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arXiv:1703.05363 · Physical review. E

Orthogonality catastrophe and fractional exclusion statistics

Filiberto Ares, Kumar S. Gupta, Amilcar R. de Queiroz

Published Feb 20, 2018• 9 citations
arXivHTMLPDF

Abstract

We show that the N-particle Sutherland model with inverse-square and harmonic interactions exhibits orthogonality catastrophe. For a fixed value of the harmonic coupling, the overlap of the N-body ground state wave functions with two different values of the inverse-square interaction term goes to zero in the thermodynamic limit. When the two values of the inverse-square coupling differ by an infinitesimal amount, the wave function overlap shows an exponential suppression. This is qualitatively different from the usual power law suppression observed in the Anderson's orthogonality catastrophe. We also obtain an analytic expression for the wave function overlaps for an arbitrary set of couplings, whose properties are analyzed numerically. The quasiparticles constituting the ground state wave functions of the Sutherland model are known to obey fractional exclusion statistics. Our analysis indicates that the orthogonality catastrophe may be valid in systems with more general kinds of statistics than just the fermionic type.

DOI
10.1103/physreve.97.022133

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We show that the N-particle Sutherland model with inverse-square and harmonic interactions exhibits orthogonality catastrophe. For a fixed value of the harmonic coupling, the overlap of the N-body ground state wave functions with two different values of the inverse-square interaction term goes to zero in the thermodynamic limit. When the two values of the inverse-square coupling differ by an infinitesimal amount, the wave function overlap shows an exponential suppression. This is qualitatively different from the usual power law suppression observed in the Anderson's orthogonality catastrophe. We also obtain an analytic expression for the wave function overlaps for an arbitrary set of couplings, whose properties are analyzed numerically. The quasiparticles constituting the ground state wave functions of the Sutherland model are known to obey fractional exclusion statistics. Our analysis indicates that the orthogonality catastrophe may be valid in systems with more general kinds of statistics than just the fermionic type.
Orthogonality catastrophe in fermionic systems is well known: in the thermodynamic limit, the overlap between the ground state wavefunctions with and without a single local scattering potential approaches zero algebraically as a function of the particle number NNN. Here we examine the analogous problem for bosonic systems. In the homogeneous case, we find that ideal bosons display an orthogonality stronger than algebraic: the wavefunction overlap behaves as exp[−λN1/3]{\rm exp}[-λN^{1/3}]exp[−λN1/3] in three dimensions and as exp[−λN/ln⁡2N]{\rm exp}[-λN/\ln ^2 N]exp[−λN/ln2N] in two dimensions. With interactions, the overlap becomes finite but is still (stretched-)exponentially small for weak interactions. We also consider the cases with a harmonic trap, reaching similar (though not identical) conclusions. Finally, we comment on the implications of our results for spectroscopic experiments and for (de)coherence phenomena.
We investigate the statistical orthogonality catastrophe (STOC) in single-particle and many-body localized systems by studying the response of the many-body ground state to a local quench. Using scaling arguments and exact numerical calculations, we establish that the STOC gives rise to a wave function overlap between the pre- and postquench ground states that has an exponential decay with the system size, in sharp contrast to the well-known power law Anderson orthogonality catastrophe in metallic systems. This exponential decay arises from a statistical charge transfer process where a particle can be effectively ``transported'' to an arbitrary lattice site. In a many-body localized phase, this nonlocal transport and the associated exponential STOC phenomenon persist in the presence of interactions. We study the possible experimental consequences of the exponential STOC on the Loschmidt echo and spectral function, establishing that this phenomenon might be observable in cold atomic experiments through Ramsey interference and radio-frequency spectroscopy.
One-dimensional fractional statistics is studied using the Calogero-Sutherland model (CSM) which describes a system of non-relativistic quantum particles interacting with inverse-square two-body potential on a ring.The inversesquare exchange can be regarded as a pure statistical interaction and this system can be mapped to an ideal gas obeying the fractional exclusion and exchange statistics.The details of the exact calculations of the dynamical correlation functions for this ideal system is presented in this paper.An effective low-energy one-dimensional "anyon" model is constructed; and its correlation functions are found to be in agreement with those in the CSM; and this agreement provides an evidence for the equivalence of the first-and the second-quantized construction of the 1D anyon model at least in the long wave-length limit.Furthermore, the finite-size scaling applicable to the conformally invariant systems is used to obtain the complete set of correlation exponents for the CSM.
One-dimensional model of nonrelativistic particles with inverse-square interaction potential known as the Calogero-Sutherland model is shown to possess fractional statistics. Using the theory of Jack polynomials the exact dynamical density-density correlation function and the hole propagator part of one-particle Green's function at any rational interaction coupling constant are obtained and used to show clear evidences of the fractional exclusion statistics in the sense of Haldane's "generalized Pauli exclusion principle." This model is also endowed with the corresponding natural exchange statistics.
According to Anderson's orthogonality catastrophe, the overlap of the N -particle ground states of a free Fermi gas with and without an (electric) potential decays in the thermodynamic limit. For the finite one-dimensional system various boundary conditions are employed. Unlike the usual setup the perturbation is introduced by a magnetic (vector) potential. Although such a magnetic field can be gauged away in one spatial dimension there is a significant and interesting effect on the overlap caused by the phases. We study the leading asymptotics of the overlap of the two ground states and the two-term asymptotics of the difference of the ground-state energies. In the case of periodic boundary conditions our main result on the overlap is based upon a well-known asymptotic expansion by Fisher and Hartwig on Toeplitz determinants with a discontinuous symbol. In the case of Dirichlet boundary conditions no such result is known to us and we only provide an upper bound on the overlap, presumably of the right asymptotic order.
We address the phenomenon of statistical orthogonality catastrophe in insulating disordered systems. In more detail, we analyse the response of a system of non-interacting fermions to a local perturbation induced by an impurity. By inspecting the overlap between the pre- and post-quench many-body ground states we fully characterise the emergent statistics of orthogonality events as a function of both the impurity position and the coupling strength. We consider two well-known one-dimensional models, namely the Anderson and Aubry–André insulators, highlighting the arising differences. Particularly, in the Aubry–André model the highly correlated nature of the quasi-periodic potential produces unexpected features in how the orthogonality catastrophe occurs. We provide a quantitative explanation of such features via a simple, effective model. We further discuss the incommensurate ratio approximation and suggest a viable experimental verification in terms of charge transfer statistics and interferometric experiments using quantum probes.