Renormalization of transition matrix elements of particle number operators due to strong electron correlation
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Renormalization of transition matrix elements ofparticle number operators due to strong electroncorrelation
Noboru Fukushima
Motomachi 13-23, Sanjo, Niigata, 955-0072 JapanE-mail: [email protected]
Abstract.
Renormalization of non-magnetic and magnetic impurities due to electrondouble occupancy prohibition is derived analytically by an improved Gutzwillerapproximation. Non-magnetic impurities are effectively weakened by the samerenormalization factor as that for the hopping amplitude, whereas magnetic impuritiesare strengthened by the square root of the spin-exchange renormalization factor, incontrast to results by the conventional Gutzwiller approximation. We demonstrate itby showing that transition matrix elements of number operators between assumedexcited states and between an assumed ground state and excited states arerenormalized differently than diagonal matrix elements. Deviation from such simplerenormalization with a factor is also discussed. In addition, as related calculation, wecorrect an error in treatment of renormalization of charge interaction in the literature.Namely, terms from the second order of the transition matrix elements are stronglysuppressed. Since all these results do not depend on the signs of impurity potential orcharge interaction parameter, they are valid both in attractive and repulsive cases.PACS numbers: 71.10.Fd, 71.27.+a, 74.72.-h, 74.81.-g enormalization of transition matrix elements of particle number operators
1. Introduction
In this paper, we discuss renormalization of impurities due to strong electron correlation.Such renormalization may be intuitive in the case of the Hubbard model, where eachsite has onsite electron repulsion. Namely, sites with higher potential energy have lowerelectron occupancy, and consequently have less chance of double occupancy. Then,the total energy loss from the impurity potential and the repulsive interaction shouldbe more uniform than in the system without the electron correlation; we can call itrenormalization of impurities. However, when the repulsion is very strong, we need toconsider much smaller energy scales. That is, if double occupancy does not occur, theabove argument cannot be applied, and thus renormalization of impurities within thelower Hubbard band is not so trivial.When electron double occupancy is prohibited at every site, a system with quitedensely packed electrons has a good chance to have one electron with spin up or downat each site. If the system has a tendency toward phase separation, small perturbationby an impurity may produce a large effect to separate a system into hole-rich regionsand electron-rich regions; it may appear in close vicinity of the half filling in the t - J –type models, where effective hopping is negligibly small compared to effective exchangeinteraction. In contrast, what we focus on in this paper are systems not that close tothe half filling or systems with relatively weak exchange interaction. Then, electronsare more mobile. Near the half filling, since there is little freedom left to change chargedistribution and sudden spatial change of particle number distribution around impuritiesis not favorable for the kinetic energy, non-magnetic impurity potentials may have littleeffect on low-energy eigenstates and only shift their eigenenergies quite uniformly. Inother words, impurity potentials can be renormalized by electron correlation even withinthe lower Hubbard band.In previous papers [1, 2], such renormalization of non-magnetic impurity potentialswas investigated numerically. That is, (i) to estimate perturbation from an impuritypotential, the variational Monte Carlo method was applied to calculation of itsmatrix elements with respect to assumed excited states in the uniform systems;(ii) inhomogeneous systems with an impurity or impurities were investigated by aBogoliubov-de Gennes equation with the double-occupancy prohibition treated by akind of mean-field approximation called the Gutzwiller approximation (GA) generalizedto inhomogeneous systems.Both of (i) and (ii) manifested strong renormalization of the impurity potential,and its renormalization factor (ratio between corresponding quantities in systems withand without the double-occupancy prohibition) seems approximately proportional to g t ≡ x/ (1 + x ), which is the renormalization factor of hopping amplitude obtained bythe GA as a function of hole concentration x . Since the double-occupancy prohibitioninhibits hopping, g t is less than unity and goes to zero as x →
