Full Self-Consistent Projection Operator Approach to Nonlocal Excitations in Solids
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Full Self-Consistent Projection Operator Approach to Nonlocal Excitationsin Solids
Yoshiro
Kakehashi ∗ , Tetsuro Nakamura , and Peter Fulde , Department of Physics and Earth Sciences, Faculty of Science, University of the Ryukyus,1 Senbaru, Nishihara, Okinawa, 903-0213, Japan Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Str. 38, D-01187 Dresden, Germany Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790-784, Korea
A self-consistent projection operator method for single-particle excitations is developed.It describes the nonlocal correlations on the basis of a projection technique to the retardedGreen function and the off-diagonal effective medium. The theory takes into account long-range intersite correlations making use of an incremental cluster expansion in the medium. Ageneralized self-consistent coherent potential is derived. It yields the momentum-dependentexcitation spectra with high resolution. Numerical studies for the Hubbard model on a sim-ple cubic lattice at half filling show that the theory is applicable in a wide range of Coulombinteraction strength. In particular, it is found that the long-range antiferromagnetic corre-lations in the strong interaction regime cause shadow bands in the low-energy region andsub-peaks of the Mott-Hubbard bands.
KEYWORDS: non-local excitations, momentum-dependent self-energy, Hubbard model,ARPES
1. Introduction
Single-particle excitations play an important role in condensed matter physics. They de-termine among others basic properties of solids such as the metal-insulator (MI) transition,magnetism, and superconductivity. Recently developed angle-resolved photoemission spec-troscopy allows to observe details of the excitation spectra in various materials.
2, 3
The exci-tations are usually strongly influenced by electron correlations. Therefore various approachesto treat the correlations have been developed. Hubbard,
4, 5 for example, was the first whoproposed a theory of the MI transition on the basis of the retarded Green function. He de-rived lower and upper Mott-Hubbard incoherent bands caused by strong electron correlations.Cyrot extended the theory to finite temperatures by using the functional integral method.Penn and Liebsch developed a Green function theory starting from the low-density limitusing the t-matrix approximation. Fulde et al. proposed methods which use the projectiontechnique. The latter describes the dynamics of electrons by means of the Liouville operatoron the basis of the retarded Green functions. ∗ E-mail address: [email protected] 1/24 . Phys. Soc. Jpn.
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In the past two decades single-site theories of excitations with use of an effective mediumhave extensively been developed. Progress was made for the MI transition in infinite di-mensions where the self-energy of the Green function becomes independent of momentum. Several authors determined the self-energy ( i.e. , an effective medium) self-consistently soas to be identical with the local self-energy of an impurity embedded in a medium. The theory,called dynamical mean-field theory (DMFT), can be traced back to a many-body coherentpotential approximation (CPA) in disordered alloys and is equivalent to the dynamicalCPA
20, 21 used in the theory of magnetism. The DMFT has clarified the MI transition ininfinite dimensions. One of its new features is that it can describe both quasiparticle statesnear the Fermi level and the Mott-Hubbard incoherent bands.In the DMFT as well as the dynamical CPA one usually deals with the temperatureGreen function. Therefore one needs to perform numerical analytic continuations at finitetemperatures, which often causes numerical difficulties in particular at low temperatures.We recently proposed an alternative method which directly starts from the retarded Greenfunction. The method, which is called the projection operator method CPA (PM-CPA), has been shown to be equivalent to the dynamical CPA and the DMFT. In the PM-CPA,the Loiuville operator is approximated by an energy-dependent Liouvillean for an effectiveHamiltonian with a coherent potential. The latter is determined by a self-consistent CPAcondition. By solving an impurity problem embedded in a medium with use of the renormalizedperturbation theory (RPT), we obtained an interpolation theory for the MI transition.Although one can treat the excitations from the weak to the strong Coulomb interactionregime with use of the single-site theories mentioned above, the single-site approximation(SSA) neglects the intersite correlations which often play an important role in real systemssuch as the Cu oxcides and Fe-pnictides high-temperature superconducting compounds. Infact, recent photoelectron spectroscopy found a pseudo gap
25, 26 and a kink structure
27, 28 incuprates which can not be explained by a SSA.Because of the reasons mentioned above, we have recently proposed a nonlocal theory ofexcitation spectra called the self-consistent projection operator method (SCPM). It is basedon the projection technique
30, 31 and the incremental cluster expansion method.
In thistheory, all the off-diagonal self-energy matrix elements are calculated by means of an incre-mental cluster expansion from a diagonal effective medium ˜Σ iiσ ( z ). The latter is determinedby the CPA condition. We call this method here and in the following the SCPM-0, The theorytakes into account long-range intersite correlations, which are missing in the other nonlocaltheories such as the dynamical cluster theory
35, 36 and the cluster DMFT.
37, 38
Moreover, thetheory can describe the momentum dependent excitation spectrum with high resolution be-cause it is based on the retarded Green function and because the intersite correlations aretaken into account up to infinity at each order of the incremental cluster expansion. Using the . Phys. Soc. Jpn.
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SCPM-0, we investigated the excitation spectra of the Hubbard model on the simple cubiclattice and the square lattice. Especially in the two-dimensional system, we found amarginal Fermi liquid behavior and a kink structure
40, 41 in the quasiparticle state in theunderdoped region. The former was proposed in a phenomenological theory and the latterwas found in the photoemission experiments in cuprates.
