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arXiv:2608.24864

Dynamical Consequences of Nontrivial Topology of Molecular Conical Intersections

Indranil Ghosh, Kush Banker, Gregory S. Engel

physics.chem-phquant-phSubmitted Aug 25, 2026• Announced Aug 26, 2026
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Abstract

The topology of the electronic structure for an avoided crossing and a conical intersection (CI) is different and is characterized by the presence of the geometric phase in the electronic wavefunction in the latter. Using the linear Jahn-Teller model, we show that an avoided crossing can be created from the conical intersection while preserving the nontrivial topology of the latter by adding a Pauli $σ_y$ term to the Hamiltonian. Analogously to solid state systems, we derive a half-integer topological invariant as the integral of the Berry curvature over the CI nuclear branching space. We investigate the influence of electronic topology on chemical dynamics by conducting fewest-switches surface hopping simulations and find distinct hopping rates on identical eigensurfaces but with different topologies. Our work extends the influence of topology on molecular excited state dynamics beyond the Berry phase that can be practically realized through electron-nuclear and spin-orbit coupling.

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The topology of the electronic structure for an avoided crossing and a conical intersection (CI) is different and is characterized by the presence of the geometric phase in the electronic wavefunction in the latter. Using the linear Jahn-Teller model, we show that an avoided crossing can be created from the conical intersection while preserving the nontrivial topology of the latter by adding a Pauli σyσ_yσy​ term to the Hamiltonian. Analogously to solid state systems, we derive a half-integer topological invariant as the integral of the Berry curvature over the CI nuclear branching space. We investigate the influence of electronic topology on chemical dynamics by conducting fewest-switches surface hopping simulations and find distinct hopping rates on identical eigensurfaces but with different topologies. Our work extends the influence of topology on molecular excited state dynamics beyond the Berry phase that can be practically realized through electron-nuclear and spin-orbit coupling.
The Born-Oppenheimer electronic wave function ${\mathrm{\ensuremath{\Phi}}}_{R}^{\text{BO}}(r)$ picks up a topological phase factor $\ifmmode\pm\else\textpm\fi{}1$, a special case of Berry phase, when it is transported around a conical intersection of two adiabatic potential energy surfaces in RRR space. We show that this topological quantity reverts to a geometric quantity ${e}^{i\ensuremath{\gamma}}$ if the geometric phase $\ensuremath{\gamma}=\ensuremath{\oint}\mathrm{Im}\ensuremath{\langle}{\mathrm{\ensuremath{\Phi}}}_{R}|{\mathbf{\ensuremath{\nabla}}}_{\ensuremath{\mu}}{\mathrm{\ensuremath{\Phi}}}_{R}\ensuremath{\rangle}\ifmmode\cdot\else\textperiodcentered\fi{}d{\mathbf{R}}_{\ensuremath{\mu}}$ is evaluated with the conditional electronic wave function ${\mathrm{\ensuremath{\Phi}}}_{R}(r)$ from the exact electron-nuclear factorization ${\mathrm{\ensuremath{\Phi}}}_{R}(r)\ensuremath{\chi}(R)$ instead of the adiabatic function ${\mathrm{\ensuremath{\Phi}}}_{R}^{\text{BO}}(r)$. A model of a pseudorotating triatomic molecule, also applicable to dynamical Jahn-Teller ions in bulk crystals, provides examples of nontrivial induced vector potentials and molecular geometric phase from the exact factorization. The induced vector potential gives a contribution to the circulating nuclear current that cannot be removed by a gauge transformation. The exact potential energy surface is calculated and found to contain a term depending on the Fubini-Study metric for the conditional electronic wave function.
Conical intersections are topologically protected crossings between the potential energy surfaces of a molecular Hamiltonian, known to play an important role in chemical processes such as photoisomerization and non-radiative relaxation. They are characterized by a non-zero Berry phase, which is a topological invariant defined on a closed path in atomic coordinate space, taking the value π when the path encircles the intersection manifold. In this work, we show that for real molecular Hamiltonians, the Berry phase can be obtained by tracing a local optimum of a variational ansatz along the chosen path and estimating the overlap between the initial and final state with a control-free Hadamard test. Moreover, by discretizing the path into N points, we can use N single Newton-Raphson steps to update our state non-variationally. Finally, since the Berry phase can only take two discrete values (0 or π ), our procedure succeeds even for a cumulative error bounded by a constant; this allows us to bound the total sampling cost and to readily verify the success of the procedure. We demonstrate numerically the application of our algorithm on small toy models of the formaldimine molecule ( H 2 C = N H ).
