Effect of Fermi Surface Curvature on Low-Energy Properties of Fermions with Singular Interactions

Abstract

We discuss the effect of Fermi surface curvature on long-distance/time asymptotic behaviors of two-dimensional fermions interacting via a gapless mode described by an effective gauge field-like propagator. By comparing the predictions based on the idea of multi-dimensional bosonization with those of the strong-coupling Eliashberg approach, we demonstrate that an agreement between the two requires a further extension of the former technique.In recent years, the behavior of fermions coupled via singular (long-range and/or retarded) interactions has been at the forefront of theoretical research in condensed matter physics. Such singular interactions are often associated with the ground state instabilities and concomitant non-Fermi-liquid behaviors which might occur even in those systems whose microscopic Hamiltonians involve only short-range couplings.In a close proximity to the corresponding quantum phase transition, an effective singular coupling is mediated by (nearly gapless) collective excitations of the emergent order parameter of either charge or spin nature. Important examples include such extensively studied problems as antiferromagnetic 1,2 and charge 3 ordering transitions in hole-doped cuprates, quantumcritical ferro-4 and antiferro-5 magnetic instabilities in heavy fermion materials, compressible Quantum Hall Effect (QHE) 6 , and Pomeranchuk transitions in lowdimensional electron gases. The Pomeranchuk transition has been originally discussed in relation to the transport anisotropies reported in QHE systems at large halfinteger filling factors 7 . In a more general setting, the idea of a spontaneous Pomeranchuk-like distortion of the Fermi surface (FS) associated with the transition to a rotationally anisotropic "nematic" state in a generic fermion system was put forward in Refs.8,9,10 . Despite their different physical nature, the systems studied in Refs. 1,3,4,5,6,8 conform to the model of a finite density gas of two-dimensional fermions coupled via a collective bosonic mode, whose own dynamics is described by the (transverse) gauge field-like propagatorA singular nature of the effective interaction (1) manifests itself in singular corrections to the fermion propa-To first order, the fermion self-energy defined aswhere. At energies ω < ω 0 the one-loop self-energy (2) exceeds the linear in energy term in the bare propagator. Therefore, it can no longer be treated as a perturbation, and the higher-order contributions must be considered as well.If one chooses to completely neglect the FS curvature altogether, a naive perturbation series expansion for the self-energy appears to produce increasingly more and more divergent terms, the n th order term behaving as Σ n ∝ ω 1−n/3 (Refs. 11,12 ). However, a finite FS curvature provides a regularization of such spurious divergences 9,11,12,13 . Namely, by treating the bosonic mode governed by the propagator (1) as a slow subsystem and invoking the generalized Migdal theorem, one finds that the vertex corrections are controled by a (inverse) FS curvature and app...

Quantum critical behavior in itinerant electron systems: Eliashberg theory and instability of a ferromagnetic quantum critical point

“…If we neglected the Eliashberg self-energy (i.e., expanded around free fermions), we would obtain ω log ω correction to the DOS. This last result agrees with the one obtained by [21] using the same technique as in [12].…”

“…We find that such corrections are dangerous if the expansion around the Eliashberg solution holds in powers of terms of order one. The same expansion around free fermions yields terms which formally diverge as powers of ω −1/3 if one neglects the curvature of the Fermi surface [12,21]. We show, in agreement with [2], that in this situation, the way to construct a fully controllable perturbation expansion around the Eliashberg theory at the QCP is to either assume that the curvature of the Fermi surface is large, or extend the theory to a large number of fermionic flavors, N .…”

“…We see that, when the curvature of the fermionic dispersion is included, that the two-loop self-energy turns out to be small compared to its one-loop counterpart, at low energy. In a separate study [21], one of us (A.C.) and D. Khveshchenko reconsidered the bosonization procedure in the presence of the curvature and obtained the same results as in (3.12).…”

Fermionic propagators for two-dimensional systems with singular interactions

“…In 2D, one-loop self-energy ω D m becomes ω 2/3 m . It has long been the issue 2,5,9,12,13 whether ω 2/3 m form is the exact expression for a non-Fermi liquid fermionic propagator. The answer to this question is still lacking.…”

Dinámica fuera de equilibrio de sistemas cuánticos aislados

“…alcance o retardadas), un modelo que presenta comportamientos que no son LF y que puede servir para describir diferentes situaciones tales como transiciones magnéticas en los cupratos, efecto Hall cuántico y la transición de Pomeranchuk (sistemas que poseen sólo interacciones de corto alcance en el modelo microscópico). Sin embargo, la validez de los resultados de la bosonización en estos casos más desafiantes es objeto de debates[34].3.3. Quenches en la interacción y bosonizaciónEn esta sección describiremos cómo hay que modificar el enfoque de la bosonización en equilibrio, descripto antes, para poder calcular funciones de correlación después de un quench de interacción en un gas de fermiones en D > 1.…”