We present a formal derivation of the many-body perturbation theory for a system of electrons and bosons subject to a nonlinear electron-boson coupling. The interaction is treated at an arbitrary high order of bosons scattered. The considered Hamiltonian includes the well-known linear coupling as a special limit. This is the case, for example, of the Holstein and Fr\"ohlich Hamiltonians. Indeed, whereas linear coupling has been extensively studied, the scattering processes of electrons with multiple bosonic quasiparticles are largely unexplored. We focus here on a self-consistent theory in terms of dressed propagators and generalize the Hedin's equations using the Schwinger technique of functional derivatives. The method leads to an exact derivation of the electronic and bosonic self-energies, expressed in terms of a new family of vertex functions, high-order correlators, and bosonic and electronic mean-field potentials. In the electronic case we prove that the mean-field potential is the $n\mathrm{th}$ -order extension of the well-known Debye-Waller potential. We also introduce a bosonic mean-field potential entirely dictated by nonlinear electron-boson effects. The present scheme, treating electrons and bosons on an equal footing, demonstrates the full symmetry of the problem. The vertex functions are shown to have purely electronic and bosonic character as well as a mixed electron-boson one. These four vertex functions are shown to satisfy a generalized Bethe-Salpeter equation. Multibosons response functions are also studied and explicit expressions for the two and the three bosons case are given.We present a formal derivation of the many-body perturbation theory for a system of electrons and bosons subject to a nonlinear electron-boson coupling. The interaction is treated at an arbitrary high order of bosons scattered. The considered Hamiltonian includes the well-known linear coupling as a special limit. This is the case, for example, of the Holstein and Fr\"ohlich Hamiltonians. Indeed, whereas linear coupling has been extensively studied, the scattering processes of electrons with multiple bosonic quasiparticles are largely unexplored. We focus here on a self-consistent theory in terms of dressed propagators and generalize the Hedin's equations using the Schwinger technique of functional derivatives. The method leads to an exact derivation of the electronic and bosonic self-energies, expressed in terms of a new family of vertex functions, high-order correlators, and bosonic and electronic mean-field potentials. In the electronic case we prove that the mean-field potential is the $n\mathrm{th}$-order extension of the well-known Debye-Waller potential. We also introduce a bosonic mean-field potential entirely dictated by nonlinear electron-boson effects. The present scheme, treating electrons and bosons on an equal footing, demonstrates the full symmetry of the problem. The vertex functions are shown to have purely electronic and bosonic character as well as a mixed electron-boson one. These four vertex functions are shown to satisfy a generalized Bethe-Salpeter equation. Multibosons response functions are also studied and explicit expressions for the two and the three bosons case are given.

arXiv:1807.06949Physical review. B./Physical review. BAug 3, 201815 citations

Quantum theory of interacting bosons

Eugene P. Gross

Annals of PhysicsFeb 1, 196064 citations

Instability of the critical Ngai's coupling and two-boson mechanism in metals

Nguyen, Phu, Vladyslav Kozii

We study the properties of a Fermi liquid coupled to a quantum critical boson via the two-boson interaction known as Ngai's coupling. We find that the original quantum critical point is generally unstable, resulting in a finite-momentum spatially modulated state unless two conditions are satisfied: (i) the critical boson is polar and transverse, and (ii) the ratio of the Fermi velocity to the transverse-boson velocity is sufficiently large. If these conditions hold and the uniform state remains stable, we demonstrate that the system enters a strong-coupling regime below a certain energy scale. In this regime, we discuss a self-consistent solution at criticality in two-dimensional systems and show that the critical boson field develops a nontrivial anomalous dimension, $η=1/2$ . Our findings highlight the significant role of two-boson coupling in critical theories, challenging the conventional view that its effects are subdominant to linear coupling.We study the properties of a Fermi liquid coupled to a quantum critical boson via the two-boson interaction known as Ngai's coupling. We find that the original quantum critical point is generally unstable, resulting in a finite-momentum spatially modulated state unless two conditions are satisfied: (i) the critical boson is polar and transverse, and (ii) the ratio of the Fermi velocity to the transverse-boson velocity is sufficiently large. If these conditions hold and the uniform state remains stable, we demonstrate that the system enters a strong-coupling regime below a certain energy scale. In this regime, we discuss a self-consistent solution at criticality in two-dimensional systems and show that the critical boson field develops a nontrivial anomalous dimension, $η=1/2$. Our findings highlight the significant role of two-boson coupling in critical theories, challenging the conventional view that its effects are subdominant to linear coupling.

