Chapter 2 Results 46
2.3 Attempts to extract exchange interactions at finite temperature
2.3.1 SCE-RDLM method below the Curie temperature
2.3 Attempts to extract exchange interactions at fi-
functions, for which
Z
Yµ1(e)∗Yµ1(e) d2e=δµ1,µ2. (2.3.4) One main difference between spherical harmonics and the generalized functions is, how- ever, that while the former are orthonormal with respect to a weight functionw(e) = 1, the latter are defined using a probability density, so their weight function isw(e) =P(e) = 4π1 . A natural specification for the basis functions is thus
u0(e) := const. =c, (2.3.5)
but then normalization implies
1 =hu0, u0i= Z
|c|2P(e) d2e =|c|2, (2.3.6) so let us choose
u0 = 1. (2.3.7)
Given the developed paramagnetic SCE-RDLM theory, it seems most straightforward to construct the new single-site functions based directly on spherical harmonics, so that the T → ∞ limit corresponds clearly to the usual paramagnetic case of the theory.
The Gram–Schmidt process provides a straightforward algorithm to construct the basis functions, but along the process the definition in Eq. (2.3.2) has to be used instead of the usual L2 scalar product.
Assuming that we are equipped with a set of single-site functions {uµ} for which u0 = 1 and Eq. (2.3.2) holds, we may carry on with the development of the spin-cluster expansion. Since the theory mainly needs only the existence of a scalar product on the single-site basis functions, the paramagnetic theory detailed in Chapter 1 may be generalized straightforwardly.
For an arbitrary set of quantum numbers{µ}={µ1, µ2, . . . , µN}(N being the number of spins in the system), due to the property Eq. (2.3.7) we may write
uµ1(e1)uµ2(e2)· · ·uµN(eN) = Y
ik∈α
uµik(eik), (2.3.8)
where α = {i1, i2, . . . , in} is an index set collecting the n sites for which the quantum numberµikis nonzero, i.e., the corresponding single-site function is nontrivial. Conversely, for any n-spin cluster α (n ≥ 1) and nonzero quantum number set µ ={µ1, . . . , µn} we
may define the cluster expansion functions
Φαµ({e}) :=uµ1(ei1)· · ·uµn(ein) (2.3.9) along with the empty cluster function
Φ0({e}) := 1. (2.3.10)
By naturally extending the definition of the scalar product to functions of the entire spin configuration {e} as
hf({e}), g({e})i:=
Z
f({e})∗g({e}) YN
k=1
Pk(ek) d2ek
, (2.3.11)
the resulting cluster expansion functions are orthonormal, hΦ0,Φ0i= 1
hΦ0,Φαµi= 0 hΦαµ,Φβνi=δαβδµν.
(2.3.12)
We can then expand any spin-dependent function (in particular, the adiabatic mag- netic energy surface) to disjoint terms with respect to n-spin clusters α with index set µ(α) containing only nonzero quantum numbers,
Ω({e}) = Ω0Φ0+X
α
X
µ(α)
JαµΦαµ({e}), (2.3.13)
where the coefficients can be calculated in terms of projections onto the cluster expansion functions due to the orthogonality property Eq. (2.3.12) as
Ω0 =hΦ0, Ωi= Z
Ω({e}) YN
k=1
Pk(ek) d2ek
=
Ω({e})
, (2.3.14)
Jαµ=hΦαµ, Ωi= Z Yn
m=1
(uµm(eim)∗) Ω({e}) YN
k=1
Pk(ek) d2ek
=Z
Ω({e})
α
Yn
m=1
uµm(eim)∗Pim(eim) d2eim
. (2.3.15)
Here we introduced the thermodynamic average and thermodynamic restricted average Ω({e})
:=
Z
Ω({e}) YN
k=1
Pk(ek) d2ek
(2.3.16)
Ω({e})
α :=
Z
Ω({e})Y
ik6∈α
Pik(eik) d2eik
. (2.3.17)
The spin-cluster expansion is then an infinite series expansion of the energy with terms describing interactions over clusters of increasing size,
Ω({e}) = Ω0+X
i
X
µ6=0
Jeiµuµ(ei) + 1 2
X
i6=j
X
µ,µ06=0
Jeijµµ0uµ(ei)uµ0(ej) +. . . , (2.3.18)
where the coefficients can be readily calculated as Ω0 =
Ω({e})
, (2.3.19)
Jeiµ =Z
Ω({e})
{i}uµ(ei)∗Pi(ei) d2ei, (2.3.20) Jeijµµ0 =Z
Ω({e})
{i,j}uµ(ei)∗uµ0(ej)∗Pi(ei)Pj(ej) d2eid2ej, (2.3.21) ...
