable in an energy eigenstate is the same as the thermal value. This must be revised if the observable is an order parameter for a spontaneously broken symmetry, which has multiple thermal values. We propose that in this case the system is unstable towards forming nearby eigenstates which yield each of the allowed thermal values. We provide strong evidence for this from a numerical study of the 2D transverse-field quantum Ising model.The eigenstate thermalization hypothesis (ETH) can explain how an isolated, quantum many-body system in an initial pure state can come to thermal equilibrium (as determined by measurements of a specified set of observables) in finite time [1][2][3]. ETH is expected to hold in systems without disorder that are sufficiently far (in parameter space) from points of integrability, for observables that are sufficiently simple (e.g. local) functions of the fundamental degrees of freedom. In recent years ETH has been the subject of intensive analytic and numerical investigations, e.g. [3][4][5][6][7][8][9][10][11][12][13][14]; see [15] for an overview including the connection to experimental results in cold atoms and other systems.The key statement of ETH is that expectation values of a relevant observable M in an energy eigenstate |α (of the full many-body hamiltonian H) take the formwhere M(E) is a smooth function of E and E α is the energy eigenvalue. In a system with N 1 degrees of freedom, this is enough information to show that M(E) is equal, up to O(N −1/2 ) corrections, to the canonical thermal average of the operator M ,where the temperature T is implicitly determined as a function of energy E by the usual relationA second key statement of ETH is that the off-diagonal matrix elements of M in the energy basis, α|M |β with α = β, are exponentially small in N . This is needed to explain why the diagonal matrix elements of eq. (1) dominate the instantaneous expectation value of M (in a generic time-dependent state) at almost all times, which in turn is necessary for thermal equilibrium to be maintained once it has been achieved. However this aspect of ETH will not be our focus.The ETH paradigm must be revisited for a system that is capable of exhibiting spontaneous symmetry breaking (SSB). Suppose that the observable M is an order parameter for a global symmetry. At energies corresponding to the broken-symmetry phase, and in the infinite-volume limit, we expect the system to have states with the same energy but with different values of M (that are related by the symmetry). In this case, eq. (1) cannot hold as written. We conjecture that, instead, the single smooth function M(E) is replaced by a multivalued function, with one branch for each allowed value of the order parameter.We note that the compatibility of ETH and SSB was assumed to hold in [16], in which the tunnelling dynamics of the order parameter were studied in "Schrodinger cat" ...

The branched pattern typically observed through the scanning gate microscopy (SGM) of two dimensional electron gases in the presence of weak, smooth disorder has recently been found to be robust against a very large shift in the Fermi energy of the electron gas. We propose a toy model, where the potential landscape reduces to a single localized feature, that makes it possible to recast the understanding of branch formation through the effect of caustics in an appropriate set of classical trajectories, and it is simple enough to allow for a quantitative analysis of the energy and spatial dependence of the branches. We find the energy stability to be extremely generic, as it rests only upon the assumptions of weak disorder, weak scattering, and the proportionality of the SGM response to the density of classical electron trajectories. Therefore, the robustness against changes of the electron's Fermi energy remains when adopting progressively realistic models of smooth disorder.Journal reference: Phys. Rev. B 100, 155435 (2019) arXiv:1904.07777v2 [cond-mat.mes-hall]