Subir Sachdev -...

The Landscape of theHubbard model

HARVARDTalk online: sachdev.physics.harvard.edu

Subir Sachdev

TASI lectures, June 2010, Boulder

Monday, June 7, 2010

1. Introduction to the Hubbard model Superexchange and antiferromagnetism

2. Coupled dimer antiferromagnet CFT3: the Wilson-Fisher fixed point

3. Honeycomb lattice: semi-metal and antiferromagnetism CFT3: Dirac fermions and the Gross-Neveu model

4. Hubbard model as a SU(2) gauge theorySpin liquids, valence bond solids: analogies with SQED and SYM

5. Quantum critical dynamics AdS/CFT and the collisionless-hydrodynamic crossover

Outline


6. Square lattice: Fermi surfaces and spin density waves Quantum critical theory of hot spots

7. Square lattice: d-wave superconductivity Instabilities to pairing and other orders

Outline







Outline


H = −�

i<j

tijc†iαcjα + U

�

i

�ni↑ −

1

2

��ni↓ −

1

2

�− µ

�

i

c†iαciα

tij → “hopping”. U → local repulsion, µ → chemical potential

Spin index α =↑, ↓

niα = c†iαciα

c†iαcjβ + cjβc

†iα = δijδαβ

ciαcjβ + cjβciα = 0

The Hubbard Model

Will study on the honeycomb and square lattices


The Hubbard Model

H = −�

i,j

tijc†iαcjα + U

�

i

�ni↑ −

1

2

��ni↓ −

1

2

�− µ

�

i

c†iαciα

In the limit of large U , and at a density of one particle per site,

this maps onto the Heisenberg antiferromagnet

HAF =

�

i<j

JijSai S

aj

where a = x, y, z,

Sai =

1

2ca†iασ

aαβciβ ,

with σathe Pauli matrices and

Jij =4t

2ij

U


5.1

Eff

ecti

veH

am

ilto

nia

nm

ethod

69

ther

efor

eal

way

sap

pro

pri

ate,

and

only

the

crit

icalp

oin

tg

=g c

has

gen

uin

ely

diff

eren

tp

rop

erti

esatT

=0.

5.1

Eff

ecti

ve

Ham

ilto

nia

nm

eth

od

We

con

sid

era

Ham

ilto

nia

nof

the

form

H=H

0+H

1,

wh

ere

the

eigen

state

s

ofH

0ar

eea

sily

det

erm

ined

,an

dw

ear

ein

tere

sted

ind

escr

ibin

gth

ein

flu

ence

ofH

1in

per

turb

atio

nth

eory

.F

urt

her

,w

eas

sum

eth

at

the

eigen

valu

esofH

0

are

sep

arat

edin

tod

isti

nct

grou

ps

ofcl

osel

y-s

pace

dle

vels

,su

chth

at

the

ener

gyse

par

atio

nb

etw

een

two

leve

lsw

ith

inth

esa

me

gro

up

isalw

ays

mu

ch

smal

ler

than

the

separ

atio

nb

etw

een

two

leve

lsin

dis

tin

ctgro

up

s.W

eu

seth

e

sym

bol

sα,β,...

tod

enot

eth

egr

oup

s,an

di,j,...

tod

enote

leve

lsw

ith

in

each

grou

p.

Thu

sth

eei

gen

stat

esofH

0ar

e|i,α〉,

wit

hei

gen

ener

gie

sEiα

,

and

so

|Eiα−Ejα|�|E

iα−Ejβ|

forα6=β

(5.2

)

We

are

ofte

nin

tere

sted

inth

est

ruct

ure

ofth

ele

vels

wit

hin

agiv

engro

up

,

and

wou

ldli

keto

un

der

stan

dth

eir

beh

avio

rw

ith

ou

tre

fere

nce

tole

vels

in

oth

ergr

oup

s.H

owev

er,

inge

ner

al,H

1w

ill

hav

en

on

-zer

om

atr

ixel

emen

ts

bet

wee

nst

ates

bel

ongi

ng

tod

iffer

ent

grou

ps:

con

sequ

entl

ya

conve

nti

on

al

per

turb

ativ

ean

alysi

sw

ill

requ

ire

rep

eate

dre

fere

nce

tost

ate

sly

ing

ou

tsid

e

the

grou

pof

inte

rest

.T

he

idea

ofth

eeff

ecti

ve

Ham

ilto

nia

nm

eth

od

isto

per

form

au

nit

ary

tran

sfor

mat

ion

wh

ich

elim

inate

sth

ese

inte

r-gro

up

matr

ix

elem

ents

.A

fter

the

un

itar

ytr

ansf

orm

atio

n,

we

wil

lob

tain

an

ewH

am

ilto

-

nia

nH

effw

hic

hh

asn

on-z

ero

mat

rix

elem

ents

on

lyw

ith

inea

chgro

up

.

