Breakdown of the Migdal-Eliashberg theory for electron-phonon systems. Role of polarons/bi-polarons

arXiv (Cornell University) · 2026-04-15

The Migdal-Eliashberg theory (MET) describes electrons interacting with phonons in the adiabatic limit when the phonon Debye frequency is much smaller than the Fermi energy. A conventional belief is that MET holds even at strong coupling, when electron self-energy is large, and breaks down only near the point where the dressed phonon spectrum softens to near zero. We analyze numerically and analytically a different option -- collapse to a polaronic/bipolaronic ground state. The last scenario has never been analyzed in precise quantitative terms for a generic electron density. Using variational considerations, we establish rigorous upper bounds on the coupling $λ$, at which a FL state transforms into the bipolaron/polaron state. We show that at small and near-maximum densities, this happens well before a dressed phonon softens. This is true both in 2D and 3D systems; in the latter the upper bound on $λ$ tends to zero in the limit of small or near-full density. We present analytical reasoning for this behavior based on hints extracted from exact diagrammatic treatment of the on-site Holstein model for the spin polarized case and argue that polarons are produced by fermions with energies comparable to the bandwidth; i.e., polaron formation is outside the realm of MET. Closer to half-filling, the leading instability upon increasing $λ$ is towards a charge-density-wave state (CDW), and there exists a strong coupling regime of MET near this instability, while the polaron/bipolaron state develops at larger $λ$ out of a CDW-ordered state and inherits a CDW order over some range of coupling.

Limits of validity for Migdal-Eliashberg theory: role of polarons/bi-polarons

arXiv (Cornell University) · 2026-04-15

It is widely believed that in an adiabatic limit a Fermi liquid state of an electron-phonon system described by Migdal-Eliashberg theory remains stable before a dressed phonon softens. Using Holstein model as a prototypical example and variational/analytic considerations we demonstrate that in a wide range of fillings both in 3D and 2D, a polaronic/bi-polaronic state emerges before phonon softening; at small filling in 3D this happens already at weak coupling. We show that a polaronic/bi-polaronic state emerges, upon increasing coupling, via an intermediate pseudogap-type mixed state, in which some fermions regain Fermi liquid behavior, yet Luttinger theorem is broken. At even larger couplings the density of states gradually approaches its form in the atomic limit.

Kinetics of Bose–Einstein condensation of magnons in Yttrium Iron Garnet films

Academia Nano Science Materials Technology · 2026-02-10

In this article, we resolve the apparent contradiction between recent experiments and earlier theoretical studies predicting strongly asymmetric condensates resulting in an attractive interaction between condensate magnons. We show that the relaxation time required to achieve equilibrium of the two condensates at the two energy minima exceeds the experiment duration. Therefore, the system is inherently out of balance and must be described by Boltzmann’s kinetic equations. We develop an appropriate kinetic theory and derive the relation between the critical pumping power and the effective temperature of over-condensate magnons.

Limits of validity for Migdal-Eliashberg theory: role of polarons/bi-polarons

ArXiv.org · 2026-04-15

It is widely believed that in an adiabatic limit a Fermi liquid state of an electron-phonon system described by Migdal-Eliashberg theory remains stable before a dressed phonon softens. Using Holstein model as a prototypical example and variational/analytic considerations we demonstrate that in a wide range of fillings both in 3D and 2D, a polaronic/bi-polaronic state emerges before phonon softening; at small filling in 3D this happens already at weak coupling. We show that a polaronic/bi-polaronic state emerges, upon increasing coupling, via an intermediate pseudogap-type mixed state, in which some fermions regain Fermi liquid behavior, yet Luttinger theorem is broken. At even larger couplings the density of states gradually approaches its form in the atomic limit.

Breakdown of the Migdal-Eliashberg theory for electron-phonon systems. Role of polarons/bi-polarons

ArXiv.org · 2026-04-15

The Migdal-Eliashberg theory (MET) describes electrons interacting with phonons in the adiabatic limit when the phonon Debye frequency is much smaller than the Fermi energy. A conventional belief is that MET holds even at strong coupling, when electron self-energy is large, and breaks down only near the point where the dressed phonon spectrum softens to near zero. We analyze numerically and analytically a different option -- collapse to a polaronic/bipolaronic ground state. The last scenario has never been analyzed in precise quantitative terms for a generic electron density. Using variational considerations, we establish rigorous upper bounds on the coupling $λ$, at which a FL state transforms into the bipolaron/polaron state. We show that at small and near-maximum densities, this happens well before a dressed phonon softens. This is true both in 2D and 3D systems; in the latter the upper bound on $λ$ tends to zero in the limit of small or near-full density. We present analytical reasoning for this behavior based on hints extracted from exact diagrammatic treatment of the on-site Holstein model for the spin polarized case and argue that polarons are produced by fermions with energies comparable to the bandwidth; i.e., polaron formation is outside the realm of MET. Closer to half-filling, the leading instability upon increasing $λ$ is towards a charge-density-wave state (CDW), and there exists a strong coupling regime of MET near this instability, while the polaron/bipolaron state develops at larger $λ$ out of a CDW-ordered state and inherits a CDW order over some range of coupling.

