We present a numerical scheme for efficiently extracting the higher-order moments and cumulants of various operators on spin systems represented as tensor product states, for both finite and infinite systems, and present several applications for such quantities. For example, the second cumulant of the energy of a state, $\ensuremath{\langle}\mathrm{\ensuremath{\Delta}}{H}^{2}\ensuremath{\rangle}$, gives a straightforward method to check the convergence of numerical ground-state approximation algorithms. Additionally, we discuss the use of moments and cumulants in the study of phase transitions. Of particular interest is the application of our method to calculate the so-called Binder cumulant, which we use to detect critical points and study the critical exponent of the correlation length with only small finite numerical calculations. We apply these methods to study the behavior of a family of one-dimensional models (the transverse Ising model, the spin-1 Ising model, and the spin-1 Ising model in a crystal field), as well as the two-dimensional Ising model on a square lattice. Our results show that in one dimension, cumulant-based methods can produce precise estimates of the critical points at a low computational cost. The method shows promise for two-dimensional systems as well.
We present general quantum mechanical expressions for higher-order moments, cumulants, and spectra of continuously measured quantum systems with applications in spin noise spectroscopy, quantum transport, and measurement theory in general. Starting from the so-called stochastic master equation of continuous measurement theory, we find that the leading orders of the fluctuating detector output
z(t) with respect to the measurement strength $\ensuremath{\beta}$ are a white shot noise background, a constant measurement offset, and the leading order quantum noise of the measured operator
A. Starting from quantum expressions for the multitime moments $\ensuremath{\langle}z({t}_{n})\ensuremath{\cdots}z({t}_{1})\ensuremath{\rangle}$ we derive three- and four-time cumulants that are valid in all orders of $\ensuremath{\beta}$ covering the full regime between the weak- and strong-measurement limit (Zeno limit). Intriguingly, quantum expressions for the cumulants were found that exhibit the same simple structure as those for the moments after introduction of only a slightly modified system propagator. Very compact expressions for the cumulant-based third- and fourth-order spectra (bispectrum and trispectrum) follow naturally. We illustrate the usefulness of higher-order spectra by treating a real-world two-spin system with strong hyperfine interaction. Moreover, spin noise spectroscopy is shown to have the potential for investigating the transition from weak measurements to the famous quantum Zeno regime for realistic probe laser intensities.
A new formulation of the renormalization-group in real space, suitable for quantum spin systems is proposed. The method is applied to the two-dimensional spin-
21 $X\ensuremath{-}Y$ model on a triangular lattice, the renormalization-group transformation being evaluated up to second order in an appropriate cumulant expansion. To first order an unstable fixed point of the transformation is found, corresponding to a critical temperature and critical indices in satisfactory qualitative agreement with present high-temperature series expansion estimates. In the second-order calculation, however, this fixed point disappears, thus throwing some doubt on the conventional picture of criticality as furnished by high-temperature series. The free energy of the model is also computed. For relatively small values of the nearest-neighbor coupling it is in good agreement with that found by high-temperature series analysis.
Nonlocality is a peculiar nature of quanta and it stands as an important quantum resource in application. Yet mere linear property of it, viz. the first order in moment, has been explored through various inequalities. Noticing the vast higher-order regime unexplored, in this study we investigate the higher-order quantum correlations in entangled hyperon-antihyperon system, which may be generated massively in charmonium decays. A new type of Clauser-Horne inequality for statistical cumulants and central moments is formulated. We find that a significant violation of the third-order constraint, indicating the existence of higher-order correlation, exists in hyperon-antihyperon system and can be observed in high energy physics experiments, like BESIII and Belle II. Notably, the violation manifests more in higher energy systems of the
ΛΛˉ pair against the kinematic contamination of timelike events.