Numerical Calculation of the Fidelity for the Kondo and the Friedel-Anderson Impurities
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p Numerical Calculation of the Fidelity for the Kondo andthe Friedel-Anderson Impurities
Gerd Bergmann and Richard S. ThompsonDepartment of Physics & AstronomyUniversity of Southern CaliforniaLos Angeles, California 90089-0484e-mail: [email protected] 6, 2018
Abstract
The fidelities of the Kondo and the Friedel-Anderson (FA) impurities are calculatednumerically. The ground states of both systems are calculated with the FAIR (Friedelartificially inserted resonance) theory. The ground state in the interacting systems iscompared with a nullstate in which the interaction is zero. The different multi-electronstates are expressed in terms of Wilson states. The use of N Wilson states simulatesthe use of a large effective number N eff of states. A plot of ln( F ) versus N ∝ ln ( N eff )reveals whether one has an Anderson orthogonality catastrophe at zero energy. Theresults are at first glance surprising. The ln ( F ) − ln ( N eff ) plot for the Kondo impuritydiverges for large N eff . On the other hand, the corresponding plot for the symmetricFA impurity saturates for large N eff when the level spacing at the Fermi level is of theorder of the singlet-triplet excitation energy. The behavior of the fidelity allows one todetermine the phase shift of the electron states in this regime.PACS: 75.20.Hr, 71.23.An, 71.27.+a , 05.30.-d Introduction
In the process of modeling a complicated physical state by a simplified model it is of greatinterest how well the model agrees with the real state. To measure this agreement onecompares the two states with each other. If the states are electronic wave functions thenthe comparison can be performed as a scalar product between the two wave functions. Theresult is called the fidelity, and it is defined as F = | h Ψ model | Ψ real i | (1)It turns out that this concept is also useful when the system (for example the Hamiltonian)depends on a parameter λ . Then one can define the fidelity as F = |h Ψ λ | Ψ i| This definition is slightly different from the definition of the differential fidelity F ( λ, dλ ) F ( λ, dλ ) = |h Ψ λ | Ψ λ + dλ i| = 1 − G ( δλ ) (2)where G is the fidelity susceptibility [1], [2], [3], [4], [5].If for example a potential in the Hamiltonian is given by λV, then when the potential λV acts in the whole volume (as for example in the periodic Hubbard model) the fidelity sus-ceptibility is generally proportional to the number of band electrons. (For phase transitionssuch as quantum critical points it can increase faster than linearly with the number of bandelectrons). Therefore it is of interest how the fidelity of a system depends on the number ofconduction electrons.The definition of the fidelity is connected with the Anderson orthogonality catastrophe(AOC) as introduced by Anderson [6]. Anderson showed that the ground state of a systemof N fermions is orthogonal to the ground state in the presence of a finite-range scatteringpotential, as N approaches infinity ln ( F ) ∝ − ln ( N ). This AOC has been intensivelystudied in connection with the Kondo effect [7], [8], [9], [10], [11], [12], [13] where a magneticd-impurity interacts with the conduction electrons through an exchange interaction J s · S , where s and S are the spins of the conduction electrons and the d-impurity.In this paper we study the fidelity for the Kondo and the Friedel-Anderson (FA) im-purities. Both systems are known to possess a singlet ground state. For sufficiently largeCoulomb repulsion between the spin-up and down impurity state the FA impurity showsa behavior that is very similar to the Kondo impurity. Schrieffer and Wolff [14] showedthat in the range of a local moment the FA Hamiltonian can be transformed into a KondoHamiltonian plus a number of