0. To explain the impurityrenormalization factor close to g t , we pointed out the similarity between the impuritypotential and the hopping in the real space, i.e., the Fourier-transformed impurity enormalization of transition matrix elements of particle number operators k -space. If electrons are densely packed inthe lattice, it must be difficult even in the k -space to hop from k to a different k ′ .However, it is a speculation and may not be trivial because the double occupancyis prohibited in the real space rather than in the k -space. In addition, we do notreally know how general the numerical results are because the calculation was done onlyfor limited parameter sets. To complement this argument, an analytic approximationis adopted in this paper, namely, (i) is redone using the GA to derive dominant g t dependence and deviation from g t explicitly. In fact, however, the conventional GA[3, 4] fails to derive this renormalization. It compares mean weights of configurationsrelevant to operators of interest with and without the electron repulsion in calculatingthe renormalization factors. Then, the renormalization factor for the particle numberoperators is actually unity, i.e., they are not renormalized. The spin rotation invariantslave-boson mean-field theory [5] is known to be equivalent to the conventional GA; thesaddle-point approximated boson fields play a role of the weights in the GA. Therefore,we speculate that it may have the same problem as the conventional GA. In addition,we believe that the slave-boson mean-field theory with only one boson often used forthe t - J model can be even less accurate because it does not yield renormalization of theexchange interaction, which may be an artifact from the lost boson hard-core property.Let us recall that the GA corresponds to taking the leading order of the Wickexpansion with respect to the intersite contractions of creation/annihilation operators[6, 7]. In fact, the weights of configurations in the conventional GA are likely to becalculated with the focus only on the lowest order; apparently it breaks down whenthe lowest order vanishes or when the next lowest order is of interest. An example isa particle number operator as shown in this paper. Although the lowest order is theaverage particle number, when we discuss transition matrix elements with excited states,this lowest order does not contribute, and the next lowest order is relevant. We willdemonstrate that off-diagonal matrix elements between an assumed ground state andexcited states as well as between different excited states are renormalized differentlythan diagonal matrix elements.Furthermore, by slightly modifying the non-magnetic impurity, i.e., by subtractionbetween up- and down-spin particle number operators, we also consider a simplemagnetic impurity. In this case, the direction of the renormalization is reversed, namely,the impurity is strengthened by the electron correlation in contrast to the non-magneticimpurity. It must be physically reasonable because electron repulsion increases singleoccupancy. As calculation related to the non-magnetic impurity, renormalization ofcharge interaction is discussed to correct an error in its treatment in the literature.That is, terms relevant to the mean-field approximation are actually the second orderof the transition matrix elements, and they are weakened by very small renormalizationfactor ( g t ) although treated usually as not being renormalized. enormalization of transition matrix elements of particle number operators
2. Model
What we have in mind is t - J –type models with impurities, namely, H ≡ P G − X i,j,σ t ij c † iσ c jσ + X i,j J ij (cid:18) S i · S j −
14 ˆ n i ˆ n j (cid:19) + H imp P G , (1)where c † iσ ( c iσ ) is the creation (annihilation) operator of the electron with site i and spin σ , and S i is the spin operator at site i . In addition,ˆ n i ≡ ˆ n i ↑ + ˆ n i ↓ , ˆ n iσ ≡ c † iσ c iσ . (2)Gutzwiller projection operator P G ≡ Q i (1 − ˆ n i ↑ ˆ n i ↓ ) prohibits electron double occupancyat each site and represents strong Coulomb repulsion. In this paper, we do not use anyexplicit form of t ij and J ij although they are implicitly included in assumed variationalground/excited states. Our main focus here is on the impurity term H imp .In sections 3, 4 and 5, our target is renormalization of a single non-magnetic δ -function impurity potential located at i = I , H imp = V I ˆ n I = V I (ˆ n I ↑ + ˆ n I ↓ ) . (3)Then, in section 6, we discuss renormalization of a simple magnetic impurity, H imp = − h I S zI = − h I n I ↑ − ˆ n I ↓ ) . (4)In addition, the focus in section 7 is not on H imp but on charge interaction ˆ n i ˆ n j inHamiltonian (1).