We

skip

the

stra

ightf

orw

ard

,b

ut

ted

iou

s,an

aly

sis

nee

ded

toob

tainH

eff

ord

er-b

y-o

rder

inH

1.

We

only

qu

ote

her

eth

efi

nal

resu

ltto

seco

nd

ord

erin

H1.

Th

en

ewH

amil

ton

ianH

effis

defi

ned

by

the

foll

owin

gn

on

-zer

om

atr

ix

elem

ents

bet

wee

nan

ytw

ole

vels

,|i,α〉a

nd|j,α〉b

elon

gin

gto

the

sam

egro

up

α:

〈i,α|H

eff|j,α〉=

Eiαδ ij

+〈i,α|H

1|j,α〉+

(5.3

)∑k,β6=α

〈i,α|H

1|k,β〉〈k,β|H

1|j,α〉

2

(1

Eiα−Ekβ

+1

Ej,α−Ekβ

)

Nat

ura

lly,

we

hav

e〈i,α|H

eff|j,β〉

=0

for

allα6=β

,en

suri

ng

thatH

effis

blo

ckd

iago

nal

,an

dw

eca

nw

ork

ind

epen

den

tly

wit

hin

each

gro

upα

.






Outline


Ground state has long-range Néel order

Square lattice antiferromagnet

H =�

�ij�

Jij�Si · �Sj

Order parameter is a single vector field �ϕ = ηi�Si

ηi = ±1 on two sublattices

��ϕ� �= 0 in Neel state.



H =�

�ij�

Jij�Si · �Sj

J

J/λ

Weaken some bonds to induce spin entanglement in a new quantum phase



H =�

�ij�

Jij�Si · �Sj

J

J/λ

Ground state is a “quantum paramagnet”with spins locked in valence bond singlets

=1√2

��↑↓�−

�� ↓↑��


λλc

M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev.B 65, 014407 (2002).

Quantum critical point with non-local entanglement in spin wavefunction


λλc

Excitation spectrum in the paramagnetic phase


λλc


Sharp spin 1 particle excitation above an energy gap (spin gap)


λλc

Excitation spectrum in the Neel phase


λλc


Spin waves


12B

asi

cC

on

cepts

wh

ere|0〉

isei

ther

ofth

egr

oun

dst

ates

obta

ined

from|↑〉

or|↓〉

by

per

tur-

bat

ion

theo

rying,

andN

06=

0is

the

“sp

onta

neo

us

magn

etiz

ati

on

”of

the

grou

nd

stat

e.T

his

iden

tifi

cati

onis

mad

ecl

eare

rby

the

sim

ple

rst

ate

men

t

⟨ 0∣ ∣ σz i∣ ∣ 0

⟩ =±N

0,

(1.1

3)

wh

ich

also

foll

ows

from

the

per

turb

atio

nth

eory

ing.

We

hav

eN

0=

1fo

r

g=

0,b

ut

qu

antu

mfl

uct

uat

ion

sat

smal

lg

red

uce

N0

toa

small

er,

bu

t

non

zero

,va

lue.

Now

we

mak

eth

esi

mp

leob

serv

atio

nth

atit

isn

ot

poss

ible

for

state

sth

at

obey

(1.9

)an

d(1

.12)

totr

ansf

orm

into

each

oth

eran

aly

tica

lly

as

afu

nct

ion

ofg.

Th

ere

mu

stb

ea

crit

ical

valu

eg

=g c

atw

hic

hth

ela

rge|xi−xj|l

imit

ofth

etw

o-p

oint

corr

elat

orch

ange

sfr

om(1

.9)

to(1

.12

)–

this

isth

ep

osi

tion

ofth

equ

antu

mp

has

etr

ansi

tion

,w

hic

hsh

all

be

the

focu

sof

inte

nsi

vest

ud

y

inth

isb

ook

.O

ur

argu

men

tsso

far

do

not

excl

ud

eth

ep

oss

ibil

ity

that

ther

e

cou

ldb

em

ore

than

one

crit

ical

poi

nt,

bu

tth

isis

kn

own

not

toh

ap

pen

for

HI,

and

we

wil

las

sum

eh

ere

that

ther

eis

only

on

ecr

itic

al

poin

tatg

=g c

.

For

g>g c

the

grou

nd

stat

eis

,as

not

edea

rlie

r,a

quan

tum

para

magn

et,

and

(1.9

)is

obey

ed.

We

wil

lfi

nd

that

asg

ap

pro

ach

esg c

from

ab

ove,

the

corr

elat

ion

len

gth

,ξ,

div

erge

sas

in(1

.2).