27, 28
The SCPM-0 makes use of a diagonal effective medium to describe the on-site correlationsat surrounding sites. The self-consistency between the self-energy and the medium is achievedonly for the diagonal site. Such a treatment results in a limited range of applications. In thispaper, we extend the SCPM-0 by introducing a new off-diagonal effective medium ˜Σ ijσ ( z ).All calculated off-diagonal self-energy matrix elements are consistent with all those of themedium. Such a full self-consistent projection operator method (FSCPM) should allow foran improved quantitative description of nonlocal excitations in solids, and should play animportant role in the phenomena with long-range charge and spin fluctuations, in particular,in low dimensional systems.In the following subsection 2.1, we briefly review the retarded Green function and pro-jection technique. In the § § §
2. Full Self-Consistent Projection Operator Method
We adopt the tight-binding Hubbard model consisting of the Hartree-Fock independentparticle Hamiltonian H and the residual interactions with intraatomic Coulomb interactionparameter U as follows: H = H + U X i δn i ↑ δn i ↓ , (1) . Phys. Soc. Jpn. Full Paper H = X i,σ ǫ σ n iσ + X i,j,σ t ij a † iσ a jσ . (2)Here ǫ σ = ǫ − µ + U h n i − σ i is the Hartree-Fock atomic level. Note that h· · · i denotes a thermalaverage. The quantities ǫ , µ , and t ij are the atomic level, the chemical potential, and thetransfer integral between sites i and j , respectively. Furthermore a † iσ ( a iσ ) denotes the creation(annihilation) operator for an electron with spin σ on site i , while n iσ = a † iσ a iσ is the electrondensity operator for spin σ , and δn iσ = n iσ − h n iσ i .The excitation spectra for electrons are obtained from a retarded Green function definedby G ijσ ( t − t ′ ) = − iθ ( t − t ′ ) h [ a iσ ( t ) , a † jσ ( t ′ )] + i . (3)Here θ ( t ) is the step function, a iσ ( t ) is the Heisenberg representation of a iσ defined by a iσ ( t ) = exp( iHt ) a iσ exp( − iHt ). Furthermore, [ , ] + denotes the anti-commutator betweenthe operators.The Fourier representation of the retarded Green function is written as follows. G ijσ ( z ) = (cid:16) a † iσ (cid:12)(cid:12)(cid:12) z − L a † jσ (cid:17) . (4)Here the inner product between the operators A and B is defined by ( A | B ) = h [ A + , B ] + i .Also z = ω + iδ with δ being an infinitesimal positive number, and L is a Liouville operatordefined by LA = [ H, A ] for an operator A . Note that [ , ] is the commutator between theoperators.Using the projection technique, we obtain the Dyson equation for the retarded Greenfunction as G ijσ ( z ) = [( z − H − Λ ( z )) − ] ijσ . (5)Here the matrices are defined by ( H ) ijσ = ǫ σ δ ij + t ij , andΛ ijσ ( z ) = ( Λ ( z )) ijσ = U G ijσ ( z ) . (6)The reduced memory function G ijσ ( z ) is defined by G ijσ ( z ) = (cid:16) A † iσ (cid:12)(cid:12)(cid:12) z − L A † jσ (cid:17) . (7)The operator A † iσ is given by A † iσ = a † iσ δn i − σ , (8)and the Liouville operator L acting on A † jσ is defined by L = QLQ , Q being Q = 1 − P . Theprojection operator P projects onto the original operator space {| a † iσ ) } , P = X iσ (cid:12)(cid:12) a † iσ (cid:1) (cid:0) a † iσ (cid:12)(cid:12) . (9)In a crystalline system, the Green function G ijσ ( z ) is obtained from its momentum rep- . Phys. Soc. Jpn. Full Paper resentation G kσ ( z ) as G ijσ ( z ) = 1 N X k G kσ ( z ) exp( i k · ( R i − R j )) . (10)Here N is a number of site. The momentum-dependent Green function is given by G kσ ( z ) = 1 z − ǫ kσ − Λ kσ ( z ) . (11)Here ǫ kσ = ǫ σ + ǫ k is the Hartree-Fock one-electron energy eigenvalue, ǫ k is the Fouriertransform of t ij , and Λ kσ ( z ) = X j Λ j σ ( z ) exp( i k · R j ) . (12)It should be noted that the momentum-dependent excitation spectrum is obtained from ρ kσ ( ω ) = − π Im G kσ ( z ) . (13)The local density of states (DOS) is given by ρ iσ ( ω ) = − π Im G iiσ ( z ) . (14)which is identical with the average DOS per atom, ρ σ ( ω ) = 1 /N P k ρ kσ ( ω ) when all sites areequivalent to each other. In the full self-consistent projection method, we introduce an energy-dependent Liouvilleoperator ˜ L ( z ). Its corresponding Hamiltonian is that of an off-diagonal effective medium˜Σ ijσ ( z ). It is ˜ L ( z ) A = [ ˜ H ( z ) , A ] , (15)for arbitrary operator A and˜ H ( z ) = H + X ijσ ˜Σ ijσ ( z ) a † iσ a jσ . (16)Defining an interaction Liouville operator L I ( z ) such that L I ( z ) A = [ H I ( z ) , A ] , (17)with H I ( z ) = U X i δn i ↑ δn i ↓ − X ijσ ˜Σ ijσ ( z ) a † iσ a jσ , (18)we can rewrite the original Liouville operator L as follows. L = ˜ L ( z ) + L I ( z ) . (19)It should be noted that the interaction H I ( z ) contains the off-diagonal components˜Σ ijσ ( z ) ( i = j ) in addition to the diagonal ones ˜Σ iiσ ( z ). Accordingly, we divide here theinteraction Liouvillean L I ( z ) into single-site terms and pair-site ones. Furthermore we intro- . Phys. Soc. Jpn. Full Paper duce site-dependent prefactors { ν i } which are either 1 or 0. L I ( z ) = X i ν i L ( i )I ( z ) + X ( i,j ) ν i ν j L ( ij )I ( z ) , (20) L ( i ) I ( z ) A = h U δn i ↑ δn i ↓ − X σ ˜Σ iiσ ( z ) n iσ , A i , (21) L ( ij ) I ( z ) A = h − X σ ( ˜Σ ijσ ( z ) a † iσ a jσ + ˜Σ jiσ ( z ) a † jσ a iσ ) , A i . (22)After substituting the Liouvillean (19) into Eq. (7), we can expand the resolvent ( z − L ) − with respect to L I ( z ) as follows.