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Computational Materials Science xxx (xxxx) xxx

Please cite this article as: Evgeny A. Plekhanov, Andrei L. Tchougrée, Computational Materials Science,

https://doi.org/10.1016/j.commatsci.2020.110140

0927-0256/© 2020 Elsevier B.V. All rights reserved.

Efcient magnetic superstructure optimization with ΘΦ

Evgeny A. Plekhanov

a

,

b

,

*

, Andrei L. Tchougr´

eeff

b

a

King’s College London, Theory and Simulation of Condensed Matter (TSCM), The Strand, London WC2R 2LS, UK

b

A.N. Frumkin Institute of Physical Chemistry and Electrochemistry of RAS, Moscow, Russia

ARTICLE INFO

Keywords:

Keywords: Hubbard model

Heisenberg model

Spin-density waves

Strongly-correlated systems

ABSTRACT

Simulating the incommensurate spin density waves (ISDW) states is not a simple task within the standard ab initio

methods. Moreover, in the context of new material discovery, there is a need for fast and reliable tool capable to

scan and optimize the total energy as a function of the pitch vector, thus allowing to automatize the search for

new materials. In this paper we show how the ISDW can be efciently obtained within the recently released ΘΦ

program. We illustrate this on an example of the single orbital Hubbard model and of γ-Fe, where the ISDW

emerge within the mean-eld approximation and by using the twisted boundary conditions. We show the

excellent agreement of the ΘΦ with the previously published ones and discuss possible extensions. Finally, we

generalize the previously given framework for spin quantization axis rotation to the most general case of spin-

dependent hopping matrix elements.

1. Introduction

Incommensurate spin structures (ISS) appear in various contexts of

condensed matter physics: frustrated spin systems[1,2], cuprates high-

temperature superconductors[3–5] and strongly correlated electronic

systems in general. In particular, the question of how the ISS appear as a

consequence of strong electron repulsion, or spin exchange is very

interesting, since the ISS come out as a result of a subtle balance between

the kinetic energy loss and potential energy gain. In Ref. [6], it was

shown how the ISS emerge out of the on-site repulsion, at least within

the mean-eld approximation. On the other hand, in Refs. [6–9], it was

shown how an arbitrary ISS pitch vector can be treated without

increasing the unit vector size by introducing the twisted boundary

conditions instead of the periodic ones. It was shown therein, that at

least at the mean-eld level, ISS emerge as the most stable phase at the

intermediate interaction strength and away from half-lling.

On the other hand, recently, our program ΘΦ[10,11], allows for ISS,

superconductivity of arbitrary order and Resonating Valence Bond

(RVB) states in multi-orbital electronic systems at nite temperature. In

addition, ΘΦ is capable to import the hopping parameters from major ab

initio codes by means of wannier90[12] and LOBSTER[13–15] programs,

which makes it possible to perform practically ab initio strongly corre-

lated magnetic or superconducting calculations.

The scope of this paper is twofold. Firstly, we test and benchmark the

capabilities of ΘΦ applying it to the ISS in single orbital Hubbard model

and in γ-Fe and compare our ndings with those obtained previously and

independently within the approaches and programs different from ΘΦ.

Secondly, in this paper, we present a general framework for the spin

quantization axis rotation in a general case when no assumptions are

made on the spin dependence of the Hamiltonian’s hopping matrix el-

ements. This general framework extends the one already presented in

the Ref. [11] and allows to treat the cases when the hopping is spin-

dependent and contains the spin-ip terms, thus permitting simula-

tions of systems with spin–orbit coupling and explicit break of time-

reversal symmetry.

This paper is organized as follows: we present the methods used in

Section 2 and Appendix A, the results of our ΘΦ calculations are shown

in Section 3, while Discussion and Conclusions are given in Section 4.

