1.3 Spin models
1.3.3 Spin-cluster expansion
whereα:={i1, . . . , in}denotes the n-spin cluster for which all theµi indices are nonzero, i.e. all the corresponding single-spin basis functions are nontrivial.
It is thus worth defining cluster expansion functions Φαµ for every n-spin cluster con- taining the spinsα ={i1, . . . , in}, with the set of quantum numbers µ={µ1, . . . , µn}, all of which are nonzero,
Φαµ(ei1, . . . ,ein) := 1
(4π)N−n2 Θµ1(ei1) Θµ2(ei2)· · ·Θµn(ein). (1.3.21) Together with the constant cluster function of index zero,
Φ0 := 1
(4π)N2 , (1.3.22)
these form an orthonormal basis set, Φαµ
Φβν
=δαβδµν, (1.3.23)
furthermore, they are complete in the sense that for any fixed clusterα (of size n), Φ0Φ0+X
γ⊆α
X
µ(γ)
Φ∗γµ(ei1, . . . ,ein)Φγµ e0i1, . . . ,e0in
=
= 1
(4π)N−n Y
ik∈α
δ eik −e0ik
. (1.3.24)
Equation (1.3.24) can be easily verified by noting that the left hand side is just a refor- mulation of
1 (4π)N−n
X∞
µ1=0
X∞
µ2=0
. . . X∞
µn=0
Θ∗µ1(ei1) Θµ1 e0i1
. . .Θ∗µn(ein) Θµn e0in!
, (1.3.25)
which, along with the single-site completeness relation Eq. (1.3.18), leads directly to Eq. (1.3.24). Writing this completeness relation for the special cluster {1, . . . , N} com- prising the entire system (in other words, for the cluster of sizen =N),
Φ0Φ0+X
α
X
µ(α)
Φ∗αµ({e}) Φαµ({e0}) = YN
i=1
δ(ei−e0i), (1.3.26)
where in the summation, as usual, µ(α) always contains only nonzero quantum numbers over the entire clusterα.
It should be noted that all we have done is a special regrouping of the basis functions
created from the product of single-site basis functions. The utility of the procedure unfolds when a spin-dependent function, say, the grand potential is expanded over this basis set,
Ω({e}) = Ωe0Φ0+X
α
X
µ
JαµΦαµ({e}). (1.3.27)
Thanks to the method of the expansion, now the terms in the summation for n-spin clusters α depend exactly on n spins from the entire system. Thus n-spin contributions for n = 1,2, . . .∞ are neatly separated in Eq. (1.3.27), and what is more, orthogonality of the SCE functions (i.e., Eq. (1.3.23)) immediately gives us the coefficients,
Ωe0 = Φ0
Ω({e}) Jαµ =
Φαµ
Ω({e})
. (1.3.28)
A convenient choice of Θµsingle-site basis functions is the system of real spherical har- monicsYL(e) (L= 0,1, . . .). Note that complex spherical harmonics would be an equally good choice, then L would denote a composite index L = (`, m). Using Eq. (1.3.28), we may derive the connection between the grand potential (obtainable from an ab initio theory) and coefficients of the SCE. The coefficient of the constant term reads as
Ωe0 = Φ0
Ω({e})
= Z
· · ·
Z 1
(4π)N2 Ω({e}) d2e1· · ·d2eN
= (4π)N2 hΩ({e})i, (1.3.29) where hfi denotes the directional average of a function f with respect to every spin in the system,
hfi:= 1 (4π)N
Z
· · · Z
f({e}) d2e1· · ·d2eN. (1.3.30) Together with the constant cluster function, we are pleased to recover the expected con- stant term of the SCE, Eq. (1.3.27),
Ω0 :=Ωe0Φ0 = (4π)N2 hΩ({e})i · 1
(4π)N2 =hΩ({e})i. (1.3.31) For a generaln-spin term of the expansion, with a cluster α={i1, . . . , in} and indices
µ={L1, . . . , Ln}, Jαµ=
Φαµ
Ω({e})
= Z
· · ·
Z 1
(4π)N−n2 YL1(ei1)· · ·YLn(ein) Ω({e}) d2e1· · ·d2eN, (1.3.32) Jαµ= (4π)N−n2
Z
· · · Z
YL1(ei1)· · ·YLn(ein)hΩ({e})iei1...ein d2ei1· · ·d2ein, (1.3.33) where we introduced the notation
hfiei1...ein := 1 (4π)N−n
Z
· · · Z
f({e}) Y
ik∈α/
d2eik (1.3.34)
for the restricted directional average of a functionf, meaning that the averaging is carried out with the direction of the spins ei1, . . . ,ein kept fixed. Again, pairing this coefficient with its expansion function in Eq. (1.3.27) results in a more transparent form,
JαµΦαµ=YL1(ei1)· · ·YLn(ein)
× Z
· · · Z
YL1(ei1)· · ·YLn(ein)hΩ({e})iei1...ein d2ei1· · ·d2ein. (1.3.35) The final form of the expansion can then be rewritten as
Ω({e}) = Ω0+X
i
X
L6=0
JiLYL(ei) + 1 2
X
i6=j
X
L,L06=0
JijLL0YL(ei)YL0(ej) +· · ·, (1.3.36)
by which arbitrarily complex spin interactions may be taken into consideration. According to Eq. (1.3.35) the parameters in Eq. (1.3.36) are calculated as
JiL= Z
hΩieiYL(ei) d2ei
JijLL0 = Z Z
hΩieiejYL(ei)YL0(ej) d2eid2ej
...