In order to relate the generalized SCE to the usual PM SCE (which can be inter- preted more easily in terms of spin models), we have to construct the spherical harmonic expansion of the single-site basis functions,
u0(e) = 1 =√
4πY0(e) (2.3.22)
uµ(e) =X
L6=0
CiµLYL(e), (µ6= 0) (2.3.23)
which can either be determined during their construction, or a posteriori by using the orthogonality of spherical harmonics leading to
CiµL = Z
uµ(ei)YL∗(ei) d2ei. (2.3.24) Note that the transformation matrices CiµL are not necessarily unitary, since spherical harmonics and the ui,µ basis functions are orthonormal with respect to two separate metrics, so the orthogonalization process does not correspond to a change of basis. Using
Eq. (2.3.24) the general expansion Eq. (2.3.18) can be rewritten as Ω({e}) = Ω0+X
i
X
L6=0
X
µ6=0
JeiµCiµL
!
| {z }
JiL
YL(ei)
+ 1 2
X
i6=j
X
L6=0
X
L06=0
YL(ei) X
µ,µ06=0
Jeijµµ0CiµLCjµ0L0
!
| {z }
JijLL0
YL0(ej) +. . . , (2.3.25)
leading to the matching relations
JiL =X
µ6=0
JeiµCiµL JijLL0 = X
µ,µ06=0
Jeijµµ0CiµLCjµ0L0. (2.3.26)
The RDLM electronic structure input into the partial averages of the grand potential starts from the same point as before, the adiabatic magnetic energy for a fixed configu- ration:
Ω({e})≈Ωc+IX
i
ln detDi(ei) +I
X∞
k=2
1 k
X
i16=i26=...6=ik6=i1
Tr
Xi1(ei1)τc,i1i2Xi2(ei2). . . Xik(eik)τc,iki1 . (2.3.27) The CPA condition, a simplified form of Eq. (1.2.15), now reads as
hXi(ei)i= Z
Pi(ei)Xi(ei) d2ei = 0, (2.3.28) implying that every term in the last sum of Eq. (2.3.27) which contains at least one site only once has vanishing partial averages.
If we again neglect backscattering terms in the partial averages of the grand potential, i.e. contributions for which every fluctuating site index appears at least twice in the final term of Eq. (2.3.27), then formally the same expressions as the PM SCE (cf., for instance, Eq. (1.3.43)) can be obtained for the generalized SCE expansion coefficients, with the role
of expansion functions straightforwardly swapped with the basis functions:
Ω0 =
Ω({e})
= Ωc+IX
i
Z
Pi(ei) ln detDi(ei) d2ei,
Jeiµ=Z
Ω({e})
{i}uµ(ei)∗Pi(ei) d2ei
=I Z
uµ(ei)∗Pi(ei) ln detDi(ei) d2ei,
Jeijµµ0 =−I ZZ
uµ(ei)∗uµ0(ej)∗Pi(ei)Pj(ej) Tr ln
I−Xi(ei)τc,ijXj(ej)τc,ji
d2eid2ej. (2.3.29) Note that from a computational point of view, it is quite useful to combine the match- ing (2.3.26) with the obtained interaction parameters of Eq. (2.3.29), to arrive at a direct expression for the PM SCE interaction parameters,
JijLL0 =−IZZ X
µ6=0
uµ(ei)∗CiµL
! X
µ06=0
uµ0(ej)∗Cjµ0L0
!
