Hybrid Gauge with Position-Dependent Phases

• The paper presents a hybrid gauge framework that integrates local, position-dependent phase factors to achieve full gauge invariance in quantum-classical and synthetic lattice models.
• It details the mathematical structure—including Berry connections and covariant derivatives—to preserve symmetry and maintain accurate physical representations of dynamic systems.
• The approach yields significant numerical improvements, reducing errors in TDDFT and lattice models, ensuring better optical and electronic response predictions.

A hybrid gauge incorporating position-dependent phases refers to a gauge framework in quantum, quantum-classical, or synthetic lattice systems, in which local (real-space or phase-space) unitary phase factors are systematically included to ensure both accurate physical representation and gauge invariance beyond conventional global or uniform-gauge protocols. This concept is realized in various contexts: hybrid quantum-classical wave equations, lattice models with spatially varying Peierls phases, and electron dynamics under external fields, notably in time-dependent density functional theory (TDDFT) using numerical atomic orbitals (NAO). The hybrid gauge approach ensures the proper transformation properties of wavefunctions and operators, preserves necessary symmetries, and resolves errors arising from incomplete phase information in conventional gauge choices.

1. Mathematical Foundation of Hybrid Gauges and Local Phases

Hybrid gauge theories introduce local, position- (or phase-space-) dependent phase factors to encode the full gauge structure inherent in quantum or hybrid quantum-classical systems. In the hybrid quantum-classical wavefunction setting, the state

$$\Psi(q,p,x,t)\in L^2(\mathbb{R}^2_{q,p}\!\times\!\mathbb{R}^n_x)$$

is subject to variational principles that are covariant under local phase transformations. For classical variables, this reads

$$\psi(q,p)\mapsto \psi'(q,p)=e^{i\alpha(q,p)/\hbar}\,\psi(q,p),\quad A(q,p)\mapsto A+\nabla\alpha,$$

expressing the unobservability of the classical phase. In TDDFT, the hybrid gauge is constructed by embedding position-dependent phases into each localized atomic orbital: $$\phi_\mu(r) \rightarrow \bar{\phi}_\mu(r,t) = e^{i\theta_\mu(r,t)} \phi_\mu(r),$$ where $\theta_\mu(r,t) = -\mathbf{A}(t)\cdot(r-\tau_\mu)$ for uniform vector potential $\mathbf{A}(t)$ and orbital center $\tau_\mu$ (Zhao et al., 21 Jan 2025, Ji et al., 21 Dec 2025). In tight-binding models, the hopping term is multiplied by $e^{i\phi_j}$, with $\phi_j$ encoding a spatially varying phase on link $j$ (Das et al., 2019).

2. Phase-Space Berry Connections and Gauge Structure

A distinguishing feature of the hybrid gauge is the emergence of geometric (Berry) connections in phase space, leading to a nonlinear, gauge-invariant hybrid wave equation. If the hybrid wavefunction is factorized as

$$\Psi(q,p,x,t) = \psi(q,p,t)\;\chi(x, t; q,p),\qquad\|\chi(\cdot; q,p)\|_{L^2_x}=1,$$

with $\psi(q,p,t) = \sqrt{D(q,p,t)}\,e^{iS(q,p,t)/\hbar}$, then substitution into the action yields a Berry connection: $$\mathcal{A}(q,p,t) = \langle \chi, -i\hbar \nabla_{(q,p)} \chi\rangle_x$$ and a total phase-space gauge potential $A^{\rm tot} = \nabla S + \mathcal{A}$ (Gay-Balmaz et al., 2021). In adiabatic quantum-electrodynamical models, the geometric phase for a hybrid system is

$$\gamma_n = i\sum_{\alpha=1}^N \oint df_\alpha^i\,\langle n; f, a | \partial/\partial f_\alpha^i | n; f, a\rangle,$$

with a Berry connection $\mathcal{A}_{\alpha,i}(f,a)$ dependent on both classical paths and quantum field configurations (Koide, 2016).

3. Gauge-Invariant Hybrid Equations: Construction and Nonlinearity

Enforcing gauge invariance requires redefining covariant derivatives and dynamics in terms of the hybrid gauge connections. For instance, in the quantum-classical hybrid theory,

$$\nabla_{(q,p)}^A = \nabla_{(q,p)} - \frac{i}{\hbar}\mathcal{A}(q,p)$$

with the dynamics governed by a wave equation of the form

$$i\hbar D_t \Psi + i\hbar (\nabla_{(q,p)}\cdot X)\Psi = \widehat{H}\Psi - (\mathcal{A}\cdot X)\Psi - (X\cdot A^{\rm tot})\Psi,$$

where the classical flow $X$ contains fluctuation corrections beyond Ehrenfest, rendering the system highly nonlinear (Gay-Balmaz et al., 2021). In NAO-based TDDFT, the hybrid Hamiltonian incorporates both electric field and vector potential: $$\bar{H}(t) = U_{\rm hyb} H_L U_{\rm hyb}^\dagger - i U_{\rm hyb} (\partial_t U_{\rm hyb}^\dagger)$$ with matrix elements

$$\langle \bar{\phi}_\mu | \bar{H} | \bar{\phi}_\nu \rangle = e^{-i\mathbf{A}(t)\cdot\Delta\tau_{\mu\nu}}\left[\langle \phi_\mu|H_0|\phi_\nu\rangle + \mathbf{E}(t)\cdot\langle \phi_\mu|r-\tau_\nu|\phi_\nu\rangle\right],$$

ensuring the full position-dependent phase information is preserved (Zhao et al., 21 Jan 2025).