additional terms b H i . Therefore it suggestive that the fidelitiesof the two systems should behave similarly. That is the reason why we choose both systemsfor our investigation.For the calculation of the fidelity we divide the Hamiltonian into two parts: there is apart b H λ =0 that is kept constant and a second part b H λ that is varied during the calculation.In many cases the Hamiltonian b H λ depends on several parameters. The FA impurity is an2xample. In this case one can choose different paths in the parameter space. The paths arecalled a fidelity paths in which the parameters λ describe the position on the paths.For the numerical evaluation we use the ground state which we obtain with the FAIR(Friedel Artificially Inserted Resonance) theory [15], [16], [17], [18], [19], [20], [21], [22], [23]. The Kondo system consists of a band of free s-electrons and a d-impurity with spin S = 1 / b H K = b H + b H ex with b H λ =0 = b H = N X ν =1 ε ν b c † ν,σ b c ν,σ For the Kondo impurity we replace λ by J . b H ex = b H J = v a J X α,β b Ψ † α (0) s α,β b Ψ β (0) ! · S = v a J (cid:16) S + b Ψ †↓ (0) b Ψ ↑ (0) + S − b Ψ †↑ (0) b Ψ ↓ (0) (cid:17) + S z (cid:16) b Ψ †↑ (0) b Ψ ↑ (0) − b Ψ †↓ (0) b Ψ ↓ (0) (cid:17) (3)where S + , S − , S z are the spin operators of the impurity with spin S = 1 / , b Ψ † α (0) and b Ψ β (0) represent field operators of the conduction electrons and s α,β are the components ofthe Pauli operators σ divided by two. The product v a J b Ψ † σ (0) b Ψ σ ′ (0) yields an energy since b Ψ † σ (0) b Ψ σ ′ (0) has the dimension of a density. The operators b c † ν,σ and b c ν,σ are the creationand annihilation operators for the Wilson states of free electrons (see appendix)The FAIR ground state for the Kondo Hamiltonian isΨ K = h B b a † , ↑ b d †↓ + C b d †↑ b b † , ↓ i n − Q i =1 b a † i, ↑ n − Q j =1 b b † j, ↓ Φ (4)+ h C ′ b b † , ↑ b d †↓ + B ′ b d †↑ b a † , ↓ i n − Q i =1 b b † i, ↑ n − Q j =1 b a † j, ↓ Φ Here the states b a † and b b † are two artificial resonance states. The second part of the state(lower line) is essentially the spin-reversed first part (after it is spin-ordered). In the groundstate one has B ′ = B and C ′ = C and one of the coefficients, for example B , is much largerthan the other so that the relative occupations differ by a factor of about 100. Therefore theFAIR state b a † is roughly the always quoted s-electron state that forms a singlet state withthe d-impurity. Details of the ground state energy and the spatial polarization and densityis discussed in [19], [20], [21] . 3 .2 Friedel-Anderson impurity For real d-electrons one has an on-site Coulomb repulsion of the d-electrons among eachother. This is described for the Friedel-Anderson (FA) impurity by a simplified Hamiltonian b H ′ F A = P σ ( N − X ν =0 ε ν b c † νσ b c νσ + N − X ν =0 V sdν [ b d † σ b c νσ + b c † νσ b d σ ] + E d b d † σ b d σ ) + U n d ↑ n d ↓ (5)The fact that a d-impurity has five different orbital states is simplified into the non-degeneratecase with only one d-state with spin up and another one with spin down. The b c † ν,σ are thecreation operators for conduction electrons with spin σ and b d † σ is the corresponding operatorfor the d-electron with spin σ . Further V sdν is the matrix element for a transition betweenthe conduction electron b c † ν,σ and the d-electron b d † σ . The most intensively studied case is thesymmetric FA impurity where E d = − U/ SS = h A b a † , ↑ b b † , ↓ + B b a † , ↑ b d †↓ + C b d †↑ b b † , ↓ + D b d †↑ b d †↓ i n − Y i =1 b a † i, ↑ n − Y i =1 b b † i, ↓ Φ (6)+ h A ′ b b † , ↑ b a † , ↓ + C ′ b b † , ↑ b d †↓ + B ′ b d †↑ b a † , ↓ + D ′ b d †↑ b d †↓ i n − Y i =1 b b † i, ↑ n − Y i =1 b a † i, ↓ Φ In the ground state the coefficients X ′ = X where X stands for A, B, C, D . Again the secondline is essentially the first line with reversed spins.
