4. KPZ behavior in quantum spin chains
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Problem 4.1.
Compute the large-time current fluctuations for the XXZ spin-$1/2$ chain on a 1D lattice.
Some of the progress is the following:
The edge fluctuations for $\Delta=0$ with domain wall initial conditions were computed in [MR4573942] and shown to have Tracy-Widom GUE distribution. Partial asymptotic computations for $\Delta\neq 0$ were carried out, resulting on a precise conjecture for the fluctuations.
The edge fluctuations for $\Delta \gg 1$ with alternating domain wall initial conditions were considered in [arXiv:2412.20147 ]. The authors gave strong numerical evidence that the edge fluctuations have Tracy-Widom GUE distribution. In the limit $\Delta \rightarrow \infty$, the authors show that the edge fluctuations have Tracy-Widom GUE fluctuations.
Other current observables were also considered in [arXiv:2406.07150]. By extensive numerical analysis, the authors show that the characteristic length of the system scales as $t^{2/3}$, in agreement with models in the KPZ universality class.
Cite this as: AimPL: All roads to the KPZ universality class, available at http://aimpl.org/roadtokpz.