4. Physical and Numerical Implications: Fidelity and Symmetry

The use of hybrid gauges incorporating position-dependent phases achieves several key outcomes:

• Restoration of Physical Observables: In TDDFT, current calculations and excitation energies become accurate to $\lesssim 1\%$ once position-dependent phases are incorporated, as opposed to errors of $10-20\%$ in current and tens of meV/atom in excitation energies under the velocity gauge for incomplete basis sets (Zhao et al., 21 Jan 2025, Ji et al., 21 Dec 2025).
• Symmetry Preservation: Translational symmetry is maintained in periodic systems, and both quantum and classical density matrices maintain positivity for all time under the hybrid quantum-classical equation (Gay-Balmaz et al., 2021).
• Resolution of Unphysical Artifacts: The hybrid gauge removes unphysical divergences in optical response functions—e.g., the low-frequency divergence in $\epsilon(\omega)$ observed under incomplete phase encodings (Zhao et al., 21 Jan 2025).

Problem	Velocity Gauge	Hybrid Gauge (with Local Phase)
Current error (NAO TDDFT)	≈10–20%	<1%
Excitonic peak intensity (Si)	Underestimated by ×2	Matches length-gauge, physical value
Positivity of densities	Not ensured	Strictly positive (quantum + classical)

5. Synthetic and Lattice Implementation: Position-Dependent Peierls Phases

Hybrid gauge structures are also realized in discrete lattice systems using spatially varying Peierls phases. In a ring-lattice Hamiltonian,

$$H = -J\sum_{j=1}^L \left[e^{i\phi_j}a_j^\dagger a_{j+1} + \text{h.c.}\right] + \sum_{j=1}^L V_j a_j^\dagger a_j,$$

parameters $\phi_j$ encode position-dependent phases, yielding an effective spatial gauge field $A(x_j) = \phi_j / \Delta x$ (Das et al., 2019). This framework generalizes to arrays with site- and direction-dependent phases, enabling the realization of intricate $(x,y)$-dependent gauge patterns (Abelian and non-Abelian). The associated Wilson loop

$$W = \exp\left(i \sum_{j=1}^L \phi_j\right) = \prod_{j=1}^L e^{i\phi_j}$$

captures the net phase around the closed lattice, providing direct observables of the implemented gauge structure.

6. Applications, Extensions, and Limitations

Hybrid gauges with position-dependent phases have demonstrated significant improvements in:

• Real-time TDDFT with hybrid functionals: Accurate modeling of ultrafast excitonic dynamics and optical responses in periodic solids, eliminating spurious artifacts and enabling robust description with NAO basis sets (Ji et al., 21 Dec 2025).
• Quantum-classical hybrid dynamics: Hamiltonian, strictly gauge-invariant modeling preserving positivity of densities, and inclusion of nonlinear back-reaction, surpassing mean-field and Ehrenfest approximations (Gay-Balmaz et al., 2021).
• Cold atom and synthetic field experiments: Engineering of arbitrarily patterned Abelian and non-Abelian gauge fields and the measurement of geometric phases via Wilson loop interference (Das et al., 2019).

Limitations include the computational overhead from phase bookkeeping in large supercells, the need for sufficiently complete basis sets to maintain gauge invariance, and the current restriction to spatially uniform vector potentials in the dipole limit for TDDFT. Extensions to inhomogeneous (finite-$\mathbf{q}$) fields and coupling to pseudospin or magnetic degrees of freedom remain active areas of investigation (Ji et al., 21 Dec 2025).

7. Connections to Geometric Phase and Quantum-Classical Backreaction

A critical insight is that position-dependent phases in hybrid gauge constructions naturally encode geometric (Berry) phases associated with adiabatic evolution in complex quantum or quantum-classical systems. In hybrid quantum electrodynamics models, the geometric phase accumulated by adiabatically moving charged particles reflects both parametric dependence and quantum backreaction,

$$\gamma_n = i\sum_{\alpha=1}^N \oint_{C_\alpha(\{a\})} df_\alpha^i\,\langle n;f(t),a|\frac{\partial}{\partial f_\alpha^i}|n;f(t),a\rangle,$$

and reduces to the standard Berry phase in the absence of backreaction. This geometric perspective underpins many observable consequences in synthetic gauge structures and underscores the role of hybrid gauges as fundamental tools for accurately capturing quantum and classical coupling, geometric effects, and nontrivial holonomy in a broad spectrum of physical systems (Koide, 2016, Gay-Balmaz et al., 2021, Das et al., 2019).