# Classical fractional time series from quantum XXZ spin chains

## TL;DR

This paper investigates how quantum correlations in the XXZ spin chain can generate classical fractional stochastic processes with varying roughness, revealing the influence of quantum symmetries on the Hurst exponent.

## Contribution

It demonstrates the connection between quantum symmetries in the XXZ chain and the fractional properties of classical processes derived from it, using numerical quantum methods.

## Key findings

- Processes with quantum symmetries tend to have H=0 with logarithmic scaling.
- Symmetry-breaking processes can produce H≥0.5, but not likely H<0.5.
- Numerical methods like MERA and TEBD support these results.

## Abstract

Entangled quantum mechanical states in one dimension can be used to represent and simulate classical stochastic processes with nontrivial statistical properties. Long-range quantum correlations translate into fractional processes with their asymptotic Hurst exponents characterizing roughness and persistence. We explore this analogy in the case of the spin-1/2 XXZ chain and investigate properties of four different classical two-state processes that this quantum system can generate. These processes show fractional characteristics with varying Hurst exponents. We argue that the continuous quantum symmetries such as U(1) or SU(2) of the XXZ chain give rise to $H=0$ with logarithmic scaling. Processes generated without these symmetries can produce $H \geq0.5$ but likely not $H < 0.5$ unless the dominant term responsible for $H=0.5$ gets canceled. This does not seem to happen for the XXZ model. We use standard quantum methods, including MERA and TEBD, to numerically substantiate our findings.