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

r

fI

f

Ii f r

ti w II i

I hi I

E i b c

2 Cj 21 en

sa se u z a uTn

invariance

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

it

Ef I o

y

Ch c o

3EIicuiiuito.a

equivalent Ck Ctk

iiiii

T t3yt3z L

Toric

0 code