University of Pennsylvania ScholarlyCommons
Department of Physics Papers Department of Physics
5-1-1996
Magnetic Anisotropies and General On-Site Coulomb Interactions in the Cuprates
Ora Entin-Wohlman
A. Brooks Harris University of Pennsylvania, [email protected]
Amnon Aharony
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Recommended Citation Entin-Wohlman, O., Harris, A., & Aharony, A. (1996). Magnetic Anisotropies and General On-Site Coulomb Interactions in the Cuprates. Physical Review B, 53 (17), 11661-11670. http://dx.doi.org/10.1103/ PhysRevB.53.11661
At the time of publication, author A. Brooks Harris was also affiliated withel T Aviv University, Tel Aviv, Israel. Currently, he is a faculty member in the Physics Department at the University of Pennsylvania.
This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/347 For more information, please contact [email protected]. Magnetic Anisotropies and General On-Site Coulomb Interactions in the Cuprates
Abstract This paper derives the anisotropic superexchange interactions from a Hubbard model for excitations within the copper 3d band and the oxygen 2p band of the undoped insulating cuprates. We extend the recent calculation of Yildirim et al. [Phys. Rev. B 52, 10 239 (1995)] in order to include the most general on-site Coulomb interactions (including those which involve more than two orbitals) when two holes occupy the same site. Our general results apply when the oxygen ions surrounding the copper ions form an octahedron which has tetragonal symmetry (but may be rotated as in lanthanum cuprate). For the tetragonal cuprates we obtain an easy-plane anisotropy in good agreement with experimental values. We predict the magnitude of the small in-plane anisotropy gap in the spin-wave spectrum of YBa2Cu3O6.
Disciplines Physics
Comments At the time of publication, author A. Brooks Harris was also affiliated withel T Aviv University, Tel Aviv, Israel. Currently, he is a faculty member in the Physics Department at the University of Pennsylvania.
This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/347 PHYSICAL REVIEW B VOLUME 53, NUMBER 17 1 MAY 1996-I
Magnetic anisotropies and general on-site Coulomb interactions in the cuprates
O. Entin-Wohlman School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
A. B. Harris School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel and Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6369
Amnon Aharony School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel ͑Received 12 June 1995͒ This paper derives the anisotropic superexchange interactions from a Hubbard model for excitations within the copper 3d band and the oxygen 2p band of the undoped insulating cuprates. We extend the recent calculation of Yildirim et al. ͓Phys. Rev. B 52, 10 239 ͑1995͔͒ in order to include the most general on-site Coulomb interactions ͑including those which involve more than two orbitals͒ when two holes occupy the same site. Our general results apply when the oxygen ions surrounding the copper ions form an octahedron which has tetragonal symmetry ͑but may be rotated as in lanthanum cuprate͒. For the tetragonal cuprates we obtain an easy-plane anisotropy in good agreement with experimental values. We predict the magnitude of the small
in-plane anisotropy gap in the spin-wave spectrum of YBa 2Cu 3O 6 .
I. INTRODUCTION formation about the Dzyaloshinskii-Moriya interaction,18,19 did not provide a satisfactory basis for understanding The magnetic anisotropies of the family of compounds the easy-plane anisotropy. Recognizing this fact, Yildirim 6,7 with structures similar to that of La 2CuO 4 ͑LCO͒ has been a et al. undertook an investigation of a model designed to subject of much current interest.1–8 These materials are calculate the anisotropy of the superexchange interaction so antiferromagnets,9 with a dominant easy-plane anisotropy.10 as to account correctly for the tetragonal symmetry of the The parent compound, LCO, is orthorhombic in the regime lattice. In their treatment all five crystal-field states of the of interest. In this structure the oxygen octahedra surround- copper 3d band ͑with a single hole on each copper ion͒ were ing each copper rotate by a small angle relative to their ori- taken into account, as well as all the ͑occupied͒ 2p states of entation in the tetragonal phase. This phase was found to the oxygen ions. As a result, the exchange interaction asso- have an antisymmetric Dzyaloshinskii-Moriya interaction, as ciated with each Cu-Cu bond had biaxial exchange anisot- allowed by the lack of inversion symmetry about the center ropy. ͑I.e., all three diagonal components of the exchange of a Cu-Cu bond.11 More recently, members of this family tensor were in general different.͒ In that work, a model of 12 which remain tetragonal, such as Sr 2CuO 2Cl 2, have Coulomb interactions was used which contained more terms been studied. In many such compounds, for example, than usual, but was still not completely general. Here we 13 14 15 Sr2CuO2Cl2, Nd2CuO4, and Pr2CuO4, the gap in the carry out the calculation with completely general on-site spin-wave spectrum at zero wave vector due to the easy- Coulomb interactions necessary to treat excited states with plane anisotropy has been found to be about 5 meV, just as two holes on a single Cu ion, or on a single oxygen ion, in LCO.16 assuming tetragonal site symmetry. As is known,20 such in- Since these systems involve copper ions in a 3d9 configu- teractions can be parametrized in terms of only three param- ration which have spin 1/2, the usually dominant mechanism eters, the Racah parameters, A, B, and C, with AӷB and of single-ion anisotropy does not come into play. Instead, AӷC. In Refs. 6 and 7 it was shown that for tetragonal anisotropy must be due to the anisotropy of the superex- symmetry the exchange anisotropy vanishes if BϭCϭ0. change interaction. Microscopic