Hubbard-Like Hamiltonians for Interacting Electrons in S, P, and D Orbitals

PHYSICAL REVIEW B 93, 075101 (2016)

Hubbard-like Hamiltonians for interacting electrons in s, p,andd orbitals

M. E. A. Coury,1,2,* S. L. Dudarev,2 W. M. C. Foulkes,1 A. P. Horsﬁeld,1,† Pui-Wai Ma,2 and J. S. Spencer1 1Imperial College London, London SW7 2AZ, United Kingdom 2CCFE, Culham Science Centre, Abingdon, Oxfordshire OX14 3DB, United Kingdom (Received 20 July 2015; revised manuscript received 2 November 2015; published 1 February 2016)

Hubbard-like Hamiltonians are widely used to describe on-site Coulomb interactions in magnetic and strongly- correlated solids, but there is much confusion in the literature about the form these Hamiltonians should take for shells of p and d orbitals. This paper derives the most general s, p,andd orbital Hubbard-like Hamiltonians consistent with the relevant symmetries, and presents them in ways convenient for practical calculations. We use the full conﬁguration interaction method to study p and d orbital dimers and compare results obtained using the correct Hamiltonian and the collinear and vector Stoner Hamiltonians. The Stoner Hamiltonians can fail to describe properly the nature of the ground state, the time evolution of excited states, and the electronic heat capacity.

DOI: 10.1103/PhysRevB.93.075101

I. INTRODUCTION J . Unfortunately, this extra term is not rotationally invariant in spin space. The starting point for any electronic structure calculation is In 1966, Roth [11] also considered two orbitals and the Hamiltonian that describes the dynamics of the electrons. two parameters, producing a Hamiltonian that is rotationally Correlated electron problems are often formulated using invariant in spin space. In 1969 Caroli et al. [12] corrected Hamiltonians that involve only a small number of localized Anderson’s Hamiltonian for two orbitals, again making it atomic-like orbitals and use an effective Coulomb interaction rotationally invariant in spin space. However, the Hamiltonians between electrons that is assumed to be short ranged, and of Roth and Caroli et al. do not satisfy rotational invariance thus confined to within atomic sites. However, there is in orbital space. This was corrected by Dworin and Narath in widespread variation in the literature over the form chosen 1970 [13], who produced a Hamiltonian that is rotationally for these Hamiltonians, and they often do not retain the invariant in both spin and orbital space, equivalent to the correct symmetry properties. Here we derive the most general vector Stoner Hamiltonian. They generalized the multiorbital Hamiltonians of this type that are rotationally invariant and Hamiltonian to include all 5 d orbitals, although they only describe electrons populating s, p, and d orbitals. included two parameters: the on-site Hartree integral, U, and Despite often missing terms, multiorbital Hamiltonians the on-site exchange integral, J . with on-site Coulomb interactions have been used successfully The next contribution to the multiorbital Hamiltonian was to describe strong correlation effects in systems with narrow by Lyon-Caen and Cyrot [14], who in 1975 considered a two- bands near the Fermi energy. Applications include studies orbital Hamiltonian for the eg d orbitals. They introduced U0 − of metal-insulator transitions [1], colossal magnetoresistant U = 2J , where U0 = Vαα,αα is the self-interaction term, U = manganites [2], and d band superconductors such as copper Vαβ,αβ is the on-site Hartree integral, and J = Vαβ,βα is the on- oxides [3] and iron pnictides [4]. The Hubbard Hamiltonian has site exchange integral, important for linking these parameters. also been used to simulate graphene [5,6], where both theory Here, V is the Coulomb interaction between electrons, and α and experiment have found magnetic ordering in nanoscale and β are indices for atomic orbitals. Lyon-Caen and Cyrot structures [7,8]. However, the lack of consistency in the forms † † also included the pair excitation, J cÎ1,↑cÎ1,↓cÎ2,↓cÎ2,↑, where used for the Hamiltonians is problematic. Indeed, according to † cˆ and cˆ are fermionic creation and annihilation operators. Dagotto [2], “the discussions in the current literature regarding The new term moves a pair of electrons from a completely this issue are somewhat confusing.” occupied orbital, 2, on site I, to a completely unoccupied The Hubbard Hamiltonian [9] is both the simplest and best orbital, 1, on site I. Such double excitations are common in known Hamiltonian of the form we are considering, and is the literature [15–18]. valid for electrons in s orbitals. Hubbard was not the first In 1978, Castellani et al. [19] wrote down the three-orbital to arrive at this form; two years earlier, in 1961, Anderson 1 Hamiltonian for the t2g d orbitals in a very clear and concise wrote down a very similar Hamiltonian [10] for electrons in way: its form is exactly what we find here for p orbital s orbitals with on-site Coulomb interactions. Furthermore, in symmetry. They too make note of the equation U0 − U = 2J the appendix of the same paper, the Hamiltonian was extended and include the double excitation terms. to two orbitals, introducing the on-site exchange interaction, The next major contribution was by Oles´ and Stollhoff [20], who introduced a d orbital Hamiltonian with three independent parameters. The third parameter, J , represents the difference * [email protected] between the t2g and the eg d orbital exchange interactions; they † Corresponding author: a.horsfi[email protected] estimated J to be 0.15J . In our discussion of d orbitals, we

Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of 1 this work must maintain attribution to the author(s) and the published In this notation, t2g refers to cubic harmonic d orbitals dxy,dyz and article’s title, journal citation, and DOI. dxz and eg refers to cubic harmonic d orbitals d3z2−r2 and dx2−y2 .

2469-9950/2016/93(7)/075101(14) 075101-1 Published by the American Physical Society M. E. A. COURY et al. PHYSICAL REVIEW B 93, 075101 (2016) adopt their notation. Unfortunately, their Hamiltonian is not vector Stoner Hamiltonian. In Sec. III, we show the ground rotationally invariant in orbital space. state results for both our Hamiltonian and the vector Stoner In 1989, Nolting et al. [21] referenced the Hamiltonian Hamiltonian for p and d orbital dimers. We find that, although proposed by Oles´ and Stollhoff but discarded the double the results are often similar, there are important qualitative and excitation terms and ignored the J term. Their Hamiltonian quantitative differences. In Sec. IV, we show the differences will be called the vector Stoner Hamiltonian in this paper. that emerge when evolving a starting state for a p orbital dimer Since then, most papers have used a Hamiltonian similar under our Hamiltonian and the vector Stoner Hamiltonian. to that of Nolting et al., with only two parameters; see, We also calculate a thermodynamic property, the electronic for example, Refs. [22–24]. The vector Stoner Hamiltonian heat capacity of a p orbital dimer, and obtain significantly is often simplified by replacing the rotationally invariant different results when using our Hamiltonian and the vector 2 = · 2 moment-squared operator, mˆ mˆ mˆ ,bymˆ z, yielding the Stoner Hamiltonian. The eigenstates of all Hamiltonians are collinear Stoner Hamiltonian. found using the full configuration interaction (FCI) method as The multiorbital Hamiltonians derived here consist of one- implemented in the HANDE code [31]. particle hopping (intersite) integrals and two-electron on-site Coulomb interactions. The on-site Coulomb interactions for II. THEORY s, p, and d orbital symmetries are presented in a clear tensor notation, as well as in terms of physically meaningful, A. General form of the Hamiltonian rotationally invariant (in both orbital and spin space), operators We start with the general many-body Hamiltonian of that include all of the terms from the previous papers as well electrons in a solid, expressed in second-quantized form as additional terms of the same order of magnitude that have using a basis of orthonormal localized one-electron orbitals not been previously considered. The d orbital Hamiltonian and ignoring terms that do not include electronic degrees of presented in section II D corrects the d orbital Hamiltonian freedom [32]: proposed by Oles´ and Stollhoff [20] by restoring full rotational ˆ = † invariance in orbital space. For an s orbital Hamiltonian, H tIαJβcÎα,σcˆJβ,σ the on-site Coulomb interactions may be described using IαJβ σ one independent parameter. Two independent parameters are + † † required for p orbital symmetry, and three for d orbital VIαJβ,KχLγcÎα,σcˆJβ,ξcˆLγ,ξ cˆKχ,σ. (1) symmetry. Furthermore, the method presented can be extended IJKLαβχ γ σξ to f and g orbital symmetry and generalized to atoms with † valence orbitals of multiple different angular momenta. Here, cÎα,σ and cÎα,σ are creation and annihilation operators, The use of a restricted basis set and the neglect of the two- respectively, for an electron in orbital α on site I with spin electron intersite Coulomb interactions means that screening σ . Upper case Roman letters such as I, J, K, and L refer effects are missing. Thus, even though the parameters that to atomic sites, the lower case Greek letters α, β, χ, and γ define the s, p, and d Hamiltonians appear as bare on-site represent localized orbitals, and σ and ζ indicate spin (either up Coulomb integrals, screened Coulomb integrals should be used or down). The one-electron contributions to the Hamiltonian when modeling real systems; this significantly reduces the are encapsulated in the tIαJβ term that includes the electronic values of the parameters. kinetic energy and electron-nuclear interaction. The electron- Although it is possible to find the on-site Coulomb integrals electron interaction terms are represented by the Coulomb using tables of Slater-Condon parameters [25,26] or the closed matrix elements VIαJβ,KχLγ. forms for the integrals of products of three spherical harmonics We now make two approximations: we only retain electron- found by Gaunt [27] and Racah [28], the point of this electron Coulomb interactions on-site and we restrict the basis paper is to remove that layer of obscurity and present model to a minimal set of localized angular momentum orbitals. Hamiltonians written in a succinct form with clear physical Thus we have just one s orbital per site for the s-symmetry meaning. The Slater-Condon parameters and Gaunt integrals Hamiltonian, three p orbitals per site for the p-symmetry were not used in the derivations of the on-site Coulomb Hamiltonian, and five d orbitals per site for the d-symmetry Hamiltonian. The Hamiltonian can then be written as follows: interaction Hamiltonians presented here; however, they did provide an independent check to confirm that the tensorial ˆ ≈ † H tIαJβcÎα,σcˆJβ,σ forms derived here are correct. The link between the spherical IαJβ σ harmonics Coulomb integrals, the Slater-Condon parameters 1 † † and the Racah parameters is explained clearly in Ref. [29]. The + V I cˆ cˆ cˆ cˆ , (2) 2 αβ,χ γ Iα,σ Iβ,ξ Iγ,ξ Iχ,σ transformation to cubic harmonics is straightforward; an ex- I αβχ γ σξ ample of which has been included for the p orbitals in Sec. II C. I = We note that Rudzikas [30] has derived rotationally where Vαβ,χ γ VIαIβ,IχIγ. To improve readability, from this invariant Hamiltonians for p and d orbital atoms. However, point onwards we drop the site index, I, from the Coulomb his derivation is different from ours and his d orbital integrals and creation and annihilation operators as we shall Hamiltonian is partially expressed in terms of tensors; the always be referring to on-site Coulomb interactions. form we present in this paper is written in terms of physically The on-site Coulomb interaction, Vαβ,χ γ , is a rotationally meaningful operators. invariant tensor if the atomic-like orbitals are angular momen- The p and d symmetry Hamiltonians are described in Sec. II tum eigenstates. To demonstrate this we note that applying (see Appendix B for full derivations) and compared with the the rotation operator Rˆ to an atomic-like orbital φα of angular

075101-2 HUBBARD-LIKE HAMILTONIANS FOR INTERACTING . . . PHYSICAL REVIEW B 93, 075101 (2016) momentum gives For completeness we expand the normal ordered magnetic moment squared, : mˆ 2 :, ˆ | = | R φα φβ dβα(R), (3) † † 2 = σ σ β : mˆ : : cˆα,ζ ζζ cˆα,ζ .cˆβ,ξ ξξ cˆβ,ξ : , αβ ζζξξ where d (R)isthe(2 + 1) × (2 + 1) matrix that corre- βα = † † − † † sponds to the rotation Rˆ in the irreducible representation :2cˆα,ζ cˆα,ξ cˆβ,ξ cˆβ,ζ cˆα,ζ cˆα,ζ cˆβ,ξ cˆβ,ξ : , αβ ζξ of angular momentum . By making the change of variable = ˆ = ˆ † † † † x Rr and x Rr , it is straightforward to show that =− 2cˆ cˆ cˆ cˆ + cˆ cˆ cˆ cˆ , (9) α,ζ β,ξ α,ξ β,ζ α,ζ β,ξ β,ξ α,ζ ∗ ∗ αβ ζξ φ (r)φ (r )φ (r)φ (r ) V = drdr ασ βσ χσ γσ , αβ,χ γ |r − r| where we have used the deﬁnition ∗ ∗ = σ .σ = δ δ − δ δ . dαα(R) dββ (R) Vαβ,χ γ dχ χ (R)dγ γ (R), (4) ζζ ξξ 2 ζ ξ ζξ ζζ ξξ If the normal ordering operator is not used in Eq. (7) then which demonstrates that Vαβ,χ γ is indeed a rotationally invariant tensor. additional one electron terms have to be subtracted to remove the self-interaction. The mean-ﬁeld form of this Hamiltonian has been included in Appendix A1. B. The one band Hubbard model: s orbital symmetry

The Hubbard Hamiltonian is applicable only to one band C. The multiorbital model Hamiltonian: p orbital symmetry models where the one orbital per atom has s symmetry. This gives a simple form for the on-site Coulomb interaction tensor, Consider the case where the local orbitals, α, β, χ, and γ appearing in Eq. (4) are real cubic harmonic p orbitals U0 = Vαα,αα; as there is only one type of orbital, there is only one matrix element. We can use this result to simplify the with angular dependence x/r, y/r, and z/r. In this case, the × Coulomb interaction part of Eq. (2) and express it in terms of rotation matrices, dαα(R), are simply 3 3 Cartesian rotation the electron number operator or the magnetic moment vector matrices. This makes Vαβ,χ γ a rotationally invariant fourth- operator, defined as follows: rank Cartesian tensor, the general form of which is [33] Vαβ,χ γ = Uδαχ δβγ + Jδαγ δβχ + J δαβ δχγ , (10) = † nˆ cˆα,ζ cˆα,ζ , (5) where U = Vαβ,αβ ,J = Vαβ,βα , and J = Vαα,ββ, with α = β. αζ By examination of the integral in Eq. (4), we note that the = † σ mˆ cˆα,ζ ζζ cˆα,ζ , Coulomb tensor has an additional symmetry when the orbitals αζζ are real (which the cubic harmonic p orbitals are), namely (6) x y z Vαβ,χ γ = Vχβ,αγ and Vαβ,χ γ = Vαγ,χβ, and hence J must be σ = σ ,σ ,σ , ζζ ζζ ζζ ζζ equal to J . Thus we find where σ x , σ y , and σ z are the Pauli spin matrices, and the sum Vαβ,χ γ = Uδαχ δβγ + J (δαγ δβχ + δαβ δχγ ). (11) over α has only one term as there is only one spatial orbital for the s case. Hence This recovers the well known equation U0 = U + 2J , where U0 = Vαα,αα [14,19], and shows that the most general cubic 1 † 1 V cˆ† cˆ cˆ cˆ = U : nˆ2 : , (7) p orbital interaction Hamiltonian is defined by exactly two 2 αβ,χ γ α,σ β,ξ γ,ξ χ,σ 2 0 αβχ γ σξ independent parameters [34]. In Appendix B1,wegiveafull derivation of Eq. (11) using representation theory. where we have use the normal ordering operator, ::, to remove In passing, we observe that symmetric fourth-rank isotropic self-interactions. The action of this operator is to rearrange the tensors can be found in other areas of physics, such as the creation and annihilation operators such that all the creation stiffness tensor in isotropic elasticity theory [35] operators are on the left, without adding the anticommutator terms that would be required to leave the product of operators Ciklm = λδikδlm + μ(δilδkm + δimδkl), (12) unaltered; if the rearrangement requires an odd number of where λ is the Lame´ coefficient and μ is the shear modulus. flips, the normal ordering also introduces a sign change. For Transforming Eq. (11) into a basis of complex spher- example, ical harmonic p orbitals with angular dependence of the † † = 2 = form Y m(θ,φ) is straightforward. We write Vmm ,m m : nˆ : : cˆα,ζ cˆα,ζ cˆβ,σ cˆβ,σ : , ∗ ∗ l l lmχ lmγ Vαβ,χ γ , where lmα is defined by αβσ ζ αβχ γ mα m β † † sph = = 2 − p = lmαpα, (13) cˆα,ζ cˆβ,σ cˆβ,σ cˆα,ζ nˆ n.ˆ (8) m αβσ ζ α and has the following values: For the s case the normal ordered number operator squared, ⎛ ⎞ : nˆ2 :, √1 − √i 0 ⎜ 2 2 ⎟ = 2 1 2 [lmα] ⎝ 001⎠. (14) : nˆ := nˆ nˆ− =− : mˆ : . σ σ 3 − √1 − √i 0 σ 2 2