2. Methods

Recently proposed program ΘΦ [11] allows to fulll two of the most

important extensions of the essentially single-particle mean-eld para-

digm of the computational solid state physics: the admission of the

Bardeen–Cooper–Schrieffer electronic ground state and allowance of the

magnetically ordered states with an arbitrary superstructure (pitch)

wave vector. Both features are implemented in the context of multi-band

systems and permit the interface with the solid state quantum physics

packages eventually providing access to the rst-principles estimates of

the relevant matrix elements of the model Hamiltonians derived from

* Corresponding author at: King’s College London, Theory and Simulation of Condensed Matter (TSCM), The Strand, London WC2R 2LS, UK.

E-mail address: evgeny.plekhanov@kcl.ac.uk (E.A. Plekhanov).

Contents lists available at ScienceDirect

Computational Materials Science

journal homepage: www.elsevier.com/locate/commatsci

https://doi.org/10.1016/j.commatsci.2020.110140

Received 22 July 2020; Received in revised form 21 September 2020; Accepted 20 October 2020

Computational Materials Science xxx (xxxx) xxx

2

the standard DFT calculations. In the Ref. [11, we have presented a

general approach to the spin quantization axis rotation, which correctly

works in the case of multi-orbital spin-independent hopping matrix el-

ements. In the present paper, we extend the spin quantization axis

rotation method to a general case of arbitrary (Hermitian) hopping. The

details of the derivation are given in the Appendix A, while the brief

summary is shown below:

•At a nite pitch vector Q, a hoping matrix tij(k,k′)δ(k−k′)is trans-

formed into a new matrix tij(k,k′)δ(k−k′±Q)in which the states

with momentum k and spin e.g. “up” are connected to the states with

the momentum k+Q and the spin “down” and vice versa.

•The translational invariance, “broken” by imposing the incommen-

surate spin spiral with the pitch vector Q, can be restored if we shift

the “down” states by Q prior to the solution of the secular equation.

•The interaction terms like Coulomb repulsion and Heisenberg ex-

change, which are typically fourth rang tensors, have to be trans-

formed according to the formulas given in Ref. [11], and this task is

facilitated by the fact that the most important contributions to them

are local i.e. do not extend outside of the unit cell.

•In the case when the hopping term is spin-independent, the

Q-dependence only comes from the interaction term, and the ISS

state is stabilized if the energy gain from the “spiralizing” the

interaction is greater than the energy lost by the kinetic energy.

•In the case when the hopping term is spin-dependent, the kinetic

energy becomes Q-dependent, although the main contribution to the

stabilization of the ISS is still expected to come from the interaction

terms.

The mean-eld self-consistency workow proceeds as usual, with the

modied Hamiltonian, and, at self-consistency, the internal energy

E(Q)as a function of Q is obtained.

3. Results and benchmarks

3.1. Incommensurate spin spirals in Hubbard model

The single orbital Hubbard model considered in this paper has the

following Hamiltonian:

H= − t

〈mn〉c†

n

σ

cm

σ

+H.c.+U

n

nn↑nn↓−

μ

n

σ

nn

σ

(0.1)

Here cn

σ

is the electron annihilation operator on site n with spin

σ

,

nn

σ

=c†

n

σ

cn

σ

is the electron occupation operator, t is the nearest

neighbor hopping parameter, U is the onsite Coulomb repulsion, while

μ

is the chemical potential. In the Hamiltonian (0.1), the electrons move

on a square lattice and only nearest neighbor hopping is considered,

which is emphasized by the notation 〈nm〉. Finally,

μ

is the system’s

chemical potential, which enforces the correct number of particles in the

system. The calculations are performed at a nite temperature T which

was typically kept low enough (T=10−4t) and on a dense 2D k-point

grid of 400 ×400 points.

The Hamiltonian (0.1) cannot be solved exactly as it is, however, in

the atomic limit (t=0) it does allow for the exact solution, revealing a

rich phase diagram with charge and spin orderings, which rapidly

evolve as a function of temperature[16–19].