( z − L ) − = G + G T G , (23) T = L I + L I G L I + L I G L I G L I + · · · . (24)Here G = ( z − L ( z )) − and L ( z ) = Q ˜ L ( z ) Q , while L I ( z ) = QL I ( z ) Q .The T matrix operator may be expanded with respect to different sites as T = X i ν i T i + X ( i,j ) ν i ν j T ij + X ( i,j,k ) ν i ν j ν k T ijk + · · · . (25)The single-site T i , two-site T ij , and three-site T ijk matrix scattering operators are obtainedby setting the indices as ( ν i = 1 , ν l ( = i ) = 0), ( ν i = ν j = 1 , ν l ( = i,j ) = 0), ( ν i = ν j = ν k =1 , ν l ( = i,j,k ) = 0), and so on. It is T i = T ( i ) , (26) T ij = T ( ij ) − T i − T j , (27) T ijk = T ( ijk ) − T ij − T jk − T ki − T i − T j − T k . (28)The operator T (c) (c = i, ij, ijk, · · · ) at the right-hand-side (r.h.s.) of the above equations isthe T matrix operators for the cluster c, i.e. , T (c) = L (c) I (1 − G L (c) I ) − . (29)Here L (c) I = QL (c)I ( z ) Q and L (c)I ( z ) is the interaction Liouvillean for a cluster c: L (c)I ( z ) = X i ∈ c L ( i )I ( z ) + X ( i,j ) ∈ c L ( ij )I ( z ) . (30)Note that the sums in the above equation are taken over the sites or pairs belonging to thecluster c.Substituting Eq. (25) into Eq. (23) we have G ijσ ( z, { ν l } ) = (cid:16) A † iσ (cid:12)(cid:12)(cid:12) ( G + X l ν l G T l G + X ( l,m ) ν l ν m G T lm G + · · · ) A † jσ (cid:17) . (31) . Phys. Soc. Jpn. Full Paper
In the incremental method, we first consider the self-energy contribution due to intra-atomic excitations, i.e. , G ( i ) iiσ ( z ) = G iiσ ( z, ν i = 1 , ν l ( = i ) = 0) = (cid:0) A † iσ (cid:12)(cid:12) ( G + G T i G ) A † iσ (cid:1) . (32)Next we consider the scattering contribution due to a two-site increment in Eq. (31), G ( il ) iiσ ( z ) = G iiσ ( z, ν i = ν l = 1 , ν m ( = i,l ) = 0) = G ( i ) iiσ ( z ) + (cid:0) A † iσ (cid:12)(cid:12) G ( T l + T li ) G (cid:12)(cid:12) A † iσ (cid:1) . (33)This defines two-site increment to the diagonal matrix element as∆ G ( il ) iiσ ( z ) = G ( il ) iiσ ( z ) − G ( i ) iiσ ( z ) . (34)In the same way, we consider G ( ilm ) iiσ ( z ) = G ( i ) iiσ ( z ) + ∆ G ( il ) iiσ ( z ) + ∆ G ( im ) iiσ ( z ) + (cid:0) A † iσ (cid:12)(cid:12) G ( T lm + T ilm ) G (cid:12)(cid:12) A † iσ (cid:1) . (35)Then, we define the increment for a three-site contribution.∆ G ( ilm ) iiσ ( z ) = G ( ilm ) iiσ ( z ) − ∆ G ( il ) iiσ ( z ) − ∆ G ( im ) iiσ ( z ) − G ( i ) iiσ ( z ) . (36)The memory function G iiσ ( z ) in Eq. (31) is then expanded as follows. G iiσ ( z ) = G ( i ) iiσ ( z ) + X l = i ∆ G ( il ) iiσ ( z ) + 12 X l = i X m = i,l ∆ G ( ilm ) iiσ ( z ) + · · · . (37)It should be noted that G (c) iiσ ( z ) (c = i, ij, · · · ) in Eqs. (32), (33), and (35) is obtained fromthe T -matrix T (c) given by Eq. (29) as G (c) ijσ ( z ) = (cid:0) A † iσ (cid:12)(cid:12) ( z − L (c) ( z )) − A † jσ (cid:1) . (38)Here the cluster Liouvillean L (c) ( z ) is defined by L (c) ( z ) = L ( z ) + L (c)I ( z ) . (39)In the same way, the off-diagonal memory function is obtained as follows. G ijσ ( z ) = G ( ij ) ijσ ( z ) + X l = i,j ∆ G ( ijl ) ijσ ( z ) + 12 X l = i,j X m = i,j,l ∆ G ( ijlm ) ijσ ( z ) + · · · , (40)with ∆ G ( ijl ) ijσ ( z ) = G ( ijl ) ijσ ( z ) − G ( ij ) ijσ ( z ) , (41)∆ G ( ijlm ) ijσ ( z ) = G ( ijlm ) ijσ ( z ) − ∆ G ( ijl ) ijσ ( z ) − ∆ G ( ijm ) ijσ ( z ) − G ( ij ) ijσ ( z ) . (42)When we take into account all the terms on the r.h.s. of Eqs. (37) and (40), the memoryfunction G ijσ ( z ) does not depend on the effective medium ˜Σ ijσ ( z ). However, it is not possiblein general to calculate the terms up to higher orders; we have to truncate the incrementalexpansion at a certain stage. In that case the memory function G ijσ ( z ) depends on the medium . Phys. Soc. Jpn. Full Paper ˜Σ ijσ ( z ). We determine the latter from the following self-consistent equation,˜Σ ijσ ( z ) = Λ ijσ ( z ) . (43)Note that the off-diagonal effective medium ˜Σ ijσ ( z ) is the self-energy for the energy-dependentLiouvillean ˜ L ( z ), i.e. , ˜Σ ijσ ( z ) = U (cid:0) A † iσ (cid:12)(cid:12) ( z − L ( z )) − A † jσ (cid:1) = U (cid:0) A † iσ (cid:12)(cid:12) G A † jσ (cid:1) . Thus theself-consistent equation (43) is equivalent to the condition that the T -matrix describing thescattering from the medium vanishes according to Eq. (23): (cid:0) A † iσ (cid:12)(cid:12) G T G A † jσ (cid:1) = 0 . (44)This is a generalization of the CPA.The present theory reduces to the previous version of the nonlocal excitations (SCPM-0) when the off-diagonal media ˜Σ ijσ ( z ) ( i = j ) are omitted and only the self-consistency ofthe diagonal part is taken into account in Eq. (43). When in the SCPM-0 only the diagonalself-energy Λ iiσ ( z ) is taken into account ( i.e. , when we make the SSA), the result reduces tothe PM-CPA which we previously proposed. The incremental cluster expansion scheme given in the last subsection can be performedwhen the cluster memory function G (c) ijσ ( z ) defined by Eq. (38) is known. We obtain here anexplicit expression for it. The Hamiltonian H (c) ( z ) to the cluster Liouvillean L (c) ( z ) in (38)is given by H (c) = H + X ijσ ˜Σ ijσ ( z ) a † iσ a jσ − X ij ∈ c X σ ˜Σ ijσ ( z ) a † iσ a jσ + U X i ∈ c δn i ↑ δn i ↓ . (45)Introducing parameters λ ijσ (0 ≤ λ ijσ ≤ H (c) as H (c) ( z ) = ˜ H (c) ( z ) + H (c)I ( z ) , (46)˜ H (c) ( z ) = H + X ijσ ˜Σ ijσ ( z ) a † iσ a jσ − X ij ∈ c X σ λ ijσ ˜Σ ijσ ( z ) a † iσ a jσ , (47) H (c)I ( z ) = U X i ∈ c δn i ↑ δn i ↓ − X ij ∈ c X σ λ ijσ ˜Σ ijσ ( z ) a † iσ a jσ . (48)Here λ ijσ = 1 − λ ijσ . Note that parameters λ ijσ control the partition ratio of the mediumpotentials ˜Σ ijσ in the cluster between the noninteracting part ˜ H (c) ( z ) and the interactingpart H (c)I ( z ) (see Fig. 1). The Hamiltonian ˜ H (c) ( z ) denotes a system with a uniform effectivemedium ˜Σ ijσ ( z ) when λ ijσ = 1 ( i, j ∈ c). When λ ijσ = 0 ( i, j ∈ c), ˜ H (c) ( z ) denotes a refer-ence system with a cluster cavity in the effective medium and H (c)I ( z ) denotes the Coulombinteractions on the cluster sites.According to the partition of the Hamiltonian (46), we introduce a noninteracting Liou- . Phys. Soc. Jpn.
