rows correspond to irreducible representations

ZG
◁ G

such, All Irreps of any Abelian Group, e.g., SO(2), are 1-dimensional

1d subspace

{ρg ·⃗v, ∀⃗v ∈ V (i)} = V (i)

Vector Space V ∼= Fd, an

Aut(H) < Sym(H)

x ⋄ y−1

be a subset of its actions

all (continuous) Permutations

homomorphism that defines how a group acts-on aset, and the set of actions on the Se

antisymmetric outer product

to avoid the formation ofvortices in the SC state.

e ac magnetic field HiSHE generatedvia the iSHE induced charge current JiSHEq is inductively

suppression of the dampinglike torque generated in the Ptlayer by the inverse spin Hall effect, which can be understood by the changes in spin current transport in thesuperconducting NbN layer.

supercurrent is induced along the x axis by enforcing the pairpotential to have a constant phase in a small region close toeach of the two boundaries along x

pair potential i = |i| exp(iφi).

With respect to thisreference frame, a rotation of the axes by π/2 degrees aboutthe z axis leads to a sign change of ˜αD , while a rotation of π/4degrees changes the tensor to ηso = ˜αD σz.

m = −γ m × [Heff + Hso ] + αGm × ˙

textured ferromagnets

as been observed in systems with broken spatialinversion symmetry and strong spin-orbit coupling (SOC)

he maximum achievable spin-orbit torque field is estimated to be on the order of 0.16 mT

such that the latter induces a Group Action on theVector Space, V , such that the latter induces a Group Action on the Vector Space, V ,

group G acts-on itself by conjugation

The set of all (continuous) Permutations of the Set, X = Rd, form a Group,Symc(Rd), the Symmetric Group of the set

G −→ Sym(X)

Group Action on the underlying Vector Space,

Rep of a Group, g ∈ G, on Vector Space

b) HomF(V, W )G

L ∈ HomF(V, W )

simple proportion (gain or attenuation), an integral (low-pass filter) and/or derivative (high-pass filter).

signal of strength M2

A general structure shouldbe regarded as chiral (Fig. 1e), as long as the total twist is non-zero atany point.

The structural chirality is not equivalent to thebreaking of inversion

bsence of any M, there can be an AFM CD unrelatedto M but proportional to the AFM order L in 𝒫𝒫𝒫𝒫-symmetric AFMs asreported in Cr2O3 (

ferromagnetic spintronics

ptical control of chirality andmagnetization,

tead a clean signal of a precessing spin-waveexcitation.

high field regime, the spin-transfer effectcannot produce a full reversal of the thin-layer moment

which involvedcontinuous ferromagnetic layers

pin-polarized electrons flowing from the thinCo layer to the thick layer can switch the moment of thethin layer antiparallel to the thick-layer moment,

the concept of reciprocity to understand Kerr effect, the useof fluctuation–dissipation theorem, which is another principalresult of linear response theory (that is, the study of irreversiblethermodynamics associated with linear processes that are referredto the equilibrium state of a system) yields an identical result

Reciprocity then requiresthat

Onsager reciprocity relations and fluctuation–dissipationtheory rely on the assumption that near equilibrium, macroscopicresponse and decay process occur in the same manner as thedecay of equilibrium fluctuations

ε ω ε ω′ = ′r r r r( , ; ) ( , ;

V r r( , ; )

ielectric function

spin–orbit coupling and exchange splitting

he fiber is highly birefringent and the diode light source has8-μm coherence length, only light that couples, after reflectingfrom the sample, between different axes in the fiber willtraverse optical path lengths that differ by less than a coherencelength and interfere coherently at the polarizer

tements of the symmetries of the electromagneticfield and its measurement entail that the reflection amplitudesbe considered quantum mechanically

polarization state ± near the source, if not as a plane wave.

gen-eral optically active

the reciprocity theorem

becomes larger than two

m through long range dipole-dipole couplin

readout of magnon density and propagation using photons of visible ligh

Samples swept with circularly polarized beams

OKE response measures primarily the TM sublattice.

