The temperature dependent geometric phase

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.

• The paper establishes a rigorous formalism showing that geometric phases in adiabatic quantum systems become intrinsically temperature-dependent.
• It employs a Born-Oppenheimer-like approach and Hartree approximation to link environmental temperature to the gauge potential and effective molecular potential.
• Numerical analysis of the H₂⁺ model reveals that temperature modulates bond lengths and phase shifts, impacting precision spectroscopy and quantum control.

Temperature-Dependent Geometric Phases in Adiabatic Quantum Systems

Background and Motivation

The geometric phase, first introduced by Berry, has had extensive impact across quantum mechanics, condensed matter, and molecular physics. Traditionally, investigations focus on pure states in isolated systems, neglecting the role of finite temperature and system-environment interactions. Extensions to mixed states via purification methods (e.g., Uhlmann phase) embed temperature dependence in the statistical weights, not in the geometric phase itself. This study systematically develops a formalism for geometric phases that are intrinsically temperature-dependent, arising from adiabatic interactions between a quantum system and an equilibrated environment.

Theoretical Framework

The analysis begins with a coupled quantum system and environment. Core assumptions are:

• The environment has rapid relaxation compared to the system, reaches equilibrium, and can be described via local Boltzmann statistics.
• The system-environment coupling allows the separation of variables, mirroring Born-Oppenheimer (BO) approximations.

The environment's wavefunction is determined under the Hartree approximation, and its amplitude $A(r)$ is explicitly temperature dependent:

$A(r) = n_0 e^{-\epsilon/k_B T(r)}$

Adiabatic evolution of the system induces an Abelian gauge potential $\mathcal{A} = i\langle\Psi|\nabla_R\Psi\rangle$, which defines the geometric phase $\gamma = \int \mathcal{A} \cdot dR$. Crucially, both $\mathcal{A}$ and $\gamma$ are functions of the environmental temperature $T(r)$ due to the temperature dependence of $A(r)$.

The effective potential governing system evolution, $V_\text{eff}(R)$, contains contributions from the gauge potential, so it inherits temperature dependence. Observables in the composite system are calculated from a density matrix evolving via a Liouville equation; the system is not necessarily at equilibrium.

Numerical Analysis: H₂⁺ Model

The H₂⁺ molecular ion is used as a prototypical system. The gauge potential and geometric phase for the even parity ground state show strong temperature dependence:

• The geometric phase $\gamma$ decreases monotonically with increasing temperature above 100 K.
• The gauge potential's peak diminishes as temperature increases, vanishing at large internuclear separations regardless of temperature.

The effective potential minimum shifts slightly with temperature, changing the equilibrium bond length; calculated minima at 100K–300K vary from 2.10 to 2.15 atomic units, close to the experimental value of 2.004 a.u. These results quantitatively confirm that temperature modulates both geometric phase and effective potential in adiabatic molecular dynamics.

Implications

This work formally establishes that geometric phases, traditionally viewed as fundamentally temperature-independent, can acquire strong temperature dependence when the system interacts with an equilibrated environment adiabatically. The temperature-dependent geometric phase arises not from statistical mixing or weights, but directly from the environmental-induced gauge potential within the BO-like formalism.

Practical Implications

• Temperature influence on molecular bond lengths and spectra via geometric phase shifts is relevant for precision spectroscopy and quantum control at finite temperature.
• Adiabatic quantum devices interacting with engineered environments may have tunable geometric phase behavior depending on reservoir temperature.

Theoretical Implications

• The geometric phase's topological stability is altered by thermal effects; this may affect robustness in quantum computation protocols utilizing geometric phases.
• The results provide foundation for generalizing geometric phase concepts to non-equilibrium and non-adiabatic regimes, pending future investigation.

Future Directions

Extension to non-adiabatic and non-equilibrium environments is explicitly stated as an open question. Quantum systems in baths with slow relaxation or strong coupling may require new theoretical tools capturing the interplay between dynamics, temperature, and phase holonomy.

Conclusion

The manuscript presents a rigorous ab initio demonstration that geometric phases and gauge potentials in quantum systems interacting adiabatically with an environment are inherently temperature dependent. Both theoretical derivations and quantitative modeling (H₂⁺) reveal that environmental temperature imprints directly onto quantum holonomy. This finding fundamentally expands the geometric phase concept beyond zero-temperature, isolated settings, with practical and theoretical repercussions for quantum dynamics and control (2604.26639).