Since any solution of the Kondo or FA impurity has to include states with very small energy(less than the Kondo energy with typical values of 10 − or 10 − in units of the bandwidth)we use Wilson states (see appendix) as the basis of our calculation. The smallest levelseparation at the Fermi level for N Wilson states is δE = 2 ∗ − N/ . This energy is essentialin the fidelity calculation. A spectrum with equidistant levels would contain N eff = 2 /δE states. For N = 48 the effective number of states N eff would be N eff = 2 /δE = 2 N/ whichis 2 ≈ . × . This shows that with a moderate number of Wilson states one simulatesa large number of band electrons. For the Kondo impurity the Hamiltonians b H λ =0 and b H λ have the form b H λ =0 = N X ν =1 ε ν b c † ν,σ b c ν,σ H λ = λ b H ex For comparison the state with λ = 0 is required. We call this state the nullstate. We choosefor the nullstate Ψ λ =0 = 1 √ (cid:16)b c † n, ↑ b d †↓ + b d †↑ b c † n, ↓ (cid:17) n − Y ν =1 b c † ν, ↑ n − Y ν =1 b c † ν, ↓ Φ with n = ( N/ b d † and the first electron stateabove the Fermi level. We call it a pseudo-singlet state because there is no coupling between b c † n and b d † since λJ is zero. The two components are two degenerate ground states, and theircombination represents the symmetry of the Kondo ground state.In the next step the numerical FAIR ground states are calculated for a given value of λJ for a total number of Wilson states of N = 20, 24, 28, 32, 36, 40, 44 and 48. Then thescalar product (the fidelity F ) between the nullstate and the FAIR ground state of the FAimpurity is calculated. In Fig.1 the logarithm of the fidelity ln ( F ) is plotted for different J asa function of the number of Wilson states N . As we pointed out above the number of Wilsonstates N corresponds to an effective number of electrons N eff . With N = 2 ∗ log N eff aplot of ln ( F ) versus N corresponds to log-log-plot between F and N eff .
20 30 40 50-2.5-2.0-1.5-1.0-0.50.0 l n ( F ) N J 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.12 0.15
Kondo
R5JxxNxx_a
Fig.1: The logarithm of the fidelity ln ( F ) for a Kondo impurity is plottedversus the number of Wilson states N . The nullstate for J = 0 is describedin the text. (The arrows are explained in the discussion).One obtains a set of curves that show in principle a linear dependence of ln ( F (0 , J )) on N at large values of N . For the J = 0 .
15 and 0 .
12 curves the linear behavior is dominant for5ost of the region shown. With decreasing values of J the onset of the linear range movesto larger values of N . For J = 0 .
05 and 0 .
04 the linear part is outside of the calculated anddrawn regime. In the linear regime all curves show the same slope of m = 0 . F (0 , J )) ∝ − . N ≈ − . ∗ ( N eff ) ≈ − .
25 ln ( N eff )or F (0 , J ) ∝ N / eff If we consider the differential fidelity between J = 0 .
09 and 0 .
10 then one obtains aninteresting result that is shown in Fig.2. For small numbers of Wilson states F (0 . , . N between 25 and 35 and assumes a constant value ofabout 0 .
95 for larger N . Fig.2 demonstrates very nicely that the slopes of the ln ( F ) versus N curves are the same at sufficiently large N (for J = 0 .
09 and 0 . J -values experiences arelative change at N ≈
30. Below we discuss that this is in the range of the singlet-tripletexcitation energy for the two J -values.
20 30 40 500.960.981.00
R5J10_09Nxx_a
R5J09NxxR5J10Nxx F ( . , . ) N Kondo< >
Fig.2: The relative fidelity F between Kondo impurities with J = 0 .
09 and J = 0 .
10 is plotted as a function of N . The FA impurity is described by several independent parameters, the s-d-hopping matrixelement (cid:12)(cid:12) V sd (cid:12)(cid:12) , the energy of the d-state E d and the exchange energy U . Therefore one6an choose many different fidelity paths. In this investigation we consider essentially twodifferent paths; (i) the symmetric FA impurity case with (cid:12)(cid:12) V sd (cid:12)(cid:12) = 0 . U = λU ( U = 1)and E d = − λU , (ii) the asymmetric FA impurity with constant (cid:12)(cid:12) V sd (cid:12)(cid:12) = 0 .
05 and U = 1and varying E d in the range ( − < E d < Symmetric case:
Here we use for the nullstate the parameters E d = 0 and U = 0. Thisrepresents the Friedel resonance with the d-energy at the Fermi level. Fig.3 shows a severalexamples of the fidelity for E d = − . U = 1 where | V sd | takes the values 0 . , . , . .