Pre

cise

lyatg

=g c

,n

eith

er(1

.9)

nor

(1.1

2)is

obey

ed,an

dw

efi

nd

inst

ead

ap

ower

-law

dep

enden

ceon|xi−xj|

atla

rge

dis

tance

s.T

he

resu

lt(1

.12)

hol

ds

for

allg<g c

,w

hen

the

gro

un

d

stat

eis

magn

etic

all

yord

ered

.T

he

spon

tan

eou

sm

agnet

izati

on

of

the

gro

un

d

stat

e,N

0,

van

ish

esas

ap

ower

law

asg

app

roach

esg c

from

bel

ow.

Fin

ally

,w

em

ake

aco

mm

ent

abou

tth

eex

cite

dst

ate

sofHI.

Ina

fin

ite

latt

ice,

ther

eis

nec

essa

rily

an

onze

roen

ergy

sep

ara

tin

gth

egro

un

dst

ate

an

d

the

firs

tex

cite

dst

ate.

How

ever

,th

isen

ergy

spaci

ng

can

eith

erre

main

fin

ite

orap

pro

ach

zero

inth

ein

fin

ite

latt

ice

lim

it,th

etw

oca

ses

bei

ng

iden

tifi

edas

hav

ing

aga

pp

edor

gap

less

ener

gysp

ectr

um

resp

ecti

vely

.W

ew

ill

fin

dth

at

ther

eis

anen

ergy

gap

∆th

atis

non

zero

for

allg6=g c

,b

ut

that

itva

nis

hes

up

onap

pro

ach

ingg c

asin

(1.1

),p

rod

uci

ng

agap

less

spec

tru

matg

=g c

.

1.4

.2Q

uan

tum

Roto

rM

odel

We

turn

toth

eso

mew

hat

less

fam

ilia

rquan

tum

roto

rm

od

els.

Ele

men

tary

qu

antu

mro

tors

do

not

exis

tin

nat

ure

;ra

ther

,ea

chqu

antu

mro

tor

isan

effec

tive

qu

antu

md

egre

eof

free

dom

for

the

low

ener

gy

state

sof

asm

all

nu

mb

erof

elec

tron

sor

atom

s.W

ew

ill

firs

td

efin

eth

equ

antu

mm

ech

an

ics

ofa

sin

gle

roto

ran

dth

entu

rnto

the

latt

ice

qu

antu

mro

tor

mod

el.

Th

e

con

nec

tion

toth

eex

per

imen

tal

mod

els

intr

od

uce

din

Sec

tion

1.3

wil

lb

e

subir

Rectangle

1.4

Theo

reti

cal

Mod

els

13

des

crib

edb

elow

inS

ecti

on1.

4.3.

Fu

rth

erd

etail

sof

this

con

nec

tion

ap

pea

r

inC

hap

ters

9,19

.

Eac

hro

tor

can

be

vis

ual

ized

asa

par

ticl

eco

nst

rain

edto

mov

eon

the

surf

ace

ofa

(fict

itio

us)

(N>

1)-d

imen

sion

alsp

her

e.T

he

ori

enta

tion

of

each

roto

ris

rep

rese

nte

dby

anN

-com

pon

ent

un

itve

ctor

ni

wh

ich

sati

sfies

n2

=1.

(1.1

4)

Th

eca

ret

onni

rem

ind

su

sth

atth

eor

ienta

tion

of

the

roto

ris

aqu

antu

m

mec

han

ical

oper

ator

,w

hil

ei

rep

rese

nts

the

site

on

wh

ich

the

roto

rre

sid

es;

we

wil

lsh

ortl

yco

nsi

der

anin

fin

ite

num

ber

of

such

roto

rsre

sid

ing

on

the

site

sof

ad-d

imen

sion

alla

ttic

e.E

ach

roto

rh

as

am

om

entu

mpi,

an

dth

e

con

stra

int

(1.1

4)im

pli

esth

atth

ism

ust

be

tan

gen

tto

the

surf

ace

of

the

N-d

imen

sion

alsp

her

e.T

he

roto

rp

osit

ion

and

mom

entu

msa

tisf

yth

eu

sual

com

mu

tati

onre

lati

ons

[nα,pβ]

=iδαβ

(1.1

5)

onea

chsi

tei;

her

eα,β

=1...N

.(H

ere,

and

inth

ere

main

der

of

the

book,

we

wil

lal

way

sm

easu

reti

me

inu

nit

sin

wh

ich

~=

1,(1

.16)

un

less

stat

edex

pli

citl

yot

her

wis

e.T

his

isal

soa

good

poin

tto

note

that

we

wil

lal

sose

tB

oltz

man

n’s

con

stan

t kB

=1

(1.1

7)

by

abso

rbin

git

into

the

un

its

ofte

mp

erat

ure

,T

.)W

ew

ill

act

ually

fin

dit

mor

eco

nve

nie

nt

tow

ork

wit

hth

eN

(N−

1)/

2co

mp

on

ents

of

the

roto

r

angu

lar

mom

entu

m

Lαβ

=nαpβ−nβpα.

