Markov Inequality as a Tool for Linear-Scaling Estimation of Local Observables

TL;DR

This work tackles the challenge of mapping spatially resolved quantum observables in large disordered lattices by introducing a linear-scaling stochastic method for positive local operators. It uses a positive-definite estimator together with Markov-inequality based bounds to achieve site-wise error control and homogeneous convergence, addressing the lack of self-averaging that plagues conventional stochastic traces. The approach extends to off-diagonal observables via local unitary rotations and is validated on a disordered 2D π-flux model, yielding accurate LDoS maps that capture power-law tails and high-resolution local-current patterns with vortex structures. The method promises scalable, high-precision real-space simulations of mesoscopic phenomena and could enhance self-consistent mean-field calculations in realistically large lattices.

Abstract

We introduce a linear-scaling stochastic method to compute real-space maps of any positive local spectral operator in a tight-binding model. By employing positive-definite estimators, the sampling error at each site can be rigorously bounded relative to the mean via the Markov inequality, overcoming the lack of self-averaging and enabling accurate estimates even under strong spatial fluctuations. The approach extends to non-diagonal observables, such as local currents, through local unitary transformations and its effectiveness is showcased by benchmark calculations in the disordered two-dimensional (2D) $π$-flux model, where the LDoS and steady-state current maps are computed. This method will enable simulations of disorder-driven mesoscopic phenomena in realistically large lattices and accelerate real-space self-consistent mean-field calculations.

Figures (4)

• Figure 1: Schematic representation of a bipartite squared lattice. The blue ellipses represent connected pairs. These form a complete set, and for each pair we apply the change of basis matrix shown in the red box.
• Figure 2: (a) Longitudinal cut across a single central vacancy on a $4096\times4096$ supercell computed with the positive-definite estimator ($32$ random vectors), showing the expected power-law decay as $\varepsilon\to0$. (b) Same cut but computed with the conventional estimator, which yields both non-positive values and misses small amplitudes. Bottom panels: LDoS maps for a $2048\times2048$ supercell with 0.5% vacancies at three representative energies and fixed spectral resolution of $100 µeV$.
• Figure 3: Probability density functions of the local relative error $\Delta_{\mathbf{r}}$ for the conventional (shades of blue) and positive-definite (shades of red) stochastic estimators, as a function of the number of random vectors, $R$. The latter reduces the mode of $\Delta_{\mathbf{r}}$ by approximately two orders of magnitude, while scaling as $R^{-0.5}$. The inset is a scatter plot of stochastic and exact LDoS for $256$ random vectors.
• Figure 4: Square-root LDoS and steady-state local currents for a $512\times512$ sample in the two-terminal setup with 0.5% concentration of long-range impurities ($W=1.1 eV$, $\kappa=9.0$). Top: square-root LDoS map ($500$ random vectors and spectral resolution $100 µeV$). Magnified regions demonstrate the local currents steady state fields. Intricate patterns in the real-space distributions of currents is evident with vortex–anti-vortex pairs being visible.