[challenge]: Diagnosing ergodicity in Constrained Path Auxiliary Field Quantum Monte Carlo

Released by

Mingpu Qin

Contact email

qinmingpu@sjtu.edu.cn

Method

Quantum Monte Carlo

Challenge issue

In the Constrained Path Auxiliary Field Quantum Monte Carlo (CP-AFQMC) method [1], a trial wave function $|\psi_T\rangle$ is employed to control the fermionic sign problem. During a random walk through Slater determinant space, a walker—represented by a single Slater determinant $|\psi\rangle$—is discarded if it intersects the nodal surface of the trial wave function (i.e., $\langle \psi | \psi_T \rangle = 0$).

This algorithm is subject to multiple sources of bias. One such bias, which remains poorly understood, is the ergodicity problem. This issue has already manifested in systems where the sign problem is absent (see the results in Table 1 in Ref. [2]). Fundamentally, it is unclear whether the entire allowed space ($\langle \psi | \psi_T \rangle > 0$) is adequately explored during the random walk. The topology of the allowed sampling space is typically highly complex, potentially fragmenting the space into disconnected domains. Transitioning between these domains may require crossing the nodal surface or traversing the disallowed space ($\langle \psi | \psi_T \rangle < 0$).

The primary challenge lies in visualizing the random walk process and determining the conditions under which the entire allowed space is ergodically accessible. To address this issue, we must first develop a method to visualize the high-dimensional Slater determinant space.

[1] S. Zhang, Auxiliary-Field Quantum Monte Carlo for Correlated Electron Systems, Vol. 3 of Emergent Phenomena in Correlated Matter: Modeling and Simulation, Ed. E. Pavarini, E.Koch, and U. Schollwock (Verlagdes ForschungszentrumJulich, 2013)

[2] M. Qin, H. Shi, and S. Zhang, Phys. Rev. B 94, 085103 (2016)

No implementation files, tests, or entry points are named. Start by locating the Python code for the CP-AFQMC random walk and Slater determinant representation, then determine what state can be recorded for visualization. Done would mean a documented visualization method and an analysis of whether the allowed space is explored ergodically, but the issue does not define a specific deliverable.