075101-3 M. E. A. COURY et al. PHYSICAL REVIEW B 93, 075101 (2016) ⎛ ⎞ The result of applying this transformation is 001 ⎜ 2 ⎟ [ξ ] = ⎝000⎠, Vmm,mm =Uδmm δmm + Jδmm δmm 2ab 1 m+m 2 00 + (−1) Jδ−mm δ−mm , (15) ⎛ ⎞ 000 ⎜ ⎟ where m, m ,m , and m are spherical harmonics indices = ⎝ 1 ⎠ [ξ3ab] 00 2 , (21) going from −1to+1,U = V10,10, and J = V10,01. 1 Using Eq. (11) as a convenient starting point, it is straight- 0 0 ⎛ 2 ⎞ forward to rewrite the on-site Coulomb Hamiltonian in terms 0 1 0 of rotationally invariant operators [30]: ⎜ 2 ⎟ [ξ ] = ⎝ 1 00⎠, 4ab 2 1 † V cˆ† cˆ cˆ cˆ 000 2 αβ,χ γ α,σ β,ξ γ,ξ χ,σ ⎛ ⎞ αβχ γ σξ 1 00 ⎜ 2 ⎟ 1 2 2 2 [ξ ] = ⎝ − 1 ⎠. = (U − J ):nˆ : −J : mˆ : −J : L : , (16) 5ab 0 2 0 2 000 where the vector angular momentum operator for a set of three We use the convention that index numbers (1,2,3,4,5) p orbitals on site I takes the form (see Appendix C1): 2 − 2 2 − 2 correspond to the d orbitals (3z r ,zx,yz,xy,x y ). Lˆ = i ( , , )cˆ† cˆ , (17) See Appendix B2 for a derivation of Eq. (20) using rep- 1βα 2βα 3βα α,σ β,σ resentation theory. We have chosen U to be the Hartree αβσ term between the t2g orbitals, J to be the average of and μβα is the three-dimensional Levi-Civita symbol. An the exchange integral between the eg and t2g orbitals, equivalent expression is and J to be the difference between the exchange in- = = tegrals for eg and t2g. That is, U V(zx)(yz),(zx)(yz),J 1 † † 1 + = V cˆ cˆ cˆ cˆ (V(zx)(yz),(yz)(zx) V(3z2−r2)(x2−y2),(x2−y2)(3z2−r2)), and J 2 αβ,χ γ α,σ β,ζ γ,ζ χ,σ 2 − αβχ γ σζ V(3z2−r2)(x2−y2),(x2−y2)(3z2−r2) V(zx)(yz),(yz)(zx). This choice of parameters is the same as was used by Oles´ and Stollhoff [20]; 1 1 1 = U − J : nˆ2 : − J : mˆ 2 : +J :(nˆ )2 : , Eq. (20) generalizes their result and restores rotational in- 2 2 2 αβ αβ variance in orbital space. Note that, in the mean field, the (18) on-site block of the density matrix in a cubic solid is diagonal; this will cause the mean-field version of Eq. (22) below where the final term corresponds to on-site electron hop- to reduce to that of Oles´ and Stollhoff [20]. However, the ping [36], presence of interfaces, vacancies, or interstitials will break the cubic symmetry and will make the on-site blocks of the = † nˆαβ cˆα,σ cˆβ,σ . (19) density matrix nondiagonal. Therefore the use of the complete σ Hamiltonian is recommended in mean-field calculations for Equation (16) embodies Hund’s rules for an atom. By making systems that do not have cubic symmetry. the substitution mˆ = 2Sˆ, we see that the spin is maximized first Rewriting Eq. (20) in terms of rotationally invariant (prefactor −2J ) and then the angular momentum is maximized operators gives − 1 (prefactor 2 J ). The mean-field form of the above p orbital Hamiltonian is given in Appendix A2. 1 1 1 Vˆ = U − J + 5J : nˆ2 : − (J − 6J ) : mˆ 2 : tot 2 2 2 D. The multiorbital model Hamiltonian: d orbital symmetry 2 + (J − 6J ) :(nˆ )2 : + J : Qˆ 2 : , (22) The on-site Coulomb interaction for cubic harmonic d αβ 3 αβ orbitals can be expressed as where Qˆ 2 is the on-site quadrupole operator squared. The 1 5 Vαβ,χ γ = Uδαχ δβγ + J + J (δαγ δβχ + δαβ δγχ) quadrupole operator for a set of five d orbitals on site I takes 2 2 the form (see Appendix C2): − 48J ξαabξβbdξχdtξγta , (20) abdt ˆ =− + Qμν 2 (ξανkξβkμ ξαμkξβkν) αβσ k where ξ is a five-component vector of traceless symmetric 3 × 3 transformation matrices defined as follows: + † ⎛ ⎞ 4 ξαmnξβj k νj mμkn cˆασ cˆβσ . (23) √1 − 00 mnjk ⎜ 2 3 ⎟ [ξ ] = ⎝ 0 − √1 0 ⎠, 1ab 2 3 More detail on the Qˆ 2 operator is included in Appendix C. 00√1 3 Equation (22) is similar in form to that found in Rudzikas [30],