20 30 40 50-2.0-1.5-1.0-0.5 U = 1E d = -0.5 l n ( F ) R3VxxU100En50Nxx_a
Friedel-Anderson |V sd | N Fig.3: The logarithm of the fidelity ln ( F ) for the symmetricFriedel-Anderson impurity is plotted versus the number of Wilsonstates N . The nullstate for U = 0 is the symmetric Friedel impuritywith the | V sd | and E d = 0.Fig.3 shows that the fidelity is essentially constant for | V sd | = 0 .
05 but decreases for | V sd | = 0 .
025 with increasing N . However, for large N the fidelity approaches a constantvalue. (The arrows in Fig.3 are explained in the discussion).The values of the fidelity for | V sd | = 0 . U = 1 and E d = − . ≤ N ≤
48 by less than 2%. This independence of the fidelity of the numberof Wilson states is observed in the whole range 0 < λ < U = λU and E d = − λ U ( U = 1). In addition the fidelity shows a quadratic dependence on λ as is shown in Fig.4for N = 32. We observe the relationship F ( λ ) = 1 − Gλ G the fidelity susceptibility has the value of G = 0 .
63. There is no unusual or singularbehavior of the fidelity in the symmetric case. F ( , ) /2 /2)|V sd | = 0.05 U = *1 E d = - *1/2 N = 32 R3V05UxxEnxxN32_a
Friedel-Anderson
Fig.4: The fidelity for N = 32 along the path E d = − λ U , U = λU with U = 1 and | V sd | = 0 .
05 for a symmetric FA impurity as afunction of λ /
2. The nullstate is again a Friedel impurity with E d = 0 and | V sd | = 0 . Asymmetric case:
Next we study the fidelity of the FA impurity along the path with (cid:12)(cid:12) V sd (cid:12)(cid:12) = 0 . U = 1 while E d is varied between − (cid:12)(cid:12) V sd (cid:12)(cid:12) = 0 . U = 1 and E d = − .
5. Fig.5 shows a typical diagramof the fidelity ln ( F ). It shows a relatively small reduction of ln ( F ) with N . The smalldeviation at N = 48 is due to the fact that the calculation of the FA ground state requires avery large number of iterations to optimize the low energy states close to the Fermi energy.The ground-state energy has to be optimized up to an accuracy better than 10 − . This high8ccuracy is normally not needed for any other physical properties.
20 30 40 50-0.10-0.08-0.06-0.04-0.02 |V sd | = 0.05 U = 1 E d = -0.25 (-0.50) N l n ( F ) R3V05U100En50_xxN_b
Friedel-Anderson
Fig.5: The logarithm of the fidelity ln ( F ) between twoFA impurities as a function of N . The impurities possess the same | V sd | = 0 .
05 and U = 1 but possess different values of E d = − . E d = − . d (ln F ) /dN is shown in Fig.6 as a function of E d . This slope is, of course, zeroat E d = − . N eff but the effect is much smaller than in the Kondo impurity. -1.0 -0.8 -0.6 -0.4 -0.2 0.0-0.010-0.0050.0000.0050.010 f( E d )=.01*( E d ) -0.35*( E d ) |V sd | = 0.05 U = 1 E d0 = -0.5 s l ope E d R3V05U100En50_xxN_a
Friedel-Anderson
Fig.6: The slope as shown in Fig.5 of ln ( F ) versus N as a functionof E d of the fidelity state for the asymmetric FA impurity. The FA impurity is defined by three parameters. Its fidelity is independent of N along thepath U = λU , E d = − λ U and (slightly) singular along other paths, for example along thepath where only E d is varied. This raises the question whether the singular behavior ofln ( F ) is a consequence of the Coulomb interaction and the resulting Kondo ground stateor whether it is a trivial result of the single particle potentials V sd and E d . Therefore it isan obvious necessity to check this question. Such a check is easy done by investigating thesimple spinless Friedel impurity which is defined by the two parameters | V sd | and E d . In the first Friedel investigation we choose for the nullstate the parameters | V sd | = 0 . E d, = 0. Then a series of fidelity series are performed with the same value for | V sd | and values for E d between − . .