In this work, we consider the incommensurate spin density waves

with the pitch vector dened as:

Q= (

π

,

π

)+δ(1,1).(0.2)

This phase is referred to as the (1,1)phase in Ref. [6] and it is assumed

that δ≪1.

Within ΘΦ, the calculations are performed by using the extended

density matrix

ρ

as the self-consistency procedure variable, so that the

internal energy E(Q)at each pitch vector Q is at self-consistency. In ΘΦ,

the density matrix is of the most general form, allowing for incom-

mensurate spin-density waves, superconductivity, Resonating Valence

Bonds (RVB). The extended density matrix is dened through the basic

fermionic operators as follows:

ρ

n,m(

τ

) = δ

τ

,0δn,m−Ψ†

n,RΨ†

m,R+

τ

,(0.3)

and the operatorial basis Ψ†

R for a single-orbital model is dened as:

Ψ†

R=c†

R↓,c†

R↑,c†

R↓,c†

R↑.(0.4)

In the particular case of a single band Hubbard model with spin-density

waves and without superconductivity and RVB, the structure of

ρ

is of

the following form:

ρ

=

n−

σ

20 0 0

0n+

σ

20 0

0 0 1 −n−

σ

20

0 0 0 1 −n+

σ

2

.(0.5)

Here, n is the average site occupation, while

σ

is the average site

magnetization. One can see from Eq. (0.5) that this type of

ρ

describes a

site with occupation imbalance between ‘up’ and ‘down’ channels. This

would normally correspond to the ferromagnetic order, however, thanks

to the method of spin quantization axis rotation, proposed in Ref. [6,7]

and implemented in ΘΦ, an arbitrary spin-density wave pitch vector can

be introduced and treated at the same computational cost as the simplest

ferromagnetic phase.

The multidimensional optimization built in ΘΦ (simplex algorithm)

allows to optimize efciently the internal energy of the system E(Q)as a

function of the pitch vector Q. A typical internal energy prole and the

progress of the minimization is shown in Fig. 1. We notice that the

commensurate antiferromagnetism with the pitch vector (

π

,

π

)corre-

sponds to a local maximum of E(Q), and the minimum of E(Q)is

twofold degenerate with the minima located symmetrically with respect

to (

π

,

π

). The optimization procedure shown in Fig. 1 corresponds to a

single set of the Hamiltonian parameters U and δ. We have performed a

scan in this parameter space in order to benchmark our implementation

against the data published in Ref. [6]. The comparison can be seen in

Fig. 3. We notice a very good overall agreement between the two results

-0.820

-0.800

-0.780

-0.760

-0.740

-0.720

-0.700

0 1 2 3 4 5 6 7

Q

E(Q)

Fig. 1. A typical internal energy prole E(Q)U=5t, δ=0.15,T=0.0001t.

E(Q)has two local minima symmetric with respect to (

π

,

π

), which is a local

maximum. The blue line shows the simplex optimization within ΘΦ.

E.A. Plekhanov and A.L. Tchougr´

eeff

Computational Materials Science xxx (xxxx) xxx

3

which validates the use of ΘΦ in the context of ISS. At largest t/U

considered (t/U=0.2), the offset as a function δ starts linearly from

zero and reaches saturation around δ=0.25. At a smaller value of t/U=

0.1 (U=10t), the offset monotonically grows from zero to approxi-

mately aΔQ/

π

=0.8.

In order to demonstrate the computational efciency of ΘΦ, we

report in Fig. 2 the nearly perfect scaling of our program’s computa-

tional time as a function of the number of k-points in the Brillouin zone.

The tests were performed for the single orbital Hubbard model on a 2D

square lattice. The slope in logarithmic scale implies a quadratic

dependence in the range of Nk shown, as it should be for the integration

in 2D.

