anti unitary operatortanmoydas.com/course_ph364/homework1_ph364_symmetry.pdf · daura symmetry...
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Daura Symmetry Total marks 130
I Show that H AT O does not mean
that A is conserved when It is an
anti unitary operatorUnder inversion parity the Cartesian coordinates
transforms as Cn s z Is C se y z
a How spherical coordinates r 0,0 transform under
parity 7b How cyllindrical co ordinates r z transform
under parity
2 Show how the folitioning terms transform undertime reversal and pain symmetries
Zeeman term He B E Ig external mag fieldSpinHeisenber term H Is Tri Sj is spin at r position
Spin orbit confting H I I I s are orbital1 spin angular momentum
Rashba spin or i coupling i It E FH.iswhere E'is electric field and I linearmomentum
Dyzoloshinskii Moriya interactionIt E CSI ID
where E is electric field and SI is Hsin att
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3Cay het us say F is the total angular moment urnof the system and it is odd uncle time s sod
TJ Tt
Let us Say I I m are the eigenvalues fTt as
jzlj.my mh IJ.ms
where m j j and
J II my jcj.it h 2lj
myShowthntT1imy LeDmli my
Hint Jt Ijm TKtm 146in I I m
fission iiianti'I
b Find the time reverse a operators forC 5 1 8 92 5
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4 Check if the followingequations are in variantunder Parity time reversal change conjugationand chiral symmetriesHall effect In any Es oxy is constantHall effect Streda formericay
Off on df.BEt
SCB S B D Begg Bz
b Continuity equation08Ft T T T o
is in variant or not unclearbarity time reversalchange conjugation and chiral symi.mntny
c Let us define chiral chare SIomdchiral current To as
So L Kr Sc Joe's Fr TTwhere R I 4 means right and leftmoving such as A B sub lattice etc but not
9 C
spin Assume Hg I have same sign of charge andthey dont rotate underparity ie R underbarity
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chiral imagineto electric effectTo gets Ape BT 50
where ape Mp Mc che al Hotdifference between Tes ie particles
Anomalous Hall effectF s E apex E 50C
g a 2
Where AP is the momentum Nfacereparation between the two chiralstats TN Te E is electric fieldi Anomalous charge or Wittencharge
go easy EE B
B is magnetic field and Ap issame as above a
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cas continuity equations ofchime charge and chiral current
d Econ
8 OTIS E Eawhere RI En F is the external electric field
Ci thereedininonUnder external electric magneticfields I B parallel
off F Fe he E is
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Top scal n
of a rector T transforms the same underinversion and Mirror its a vector likevelocity Sfa rector is even under inversion
and oddun rer symmetry its calledpseudo or axial vector They are usual kg genera idby a rotationshow that Monogenetic field CB spin LST andangular momentum f T are pseudo vector
of a scalar can be written as a dot fowduedof a rector and a pseudo vector its called apseudo scalar
Show that spin momentum coupling term
such as F I is a pseudo scalar Also themagneto electric coupling sometimes also calledaxion term E B ri also a pseudo scat.ae
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6 Find the fo fine reversalchare e conjugation and chiral
operators for the following two Hamiltonians
C Hey Serik't Sui kxtisiik.gl3D Seiki isinks sik I
n
when the basis we conhideared to be
he new fiA B two sub lattices 9,4 spin 42
200Ci 0 du Dia O
Areµ O o Dk
o iii aO
YaµAn
by Dh Do D K2
3 Ydo Kxtiks
too Do D are real
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I
j show that the Hamiltonian ha two sold
degeneracy at all R pointsShow that the Hamiltonian hes particlehole symmetric eigenvalues at all k points