derivations of the anisotropy Here we obtain results which include all contributions to the energies have been carried out1–8 on the basis of Anderson’s exchange anisotropy correct to first order in both B and C. theory17 of kinetic superexchange, and Moriya’s extension18 Compared to the previous work, we find an additional con- to incorporate spin-orbit interactions. These have been tribution which slightly increases the biaxiality of the anisot- mostly confined to the consideration of the orthorhombic ropy, without sensibly changing the easy-plane anisotropy, phase and were based on terms requiring the existence of a which still agrees quite well with experimental results. Under distortion. As a result, these calculations produce anisotropy certain approximations, our results can be applied to the energies which are proportional to the distortion angle and orthorhombic phase of LCO. We should emphasize, how- which therefore vanish in the tetragonal phase. However, ever, that our aim in this paper is to calculate only the an- since the experiments cited above13–16 indicate that the mag- isotropy of the exchange interaction. This point is discussed nitude of the easy-plane anisotropy is independent of the at the end of Sec. III. structural distortion, these theories, while giving correct in- Briefly, this paper is organized as follows. In Sec. II we
0163-1829/96/53͑17͒/11661͑10͒/$10.0053 11 661 © 1996 The American Physical Society 11 662 O. ENTIN-WOHLMAN, A. B. HARRIS, AND AMNON AHARONY 53 describe the model for which we calculate the exchange an- in the notation21 of Table A26 of Ref. 20. The second term in isotropy. In Sec. III we describe the perturbative calcula- Eq. ͑1͒ gives the Hamiltonian of the oxygen ions. We assume tions, Sec. IV contains specific results for tetragonal symme- that the spin-orbit interaction on the oxygen is much smaller try, and in Sec. V we discuss and briefly summarize our than that on the copper and may be neglected. Hence we results. We will attempt to make this paper self-contained, write but readers wishing more details on the approach or the his- tory of this problem are advised to consult Ref. 7, Yildirim, † HOϭ ⑀npknpkn Harris, Aharony, and Entin-Wohlman ͑YHAE͒. That refer- kn͚ ence also contains the details of the spin-wave spectrum which results from a calculation of the exchange anisotropy. 1 † † ϩ Un n n n pkn p pkn pkn , 2 ͚ ͚ 1 2 3 4 1 kn2Ј 3 Ј 4 n1n2n3n4 Јk II. THE MODEL ͑4͒
We start by considering the ground state of the CuO 2 † plane, when the kinetic energy is completely neglected. In in which pkn creates a hole in one of the three p orbitals that case, there is a single 3d hole on each copper ion and the ͑denoted by n) on the oxygen at site k. Here the Coulomb oxygen 2p band is completely full. Since the spin of the Cu matrix element is obtained analogously to Eqs. ͑3͒. Finally, hole is arbitrary, this ground state is 2N-fold degenerate, Hhop describes the hopping between two neighboring oxy- where N is the number of Cu ions. The kinetic superex- gen and copper ions: change interaction is obtained as the effective interaction within this degenerate manifold when the kinetic energy is ik † Hhopϭ t␣ndi␣pknϩH.c., ͑5͒ treated perturbatively. For this purpose we write the Hamil- i͚␣ kn͚ tonian as ik in which t␣n is the hopping matrix element and H.c. denotes HϭHCuϩHOϩHhop . ͑1͒ the Hermitian conjugate of the preceding terms. As men- tioned, it is H that lifts the degeneracy of the 2N-fold Here, H (H ) describes the Hamiltonian of the copper hop Cu O degenerate ground state. ͑oxygen͒ ions and H is the kinetic energy ͑hopping be- hop In order to derive the effective magnetic Hamiltonian it is tween Cu and O ions͒. To describe the Cu ions we work in a convenient to start from the unperturbed Hamiltonian which representation in which the crystal-field Hamiltonian is diag- contains the spin-orbit interaction exactly. To achieve this, onal. Diagonalization of the crystal field potential yields five we introduce the unitary transformation which diagonalizes spatial d states, denoted by ͉␣͘, with site energies ⑀ . Each ␣ the single-particle part of H . As noted previously,6,7 the such energy is doubly degenerate due to spin ϭϮ1. In the Cu spin dependence of this transformation is fixed by tetragonal systems under consideration, the five d-states ͉␣ are deter- ͘ symmetry, so that we can write mined by the tetragonal symmetry. Even in the orthorhombic phase the site symmetry of the copper ions can be taken to be tetragonal, since the crystal-field is primarily generated by di␣ϭ͚ m␣a͓͑␣͔͒ЈciaЈ, ͑6͒ the neighboring oxygen ions, which, to a good approxima- aЈ tion, still form an octahedron, albeit a rotated one. For tetrag- onal symmetry we label these crystal field states as where ciaЈ destroys a hole in the exact eigenstate of the ͉0͘ϳx2Ϫy 2, ͉1͘ϳ3z2Ϫr2, ͉z͘ϳxy, ͉x͘ϳyz, and Hamiltonian which consists of the crystal-field and spin-orbit interactions. These states have a site label i, a state label ͑for ͉y͘ϳzx, where the z axis is perpendicular to the CuO 2 plane and ͉0͘ is the lowest energy single-particle state. Thus which we use roman letters͒, and a pseudospin index Ј.In Eq. ͑6͒ we define (␣) for each crystal-field state ͉␣͘ as follows: (␣)ϭ␣ is the Pauli matrix for ␣ϭx,y,z and H ϭ ⑀ d† d ϩ L ͓ជ͔ d† d Cu ͚ ␣ i␣ i␣ 2 ͚ ͚ ␣• Ј i␣ iЈ (␣)ϭI is the unit matrix, for ␣ϭ0,1. The 5 ϫ 5 matrix i␣ i␣ Ј m is the solution to