(1.3.37)
Clearly the key quantities are the restricted directional averages of the grand potential.
In principle, it is possible to perform a series of ab initio calculations to perform the averages of the first-principles grand potential manually, but it is obvious that a tractable method of obtaining averages overN −n spin variables is necessary. It should be noted at this point that the SCE by itself is a clear mathematical expansion of functions defined over the 2N-dimensional space spanned by the classical spin variables of the system, and
that the directional averages appearing above are due to the orthogonality properties of the single-spin basis functions. These averages are not in any way related to a possible paramagnetic state of the system, all information regarding the magnetic behaviour is encoded in the directional averages of the grand potential. Whether the parametrization is suitable for a given system (magnetically ordered or not) only depends on the goodness of the guess forhΩiei...and the choice of termination of the infinite SCE series in Eq. (1.3.36).
Combining the SCE with paramagnetic RDLM by computing the restricted averages of the grand potential within the framework of RDLM makes for an efficient mapping procedure providing spin model parameters directly from the electronic structure. For this we only need to evaluate the directional averages of Eq. (1.2.29) up to two restricted sites, and note that these averages are exactly in accordance with the RDLM single-site probabilities in a uniform paramagnetic state. During the evaluation of these averages, we will neglect the so-called backscattering contributions, i.e. terms in which every site outside the fixed cluster appears at least twice. This approximation is consistent with the single-site CPA [53].
The constant term of the SCE, Eq. (1.3.31), is given by the average of the grand potential,
Ω0 =hΩ({e})i= Ωc+X
i
I Z 1
4πln detDi(ε;ei) d2ei. (1.3.38) The single-site restricted average reads as
hΩ({e})iei = Ωc+Iln detDi(ε;ei) +IX
j(6=i)
1 4π
Z
ln detDj(ε;ej) d2ej, (1.3.39)
leading to the formula for the single-site SCE coefficients (through Eq. (1.3.37)), JiL=
Z
hΩieiYL(ei) d2ei =I Z
YL(ei) ln detDi(ε;ei) d2ei. (1.3.40) Note that after determining the impurity matrices on a sufficiently dense mesh for the spherical integrations, it is only a matter of choice at which L we stop in the expansion.
Given the matricesDi(ε;ei), we may include on-site anisotropies of arbitrary order with virtually the same amount of computational effort.
For a spin Hamiltonian containing up to two-spin interactions, only two-site restricted
averages of the grand potential are left to determine. These can be expressed as hΩ({e})ieiej = Ωc+I
ln detDi(ε;ei) + ln detDj(ε;ej)
+I X
k(6=i,j)
1 4π
Z
ln detDk(ε;ek) d2ek
+I X∞
k=1
2 2k Tr
Xi(ε;ei)τc,ij(ε)Xj(ε;ej)τc,ji(ε)k
.
(1.3.41)
Note that most of the terms in the final sum of Eq. (1.2.29) visit at least one included site only once, and these contributions are exactly zero in the restricted average due to the CPA condition, a special case of Eq. (1.2.15):
hXi(ε;ei)i= 1 4π
Z
Xi(ε;ei) d2ei. (1.3.42) The neglect of backscattering terms refers to the rest of the terms, where every site outside the two fixed spins appears at least twice. Similarly to the previous cases, orthogonality of the spherical harmonics implies that only the last term of Eq. (1.3.41) contributes to the two-site SCE coefficient,
JijLL0 = Z Z
hΩieiejYL(ei)YL0(ej) d2eid2ej
=I X∞
k=1
1 k
ZZ
YL(ei)YL0(ej)
×Tr
Xi(ε;ei)τc,ij(ε)Xj(ε;ej)τc,ji(ε)k
d2eid2ej
=−I ZZ
YL(ei)YL0(ej)
×Tr ln
I−Xi(ε;ei)τc,ij(ε)Xj(ε;ej)τc,ji(ε)
d2eid2ej. (1.3.43) While on paper more concise in its final form, the logarithm in Eq. (1.3.43) is actually computed in terms of the power series. This series converges quite rapidly, as the site off-diagonal elements of the SPO, like those of the real space Green’s function, decay with distance, and with each consecutive order of the power series two more propagations are included between sitesiand j. Correspondingly, the largest contributions come from the first order term, and higher orders can be optionally neglected to obtain the so-called simplified pair interactions. A huge technical benefit of this approximation is that the angular integrals in Eq. (1.3.43) can be performed independently, substantially reducing computational time. From a theoretical point of view, the simplified SCE-RDLM pair interactions are what correspond to our formulation of the RTM-RDLM method to be
detailed in Section 2.3.2.
