A Compact Treatment of Singular Impurities Using t

Authors: Gerd Bergmann

Publish Date: 2012/02/08

Volume: 25, Issue: 3, Pages: 609-625

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Abstract

Building on the tools that Friedel introduced in the 1950s an offspring of the Friedel resonance is developed which is called the FAIR approach for “Friedel Artificially Inserted Resonance” In the FAIR approach an arbitrary selectron state a 0dagger is cut out of the conduction band and the remaining free electron Hamiltonian is orthogonalized yielding an artificial Friedel resonance In the presence of a real Friedel dresonance state d † one can find an optimal fair state a 0dagger so that the exact nelectron ground state consists of two Slater states one containing the delectron and the other the fair state a 0dagger This separation according to the doccupation is ideal for impurities with a Coulomb interaction between delectrons The wave function of the Friedel–Anderson FA impurity in the magnetic state and the singlet state are constructed with the FAIR method using two fair states a 0dagger and b 0dagger The magnetic state consists of four and the singlet state consists of eight Slater states The latter is invariant with respect to the inversion of all spins The FAIR ground state Ψ SS for the singlet state is composed of two magnetic states Ψ MS with opposite moments which are not orthogonal to each other The degree of overlap determines the Kondo energy Because of the compactness of the wave function a number of properties can be calculated relatively quickly Comparison with the best previous ground state energy and doccupation by Gunnarson and Schoenhammer show excellent agreement A number of physical properties are calculated among them are i the Kondo cloud in a small magnetic field Two components of polarization are observed in linear response an oscillating part and a nonoscillating part ii The fidelity which represents the scalar product between the ground states of the symmetric FAimpurity and the symmetric Friedel impurity is calculated as a function of the number of electron states In the literature this has the somewhat misleading name of fidelity This calculation does not show an Anderson orthogonality catastrophe which indicates that the FA ground state has a phase shift of π/2 at the Fermi energy iii The Friedel oscillations of the FAimpurity Surprisingly these oscillations are very similar to the Friedel oscillations of a very narrow Friedel resonance at the Fermi level The amplitude Ar is close to zero at short distances and saturates at two for large distances The inversion point where Ar=1 correlates with the characteristic energies of the impurities the halfband width Γ for the Friedel impurity and the Kondo energy for the FAimpurity This raises the fascinating question how these simple properties are hidden in the multielectron wave functionThe Wilson band extends from −1 to 1 It is divided into energy cells ℭ ν which are bordered by −1/2 ν and +1/2 N−ν for positive energies In each cell a Wilson state is constructed by averaging over all states in the cell The energies are shown on the right sideActually Wilson invented the first half of the FAIR approach Each Wilson state c nudagger can be considered as a fair state for the subbasis in ℭ ν which represent all the states in the energy cell ℭ ν The second half would be to diagonalize the matrix of the free electron Hamiltonian langle c nuldaggervarPhi 0vert H 0vert c nulprimedaggervarPhi 0 rangle for ll′0 where 0ll′Z ν and c nuldagger represents the remaining orthonormal basis see 10The rotation leaves the whole basis a 0daga idag orthonormal Step 4 the diagonalization of the N−1sub Hamiltonian is now much quicker because the N−1subHamiltonian is already diagonal with the exception of the i 0row and the i 0column For each rotation plane a 0daga i 0dag the optimal a 0dag with the lowest energy expectation value is determined This cycle is repeated until one reaches the absolute minimum of the energy expectation value In the example of the Friedel resonance Hamiltonian this energy agrees numerically with an accuracy of 10−15 with the exact groundstate energy of the Friedel Hamiltonian 8 For the FA impurity the procedure is stopped when the expectation value changes by less than 10−10 during a full cycleIf the conduction electrons are described by a basis of N states then together with the dstate this yields an N+1dimensional Hilbert space ℌ N+1 The Friedel Hamiltonian is a single particle Hamiltonian and possesses in our case N+1 orthonormal eigenstates b jdag which are compositions of the N states c nudag and the one dstate d † The nelectron ground state is then the product of the n creation operator b jdag with the lowest energy applied to the vacuum state Φ 0 These n states define the ndimensional occupied subHilbert space ℌ n The remaining N+1−n eigenstates form the complementary unoccupied subHilbert space ℌ N+1−n In the following we treat the creation operators as unit vectors within the Hilbert spaceThe vector d 1 can be used as a basis vector of the N+1 Hilbert space ℌ N+1 It lies completely within the occupied subHilbert space Now we divide the occupied subHilbert space ℌ n into the onedimensional space d 1 and an n−1dimensional subspace mathfrakS n1 which is orthogonal to d 1 This subspace mathfrakS n1 is also orthogonal to the dstate vector d and is therefore built only of c ν vectors It can be decomposed into n−1 orthonormal basis vectors barmathbfa iSimilarly the subHilbert space ℌ N+1−n can be divided into d 2 and a subspace mathfrakS Nn orthogonal to d 2 which is therefore also orthogonal to d mathfrakS Nn can be expressed in terms of N−n orthonormal basis vectors barmathbfa i which consists only of vectors c ν The creation operators bara i are not yet uniquely determined That is done by diagonalizing the Hamiltonian H 0 in mathfrakS n1 and mathfrakS Nn This yields the new basis a idag 1leq iN1 It is straight forward to show that the matrix elements langle a idag vert H 0vert a iprimedag rangle for a idaginmathfrakS n1 and a iprimedaginmathfrakS Nn vanish as well We know the matrix elements of H F between any state in mathfrakS Nn and any state in mathfrakS n1 vanishes because the two subHilbert spaces are built from a different subset of eigenstates of H F Therefore langle a idag vert H Fvert a iprimedag rangle =0 if a idaginmathfrakS n1 and a iprimedaginmathfrakS Nn Since mathfrakS n1 and mathfrakS Nn are orthogonal to d † the d component of the Hamiltonian H F vanishes anyhow and the remaining part langle a idag vert H 0vert a iprimedag rangle =0 vanishes for all pairs of i and i′

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