3.2. Spiral phases of γ-Fe

In the raw of 3d materials, Fe falls close to the node separating the

antiferromagnetic metals Cr and Mn with a nearly half-lled d band

from strong ferromagnets Co and Ni with a nearly full band. It is clear,

therefore, that the magnetic ordering in Fe should be extremely sensible

to the interatomic distances, bond angles, volume packing type and

other crystallographic details. In fact, it was found experimentally[20],

that the magnetic ordering of γ-A phase of Fe precipitates in Cu is a spin-

spiral state propagating with wave vector Qexp =2

π

/a(0.1,0,1)(Carte-

sian coordinates).

On the other hand, theoretically, a number of attempts has been

made to tackle this problem (see Refs. [13–15], and the references

therein). The main theoretical conclusion was that, in general, there are

two minima: one at the ΓX and the other at the XW lines, while the

relative depth of these minima depends on the unit cell volume: at

higher volumes, the ΓX line minimum is lower, while at lower volumes,

the XW line minimum is more stable. In particular, in the Ref. [21], the

two minima have the following positions: Q1=2

π

/a(0,0,0.6)and

Q2=2

π

/a(0.2,0,1)(Cartesian coordinates). In the internal coordinates,

these points translate into: Q1=6

10 ΓX, while Q2=2

5XW. The rational

numbers in these denitions reect the fact that the Q-vector has to

belong to the same nite Brillouin zone grid as the one used in the DFT

calculations (13 ×13 ×13 Monkhorst–Pack in the Ref. [21]). Denser

grids are expected to rene these numbers to some extent.

In this section, we show how this theoretical picture can be repro-

duced by using ΘΦ. We derive the hopping parameters from VASP DFT

calculations[22] using the maximally-localized Wannier orbitals, as

implemented in wannier90 package[12]. We consider γ-Fe with fcc

lattice at several lattice constants: a=3.678 Å, a=3.583 Å, a=3.577

Å, a=3.545 Å, a=3.510 Å, and a=3.493 Å, which correspond to

decreasing volumes of V=12.439 Å

3

V=11.500 Å

3

, V=11.442 Å

3

, V

=11.138Å

3

, V=10.811 Å

3

and V=10.655 Å

3

respectively. In

extracting the hopping parameters, we utilize VASP paramagnetic DFT

calculations with PBE[23] functional at a 8 ×8×8 k-point grid, with

E=500 eV plane-wave cut-off. The quality of the wannierization pro-

cedure employed to obtain the hopping parameters is similar to the one

reported in our previous work (Ref. [11], Appendix C).

In the Refs. [24,25,21], the magnetic ordering originated within the

LSDA approximation as a consequence of the Stoner mechanism, where

the exchange correlation functional acts as a “driving force”, gaining

energy from magnetic polarization. Therefore, in this case, the exchange

correlation functional determines the scale of the total energy landscape

as a function of Q-vector. In our calculations, we use another “driving

force”, namely the local Heisenberg exchange interaction J which

transforms under spin quantization axis rotation as described in

Ref. [11]. This Heisenberg interaction is treated within the mean-eld

approach. It is clear that the scale of the relative energy gains will be

of different origin in our case and cannot be directly compared to the

above mentioned references. Nevertheless, we will show that the rela-

tive stability of the two local minima as a function of volume is correctly

reproduced by ΘΦ. We have used J= − 1eV (ferromagnetic exchange)

for Fe. This value was already successfully tested in our previous ΘΦ

calculations of Fe in the Ref. [11]. In addition, this value is constrained

from below by the fact that a value as small as J=− 0.8eV does not

stabilize at all any magnetic solution, and from above by the observation

that a value as big as J=− 1.2eV stabilizes a ferromagnetic solution,

which is always more stable than any ISS state. Finally, we used a full

19 ×19 ×19 Monkhorst–Pack k-point grid in our ΘΦ calculations. It

was pointed out in Ref. [21], that inuence of the spin–orbit coupling on

the magnetization in γ-Fe is very small, that is why we decided to neglect

it in the present work.

