[challenge]: Is the machine-discovered Burgers equation an asymptotic hydrodynamic law of the 1D Heisenberg chain?

Released by

Junkai Wang

Contact email

WangTheoPhys@outlook.com

Method

MPS Based Algorithm

Challenge issue

Here is a question from my advisor: Kharkov et al. 2021 "Discovering hydrodynamic equations of many-body quantum systems" used sparse symbolic regression on tDMRG KPZ magnetization data, from Ljubotina et al. 2019, "Kardar-Parisi-Zhang Physics in the Quantum Heisenberg Magnet", for the isotropic Heisenberg chain $\Delta=1$, initialized in a high-temperature weak domain-wall state, and found that the rescaled magnetization $u(t,x)=\frac{\langle S^z(t,x)\rangle}{\mu}$ is accurately described by a deterministic 1D viscous Burgers equation!

$$
u_t+a,u,u_x=D,u_{xx},
$$

over the available hydrodynamic window.

So where does this equation come from? Is there an analytical interpretation for this result? It should be interesting that there is an emergent fluid equation from the quantum many-body system.

Another interesting thing is that: however, the quantum data extend only to finite times, while the later apparent $x/t^{2/3}$ collapse was obtained by evolving the fitted PDE itself rather than by comparison with additional Heisenberg-chain data. The challenge is to determine whether this equation—with constant coefficients—is a genuine asymptotic hydrodynamic law derivable from the microscopic Heisenberg Hamiltonian, or only an accurate finite-time and finite-resolution closure. A successful solution should either provide a controlled microscopic derivation, including the assumptions behind the scalar closure and the origin of $a$ and $D,$ or falsify asymptotic Burgers behavior through longer-time simulations and systematic tests of the fitted coefficients against the training window, initial bias $\mu,$ coarse-graining scale, system size, and omitted higher-gradient or stochastic terms.

The issue names an MPS-based algorithm but no repository files, tests, or runnable entry point. Start by locating the implementation used for the Heisenberg-chain simulations and the fitting workflow, then compare longer-time results and fitted coefficients across the listed variables. Done means either a controlled microscopic derivation of the Burgers coefficients and closure assumptions or systematic evidence that asymptotic Burgers behavior fails.