3. Non-magnetic impurity renormalization
Let us start from a uniform system without impurities. A basic idea of variationaltheories is that the ground state of the t - J –type models may be something similar tothe BCS superconducting state | Ψ i ≡ Y k (cid:16) u k + v k c † k ↑ c †− k ↓ (cid:17) | i , (5)but somewhat modified by the electron correlation. Simple variational wave functionsadopted by most of analytic theories have a form of P G | Ψ i with something to controlthe particle number. One way to control it is to use projection P N to fixed particlenumber N . ‡ Another is to attach fugacity factors to the projector, namely, | Ψ i ≡ P | Ψ i , P ≡ Y i P i , P i ≡ λ ˆ n i ↑ i ↑ λ ˆ n i ↓ i ↓ (1 − ˆ n i ↑ ˆ n i ↓ ) . (6)The latter is adopted in this paper. The reason to control the particle number is that P G changes the average particle number of | Ψ i because states with a larger particle ‡ Many different | Ψ i correspond to | Ψ i under the projections. For example, exp( λ ˆ N ) with ˆ N thetotal particle number operator is constant under P N , and thus exp( λ ˆ N ) | Ψ i is equivalent to | Ψ i . enormalization of transition matrix elements of particle number operators h ˆ O i ≡ h Ψ | ˆ O | Ψ i , h ˆ O i ≡ h Ψ | ˆ O | Ψ ih Ψ | Ψ i , (7)for some operator ˆ O , usually it is not convenient if | Ψ i and | Ψ i are totally different,e.g., if | Ψ i has a more-than-half filled electron band. § Although our main interest hereis perturbation from the uniform state, most of derivation in this paper is valid also forinhomogeneous systems, and thus we prefer to keep general expressions with site andspin indices throughout the paper, e.g., n iσ ≡ h ˆ n iσ i , n i ≡ h ˆ n i i = n i ↑ + n i ↓ . (8)However, we use 0 = h c † i ↑ c † i ↓ i = h c † iσ c † jσ i = h c † i ↑ c j ↓ i to avoid making formulas toolengthy.Although choice of the fugacity factors is not unique especially in inhomogeneoussystems [7], yet it is convenient to define λ iσ ≡ − n iσ − n i , (9)because it satisfies h ˆ n iσ i ≈ h ˆ n iσ i , (10)for any i and σ [6, 7], neglecting terms of the “fourth order”. Here, and throughoutthis paper, if not specified, “ n -th order” represents n -th order with respect to intersitecontractions such as h c † iσ c jσ i and h c i ↓ c j ↑ i with i = j . Note that h ˆ O i of any ˆ O can be inprinciple calculated by the Wick theorem, which yields many such intersite contractions.High order terms may be neglected by recalling that onsite contractions are larger thanintersite contractions. The GA corresponds to taking the leading order only, e.g., h P i ≈ Y i h P i i , (11) h P i i = (1 − n i ↑ )(1 − n i ↓ ) + λ i ↑ n i ↑ (1 − n i ↓ ) + λ i ↓ n i ↓ (1 − n i ↑ )= (1 − n i ↑ )(1 − n i ↓ )1 − n i . (12)The terms neglected in the approximation in (11) are of the fourth order because thesecond order terms cancel out when λ iσ is defined as (9) [6, 7]. Let us show it explicitlywith a notation to treat c † and c together, c + iσ ≡ c † iσ , c − iσ ≡ c iσ , (13) § The variational Monte Carlo method does not have such restriction. For example, local magneticmoments before and after the projection are different in general, and the chemical potential in avariational mean-field Hamiltonian is a variational parameter under P N rather than a parameter tocontrol the particle number. enormalization of transition matrix elements of particle number operators P i and operators at some site(s) j, j ′ = i , h P i c τ ′ j ′ σ ′ c τjσ i = h P i i h c τ ′ j ′ σ ′ c τjσ i + (cid:16) −h c i ↑ c τ ′ j ′ σ ′ i h c † i ↑ c τjσ i + h c i ↑ c τjσ i h c † i ↑ c τ ′ j ′ σ ′ i (cid:17) [(1 − n i ↓ ) − λ i ↑ (1 − n i ↓ ) + λ i ↓ n i ↓ ]+ (cid:16) −h c i ↓ c τ ′ j ′ σ ′ i h c † i ↓ c τjσ i + h c i ↓ c τjσ i h c † i ↓ c τ ′ j ′ σ ′ i (cid:17) [(1 − n i ↑ ) + λ i ↑ n i ↑ − λ i ↓ (1 − n i ↑ )] (14)for arbitrary τ , τ ′ , σ and σ ′ . Then, the quantities in the square brackets vanish.We assume that | Ψ i is a good variational ground state, and that the excited statesare well represented by projected quasiparticles | ks i ≡ P γ † ks | Ψ i q h Ψ | γ ks P P γ † ks | Ψ i ≈ P γ † ks | Ψ i q h P i , (15)where γ ks are quasiparticles for | Ψ i , namely, γ † k ↑ = u ∗ k c † k ↑ − v ∗ k c − k ↓ , γ − k ↓ = v k c † k ↑ + u k c − k ↓ . (16)For the denominator of | ks i , we have used approximation h Ψ | γ kσ P γ † kσ | Ψ i ≈ h P i [7, 9], and errors from this approximation are of the second order.By switching on the impurity potential, these excited states should be mixed bymatrix elements V k ′ ,k N L ≡ h k ′ s | ˆ n I | ks i ≈ D γ k ′ s P ˆ n I P γ † ks E h P i , (17)with N L the number of sites. The limit of the half filling can be exactly evaluated; λ → ∞ , P ˆ n I P → P P , and thus V k ′ ,k /N L → h k ′ s | ks i = δ k ′ k . According to theBCS theory, V BCS k ′ ,k ≡ h γ k ′ s ˆ n I γ † ks i = u k ′ u ∗ k − v k ′ v ∗ k . In the previous paper [2], theauthor noted that V k ′ ,k is not renormalized with the conventional GA [4] because itoriginally comes from a particle number operator. However, more careful analysis herewill show that, although the diagonal matrix elements of the particle number operatorsare not renormalized [eg., see (10)], their off-diagonal matrix elements with respect tothe projected quasiparticle excited states are renormalized.The Wick expansion of h γ k ′ s P ˆ n Iσ P γ † ks i yields many terms, and some terms containonsite contraction of ˆ n Iσ at the center as ˆ n Iσ → n Iσ , and the others do not. Let usseparate these two groups of terms, h γ k ′ s P ˆ n Iσ P γ † ks i = n Iσ h γ k ′ s P γ † ks i + h γ k ′ s P (ˆ n Iσ − n Iσ ) γ † ks i . (18)The first term is proportional to h k ′ ↑ | k ↑i , and vanishes when k = k ′ . Namely, we canonly consider the second term.Let us first take only ˆ n I ↑ in the impurity potential term. Since the GA is carriedout in the real space, the k representation should be inverse Fourier transformed intothe real space representation. Namely, what we should calculate is h c τ ′ i ′ σ ′ P ˆ n I ↑ P c τiσ i . Letus first take the case of i = I , i ′ = I and i = i ′ , which makes dominant contribution to V k ′ k . After using P I ˆ n I ↑ P I = λ I ↑ ˆ n I ↑ (1 − ˆ n I ↓ ), we take onsite contractions for all the sitesexcept i , i ′ and I of the numerator neglecting fourth-order terms, h c τ ′ i ′ σ ′ P ˆ n I ↑ P c τiσ i h P i ≈ λ I ↑ h c τ ′ i ′ σ ′ P i ′ ˆ n I ↑ (1 − ˆ n I ↓ ) P i c τiσ i h P i ′ i h P I i h P i i . (19) enormalization of transition matrix elements of particle number operators i and i ′ , P i c † iσ = λ iσ (1 − ˆ n i ¯ σ ) c † iσ , P i c iσ = [(1 − ˆ n i ¯ σ ) + λ i ¯ σ ˆ n i ¯ σ ] c iσ . (20)For the moment, we take the onsite contractions for i ¯ σ and i ′ ¯ σ ′ neglecting intersitecontractions between I ↑ or I ↓ and them; the terms neglected here are of the thirdorder and will be calculated in the next section. Accordingly, using λ iσ (1 − n i ¯ σ ) h P i i = (1 − n i ¯ σ ) + λ i ¯ σ n i ¯ σ h P i i = 1 , (21)(19) can be approximated as h c τ ′ i ′ σ ′ P ˆ n I ↑ P c τiσ i h P i ≈ λ I ↑ h c τ ′ i ′ σ ′ ˆ n I ↑ (1 − ˆ n I ↓ ) c τiσ i h P I i . (22)It is convenient to define mean-value–subtracted operators here,˜ n iσ ≡ ˆ n iσ − n iσ . (23)Consequently, we obtain h c τ ′ i ′ σ ′ P ˜ n I ↑ P c τiσ i h P i ≈ h c τ ′ i ′ σ ′ ˜ n I ↑ c τiσ i − n I ↑ − n I ↓ h c τ ′ i ′ σ ′ ˜ n I ↓ c τiσ i . (24)Here, the first term and the second term in the r.h.s. are from the onsite contraction of1 − ˆ n I ↓ and ˆ n I ↓ , respectively; from the residual operators (ˆ n I ↑ and 1 − ˆ n I ↓ , respectively),their mean values are subtracted to cancel their onsite contraction.For the moment, we neglect deviation from (24) for any i and i ′ , which will bediscussed in the next section. Then, it is straightforward to Fourier transform back, h γ k ′ s P ˜ n I ↑ P γ † ks i h P i ≈ h γ k ′ s ˜ n I ↑ γ † ks i − n I ↑ − n I ↓ h γ k ′ s ˜ n I ↓ γ † ks i . (25)The formula for ˆ n I ↓ is obtained by exchanging ↑ and ↓ at site I , and these formularepresent that ˜ n Iσ is renormalized into ˜ n Iσ − ˜ n I ¯ σ n Iσ / (1 − n I ¯ σ ).In fact, the derivation above is valid also for inhomogeneous systems by replacing γ ks with Bogoliubov quasiparticles γ ℓ . A difference is that the orthogonality of theGutzwiller-projected Bogoliubov quasiparticle states is only approximately satisfied[7], i.e., errors from the GA can be larger than those in uniform systems. Therenormalization of ˆ n I in inhomogeneous systems is obtained by summing up ˆ n I ↑ andˆ n I ↓ for ℓ = ℓ ′ , h γ ℓ ′ P ˆ n I P γ † ℓ i q h γ ℓ ′ P P γ † ℓ ′ i h γ ℓ P P γ † ℓ i ≈ D γ ℓ ′ (cid:16) g tI ↑ ˜ n I ↑ + g tI ↓ ˜ n I ↓ (cid:17) γ † ℓ E , (26)where g tiσ ≡ − n i − n iσ (27)is the Gutzwiller renormalization factor for the hopping amplitude.Returning to our main target, i.e., the non-magnetic uniform system, we can set g tIs = g tI ¯ s , then V k ′ ,k = h k ′ s | ˆ n I | ks i ≈ g tIs ( u k ′ u ∗ k − v k ′ v ∗ k ) = g tIs V BCS k ′ ,k , (28) enormalization of transition matrix elements of particle number operators k -points by the variational Monte Carlo method, i.e., therenormalization factor is close to g t and insensitive to model parameters. The importantpoint here may be g t appears only after summation of up and down spins, ˆ n I ↑ + ˆ n I ↓ ,which is a difference from the hopping amplitude renormalization in the real space.According to the conventional GA [4], what is renormalized is an operator ratherthan its matrix elements, and thus diagonal and off-diagonal matrix elements have thesame renormalization factor. In fact, however, what is renormalized should be matrixelements rather than operators, and diagonal and off-diagonal matrix elements withrespect to excited states can have different renormalization factors as demonstratedabove.By exactly the same procedure as above, transition matrix elements between thevariational ground state and projected two-quasiparticle excited states can be alsocalculated. Corresponding to (26), h Ψ | γ ℓ γ ℓ ′ P ˆ n I | Ψ i q h Ψ | γ ℓ γ ℓ ′ P P γ † ℓ ′ γ † ℓ | Ψ ih Ψ | Ψ i ≈ h Ψ | γ ℓ γ ℓ ′ (cid:16) g tI ↑ ˜ n I ↑ + g tI ↓ ˜ n I ↓ (cid:17) | Ψ i . (29)