1 † † ϩ U␣␥␦di␣di di␥ di␦ , ͑2͒ 2␣␥͚␦ ͚ Ј Ј iЈ ⑀␣m␣bϩ L␣mbϭEbm␣b ͑7͒ ͚ † where di␣ creates a hole in the crystal-field state ␣ at site i with spin . Here the first term is the crystal-field Hamil- with ͚␣m␣*am␣bϭ␦ab , where ␦ is the Kronecker delta func- tonian. The second term is the spin-orbit interaction, where tion. Here L␣ are related to the matrix elements of the is the spin-orbit coupling constant and L␣ denotes the orbital angular momentum vector and are given by matrix elements of the orbital angular momentum vector be- tween the crystal-field states ␣ and . The last term is the L0zϭϪi, L0xϭL0yϭi/2, L1xϭϪL1yϭiͱ3/2, Coulomb interaction, where L ϭL ϭL ϭ/2. ͑8͒ e2 zx zy xy U␣␥␦ϭ͵ dr1 ͵ dr2␣͑r1͒͑r2͒ ␥͑r2͒␦͑r1͒ ͑3a͒ Matrix elements not listed and not obtainable using r12 L ϭL* are zero. When 0, each state ͉a͘ approaches ␣ ␣ → ϵ͑␣␦͉␥͒ ͑3b͒ one of the states ͉␣͘. Using this identification, the indices 53 MAGNETIC ANISOTROPIES AND GENERAL ON-SITE . . . 11 663
TABLE I. Values ͑in eV͒ of the parameters used to calculate the ¯ anisotropic exchange. For a discussion of these values, see YHAE. ⌬Uss s sЈ͑abcd͒ϭ ⌬U␣␥␦m␣*am*bm␥cm␦d Ј 1 1 ␣␥͚␦
AB C(pd) ⑀1 ⑀xϭ⑀y ⑀z ⑀p ⑀p ⑀p x y z ϫ͓͑␣͒͑␦͔͒ssЈ͓͑͒͑␥͔͒s s . ͑14͒ 0.1 7.0 0.15 0.58 1.5 1.8 1.8 1.8 3.25 3.25 3.25 1 Ј 1 ⌬U involves only the small Racah coefficients, B and C. The effective magnetic interaction H(i, j) between mag- a also run over the values 0,1,z,x, and y. We use the values netic ions i and j is found by perturbation expansion with of the parameters which are listed in Table I and are dis- respect to H1 . All the perturbation contributions to cussed in detail in YHAE. H(i, j) should involve products of matrix elements which begin and end within the 2N-fold degenerate ground-state III. PERTURBATION EXPANSION manifold of H0 , each state of which has one hole at each copper site, with arbitrary spin . Denoting such a ground We now divide the total Hamiltonian H into an unper- state by ͉ ͘, and concentrating on perturbation terms which turbed part, H , and a perturbation term H . The part 0 0 1 involve only two coppers i and j and one oxygen between H0 contains the single-particle Hamiltonians on the coppers them, it is convenient to replace ͉0͘ by and on the oxygens, and the leading on-site Coulomb poten- † † ͚ c c cj0 ci0 ͉0͘. This ensures that ͉0͘ indeed tials, 1 i0 j01 1 has exactly one hole on each copper i and j. Clearly the lowest-order contributions to the energy are of order ˜t4. U † 0 † † There are two possible channels in this order, which we de- H0ϭ͚ Eaciaciaϩ ͚ ciacib cibЈcia ia 2 iab Ј note by a and b. In channel a, the hole is transferred from Ј one of the coppers to the oxygen, then to the second copper. Afterwards, one of the holes on this second copper returns to UP ϩ ⑀ p† p ϩ p† p† p p . the empty copper. Hence in this channel there are two holes ͚ n kn kn 2 ͚ kn knЈЈ knЈЈ kn kn knnЈ on the copper in an intermediate state. In channel b, the hole Ј is transferred from one of the coppers to the oxygen, and ͑9͒ then a second hole is taken from another copper to the same oxygen. Then the two holes hop back, each to one of the For tetragonal site symmetry, we choose U0ϵU␣␣␣␣ initial coppers. Thus in channel b there are two holes on the ϭAϩ4BϩC and U PϵUnnnn . The perturbation Hamil- oxygen in an intermediate state. When the perturbation con- tonian includes the kinetic energy and the remaining Cou- tributions coming from the Coulomb potential ⌬H ͓Eqs. lomb interactions. Because of the transformation 6 , the C ͑ ͒ ͑10͒, ͑13͒ and ͑14͔͒ are included, the on-site interactions on hopping becomes spin dependent. Thus we have the copper are effective in channel a, and those on the oxy- gen appear in channel b. These contributions are of order ˜4 H1ϭHhopϩ⌬HC , ͑10͒ t ⌬HC . The perturbation contributions to order˜t4 are the same as in which those given by YHAE and are
† ϭ ˜ik ϩ ͑1͒ ik kj jk ki Hhop ͑tan͒ cia pkn H.c., ͑11͒ H ͑i, j͒ϭ gnn Tr͕ជ Si˜t ˜t ជ Sj˜t ˜t ͖, ͑15͒ ͚ia ͚ Ј Ј ͚ Ј • 0n n0 • 0nЈ nЈ0 knЈ nnЈ with the 2ϫ2 matrix in which
1 ˜t ik ϭ tik m* , 12 † ជ an ͚ ␣n ␣a͑␣͒ ͑ ͒ Siϭ ci0͑͒ ci0 , ͑16͒ ␣ 2 ͚ Ј Ј Ј and ⌬HC describes the perturbation parts of the on-site is the spin on the copper at site i in the orbital state ͉0͘, Coulomb potentials, ˜ik t 0n is the 2ϫ2 matrix given by Eq. ͑12͒ and
1 2 ¯ † † 2 1 1 1 1 ⌬HCϭ ⌬Uss s s ͑abcd͒ciasc cics cids 2 ͚ Ј 1 1Ј ibsЈ 1 1Ј gnnЈϭ ϩ ϩ . ͑17͒ ss s s U ⑀ ⑀ U ϩ⑀ ϩ⑀ ͩ⑀ ⑀ ͪ Ј 1 1Ј 0 n nЈ P n nЈ n nЈ iabcd The first term in gnnЈ results from channel a, while the sec- 1 † † ond arises from channel b. In deriving expression ͑15͒ we ϩ ⌬Un n n n pkn p pkn pkn , 2 ͚ 1 2 3 4 1 kn2Ј 3 Ј 4 have assumed that the single-particle energy E on the kЈ aϭ0 n1n2n3n4 copper is equal to zero. (1) ͑13͒ From the form of H (i,j) it is clear that it may yield anisotropic magnetic interactions only when the effective with hopping between the coppers, ˜tik˜tkj, involves spin flips. 11 664 O. ENTIN-WOHLMAN, A. B. HARRIS, AND AMNON AHARONY 53
When the hopping conserves the spin, the ˜t products are Dzyaloshinskii-Moriya interaction, as well as symmetric proportional to the 2ϫ2 unit matrix and the trace in Eq. ͑15͒ magnetic anisotropies.2,3 gives a result proportional to Si•Sj . This is indeed the case We now turn to the contributions of the Coulomb poten- for tetragonal symmetry, as we discuss in the next section.6,7 tial. In particular, we consider contributions to the magnetic 4 In the orthorhombic phase of La 2CuO 4 , however, the effec- energy which are of order ˜t and first order in ⌬HC . Fol- tive hopping between the coppers is accompanied by spin lowing the approach of YHAE one finds that the processes flip, and consequently H(1)(i,j) includes the antisymmetric from channel a ͑indicated by a subscript ‘‘a’’͒ yield
˜ik˜kj ˜jk ˜ki ⌬U␣␥␦ t0ntnb tan tn 0 H͑2͒͑i, j͒ϭ Tr ជ S ͑␣͒͑␦͒ Ј Ј Tr͕ជ S ͑͒͑␥͖͒m* m* m m a ͚ ͑U ϩE ͒͑U ϩE ͒ • i͚ ⑀ ͚ ⑀ • j ␣b 0 ␥0 ␦a ͫ a,b,Þ0 0 a 0 b ͩ n n nЈ nЈ ␣␥␦ ͭ ͮ ˜ik˜kj ˜jk ˜ki t0ntnb tan tn 0 ϪTr ជ S ͑␣͒͑␦͒ជ S ͑͒͑␥͒ Ј Ј m* m* m m ϩ͑i j͒ , ͑18͒ • i͚ ⑀ • j ͚ ⑀ ␣b 0 ␥a ␦0 ͭ n n nЈ nЈ ͮ ͪ ↔ ͬ
where i j denotes the sum of previous terms with i and j by U(2) , both products (␣)(␦) and ()(␥) are pro- ↔ ␣␥␦ interchanged. Here it is convenient to classify the Coulomb portional to , ϭx,y,z. In treating this contribution it is matrix elements into two classes, by writing convenient to use the identity