strongertemperature dependence and the T

This results in a narrow composition rangewhere the two sublattice

lting in a net magnetization equal to zero

ces. In the case of thelight RE (4f electrons < 7) the two sublattices are exchange-coupledferromagnetically whereas for heavy RE (4f electrons ≥ 7) the twosublattices are exchange-coupled antiferromagnetically, forming aferrimagnet.

describes deterministic magnetization reversal ofthe material under the beam with no external magnetic field.

f GdFeCo (ref. 3), TbCo

0 fJ is expected to be sufficient

exhibiting magnetization compensation (andtherefore angular momentum compensation)

0.4 nm Irinterlayers

ferromagnetic sample

magnetization switching usingfemto- or picosecond pulsed lasers3

challenge present theories of AO-HDS

all-optical helicity-dependent switching (AO-HDS)

manipulating magnetic systems without applied magnetic fields have attracted growing attention

On the other hand,SOT is orthogonal to the magnetization of the free layer, which isexpected to provide “instant-on” switching torque.

examine the nature of a ∧ b, consider the formula ( a ⋅ b ) 2 − ( a ∧ b ) 2 = a 2 b 2 , {\displaystyle (\mathbf {a} \cdot \mathbf {b} )^{2}-(\mathbf {a} \wedge \mathbf {b} )^{2}=\mathbf {a} ^{2}\mathbf {b} ^{2},}

here exist explicit cut-and-dried algorithms for calculating the Hodge dual of B, especially if B is known in terms of components in some basis. See the discussion in reference 13, or see the actual code in reference 14.

dot-multiplying by a vector lowers the grade by 1

As another example, in d=3, it converts a vector to a certain “corresponding” pseudoscalar. Meanwhile, in d=4, it converts a vector into the “corresponding” pseudovector (not pseudoscalar)

Spinors (aka Pauli spin matrices)

The matrix representation is particularly efficient if you have one particular rotation and wish to apply it repeatedly, using it to rotate a large number of vectors. The advantage is most conspicuous in four or more dimensions. See section 7.

the standard Schroedinger wave function is a solution the Schroe-dinger equation (107)

definea rotor D by assuming that it satisfies the equationdDdt = 12( emc iB)D

We can generalize (96) to arbitrary states by interpretingρE = 〈∂t ψ iσ3¯hψ†

For a stationary solution with B × s = 0,

we can writeψ = ρ 12 U, where U U † = 1

where ψ′ = ψC† is the wave function relative to the alternative quantizationaxis σ′3. The matrix analog of this transformation is a change in matrix repre-sentation for the column spinor Ψ

i is now the unit pseudoscalar

Now write Ψ in the formΨ = ψu,

with complete generality that ψ in (80) is areal even multivector. Now we can reinterpret the σk in ψ as vectors in GAinstead of matrices. Thus, we have established a one-to-one

where HS is the Schroedinger hamiltonian

a column matrix Ψ

σ · a σ · b = a · b I + iσ · (a × b).

Indeed, (72) is the matrix representation of the GAproduct formula (29),

erates as a rotor in essentially the same way as rotors inclassical mechanics. This suggests that the bilinear dependence of observableson the wave function is not unique to quantum mechanics — it is equally naturalin classical mechanics for geometrical reasons

e have seen, every √−1 has a geometric meaning

be reduced to the form U = a + iβif n is an odd integer, or U = α + ib if n is even.

oducts with a common factor,U1 = ca, U2 = bc. (46)Hence (44) gives usU3 = U2U1 = (bc)(ca) = ba,

U1 = a and U2 = b produces a rotation U3 = ba.

x‖U † = U x‖.

Note that, respectively, the two components commute (anticommute) with i,

unit bivector for the b ∧ a-plane by i.