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ΣλΣ λΣλΣλΣ = + H I (z) (c) Fig. 1. Partition of the medium on the cluster sites. The cluster Hamiltonian H (c) ( z ) with on-siteCoulomb interaction embedded in the off-diagonal medium ˜Σ (the left-hand-side) can be split intoan effective Hamiltonian ˜ H (c) ( z ) with cluster potential λ ˜Σ and the remaining interaction H (c)I ( z ).The parameter λ can control the magnitude of the potential ˜Σ on the cluster sites. Note that eachsite in the cluster may be away from the other sites of the cluster. villean L (c)0 ( z ) = QL (c)0 ( z ) Q and an interacting Liouvillean L (c) I ( z ) = QL (c) I ( z ) Q as follows. L (c)0 ( z ) A = [ ˜ H (c) ( z ) , A ] , (49) L (c)I ( z ) A = [ H (c)I ( z ) , A ] . (50)Then we have L (c) ( z ) = L (c)0 ( z )+ L (c)I ( z ), and the cluster memory function G (c) ijσ ( z ) is expressedas G (c) ijσ ( z ) = (cid:0) A † iσ (cid:12)(cid:12) ( z − L (c)0 ( z ) − L (c)I ( z )) − A † jσ (cid:1) . (51)Note that L (c)I ( z ) in the above expression has been redefined by Eq. (50), i.e. , Eq. (30) inwhich ˜Σ ijσ ( z ) has been replaced by λ ijσ ˜Σ ijσ ( z ).The interaction Liouvillean L (c)I ( z ) expands the operator space to L (c)I ( z ) | A † lσ ) = h U (1 − h n l − σ i ) − λ llσ ˜Σ llσ ( z ) i | A † lσ ) − X ( i,j ) ∈ c (cid:16) δ lj | ˜ B † liσ ) + δ li | ˜ B † ljσ ) (cid:17) , (52) | ˜ B † ljσ ) = λ ljσ ˜Σ ljσ ( z ) | a † jσ δn l − σ )+ λ lj − σ ˜Σ lj − σ ( z ) | a † lσ δ ( a † j − σ a l − σ )) + λ lj − σ ˜Σ lj − σ ( z ) | a † lσ δ ( a † l − σ a j − σ )) . (53)Therefore we introduce the projection operator P = X iσ (cid:12)(cid:12) A † iσ (cid:1) χ − iσ (cid:0) A † iσ (cid:12)(cid:12) , (54)where χ iσ = h n i − σ i (1 − h n i − σ i ). It projects onto the space {| A † iσ ) } , while Q = 1 − P eliminatesthe space {| A † iσ ) } . Making use of these operators one can divide the interaction Liouvillean L (c)I ( z ) into two parts. L (c) I ( z ) = P L (c) I ( z ) P + L (c) IQ ( z ) , (55) . Phys. Soc. Jpn. Full Paper L (c) IQ ( z ) = Q L (c) I ( z ) P + L (c) I ( z ) Q . (56)The first term at the r.h.s. of Eq. (55) acts in the subspace {| A † iσ ) } , while the second termexpands the operator space beyond {| A † iσ ) } . It should be noted that the first term at the r.h.s.of Eq. (56) does not vanish in general when we take into account the off-diagonal characterof the effective medium. This point differs from the case of the SSA. Making use of Eq. (52), we obtain the expression of
P L (c) I ( z ) P in Eq. (55) as P L (c) I ( z ) P = X ijσσ ′ (cid:12)(cid:12) A † iσ (cid:1) ( L (c) I ) iσjσ ′ (cid:0) A † jσ ′ (cid:12)(cid:12) , (57) (cid:0) L (c) I (cid:1) iσjσ ′ = (cid:2) U (1 − h n i − σ i ) − λ iiσ ˜Σ iiσ ( z ) (cid:3) δ σσ ′ − χ − iσ P l ( = i ) ∈ c ( A † iσ | ˜ B † ilσ ′ ) χ iσ δ ij − ( A † jσ | ˜ B † jiσ ′ ) χ iσ χ jσ ′ (1 − δ ij ) . (58)Substituting Eq. (55) into Eq. (51), we obtain G (c) ijσ ( z ) = h G (c)0 · (1 − L (c) I · G (c)0 ) − i ijσ . (59)Here the screened memory function G (c)0 ijσ ( z ) = ( G (c)0 ) iσjσ is defined by G (c)0 ijσ ( z ) = (cid:0) A † iσ (cid:12)(cid:12)(cid:0) z − L ( z ) − L (c) IQ ( z ) (cid:1) − A † jσ (cid:1) . (60)The expression of the cluster memory function (59) is exact. The simplest approximationis to neglect the interaction Liouville operator L (c) IQ in the screened cluster memory function G (c)0 ijσ , which we called the zeroth renormalized perturbation theory (RPT-0). The approx-imation yields in the weak interaction limit the correct result of second-order perturbationtheory, as well as the exact atomic limit.An explicit expression of the screened cluster memory function in the RPT-0 can beobtained approximately as shown in Appendix. G (c)0 ijσ ( z ) = A ijσ Z dǫdǫ ′ dǫ ′′ ρ (c) ijσ ( λ, ǫ ) ρ (c) ij − σ ( λ, ǫ ′ ) ρ (c) ji − σ ( λ, ǫ ′′ ) χ ( ǫ, ǫ ′ , ǫ ′′ ) z − ǫ − λ σ ˜Σ σ ( ǫ, z ) − ǫ ′ − λ − σ ˜Σ − σ ( ǫ ′ , z ) + ǫ ′′ + λ − σ ˜Σ − σ ( ǫ ′′ , z ) , (61) χ ( ǫ, ǫ ′ , ǫ ′′ ) = (1 − f ( ǫ ))(1 − f ( ǫ ′ )) f ( ǫ ′′ ) + f ( ǫ ) f ( ǫ ′ )(1 − f ( ǫ ′′ )) . (62)Here 0 ≤ λ σ ≤
1, and f ( ω ) is the Fermi distribution function.The prefactor A ijσ in Eq. (61) has been introduced to recover the exact second momentof the spectrum. A ijσ = χ iσ h n i − σ i c (1 − h n i − σ i c ) δ ij + 1 − δ ij . (63)Here the electron number for a cavity state h n iσ i c is defined by h n iσ i c = Z dǫρ (c) iiσ ( λ, ǫ ) f ( ǫ ) . (64) . Phys. Soc. Jpn. Full Paper
