Gauge-invariant Green function dynamics: A unified approach
Introduction
While at a fundamental level the interaction of the electromagnetic field with charges is described by the minimal coupling Hamiltonian, for applications in atomic and molecular physics it is usually more convenient to use the transformed Hamiltonian developed by Power, Zienau, and Wooley (PZW) [1], and elaborated and clarified by Healy [2]. In the PZW transformation the interaction Hamiltonian is converted to one involving polarization and magnetization fields interacting directly with the electric and magnetic fields, which is thus gauge invariant. Often one then expands the polarization and magnetization fields in terms of electric and magnetic multipole moments, yielding an approximate Hamiltonian that takes into account the variations of the electric and magnetic fields over the atom or molecule to the order required. Even before such an expansion is undertaken, a “special point” must be chosen to effect the PZW transformation. For an atom this special point is often taken to be the position of the nucleus, idealized as fixed; more generally it can be taken to be the position of the center of mass of an atom or molecule [2].
Another strategy for constructing gauge-invariant quantum dynamical equations is the approach of Levanda and Fleurov (LF) [3], [4]. They showed that by concentrating on the equations of motion for the Green functions, rather than modifying the Hamiltonian formulation of those equations, a gauge-invariant description of the dynamics is possible. Their approach employs a generalized Peirels phase that involves an integration along a straight line in space–time between the two Green function variables; see also [5] for a discussion and [6] for the extension to superconductors. The LF approach is attractive because it does not rely on a special point, as does the PZW transformation. Thus this kind of approach, or a related one, could provide a basis for gauge-invariant descriptions of the interaction of radiation with charges in extended systems, such as solids, where the use of a special point does not seem to make physical sense. Gauge-invariant descriptions would be useful because the straight-forward application of the minimal coupling Hamiltonian often requires the identification of sum rules to eliminate apparent divergences, even in problems as simple as the linear and nonlinear optical response of clean, cold semiconductors, with electrons treated in the independent particle approximation [7], [8], [9], [10].
These apparent divergences are associated with the difference between the mechanical momenta of the charges and their canonical momenta, described by a term involving the vector potential, and they seem to be avoided whenever a gauge-invariant description of the quantum dynamics can be constructed. There is a long tradition of work on special cases where this has been implemented. For situations where the response of the medium can be approximated as responding locally to the electric field, taken as uniform, an approach originally pioneered by Adams and Blount [11] can be applied. A quantity that plays the role of the matrix element of a dipole moment operator between Bloch states can be introduced, involving interband and intraband contributions [12]. The approach of the “modern theory of polarization” [13], where polarization and magnetization are associated with the appropriate moments of Wannier functions [14], [15], or extensions necessary in materials with nonzero Chern numbers [16], can be seen to be closely related to this, for these moments can be written in terms of the effective matrix elements of Adams and Blount and their generalizations. But to date these programs are restricted to the assumption of uniform or nearly uniform applied fields.
So an exploration of strategies based on the LF approach to treat the response of solids to more arbitrarily varying electromagnetic fields seems promising. A natural first step would be to link that approach with the PZW transformation, which is to date probably the most successful gauge-invariant description of the response of charges to the electromagnetic field. Yet when both methods are applied to an atom, the equivalence can be seen only after much manipulation, even when the electromagnetic field varies little over the atom. The first goal of this paper is to clarify the link between the gauge-invariant descriptions of LF and PZW.
To do this we find it is necessary to extend the LF approach to treat a Peirels phase involving an arbitrary path in space–time connecting the two Green function variables. This is done in Section 2, where we construct a generalized LF approach (GLF). Then we can recover the LF and PZW results by the choice of particular paths. This is done in Sections 3 The link to the LF approach, 4 The link to PZW.
Establishing those connections illustrates that there are two qualitatively different types of Peirels phase that can arise in a GLF approach. In the first type, which appears in recovering the PZW results, the generalized Peirels phase is given by the difference of the values that a single variable function takes at the two space–time points in the Green function. In the second type, which appears for example in the original LF approach, it cannot be written in that form. In Section 4 we show that the Peirels phase of the first type leads to the gauge-invariant Green functions that can be understood as constructed from transformed field operators, which satisfy dynamical equations following from an effective Hamiltonian. In the special case of the path that leads to the PZW results, that transformation is precisely the usual PZW transformation, through which the polarization and magnetization fields arise.
The polarization and magnetization fields that arise in the PZW treatment of an atom or small molecule are examples of what we call ideal polarization and magnetization fields, in that they are nonzero only over distances on the order of an atomic spacing from the charge and current densities that are responsible for them. A second goal of this paper is to establish that, for systems where electrons cannot be considered as being bound to identifiable units such as atoms or molecules, it is in general impossible to construct effective Hamiltonians involving such ideal polarization and magnetization fields. This is done in Section 5. We conclude that section by arguing that for such systems a more promising strategy in a search for gauge-invariant descriptions of optical response would be to seek an approach based on a Peirels phase of the second type, at least in problems where the correlation lengths are short compared to the distance over which the electromagnetic field varies. We summarize our results in Section 6.
Section snippets
Generalized LF
In this section we extend the approach of Levanda and Fleurov [3] by generalizing the Peirels phase used in the transformation leading to gauge-invariant Green functions. We refer to the Green function formalism that results as “generalized Levanda–Fleurov” (GLF).
We begin by introducing an integral of the four-potential over a general path linking two space–time points and , We use the relativistic notation with the metric , contravariant components of
The link to the LF approach
In this section we outline the connection between the general formalism of Section 2 and the approach of Levanda and Fleurov (LF). We choose the path in space–time to be a straight line, and employ the parametrization with the parameter ranging from to . The phase factor (2) takes the form where we have changed the integration variable . We introduce the center and difference
The link to PZW
Next we identify the path in space–time that leads to Green function dynamics associated with the usual PZW results [1], [19]. We choose a special point , which for an atom could be the position of a nucleus, considered fixed in this simple example.
Parts of the paths in space–time we consider here will be paths in space at fixed time. We introduce continuous, time-independent functions to describe paths in three-space going from to as ranges from to . We also
Hamiltonians and Green functions
In their applications to the physics of atoms and small molecules, the polarization and magnetization fields (51), (52) are characterized by the property that they are nonzero only on the order of a few atomic spacings from the charge and current densities that are responsible for them. We generally call polarization and magnetization fields characterized by this property ideal.
The problem is qualitatively different if we consider a crystal or other extended system, where the charges are not
Summary
We have shown that two seemingly distinct methods for introducing gauge-invariant forms of the interaction of the electromagnetic field with matter can be seen as special cases of a general formalism. The first method is the approach of Levanda and Fleurov for the introduction of gauge-invariant Green function dynamics. Here the gauge-invariant Green function is introduced using a generalized Peirels phase involving a straight-line path in space–time. The second method, due to Power, Zienau,
Acknowledgment
The authors acknowledge support from the Natural Sciences and Engineering Research Council of Canada, including the award of a CGS-M scholarship to S.D.S.
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