Taking into account the above considerations, the Hamiltonian used

in the present section assumes the form:

HFe =

i,j,

τ

,

α

,β,R

c†

R,i,

α

t

α

β

ij

τ

cR+

τ

,j,β(0.6)

+

i,j,

τ

,

α

,β,R

J

α

,β

i,j

τ

S

α

i,RSβ

j,R+

τ

,(0.7)

where the hopping matrix t

α

β

ij (

τ

)is obtained from the wannierization

procedure and is transformed under spin quantization axis rotations as

described in the Appendix, S

α

i,R is the

α

-component of the spin operator

on orbital i in the cell R, while the Heisenberg tensor J

α

,β

i,j(

τ

) = δ(

τ

)δ

α

,β×

J, with J= − 1eV, and the i and jindices are restricted to the d-shell

orbitals.

On the other hand, the density matrix in this case has the form

summarized in Table 1 (only the relevant diagonal part of the parti-

cle–particle

τ

=0 density matrix component is show).

Here, for the sake of brevity, 18 diagonal matrix elements are ar-

ranged in a 2 ×9 table with the orbital indices running along the col-

umns and the spin components running along the rows. The order of the

orbitals corresponds to the one established in wannier90. Finally, the

matrix elements are expressed in terms of the parameters ni and mi, so

that, for example, ns and ms are the occupation and the magnetization

of the sorbital respectively.

ΘΦ results for γ-Fe are shown in Fig. 3. The total energy dependence

in our calculations is, indeed, very sensible to the unit cell volume.

Starting from a=3.493 Å, the relative energy gain of the Q1 spin spiral

gradually grows up to a=3.545 Å, and then, as the volume increases

further, starts to decrease until, at a=3.678 Å, a situation realizes when

the ferromagnetic state at Q=0 becomes the most stable. The

101

102

103

104

105

400 800 1600 3200 6400

time, (sec.)

Nk

Fig. 2. A benchmark of ΘΦ, showing the dependence of the time for single self-

consistency solution on the number of k-points along each of the dimensions of

a 2D grid in the Brillouin zone. Note the logarithmic scale on both axes. Here,

N

k is the number of k-points along each of the dimensions of a 2D grid in the

Brillouin zone, so that in each grid there are Nk×Nk points. U=10t,T=

0.01t, ISS phase at Q=0.

E.A. Plekhanov and A.L. Tchougr´

eeff

Computational Materials Science xxx (xxxx) xxx

4

experimental volume V=11.442 Å

3

, corresponding to a=3.577 Å, is

close to the maximum energy gain tested in the present paper: 0.1eV.

The Q2 spin spiral is not reproduced in our calculations in the sense that

the broad minimum of the energy in the XW interval is realized at the W

point or in its vicinity, although the energy difference between the Q1

spin spiral and the whole XW line at the experimental volume is

extremely small.

In our calculations, the d-shell magnetic moment of the spiral phase

Table 1

Density matrix table for γ-Fe.

s pz px py dz2 dxy dyz dx2−y2 dxy

↑ ns+ms

2

npz +mpz

2

npx +mpx

2

npy +mpy

2

ndz2+mdz2

2

ndxz +mdxz

2

ndyz +mdyz

2

ndx2+mdx2

2

ndxy +mdxy

2

ns−ms

2

npz −mpz

2

npx −mpx

2

npy −mpy

2

ndz2−mdz2

2

ndxz −mdxz

2

ndyz −mdyz

2

ndx2−mdx2

2

ndxy −mdxy

2

0

0.05

0.1

0.15

0.2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0

0.2

0.4

0.6

0.8

1

t/U

δ

a

∆Q/π

Fig. 3. A benchmark of ΘΦ ISS phase diagram against the results of Ref. [6]: (left panel) Ref. [6], (right panel) ΘΦcalculations. The axes represent inverse interaction

t/U, doping δ and pitch vector offset from (

π

,

π

)normalized to

π

.