The densities of states for the cavity state are given by ρ (c) ijσ ( λ, ǫ ) = − π Im [( F c ( λ, z ) − + λ σ ˜ Σ (c) ( z )) − ] ijσ , (65)( F c ( λ, z )) ijσ = F ijσ ( z ) = X k h i | k ih k | j i z − ǫ kσ − λ σ ˜Σ kσ ( z ) . (66)Furthermore, λ σ = 1 − λ σ , ( ˜ Σ (c) ( z )) ijσ = ˜Σ ijσ ( z ) ( i, j ∈ c), and ˜Σ kσ ( z ) is the Fouriertransform of ˜Σ ijσ ( z ); ˜Σ kσ ( z ) = P j ˜Σ j σ ( z ) exp( i k · R j ).The simplified self-energy ˜Σ σ ( ǫ, z ) in Eq. (61) is calculated from ˜Σ ijσ ( z ) as˜Σ σ ( ǫ, z ) = X i ρ ij ( ǫ − ǫ σ ) ρ ( ǫ − ǫ σ ) ˜Σ ijσ ( z ) , (67)where ρ ij ( ǫ ) is the DOS of the noninteracting system defined by ρ ij ( ǫ ) = P k h i | k i δ ( ǫ − ǫ k ) h k | j i and ρ ( ǫ ) is defined by ρ ( ǫ ) = 1 /N P k δ ( ǫ − ǫ k ).Finally the simplified cluster memory function in the RPT-0 is given by Eq. (59), Eq.(58) in which { λ ijσ } has been replaced by { λ σ } , and Eq. (61). In the present scheme, we firstassume ˜Σ ijσ ( z ). Then we calculate the coherent Green function F ijσ ( z ) (Eq. (66)), the screenedcluster memory function G ( c )0 ijσ ( z ) according to Eq. (61), as well as the atomic frequencymatrix (cid:0) L (c) I (cid:1) iσjσ ′ given by Eq. (58). Using these matrices, we calculate the cluster memoryfunction G ( c ) ijσ ( z ) (Eq. (59)). Note that the static quantities ( A † iσ ′ | ˜ B † jlσ ′ ) in Eq. (58) have to becalculated separately, for example, by means of the local ansatz wavefunction method. Afterhaving calculated cluster memory functions, we can obtain the memory functions (37) and(40) according to the incremental scheme. Then we calculate the diagonal and off-diagonalself-energies (6). If the self-consistent condition (43) is not satisfied, we repeat the above-mentioned procedure renewing the medium ˜Σ ijσ ( z ) until the self-consistency (43) is achieved.When we obtain the self-consistent solution ˜Σ ijσ ( z ), we can calculate the excitation spectrum ρ kσ ( ω ) from the Green function (11) and the DOS ρ σ ( ω ) from Eq. (14).
3. Nonlocal Excitations on a Simple-Cubic Lattice
We present here the numerical results of excitation spectra of the Hubbard model on asimple cubic lattice at half-filling in the paramagnetic state in order to examine the nonlocalcorrelations in the FSCPM. As in our previous calculations, we choose the parameters λ σ = 0in Eq. (61); we start from the cavity cluster state ( λ ij = 0) for the calculation of the memoryfunction. The form (61) with λ σ = 0 reduces to the iterative pertubation scheme at half-filling in infinite dimensions. Note that we need not to calculate ˜Σ σ ( ǫ, z ) defined by Eq. (67)as well as ( A † jσ | ˜ B † ilσ ) in Eq. (58) when λ σ = 0.We adopt the nearest-neighbor transfer integral t here and in the following, choosethe energy unit as | t | = 1. The Fourier transform of the transfer integrals is given by ǫ = − k x + cos k y + cos k z ) in the unit of lattice constant a . Furthermore, in Eqs. (37) and . Phys. Soc. Jpn. Full Paper -10-8-6-4-202468-20 -15 -10 -5 0 5 10 15 20 S e l f - ene r g y Λ ( z ) Energy
U=10
FSCPM SCPM-0SSA
Fig. 2. Diagonal self-energies for U = 10 in the FSCPM (thick curves), the SCPM-0 (middle-sizecurves), and the SSA (thin curves). The real (imaginary) part in each method is drawn by thesolid (dotted) curve. (40) we take into account single-site and pair-site terms only, but the latters up to the 10thnearest neighbors. In the FSCPM, we have to perform a numerical k integration to obtainthe coherent Green function F ijσ (see Eq. (66)). We have adopted a 80 × ×
80 mesh in thefirst Brillouin zone for such integration.In the numerical calculations of the screened memory function (61), we applied the methodof Laplace transformation. This reduces the 3-fold integrals with respect to energy into theone-fold integral with respect to time.Starting from an initial set of values { ˜Σ j ( z ) = 0 } ( j = 0 , , · · · , M ), we calculated { F ij ( z ) } (Eq. (66)), { ρ ( c ) ij ( z ) } (Eq. (65)), { G (c)0 ij ( z ) } (Eq. (61)), { G (c) ij ( z ) } (Eq. (59)), and { G ij ( z ) } (Eqs.(37) and (40)), and finally obtained Λ ijσ ( z ) = U G ij ( z ). We have repeated the same procedureuntil self-consistency (43) was achieved.We show in Fig. 2 the self-consistent diagonal self-energy at U = 10 and compare theresults with the SSA and SCPM-0 calculations. The SCPM-0 shifts the spectral weight of theSSA towards the higher energy region. The FSCPM suppresses the amplitude of the SCPM-0self-energy in the high energy region. In the low energy region, we find that the imaginarypart is close to that of the SSA, while the real part is in-between the SSA and the SCPM-0.The off-diagonal self-energies are shown in Fig. 3. We find again that self-energies aresuppressed by the full self-consistency. Furthermore the self-energies rapidly damp with in-creasing interatomic distance. The fourth-nearest neighbor contribution and the contributionfrom more distant pairs can be neglected when U = 10, though we took into account the . Phys. Soc. Jpn. Full Paper -0.4-0.3-0.2-0.100.10.20.30.4-20 -15 -10 -5 0 5 10 15 20 S e l f - ene r g y EnergyU=10
Re[ Λ (1,z)] Re[ Λ (2,z)] Re[ Λ (3,z)] Re[ Λ (4,z)] Im[ Λ (1,z)] Im[ Λ (2,z)] Im[ Λ (3,z)] Im[ Λ (4,z)] Fig. 3. Off-diagonal self-energy Λ( n, z ) for the n -th nearest neighbor pairs ( n = 1 , , , off-diagonal ones up to 10th nearest neighbors.The momentum-dependent self-energies Λ kσ ( z ) are also suppressed by the full self-consistency. As shown in Fig. 4, in the low energy region the imaginary parts of Λ kσ ( z )hardly depend on momentum k and are close to those of the SSA. In the incoherent region | ω | &
2, both real and imaginary parts show considerable k dependence. For example, theimaginary part of Λ kσ ( z ) shows at the Γ point a minimum at ω ≈ ω ≈ −