075101-4 HUBBARD-LIKE HAMILTONIANS FOR INTERACTING . . . PHYSICAL REVIEW B 93, 075101 (2016) the difference being that our Hamiltonian is expressed in terms the values of Paxton and Finnis [39]: ppσ : ppπ = 2:−1for of physically meaningful operators. The mean-field form of p orbitals, and ddσ : ddπ : ddδ =−6:4:−1ford orbitals. this Hamiltonian is given in Appendix A3. The directionally averaged magnetic correlation between the two sites has been calculated for the ground state. If positive E. Comparison with the Stoner Hamiltonian it shows that the atoms are ferromagnetically correlated (spins parallel on both sites) and if it is negative it shows that the The Stoner Hamiltonian for p and d orbital atoms, as atoms are antiferromagnetically correlated (spins antiparallel generally defined in the literature, has the following form for between sites). The correlation between components of spin the on-site Coulomb interaction [24]: projected onto a direction described by angles θ and φ,in Vˆ = 1 U − 1 J : nˆ2 : − 1 J : mˆ 2 : , (24) spherical coordinates, is defined as follows: Stoner 2 2 4 z = θφ θφ where J , in this many-body context, may be interpreted as the Cθφ : mˆ 1 mˆ 2 : , (27) Stoner parameter, often denoted by I. However, in the mean where the :: is the normal ordering operator, used to remove field, the Stoner parameter should not be taken to be equal self-interaction, and to J due to the self-interaction error; see reference [37]. This θφ = θφ † form, which is identical to Eq. (4)ofRef.[24], clearly breaks mˆ I σξξ cÎα,ξcÎα,ξ , rotational symmetry in spin space, which can be restored by αξξ 2 2 (28) substituting : mˆ z : with : mˆ :, θφ = z + x + y σξξ σξξ cos θ σξξ sin θ cos φ σξξ sin θ sin φ. ˆ = 1 − 1 2 − 1 2 Vmˆ 2Stoner 2 U 2 J : nˆ : 4 J : mˆ : . (25) The magnetic correlation averaged over a solid angle is 1 A direct calculation shows that the above Eq. (25) is in fact Cavg = : mˆ 1.mˆ 2 :. (29) identical to the Hamiltonian proposed by Dworin and Narath, 3 Eq. (1)ofRef.[13]. In this paper, we will compare our B. Ground state: p orbital dimer Hamiltonians, Eqs. (18) and (22), with Eq. (25), which we shall call the vector or mˆ 2 Stoner Hamiltonian. By starting The ground states of our Hamiltonian and the vector Stoner with the vector Stoner Hamiltonian and working backwards, Hamiltonian have been found to be rather similar for 2, 6, and it is possible to find the corresponding tensorial form of the 10 electrons split over the p orbital dimer, but qualitatively Coulomb interaction: different for 4 and 8 electrons. The magnetic correlation between the two atoms for 4 electrons is shown in Fig. 1. mˆ 2Stoner = + Vαβ,χ γ Uδαχ δβγ Jδαγ δβχ. (26) The symmetries of the wave functions [40] are indicated on ± 2S+1 ± This is very similar to the tensorial form for the p-case the graph by the notation u/g :the is only used for = Coulomb interaction, Eq. (11), except that it is missing a Lz 0 (i.e. states) and corresponds to the sign change after a reflection in a plane parallel to the axis of the dimer; term, Jδαβ δγχ. The lack of this term means that the vector Stoner Hamiltonian does not respect the symmetry between the u/g term refers to ungerade (odd) and gerade (even) pairs αχ and βγ evident from the form of the integral in and corresponds to a reflection through the midpoint of the 2 2 dimer; is the symbol corresponding to the total value for Eq. (4), i.e., V mˆ Stoner = V mˆ Stoner. As we shall see later, this χβ,αγ αβ,χ γ L (e.g., L = 1is,L = 2is,L = 3is,L = 4is); omission changes the computed results. z z z z z and S is the total spin. The differences between the ground states of our Hamiltonian and the vector Stoner Hamiltonian III. GROUND-STATE RESULTS shown in Fig. 1 are as follows: the ground-state wave function 3 − A. Simulation setup of the vector Stoner Hamiltonian has a region with g symmetry extending far up the J axis, which is not present Here, we investigate the eigenstates of dimers using both for our Hamiltonian; the region with 1 symmetry is doubly our Hamiltonians, Eqs. (18) and (22), and the vector Stoner g degenerate for our Hamiltonian, as total Lz =±2, whereas Hamiltonian, Eq. (25). We employ a restricted basis with for the Stoner Hamiltonian it is triply degenerate, as it is also a single set of s, p, and d orbitals for the s, p, and d 1 + degenerate with the state of symmetry g (this state appears Hamiltonians, respectively. We perform FCI calculations using at a higher energy for our Hamiltonian). the HANDE [31] computer program to find the exact eigenstates of the model Hamiltonians. The Lanczos algorithm was used to calculate the lowest 40 wave functions for our Hamiltonian, C. Ground state: d orbital dimer more than sufficient to find the correct ground-state wave The ground states of our Hamiltonian and the vector Stoner function, while the full spectrum was calculated for the vector Hamiltonian have been found to be rather similar for 2, 4, Stoner Hamiltonian. 6, 14, 16, and 18 electrons split over the d orbital dimer The single-particle contribution to the Hamiltonian for all when J is small. The simulations for 10 electrons were models (the hopping matrix) is defined for a dimer aligned not carried out as they are too computationally expensive. along the z axis: in this case it is only nonzero between orbitals Qualitative differences were found for 8 and 12 electrons; for of the same type, whether on the same site or on different sites. an example see Fig. 2, which shows the magnetic correlation The sigma hopping integral, |tσ |, was set to 1 to define the between two atoms in the ground state of the d-shell dimer energy scale. The other hopping matrix elements follow the with 6 electrons per atom. From Fig. 2, we see that the results canonical relations suggested by Andersen [38], and we use for our Hamiltonian with J/|t|=0.0 (top graph) and with a

075101-5 M. E. A. COURY et al. PHYSICAL REVIEW B 93, 075101 (2016)

FIG. 1. The magnetic correlation between two p-shell atoms, each with two electrons, for a large range of parameters U/|t| and J/|t|,wheret is the sigma bond hopping. The different regions of the graph are labeled by the symmetry of the ground state. The top graph is generated from our Hamiltonian and the bottom from the vector Stoner Hamiltonian. The bottom graph has a region with symmetry 3 − g extending a long way up the J axis, which is not present in the ground state of our Hamiltonian. The bottom graph also includes 1 1 + a region with two degenerate states with symmetries g and g ; this degeneracy is broken when our Hamiltonian is used.

small value of J/|t|=0.1 (middle graph) are qualitatively different from those for the vector Stoner Hamiltonian (bottom + graph). The vector Stoner Hamiltonian makes the 1 and 1 g g FIG. 2. The magnetic correlation between two d-shell atoms, states degenerate, which is not the case for our Hamiltonian. each with six electrons, for a large range of parameters U/|t| and J/|t|= . The largest differences are between the 0 1 graph J/|t|,wheret is the sigma bond hopping. The different regions of the and the vector Stoner Hamiltonian: new regions with symmetry 1 + 3 5 − graph are labeled by the symmetry of the ground state. The top graph g , g, and g appear, and the region with symmetry is generated using our d-shell Hamiltonian with J/|t|=0.0; the 3 − g almost disappears. This shows that the inclusion of the middle graph is generated using our Hamiltonian with a small value quadrupole term in Eq. (22) can make a qualitative difference of J/|t|=0.1; and the bottom graph is generated using the vector to the ground state. Stoner Hamiltonian.

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IV. EXCITED STATE RESULTS 1x Differences between our Hamiltonian and the vector Stoner 2.0 1y Hamiltonian are also observed for excited states. Here we 1z present three examples, one demonstrating explicitly the effect 2x of the pair hopping, one showing more general differences 1.5 2y in the excited states through a calculation of the electronic 2z

heat capacity as a function of temperature, and one showing α

I .

a difference in the spin dynamics of a collinear and a ˆ 1 0 n noncollinear Hamiltonian. The full spectrum of eigenstates was calculated for all of these examples. 0.5 A. Pair hopping † † = The pair hopping term, J αβσ σ cˆα,σ cˆα,σ cˆβ,σ cˆβ,σ 0.0 2 0 2 4 6 8 10 12 14 16 J αβ :(nˆαβ ) :, is found in both our p and d orbital Hamil- time(h¯/|t|) tonians, but is absent from the vector Stoner Hamiltonian. To demonstrate the effect of the pair hopping term, we initialize 1x the wave function in a state with two electrons in the x orbital 2.0 1y on site 1 of a p-atom dimer. Figure 3 shows the evolution 1z of the wavefunction in time using our Hamiltonian and the 2x vector Stoner Hamiltonian, with U = 5.0|t| and J = 0.7|t|. 1.5 2y For our Hamiltonian, the pair of electrons in orbital x on atom 2z 1 hops to the y and z orbitals on atom 1 (mediated by the pair

hopping term) as well as between the two atoms (mediated by α I .