Fig. 4. Left panel: A scan on pitch vector along the Γ→X→W path of the spin spiral state energy gain with respect to the ferromagnetic state for different unit cell

volumes to compare against the results of Ref. [21]. Right panel: The total d-shell magnetization along the same path for different volumes. The black lines

correspond to the experimental volume.

E.A. Plekhanov and A.L. Tchougr´

eeff

Computational Materials Science xxx (xxxx) xxx

5

decreases as a function of Q at ΓX line and increases at XW line, having a

minimum at X, as shown in Fig. 4.The absolute value of the moment,

approximately one

μ

B, is similar to the values reported in the Ref. [21].

The energy gain of the Q1 spin spiral phase with respect to the Q=0

phase is approximately twice larger than the one reported in the Refs.

[24,21], and this difference can be ascribed to a completely different

mechanism which stabilizes the magnetic phase, as mentioned above.

4. Discussion and conclusions

In the present paper, we have shown how ΘΦ program can be easily

used to calculate the properties of incommensurate spin density waves.

We have performed the benchmark of our results against those of

Ref. [6] and Ref. [21] and shown the excellent agreement. In particular,

for the single-orbital Hubbard model we have shown that the ISS state

can be stabilized in a wide range of the on-site repulsion U, and the pitch

vector off-set from the commensurability varies from zero to one as a

function of U and doping δ. In addition, the ΘΦ minimization procedure

stably locates the total energy minimum as a function of the pitch vector

Q.

In the case of the ISS in γ-Fe, we have shown that it can be success-

fully stabilized and has the energy lower than the ferromagnetic state, at

least along the path Γ→X→W. Although in our calculations the inter-

action term, was completely different from the Refs. [24,21] our results

are rather similar to the ones reported in there. Indeed, in DFT the

interaction comes from the exchange functional, which, in turn, is ob-

tained by tting the homogeneous electron gas Monte Carlo simulations,

while in our calculations, the full (x,y,z-components) Heisenberg

interaction on d-orbitals, supplied with the spin quantization axis rota-

tion formulas was used. The only parameter in our calculations – The

Heisenberg exchange J– is close to the values routinely used for iron and

is constrained from below and above by the absence of magnetic solu-

tion and the instability with respect to the ferromagnetic phase

respectively. Taking into account these differences, our results are in

surprisingly good agreement with the previously published ones: the

sensitivity to the unit cell volume, the stability of the Q1 state, the order

of magnitude of the energy gain, the value of the magnetic moment. The

only feature not reproduced in our calculations is the absolute stability

of the Q2 state, although the energy difference between the Q1 and the

Q2 states is very small. This discrepancy is currently under investigation

and will be explained in a later work.

Additionally, in this paper, we have proposed a general approach for

the spin quantization axis rotation of the most problematic Hamiltonian

part – kinetic energy. This approach only requires the hopping matrix to

be Hermitian, without demanding that the off-diagonal spin-ip terms

like spin–orbit coupling be small and treated as perturbations. Such an

approach will allow us to efciently calculate the ISS in heavy elements

with sizable spin–orbit coupling like lanthanides and actinides.

Within ΘΦ, both single-orbital case and a general multi-orbital one

can be routinely treated as shown here and in Refs. [10,11]. This paves

the way to the computationally cheap calculations of ISS in complex

multi-orbital systems with the ‘ab initio’ predictive power. These cal-

culations can be further combined with the large-scale material search

codes (see e.g. Ref. [26]) in order to perform the large scale material

search with the given functional properties.

CRediT authorship contribution statement

Evgeny A. Plekhanov: Methodology, Software, Data curation,

Writing - original draft. Andrei L. Tchougr´

eeff: Conceptualization,

Validation, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing nancial

interests or personal relationships that could have appeared to inuence

the work reported in this paper.

Acknowledgement

Authors gratefully acknowledge the support of RSF (grant 19-73-

10206).

Appendix A. Rotating the local quantization axes

In this Appendix, we briey review the method of local quantization axes rotation, originally formulated in Refs. [6,8,9,11,27].