arXiv:2504.11523Apr 15, 20250 citations

Hamiltonian Conditions for Dark Modes in Multimode Bosonic Systems

Zuxin Wu, Shengshi Pang

Dark modes arise when destructive interference prevents selected bosonic degrees of freedom from coupling to environmental channels. We formulate a Hamiltonian criterion for identifying such modes in multimode bosonic systems by separating two requirements: the candidate mode must be invisible to the direct system--environment coupling, and its generated operator space must remain invariant under the intrinsic system dynamics. For linear environment coupling and quadratic system Hamiltonians, the criterion is reduced to the the familiar null-space and invariant-subspace conditions of passive linear dark-mode theory. We then extend the analysis to nonlinear scenarios. For a two-photon conversion channel coupled to an auxiliary environmental mode, interference among nonlinear conversion pathways can reduce the environment coupling to a single collective two-photon channel, leaving a complementary bosonic mode decoupled from the environment. We show that preserving this mode under nonlinear intrinsic dynamics generally requires more than conventional Kerr-type quartic interactions: correlated four-boson conversion processes are needed to cancel mixed nonlinear conversion between the dark and environment-coupled collective modes. Finally, we show that the Bogoliubov dark mode of a parametrically driven optomechanical satisfies the same Hamiltonian criterion through an active canonical transformation. These results provide a unified Hamiltonian framework for identifying and engineering dark modes in linear, nonlinear, and driven bosonic systems.Dark modes arise when destructive interference prevents selected bosonic degrees of freedom from coupling to environmental channels. We formulate a Hamiltonian criterion for identifying such modes in multimode bosonic systems by separating two requirements: the candidate mode must be invisible to the direct system--environment coupling, and its generated operator space must remain invariant under the intrinsic system dynamics. For linear environment coupling and quadratic system Hamiltonians, the criterion is reduced to the the familiar null-space and invariant-subspace conditions of passive linear dark-mode theory. We then extend the analysis to nonlinear scenarios. For a two-photon conversion channel coupled to an auxiliary environmental mode, interference among nonlinear conversion pathways can reduce the environment coupling to a single collective two-photon channel, leaving a complementary bosonic mode decoupled from the environment. We show that preserving this mode under nonlinear intrinsic dynamics generally requires more than conventional Kerr-type quartic interactions: correlated four-boson conversion processes are needed to cancel mixed nonlinear conversion between the dark and environment-coupled collective modes. Finally, we show that the Bogoliubov dark mode of a parametrically driven optomechanical satisfies the same Hamiltonian criterion through an active canonical transformation. These results provide a unified Hamiltonian framework for identifying and engineering dark modes in linear, nonlinear, and driven bosonic systems.

quant-pharXiv:2607.10118Jul 14, 20260 citations

Unified Theory of Interacting Bosons

Eugene P. Gross

Physical ReviewApr 1, 1957190 citations

n\mathrm{th}

-order extension of the well-known Debye-Waller potential. We also introduce a bosonic mean-field potential entirely dictated by nonlinear electron-boson effects. The present scheme, treating electrons and bosons on an equal footing, demonstrates the full symmetry of the problem. The vertex functions are shown to have purely electronic and bosonic character as well as a mixed electron-boson one. These four vertex functions are shown to satisfy a generalized Bethe-Salpeter equation. Multibosons response functions are also studied and explicit expressions for the two and the three bosons case are given.

η=1/2

. Our findings highlight the significant role of two-boson coupling in critical theories, challenging the conventional view that its effects are subdominant to linear coupling.