4. Corrections to the simple g t renormalization In the cases of i = I = i ′ , i ′ = I = i and i = i ′ = I , we obtain formulas equivalent to(24). However, for i = i ′ = I , we have h c τ ′ iσ ′ P ˆ n I ↑ P c τiσ i h P i ≈ λ I ↑ h ˆ n I ↑ (1 − ˆ n I ↓ ) c τ ′ iσ ′ P i c τiσ i h P I i h P i i , (30)where c τ ′ iσ ′ P i c τiσ can be explicitly written as c iσ P i c † iσ = λ iσ (1 − ˆ n i ¯ σ )(1 − ˆ n iσ ) , c † iσ P i c iσ = [(1 − ˆ n i ¯ σ ) + λ i ¯ σ ˆ n i ¯ σ ]ˆ n iσ , (31)because the other combinations of c τ ′ iσ ′ and c τiσ yield zero or very small quantities. Then,although the onsite contraction of i ¯ σ with intersite contractions between iσ and I yieldsa formula equivalent to (24), the onsite contraction of iσ with intersite contractionsbetween i ¯ σ and I additionally yields the same order of contribution. To compactlywrite them, let us define κ + iσ ≡ − − n i ¯ σ , κ − iσ ≡ n iσ (1 − n i ↑ )(1 − n i ↓ ) , (32)as well as η i ′ σ ′ ,iσ ≡ h ˆ n i ′ σ ′ ˆ n iσ i − n i ′ σ ′ n iσ , (33)which extracts only intersite contractions in h ˆ n i ′ σ ′ ˆ n iσ i . Then, h ˆ n i ′ σ ′ (1 − ˆ n iσ ) i − n i ′ σ ′ (1 − n iσ ) = − η i ′ σ ′ ,iσ , and h (1 − ˆ n i ′ σ ′ )(1 − ˆ n iσ ) i − (1 − n i ′ σ ′ )(1 − n iσ ) = η i ′ σ ′ ,iσ . More explicitly, η i ′ σ,iσ = − (cid:12)(cid:12)(cid:12) h c † i ′ σ c iσ i (cid:12)(cid:12)(cid:12) , η i ′ ¯ σ,iσ = (cid:12)(cid:12)(cid:12) h c † i ′ ¯ σ c † iσ i (cid:12)(cid:12)(cid:12) . (34)Using these notations, h c ¯ τiσ P ˜ n I ↑ P c τiσ i h P i ≈ ¯ τ η I ↑ ,iσ − n I ↑ − n I ↓ η I ↓ ,iσ ! + κ τiσ h c ¯ τiσ c τiσ i η I ↑ ,i ¯ σ − n I ↑ − n I ↓ η I ↓ ,i ¯ σ ! . (35) enormalization of transition matrix elements of particle number operators n I ↑ and ˆ n I ↓ , h c ¯ τiσ P (˜ n I ↑ + ˜ n I ↓ ) P c τiσ i h P i ≈ ¯ τ ( g tI ↑ η I ↑ ,iσ + g tI ↓ η I ↓ ,iσ ) + κ τiσ h c ¯ τiσ c τiσ i (cid:16) g tI ↑ η I ↑ ,i ¯ σ + g tI ↓ η I ↓ ,i ¯ σ (cid:17) . (36)Since i = i ′ occurs more often than i = i ′ , the third-order terms neglected in theprevious section for the case of i = I , i ′ = I , and i = i ′ may have larger contributionthan the newly derived terms above. Such terms are derived by taking into accountintersite contraction including i ¯ σ and i ′ ¯ σ ′ . However, if intersite contractions are takenbetween i ¯ σ and i ′ ¯ σ ′ and the onsite contractions are taken for I , then such terms do notcontribute as explained around (18). Using the notation above, (20) is rewritten as P i c τiσ h P i i = (1 + κ τiσ ˜ n i ¯ σ ) c τiσ . (37)Then, for τ ′ = − τ σσ ′ ( ↑ , ↓ and +1 , − h c τ ′ i ′ σ ′ P (˜ n I ↑ + ˜ n I ↓ ) P c τiσ i h P i ≈ g tI ↑ h c τ ′ i ′ σ ′ ˜ n I ↑ c τiσ i + g tI ↓ h c τ ′ i ′ σ ′ ˜ n I ↓ c τiσ i + h c τ ′ i ′ σ ′ c τiσ i h κ τiσ (cid:16) g tI ↑ η I ↑ ,i ¯ σ + g tI ↓ η I ↓ ,i ¯ σ (cid:17) + κ ¯ τ ′ i ′ σ ′ (cid:16) g tI ↑ η I ↑ ,i ′ ¯ σ ′ + g tI ↓ η I ↓ ,i ′ ¯ σ ′ (cid:17)i + σκ τiσ h c τ ′ i ′ σ ′ c ¯ τi ¯ σ i (cid:16) g tI ↑ h c † I ↑ c τi ¯ τ i h c I ↑ c τiτ i − g tI ↓ h c † I ↓ c τiτ i h c I ↓ c τi ¯ τ i (cid:17) + σ ′ κ ¯ τ ′ i ′ σ ′ h c ¯ τ ′ i ′ ¯ σ ′ c τiσ i (cid:16) g tI ↑ h c † I ↑ c τ ′ i ′ ¯ τ ′ i h c I ↑ c τ ′ i ′ τ ′ i − g tI ↓ h c † I ↓ c τ ′ i ′ τ ′ i h c I ↓ c τ ′ i ′ ¯ τ ′ i (cid:17) . (38)Although all the new terms in (36) and (38) contain the g t factors, they arenot so simple as (24) and inhibit the straightforward analytical transform back to k -representation. In other words, they cause k -dependence of the renormalization. Sincethe ratio between the leading order and the corrections calculated in (38) is only of thefirst order, the influence from the corrections may be larger than those in the GA forthe real-space hopping amplitude, where the ratio is of the second order.Other corrections are the terms neglected in (15). We expect that they only slightlychange the magnitude of the leading order, and that their contribution is probably notvery important.