The most general form of spin-dependent hopping between an orbital i and an orbital j is:

tij

τ

=

t↑↑

ij

τ

t↑↓

ij

τ

t↓↑

ij

τ

t↓↓

ij

τ

.(8)

It obeys the hermiticity condition:

tij

τ

=t†

ji−

τ

.

On the other hand, the rotation operator (around y-axis) reads as:

Ω

τ

=e−i(Q,

τ

)

2

σ

y.

Here, we rotate the spin-quantization axes around y-axis, that is why

σ

y appears in Ω

τ

. The other choices would be rotations around x- and z-axis and

could be simply accounted for by substituting y→x or y→z. For the moment, we leave apart the individual orbital rotations Ωi (which are also

implemented in ΘΦ) since they do not interfere with the lattice translations, and we will take them into account in the nal answer. The rotated-by-

the-super-structure-vector-Q kinetic energy reads as:

T=

i,j,

τ

,

α

,β,R

c†

R,i,

α

ei(Q,R)

2

σ

ytij

τ

e−i(Q,(R+

τ

))

2

σ

y

α

,β

cR+

τ

,j,β.(9)

E.A. Plekhanov and A.L. Tchougr´

eeff

Computational Materials Science xxx (xxxx) xxx

6

We convert the summation on R into the summation on k and k′:

T=1

N

i,j,

τ

,

α

,β,R,k,k′

e−ik′

τ

ei(k−k′)Rc†

k,i,

α

(10)

×

ei(Q,R)

2

σ

ytij

τ

e−i(Q,(R+

τ

))

2

σ

y

α

,β

ck′,j,β(11)

If we dene the matrix Lij (k−k′,

τ

):

Lij

k−k′,

τ

=1

N

R

ei(k−k′)Rei(Q,R)

2

σ

ytij

τ

e−i(Q,R)

2

σ

y,(12)

then the kinetic energy expression simplies as follows:

T=

i,j,

τ

,

α

,β,k,k′

e−ik′

τ

c†

k,i,

α

Lij

k−k′,

τ

e−i(Q,

τ

)

2

σ

yck′,j,β=

i,j,

τ

,

α

,β,k,k′

e−ik

τ

c†

k,i,

α

e−i(Q,

τ

)

2

σ

yLij

k−k′,

τ

ck′,j,β.(13)

Let us consider the matrix Lij(k−k′,

τ

). First of all, if the matrix tij(

τ

)is proportional to identity, or to

σ

y then the rotations on the left and right hand side

cancel each other and Lij(k−k′,

τ

) = δ(k−k′)tij(

τ

)and we obtain the result outlined in the Ref. [11]. Although, there is still a room for the non-trivial

results, the most physically interesting situation is when tij(

τ

)does not commute with

σ

y, spin–orbit coupling being an important example. In the

general case, we can write:

Ω

τ

=cos (Q,

τ

)

2−i

σ

ysin (Q,

τ

)

2,

so that after some simplication we have:

Lijk−k′,

τ

=1

2N

R

ei(k−k′)R×tij

τ

+

σ

ytij

τ

σ

y+tij

τ

−

σ

ytij

τ

σ

ycosQ,R+i

σ

y,tij

τ

sinQ,R.(14)

At this point, we can do the Fourier transform in the above expression, but the problem now is that the resulting expression will not be diagonal in k-

space, i.e. there will be terms with k and k±Q connected!