5, while it shows a minimum at ω = − ω = 5 at the R point.Calculated momentum dependent excitation spectra are shown in Figs. 5, 6, and 7. Forrather weak Coulomb interaction ( U = 6), we find that the quasiparticle band is reduced byabout 30% in width as compared with that of the noninteracting band. As seen in Fig. 5, theMott-Hubbard incoherent bands appear at | ω | ≈ . U = 10), the quasiparticle band widthbecomes narrower and is reduced by 25% as compared with that of the SCPM-0. The spectralweight of the quasiparticle band around the Γ and R point moves farther to the lower andupper Mott-Hubbard bands. The latters are more localized in the vicinity of the Γ and Rpoint than in the SCPM-0, and their peaks are reduced by 25% as compared with those inthe SCPM-0.In the FSCPM scheme, we can obtain the momentum-dependent spectra in the strongCoulomb interaction regime ( U = 16). This is shown in Fig. 7. In this regime, the quasiparti-cle band becomes narrower and the spectral weight becomes smaller. New excitations which . Phys. Soc. Jpn. Full Paper -10-505-20 -15 -10 -5 0 5 10 15 20 Λ k ( z ) Energy
U=10
Fig. 4. Momentum dependent self-energies at various k points along the high symmetry line Γ-X-M-R. The real (imaginary) parts are shown by solid (dotted) curves. Note that the real part at the Γpoint shows a maximum at energy ω ≈ ω ≈ −
3, while the imaginarypart shows a minimum at ω ≈ ω ≈ −
5. The results of the SSA arealso shown by the thick solid and dotted curves. appear in this regime are the subbands at | ω | ≈ . may be given by ǫ SDW ± ( k ) = ± q ˜ ǫ k + ∆ . Here ˜ ǫ k is a quasi-particle band, ∆ is an exchange splitting defined by ∆ = U eff | m | / | m | being a temporalamplitude of the magnetic moments with long-range AF correlations, and U eff is an effectiveCoulomb interaction.As seen in Fig. 7 the Mott-Hubbard bands in the strong Coulomb interaction regimeshow a weak momentum dependence around | ω | ≈ ±
11. It should be noted that the splittingbetween the upper and lower Hubbard bands is about 22, which is larger than U = 16 (avalue yielding the atomic limit). In the strong Coulomb interaction limit at half-filling, weexpect strong AF intersite correlations due to the super-exchange interaction J = 4 | t | /U .The energy to remove (add) electrons is then expected to be ǫ − z NN | J | ( ǫ + U + z NN | J | ), z NN being the number of nearest neighbors ( z NN = 6 in the present case). Thus the splitting isexpected to be U +2 z NN | J | instead of U . This formula yields the splitting 19 instead of U = 16.The former seems to be consistent with the value 22. We also find additional excitations at | ω | ≈
8. These sub-bands may be interpreted as local excitations of the lower and upperHubbard bands without AF correlations. . Phys. Soc. Jpn.
Full Paper
U=6M X Γ M R k -15 -10 -5 0 5 10 15 ω ρ (k, ω) Fig. 5. Single-particle excitation spectra along the high-symmetry lines at U = 6. The Fermi level isindicated by a bold line. U=10M X Γ M R k -15 -10 -5 0 5 10 15 ω ρ (k, ω) Fig. 6. Single-particle excitation spectra at U = 10. The total densities of states are presented in Fig. 8 for various values of U . For weakCoulomb interactions U .
6, the deviation of the spectra from those of the SCPM-0 isnegligible. When
U >
6, Mott-Hubbard bands appear and the DOS deviate from the SCPM-0. An example is shown in Fig. 9 for U = 10. We find there that the full self-consistencysuppresses the weight of the Mott-Hubbard bands and enhances the quasiparticle peaks. . Phys. Soc. Jpn. Full Paper
U=16M X Γ M R k -15 -10 -5 0 5 10 15 ω ρ (k, ω) Fig. 7. Single-particle excitation spectra at U = 16. Note that the vertical scale has been changed inorder to show detailed structure of excitations. D O S ( s t a t e s / a t o m s p i n ) Energy
U=16 U=12 U=8U=4
Fig. 8. The total DOS for U = 4 , , ,
16. The DOS for noninteracting system is shown by dottedcurve.
Resulting DOS is in-between the SSA and the SCPM-0. When
U >
10, a shadow banddevelops around | ω | = 3 as seen in Fig. 8. Furthermore, for U >
12 we find two peaks in eachMott-Hubbard band; one is due to the excitations without intersite AF correlations, anotheris due to the excitations with strong AF correlations.A remarkable point of the nonlocal excitation spectra is that the quasiparticle peak at the . Phys. Soc. Jpn.
Full Paper D O S ( s t a t e s / a t o m s p i n ) Energy
FSCPM SCPM-0SSA U=10
Fig. 9. The DOS in the FSCPM (solid curve), the SCPM-0 (dashed curve), and the SSA (dottedcurve) at U = 10. Fermi level reduces with increasing Coulomb interaction U . As shown in Fig 10, the ratio of ρ (0) /ρ SSA (0) to the SSA, monotonically decreases with increasing U , and reaches 0.59 when U = 20.Momentum-dependent effective masses m k along the high symmetry line are presented inFig. 11. In the FSCPM calculations, we obtained the self-consistent m k up to U = 20. Wefind that m k are approximately equal to those in the SCPM-0 for U .
6. But, for
U > m k as compared with those of the SCPM-0. Momentumdependence of m k becomes larger with increasing U . The mass has a minimum ( e.g. , 9.4 for U = 20) at Γ(0 , ,
0) and R ( π, π, π ) points, and has a maximum ( e.g. , 10.0 for U = 20) at( π, π/ , Z = ( N − P k m k ) − vs. U curve is shown in Fig. 10.Quasiparticle weight in the SSA monotonically decreases and vanishes for U c = 16 .