5. General estimation of higher-order terms
Let us estimate the other higher-order terms neglected above. The terms appearingin the Wick expansion can be classified into three groups by how to take contractionsof ˆ n Iσ (1 − ˆ n I ¯ σ ): (i) Onsite contractions are taken both for I ↑ and I ↓ . These termsdo not contribute to V k ′ k as explained around (18). (ii) If intersite contractions aretaken for Iσ and the onsite contraction is taken for I ¯ σ , then λ Iσ ˆ n Iσ (1 − ˆ n I ¯ σ ) / h P I i isreduced to ˜ n Iσ . Doing the same for λ I ¯ σ ˆ n I ¯ σ (1 − ˆ n Iσ ) / h P I i yields − ˜ n Iσ n I ¯ σ / (1 − n Iσ ).Then their summation is g tIσ ˜ n Iσ . Namely, all of these terms are proportional to g tIσ .(iii) For the other terms, intersite contractions are taken both for I ↑ and I ↓ . Naiveevaluation of these terms does not yield any explicit factor vanishing at half filling, andwe expect that many terms cancel out each other in some way. Instead, to derive explicit enormalization of transition matrix elements of particle number operators h k ′ s | (1 − ˆ n I ↑ )(1 − ˆ n I ↓ ) | ks i , whichis equivalent to V k ′ ,k for k = k ′ . Then we can replace as(1 − ˆ n I ↑ )(1 − ˆ n I ↓ ) h P I i = ⇒ − n I (1 − n I ↑ )(1 − n I ↓ ) ˜ n I ↑ ˜ n I ↓ , (39)i.e., all such terms contain g tIσ / (1 − n I ¯ σ ) explicitly. These considerations in (i), (ii) and(iii) above demonstrate that V k ′ ,k contains overall factor g tIσ .
6. Magnetic impurity renormalization
Let us consider a simple magnetic impurity (4), i.e., local magnetic field is applied atsite I . Its renormalization can be easily calculated by subtraction instead of summationof renormalized ˆ n I ↑ and ˆ n I ↓ using formulas above. Corresponding to (26) and (28), h γ ℓ ′ P S zI P γ ℓ i q h γ ℓ ′ P γ ℓ ′ i h γ ℓ P γ ℓ i ≈ " − n I ↑ + n I ↓ − n I ↑ h γ ℓ ′ ˜ n I ↑ γ † ℓ i − − n I ↓ + n I ↑ − n I ↓ h γ ℓ ′ ˜ n I ↓ γ † ℓ i (40) −→ − n Iσ D γ ℓ ′ S zI γ † ℓ E ( n ↑ = n ↓ ) , (41)The renormalization factor for n I ↑ = n I ↓ is (1 − n Iσ ) − , which is the square root ofthe Gutzwiller renormalization factor for the exchange interaction. Namely, in contrastto the non-magnetic impurity, the magnetic impurity is strengthened by the strongelectron correlation. It also makes a good contrast with the unrenormalized diagonalmatrix element h S zI i = h S zI i (to derive this, the limit of λ I ↑ − λ I ↓ → λ I ↑ = λ I ↓ ).In fact, also for magnetic systems ( n I ↑ = n I ↓ ), the factors appearing in (40) areequivalent to those in the renormalization of the exchange interaction derived in [7], i.e., h S zi S zj i ≈ h S zi i h S zj i + 14 X σ,σ ′ η iσ,jσ ′ σ − σ h S zi i − n iσ ! σ ′ − σ ′ h S zj i − n jσ ′ ! . (42)Although it is not explicitly noted in [7], in this renormalization of the spin interaction,the first term is from onsite contractions and not renormalized (from diagonalmatrix elements of the spin- z operators), whereas the second term including intersitecontractions is enhanced by the renormalization factor (from the second order of thetransition matrix elements of the spin- z operators). In fact, as shown in the next section,charge interaction is also renormalized in a similar manner although the direction ofrenormalization is opposite.