⌬U ϭU͑1͒ ϩU͑2͒ , ͑19͒ ជ ជ ␣␥␦ ␣␥␦ ␣␥␦ •Sϭ2SϪ•S. ͑21͒ where U(1) is nonzero only if (2) ␣␥␦ We see that the matrix elements U␣␥␦ give rise to magnetic interactions which depend on the Cartesian index . Rear- ϭ ϭ ͑␣͒ ͑␦͒, and ͑͒ ͑␥͒, ͑20a͒ ranging the terms in ͑18͒ and defining while U(2) is nonzero only if ␣␥␦ U͑2͒ ϪU͑1͒ ␣␥␦ ␣␥␦ * * Qabϭ ͚ m␣bm0m␥0m␦a , ͑␣͒͑␦͒ϭC1ϭC2͑͒͑␥͒, for ϭx,y,z, ␣␥␦ ͑U0ϩEa͒͑U0ϩEb͒ ͑20b͒ ͑22a͒ where C and C are constants which may be imaginary. 1 2 U͑2͒ This classification will be of immediate use. For the elements ␣␥␦ K ϭ ͑m* m* Ϫm* m* ͒ (1) ab ͚ U ϩE U ϩE ␣b 0 b ␣0 denoted U␣␥␦ , the products (␣)(␦) and ()(␥) are ␣␥␦ ͑ 0 a͒͑ 0 b͒ both proportional to the unit matrix. Therefore the first term in the square brackets of Eq. ͑18͒ vanishes, and one is left ϫ͑m␦am␥0Ϫm␦0m␥a͒, ͑22b͒ with the second term alone. For the matrix elements denoted we obtain
˜ik˜kj ˜jk ˜ki ˜ik˜kj ˜jk ˜ki t0ntnb tan tn 0 t0ntnb tan tn 0 H͑2͒͑i, j͒ϭ Q Tr ជ S ជ S Ј Ј ϩ K S Tr ជ S Ј Ј ϩ͑i j͒ . a ͚ ab • i͚ ⑀ • j͚ ⑀ ͚ ab j • i͚ ⑀ ͚ ⑀ a,bÞ0 ͫ ͭ n n nЈ nЈ ͮ ͭ n n nЈ nЈ ͮ ↔ ͬ ͑23͒ Note that the sum over state labels in Eqs. ͑22a͒ and ͑22b͒ is restricted by the conditions of Eqs. ͑20a͒ and ͑20b͒. One notes that the first term in Eq. ͑23͒ has a structure similar to that of H(1)(i,j) in Eq. ͑15͒ and therefore has the same magnetic symmetries. The second term, however, leads to magnetic anisotropy even for tetragonal symmetry, for which the effective hopping between the coppers is spin independent. This is elaborated upon in the next section. Finally we consider the processes in channel b, in which the two holes are on the oxygen in an intermediate state and thus experience the Coulomb interaction on the oxygen. These processes yield
1 1 1 1 ⌬Un nЈnnЈ ͑2͒ 1 1 ជ ˜ik ˜ki ជ ˜jk ˜kj Hb ͑i, j͒ϭ ͚ ϩ ϩ ͓Tr͕•Sit0n tn 0͖Tr͕•Sjt0n tn0͖ ⑀n ⑀n ⑀n ⑀n ͑UPϩ⑀nϩ⑀n ͒͑UPϩ⑀n ϩ⑀n ͒ 1 Ј 1Ј nn n nЈ ͩ Јͪͩ 1 1Јͪ Ј 1 1Ј Ј 1 1 ជ ˜ik ˜kj ជ ˜jk ˜ki ϪTr͕ Sit0n tn 0 Sjt0n tn0͖͔. ͑24͒ • 1 Ј • 1Ј 53 MAGNETIC ANISOTROPIES AND GENERAL ON-SITE . . . 11 665
To the order we work, the total magnetic interaction between Investigation of Eqs. ͑15͒, ͑23͒, and ͑24͒ reveals that one a pair of copper ions is given by can define an effective hopping between the copper ions, which is generally given by the product ˜tik˜tkj . The inter- an nЈb ͕ action H(2)(i,j) ͓Eq. ͑24͔͒ requires also the cases iϭ j.͖ We ͑1͒ ͑2͒ ͑2͒ b H͑i, j͒ϭH ͑i, j͒ϩHa ͑i, j͒ϩHb ͑i, j͒. ͑25͒ therefore start by examining this quantity. To this end we note that the nonzero hopping matrix elements between the tetragonal states ͉␣͘ on the coppers and the states ͉n͘ on the In the next section we examine H(i, j) in the tetragonal oxygen are t , t , t , and t for a bond along the x phase. In the orthorhombic phase the situation is more com- 0px 1px ypz zpy direction in the CuO plane, and analogously t , t , plicated. In principle, one should start by replacing the te- 2 0py 1py t , and t for a bond along y.6,7 Using now Eq. ͑6͒ we tragonal crystal-field states by those which are calculated in xpz zpx the presence of the orthorhombic distortion. A reasonable find approximation is to take account of this distortion by consid- ering the crystal-field states of the octahedron of oxygen ions assuming this octahedron to be rotated rigidly away from its orientation in the tetragonal phase. Then, to the extent that ˜ ˜ * t antnЈbϭ͚ t␣ntnЈm␣amb͑␣͒͑͒, ͑27͒ the crystal field is wholly determined by the octahedron of ␣ oxygen neighbors, it will be tetragonal in the rotated coordi- nate system fixed by the shell of oxygen neighbors. Then the where we have omitted the site indices for convenience. It result of Eq. ͑25͒ can be used. The major complication is that therefore follows that in the case nϭnЈ the quantity in Eq. the hopping matrix elements are those between rotated orbit- ͑27͒ is proportional to the 2ϫ2 unit matrix, namely, the als, as will be detailed elsewhere.22 effective hopping between the copper ions is not accompa- Some comments concerning the applicability of our re- nied by spin flip. This means that in the expressions for the sults should be made. As discussed in YHAE, we believe (1) (2) interactions H (i,j) and Ha (i,j) ͓Eqs. ͑15͒ and ͑23͔͒ we that the use of perturbation theory is justified for the calcu- may take all ˜t’s outside the trace. Consequently, the contri- lation of the anisotropy of the superexchange Hamiltonian. bution to the magnetic energy from Eq. ͑15͒ is isotropic, and 23 In contrast, for the isotropic terms it has been shown that so is that from the first term in Eq. ͑23͒, proportional to perturbation theory is not reliable, mainly because the ener- (2) Qab . The second term in Ha (i,j), which arises from gies of the excited states of the oxygen levels are not very U(2) , leads to an anisotropic magnetic interaction of the large in comparison to t. However, since the spin-orbit in- ␣␥␦ form of Eq. ͑26͒, with teraction takes place on the copper ions, this effect is much reduced in the anisotropic terms. In addition, there are con- tributions to the isotropic interactions which cannot be ob- ˜ ˜ ˜ ˜ tained by consideration of a single bond. In one of these, a t0ntnb tanЈtnЈ0 hole hops from a copper to one nearest-neighboring oxygen, Jϭ4 ͚ Kab͚ ͚ . ͑28͒ a,bÞ0 n ⑀n ⑀n then diagonally to a different nearest-neighbor oxygen, and nЈ Ј finally back to the original copper.23 Also, the effective spin (2) Hamiltonian acquires isotropic contributions from processes We now consider the magnetic symmetry of Hb (i,j), of order ˜t8 which involve a hole hopping around a plaquette which results from the on-site Coulomb potential on the oxy- of Cu ions.24 These plaquette interactions involve both two- gen. The only nonzero matrix elements ⌬U are25 n1nЈnnЈ spin interactions ͑between nearest- and next-nearest neigh- 1 bors͒ and four-spin interactions.