0. Thecurve in the SCPM-0 deviates upwards from the SSA. The curve in the FSCPM deviatesdownwards from the SCPM-0 beyond U ≈
5, and is between the SSA and the SCPM-0. Thepresent result suggests that the critical Coulomb interaction for the divergence of the effectivemass is U c ( m ∗ = ∞ ) &
30. We want to mention that for the Gutzwiller wave function a criticalCoulomb interaction U c ( m ∗ = ∞ ) exists only in infinite dimensions ( i.e. , in the SSA). We have also calculated the momentum distribution as shown in Fig. 12. The distributionsshow a jump at the Fermi surface, and extend beyond the surface. The basic behavior of thedistribution is similar to that in the SCPM-0. For U .
6, the curves h n k i agree with those ofthe SCPM-0. When U >
6, the FSCPM reduces h n k i of the SCPM-0 below the Fermi level, and . Phys. Soc. Jpn. Full Paper Z U ρ (0) / ρ SSA (0)FSCPM SCPM-0SSA
Fig. 10. The average quasiparticle weight vs. Coulomb interaction curves in the FSCPM (solidcurves), the SCPM-0 (dashed curve), and the SSA (thin solid curve). The DOS divided by theSSA one at the Fermi level is shown by dotted curve. Note that ρ SSA (0) equals the noninteractingDOS ρ (0). ρ (0) /ρ SSA (0) beyond U c (SSA) = 16 . ρ (0) /ρ (0). enhances it above the Fermi level suggesting increased localization due to full self-consistency.
4. Summary and Discussions
We have developed a fully self-consistent projection operator method (FSCPM) for non-local excitations by using the projection technique for the retarded Green function and theeffective medium. The method makes use of an energy-dependent Liouville operator ˜ L ( z )with a corresponding Hamiltonian for an off-diagonal effective medium ˜Σ ijσ ( z ). It allows usto calculate the nonlocal self-energy Λ ijσ ( z ) by using the incremental cluster expansion fromthe off-diagonal medium. Each term of the expansion is calculated from the memory functionfor clusters in a nonlocal effective medium. The latters are obtained by the renormalized per-turbation theory within the RPT-0. The off-diagonal effective medium ˜Σ ijσ ( z ) is determinedfrom a fully self-consistent condition { ˜Σ ijσ ( z ) = Λ ijσ ( z ) } . It is a generalization of the CPAas given in Eq. (44), i.e. , (cid:0) A † iσ (cid:12)(cid:12) G T G A † jσ (cid:1) = 0 where T denotes a scattering T -matrix fromthe off-diagonal medium and G is a resolvent for the Liouville operator L correspondingto the medium. In this way we can take into account the long-range intersite correlations asextensive as we want in each order of expansion. We obtain momentum-dependent spectra ofhigh resolution from the weak to the strong Coulomb interaction regime.The present theory reduces to the PM-CPA when we omit all off-diagonal matrix elements . Phys. Soc. Jpn. Full Paper Γ M R m k U=24681012141618U=20
Fig. 11. The momentum-dependent effective mass curves along the high-symmetry lines for variousCoulomb interaction U . The curves in the SCPM-0 are also shown by the dotted curves for U = 2 , , , ,
10, and 12 from the bottom to the top. of the effective medium ˜Σ ijσ ( z )( i = j ) and all the off-diagonal self-energy contributions. Thetheory reduces to a self-consistent theory (SCPM-0) when we omit the off-diagonal medium˜Σ ijσ ( z )( i = j ), but take into account all the off-diagonal self-energy contribution Λ ijσ ( z ).We have performed numerical calculations of the excitation spectra for the half-filledHubbard model on a simple cubic lattice by using the FSCPM within the two-site approxima-tion. We have obtained the self-consistent nonlocal self-energy up to the Coulomb interaction U ≈
20. We found that the FSCPM suppresses the amplitudes of local and nonlocal self-energy Λ ijσ ( z ) for an intermediate strength of Coulomb interactions, reduces the weight ofthe Mott-Hubbard bands, and enhances the quasiparticle peaks at the Fermi level in the aver-age DOS when they are compared with those in the SCPM-0. Moreover the FSCPM enhancesthe momentum-dependent effective mass m k as compared with the SCPM-0. Thus the Z − U curve is located between the SSA and the SCPM-0. These results indicate that the full self-consistency tends to suppress the nonlocal effects found in the SCPM-0. We suggest a criticalCoulomb interaction U c ( m ∗ = ∞ ) &
30 for the simple cubic lattice, which is much largerthan the SSA value U c = 16. In order to obtain the explicit value of U c , we have to take intoaccount larger clusters embedded in the off-diagonal medium.The FSCPM enables us to investigate the nonlocal excitation spectra in the stronglycorrelated region. There the quasiparticle bands becomes narrower and their weight becomessmaller. We found shadow band excitations at | ω | ≈ | ω | ≈ U/ . Phys. Soc. Jpn. Full Paper Γ M R 〈 n k 〉 U=481216U=20
Fig. 12. Momentum distribution along the high-symmetry lines in the FSCPM (solid curves). Thecurves in the SCPM-0 are also drawn by dotted curves for U = 4 , , that strong AF correlations can shift the Mott-Hubbard bands towards higher energies.In the present calculations, we investigated the effects of nonlocal correlations assuming theparamagnetic state. One has to extend the calculations to the AF case in the next step becausethe ground state of the three dimensional Hubbard model is believed to be antiferromagneticin general. Furthermore, the numerical results of calculations in the strongly correlated regionpresented here should be extended by taking into account the contributions from larger clustersin the incremental cluster expansion. Improvements of the self-energies for the clusters in theoff-diagonal effective medium are also left for future investigations towards more quantitativecalculations of the nonlocal excitations. Acknowledgment
This work was supported by Grant-in-Aid for Scientific Research (19540408). Numericalcalculations have been partly carried out with use of the Hitachi SR11000 in the Supercom-puter Center, Institute of Solid State Physics, University of Tokyo. . Phys. Soc. Jpn.
