Study of resonant inelastic light scattering in Keldysh-Schwinger functional integral formalism

Abstract: The scattering cross section of the resonant inelastic light scattering is represented as a correlation function in the Keldysh-Schwinger functional integral formalism. The functional integral approach enables us to compute the cross section in the Feynman diagram perturbation theory where many-body effects can be fully incorporated. This approach is applied to the one G-phonon Raman scattering of graphene, and the result is shown to agree with the one previously obtained by the conventional Fermi golden rule formula. Also, this approach is generalized to the systems in non-equilibrium conditions.

• The paper reformulates resonant ILS as a four-current correlation function, bypassing the need to sum over intermediate states.
• It applies the Keldysh-Schwinger formalism to calculate one-phonon Raman scattering in graphene, validating the approach against traditional methods.
• The method naturally extends to non-equilibrium systems, enabling systematic inclusion of many-body effects like self-energy and vertex corrections.

Resonant Inelastic Light Scattering via Keldysh-Schwinger Functional Integral Approach

Introduction and Motivation

This work develops a field-theoretic approach to resonant inelastic light scattering (ILS) using the Keldysh-Schwinger (KS) closed time path formalism, recasting the scattering cross section as a correlation function accessible to standard functional integral and diagrammatic methods. This formalism addresses a critical gap in the theoretical treatment of resonant ILS: enabling precise inclusion of many-body effects without explicit reference to usually intractable intermediate states, and extending applicability to non-equilibrium systems. The approach is concretely demonstrated on one-phonon Raman scattering in graphene and generalized to systems under non-equilibrium drive.

From Golden Rule to Correlation Function Formulation

The traditional calculation of ILS cross sections utilizes the Fermi golden rule and, in the context of resonant processes, the Kramers-Heisenberg formula, which involves a summation over intermediate many-body eigenstates. This summation is computationally prohibitive and, for strongly correlated systems, typically inaccessible. The core methodological advance in this work is expressing the resonant contribution to the cross-section as a four-point correlation function of electric current operators, circumventing the need to enumerate intermediate states.

Crucially, the formalism maintains both time-ordering and anti-time ordering, as necessitated by the structure of transition probabilities in quantum mechanics. This naturally motivates the use of the KS closed time contour—the foundation of non-equilibrium field theory—which accommodates forward and backward time evolution in a unified path-integral framework.

Keldysh-Schwinger Functional Integral Formalism

The KS approach expresses the transition probability in terms of time-ordered, anti-time-ordered, and mixed correlation functions. The cross-section is recast as an integral over four electronic current operators, with time-ordering structure completely disentangled and encoded in the functional integral over the KS contour.

A general result for correlation functions of the form

$$\Pi(\tau_1, \tau_2, \tau_3, \tau_4) = \mathrm{Tr}\left[\tilde{\mathcal{T}}[\widehat{A}(\tau_1)\widehat{B}(\tau_2)] \mathcal{T}[\widehat{C}(\tau_3)\widehat{D}(\tau_4)] \hat{\rho}\right]$$

is established, and a precise mapping between operator expressions and their representation as functionals of coherent state variables on the KS contour is constructed. This result enables systematic diagrammatic expansion using standard tools of many-body field theory.

Application: G-Phonon Raman Scattering in Graphene

The method is applied to the problem of one-phonon Raman scattering in graphene. Utilizing the specific lattice and band structure, the four-current correlation function formalism allows explicit construction of the relevant Feynman diagrams. The authors demonstrate that the result for the Raman intensity derived via the KS functional integral method is in complete agreement with the result obtained from the Fermi golden rule, as previously reported in "Theory of resonant multiphonon Raman scattering in graphene" by Basko [basko]. This serves as a non-trivial validation of the formalism.

The approach rigorously elucidates the connection between the Feynman diagrammatics of the correlation function and the energy conservation constraints implicit in the Kramers-Heisenberg formula. The formalism clarifies which diagrams encode resonant processes and which are suppressed—such as those not satisfying overall energy conservation—permitting efficient diagram selection.

The finite-temperature generalization is naturally achieved within the KS framework, and the Stokes/anti-Stokes symmetry emerges intrinsically. The inclusion of Keldysh Green’s functions ensures all quantum statistical effects are incorporated in a consistent manner.

Non-Equilibrium Generalization

A significant extension is provided for non-equilibrium systems, e.g., semiconductors under intense pump-probe fields. The formalism is adapted to accommodate time-dependent Hamiltonians via the explicit inclusion of drive (pump) terms in the KS action, allowing for treatment of pump-probe ILS and time-resolved Raman processes. The initial state preparation under non-equilibrium conditions is rigorously handled by forward and backward time evolution with the pump field, and the functional integral prescription remains valid, with modifications to the correlation and response functions as dictated by the non-equilibrium Green’s functions.

Many-Body Effects and Diagrammatics

A central technical implication of the formalism is its capacity to systematically include many-body effects, such as self-energy and vertex corrections, which appear in higher-order Feynman diagrams. The formalism facilitates the computation of full interacting Green’s functions and their impact on observable quantities such as linewidths, phonon frequency shifts, and nontrivial lineshape modifications.

Notably, the logarithmic corrections expected in electron (and phonon) self-energies, which are known to be essential for the accurate interpretation of experimental ILS data (e.g., in graphene), can be incorporated straightforwardly [aleiner]. The vertex corrections, typically intractable in traditional golden rule-based approaches, are likewise accessible.

Comparison with Conventional Formalisms

A salient theoretical point is that the KS-based approach contains, and in fact goes beyond, the Kramers-Heisenberg formula: additional diagrams appear which do not enforce strict energy conservation between initial and final states. In physical regimes where the perturbation is switched on adiabatically or for strictly stationary processes, only those diagrams consistent with conventional energy selection rules contribute dominantly, and the formalism reproduces the traditional outcome. However, in transient or quenched systems, or under strong driving, nonresonant or off-shell contributions may become non-negligible, and the present formalism captures these without modification.

Conclusion

This work establishes a robust, systematic methodology for computing resonant inelastic light scattering cross sections in both equilibrium and non-equilibrium many-body systems by recasting the problem in the language of Keldysh-Schwinger functional integrals. The formalism fully incorporates many-body effects and naturally generalizes to non-equilibrium settings, demonstrating equivalence with, and extension of, standard diagrammatic perturbation theory results such as those for G-phonon Raman scattering in graphene. This approach provides a unified framework for future studies of ILS in correlated materials, especially in dynamically driven or non-equilibrium regimes, and opens the prospect of systematic inclusion of complex many-body corrections inaccessible to prior treatments.