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arXiv:2604.26639

The temperature dependent geometric phase

Zheng-Chuan Wang

quant-phSubmitted Apr 29, 2026• Announced Apr 30, 2026• 0 citations
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Abstract

There exists a geometric phase for a quantum state during the adiabatic evolution of the system. If the adiabatic procedure happens between the system and the environment interacting with it similar to Born-Oppenheimer (BO) approximation, we can introduce a temperature into the environment, which can be regarded as in an equilibrium state. Then a temperature-dependent geometric phase can be obtained for the system, which originates from the Abelian gauge potential induced by the BO approximation. This gauge potential contributes to the effective potential of the system, which is temperature dependent, too. Finally, we demonstrate them using an example of H_2^+ ion system.

Comments
7 pages,3 figures

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There exists a geometric phase for a quantum state during the adiabatic evolution of the system. If the adiabatic procedure happens between the system and the environment interacting with it similar to Born-Oppenheimer (BO) approximation, we can introduce a temperature into the environment, which can be regarded as in an equilibrium state. Then a temperature-dependent geometric phase can be obtained for the system, which originates from the Abelian gauge potential induced by the BO approximation. This gauge potential contributes to the effective potential of the system, which is temperature dependent, too. Finally, we demonstrate them using an example of H_2^+ ion system.
Geometrical concepts have played an important role in quantum mechanics. In the Schrodinger representation, the Berry–Simon connection and the quantum geometric tensor appear as gauge potentials in the effective Hamiltonian for a heavy system under the Born–Oppenheimer approximation. On the other hand, the Berry phase should arise as the amplitude of each path in the pathintegral formalism. In their pioneering work, Kuratsuji and Iida have developed the treatment of the Born– Oppenheimer approximation in the path integral quantization, where the Berry phase of the light system naturally appears in the effective action for the heavy system. However, they did not derive the gauge potentials themselves in the effective Hamiltonian. In this short note, we show that the gauge potentials which relate to the geometry of the Hilbert space of the light system are also derived in the adiabatic scheme of path integrals. Our result presents the path-integral formalism to consider the effective dynamics of the heavy system under the adiabatic approximation. Let us consider the following Hamiltonian:
The fundamental theory of the geometric phase is summarized in a way suitable for use in molecular systems treated by the Born-Oppenheimer approach. Both Abelian and non-Abelian cases are considered. Applications discussd include the Abelian geometric phase associated with an intersection of two electronic potential-energy surfaces; screening of nuclei by the electrons from an external magnetic field; non-Abelian gauge potentials in molecular systems with Kramers degeneracy; and the coupling between different electronic levels (Born-Oppenheimer breakdown) represented as a gauge potential. Experimental tests for these systems are discussed, as well as a number of experiments on spin systems.
We introduce and analyze several low-dimensional scattering systems that exhibit geometric phase phenomena. The systems are fully solvable and we compare accurate solutions of them with those obtained in a Born-Oppenheimer projection approximation. We illustrate how geometric magnetism manifests in them, and explore the relationship between solutions obtained in the diabatic and adiabatic pictures. We provide an example, involving a neutral atom dressed by an external field, in which the system mimics the behavior of a charged particle that interacts with, and is scattered by, a ferromagnetic material. We also introduce a similar system that exhibits Aharonov-Bohm scattering. We propose possible practical applications. We provide a theoretical approach that underscores universality in the appearance of geometric gauge forces. We do not insist on degeneracies in the adiabatic Hamiltonian, and we posit that the emergence of geometric gauge forces is a consequence of symmetry breaking in the latter.
This study shows that the usual time evolution operator can act as a U(1) gauge transformation on the initial wavefuntion. When considering the spin freedom in the system, the time evolution operator corresponds to a non-Abelian SU(2) gauge transformation for spin-1/2 particles and a SU(3) gauge transformation for spin-1 particles. If we adopt the adiabatic approximation in the magnetization dynamics of a ferromagnetic system driven by a spin-polarized current, a temperature-dependent non-Abelian thermal gauge potential will appear. We can employ a temperature-dependent Landau-Lifshitz-Gilbert (LLG) equation to describe the magnetization dynamics in the semi-classical limit, a linear approximated solution for this equation is given analytically.