All that is left is to map the SCE to the usual form of the spin Hamiltonian, i.e., a second-order tensorial Heisenberg model as seen in Eq. (1.3.1). For this we have to consider the connection between the basis functions and the Cartesian coordinates. The explicit forms of the real spherical harmonics will be needed; the functions up to L = 8 (` = 2) are listed in Table 1.1.
Y0(x, y, z) = 1
√4π Y1(x, y, z) = r 3
4πx Y2(x, y, z) = r 3
4πy Y3(x, y, z) =
r 3
4πz Y4(x, y, z) = r15
4πxy Y5(x, y, z) = r15
4πxz Y6(x, y, z) =
r15
4πyz Y7(x, y, z) = r 15
16π x2−y2
Y8(x, y, z) = r 5
16π 3z2−1 Table 1.1: Real spherical harmonics up to` = 2.
In order to relate the SCE two-site terms to the usual Heisenberg couplings, we ob- viously have to match the two-site parameters of the spin Hamiltonian with the two-site SCE terms. For this we have to perform the matching to ensure that
−eiJ
ijej = X
L,L0∈L
JijLL0YL(ei)YL0(ej) (1.3.44)
for an appropriate set of indicesL. Strictly speaking, we would have to match the full SCE to a complete spin Hamiltonian (of infinite order) containing the usual model Eq. (1.3.1), determine the connection between the two sets of infinite coefficients (leading to, in gen- eral, an infinite system of linear equations connecting the two sets of coefficients), then dispose of the unnecessary coefficients that go beyond the tensorial Heisenberg model.
The reason why this isn’t necessary is because of the clear bijection between real spheri- cal harmonics for `= 1 and the linear functions x, y, z,
ei ≡ e1i, e2i, e3i
≡(xi, yi, zi) = r4π
3 Y1(ei), Y2(ei), Y3(ei)
, (1.3.45)
implying
−eiJ
ijej =− X3
αβ=1
eαi Jij
αβeβj =−4π 3
X3
L,L0=1
J
ij
LL0YL(ei)YL0(ej). (1.3.46)
Comparing this with Eq. (1.3.44), we see that the L, L0 = 1,2,3 (or` =`0 = 1) subspace of real spherical harmonics give exactly the tensorial exchange interaction, there is only a numerical factor between the two sets of coefficients. We have now arrived at the final form of the tensorial exchange coupling within the SCE-RDLM scheme:
Jij
αβ =− 3 4πJijαβ
=− 3 4πI
ZZ
Yα(ei)Yβ(ej)
×Tr ln
I−Xi(ε;ei)τc,ij(ε)Xj(ε;ej)τc,ji(ε)
d2eid2ej
=− 9 16π2I
ZZ eαieβj
×Tr ln
I−Xi(ε;ei)τc,ij(ε)Xj(ε;ej)τc,ji(ε)
d2eid2ej. (1.3.47) We note that the two-site contributions can also be extended to higher orders inL, L0 without much effort, and for many novel systems the Heisenberg model seems insufficient.
The SCE allows for the natural extension of the tensorial Heisenberg model with higher- order interactions, the simplest form of which is an isotropic biquadratic coupling between spins in the form of −12 P
i6=j
Bij(ei ·ej)2, expected to dominate among biquadratic terms.
The mapping for these terms reads as [32]
Bij =− 3 8π
X8
L=4
JijLL, (1.3.48)
clearly showing that no additional information is needed from the electronic structure as compared to the bilinear interactions. We also note that the usual tensorial Heisenberg model could also be extended with multi-spin interactions, however, these include very complex terms originating from Eq. (1.2.29), and it is not clear how such couplings could be computed using SCE-RDLM in a tractable way. For reference, the SCE coefficient of a general n-spin interaction within SCE-RDLM would be given by [43]
JiL11,i,L2,...,i2,...,Ln n =I X∞
k≥n
1 k
X
j16=j26=...6=jk6=j1 1≤j1,j2,...,jk≤n
Z
· · ·
Z Yn
l=1
YLl(el)
!
×Trh Xij
1(ej1)τc,ij
1ij2Xij
2(ej2). . . Xi
jk(ejk)τc,i
jkij1
i d2e1· · ·d2en. (1.3.49) The SCE-RDLM can be extended to the FM phase, i.e. to systems with nonzero net magnetization. This would allow the spin model to be determined at finite temperatures,
and thermal effects in the electronic structure (in particular, longitudinal spin fluctua- tions) could be naturally incorporated into the model parameters. Our attempts at this generalization are detailed in Section 2.3.1, although preliminary calculations suggest that some approximations used above might fail in the ordered phase.