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Journal of the Physical Society of Japan

Adiabatic Approximation for Path Integrals and Geometrical Potentials

Takashi Ohnishi, Naoto Nagaosa

Published Dec 25, 2006• 2 citations

Abstract

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:

DOI: 10.1143/jpsj.76.015003

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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:

Generalizing the concept of the Berry phase, I show that an additional gauge structure associated with a time-component vector potential

{A}_{0}

is present in the adiabatic evolution of a quantum system. The gauge structure, which generates the usual dynamical phase, is revealed by a formalism in which the time variable is treated on an equal footing with other parameters on which the Hamiltonian depends. The invariant curvature associated with

{A}_{0}

, an electriclike field, determines the phase difference between low- and high-energy paths in a generalized phase-space and time picture. The covariance of the Born-Oppenheimer approximation illustrates the results.

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 adiabatic approximation usually applies to a collection of lighter particles (the ``fast'' system) and heavier particles (the ``slow'' system), where the geometric phase naturally appears. With the help of the recently introduced gauge potential in the context of adiabatic Berry phase for open paths, we establish a fluctuation-correlation theorem by relating the quantum fluctuations in the generator of the parameter change (of the ``slow'' system) to the time integral of the quantum correlation function between the projection operator and force operator acting on the ``fast'' system. By taking a cue from linear response theory we relate the quantum fluctuation in the generator to the generalized susceptibility. The relation between the open-path geometric phase, diagonal elements of the quantum metric tensor, and the force-force correlation function are provided and the classical limit of the fluctuation-correlation theorem is discussed.