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

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

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

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

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

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

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

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