7. The plots of ln ( F ) versus N yield straight lineswith a relatively small slope. These slopes are plotted in Fig.7 as a function of E d . At E d = 0 the slope is, of course, zero because both states are identical. As a whole one obtainsa bell-shaped curve for the slopes. This demonstrates that a simple change of E d yields asingular ln ( F ) for large N without any electron-electron interaction.10 R1V05E00_xxNxx_a S l ope E d Friedel |V sd | = 0.05 E d,0 = 0 Fig.7: The slope of ln ( F ) versus N for the asymmetric Friedelimpurity as a function of E d . The nullstate is a Friedel impurity with E d = 0 and the same | V sd | = 0 . | V sd | = 0 .
05 and E d, = 0. For the fidelity states the s-d-hopping matrix | V sd | is varied between 5 × − and0 .
05. For each value of | V sd | the fidelity approaches a constant value for a sufficiently largenumber of Wilson states. There is no singular behavior of ln ( F ) as a function of N . Ofcourse, the constant value of the fidelity depends on | V sd | . This dependence of F is shown11n Fig.8 as a function of ln (cid:0) | V sd | (cid:1) . -10 -8 -6 -4 -20.60.70.80.91.0 R1V05_xxE0Nxx_a F ln(V sd2 ) E d = 0 |V sd0 | = 0.05 Friedel
Fig.8: Fidelity between a two Friedel states as a function of ln (cid:0) | V sd | (cid:1) in the fidelity state while the nullstate has | V sd | = 0 .
05. Both stateshave identical values of E d = 0 , U = 0. The fidelity showsessentially no dependence on the number of Wilson states N . In our fidelity calculation we use the Wilson basis for the conduction band in the (normalized)energy range ( − describes theground state of of a system with a resonance at ε d, = − . has a resonance at ε d, = − .
4. Both resonances are sharp and have a width of ∆ = 0 .
05. To simplify thesituation we assume that the s-d-matrix element vanishes for | ε − ε d,i | > .
1. In bothsystems the conduction band is half filled (all states in the energy range ( − ≤ ε ≤
0) areoccupied). Intuitively one might assume that the wave functions of Ψ and Ψ are quitedifferent because their resonances don’t overlap. However, this is not the case. The scalarproduct (fidelity) h Ψ | Ψ i is essentially one. The reason is that in both wave functions thed-state and the band states in the range ( − ≤ ε ≤
0) are all occupied so that their wave12unctions are given by Ψ ≈ Ψ ≈ b d † Y ε< b c † ε Φ where b c † ε describes the band states with the energy ε . The different density of states farbelow the Fermi level (as well as far above) is not important for the fidelity. What counts inthe fidelity is the occupation of states close to the Fermi level. For this reason the Wilsonbasis is particularly well suited for fidelity calculations because it emphasizes the states closeto the Fermi level where it counts and it does not waste states far away from the Fermi level.In the appendix we demonstrate that it is the (smallest) level separation at the Fermi levelwhich determines the fidelity. Halving the level separation by introducing one additionalstate above and below the Fermi level has the same effect as doubling the number of states(which also halves the level spacing at the Fermi level).An important question in this investigation is whether the fidelity identifies and helpsto understand interacting electron systems. Both the Kondo and the FA impurities possessa singlet ground state. For sufficiently large Coulomb repulsion between the spin-up anddown impurity states the FA impurity shows a behavior that is very similar to the Kondoimpurity.A comparison between Fig.1 for the Kondo impurity and Fig.3 for the FA impurity showsthat the fidelities of the two systems behave very differently. For the following discussionit will be useful to calculate the singlet-triplet excitation energy for the two systems. Intable I the relaxed singlet-triplet excitation energy ∆ E st is collected for the parameters ofthe Kondo impurity investigated in [19]. The relaxed singlet-triplet excitation energy ∆ E st is obtained by optimizing the two bases nb a † i o and nb b † i o independently in the singlet stateand the triplet state.For the development of the ground state it is important that the smallest level separa-tion δE at the Fermi level (which is δE = 2 ∗ − N/ ) is less than the excitation energy∆ E st . Therefore we collect in table I also the critical number of Wilson states N st ≈ ∗ [[log (1 / ∆ E st )] + 1] that yields a level separation of about ∆ E st . In Fig.1 this criti-cal value is marked with a small arrow. One recognizes that for N < N st the fidelity isessentially constant and for N > N st the logarithm of the fidelity changes linearly with N . J ∆E st N st .
15 9 . × − .