Incidentally, the term versor was coined in the19th century for an operator that can re-verse a direction.

versor, a

Note how this differs from the field invariantF 2 = (E + iB)2 = E2 − B2 + 2i(E·B),

This enables us to write the outer product defined by (6) inthe forma ∧ b = i a × b

The mostimportant example is the expression of the electromagnetic field F in terms ofan electric vector field E and a magnetic vector field B:F = E + iB

International Conference on Clif-ford Algebras and their Applications in Mathematical Physics”

discussed in the sequel to this paper.

We share the hope of Wheeler and Feynman that some of the paradoxesof classical and quantum electrodynamics, in particular the infinite self-energy of a point charge, might be avoidable by working with adjunctfields of this kind.

B = a + ib is an arbitrary 6-component bivector (a and b are relativevectors)

The vector derivative isinvertible so tha

There is another language which has some claim to achieve usefulunifications. The use of "differential forms" became popular with physicists,particularly as a result of its use in the excellent, and deservedly influential,"Big Black Book" by Misner, Thorne, and Wheeler. (33) Differential formsare skew multilinear functions, so that, like multivectors of grade k, theyachieve the aim of coordinate independence. By being scalar-valued,however, differential forms of different grades cannot be combined in theway multivectors can in geometric algebra. Consequently, rotors andspinors cannot be so easily expressed in the language of differential forms

review the subject of monogenicfunctions;

operator precedence convention in which anouter or inner product always takes precedence over a geometric product.Thus a A b is taken before the multiplication by i

"inner product," which is necessary to a great deal ofphysics, has to be grafted into this approach through the use of the dualityoperation, and so the language of differential forms never unifies the innerand outer products in the manner achieved by geometric algebra.

Since ~,~ is a positive-definite scalar in the Pauli algebra, we canwrite¢ = p l / 2 R (3.19)Thus, the Pauli spinor 10) can be seen as a (heavily disguised) instructionto rotate and dilate.

l t h o u g h this is less evident in two dimensions, inthree dimensions it is obvious: the r o t o raR = e x p ( - ia/2) = cos(lat/2) - i~-~ sin(lat/2) (3.13)represents a r o t a t i o n of tal radians a b o u t the axis along the direction of a.

s for at'/spaces, whatever dimension.

"bilinear" transformation of a.

R -= exp( -- al 0-20/2) = cos(0/2) - a l 0"2 sin(0/2)

R--- (3.5)l a + b l , / 2 ( l + b . a

and 0-10-20-3 is thus the unit pseudoscalar for 3-dimen-sional space. In view of its properties, we give it the special symbol i:i-= 0-10-za3

Laser scanning tube lenses can be used in telecentric systems to scan a laser spot across a sample.

Note: Eyepiece designs can be used as the imaging lens to improve performance.

homotopy equivalent to Sn i

metric product

Bivectorschange their sign

∧rB

2 = (−1) d(d−1)2 +s,

I = e1 . . . ed = e1 ∧ . . . ∧ e

IR

polyvector

Clifford algebra structure may be defined in an abstract way, with aClifford product. In this case, the vector space of multivectors is simply a peculiar repre-sentation.

(bilateral) ideal

(In the special case where themetric is zero, Cℓ(V ) = ∧ V .)

One defines the Clifford (or geometrical) product of two vectors asu v ≡ u · v + u ∧ v.

The wedge product of a p−mutivector by itself, M ∧ M , is always 0 when p isodd. This is not true when p is even

form ǫαβγ.

canonical (musical) isomorphism between V and its dual V ∗, which extends to theexterior algebras ∧ V and ∧ V ∗;

an (Hodge) duality (1.2.4) in the exterior algebras

exterior algebra [of multi-vectors] ∧ V of V

⊗

e defines the vector space of all tensors

normalized)

τ (s,r)V ≡s⊗ V ∗r⊗ V

An antisymmetric [con-travariant]tensor of type (0; p) will be called a p-vector , more generally a multivector .An antisymmetric [covariant] tensor of type (p; 0) defines a p-form, more generally amultiform (more simply, a form

that the separation of the spin componentsby the magnetic interaction is counteracted by the effect ofthe Lorentz force on the moving particle.