ˆ 1 0 n the single electron hopping term). The result is that there is a ﬁnite probability of ﬁnding the pair of electrons in any of the x,y, and z orbitals on either atom. However, for the vector 0.5 Stoner Hamiltonian, the two electrons in the x orbital on site 1 are unable to hop into the y and z orbitals on atom 1; they are only able to hop to the x orbital on atom 2. This means that 0.0 there is no possibility of observing the pair of electrons in the 0 2 4 6 8 10 12 14 16 y or z orbitals on either atom. time(h¯/|t|)

FIG. 3. The time evolution under our Hamiltonian (top) and the B. Heat capacity vector Stoner Hamiltonian (bottom) of a starting state with two The heat capacity is calculated from electrons in 1x,thex orbital on atom 1, of a p-atom dimer with U = 5.0|t| and J = 0.7|t|. We see that the two electrons in 1x are C 1 able to hop into 1y and 1z when the wave function is evolved using V = ( E2(T )− E(T )2), (30) 2 our Hamiltonian but not when it is evolved using the vector Stoner kB Na Na(kB T ) Hamiltonian. where C. Spin dynamics n − εi i εi exp En(T )= kB T , (31) Here, we show how the collinear Stoner Hamiltonian can exp − εi =+ i kB T give rise to unphysical spin dynamics. We consider an Sz 1 triplet with both electrons on one of the two p-orbital atoms, T is the temperature, kB is Boltzmann’s constant, εi is a many- |=|T+1. Written as linear combinations of two-electron electron energy eigenvalue, and Na is the number of atoms. The Slater determinants, the three states in the triplet are result for both Hamiltonians as a function of temperature for = | | four electrons split over two p-shell atoms with U 5.0 t and | =√1 −| ↑ ↑ + | ↑ ↑ = | | T+1 ( 1x 1y 2x 2y ), J 0.7 t is shown in Fig. 4. There is a qualitative difference 2 between the results for our Hamiltonian (the solid line) and 1 the vector Stoner Hamiltonian (dashed line). The energies of |T−1=√ (−|1x ↓ 1y ↓ + |2x ↓ 2y ↓), the eigenstates of our Hamiltonian are more spread out than 2 those of the Stoner Hamiltonian. This is due to the inclusion of 1 |T0=−√ (|1x ↓ 1y ↑ + |1x ↑ 1y ↓) the pair hopping term, which causes states with on-site paired 2 electrons to rise in energy (by ∼J ). The unphysical reduction 1 of the heat capacity to zero as the temperature tends to inﬁnity + √ (|2x ↓ 2y ↑ + |2x ↑ 2y ↓). (32) is a consequence of the use of a restricted basis set. 2

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0.8 In contrast, using the vector Stoner Hamiltonian or our p-shell p orbital Hamiltonian Hamiltonian, the expectation value of the magnetic moment 0.7 Stoner Hamiltonian is independent of time, equal to (1,0,0) in vector format. This demonstrates that calculations of spin dynamics using . 0 6 the collinear Stoner Hamiltonian can give rise to unphysical

) . oscillations of the magnetic moments. a 0 5 N B

k 0.4 (

/ V. CONCLUSION V 0.3 C We have established the correct form of the multiorbital 0.2 model Hamiltonian with on-site Coulomb interactions for atoms with valence shells of s, p, and d orbitals. The method- 0.1 ology used may be extended to atoms with f and g shells, and to atoms with valence orbitals of several different angular 0.0 10−3 10−2 10−1 100 101 102 momenta. The results presented show that there are important kBT/|t| differences between our p- and d-shell Hamiltonians and the vector Stoner Hamiltonian. The vector Stoner Hamiltonian misses both the pair hopping term, which is present in our p- FIG. 4. The electronic heat capacity per atom of a dimer with four and d-orbital Hamiltonians, and the quadrupole term, which electrons split over two p-shell atoms with U = 5.0|t| and J = 0.7|t|. The solid line is generated by diagonalising our Hamiltonian exactly is present in our d-orbital Hamiltonian. The pair hopping term and the dashed line by diagonalising the vector Stoner Hamiltonian pushes states with pairs of electrons up in energy, whereas exactly. the magnetism term pulls states with local magnetic moments down in energy. The pair hopping term has the largest effect on the ground state for p orbitals with two and four electrons These states are simultaneously eigenstates of the collinear per atom and for d orbitals with four and six electrons per Stoner Hamiltonian, the vector Stoner Hamiltonian, and our atom, for J/|t| < 2. This is because the number of possible − p-case Hamiltonian. They are degenerate, with eigenvalue U determinants with paired electrons on site is large for these J , for both our p-case Hamiltonian and for the vector Stoner filling factors and the low lying states can be separated based Hamiltonian. For the collinear Stoner Hamiltonian the states on this. At values of J/|t| > 2, magnetism dominates the | | − | T+1 and T−1 have eigenvalue U J , whereas state T0 has selection of the ground state and the difference between eigenvalue U. This means that the degeneracy of this triplet the ground state of our Hamiltonian and that of the vector is broken in the collinear Stoner Hamiltonian. We now rotate Stoner Hamiltonian becomes small. However, the differences | in spin space so that the spins are aligned with the x axis, are rather more pronounced for the excited states. This is =+ i.e., Sx 1, evidenced by the hopping of pairs of electrons between orbitals rot 1 1 on the same site, which is allowed by our Hamiltonian but | = (|T+1+|T−1) + √ |T0. (33) 2 2 not by the vector Stoner Hamiltonian, and in differences in the electronic heat capacity as a function of temperature. We | rot − is still an eigenstate, with eigenvalue U J , of both the also find clear evidence that the collinear Stoner Hamiltonian vector Stoner Hamiltonian and our Hamiltonian, but it is no is inappropriate for use in describing spin dynamics as it longer an eigenstate of the collinear Stoner Hamiltonian. The breaks rotational symmetry in spin space. We would expect | rot wave function evolves in time as similar problems when using collinear time-dependent DFT − iHτˆ simulations to model spin dynamics. |tot(τ)=e |tot(0), ⎧ − − iJτ ⎪ − i(U J )τ 1 e ⎪e (|T+ +|T− ) + √ |T ⎪ 2 1 1 2 0 ⎨⎪ ACKNOWLEDGMENTS = collinear Stoner Hamiltonian − , The authors would like to thank Toby Wiseman for help ⎪ − i(U J )τ 1 1 ⎪ e (|T+ +|T− ) + √ |T with the representation theory and Dimitri Vvedensky, David ⎪ 2 1 1 2 0 ⎩ M. Edwards, Cyrille Barreteau, and Daniel Spanjaard for mˆ 2 Stoner and our Hamiltonian stimulating discussions. M. E. A. Coury was supported through (34) a studentship in the Centre for Doctoral Training on Theory and where τ is time. If we take the expectation value of the Simulation of Materials at Imperial College funded by EPSRC magnetic moment on each site using the collinear Stoner under Grant No. EP/G036888/1. This work was part-funded Hamiltonian, we find by the EuroFusion Consortium, and has received funding from Euratom research and training programme 20142018 under Jτ rot(τ)|mˆ |rot(τ)= rot(τ)|mˆ |rot(τ)=cos , Grant Agreement No. 633053, and funding from the RCUK 1x 2x Energy Programme (Grant No. EP/I501045). The views and opinions expressed herein do not necessarily reflect those of rot(τ)|mˆ |rot(τ)= rot(τ)|mˆ |rot(τ)=0, 1y 2y the European Commission. The authors acknowledge support rot rot rot rot (τ)|mˆ 1z| (τ)= (τ)|mˆ 2z| (τ)=0. (35) from the Thomas Young Centre under Grant TYC-101