12 1 . × − .
10 2 . × − .
09 7 . × − .
08 1 . × − .
07 2 . × − < . < − > E st for the Kondo impurity as afunction of J . The third column gives the (closest) number of Wilson states N st so that the13mallest level separation is roughly equal to the excitation energy ∆ E st .In table II the corresponding data ∆ E st and N st are collected for different values of | V sd | for the FA impurity. In Fig.3 the critical values of N st are also marked on the curves.However, now the behavior is almost reversed compared with the Kondo impurity. For theFA impurity we observe essentially a linear decrease of ln ( F ) with increasing N for N < N st and a saturation of ln ( F ) for N > N st . In particular there is no singular behavior of ln ( F )for large N . | V sd | ∆E st N st .
05 8 . × − .
04 1 . × − .
03 3 . × − .
025 2 . × − E st as a function of | V sd | . Thethird column gives the (closest) number of Wilson states N st so that the smallest levelseparation is roughly equal to the excitation energy ∆ E st .This may be rather surprising since the symmetric FA impurity approaches the Kondoimpurity asymptotically for small | V sd | /U , but this is not reflected by the fidelity behavior.Recently Weichselbaum et al . [24] calculated the fidelity of the FA impurity using thenumerical renormalization group (NRG) theory. They obtained in general a logarithmicdecrease of the fidelity. However, they used very different fidelity paths. In one examplethey varied the energy of the d-level and kept the other parameters constant. Thereforewe performed a similar calculation which is shown in Fig.5. We believe, however, that thelinear decrease of ln ( F ) with N is not a many-body effect. Therefore we have calculatedthe fidelity of the simple non-interacting Friedel impurity. Fig.7 shows that one obtains asingular behavior of ln ( F ) for large N . This is not surprising since it was derived earlierby Anderson and is known as the Anderson orthogonality catastrophe. We believe thatthe singular behavior as observed by Weichselbaum et al . is due to the change of thepotential scattering in the underlying Friedel resonance. Weichselbaum et al . use rathersmall parameters of U, E d and | V sd | such as U = 0 .
12 and Γ µ = π | V µsd | ρ µ = 0 .
01 andseveral hybridization processes µ ( ρ µ is the density in the band µ ). We did not extend oursoftware to several hybridization processes since we concluded that our two examples of theKondo and the FA impurity already illuminate the physics.We suggest the following mechanisms for the different behavior of the fidelity ln ( F ) asa function of N . In the Kondo impurity we compare the Kondo solution with the J = 0state. The latter is a homogeneous electron gas with the same density at the impurity asanywhere else. For small J the magnetic d-electron causes only a relatively small change for14mall N since the Kondo ground state has not yet developed. When N becomes larger than N st the Kondo ground state has formed and causes a phase shift of π/ J = 0) goes to zero, i.e. ln ( F ) diverges. It is analogous to the Anderson orthogonalitycatastrophe.The difference between the Kondo and the symmetric FA impurity is that we don’tcompare the latter with the free electron case but with a state that has the same s-d-potential | V sd | . If one chooses for the nullstate the symmetric Friedel impurity then allelectrons within the resonance width already have a phase shift of π/ E st also have a phase shift of π/
2. Since this is the same phase shift as in thenullstate it does not reduce the scalar product of the fidelity between the nullstate and thesinglet ground state with increasing N . The fidelity becomes asymptotically constant.Finally it is tempting to compare the ground-state wave function of a Kondo impuritywith that of a FA impurity. From tables I and II one finds that the Kondo impurity with J = 0 .
12 and the FA impurity with | V sd | = 0 . U = 1 and E d = − . . × − versus 1 . × − ). Therefore we calculatethe scalar product which yields the similarity between the wave functions for different N .This similarity (which is defined in the literature as fidelity) is plotted in Fig.9. It showsthat F is close to 1.0 and approaches a constant value of 0 .
95 for large N . This confirms thesimilarity between the Kondo and the FA impurity (for large U/ | V sd | ), and the phase shift15n both systems close to the Fermi level is essentially the same.
