Adiabatic Variational Principle for Electron-Driven Phase-Ordering Dynamics

Inhomogeneous, electron-driven symmetry-breaking states are ubiquitous across a wide range of condensed-matter systems. Theoretical descriptions of such states and their nonequilibrium evolution have traditionally relied on phenomenological phase-field approaches, exemplified by the time-dependent Ginzburg-Landau equation. Here we introduce an adiabatic variational principle (AVP), which provides a microscopic foundation for the dynamics of electron-driven order-parameter fields within the spirit of the Born-Oppenheimer approximation. By combining AVP with relaxation dynamics constrained by symmetry and conservation laws, we develop a unified framework for simulating the real-time evolution of complex electronic orders. The approach is demonstrated through the coarsening dynamics of a checkerboard charge-density-wave phase in a two-dimensional square-lattice generalized Hubbard model.