⌬U , ⌬U , ⌬U . ͑29͒ IV. TETRAGONAL SYMMETRY nnЈnЈn nnЈnnЈ nnnЈnЈ Here we study the effective magnetic Hamiltonian The first of these implies terms of the form ˜t0n˜tn0 ͓cf. Eq. H(i, j) ͓Eqs. ͑15͒ and ͑23͒–͑25͔͒ for the case of tetragonal ͑24͔͒, which are proportional to the unit matrix. As a result symmetry. For this symmetry the effective interaction which the first term in the square brackets of ͑24͒ disappears. The is bilinear in the spin operators must be of the form second, which yields an isotropic interaction, can be com- (1) bined into H (i,j) by redefining gnnЈ ͓Eq. ͑17͔͒ to be
H͑i, j͒ϭ͚ J͑i,j͒S͑i͒S͑j͒, ͑26͒ 2 1 1 1 1 2 gnnЈϭ ϩ ϩ . U0 ⑀n⑀nЈ UPϩ⌬UnnЈnЈnϩ⑀nϩ⑀nЈ ͩ⑀n ⑀nЈͪ where ϭx,y,z. Since we are interested in the anisotropic ͑30͒ part of the exchange interaction, we will drop any contribu- tions which we identify as being isotropic. Obviously, our For simplicity we have put the ⌬U in the denominator of this results cannot be used for the magnitude of an individual expression. The results of this paper are correct to first order J , but rather apply to the difference between two such in ⌬Hc . quantities. The other two matrix elements of Eq. ͑29͒ lead to 11 666 O. ENTIN-WOHLMAN, A. B. HARRIS, AND AMNON AHARONY 53
1 1 2 1 2 H͑2͒͑i, j͒ϭ ⌬U ϩ ͑Tr͕ជ S˜t ˜t ͖Tr͕ជ S˜t ˜t ͖ϪTr͕ជ S˜t ˜t ជ S˜t ˜t ͖͒ b ͚ ͫ nnЈnnЈͩ⑀ ⑀ ͪ ͩU ϩ⑀ ϩ⑀ ͪ • i 0n nЈ0 • j 0nЈ n0 • i 0n nЈ0 • j 0nЈ n0 nnЈ n nЈ P n nЈ nÞnЈ 4 1 ជ ˜ ˜ ជ ˜ ˜ ជ ˜ ˜ ជ ˜ ˜ ϩ⌬UnnЈnnЈ ͑Tr͕•Sit0ntnЈ0͖Tr͕•Sjt0ntnЈ0͖ϪTr͕•Sit0ntnЈ0•Sjt0ntnЈ0͖͒ . ͑31͒ ⑀n⑀nЈ ͑UPϩ2⑀n͒͑UPϩ2⑀nЈ͒ ͬ
˜ ˜ 2 The main point to note here is that although t0ntnЈ0 implies a in ͑22b͒ are of order provided that ͉b͘ϭ͉1͘ and spin flip in the sense that this product is proportional to one ͉a͘ϭ͉x͘ or ͉y͘. In all other cases the m products are at least of the Pauli matrices ͓cf. Eq. ͑27͔͒,itisthesame Pauli ma- of order 3. But with this choice ͓see Eqs. ͑34͔͒ the hopping trix for both products appearing in each term of Eq. ͑31͒. matrix elements provide another factor of , to render J to 3 (2) This makes the resulting interaction isotropic. For example, be of order . It follows that out of all nonzero U␣␥␦ , the ˜ ˜ ϭ 2 writing t0ntnЈ0ϰ , with x,y or z, we find that the spin ones that contribute to J up to order are dependence in each of the terms in Eq. ͑31͒ is proportional to ͑2͒ ͑2͒ ͑2͒ ͑2͒ U00ϭU00 , U11ϭU11 , ជ ជ ជ ជ Tr͕•Si͖Tr͕•Sj͖ϪTr͕•Si•Sj͖ϭ2Si•Sj , ͑32͒ ͑2͒ ͑2͒ ͑2͒ ͑2͒ U01ϭU01ϭU10ϭU10 . ͑35͒ where we have used the identity ͑21͒. Thus the perturbation We are now in position to calculate K . When both ͉a and processes for which there are two holes on the oxygen in the ab ͘ b are equal to 1 one finds intermediate state do not contribute to the magnetic anisot- ͉ ͘ ͉ ͘ ropy in tetragonal symmetry. This result holds because the 2 L1L1 ͑2͒ L0L0 ͑2͒ spin-orbit interaction on the oxygen has been discarded. K11ϭ 2 Ϫ 2 U00Ϫ 2 U11 The anisotropic exchange interactions in the tetragonal ͑U0ϩ⑀1͒ ͫ ͑⑀1Ϫ⑀͒ ⑀ phase thus come exclusively from channel a. Also, as pre- ϩ L1L0 L0L1 ͑2͒ viously noted, only the term proportional to K in Eq. ͑23͒ Ϫ U01 . ͑36͒ gives rise to anisotropy. Furthermore one notes that there is ⑀͑⑀1Ϫ⑀͒ ͬ no anisotropic contribution to J from the matrix elements When a ϭ 1 and b ϭ or vice versa, we have (2) ͉ ͘ ͉ ͘ ͉ ͘ ͉ ͘ U␣␥␦ for which ␣ϭ or ␥ϭ␦. Up to now, our results have been valid to all orders in the 2 L1 K ϭ͑K ͒*ϭ Ϫ U͑2͒ spin-orbit coupling, . We now obtain an explicit expression 1 1 ͑U ϩ⑀ ͒͑U ϩ⑀ ͒ ͫ ⑀ Ϫ⑀ 00 0 1 0 1 for the anisotropic part of J to leading order in , which 2 turns out to be O( ). To this end we use Eq. ͑7͒, from L0 Ϫ U͑2͒ , ͑37͒ which we get ⑀ 01ͬ 1 for ␣ϭa where we have kept terms up to order since the hopping 2 elements in this case will contribute another factor of . m aϭ L␣a ϩO͑ ͒, ͑33͒ ␣ for ␣Þa Finally, the case where both ͉a͘ and ͉b͘ are equal to ͉͘ ͭ ⑀aϪ⑀ 0 ␣ requires terms to order and is therefore where L for ␣Þa ͓see Eqs. ͑8͔͒ is of order . Next we ␣a 2 note that the summation indices a and b of Eq. ͑28͒ can take ͑2͒ KϭϪ 2 U00 . ͑38͒ the values ͉a͘,͉b͘ϭ͉1͘ or ͉a͘,͉b͘ϭ͉x͘,͉y͘ or ͉z͘. We there- ͑U0ϩ⑀͒ fore need to evaluate ͚n˜t0n˜tnb /⑀n for ͉b͘ϭ͉1͘ and for ͉b͘ϭ͉͘, where ϭx,y,z. Using Eqs. ͑12͒ and ͑33͒ we find Combining the results of Eqs. ͑34͒–͑38͒ we obtain ͑2͒ 2 2 2 ˜t ˜t t t 8U00 L1 t0ntn1 L0 t0nϪtn 0n n1 0n n1 2 ϭ ϩO͑ ͒, JϭϪ 2 ͚ Ϫ ͚ ͚ ͚ ͑U0ϩ⑀͒ ͯU0ϩ⑀1 n ⑀n ⑀ n ⑀n ͯ n ⑀n n ⑀n ͑2͒ 2 2 2 8U11 L0 t0ntn1 ˜t0n˜tn t0nϪt n L0 t0ntn1 L1 Ϫ 2 ͚ ͚ ϭ͚ ϩ͚ ͑U0ϩ⑀1͒ ͯ ⑀ n ⑀n ͯ n ⑀n n ⑀n ⑀ n ⑀n ⑀Ϫ⑀1 ͑2͒ 2 16U01 L0 t0ntn1 ϩO͑ ͒, ͑34͒ Ϫ ͑U ϩ⑀ ͒͑U ϩ⑀ ͒ ⑀ ͚n ⑀ where the t’s are the hopping matrix elements for the tetrag- 0 1 0 n onal states. Equations ͑33͒ and ͑34͒, in conjunction with Eq. 2 2 L0 t0nϪtn L1 t0ntn1 (2) ϫ Ϫ . 39 ͑22b͒, imply that the contributions of U␣␥␦ where more ͚ ͚ ͑ ͒ ͫ ⑀ n ⑀n U0ϩ⑀1 n ⑀n ͬ than two of the indices ␣, , ␥, and ␦ take the values x, y or z are at least of order 3. Take for example the element It remains to insert here the explicit expressions for the hop- (2) U1zxy . From Eqs. ͑22b͒ and ͑33͒ we see that the m products ping terms, Eqs. ͑34͒, in conjunction with the expressions for 53 MAGNETIC ANISOTROPIES AND GENERAL ON-SITE . . . 11 667
TABLE II. Values of (␣␦͉␥)ϭU␣␥␦ in terms of the Racah numerical evaluations of exact diagonalization within a Cu- coefficients, taken from Table A26 of Ref. 20 ͑Ref. 21͒. O-Cu cluster. For our numerical evaluation we use the pa- rameters of Table I. We note that all the nonzero hopping ␣ϭx ␣ϭy ␣ϭz matrix elements can be expressed in terms of (pd) and (␣0͉␣ 0͒ ϭ 3BϩC 3BϩCC 1 26 (pd)ϷϪ 2 (pd): (␣0͉␣1͒ϭ Ϫͱ3Bͱ3B 0 (␣1͉␣1͒ϭ BϩCBϩC4BϩC 3 t ϭϪͱ3t ϭ ͑pd͒, t ϭt ϭ͑pd͒. 0,px 1,px ͱ2 y,pz 0,py the angular momentum matrix elements, L␣ , Eqs. ͑8͒.To ͑41͒ be specific, we consider a bond along x, and use the values Thereby we find ͑in eV͒ of U␣␥␦ϵ(␣␦͉␥) as listed in Table II. One then finds
2 t t t2 2 Ϫ ϭ Ϫ ϭ 2 ͑3BϩC͒ ͱ3 0px 1px 1 0px ⌬JϵJav Jzz 30, ␦JϵJЌ Jʈ 41, ͑42͒ JxxϭϪ 2 Ϫ ϩ ͑U0ϩ⑀x͒ ͫ U0ϩ⑀1 ⑀p ⑀x ⑀p ͬ x x where Javϭ(JxxϩJyy)/2, JЌϭJyy , and JʈϭJxx . Note that 2 2 the gap in the spin-wave spectrum due to the easy-plane ϩ t0p t1p 1/2 2 ͑B C͒ 1 x x anisotropy is proportional to (⌬J) whereas, as explained Ϫ 2 ͑U0ϩ⑀1͒ ͫ⑀x ⑀p ͬ by YHAE, the gap due to the anisotropy within the basal x plane is proportional to ␦J. ͑This statement applies to sys- 2 4 ͑ϪBͱ3͒ 1 t0p t1p tems like YB Cu O .InSrCuO Cl the in-plane anisot- Ϫ x x 2 3 6 2 2 2 ͑U ϩ⑀ ͒͑U ϩ⑀ ͒ ⑀ ⑀ ropy will have contributions from dipolar interactions.͒ In 0 1 0 x x px YHAE, the terms in Jxx and Jyy involving Bͱ3 were not t2 t t included because only Coulomb matrix elements involving at 1 0px ͱ3 0px 1px ϫ Ϫ ; ͑40a͒ most two orbitals were kept. To see the effect of the addi- ͫ⑀x ⑀p U0ϩ⑀1 ⑀p ͬ x x tional terms in the present work, we give, for comparison, the perturbative results of YHAE: ⌬Jϭ30 eV and 2 t t t2 t2 2 2 ͑3BϩC͒ ͱ3 0px 1px 1 0px ypz ␦Jϭ 26 eV. ͑The results from exact diagonalization on a JyyϭϪ 2 ϩ Ϫ Cu-O-Cu cluster were ⌬Jϭ31 eV and ␦Jϭ31 eV.͒ ͑U0ϩ⑀y͒ ͫU0ϩ⑀1 ⑀p ⑀y ͩ ⑀p ⑀p ͪͬ x x z
2 t t 2 2 ͑BϩC͒ 1 0px 1px V. CONCLUSIONS Ϫ 2 ͑U0ϩ⑀1͒ ͫ⑀y ⑀p ͬ x In view of the results of Ref. 4, YHAE already demon-