Full Paper
Appendix: Derivation of Approximate Expression for the Screened ClusterMemory Function (61)
An approximate expression (61) of the screened cluster memory function in the RPT-0can be obtained as follows by using the momentum representation of the operator space.For that purpose, we first express the local operator A † iσ by means of the creation andannihilation operators in the momentum representation as A † iσ = X k,k ′ ,k ′′ a † kσ δ ( a † k ′ − σ a k ′′ − σ ) h k | i ih k ′ | i ih i | k ′′ i . (A · h k | i i = h i | k i ∗ = 1 / √ N exp( i k · R i ). The operator a † kσ δ ( a † k ′ − σ a k ′′ − σ ) in Eq. (A ·
1) is theeigen state of L ( z ) = Q ˜ L ( z ) Q , i.e. , L ( z ) | a † kσ δ ( a † k ′ − σ a k ′′ − σ )) = [ ǫ σ + ǫ k + ˜Σ kσ ( z ) + ǫ k ′ + ˜Σ k ′ − σ ( z ) − ǫ k ′′ − ˜Σ k ′′ − σ ( z )] | a † kσ δ ( a † k ′ − σ a k ′′ − σ )) . (A · kσ ( z ) is the Fourier transform of ˜Σ ijσ ( z ), which is defined by ˜Σ kσ ( z ) = P j ˜Σ j σ ( z ) exp( i k · R j ).Substituting Eq. (A ·
1) into the approximate expression G (c)0 ijσ ( z ) = (cid:0) A † iσ (cid:12)(cid:12)(cid:0) z − L ( z ) (cid:1) − A † jσ (cid:1) and making use of the relation (A · G (c)0 ijσ ( z ) = X kk ′ k ′′ k k ′ k ′′ h i | k ih i | k ′ ih k ′′ | i ih k ′′ | i i ( G (c)0 ) k k ′ k ′′ σkk ′ k ′′ σ h k | j ih k ′ | j ih j | k ′′ i , (A · G (c)0 = χ ( z − L − v c ) − , (A · χ ) k k ′ k ′′ σ ′ kk ′ k ′′ σ = ( a † k σ ′ δ ( a † k ′ − σ ′ a k ′′ − σ ′ ) | a † kσ δ ( a † k ′ − σ a k ′′ − σ )) , (A · L ) k k ′ k ′′ σ ′ kk ′ k ′′ σ = (cid:0) ǫ σ + ǫ k + ˜Σ kσ ( z ) + ǫ k ′ + ˜Σ k ′ − σ ( z ) − ǫ k ′′ − ˜Σ k ′′ − σ ( z ) (cid:1) δ k k δ k ′ k ′ δ k ′′ k ′′ δ σσ ′ , (A · v c ) k k ′ k ′′ σ ′ kk ′ k ′′ σ = X lm ∈ c h − λ lmσ ˜Σ lmσ ( z ) h k | l ih m | k i δ k ′ k ′ δ k ′′ k ′′ − λ lm − σ ˜Σ lm − σ ( z ) h k ′ | l ih m | k ′ i δ k k δ k ′′ k ′′ + λ lm − σ ˜Σ lm − σ ( z ) h k ′′ | l ih m | k ′′ i δ k k δ k ′ k ′ i δ σσ ′ . (A · ·
3) depends on the choice of { λ ijσ } . When we choose { λ ijσ = 1 } and adopt the Hartree-Fock approximation when computing the static average in . Phys. Soc. Jpn. Full Paper
Eq. (A · G (c)0 ijσ ( z ) = X k,k ′ ,k ′′ h i | k ih i | k ′ ih k ′′ | i i χ ( ǫ kσ , ǫ k ′ − σ , ǫ k ′′ − σ ) h k | j ih k ′ | j ih j | k ′′ i z − ǫ kσ − ˜Σ kσ ( z ) − ǫ k ′ − σ − ˜Σ k ′ − σ ( z ) + ǫ k ′′ − σ + ˜Σ k ′′ − σ ( z ) , (A · χ ( ǫ k , ǫ k ′ , ǫ k ′′ ) = (1 − f ( ǫ k ))(1 − f ( ǫ k ′ )) f ( ǫ k ′′ ) + f ( ǫ k ) f ( ǫ k ′ )(1 − f ( ǫ k ′′ )) . (A · ǫ kσ = ǫ σ + ǫ k is the Hartree-Fock energy, f ( ω ) is the Fermi distribution function.A way to simplify in Eq. (A ·
8) the three-fold sum with respect to k might be to introducean approximate ˜Σ kσ ( z ) whose k dependence has been projected onto the Hartree-Fock energy ǫ kσ , as follows. ˜Σ σ ( ǫ kσ , z ) = R d k ′ δ ( ǫ kσ − ǫ k ′ σ ) ˜Σ k ′ σ ( z ) R d k ′ δ ( ǫ kσ − ǫ k ′ σ ) . (A · kσ ( z ) ≈ ˜Σ σ ( ǫ kσ , z ), Eq. (A ·
8) is expressed as G (c)0 ijσ ( z ) = Z dǫdǫ ′ dǫ ′′ ρ ijσ ( ǫ ) ρ ij − σ ( ǫ ′ ) ρ ji − σ ( ǫ ′′ ) χ ( ǫ, ǫ ′ , ǫ ′′ ) z − ǫ − ˜Σ σ ( ǫ, z ) − ǫ ′ − ˜Σ − σ ( ǫ ′ , z ) + ǫ ′′ + ˜Σ − σ ( ǫ ′′ , z ) . (A · ρ ijσ ( ǫ ) is the Hartree-Fock density of states defined by ρ ijσ ( ǫ ) = P k h i | k i δ ( ǫ − ǫ kσ ) h k | j i .On the other hand, in the case of { λ ijσ = 0 } ( i.e. , { λ ijσ = 1 } ), it is not easy to obtaindirectly a simplified expression of Eq. (A ·
3) because v c remains. However, H (c)I ( z ) in Eq.(48) becomes the Coulomb interaction in this case. The second-order perturbation of thetemperature Green function yields then an approximate expression. G (c)0 ijσ ( z ) = Z dǫdǫ ′ dǫ ′′ ρ (c) ijσ ( ǫ ) ρ (c) ij − σ ( ǫ ′ ) ρ (c) ji − σ ( ǫ ′′ ) χ ( ǫ, ǫ ′ , ǫ ′′ ) z − ǫ − ǫ ′ + ǫ ′′ . (A · ρ (c) ijσ ( ǫ ) is the DOS for a cavity Green function for the Hamiltonian (47) with { λ ijσ = 1 } .Therefore we obtain a simplified expression of the screened cluster memory function, whichis an interpolation between Eq. (A ·
11) for λ ijσ = 1 and Eq. (A ·
12) for λ ijσ = 0. G (c)0 ijσ ( z ) = A ijσ Z dǫdǫ ′ dǫ ′′ ρ (c) ijσ ( λ, ǫ ) ρ (c) ij − σ ( λ, ǫ ′ ) ρ (c) ji − σ ( λ, ǫ ′′ ) χ ( ǫ, ǫ ′ , ǫ ′′ ) z − ǫ − λ σ ˜Σ σ ( ǫ, z ) − ǫ ′ − λ − σ ˜Σ − σ ( ǫ ′ , z ) + ǫ ′′ + λ − σ ˜Σ − σ ( ǫ ′′ , z ) . (A · A ijσ , the densities of states for the cavity states ρ (c) ijσ ( λ, ǫ ), anda simplified self-energy ˜Σ σ ( ǫ, z ) are given by Eqs. (63), (65), and (67), respectively. . Phys. Soc. Jpn. Full Paper
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