Constraint Phase Space Formulations for Finite-State Quantum Systems: The Relation between Commutator Variables and Complex Stiefel Manifolds

TL;DR

This work develops a generalized constraint coordinate-momentum phase-space (CPS) formulation for finite-state quantum systems, showing that each connected CPS component is a complex Stiefel manifold $V_r(\mathbb{C}^F)$ labeled by the eigenvalues of the mapping kernel, enabling exact trajectory-based dynamics in the frozen-nuclei limit that are isomorphic to the time-dependent Schrödinger equation. It introduces a $\mathrm{U}(F)$-covariant mapping framework with a kernel $\hat{K}(\mathbf{X};\gamma)$ and a commutator matrix $\mathbf{\Gamma}$, unifying prior mappings and recasting Stratonovich-type phase spaces as special cases via the parameter $\gamma$. The paper then classifies phase-space formulations into covariant-covariant (cc), covariant-noncovariant (cx), and noncovariant-covariant (xc) types, and demonstrates two covariant-kernel strategies that yield exact time-correlation functions (TCFs) in the frozen-nuclei limit: one with a covariant observable kernel and one with a covariant density kernel. To ensure physical trajectory contributions, it introduces positive semi-definite window-window TCFs (including TW, Ehrenfest, $\Lambda$-point, HWF, and (G)DTWA) and discusses their connections to DTWA/GDTWA on appropriate complex Stiefel manifolds, offering practical trajectory-based schemes for nonadiabatic and many-body quantum dynamics.

Abstract

We have recently developed the \textit{constraint} coordinate-momentum \textit{phase space} (CPS) formulation for finite-state quantum systems. It has been implemented for the electronic subsystem in nonadiabatic transition dynamics to develop practical trajectory-based approaches. In the generalized CPS formulation for the mapping Hamiltonian of the classical mapping model with commutator variables (CMMcv) method [\textit{J. Phys. Chem. A} \textbf{2021}, 125, 6845-6863], each {connected} component of the generalized CPS is the \textit{complex Stiefel manifold} labeled by the eigenvalue set of the mapping kernel. Such a phase space structure allows for exact trajectory-based dynamics for pure discrete (electronic) degrees of freedom (DOFs), where the equations of motion of each trajectory are isomorphic to the time-dependent Schrödinger equation. We employ covariant kernels {within the generalized CPS framework} to develop two approaches that naturally yield exact evaluation of time correlation functions (TCFs) for pure discrete (electronic) DOFs. In addition, we briefly discuss the phase space mapping formalisms where the contribution of each trajectory to the integral expression of the {TCF} of population dynamics is strictly positive semi-definite. The generalized CPS formulation also indicates that the equations of motion in phase space mapping model I of our previous work [\textit{J. Chem. Phys.} \textbf{2016}, 145, 204105; \textbf{2017}, 146, 024110; \textbf{2019}, 151, 024105] lead to a complex Stiefel manifold $\mathrm{U}(F)/\mathrm{U}(F-2)$. It is expected that the generalized CPS formulation has implications for simulations of both nonadiabatic transition dynamics and many-body quantum dynamics for spins/bosons/fermions.

Figures (2)

• Figure 1: The phase space structure is determined by the corresponding mapping kernels. Beginning at point $\mathbf{X}_0$ on the phase space, the associated mapping kernel $\hat{K}(\mathbf{X}_0)$ is transformed to $\hat{K}(\mathbf{X}_t)=\hat{U}(t)\hat{K}(\mathbf{X}_0)\hat{U}^{-1}(t)$ through the unitary time evolution operator $\hat{U}(t)$. The phase trajectory connecting $\mathbf{X}_0$ and $\mathbf{X}_t$ corresponds to the unitary transformation.
• Figure 2: The complex Stiefel manifold $V_2(\mathbb{C}^F)$ is depicted, where each point on the primary sphere $S^{2F-1}$ is associated with an attached sphere $S^{2F-3}$. These attached spheres are in a non-parallel, twisted configuration, emphasizing the intricate structure of the manifold. Similar patterns also apply to higher order complex Stiefel manifolds, $V_r(\mathbb{C}^F)$.