2 2 2 strated that the out-of-plane anisotropy in the superexchange 4 ͑Bͱ3͒ 1 t0p t1p 1 t0p typ Ϫ x x x Ϫ z interaction between Cu ions is dominated by Coulomb ex- ͑U0ϩ⑀1͒͑U0ϩ⑀y͒ ⑀y ⑀p ͫ⑀y ͩ ⑀p ⑀p ͪ change terms. Therefore, the simplified Coulomb interaction x x z of Eq. ͑9͒, used widely in the literature, is insufficient to t t ͱ3 0px 1px explain this anisotropy. YHAE then considered the anisot- ϩ ; ͑40b͒ ropy due to the simplest exchange terms, like U␣␣ , which U0ϩ⑀1 ⑀p ͬ x involve only two orbitals, and thereby obtained an anisotropy in the superexchange interaction whose value agreed with 2 t2 t2 2 8 C 1 0px zpy experiments. JzzϭϪ 2 Ϫ ͑U0ϩ⑀z͒ ͫ⑀z ͩ ⑀p ⑀p ͪͬ In the present paper we included all the Coulomb terms x y which are allowed by tetragonal site symmetry, and found 2 t t 2 that the additional terms, involving (␣0 ␣1), practically do 8 ͑4BϩC͒ 1 0px 1px ͉ Ϫ 2 . ͑40c͒ not affect the out-of-plane anisotropy ⌬J. The in-plane ͑U0ϩ⑀1͒ ͫ⑀z ⑀p ͬ x single-bond anisotropy, ␦J, is somewhat larger than before. An alternative derivation of these results is given in the Ap- YHAE showed that the in-plane gap in the spin-wave spec- pendix. trum is a manifestation of quantum zero-point fluctuations One should keep in mind that Eq. ͑25͒ holds for a single and is proportional to ␦J. Using the results of our present bond. To obtain the effective magnetic Hamiltonian of the calculation in Eq. ͑82͒ of YHAE we estimate the in-plane entire CuO 2 plane, one has to sum the magnetic interaction gap in the spin-wave spectrum to be about 33 eV ϳ 0.27 Ϫ1 H(i, j) over all bonds, allowing for the crystal symmetry. cm . The direct observation of this gap would provide an Within a classical approximation the resulting exchange interesting and significant test of our calculations. Hamiltonian of the crystal has only an easy-plane anisotropy. To obtain the fourfold anisotropy within the easy plane re- ACKNOWLEDGMENTS quires a consideration of the spin-wave zero-point motion.7 Now we evaluate these results numerically. In this con- We acknowledge support from the U.S.-Israel Binational nection it is useful to emphasize that for their less general Science Foundation. A.B.H. was also supported in part by model YHAE have shown that the perturbative results for the the U.S. Israel Education Foundation and by the National anisotropy in the J’s agrees to within about 10% with the Science Foundation under Grant No. DMR-91-22784. 11 668 O. ENTIN-WOHLMAN, A. B. HARRIS, AND AMNON AHARONY 53
APPENDIX: ALTERNATIVE CALCULATION the Hermitian conjugate of type two and therefore are subject OF THE EXCHANGE ANISOTROPY to the same argument. Thus all the anisotropic terms are contained in the expression in Eq. ͑A2͒. In this appendix we obtain the result for J for tetrago- If T denotes hopping from i to j, we can write Eq. ͑A2͒ nal symmetry by direct application of perturbation theory. ij as We take the unperturbed Hamiltonian to be the same as that of Eq. ͑9͒, except that we do not include in it the spin-orbit † 1 1 interaction on the copper ions. ͓See Eq. ͑2͒.͔ That term is H1ϭ2 Q␣ C Q␣ , ͑A3͒ ␣ϭ͚x,y,z E E now to be included into H1 . For simplicity we only con- sider the contribution to the anisotropy from channel a.A where similar calculation to that given here shows that channel b gives no anisotropy. Since we only consider channel a,we 1 1 Q ϭ V T ϩT V . ͑A4͒ can eliminate the oxygens completely from the problem by ␣ ͩ ␣ E ij ij E ␣ͪ ¯ introducing an effective hopping matrix element t ␣ between crystal-field states on near-neighboring copper ions. For in- Here V␣ϭ͚hL␣(h)s␣(h), where the sum is over the two stance, holes, h. Acting on the ground state, the operator Q␣ pro- duces two final states, depending on whether the orbital ͉0͘ ¯t ϭt2 /⑀ , ¯t ϭt t /⑀ , ¯t ϭ0, or ͉1͘ is occupied. So we define 00 0,px px 01 0,px 1,px px xx 1 ¯ 2 ¯ 2 ͑␥͒ t yyϭty,p /⑀p , t zzϭtz,p /⑀p , ͑A1͒ ͓Q ͔ ϭ͗0͉d d V T z z y y ␣ ,;Ј,Ј i,␥, i,␣,ͩ ␣ E ij as in Eq. ͑53͒ of YHAE. Accordingly, we need to work to 1 fifth order in perturbation theory, where the perturbations are † † ϩTij V␣ d d ͉0͘, ͑A5͒ V, the spin-orbit interaction, which we take to second order, E ͪ i,0,Ј j,0,Ј HhopϵT , the hopping between copper ions, which we take where ␥ can be 0 or 1. Then to second order, and ⌬HCϭC on the copper ions, which we take to first order. We also use the fact that U␣␥␦ is only H ϭ2 Q͑␥Ј͒ † ␥Ј␥ Q͑␥͒ ⌬Ϫ1⌬Ϫ1 , ͑A6͒ nonzero when (␣)()ϳ(␥)(␦). 