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arXiv:2604.12475 · Physical Review A

Explicit proof of Anderson's orthogonality catastrophe for the one-dimensional Fermi polaron with attractive interaction

Giuliano Orso

Published Aug 20, 2026• 0 citations
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Abstract

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.

DOI
10.1103/tgtd-3s4k

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Journal of Statistical Mechanics Theory and Experiment•Jun 14, 2007•21 citations
We provide a fully analytical derivation of Anderson's orthogonality catastrophe for the one dimensional Fermi polaron integrable model, describing a system of NNN spin-up fermions, with fixed density n=N/Ln=N/Ln=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 ZZZ decays algebraically, Z=WN−θZ=W N^{-θ}Z=WN−θ. We confirm that the Anderson exponent θθθ is equal to 2δF2/π22δ_F^2/π^22δF2​/π2, where δFδ_FδF​ is the Bethe-ansatz phase shift at the Fermi edge. We also derive the asymptotic behavior of the prefactor WWW at strong coupling and compare it against exact numerics.
We study the correlation functions in a one-dimensional attractive Fermi polaron system of spin-1/2 Fermi gas with one flipped spin as an impurity immersed in the fully polarized spin-up media. By means of the exact Bethe ansatz wave function with string solutions for the momentum, we manage to explicitly deduce the asymptotic form of the correlation functions in the infinite attractive limit and show that the system undergoes a polaron-molecule crossover when the attraction grows. The correlation functions in the infinite attractive limit can be decomposed into the free fermion term with the number of particles reduced by one and the molecular term indicated by the emergence of a sharp peak or dip in the central area of the correlation. These analytical results are checked by the numerical Monte Carlo methods for multiple integrals. The calculation is further extended to the excited states, i.e., the super-Tonks-Girardeau gas, and there is no peak or dip found in the vicinity of the correlation center. Tan contact is found to be proportional to the cubic of the interaction strength and the total energy is verified to satisfy the Tan adiabatic theorem in the strong attraction regime.
We consider a mobile impurity coupled to an ideal Fermi gas in one spatial dimension through an attractive contact interaction. We calculate the quasi-particle residue ZZZ exactly, based on Bethe Ansatz and diagrammatic Monte Carlo methods, and with varational Ansatz up to one particle-hole excitation of the Fermi sea. We find that the exact quasi-particle residue vanishes in the thermodynamic limit as a power law in the number of particles, consistent with the Luttinger-liquid paradigm and the breakdown of Fermi-liquid theory. The variational Ansatz, however, predicts a finite value of ZZZ, even in the thermodynamic limit. We also study how the presence of the impurity affects the density of the spin-up sea by calculating the pair correlation function. Subtracting the homogeneous background and integrating over all distances gives the charge QQQ. This charge turns out to grow continuously from 0 at zero coupling to 1 in the strong-coupling limit. The varational Ansatz predicts Q=0Q=0Q=0 at all couplings. So, although the variational Ansatz has been shown to be remarkably accurate for the energy and the effective mass, it fails even qualitatively when predicting ZZZ and the pair correlation function in the thermodynamic limit.
The goal of this thesis is a mathematical understanding of a phenomenon called Anderson's Orthogonality in the physics literature. Given two non-interacting fermionic systems which differ by an exterior short-range scattering potential, the scalar product of the corresponding ground states is predicted to decay algebraically in the thermodynamic limit. This decay is referred to as Anderson's orthogonality catastrophe in the physics literature and goes back to P.W.Anderson [Phys. Rev. Lett. 18:1049--1051] and is used to explain anomalies in the X-ray absorption spectrum of metals. We call this scalar product SNLS_N^LSNL​, where NNN refers to the particle number and LLL to the diameter of the considered fermionic system. This decay was proven in the works [Commun. Math. Phys. 329:979--998] and [arXiv:1407.2512] for rather general pairs of Schrödinger operators in arbitrary dimension d∈Nd\in\Nd∈N, i.e. ∣SNL∣2≤L−γ|S_N^L|^2\le