Till now, the formalism is completely general and other cases of rotation around e.g. x or y axes can be performed by substituting y→x, or y→z. The

case of rotation around z axis is particularly instructive since

σ

z is diagonal. In the most general case we can present:

tij

τ

=aij

τ

cij

τ

dij

τ

bij

τ

=t‖

ij +t⊥

ij (15)

=aij

τ

0

0bij

τ

+0cij

τ

dij

τ

0.(16)

The rst matrix will commute with

σ

z and will only contribute to the rst term (independent on Q) in Lij(k−k′,

τ

). For t⊥

ij we can explicitly work out:

σ

zt⊥

ij

τ

σ

z= − t⊥

ij

τ

(17)

σ

z,t⊥

ij

τ

=20cij

τ

−dij

τ

0.(18)

Therefore, Lij(k−k′,

τ

)will become:

Lijk−k′,

τ

=t‖

ij

τ

δk−k′+t⊥

ij

τ

δ(k−k′−Q)0

0δ(k−k′+Q).(19)

At this point we can introduce the Fourier transforms:

τ

tij

τ

e−ik

τ

=aijkcij k

c☆

ij kbijk,

where we have used the hermiticity condition:

E.A. Plekhanov and A.L. Tchougr´

eeff

Computational Materials Science xxx (xxxx) xxx

7

τ

dij

τ

e−ik

τ

=

τ

c☆

ij −

τ

e−ik

τ

.

Substituting Lij(k−k′,

τ

)into Eq.(13)changing

σ

y→

σ

z, we obtain, after simplications:

TzQ=

i,j,kc†

k,i,↑,c†

k+Q,j,↓×(20)

aijk+Q

2cijk+Q

2

c☆

ij k+Q

2bijk+Q

2

ck,i,↑

ck+Q,j,↓.(21)

The rotation around y-axis can be evaluated similarly, although the calculations in that case are more cumbersome. We report below the nal answer:

TyQ=

i,j,kc†

k,i,↑,c†

k+Q,j,↓×(22)

α

ijk+Q

2γijk+Q

2

γ☆

ij k+Q

2βijk+Q

2

ck,i,↑

ck+Q,j,↓.(23)

where:

α

ijk=aij k+bijk

2+Im cijk(24)

βijk=aij k+bijk

2−Im cijk(25)

γijk= − aijk+bij k

2−iRe cijk(26)

It is easy to see that

Tz(Q)and

Ty(Q)are different representations of the same operator and, indeed, are related by a unitary transformation S:

S=1

2

√i−i

1 1 ,

so that:

S†aijkcij k

c☆

ij kbijkS=

α

ijkγij k

γ☆

ij kβijk.

It can be seen, that the two ways to rotate the quantization axis are fully equivalent. Since rotation around z-axis features somehow simpler formulas,

we will carry out further calculations assuming z-axis rotation.

Putting back the individual orbital rotations Ωi, we can assemble the nal rotation formula for the multi-orbital case as follows:

Tz

Q

=

i,j,kc†

k,i,↑,c†

k+Q,j,↓Ω†

i

aijk+Q

2cijk+Q

2

c☆

ij k+Q

2bijk+Q

2

Ωjck,i,↑

ck+Q,j,↓.(27)

In this formula, the states with different k-vectors are mixed. This breaks the translational invariance of the crystal and makes it impossible to

effectively treat ISS states. The way out of this situation is to shift the down-spin electronic states by the vector Q so that in the new notation the

Fockian matrix will be diagonal in k-space:

TzQ=

i,j,kc†

k,i,↑,c†

k,j,↓Ω†

i×(28)

E.A. Plekhanov and A.L. Tchougr´

eeff

Computational Materials Science xxx (xxxx) xxx

8

aijk+Q

2cijk+Q

2

c☆

ij k+Q

2bijk+Q

2

Ωjck,i,↑

ck,j,↓,(29)

where

ck,j,↓=ck+Q,j,↓.

For what regards the interaction term, we consider here only the generalized Heisenberg term, as dened in the Ref. [11]:

HJ=

i,j,R,

τ

,

α

,β

J

α

,β

i,j

τ

S

α

i,RSβ

j,R+

τ

,

where S

α

i,R=1

2s,s′c†

R,i,s

σα

s,s′c†

R,i,s′is the operator of the

α

’s component of the spin of the orbital i in the cell R. The transformation of the exchange

coupling J

α

,β

i,j(

τ

)under the spin quantization axis rotation is reported in the Ref. [11].

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