7. Charge interaction renormalization
The conventional GA [4] relates h ˆ O i to h ˆ O i for an operator ˆ O using a renormalizationfactor. By following this procedure, the renormalization factor is unity for the chargeinteraction, namely, h ˆ n i ˆ n j i ? ≈ h ˆ n i ˆ n j i = n i n j + X σ,σ ′ η iσ,jσ ′ (43) enormalization of transition matrix elements of particle number operators n i n j and therenormalization factor is likely to be derived by taking only the lowest order into account.Using a procedure similar to that for the non-magnetic impurity, more careful analysiscan be carried out, i.e., h ˆ n i ˆ n j i ≈ X σ,σ ′ λ iσ λ jσ ′ h ˆ n iσ (1 − ˆ n i ¯ σ )ˆ n jσ ′ (1 − ˆ n j ¯ σ ′ ) i h P i i h P j i ≈ n i n j + X σ,σ ′ η iσ,jσ ′ − n iσ − n i ¯ σ η i ¯ σ,jσ ′ − n jσ ′ − n j ¯ σ ′ η iσ,j ¯ σ ′ + n iσ − n i ¯ σ n jσ ′ − n j ¯ σ ′ η i ¯ σ,j ¯ σ ′ ! = n i n j + X σ,σ ′ g tiσ g tjσ ′ η iσ,jσ ′ . (44)At the half filling, any state is an eigenstate of ˆ n i ˆ n j with the eigenvalue unity bydefinition because every site is occupied by one electron and there is no particle numberfluctuation, which contradicts (43) but is consistent with (44). In fact, the second termof r.h.s. of (44) is the second order of (29), namely, it comes from a process in which ˆ n j creates two quasiparticles and ˆ n i annihilates them.To our knowledge, every calculation in the literature on the GA is using (43) insteadof (44) including the calculation by the author himself, and probably this error is pointedout for the first time here. However, this charge interaction usually does not give veryimportant contribution in t - J –type models, and this correction is likely to make onlyminor modification to numerical values. Therefore, we expect that main conclusions arenot drastically changed by this correction. Following this correction, equations in [7]should be modified, namely, (3 g sij −
1) and (3 g sij + 1) in (14) and (15) should be replacedby (3 g sij − g tii g tjj ) and (3 g sij + g tii g tjj ), respectively, and derivative of g tii should be alsoconsidered for (16).
8. Conclusion
Since the Gutzwiller approximation is formulated to (almost) conserve the particlenumber at the Gutzwiller projection, one may consider that quantities related toparticle number operators are not renormalized. However, since the particle numberis an expectation value with respect to an assumed ground state, the constraint of itsconservation does not restrict transition matrix elements with excited states. Our resultshere correct description by the conventional Gutzwiller approximation in the literature,where such renormalization factors are calculated with a focus on diagonal matrixelements or lowest-order terms and regarded as unity. The results in this paper aregeneral and do not depend on parameters. Namely, they are valid both for attractive andrepulsive impurity potentials and both for attractive and repulsive charge interactions.The Fourier-transformed impurity potential has a form of hopping in the k -space.We have derived similarities and differences between this “hopping” in the k -spaceand in the real space under real-space electron double-occupancy prohibition. As asimilarity, they are strongly renormalized to decrease with hole concentration x , and enormalization of transition matrix elements of particle number operators g t = 2 x/ (1 + x ) in uniform non-magnetic systems. Inaddition, the higher order terms also contain g t . It should represent that not manyavailable seats to hop are left because of the electron repulsion. A difference is,however, h c † iσ c jσ i of each σ is renormalized in the real space, whereas renormalizationof P σ h c † k ′ σ c kσ i appears only after the summation over spin σ = ± in the k -space. If thissummation is replaced by subtraction, which corresponds to a magnetic impurity in thereal space, then the direction of the renormalization is reversed, i.e., the renormalizationfactor is larger than unity and equivalent to the square root of that for the exchangeinteraction. As another difference, the corrections to the leading order term in the k -space can be larger and have more complicated expression than those in the real space.As related calculation, renormalization of charge interaction has been also derived.The leading order is rather trivial and unrenormalized, i.e., it is the product of particledensities at the two relevant sites. The next leading order term is the second orderof transition matrix elements of the number operators with excited states. Sincethe transition matrix elements are renormalized by g t , these second order terms arerenormalized by ( g t ) , namely, strongly reduced. These terms include hopping andpairing amplitude and are relevant to the mean-field approximation. Similar relationis found also in the z -component of the exchange interaction. Namely, the leadingorder is the product of spin- z densities at the two relevant sites. The next term is thesecond order of transition matrix elements of the spin- z operator, which is strengthenedby the electron repulsion. At the half filling, any state is an eigenstate of ˆ n i ˆ n j , withthe eigenvalue unity. In fact, (44) satisfies it even in magnetic systems, which maydemonstrate that the choice of fugacity factors by (9) is reasonable. Other choices offugacity factors also discussed in [7] do not seem to satisfy it in magnetic systems, andtheir use is likely to be restricted in systems with small magnetic moments. References [1] Fukushima N, Chou C-P and Lee T K 2008
J. Phys. Chem. Solids Phys. Rev. B Phys. Rev.
A1726-35[4] Zhang F C, Gros C, Rice T M and Shiba H 1988
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