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APPENDIX A: MEAN-FIELD HAMILTONIANS the following: 1. The one-band Hubbard model: s orbital symmetry 1 J VˆMF = U − J + 2J nˆnˆ − mˆ .mˆ Application of the mean-field approximation [41]tothe 2 2 on-site Coulomb interaction part of the Hubbard Hamiltonian + (J − 6J ) ( nˆ nˆ + nˆ nˆ ) with s orbital symmetry, Eq. (7), yields the following: αβ βα αβ αβ αβ † 2 Vˆ =U nˆnˆ − cˆ† cˆ cˆ cˆ . (A1) + J Qˆ Qˆ MF 0 σ ζ ζ σ 3 μν νμ σζ μν − † † + † † The total energy is (U cˆασ cˆβζ cˆβζ cˆασ J cˆασ cˆβζ cˆαζ cˆβσ ) αβσ ζ ⎛ ⎞ † † 1 † + 48J ξ ξ ξ ξ cˆ cˆ cˆ cˆ . ECoulomb = U ⎝ nˆ2 − cˆ† cˆ cˆ cˆ ⎠. (A2) αst βtu χuv γvs ασ γζ βζ χσ MF 2 0 σ ζ ζ σ αβγ χ σ ζ stuv σζ (A7) Equivalently, applying the mean-field approximation yields The total energy is the following: 1 1 J Coulomb = − + 2 − 2 EMF U J 2J nˆ mˆ ˆ =1 − − † † 2 2 4 VMF U0 2 nˆ nˆ mˆ .mˆ cˆσ cˆζ cˆζ cˆσ . (A3) 3 J − 6J σζ + ( nˆ nˆ + nˆ 2) 2 αβ βα αβ αβ The total energy is + 1 ˆ ˆ J Qμν Qνμ 1 † † 3 ECoulomb = U 2 nˆ2 − mˆ 2 − cˆ cˆ cˆ cˆ . (A4) μν MF 6 0 σ ζ ζ σ σζ U † J † − cˆ† cˆ cˆ cˆ + cˆ† cˆ cˆ cˆ 2 ασ βζ βζ ασ 2 ασ βζ αζ βσ αβσ ζ 2. The multiorbital model Hamiltonian: p orbital symmetry + Application of the mean-field approximation [41]tothe 24J ξαstξβtuξχuvξγvs model Hamiltonian with p orbital symmetry, Eq. (18), yields αβγ χ σ ζ stuv † the following: × cˆ† cˆ cˆ cˆ . (A8) ασ γζ βζ χσ J J VˆMF = U − nˆnˆ − mˆ .mˆ APPENDIX B: REPRESENTATION THEORY 2 2 + + 1. p orbital symmetry J ( nˆαβ nˆβα nˆαβ nˆαβ ) αβ As a precursor to our treatment of the d case in Appendix B2, we use representation theory [42] to derive − † † + † † (U cˆασ cˆβζ cˆβζ cˆασ J cˆασ cˆβζ cˆαζ cˆβσ ). (A5) Eq. (11) for cubic harmonic p orbitals. First, we define αβσ ζ the following irreducible objects: “0” is a scalar, “1” is a three-dimensional vector, “2” is a 3 × 3 symmetric traceless The total energy is second-rank tensor, “3” is a third-rank tensor consisting of three “2”s, and “4” is a fourth-rank tensor consisting of three 1 J J ECoulomb = U − nˆ2 − mˆ 2 “3”s. The Coulomb interaction Vαβ,γ δ transforms as a tensor MF 2 2 4 product of four “1”s, 1 ⊗ 1 ⊗ 1 ⊗ 1, but we know also that it J must be isotropic and that from the above irreducible objects + ( nˆ nˆ + nˆ 2) only the “0” is isotropic. 2 αβ βα αβ αβ Objects such as 1 ⊗ 1, represented as A , may be expanded ij just as in the addition of angular momentum: U † − cˆ† cˆ cˆ cˆ 2 ασ βζ βζ ασ 1 ⊗ 1 = 0 ⊕ 1 ⊕ 2, (B1) αβσ ζ where “0” is a scalar, s = A δ , “1” is a vector, J † † ij ij ij + cˆ cˆ cˆ cˆ . (A6) = 2 ασ βζ αζ βσ vi jk Ajkij k , and “2” is a traceless symmetric tensor, = 1 + − 1 Mij 2 (Aij Aji) 3 δij ( kl Aklδkl). The reader may be more familiar with combinations of angular momentum. In the 3. The multiorbital model Hamiltonian: d-orbital symmetry language of angular momentum, the transformation properties Application of the mean-field approximation [41]tothe of a tensor product of two objects of angular momentum model Hamiltonian with d-orbital symmetry, Eq. (22), yields with l = 1 are described by a 9 × 9 matrix, which is block

075101-9 M. E. A. COURY et al. PHYSICAL REVIEW B 93, 075101 (2016) diagonalizable into a 1 × 1 matrix (which describes the found from symmetric and antisymmetric contractions of Tij kl . transformations of an l = 0 object), a 3 × 3matrix(which We are interested in the form of the p orbital on-site Coulomb describes the transformations of an l = 1 object), and a interactions, Vαβ,χ γ , which have an additional symmetry, such 5 × 5 matrix (which describes the transformations of an l = 2 that they remain unchanged under exchange of α and χ object). In the angular momentum representation, the matrix and under exchange of β and γ . We therefore only require elements are complex. Here we are using cubic harmonics and the symmetric scalars that arise from the 0 ⊗ 0 and 2 ⊗ 2 the matrix elements are real. Similarly, 1 ⊗ 1 ⊗ 1 ⊗ 1 may be contractions to describe Vαβ,χ γ : expanded out as Vαβ,χ γ = s0δαχ δβγ + s2(δαγ δβχ + δαβ δχγ ), (B3) 1 ⊗ 1 ⊗ 1 ⊗ 1 which is symmetric under exchange of α and χ or β and γ ; = (1 ⊗ 1) ⊗ (1 ⊗ 1), where = ⊕ ⊕ ⊗ ⊕ ⊕ (0 1 2) (0 1 2), s0 = Vαβ,χ γ δαχ δβγ , = (0 ⊗ 0) ⊕ (0 ⊗ 1) ⊕ (0 ⊗ 2) ⊕ (1 ⊗ 0) ⊕ (1 ⊗ 1) χγ (B4) ⊕(1 ⊗ 2) ⊕ (2 ⊗ 0) ⊕ (2 ⊗ 1) ⊕ (2 ⊗ 2). (B2) s2 = Vαβ,χ γ δαγ δβχ = Vαβ,χ γ δαβ δχγ . The only isotropic, “0”, objects arise from 0 ⊗ 0, 1 ⊗ 1, χγ χβ and 2 ⊗ 2. A general isotropic three-dimensional fourth-rank This is equivalent to Eq. (11), where s0 = U and s2 = J . tensor, Tij kl , therefore has three independent scalars that are

2. d orbital symmetry To find the isotropic five-dimensional fourth-rank tensor that describes the on-site Coulomb interactions for d orbital symmetry, we find it convenient to map from a five-dimensional fourth-rank tensor to a three-dimensional eighth-rank tensor. We do this by replacing the five-dimensional vector of d orbitals by the three-dimensional traceless symmetric B matrix of the d orbitals: ⎛ ⎞ 2 2 − 1 2 + 2 3 x 3 (y z ) xy xz = ⎝ 2 2 − 1 2 + 2 ⎠ B xy 3 y 3 (x z ) yz . (B5) 2 2 − 1 2 + 2 xz yz 3 z 3 (x y )

