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LIZ KITAEVSPINLIQUIDST
The idea that quantum zero point fluctuations might
destroy antiferromagnetic order was probably firstentertained
by Landau Pomeranchuk who actually speculatedthat
antiferromagnehinwould never be stable against these
fluctuations
By the early 19505 it became clear that in higher
dimensions Siriple artiferromagnett are stableeven for 5 42
In 1973 Anderson Fazekas proposed that in a frustrated
triangular lattice the ground state wouldnot be AFM ordered
but would instead be a linear combination of resonating
valence bonds RUB
F 9,2 Hz It It 147117I 2
147 ftp.T i Pi
RUB Q5L
112
A E Ji f5i 5
1987 S C Lasco RUB Mother state of cuprate S.c
In conventional magnetsthe excitation's cary
8 1
req FM E9 AFM
ut 19
These spin wave spectra can be seen using neutron scattering
spin wave7
rHowever in a spin liquid a spin flip can break up
or fracture into mobile 5 42 quasiparticlesor
spinors
FRACTIONALIZATION
IN HEISENBERG cnn.inH J Jit Ii
Expt
BetheAnsate
4 11
fIfEgpfp Spin breaks up into
Spinors
INAfD
Bethe c 1931
147 Po If htt f III 107Po 17 n G ni
HIGHER DIMENSIONAL SPIN LIQUID CANDIDATES
Organic charge transfer saltseof k
BEDT TTF z
Herbertsmithite
2ncus OM Cla
Kagome lattice of Cu 5 42moments
KITAEV SPIN LIQUIDS
Kitaey 2006 but workedout around 2002
Ann Phys 321 2 Ill 2006
j 312
ToJacketit Khalanin nIridates
PRL 102 017205 2009 tzghole k
TopologicalPhysics
O i
Em aem
KITAEV HONEYCOMB MODEL
F
If I
x y
HkHkHkwp.o.xoioo.xotoE H
tt 3xE.E.io Ty oil 3 O
Up h 0 commute Wp Il constants VORTICES
124 H Iria JyoioA Wp Otoko
oxo.tt z oYHzoeY ttHzzoi
oozzY0zYosYHyo Hyo t Myology
It vortices
0 0 off
147 I Wp wpl47 Up147Wp II
MAJORANA FERMIONS BRIEF ASIDE
Conventional complex DIRAC fermions can be divided
into a real imaginary part
D at its a at dtdt MAJORANA
FERMIONS2 b bt cd
dtla.ba
ddtYi
Mary Majorana Ra Nb 2dab
e.g I H E dtd E E I Elz
c Kazbtatib E a b
f f ab
ie ab jab 1Antisymmetric
2
d d 3 1
d tf.o
dta b2
0iab dtd 1
DE d eit
die die See
dat d Lives in Half Brillouin Zone
MAJORANASPINREPRESENTATONI
Ex ib c
g y i b c
Yy fE't i b't c
HLike Pauli operators they squareto unity and they
articommute
12 1 8 80 878
But they do not oahity the correct communion relations
across the entire Fock space to do this requires a constraint
D 47 147 D b b'bite
D Ea 0 so this commutes Lith the spinoperator
Check 8 89 Et i
Eto Et I bX c b c b e i b b b't c ID
Ea Eb Zi DEE ncorrect properties at D t
E Ii'Ectbici'E Yi I
2 gauge inv
Cj 7 Cell q bf fIl b9
11
Mmmm
8 09 IDEZ Ex 07 2iDEZ
On the lattice we impose Dj 147 147 H j
µFock spaceof Majorana
Physical space
Petit l
Ja j of Ja ibd.cl ibfcr
JacjckfbjdbI
Tdcj Ck Ijk
where ujk BIBI 2 25,7
Ilija HI o Tyr147 4 147
Ugk Ii
22 gauge field
E og rk 3 2 0 0 3g of oil
D I T o Tz GEEZ
Mr
2 a gauge inv
Cj Cj
Ugk 7 Uj k
IY
isx G y
wp oiotofo.IO oE 1
CbicHbicdCbEg1lbIcallbsst big
bibber I b k bYbZ
hip bibillbE b fjb.it bYcbE lbIbsYfb bi
UlzU23U34U45U5G 61
Ttu k Wp BaikTik Edp Cj Edp
bjc b b 1
bjc bibbib
ADj j i j i
pyr
Lowest energy stateno vorticity Choose
uniform phase Ujk E in c X ty ez directions
if fifi fi ji yiyii
II I Ajk Cyck
A LIK
Aik 2J y kn r
N M I Aerie cetace th eI I 4w a I.FI
C oddsubjaltice
Ugk Uk
EE a eEE
ti ei
na f ifCCodd
nCta
Let CE Eg eihre
CE In Ece'E Ei
Ii E ri th Rie
cnn.caty In Elsieele SEE
Cj If queE ri
A t.EEaAe a eeiE etrieleiE Eeha
th c
az2Jz 2Jxeikri zsye'h
niciIch
the
ii.ciIIf.i I'Ifi
j e
th
ujkd
i 2i in t
I
E YEICH4kKE'TBZ