L^{-\gamma}∣SNL​∣2≤L−γ in the thermodynamic limit N/Ld→ρ>0N/L^d\to \rho>0N/Ld→ρ>0 approaching a positive particle density. In the general case, the biggest found decay exponent is given by $\gamma=\frac 1 {\pi^2} \norm{\arcsin|T/2|}_{\text{HS}}$, where T refers to the scattering T-matrix. In this thesis, we prove such upper bounds in more general situations than considered in both [Commun. Math. Phys. 329:979--998] and [arXiv:1407.2512]. Furthermore, we provide the first rigorous proof of the exact asymptotics Anderson predicted. We prove that in the 333-dimensional toy model of a Dirac-δ\deltaδ perturbation that the exact decay exponent is given by ζ:=δ2/π2\zeta:= \delta^2/ \pi^2ζ:=δ2/π2. Here, δ\deltaδ refers to the s-wave scattering phase shift. In particular, this result shows that the previously found decay exponent γ\gammaγ does not provide the correct asymptotics of §NL\S_N^L§NL​ in general. Since the decay exponent is expressed in terms of scattering theory, these bounds depend on the existence of absolutely continuous spectrum of the underlying Schrödinger operators. We are able to deduce a different behavior in the contrary situation of Anderson localization. We prove the non-vanishing of the expectation value of the non-interacting many-body scalar product in the thermodynamic limit. Apart from the behavior of the scalar product of the non-interacting ground states, we also study the asymptotics of the difference of the ground-state energies. We show that this difference converges in the thermodynamic limit to the integral of the spectral-shift function up to the Fermi energy. Furthermore, we quantify the error for models on the half-axis and show that higher order error terms depend on the particular thermodynamic limit chosen.
Using the Bethe-ansatz analysis as reformulated by Edwards, we calculate the spectral properties of a particle interacting with a bath of fermions in one dimension for the case of equal particle-fermion masses. These are directly related to singularities apparent in optical experiments in one-dimensional systems. The orthogonality catastrophe for the case of an infinite particle mass survives in the limit of equal masses. We find that the exponent \ensuremath{\beta} of the quasiparticle weight Z\ensuremath{\simeq}${\mathit{N}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}}$ is different for the two cases and proportional to their respective phase shifts at the Fermi surface; we present a simple physical argument for this difference. We also show that these exponents describe the low-energy behavior of the spectral function for repulsive as well as attractive interactions.
Using exact Bethe ansatz solutions, we show that a spin-down fermion immersed into a fully polarized spin-up Fermi sea with a weak attraction is dressed by the surrounding spin-up fermions to form the one-dimensional analog of a polaron. As the attraction becomes strong, the spin-down fermion binds with one spin-up fermion to form a tightly bound molecule. Throughout the whole interaction regime, a crossover from the polaron to a molecule state is fully demonstrated through exact results of the excitation spectrum, the effective mass, binding energy and kinetic energy. Furthermore, a clear distinction between the polaron and molecule is conceived by the probability distribution, single-particle reduced density matrix, and density-density correlations, which are calculated directly from the Bethe ansatz wave function. Such a polaron-molecule crossover presents a universal nature of an impurity immersed into a fermionic medium with an attraction in one dimension.
The ground state of a one-dimensional δ-function attractive spin-1/2 fermions is studied by the nested Bethe ansatz method. Explicitly, the Gaudin integral equation for the system is solved in the form of power series with arbitrary spin polarization. The first few terms of the asymptotic expansions for both strong and weak coupling cases are calculated analytically. The physical quantities, such as the ground state energy and the chemical potentials, are expressed in terms of the dimensionless coupling constant and the polarization , where c is the coupling constant and are the number densities of the spin-up (down) particles. While in the limiting case P = 1, the system consists of spinless free fermions, the other limit P = 0 describes the BCS–BEC crossover in one dimension. As a function of P , the evolution is continuous and smooth.