20 30 40 500.800.850.900.95
R3V04R5J12Nxx_a N F Kondo J=0.12 versusFriedel-Anderson |V sd | = 0.04 E d = -0.5 U = 1 Fig.9: The similarity (fidelity) between the ground state ofa Kondo and a FA impurity as a function of the number ofWilson states. The parameters of the impurities are shownin the figure.The minimum of the curve is at about N = 24. This corresponds to a level separationat the Fermi energy of 2 ∗ − N/ ≈ . × − . This is of the order of the singlet-tripletexcitation energy of the two systems. 16 Conclusion
In this paper the ground states of the Kondo impurity and the Friedel-Anderson impurityare calculated for many parameters and seven different numbers N of Wilson states using theFAIR theory. The effective number of band electrons is N eff ≈ ∗ N/ . For each number ofWilson states the resulting ground states (which we denote as fidelity states) are comparedwith the corresponding ground states for zero interaction, the so-called nullstates. Thefidelity is obtained by forming the scalar product between the fidelity state and the nullstatefor each N . Then the logarithm of the fidelity ln ( F ) is plotted versus N ≈ ( N eff / N while for the Kondoimpurity the logarithm ln ( F ) diverges. This result demonstrates that the behavior of thefidelity depends as much on the choice of the simple nullstate as on the interacting fidelitystate.For the symmetric FA impurity we choose a nullstate with U = 0 and E d = 0 (tomaintain the symmetry) but leave | V sd | constant. Here the s-electrons close to the Fermienergy already have a phase shift of π/ π/ J = 0 and obtain a nullstatewhose conduction band is the free electron band that has no phase shift, and the fidelitydecreases with increasing N . It is not sufficient to turn off the interaction in the nullstate.One also has to know or investigate the phase shift of its s-electrons close to the Fermi level.In this respect the fidelity calculations yield comparative information about the s-electronsat the Fermi level. In addition a change in the slope of ln ( F ) versus N indicates at whichenergy the inner structure changes, either of the nullstate or the fidelity state.If we compare two multi-electron states then the behavior of the fidelity does not tell uswhether none, one or both are interacting electron systems. The fidelity does not correlatewith the many-body physics of the problems.Finally, we observed that the fidelity of ground states of the Kondo and the FA Hamiltoni-ans with similar Kondo temperatures does not show an Anderson orthogonality catastrophe,but on the contrary is relatively close to one and becomes constant with an increasing numberof Wilson states N. AppendixA Wilson’s states
Wilson considered an s-band with a constant density of states and the Fermi energy in thecenter of the band. By measuring the energy from the Fermi level and dividing all energiesby the Fermi energy Wilson obtained a band ranging from − ζ = 0 as accurately as possible he divided the energy interval( − ζ ν = Λ − ν . In most cases the value Λ = 2 is usedyielding − / , − / , − / , . . i.e. ζ ν = − / ν . This yields energy cells C ν with the range {− / ν : − / ν +1 } , width ∆ ν = ζ ν +1 − ζ ν = 1 / ν +1 and average energy ε ν = ( ζ ν + ζ ν − ) / ϕ k ( x ) in such a way thatonly one state within each cell C ν had a finite interaction with the impurity. Assuming thatthe interaction of the original electron states ϕ k ( x ) with the impurity is independent of k ,this interacting state in C ν had the form ψ ν ( x ) = P C ν ϕ k ( x ) / p Z ν where Z ν is the total number of states ϕ k ( x ) in the cell C ν ( Z ν = Z ( ζ ν +1 − ζ ν ) / , Z is thetotal number of states in the band). There are ( Z ν −
1) additional linear combinations of thestates ϕ k in the cell C ν but they have zero interaction with the impurity and were ignoredby Wilson as they are within this paper.The interaction strength of the original basis states ϕ k ( x ) with the d-impurity is assumedto be a constant, v sd . Then the interaction between the d-state and the Wilson states ψ ν ( x )is given by V sd ( ν ) = V sd p ( ζ ν +1 − ζ ν ) / | V sd | = P k | v sd | = P ν | V sd ( ν ) | . A.1 FAIR theory