We refer to the elements of B as Bab, and use the subscripts on the ξ matrices as they are real. The B matrix is a a, b, c, d, s, t, u, and v as the indices for irreducibles of rank three-dimensional traceless symmetric “2” and thus contains 2 or higher. The transformation between the irreducible Bab five independent terms. As we show below, it is possible to and the cubic harmonic d orbitals is represent the isotropic three-dimensional eighth-rank tensor 1 as a list of quadruple Kronecker deltas, with five independent φ (r) = N ξ B R (r)S(σ), (B6) ασ d αab ab r2 d parameters in general and three independent parameters for the ab on-site Coulomb interactions. The isotropic three-dimensional where ξ is a traceless, symmetric transformation matrix, eighth-rank tensor is then converted back into an isotropic five- dimensional fourth-rank tensor. For an independent verifica- defined in Eq. (21), N = 1 15 is a normalization factor, d 2 π tion of the form of the isotropic three-dimensional eighth-rank Rd is the radial function for a cubic d orbital, and S is the tensor, see Ref. [43]. spin function. The orbital indices α, β, χ, and γ run over the The power of representation theory is that one can follow an five independent d orbitals, whereas the irreducible indices analogous procedure for shells of higher angular momentum, a, b, c, d, s, t, u, and v run over the three Cartesian directions. using a “3” to represent the seven f orbitals, a “4” to represent The mapping between the five-dimensional fourth-rank tensor the nine g orbitals, and so on. One can also construct interaction and the three-dimensional eighth-rank tensor is as follows: Hamiltonians for atoms with important valence orbitals of ∗ ∗ φ (r1)φ (r2)φ (r1)φ (r2) several different angular momenta. V = dr dr ασ βσ χσ γσ , αβ,χ γ 1 2 |r − r | We now return to the case of a d shell and explain 1 2 the procedure used to find the general isotropic Coulomb = 4 22 22 Hamiltonian in more detail. We start by representing the dr1dr2Nd Rd (r1)/r1 Rd (r2)/r2 isotropic three-dimensional eighth-rank tensor, Vab,cd,st,uv,as ξ ∗ B∗ ξ ∗ B∗ ξ B ξ B a2⊗ 2 ⊗ 2 ⊗ 2. Proceeding as we did for the p case, we write × abcd stuv αab ab βcd cd χst st γuv uv , |r − r | 1 2 2 ⊗ 2 ⊗ 2 ⊗ 2 = (2 ⊗ 2) ⊗ (2 ⊗ 2), = ξαabξβcdξχstξγuvVab,cd,st,uv, (B7) abcd stuv 2 ⊗ 2 = 0 ⊕ 1 ⊕ 2 ⊕ 3 ⊕ 4, where this equation defines the isotropic three-dimensional ⇒ 2 ⊗ 2 ⊗ 2 ⊗ 2 = (0 ⊕ 1 ⊕ 2 ⊕ 3 ⊕ 4) eighth-rank tensor for the on-site Coulomb integrals, ⊗ (0 ⊕ 1 ⊕ 2 ⊕ 3 ⊕ 4). (B8) Vab,cd,st,uv, and we have dropped the complex conjugates

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TABLE I. Table to show the node contractions for a d shell. Every contraction is represented both diagrammatically and mathematically.

1 Δab,cd,st,uv = δadδbcδsvδtu + δacδbdδsvδtu + δadδbcδsuδtv + δacδbdδsuδtv

2 Δab,cd,st,uv = δatδbsδcvδdu + δasδbtδcvδdu + δatδbsδcuδdv + δasδbtδcuδdv

3 Δab,cd,st,uv = δavδbuδctδds + δauδbvδctδds + δavδbuδcsδdt + δauδbvδcsδdt

4 Δab,cd,st,uv = δatδbdδcvδsu + δadδbtδcvδsu + δatδbcδdvδsu + δacδbtδdvδsu +δatδbdδcuδsv + δadδbtδcuδsv + δatδbcδduδsv + δacδbtδduδsv +δasδbdδcvδtu + δadδbsδcvδtu + δasδbcδdvδtu + δacδbsδdvδtu + δasδbdδcuδtv + δadδbsδcuδtv + δasδbcδduδtv + δacδbsδduδtv

5 Δab,cd,st,uv = δavδbdδctδsu + δadδbvδctδsu + δavδbcδdtδsu + δacδbvδdtδsu +δauδbdδctδsv + δadδbuδctδsv + δauδbcδdtδsv + δacδbuδdtδsv +δavδbdδcsδtu + δadδbvδcsδtu + δavδbcδdsδtu + δacδbvδdsδtu +δauδbdδcsδtv + δadδbuδcsδtv + δauδbcδdsδtv + δacδbuδdsδtv

6 Δab,cd,st,uv = δavδbtδcuδds + δatδbvδcuδds + δauδbtδcvδds + δatδbuδcvδds +δavδbsδcuδdt + δasδbvδcuδdt + δauδbsδcvδdt + δasδbuδcvδdt +δavδbtδcsδdu + δatδbvδcsδdu + δavδbsδctδdu + δasδbvδctδdu +δauδbtδcsδdv + δatδbuδcsδdv + δauδbsδctδdv + δasδbuδctδdv

The terms in Eq. (B8) that can generate a rank 0 (a scalar) indices ab, cd, st,oruv because Bab is traceless, hence ⊗ ⊗ ⊗ ⊗ ⊗ = are 0 0, 1 1, 2 2, 3 3, and 4 4. This implies that ab δabBab 0. a general isotropic three-dimensional eighth-rank tensor is By examining the diagrammatic contractions from Ta- defined by five independent parameters. We know that, for ble I, and maintaining the symmetry between the pairs Vαβ,χ γ , there exists a symmetry between the pairs αχ and (ab)(st) and (cd)(uv), we conclude that the relevant contrac- 2 1 + 3 5 βγ. It follows that Vab,cd,st,uv has a symmetry between the tions are ab,cd,st,uv, (ab,cd,st,uv ab,cd,st,uv),ab,cd,st,uv pairs (ab)(st) and (cd)(uv). We therefore require only the 4 + 6 and (ab,cd,st,uv ab,cd,st,uv). This is evident because even contractions, 0 ⊗ 0, 2 ⊗ 2, and 4 ⊗ 4, reducing the five switching two symmetry-related pairs [that is, switching (ab) parameters to three;2 this is the same number proposed in and (st)or(cd) and (su)] in diagram 1 yields diagram 3. Chapter 4 of Ref. [34]. To get the scalars from Vab,cd,st,uv Similarly, switching two symmetry-related pairs in diagram 4 we have to contract it. There are six possible contractions yields diagram 6. We see that there are four contractions but of Vab,cd,st,uv, outlined in Table I, of which only five are we have only three parameters. This implies that out of these independent. Note that one does not contract within the four contractions there are only three linearly independent contractions. We can now write down a general interaction Hamiltonian Vab,cd,st,uv as a linear combination of three Kronecker delta products, each multiplied by one of the three independent 2By continuing this argument, one finds that four parameters coefficients: are required to describe the on-site Coulomb interactions for an f shell (arising from 0 ⊗ 0, 2 ⊗ 2, 4 ⊗ 4, and 6 ⊗ 6) and five ⊗ ⊗ ⊗ ⊗ = 2 + 1 + 3 parameters for a g shell (arising from 0 0, 2 24 4, 6 6, and Vab,cd,st,uv c0ab,cd,st,uv c1 ab,cd,st,uv ab,cd,st,uv 8 ⊗ 8), respectively. This trend continues with increasing angular + 5 momentum. c2ab,cd,st,uv. (B9)

