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Spectral and Eigenfunction correlations of finite-volume Schrödinger operators

Martin Gebert

Published Jan 1, 2015• 0 citations
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Abstract

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 $S_N^L$, where $N$ refers to the particle number and $L$ 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 $d\in\N$, i.e. $|S_N^L|^2\le L^{-\gamma}$ in the thermodynamic limit $N/L^d\to \rho>0$ 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 $3$-dimensional toy model of a Dirac-$\delta$ perturbation that the exact decay exponent is given by $\zeta:= \delta^2/ \pi^2$. Here, $\delta$ refers to the s-wave scattering phase shift. In particular, this result shows that the previously found decay exponent $\gamma$ does not provide the correct asymptotics of $\S_N^L$ 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.

DOI
10.5282/edoc.18724

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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 γ=1π2\normarcsin⁡∣T/2∣HS\gamma=\frac 1 {\pi^2} \norm{\arcsin|T/2|}_{\text{HS}}γ=π21​\normarcsin∣T/2∣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.
The asymptotics of an eigenfunction-correlation determinant for Dirac-δ\deltaδ perturbations
We prove the exact asymptotics of the scalar product of the ground states of two non-interacting Fermi gases confined to a 333-dimensional ball BLB_LBL​ of radius LLL in the thermodynamic limit, where the underlying one-particle operators differ by a Dirac-δ\deltaδ perturbation. More precisely, we show the algebraic decay of the correlation determinant ∣det⁡(⟨φjL,ψkL⟩)j,k=1,...,N∣2=L−ζ(E)+o(1)\big|\det\big(\langle\varphi_j^L, \psi_k^L\rangle\big)_{j,k=1,...,N}\big|^2= L^{-\zeta(E)+ \text{o}(1)}​det(⟨φjL​,ψkL​⟩)j,k=1,...,N​​2=L−ζ(E)+o(1), as N,L→∞N,L\to\inftyN,L→∞ and N/∣BL∣→ ρ>0N/|B_L|\to~ \rho>0N/∣BL​∣→ ρ>0, where φjL\varphi_j^LφjL​ and ψkL\psi_k^LψkL​ denote the lowest-energy eigenfunctions of the finite-volume one-particle Schr\odinger operators. The decay exponent is given in terms of the s-wave scattering phase shift ζ(E):=δ2(E)/π2\zeta(E):=\delta^2(\sqrt E)/{\pi^2}ζ(E):=δ2(E​)/π2. For an attractive Dirac-δ\deltaδ perturbation we conclude that the decay exponent 1π2∥arcsin⁡∣T(E)/2∣∥HS2\frac 1 {\pi^2}\Vert\arcsin |T(E)/2|\Vert^2_{\text{HS}}π21​∥arcsin∣T(E)/2∣∥HS2​ found in [GKMO14] does not provide a sharp upper bound on the decay of the correlation determinant.
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.
We prove the exact asymptotics of the scalar product of the ground states of two non-interacting Fermi gases confined to a 3-dimensional ball BL of radius L in the thermodynamic limit, where the underlying one-particle operators differ by a Dirac-δ perturbation. More precisely, we show the algebraic decay of the correlation determinant det〈φjL,ψkL〉j,k=1,…,N2=L−ζ(E)+o(1), as N, L → ∞ and N/BL→ρ>0, where φjL and ψkL denote the lowest-energy eigenfunctions of the finite-volume one-particle Schrödinger operators. The decay exponent is given in terms of the s-wave scattering phase shift ζ(E)≔1π2δ2(E). For an attractive Dirac-δ perturbation, we conclude that the decay exponent 1π2arcsinT(E)/2HS2 found in the work of Gebert et al. [J. Spect. Theory (to be published); e-print arXiv:1407.2512] does not provide a sharp upper bound on the decay of the correlation determinant.
Anderson orthogonality (AO) refers to the fact that the ground states of two Fermi seas that experience different local scattering potentials, say $|{G}_{\mathrm{I}}\ensuremath{\rangle}$ and $|{G}_{\mathrm{F}}\ensuremath{\rangle}$, become orthogonal in the thermodynamic limit of large particle number NNN, in that $|\ensuremath{\langle}{G}_{\mathrm{I}}|{G}_{\mathrm{F}}\ensuremath{\rangle}|\ensuremath{\sim}{N}^{\ensuremath{-}\frac{1}{2}{\ensuremath{\Delta}}_{\mathrm{AO}}^{2}}$ for $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We show that the numerical renormalization group offers a simple and precise way to calculate the exponent ${\ensuremath{\Delta}}_{\mathrm{AO}}$: the overlap, calculated as a function of Wilson chain length kkk, decays exponentially $\ensuremath{\sim}{e}^{\ensuremath{-}k\ensuremath{\alpha}}$, and ${\ensuremath{\Delta}}_{\mathrm{AO}}$ can be extracted directly from the exponent $\ensuremath{\alpha}$. The results for ${\ensuremath{\Delta}}_{\mathrm{AO}}$ so obtained are consistent (with relative errors typically smaller than 1%) with two other related quantities that compare how ground-state properties change upon switching from $|{\mathrm{G}}_{\mathrm{I}}\ensuremath{\rangle}$ to $|{\mathrm{G}}_{\mathrm{F}}\ensuremath{\rangle}$: the difference in scattering phase shifts at the Fermi energy, and the displaced charge flowing in from infinity. We illustrate this for several nontrivial interacting models, including systems that exhibit population switching.
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.
When noninteracting fermions are confined in a DDD-dimensional region of volume O(LD)O({L}^{D})O(LD) and subjected to a continuous (or piecewise-continuous) potential VVV which decays sufficiently fast with distance, in the thermodynamic limit, the ground-state energy of the system does not depend on VVV. Here, we discuss this theorem from several perspectives and derive a proof for radially symmetric potentials valid in DDD dimensions. We find that this universality property holds under a quite mild condition on VVV, with or without bounded states, and extends to thermal states. Moreover, it leads to an interesting analogy between Anderson's orthogonality catastrophe and first-order quantum phase transitions.