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arXiv:2401.15155 · Physical review. B./Physical review. B

Polaron spectra and edge singularities for correlated flat bands

Dimitri Pimenov

Published May 16, 2024• 9 citations
arXivHTMLPDF

Abstract

Single- and two-particle spectra of a single immobile impurity immersed in a fermionic bath can be computed exactly and are characterized by divergent power laws (edge singularities). Here we present the leading lattice correction to this canonical problem by embedding both impurity and bath fermions in bands with nonvanishing Bloch band geometry, with the impurity band being flat. By analyzing generic Feynman diagrams, we pinpoint how the band geometry reduces the effective interaction which enters the power laws; we find that for weak lattice effects or small Fermi momenta, the leading correction is proportional to the Fermi energy times the sum of the quantum metrics of the bands. When only the bath fermion geometry is important, the results can be extended to large Fermi momenta and strong lattice effects and cross-validated by analysis of $S$-matrix eigenvalues. We numerically illustrate our results on the Lieb lattice and draw connections to various spectroscopy experiments.

DOI
10.1103/physrevb.109.195153

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Physical review. B, Condensed matter•Aug 15, 1996•18 citations
Single- and two-particle spectra of a single immobile impurity immersed in a fermionic bath can be computed exactly and are characterized by divergent power laws (edge singularities). Here we present the leading lattice correction to this canonical problem by embedding both impurity and bath fermions in bands with nonvanishing Bloch band geometry, with the impurity band being flat. By analyzing generic Feynman diagrams, we pinpoint how the band geometry reduces the effective interaction which enters the power laws; we find that for weak lattice effects or small Fermi momenta, the leading correction is proportional to the Fermi energy times the sum of the quantum metrics of the bands. When only the bath fermion geometry is important, the results can be extended to large Fermi momenta and strong lattice effects and cross-validated by analysis of SSS-matrix eigenvalues. We numerically illustrate our results on the Lieb lattice and draw connections to various spectroscopy experiments.
We calculate the exact spectral function of a single impurity repulsively interacting with a bath of fermions in one-dimensional lattices, by deriving the explicit expression of the form factor for both regular Bethe states and the irregular spin-flip state and η-pairing state, based on the exactly solvable one-dimensional Hubbard model. While at low impurity momentum Q∼0 the spectral function is dominated by two power-law Fermi singularities, at large momentum we observe that the two singularities develop into two-sided distributions and eventually become anomalous Fermi singularities at the boundary of the Brillouin zone (i.e., Q=±π), with the power-law tails extending toward low energy. Near the quarter filling of the Fermi bath, we also find two broad polaron peaks at large impurity momentum, collectively contributed by many excited many-body states with non-negligible form factors. Our exact results of those distinct features in one-dimensional Fermi polarons, which have no correspondences in two and three dimensions, could be readily probed in cold-atom laboratories by trapping highly imbalanced two-component fermionic atoms into one-dimensional optical lattices.
A magnetic impurity embedded in a metal is collectively screened by Fermi-surface quasiparticles into a many-body spin-singlet ground state, forming a Kondo cloud of size ξK∼ℏvF/(kBTK)ξ_{\rm K}\sim\hbar v_F/(k_B T_{\rm K})ξK​∼ℏvF​/(kB​TK​). This kinematic picture collapses in flat bands, where vF=0v_F=0vF​=0 and the hierarchy of dispersive energy shells is absent. Here we show that the missing organizing principle is quantum geometry. A magnetic impurity coupled to an isolated flat band selects a single active bath mode: a coherent superposition of flat-band Bloch states weighted by the hybridization factor v(k)v(\mathbf{k})v(k), while all orthogonal flat-band modes remain dark. The resulting flat-band Kondo problem is a quantum geometric molecule, with an algebraic Kondo scale set by the total projected hybridization strength rather than a logarithmic-renormalization scale. In real space, the impurity-bath spin correlation defines a quantum geometric Kondo cloud. Its cloud-size tensor admits a gauge-invariant decomposition into a hybridization-weighted quantum metric, a dressed Berry-connection covariance, and a positive hybridization-gradient term, yielding the lower bound ξK2≥∑kρ(k)Tr g(k)ξ_{\rm K}^2\geq \sum_{\mathbf{k}}ρ(\mathbf{k}){\rm Tr}\,g(\mathbf{k})ξK2​≥∑k​ρ(k)Trg(k), where ρ(k)=∣v(k)∣2/∑k∣v(k)∣2ρ(\mathbf{k})=|v(\mathbf{k})|^2/\sum_{\mathbf{k}}|v(\mathbf{k})|^2ρ(k)=∣v(k)∣2/∑k​∣v(k)∣2. Our result reveals that, in flat bands, Kondo screening is governed by the quantum geometry and interference structure of the impurity-selected Bloch wave packet, rather than Fermi-surface kinematics.
