We present a detailed numerical study of the orthogonality catastrophe exponent for a one-dimensional lattice model of spinless fermions with nearest-neighbor interaction using the density-matrix renormalization-group algorithm. Keeping up to
2000 states per block we achieve a very great accuracy for the overlap, which is needed to extract the orthogonality exponent reliably. We discuss the behavior of the exponent for three different kinds of a localized impurity. For comparison we also discuss the noninteracting case. In the weak impurity limit our results for the overlap confirm scaling behavior expected from perturbation theory and renormalization-group calculations. In particular we find that a weak backward scattering component of the orthogonality exponent scales to zero for attractive interaction. In the strong impurity limit and for repulsive interaction we demonstrate that the orthogonality exponent cannot be extracted from the overlap for systems with up to 100 sites, due to finite-size effects. Neverthless we find indirect evidence that the backward scattering contribution to the exponent scales to
1/16 based on predictions of boundary conformal field theory.
We show that to understand the orthogonality catastrophe in the half-filled lattice model of spinless fermions with repulsive nearest-neighbor interaction and a local impurity in its Luttinger liquid phase one has to take into account (i) the impurity scaling, (ii) unusual finite-size
L corrections of the form
ln(L)/L, as well as (iii) the renormalization group flow of the umklapp scattering. The latter defines a length scale
Lu, which becomes exceedingly large the closer the system is to its transition into the charge density wave phase. Beyond this transition umklapp scattering is relevant in the renormalization group sense. Field theory can only be employed for length scales larger than
Lu. For small to intermediate two-particle interactions, for which the regime
L>Lu can be accessed, and taking into account the finite-size corrections resulting from (i) and (ii) we provide strong evidence that the impurity backscattering contribution to the orthogonality exponent is asymptotically given by
1/16. While further increasing the two-particle interaction leads to a faster renormalization group flow of the impurity towards the cut chain fixed point, the increased bare amplitude of the umklapp scattering renders it virtually impossible to confirm the expected asymptotic value of
1/16 given the accessible system sizes. We employ the density matrix renormalization group.
The behavior of a single impurity in a one-dimensional Luttinger liquid is numerically investigated by means of the density-matrix renormalization group. By analyzing the finite-size scaling behavior of the low-energy spectrum, we confirm the theoretical prediction of Kane and Fisher [Phys. Rev. Lett. 68, 1220 (1992)] both for attractive and repulsive interactions. Moreover, we calculate the exponent of the orthogonality catastrophe, which gives a further support to the above theoretical prediction.
We prove a simple theorem on the overlap of the wavefunctions of a manybody system with and without a single impurity and show how, and under which conditions, this leads to the ``Orthogonality Catastrophe'' (OC) described by Anderson. A compact new derivation of the known result for the OC exponent in a free electron gas, $α=2 \sum_l(2l+1) (δ_l(ε_f)/π)^2 $ %
α=2∑l[2l+1](δl(εf)/π)2, is given. We also introduce a simple way of calculating core level photoemission (XPS) lineshapes from the finite size scaling of groundstate energies, reproducing and extending results previously derived using boundary conformal field theory. Different conditions under which the OC fails are discussed, and simple physical predictions for XPS lineshapes made in each case.
The orthogonality catastrophe in a Tomonaga-Luttinger liquid with an impurity is reexamined for the case when the interaction parameter or the dimensionless conductance is
g=1/2. By transforming bosons back to fermions, the Hamiltonian is reduced to a quadratic form, which allows for explicit calculation of the overlap integral and the local density of states at the defect site. The exponent of the orthogonality catastrophe due to a backward scattering center is found to be 1/8, in agreement with previous studies using different approaches. The time dependence of the core-hole Green's function is computed numerically, which shows a clear crossover from a nonuniversal short-time behavior to a universal long-time behavior. The local density of states vanishes linearly in the low-energy limit at
g=1/2.We quantify the asymptotic vanishing of the ground-state overlap of two non-interacting Fermi gases in d -dimensional Euclidean space in the thermodynamic limit. Given two one-particle Schrödinger operators in finite-volume which differ by a compactly supported bounded potential, we prove a power-law upper bound on the ground-state overlap of the corresponding non-interacting N -Fermion systems. We interpret the decay exponent \gamma in terms of scattering theory and find \gamma = \pi^{-2}\|\mathrm {arcsin}|T_E/2|\|_{\mathrm {HS}}^2 , where T_E is the transition matrix at the Fermi energy E . This exponent reduces to the one predicted by Anderson [Phys. Rev. 164, 352–359 (1967)] for the exact asymptotics in the special case of a repulsive point-like perturbation.