Let us first consider the Friedel impurity without spin. Its Hamiltonian is b H F = N X ν =1 ε ν b c † ν b c ν + E d b d † b d σ + X σ V sdν (cid:16)b c † ν b d σ + b d † b c ν (cid:17) (7)We call this Hamiltonian sub-diagonal because it is diagonal in the states b c † ν but not between b c † ν and b d † . (We use here the creation operators to denote the corresponding states b c † ν Φ or b d † Φ , where Φ is the vacuum).By diagonalization one finds the exact eigenstates b b † j = N +1 X ν =1 β νj b c † ν + β j b d † (8)and a diagonal Hamiltonian. The ground state with n electrons is given byΨ F = n Y j =1 b b † j Φ (9)18here Φ is the vacuum state.Of course, one can reverse the process and starting from the diagonal Hamiltonian b H = X j E bj b b † j b b j extract the resonance state b d † and build an arbitrary orthonormal basis out of the b b † j which is orthogonal to b d † . The Hamiltonian will not be diagonal in this basis. So in thefinal step one sub-diagonalizes the Hamiltonian excluding the state b d † in the process.This reverse process can also be applied to the s-electron part of b H F . One can build anarbitrary state b a † = X ν α ν b c † ν . In the next step one builds a new orthonormal conductionband basis nb a † i o with ( N −
1) states which are also orthogonal to b a † . Again the Hamiltonian b H will not be diagonal and in the final step one sub-diagonalizes the Hamiltonian excludingthe state b a † in the process. Now b a † is an artificial Friedel resonance, i.e. the FAIR state.The state b a † determines the composition of the whole basis nb a † i o .This FAIR concept is rather flexible because b a † can be any combination of the s-states b c † ν . It turns out that there is one special state b a † with which one can construct the exactground state of the Friedel resonance. With this special FAIR state the Friedel ground statetakes the form Ψ F = (cid:16) A b a † + B b d † (cid:17) n − Y i =1 b a † i Φ This ground state of the Friedel resonance has the great advantage that b d † is only hy-bridized with one single s-electron b a † . The FAIR state b a † is in a way representing all others-electrons. (cid:16) A b a † + B b d † (cid:17) forms a composed state which shifts the energy of all the otherelectron states (introducing a phase shift). It is the building block for the compact groundstate of the FA and the Kondo impurity. B Relation between Wilson state number N and effec-tive number of electrons N ef f The Wilson states are defined by the ratio Λ. However, for some physical properties thissub-division of the energy band is too coarse. We observed an error in the amplitude ofthe Friedel oscillation of about 10% for Λ = 2 which became of the order of 1% for whenthe intervals where sub-divided twice (corresponding to Λ = √ E d = 0 and E d = −
1. In both cases the s-d-coupling is | V sd | = 0 .
05. InFig.10 the squares give the plot of ln ( F ) versus the number of Wilson states for Λ = 2,which is equivalent to all the plots in this paper. Then we subdivided each cell into twoequal subcells (full circles) and again each subcell into two new subcells (full triangles). The19umber of states increased each time by a factor two but we plotted the newly calculatedln ( F ) as a function of the original number N of Wilson states. First we observe that theresulting straight lines are perfectly parallel. Secondly the two sub-divisions into equalsubcells reduced the smallest energy δE at the Fermi level by a factor of 4 = 2 . This meansthat after two subdivisions the smallest δE for N Wilson states is equal to the original δE for ( N + 4) Wilson states. For example, the plot shows that the square at N = 36 has thesame value as the triangle at N = 32. If one would plot ln ( F ) versus 2 log (1 /δE ) all pointswould fall on one straight line (the one with the squares), although the number of states usedin the calculation are varied by a factor four. This demonstrates that the fidelity dependsessentially on the smallest energy δE at the Fermi energy and not on the total number ofstates.
20 30 40 50-2.0-1.5-1.0-0.5
R1V05En100Nxx_d N l n ( F ) *1 *2 *4 Friedel |V sd | = 0.05 E d = 0 (-1) Fig.10: The logarithm of the fidelity between two Friedel impuritieswith different d-level energies. The squares are for a regular Wilsonspectrum. For the circles each Wilson energy cell is divided into twocells increasing the number of states to 2 N . For the triangles theoriginal energy cells are divided into four cells yielding 4 N states.Therefore the new states are essentially a factor 2 and 4 closer, andthe smallest energy separation is smaller by a factor 2 and 4. Thefidelity is plotted in all cases versus the original number of Wilsonstates. The straight lines are perfectly parallel. In addition a triangleat N = 36 has the same smallest energy as a square at N = 40, andindeed they have the same fidelity.20 eferenceseferences