We study the spectrum of an impurity coupled to a Fermi sea (e.g., minority atom in an ultracold gas, exciton in a solid) by attraction strong enough to form a molecule/trion. We introduce a diagrammatic scheme which allows treating a finite mass impurity while reproducing the Fermi-edge singularity in the immobile limit. For large binding energies the spectrum is characterized by a semicoherent repulsive polaron and an incoherent molecule-hole continuum, which is the lowest-energy feature in the single-particle spectrum. The previously predicted attractive polaron seems not to exist for strong binding.
Abstract By using the exact Bethe wavefunctions of the one-dimensional Hubbard model with N spin-up fermions and one spin-down impurity, we derive an analytic expression of the impurity form factor, in the form of a determinant of a (N+1)(N+1)(N+1) ( N + 1 ) by (N+1)(N+1)(N+1) ( N + 1 ) matrix. This analytic expression enables us to exactly calculate spectral properties of one-dimensional Fermi polarons in lattices, when the masses of the impurity particle and the Fermi bath are equal. We present the impurity spectral function as functions of the on-site interaction strength and the filling factor of the Fermi bath, and discuss the origin of Fermi singularities in the spectral function at small momentum and the emergence of polaron quasiparticles at large momentum near the boundary of Brillouin zone. Our analytic expression of the impurity form factors pave the way to exploring the intriguing dynamics of a particle interacting with a Fermi bath. Our exact predictions on the impurity spectral function could be directly examined in cold-atom laboratories by using the radio-frequency spectroscopy and Ramsey spectroscopy.
We consider the behavior of a small density of mobile impurities (polarons) immersed in a quantum gas, a generic scenario that can be realized in cold atomic gases, liquid helium mixtures, and doped semiconductors. We present a unified theoretical framework for understanding polaron quasiparticles beyond the single-impurity limit, and we identify two key factors that control the polaron-polaron interactions: (i) the statistics of the impurities, including whether or not they are degenerate, and (ii) the constraints on the medium response, i.e., whether the medium density or chemical potential is held fixed. By constructing wave functions for two bosonic, fermionic, or distinguishable impurities immersed in a Bose or Fermi gas, we derive rigorous results for the polaron interactions in the limit of weak impurity-medium coupling. We furthermore obtain an exact relationship between the polaron interactions at fixed medium density and at fixed chemical potential, a result which is valid for arbitrary interaction strength. Our work provides an important guide for understanding experiments, and it acts as a starting point for future strong-coupling theories of polaron interactions that capture all of the effects identified in this work.
Effects of long-range Coulomb interactions on the Fermi-edge singularity in optical spectra are investigated theoretically for one-dimensional spin-1/2 fermion systems with the use of the Tomonaga-Luttinger bosonization technique. Low-energy excitation spectrum near the Fermi level shows that dispersion of the charge-density fluctuation remains gapless but is nonlinear when the electron-electron (e-e) Coulomb interaction is of the ${\mathit{x}}^{\mathrm{\ensuremath{-}}1}$ type (i.e., an infinite force range). Temporal behavior of the current-current correlation function is calculated analytically for arbitrary force ranges, ${\ensuremath{\lambda}}_{\mathit{e}}$ and ${\ensuremath{\lambda}}_{\mathit{h}}$, of the e-e and the electron-hole (e-h) Coulomb interactions. (i) When both the e-e and the e-h interactions have large but finite force ranges (${\ensuremath{\lambda}}_{\mathit{e}}$\ensuremath{\infty} and ${\ensuremath{\lambda}}_{\mathit{h}}$\ensuremath{\infty}), the correlation function yields a power-law decay only for a long-time regime, which is determined by the force ranges as t>max[${\ensuremath{\lambda}}_{\mathit{e}}$,${\ensuremath{\lambda}}_{\mathit{h}}$]/vF{\mathit{v}}_{\mathit{F}}vF​. Corresponding optical spectrum near the Fermi edge (within an energy range of \ensuremath{\Elzxh}vF{\mathit{v}}_{\mathit{F}}vF​/max[${\ensuremath{\lambda}}_{\mathit{e}}$,${\ensuremath{\lambda}}_{\mathit{h}}$]) exhibits the power-law divergence or the power-law convergence, which is an ordinary Fermi-edge singularity. (ii) When either the e-e or the e-h interaction is of the ${\mathit{x}}^{\mathrm{\ensuremath{-}}1}$ type (i.e., ${\ensuremath{\lambda}}_{\mathit{e}}$\ensuremath{\rightarrow}\ensuremath{\infty} and/or ${\ensuremath{\lambda}}_{\mathit{h}}$\ensuremath{\rightarrow}\ensuremath{\infty}), an exponent of the correlation function is dependent on time to lead the faster decay than that of any power laws. Then the optical spectra show no power-law dependence and always converge (become zero) at the Fermi edge, which is in striking contrast to the ordinary power-law singularity. Relations between the spectral shape and a measurement time (frequency resolution) are also clarified. \textcopyright{} 1996 The American Physical Society.