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Register a new userSensitivity of the spectral form factor to short-range level statistics
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The spectral form factor is a dynamical probe for level statistics of quantum systems. The early-time behaviour is commonly interpreted as a characterization of two-point correlations at large separation. We argue that this interpretation can be too restrictive by indicating that the self-correlation imposes a constraint on the spectral form factor integrated over time. More generally, we indicate that each expansion coefficient of the two-point correlation function imposes a constraint on the properly weighted time-integrated spectral form factor. We discuss how these constraints can affect the interpretation of the spectral form factor as a probe for ergodicity. We propose a new probe, which eliminates the effect of the constraint imposed by the self-correlation. The use of this probe is demonstrated for a model of randomly incomplete spectra and a Floquet model supporting many-body localization.
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We numerically study the level statistics of the Gaussian $beta$ ensemble. These statistics generalize Wigner-Dyson level statistics from the discrete set of Dyson indices $beta = 1,2,4$ to the continuous range $0 < beta < infty$. The Gaussian $beta$ ensemble covers Poissonian level statistics for $beta to 0$, and provides a smooth interpolation between Poissonian and Wigner-Dyson level statistics. We establish the physical relevance of the level statistics of the Gaussian $beta$ ensemble by showing near-perfect agreement with the level statistics of a paradigmatic model in studies on many-body localization over the entire crossover range from the thermal to the many-body localized phase. In addition, we show similar agreement for a related Hamiltonian with broken time-reversal symmetry.
We theoretically study correlations present deep in the spectrum of many-body-localized systems. An exact analytical expression for the spectral form factor of Poisson spectra can be obtained and is shown to agree well with numerical results on two models exhibiting many-body-localization: a disordered quantum spin chain and a phenomenological $l$-bit model based on the existence of local integrals of motion. We also identify a universal regime that is insensitive to the global density of states as well as spectral edge effects.
Angle-resolved photoemission spectroscopy (ARPES) is one of most powerful techniques to unravel the electronic properties of layered materials and in the last decades it has lead to a significant progress in the understanding of the band structures of cuprates, pnictides and other materials of current interest. On the other hand, its application to Mott-Hubbard insulating materials where a Fermi surface is absent has been more limited. Here we show that in these latter materials, where electron spins are localized, ARPES may provide significant information on the spin correlations which can be complementary to the one derived from neutron scattering experiments. Sr$_2$Cu$_{1-x}$Zn$_x$O$_2$Cl$_2$, a prototype of diluted spin $S=1/2$ antiferromagnet (AF) on a square lattice, was chosen as a test case and a direct correspondence between the amplitude of the spectral weight beyond the AF zone boundary derived from ARPES and the spin correlation length $xi$ estimated from $^{35}$Cl NMR established. It was found even for correlation lengths of a few lattice constants a significant spectral weight in the back-bended band is present which depends markedly on $xi$. Moreover the temperature dependence of that spectral weight is found to scale with the $x$ dependent spin-stiffness. These findings prove that ARPES technique is very sensitive to short-range correlations and its relevance in the understanding of the electronic correlations in cuprates is discussed.
We compare the critical behavior of the short-range Ising spin glass with a spin glass with long-range interactions which fall off as a power sigma of the distance. We show that there is a value of sigma of the long-range model for which the critical behavior is very similar to that of the short-range model in four dimensions. We also study a value of sigma for which we find the critical behavior to be compatible with that of the three dimensional model, though we have much less precision than in the four-dimensional case.
The many-body localization transition (MBLT) between ergodic and many-body localized phase in disordered interacting systems is a subject of much recent interest. Statistics of eigenenergies is known to be a powerful probe of crossovers between ergodic and integrable systems in simpler examples of quantum chaos. We consider the evolution of the spectral statistics across the MBLT, starting with mapping to a Brownian motion process that analytically relates the spectral properties to the statistics of matrix elements. We demonstrate that the flow from Wigner-Dyson to Poisson statistics is a two-stage process. First, fractal enhancement of matrix elements upon approaching the MBLT from the metallic side produces an effective power-law interaction between energy levels, and leads to a plasma model for level statistics. At the second stage, the gas of eigenvalues has local interaction and level statistics belongs to a semi-Poisson universality class. We verify our findings numerically on the XXZ spin chain. We provide a microscopic understanding of the level statistics across the MBLT and discuss implications for the transition that are strong constraints on possible theories.
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