Geometric theory of constrained Schrödinger dynamics with application to time-dependent density-functional theory on a finite lattice

TL;DR

This work develops a finite‑dimensional geometric framework for Schrödinger dynamics under prescribed observables, revealing a canonical variational route and a distinct purely geometric route for constraint enforcement. It shows how constrained dynamics emerge from state‑space geometry, introduces an interpolating oblique principle, and analyzes both with commuting and noncommuting observables, leading to a robust understanding of v‑representability on lattices. The theory is applied to time‑dependent density functional theory for fermions on finite lattices, yielding novel Kohn–Sham schemes where density constraints are enforced by an imaginary potential or a nonlocal Hermitian operator, and is illustrated in detail on the Hubbard dimer. These geometric TDDFT formulations open avenues for nonadiabatic approximations and provide a rigorous backbone for density–to–potential mappings in lattice systems with constrained dynamics.

Abstract

Time-dependent density-functional theory (TDDFT) is a central tool for studying the dynamical electronic structure of molecules and solids, yet aspects of its mathematical foundations remain insufficiently understood. In this work, we revisit the foundations of TDDFT within a finite-dimensional setting by developing a general geometric framework for Schrödinger dynamics subject to prescribed expectation values of selected observables. We show that multiple natural definitions of such constrained dynamics arise from the underlying geometry of the state manifold. The conventional TDDFT formulation emerges from demanding stationarity of the action functional, while an alternative, purely geometric construction leads to a distinct form of constrained Schrödinger evolution that has not been previously explored. This alternative dynamics may provide a more mathematically robust route to TDDFT and may suggest new strategies for constructing nonadiabatic approximations. Applying the theory to interacting fermions on finite lattices, we derive novel Kohn--Sham schemes in which the density constraint is enforced via an imaginary potential or, equivalently, a nonlocal Hermitian operator. Numerical illustrations for the Hubbard dimer demonstrate the behavior of these new approaches.

Figures (7)

• Figure 1: In the variational principle, an optimal trajectory $t\mapsto\psi(t)$ is by definition such that the action functional is stationary against small deformations staying on the constraining manifold $\mathcal{M}$. To first order, such deformations $h(t)$ are vectors in the tangent space $\mathcal{T}_{\psi(t)}$.
• Figure 2: In the geometric principle, an optimal trajectory $t\mapsto\psi(t)$ is by definition such that the tangent $\partial_t\psi(t)$ to the trajectory is at every time the orthogonal projection of $-iH(t)\psi(t)$ on the tangent space $\mathcal{T}_{\psi(t)}$. This projection is $\partial_t \psi(t) = -iH(t)\psi(t) + \sum_{m=1}^M w_m(t) \mathcal{O}_m(t)\psi(t)$ for some real numbers $w_m(t)$.
• Figure 3: The oblique principle continuously interpolates between the variational and geometric principles, using a parameter $\theta$ similar to an angle. The model shares the properties of the geometric principle for all $\theta\neq0$ modulo $\pi$. In the limit $\theta\to0$ modulo $\pi$ one recovers the variational principle but the limit is very singular.
• Figure 4: Bloch sphere representation of the states $\psi$ of the qubit. The matrix $S^\psi$ is invertible everywhere except at the South and North poles, corresponding to $\rho_1=0$ or $1$. The $\psi$'s of fixed density $0<\rho_1=1-\rho_2<1$ correspond to circles of latitude, on which $S^\psi$ is always invertible, hence $\mathcal{C}=\mathcal{M}$. On the other hand, the number $K^\psi=\Re(\overline{\psi_2}\psi_1)$ in \ref{['eq:qubit_K']} vanishes on the latitude circle of relative angle $\alpha_2-\alpha_1=\pi/2$ modulo $\pi$, corresponding to $\psi=e^{i\alpha}(\sqrt{\rho_1},\pm i\sqrt{\rho_2})$. The solution to the variational principle can never cross this circle, so that the Bloch sphere is split into two disconnected parts. The solutions to the geometric and oblique principles can perfectly cross the circle and only the two poles have to be avoided.
• Figure 5: Top panel: time evolution in logarithmic scale of the solution $\beta(t)$ obtained from the oblique principle for a single qubit, shown for several values of the parameter $\theta$ with $\beta(0)=1.3$ and $\rho_1=0.7$. Bottom panel: corresponding potentials $u_1^\theta(t)$. For visualization purposes, we plot the transformed quantity $g(u_1^\theta)$ with $g(x) = \operatorname{sign}(x)\log_{10}(1 + |x|)$ in order to highlight the convergence toward Dirac deltas.

Theorems & Definitions (14)

• Theorem 1: Variational principle in symplectic case
• Theorem 2: Variational principle for commuting observables
• Theorem 3: Geometric principle
• Theorem 4: oblique principle
• Theorem 5: Invertibility of the matrix $K^\psi$
• Definition 6: Irreducibility of $\gamma$
• Theorem 7: Irreducibility criterion
• proof
• Theorem 8: Irreducibility of $\gamma_\Psi$ and invertibility of $S^\Psi$
• proof