Themagnetic moment of the orbital motion of the electron in anatom is of the same order, but it can be taken to a classicalvalue by increasing the angular momentum quantum num-ber.

the situation is different for X-ray MO effects. Here the magnetic circular dichroism, whichis equivalent to the Faraday ellipticity, is a commonly used technique

Both z { and } { are thereby expressed as an angle

Various experimental techniques for detecting the Kerr rotation and ellipticity have beendiscussed in the literature (see, e.g., Robinson 1963, Jasperson and Schnatterly 1969, Suits1971, Sato 1981). Schoenes (1992) has given a classification of the available techniquesand a discussion of their respective advantages and disadvantages.

m first-principles energy-bandtheory, and even beyond, that it is feasible to make ab initio predictions of MOKE spectra

effect of Fe provides a minute description of theexperimental MOKE

UNiSn is an antiferromagnetic material

m. Daalderop etal. (1988a) predicted a large Kerr rotation of 5j for UNiSn, but did, unfortunately, notpublish a test of the computational method on simpler systems like the elemental 3i metals.

Another goal of MO research whichbecame intensively pursued in the eighties was to extracted information on the electronicstructure of, in particular, lanthanide and actinide compounds.

Most of the accumulatedexperimental MO data have recently been surveyed by Buschow (1988) and by Schoenes(1992),

measured MO spectraof many materials appeared in the last three decades.

become customary to relateMO phenomena to the materials band structure.

microwave magneto-absorption and Faradayeffect in semiconductors (Dresselhaus et al. 1955, Zwerdling and Lax 1957, Lax et al.1959).

nterest in MO recording, which since then has developed into a leading technologicalapplication of MOKE (

to visualizing surface and subsurface magnetic domains developed.

The only exception was the discovery of the MO Kerr effect in paramagneticmetals in an applied field by Majorana (1944

by Lorentz (1884), based on the idea that left- and right-circularly polarizedlight coupled differently t

MO phenomena hat become an important research topic.Quantum mechanics had not yet emerged, therefore the theoretical understanding of thephenomena was completely lacking

time reversal is preservedand inversion symmetry is absent, a nonequilibrium Kerreffect is allowed by symmetry,

Without loss of generality,

both effects only require inversionsymmetry breaking

nonequilibrium

d that the rectificationinduced by the nonlinear Hall effect

Berrycurvature dipole [5] (BCD)

a nonlinearHall effect

even in the absence of spin-orbitcoupling

nondestructive techniques

his showsthat h is 2π-periodic, and therefore of the form h(t)  g(c(t))

R if k  0,0 if k ≥ 1.Proof. This is a restatement of the Poincaré lemma,

ne can check that these operations arewell-defined

A contractible manifold is simply connected.

k , 1

µ  µM be the volume form of M.

∗dx  nµM .

α0  x · ∗dx‖x‖n .

t dxi ),

φ∗1 (α)  α and φ∗0 (α)  0,

contraction ofM onto x0

On somemanifolds all closed forms (of positive degree) are exact, on others this is true onlyin certain degrees

∫M φ∗ (α0 ) is 2π

0 ∫M×[0,1]φ∗ (dα)

ι0 (x)

g dt dxJ , then ι∗0 (α)  ι∗1 (α)  0

( f (x, 1) − f (x, 0)) dxI  ι∗1 (α) − ι∗0 (α)

It can be regarded as an application of Stokes’ theorem, but weshall give a direct proof.

by taking the piece of α involving dt and integrating it over the unit interval

exists a pointx0 in M

homotopy of loops from φ0 to φ1 in the punctured plane

a piece of string moving through the manifold N

A manifold M is said to be contractible

Suppose that φ0 and φ1 are two maps from amanifold M to a manifold N

structure constant of the Pauli algebra,

ut for mathematical reasons 2 × 2 matrices inphysics need to be unitary

The space of spinors is formally defined as the fundamental representation of the Clifford algebra. (