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We find Vαβ,χ γ by transforming this expression back to APPENDIX C: ANGULAR MOMENTUM AND the five-dimensional fourth-rank representation, as dictated QUADRUPOLE OPERATORS by Eq. (B7). For completeness, we include all six of the 1. Angular momentum transformed contractions: The angular momentum operators are most naturally 1 ξ ξ ξ ξ expressed in spherical polar coordinates, even when using ab,cd,st,uv αab βcd χst γuv them to operate on cubic harmonics. Their forms are abcd stuv = 4ξ ξ ξ ξ = δ δ , ∂ ∂ αab βba χst γts αβ χγ Lˆ x = i sin φ + cot θ cos φ , (C1) abst ∂θ ∂φ 2 ∂ ∂ ab,cd,st,uvξαabξβcdξχstξγuv Lˆ y = i − cos φ + cot θ sin φ , (C2) abcd stuv ∂θ ∂φ ∂ = 4ξ ξ ξ ξ = δ δ , Lˆ z =−i , (C3) αab χba βcd γdc αχ βγ ∂φ abcd 3 where we have set to 1 for simplicity. When applying the ab,cd,st,uvξαabξβcdξχstξγuv above operators to the on-site cubic harmonic p orbitals (we abcd stuv have dropped the site index for clarity) it is straightforward to = = show that 4ξαabξγbaξβcdξχdc δαγ δβχ, abcd ˆ = Lμpj i μj k pk. (C4) 4 k ab,cd,st,uvξαabξβcdξχstξγuv abcd stuv ˆ A general one-particle operator O may be expressed in terms = of creation and annihilation operators as follows: 16ξαabξβbdξγdvξχva, abdv ˆ = | ˆ | † O φασ O φβσ cˆασ cˆβσ . (C5) 5 αβσ σ ab,cd,st,uvξαabξβcdξχstξγuv abcd stuv Substituting Eq. (C4) into Eq. (C5) yields = 16ξαabξβbdξχdtξγta, ˆ = † Lμ i μβα cˆασ cˆβσ , (C6) abdt αβσ 6 ab,cd,st,uvξαabξβcdξχstξγuv where μ is a Cartesian direction and α and β are cubic abcd stuv harmonic p orbitals (x,y or z). The d case is slightly more = complicated, but is greatly simplified by the use of Eq. (B6). 16ξαabξχbtξβtdξγda. (B10) abdt Applying the angular momentum operators to the B matrix reveals that We find that the last three of the transformations of the Lˆ |B =i ( |B + |B ), (C7) contractions, although rotationally invariant in orbital space, μ jk μj s sk μks js are not expressible in terms of Kronecker deltas of α, β, χ, s and γ . This is to be expected as there exist terms in Vαβ,χ γ with that are nonzero and have more than two orbital indices, e.g., Rd (r) V 2− 2 or V 2− 2 2− 2 . Lˆ |φ =iN S(σ ) ξ ( |B + |B ). (3z r )(xy),(yz)(xz) (3z r )(xy),(xy)(x y ) μ ασ d r2 αj k μj s sk μks js We can now write down Vαβ,χ γ concisely, jks (C8) 5 By finding the expectation value and substituting into Eq. (C5), Vαβ,χ γ = Uδαχ δβγ + J + J (δαγ δβχ + δαβ δχγ ) 2 we obtain − ˆ = † 48J ξαabξβbdξχdtξγta, (B11) Lμ 4i μmkξαkj ξβj mcˆασ cˆβσ . (C9) abdt jkm αβσ where we have defined U to be the Hartree term between the 2. Quadrupole operator = t2g orbitals on-site, U V(zx)(yz),(zx)(yz),J to be the average We define the quadrupole operator for a single electron as of the exchange integral between the eg and t2g orbitals = 1 + 1 1 2 on-site, J 2 (V(zx)(yz),(yz)(zx) V(3z2−r2)(x2−y2),(x2−y2)(3z2−r2)), Qˆ μν = (Lˆ μLˆ ν + Lˆ ν Lˆ μ) − δμν Lˆ . (C10) and J to be the difference between the exchange in- 2 3 ˆ ˆ tegrals for eg and t2g,J = V(3z2−r2)(x2−y2),(x2−y2)(3z2−r2) − Here, we interpret Lμ and Lν in Eq. (C10) as components of V(zx)(yz),(yz)(zx). We could equally well have chosen to use the the angular momentum of a single electron. The quadrupole sum of the fourth and sixth contractions instead of the fifth operator for a system of N electrons is then a sum over ˆ = N ˆ contraction, although with different coefficients. contributions from each electron: Qμν i=1 Qμν (i). This

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makes Qˆ μν a one-electron operator, respresented in second Equation (C5) yields the corresponding operator for a system quantization as a linear combination of strings of one creation of many electrons: operator and one annihilation operator. Squaring and tracing ˆ ˆ =− + the one-electron tensor operator Qμν yields a two-electron Qμν 2 (ξανkξβkμ ξαμkξβkν) ˆ 2 = ˆ ˆ αβσ k operator Q μν Qμν Qνμ. To find the form of this operator, we start by applying the + † single-electron version of Qˆ μν to the B matrix, making use of 4 ξαmnξβj k νj mμkn cˆασ cˆβσ . (C15) mnjk Eq. (C7). Starting with Lˆ μLˆ ν We now define the normal ordered quadrupole squared as : Qˆ 2 := : Qˆ Qˆ : . (C16) ˆ ˆ | = ˆ | + | μν νμ LμLν Bjk iLμ (νj s Bsk νks Bjs ), μν s 2 = (i) (νj s (μss |Bsk+μks |Bss ) By substituting Eq. (C15) into Eq. (C16) and dropping the site ss index for clarity, one obtains the following simple formula for the quadrupole squared operator: + | + | νks(μj s Bs s μss Bjs )), ˆ 2 = − + + : Q : 3δαχ δβγ 9(δαγ δβχ δαβ δχγ ) = (−1) δjμ|Bνk+δkμ|Bjν−2δνμ|Bjk αβγ χ σ σ † † + + | − (νj s μks νksμj s ) Bss , (C11) 72 ξαstξβtuξχuvξγvs cˆασ cˆβσ cˆγσ cˆχσ. (C17) ss stuv ˆ ˆ | = − | + | − | Lν Lμ Bjk ( 1) δjν Bμk δkν Bjμ 2δμν Bjk 3. The quadrupole operator squared and the on-site Coulomb integrals + (μj s νks + μksνj s )|Bss , (C12) The quadrupole operator is of interest to us because one ss of the terms in the d-shell on-site Coulomb interaction can be represented in terms of Qˆ 2. This term is where the last equation was found by exchanging μ and ν in − † † 48J ξαstξβtuξχuvξγvscˆασ cˆβσ cˆγσ cˆχσ, (C18) the previous equation. Following the same process, the final αβχ γ σ σ stuv term in Eq. (C10) becomes which is proportional to the final term in Eq. (C17). The other d terms in Eq. (C17) also exist elsewhere in the -shell on-site interaction Hamiltonian, so introducing an explicit Qˆ 2 term is Lˆ Lˆ |B =6|B . (C13) ν ν jk jk straightforward: ν − † † 48J ξαstξβtuξχuvξγvscˆασ cˆβσ cˆγσ cˆχσ This is not a surprising result as the B matrix contains linear αβχ γ σ σ stuv combinations of spherical harmonics |l,l with l = 2, and 2 † z = ˆ 2 + † ˆ 2| = + | J : Q : 2J cˆασ cˆ cˆ cˆασ L l,lz l(l 1) l,lz . Combining Eqs. (C11)–(C13) we find 3 βσ βσ αβσ σ † † † † −6J (cˆ cˆ cˆ cˆ + cˆ cˆ cˆ cˆ ) 1 ασ βσ ασ βσ ασ ασ βσ βσ Qˆ |B =− (δ |B +δ |B +δ |B + δ |B ) αβσ σ μν jk 2 μj νk kμ jν jν μk kν jμ 2 = J : Qˆ 2 : +5J : nˆ2 : +3J : mˆ 2 : + + | (νj s μks νksμj s ) Bss . (C14) 3 ss − 2 6J :(nˆαβ ) : . (C19) αβ

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