1 ͚ ͓ ␣ ͔ ͓C ␣ ͔͓ ␣ ͔ ␥␣ ␥Ј␣ In fifth-order perturbation theory there are, a priori,30 ␣␥␥Ј different ways to order these perturbations. But obviously the where ⌬␥␣ϭ⑀␥ϩ⑀␣ϩU0 and Coulomb perturbation can only appear when the two holes are on the same site. So we have only to consider how to ␥Ј␥ ϭ 0 d d ⌬ d† d† 0 ͓C ␣ ͔ЉЉ;ЈЈ ͗ ͉ i␥ЈЉ i␣Љ HC i␣Ј i␥Ј͉ ͘ insert two powers of V into the sequence TCT. There are basically three types of terms to consider. The first is ϭ␦Љ,Ј␦Ј,Љ͗␣␥Ј͉⌬HC͉␣␥͘
1 1 1 1 1 1 Ϫ␦Љ,Ј␦Љ,Ј͗␣␥Ј͉⌬HC͉␥␣͘. ͑A7͒ H ϵ V T ϩT V C V T ϩT V , ͑A2͒ 1 ͩ E E ͪ E E ͩ E E ͪ Now we introduce the notation for direct products
͓AB͔Ј,Ј;,ϭAЈBЈ , ͑A8͒ where E denotes the appropriate energy denominator. The second type of terms are those in which the two V’s are both so that to the left of C , and the third type are terms which are the ␥Ј,␥ Hermitian conjugates of the second type, i. e., those in which C ␣ ϭ͓O ͔͗␣␥Ј͉⌬HC͉␣␥͘ϵ͓O͔͑␣␥͉␣␥Ј͒ ͑A9͒ the two V’s are to the right of . C ជ ជ That the second type of term does not lead to any anisot- in the notation of Eqs. ͑3͒. Also ͓O ͔ϭ͓IIϪ•͔/2. ropy can be established by the following argument. Suppose From Appendix H of YHAE we also take the results we show that these terms vanish when applied to any triplet ͓Q͑0͔͒ϭ͑C ϩC ͓͒I ͔ϪC ͓I ͔͓O͔ ͑A10͒ spin state. That would imply that these terms are of the form ␣ 1 2 ␣ 2 ␣ (1/4)ϪSi•Sj , which obviously gives rise to no anisotropy. and The second type of term has first ͑reading from right to left͒ ͑1͒ T acting on the triplet state. That will put the two holes, ͓Q␣ ͔ϭC3͓I ␣͔, ͑A11͒ which were initially distributed one on site i, the other on where site j, onto the same site, perforce one in state ͉0͘, the other in state ͉1 . Now apply the Coulomb perturbation. This op- ¯ ¯ ¯ ͘ ͑t␣␣Ϫt00͒L0␣ t01L1␣ erator can leave the two holes in the same states, viz. one in C ϩC ϭ ϩ ͑A12͒ 1 2 ͫ ⑀ ͑⑀ ϩU ͒ͬ ͉0͘ and the other in ͉1͘. In fact, by our observation on the ␣ 1 0 form of U␣␥␦ and by the fact that parallel spins are not and allowed in the same spatial orbital, there are no other final states. But such a diagonal matrix element of U was treated ⑀1ϩ⑀␣ϩU0 C ϭϪ¯t L . ͑A13͒ by YHAE and found to give no anisotropy in the second type 3 01 0␣ͩ ⑀ ͑⑀ ϩU ͒ ͪ ␣ 1 0 of term. So we conclude that the second type of term does not give rise to any anisotropy. Terms of the third type are Thus we obtain the result 53 MAGNETIC ANISOTROPIES AND GENERAL ON-SITE . . . 11 669
2͑␣0͉␣0͒ H͑i, j͒ϭ͚ 2 ͓C1*IIϩC2*IIϪC2*O͔͓I ␣͔͓O͔͓I ␣͔͓C1IIϩC2IIϪC2O͔ ␣ ͭ ⌬0 2͑␣1͉␣1͒ 2͑␣0͉␣1͒ ϩ 2 ͑C3*͓I␣͔͓O͔͓I ␣͔C3͒ϩ C3*͓I ␣͔͓O͔͓I ␣͔͓C1IIϩC2IIϪC2O͔ ⌬1 ⌬0⌬1 2͑␣0͉␣1͒ ϩ ͓C1*IIϩC2*IIϪC2*O͔͓I ␣͔͓O͔͓I ␣͔C3 . ͑A14͒ ⌬0⌬1 ͮ Now we use the equality
1 ͓I ͔͓O͔͓I ͔ϭ ͓IIϪ2 ϩជ ជ͔. ͑A15͒ ␣ ␣ 2 ␣ ␣ • So, dropping isotropic terms, we have21
2͑␣0͉␣0͒ 2͑␣1͉␣1͒ H͑i, j͒ϭ͚ Ϫ 2 ͓C1*IIϩC2*IIϪC2*O͔͓␣␣͔͓C1IIϩC2IIϪC2O͔Ϫ 2 C3*͓␣␣͔C3 ␣ ͭ ⌬0␣ ⌬1␣ 2͑␣0͉␣1͒ 2͑␣1͉␣0͒ Ϫ C3*͓␣␣͔͓C1IIϩC2IIϪC2O͔Ϫ ͓C1*IIϩC2*IIϪC2*O͔͓␣␣͔C3 . ͑A16͒ ⌬0␣⌬1␣ ⌬0␣⌬1␣ ͮ Now use
ជ ជ 2͓␣␣͔͓O͔ϭ2͓O͔͓␣␣͔ϭ͓•ϪII͔. ͑A17͒ Thus the terms involving O are isotropic. So the anisotropic terms are correctly given by Eq. ͑26͒ with 8 0 0 8 1 1 0 1 ͑ ͉ ͒ 2 ͑ ͉ ͒ 2 ͑ ͉ ͒ JϭϪ 2 ͉C1ϩC2͉ Ϫ 2 ͉C3͉ Ϫ16 Re C3*͑C1ϩC2͒. ͑A18͒ ⌬0 ⌬1 ⌬0⌬1 Now we use Eqs. ͑A12͒ and ͑A13͒ to write ¯ ¯ ¯ 2 ¯ 8͑0͉0͒ ͑t00Ϫt͒L0 t01L1 8͑1͉1͒ 16͑0͉1͒t10L0 J ϭ Ϫ ϩ ¯t2 L2 Ϫ ͑U ϩ⑀ ͒2 ͫ ⑀ ͑⑀ ϩU ͒ͬ ⑀2 ͑⑀ ϩU ͒2 01 0 ⑀ ͑U ϩ⑀ ͒͑⑀ ϩU ͒ 0 1 0 1 0 0 1 0 ¯ ¯ ¯ ͑tϪt00͒L0 t01L1 ϫ ϩ . ͑A19͒ ͫ ⑀ ͑⑀ ϩU ͒ͬ 1 0 This result reproduces Eqs. ͑40͒ of the text.
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