We demonstrate that Berry phases may greatly affect the dynamics of spin-orbit-coupled Bose-Einstein condensates. The effective model Hamiltonian under consideration is shown to be equivalent to the $E\ifmmode\times\else\texttimes\fi{}\ensuremath{\epsilon}$ Jahn-Teller model first introduced in molecular physics. The corresponding conical intersection is identified and the Berry phase acquired for a wave packet encircling the intersection is studied. It is found that this phase manifests itself in the density profile of the condensate, making it a directly measurable quantity via time-of-flight detection. Moreover, the non-Abelian gauge structure of the system is addressed and we verify how it affects the dynamics.
Jahn-Teller systems and the Jahn-Teller effect are discussed in terms of cavity QED models. By expressing the field modes in a quadrature representation, it is shown that certain setups of a two-level system interacting with a bimodal cavity are described by the Jahn-Teller $E\ifmmode\times\else\texttimes\fi{}\ensuremath{\epsilon}$ Hamiltonian. We identify the corresponding adiabatic potential surfaces and the conical intersection. The effects of a nonzero geometrical Berry phase, governed by encircling the conical intersection, are studied in detail both theoretically and numerically. The numerical analysis is carried out by applying a wave packet propagation method, more commonly used in molecular or chemical physics, and analytic expressions for the characteristic time scales are presented. It is found that the collapse-revival structure is greatly influenced by the geometrical phase and as a consequence, the field intensities contain direct information about this phase. We also mention the link between the Jahn-Teller effect and the Dicke phase transition in cavity QED.
Conical intersections are common in molecular physics and photochemistry, and are often invoked to explain observed reaction products. A conical intersection can occur when an excited electronic potential energy surface intersects with the ground electronic potential energy surface in the coordinate space of the nuclear positions. Theory predicts that the conical intersection will result in a geometric phase for a wavepacket on the ground potential energy surface. Although conical intersections have been observed experimentally, the geometric phase has not been observed in a molecular system. Here we use a trapped atomic ion system to perform a quantum simulation of a conical intersection. The internal state of a trapped atomic ion serves as the electronic state and the motion of the atomic nuclei are encoded into the normal modes of motion of the ions. The simulated electronic potential is constructed by applying state-dependent forces to the ion with a near-resonant laser. We experimentally observe the geometric phase on the ground-state surface using adiabatic state preparation followed by motional state measurement. Our experiment shows the advantage of combining spin and motion degrees of freedom in a quantum simulator.
In 1937 the Jahn-Teller (JT) effect addressed the instability (potential or actual) of non-linear symmetric molecules with degenerate orbital electronic states. In view of the large variety of JT activity that has taken place since then, we might broaden our perspective to look at works whose subjects fall under the more general heading of "Strong interactions between two dissimilar systems" (where one system is usually bosonic and the other fermionic). In these intervening years we find several highly important works in Physics and Chemistry that come under this heading and were either connected with, or arose from, JT systems, problems and approaches. Apart from high temperature superconductors, we mention Yang-Mills gauge-forces, symmetry breaking (in elementary particles), conical intersections in molecular potential surfaces, surface crossings between them in chemical reactions, entanglements in the quantum theory of measurements and Berry phases. We elaborate on the last two topics. We show first that the slow evolution of a Txe coupling from the weak to strong regime can model the quantum mechanical three-state measurement situation, when the positions of the nuclei acts as the measuring device. We then employ recently derived integral relations between component moduli and phases in a time dependent wave-function to demonstrate the equivalence between the state-reduction and the phase decoherence interpretations of the measurement process.