Pi(ei)Pj(ej)
×Tr ln
I−Xi(ei)τc,ijXj(ej)τc,ji
d2eid2ej
=−I ZZ
Ai,L(ei)Aj,L0(ej)Pi(ei)Pj(ej) Tr ln
I−Xi(ei)τc,ijXj(ej)τc,ji
d2eid2ej, (2.3.30) thereby saving us the trouble of computing and storing huge matrices with indices µ, µ0. This way we directly calculate the 9×9 matrix (up to`, `0 = 2)JijLL0 for each pair of sites, which implies that we do not need to store the SCE interaction matrix Jeijµµ0, nor do we have to perform a cumbersome double summation over µ indices along with the double integral over Lebedev grid points in Eq. (2.3.30), substantially reducing computational effort. Instead, we only have to calculate the auxiliary functions
Ai,L(ei) =X
µ6=0
uµ(ei)∗CiµL, (2.3.31)
involving only a single sum over µ indices, and which needs to be carried out only once for the entire calculation, albeit for a sufficiently largeµ-cutoff.
Although the above procedure seems straightforward and just as well-founded as the PM case, preliminary calculations indicate that some point of the procedure is failing.
The isotropic Heisenberg terms otained for bulk bcc-Fe with lattice constanta= 2.789 ˚A
0.0 0.2 0.4 0.6 0.8 2.61.0
2.8 3.0 3.2 3.4 3.6 3.8 4.0
J1[mRy]
SCE-RDLM RTM
0.0 0.2 0.4 0.6 0.8
−0.21.0 0.0 0.2 0.4 0.6 0.8 1.0
J2[mRy]
SCE-RDLM RTM
0.0 0.2 0.4 0.6 0.8
−0.41.0
−0.3
−0.2
−0.1 0.0 0.1
m J4[mRy]
SCE-RDLM RTM
0.0 0.2 0.4 0.6 0.8
−0.21.0
−0.1 0.0 0.1 0.2 0.3 0.4 0.5
m J5[mRy]
SCE-RDLM RTM
Figure 2.26: Preliminary results for the first few nn isotropic Heisenberg terms of bcc-Fe with a= 2.789 ˚A using the SCE-RDLM method generalized to the ordered phase.
are shown for a first few nearest neighbours in Fig. 2.26 along with the T = 0 K case of the torque method. Even though the PM limit of the calculations goes smoothly into the usual PM SCE-RDLM results, at low temperatures the computed interaction parameters seem to diverge, showing spurious intensification. For the first three nearest neighbours (only first two shown) the inconsistency is less evident, and one might be tempted to think that the apparent lack of smoothness between the low-temperature SCE-RDLM couplings and those from the RTM are due to some numerical subtlety. However, the fourth nn couplingJ4 strongly overestimates the zero-temperature limit, and the curve forJ5 shows very different behaviour from the RTM: the obtained coupling diverges towards positive values while the RTM suggests a small and negative coupling (already the curve of J5
versus magnetization starts off from the PM limit in the other direction).
In order to test the computational scheme we artificially entered a perfect isotropic Heisenberg model into the SCE formula, Eq. (2.3.21). The constrained two-site average
of an isotropic Heisenberg model is given by
*
−1 2
X
k6=l
Jklek·el
+
ei,ej
=−Jijei·ej
− X
k6=i,j
(Jikei+Jjkej)·mk−1 2
X
k6=l k,l /∈{i,j}
Jklmk·ml, (2.3.32)
with the dimensionless average magnetizationmk=heki. As the second and third terms in Eq. (2.3.32) are independent of at least one ofeiandej, these terms do not contribute to the corresponding two-site SCE coefficients due to the orthogonality of the SCE expansion functions, cf. Eqs. (2.3.2) and (2.3.21). Consequently, it suffices to input
hΩ({e})i{i,j} =−Jijei·ej (2.3.33) into the SCE solver (and in particular, Eq. (2.3.21)), and the computation should ulti- mately yield an isotropic model corresponding to the input regardless of the probability density (or, equivalently, order parameter) used in the calculation. Indeed, when carrying out this test our program perfectly reproduced the underlying isotropic Heisenberg model, even for an order parameter of m= 0.9 in which case the probability density is peak-like and the single-site SCE functions are far from spherical harmonics.
These considerations suggest that the orthogonalization procedure of the single-site functions and the mapping in Eq. (2.3.26) works properly, and that rather the ab initio input into the method is inconsistent. According to our current understanding, the most probable culprit is the neglect of backscattering terms when evaluating the RDLM par- tial averages, especially as the consequently ignored vertex corrections are likely to give significant contributions in the dilute limit of alloys (corresponding to them→1 limit of our magnetic theory).