We discuss the orthogonality catastrophe between the ground states of the unperturbed and perturbed Fermi systems due to local electron correlation U in the Anderson model. We show that the symmetric Anderson model presents a special case in which no catastrophe appears and that in the asymmetric case the catastrophe results from the self-energy shift.

Progress of Theoretical PhysicsJan 1, 19787 citations

arXiv:1301.4923 · Annales Henri Poincaré

Anderson’s Orthogonality Catastrophe for One-Dimensional Systems

Heinrich Küttler, Peter Otte, Wolfgang Spitzer

Published Oct 16, 2013• 8 citations

arXiv HTML PDF

DOI: 10.1007/s00023-013-0287-z

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We provide a fully analytical derivation of Anderson's orthogonality catastrophe for the one dimensional Fermi polaron integrable model, describing a system of

N

spin-up fermions, with fixed density

n=N/L

, interacting with a single spin-down fermion via an attractive contact potential. The proof combines the determinant representations of the norm of the many-body wave function and of its scalar product with the noninteracting ground state, obtained from the Bethe ansatz solution, with the special properties of Cauchy matrices. We derive the leading asymptotics of the two determinants in the thermodynamic limit and show that the quasi-particle residue

Z

decays algebraically,

Z=W N^{-θ}

. We confirm that the Anderson exponent

θ

is equal to

2δ_F^2/π^2

, where

δ_F

is the Bethe-ansatz phase shift at the Fermi edge. We also derive the asymptotic behavior of the prefactor

W

at strong coupling and compare it against exact numerics.

Anderson orthogonality catastrophe in

2+1

-D topological systems

(2+1)

-D topological systems at fixed points, there exists a universal topological response term in the scaling of the ground-state overlap. For Laughlin wave functions, in particular, we also find a leading term decaying faster than exponential, which is beyond AOC. Such finite-size scaling behaviors could be utilized to theoretically detect the gapless edge modes, distinguish the topology of quantum states or serve as a signature for topological phase transitions.