Superconductivity Near a Quantum Critical Point: Bounds on the Transition Temperature in the $γ$-Model

ArXiv.org · 2025-12-23

Near a quantum critical point (QCP) in a metal, strong Fermion-Fermion interactions mediated by soft collective bosons give rise to two competing phenomena: non-Fermi liquid behavior and superconductivity that deviates from conventional BCS and Migdal-Eliashberg theories. We consider the problem of obtaining closed-form analytical lower and upper bounds on transition temperatures for such systems. We focus mainly on a class of models known as the gamma-model, which generalizes the Eliashberg theory of Superconductivity where the effective interaction potential scales as V(Omega) ~ 1/|Omega|^gamma. Building on a recent reformulation of Migdal-Eliashberg theory, expressed as a classical infinite spin chain with nonlocal interactions, and employing a simple linear algebra framework, we derive rigorous closed-form expressions for upper and lower bounds on the superconducting transition temperature for any gamma > 0. While our lower bounds coincide precisely with those previously in the literature, our derivation offers a more streamlined and accessible method. Moreover, our upper bounds are substantially tighter than any existing estimates and converge rapidly enough to the numerical results from various prior studies.

Three-Dimensional Domain-Wall Membranes

ArXiv.org · 2025-09-18

Three-dimensional magnetic textures, such as Hopfions, torons, and skyrmion tubes, possess rich geometric and topological structure, but their detailed energetics, deformation modes, and collective behavior are yet to be fully understood. In this work, we develop an effective geometric theory for general three-dimensional textures by representing them as embedded two-dimensional orientable domain-wall membranes. Using a local ansatz for the magnetization in terms of membrane coordinates, we integrate out the internal domain-wall profile to obtain a reduced two-dimensional energy functional. This functional captures the coupling between curvature, topology, and the interplay of micromagnetic energies, and is expressed in terms of a small set of soft-mode fields: the local wall thickness and in-plane magnetization angle. Additionally, we construct a local formula for the Hopf index which sheds light on the coupling between geometry and topology for nontrivial textures. We analyze the general properties of the theory and demonstrate its utility through the example of a flat membrane hosting a vortex as well as a toroidal Hopfion, obtaining analytic solutions for the wall thickness profile, associated energetics, and a confirmation of the Hopf index formula. The framework naturally extends to more complex geometries and can accommodate additional interactions such as Dzyaloshinskii-Moriya, Zeeman, and other anisotropies, making it a versatile tool for exploring the interplay between geometry, topology, and micromagnetics in three-dimensional spin systems.

Thickness-driven transitions between magnetic states in ferromagnetic films

Physical review. B./Physical review. B · 2025-02-21 · 2 citations

Magnetic materials hosting stable topological spin textures have demonstrated energy efficiency and potential as information carriers in logic and memory devices, offering an alternative to magnetic tunnel junction technology. While these structures are well understood in two dimensions (2D), in 3D their stability, interactions, and topological transitions require further exploration. Here we present two thermodynamically stable topological states, termed the ``hourglass'' and ``dome,'' in centrosymmetric magnetic films of varying thickness. Crucially, we observe thickness-dependent transitions between the two states, with regions of metastability where both configurations coexist. We construct a phase diagram detailing the parameter space for their existence and transitions, provide an effective description of the interactions that mediate the transition, and discuss the implications of this type of state switching.

Kinetics of Bose-Einstein condensation of magnons in Yttrium Iron Garnet films

arXiv (Cornell University) · 2025-12-19

In this article, we explain the reason of the apparent contradiction between recent experiments [1] and [2] and earlier theoretical predictions [3] of strongly asymmetric condensate resulting in attractive interaction between the condensate magnons. We show that the relaxation time for equilibrium between two condensates at two minima of energy exceeds the time of experiment. Therefore, it should be described by Boltzmann kinetic equation. We develop the proper kinetic theory and find the relation between the critical pumping power and the effective temperature of over-condensate magnons.

Non-BCS behavior of the pairing susceptibility near the onset of superconductivity in a quantum-critical metal

Physical review. B./Physical review. B · 2025-02-21 · 3 citations

In a quantum critical metal, superconductivity out of non-Fermi liquid normal state develops only when the pairing interaction exceeds a certain threshold. The authors show here that the dynamic pairing susceptibility exhibits an unexpected behavior: it diverges at the onset of pairing only for a certain subclass of bare/test pairing amplitudes and remains nonsingular for other amplitudes, and below the instability becomes a nonunique function. The authors argue that this behavior reflects a highly non-BCS nature of the pairing at quantum criticality.