(1.1

8)

Th

ese

oper

ator

sar

eth

ege

ner

ator

sof

the

grou

pof

rota

tion

inN

dim

ensi

on

s,

den

oted

O(N

).T

hei

rco

mm

uta

tion

rela

tion

sfo

llow

stra

ightf

orw

ard

lyfr

om

(1.1

5)an

d(1

.18)

.T

he

caseN

=3

wil

lb

eof

part

icu

lar

inte

rest

tou

s:F

or

this

we

defi

neLα

=(1/2)ε αβγLβγ

(wh

ereε αβγ

isa

tota

lly

anti

sym

met

ric

ten

sor

wit

hε 1

23

=1)

,an

dth

enth

eco

mm

uta

tion

rela

tion

bet

wee

nth

eop

erato

rs

onea

chsi

tear

e

[Lα,L

β]

=iεαβγLγ,

[Lα,n

β]

=iεαβγnγ,

(1.1

9)

[nα,n

β]

=0;

the

oper

ator

sw

ith

diff

eren

tsi

tela

bel

sal

lco

mm

ute

.

14B

asi

cC

on

cepts

Th

ed

yn

amic

sof

each

roto

ris

gove

rned

sim

ply

by

its

kin

etic

ener

gy

term

;

inte

rest

ing

effec

tsw

ill

aris

efr

omp

oten

tial

ener

gy

term

sth

at

cou

ple

the

roto

rsto

geth

er,

and

thes

ew

ill

be

con

sid

ered

mom

enta

rily

.E

ach

roto

rh

as

the

kin

etic

ener

gy

HK

=Jg 2L

2,

(1.2

0)

wh

ere

1/Jg

isth

ero

tor

mom

ent

ofin

erti

a(w

eh

ave

pu

ta

tild

eov

erg

as

we

wis

hto

rese

rveg

for

ad

iffer

ent

cou

pli

ng

tob

ein

trod

uce

db

elow

).T

he

Ham

ilto

nia

nHK

can

be

read

ily

dia

gon

aliz

edfo

rgen

eralva

lues

ofN

by

wel

l-

kn

own

grou

pth

eore

tica

lm

eth

od

s.W

equ

ote

the

resu

lts

for

the

physi

call

y

imp

orta

nt

case

sofN

=2

and

3.F

orN

=2

the

eigen

valu

esare

Jg`2/2

`=

0,1,

2,...

;d

egen

eracy

=2−δ `,0.

(1.2

1)

Not

eth

atth

ere

isa

non

deg

ener

ate

grou

nd

state

wit

h`

=0,w

hil

eall

exci

ted

stat

esar

etw

o-fo

lddeg

ener

ate,

corr

esp

ond

ing

toa

left

-or

right-

mov

ing

roto

r.

Th

issp

ectr

um

wil

lb

eim

por

tant

inth

em

app

ing

top

hysi

cal

model

sto

be

dis

cuss

edin

Sec

tion

1.4.

3.F

orN

=3,

the

eigen

valu

esofHK

are

Jg`(`

+1)/2

`=

0,1,

2,...;

deg

ener

acy

=2`

+1,

(1.2

2)

corr

esp

ond

ing

toth

efa

mil

iar

angu

lar

mom

entu

mst

ate

sin

thre

ed

imen

sion

s.

Th

ese

stat

esca

nb

evie

wed

asre

pre

senti

ng

the

eigen

state

sofan

even

nu

mb

er

ofan

tife

rrom

agn

etic

ally

cou

ple

dH

eise

nb

erg

spin

s,as

wil

lb

ed

iscu

ssed

more

exp

lici

tly

inS

ecti

on1.

4.3

and

inC

hap

ter

19,

wher

ew

ill

see

that

ther

eis

age

ner

alan

dp

ower

ful

corr

esp

ond

ence

bet

wee

nqu

antu

manti

ferr

om

agn

ets

andN

=3

roto

rs.

We

are

read

yto

wri

ted

own

the

full

qu

antu

mro

tor

Ham

ilto

nia

n,

whic

h

shal

lb

eth

efo

cus

ofin

ten

sive

stu

dy

inP

arts

IIan

dII

I.W

ep

lace

asi

n-

gle

qu

antu

mro

tor

onth

esi

tes,i,

ofad-d

imen

sion

al

latt

ice,

ob

eyin

gth

e

Ham

ilto

nia

n

HR

=Jg 2

∑ i

L2 i−J∑ 〈ij〉

ni·n

j.

(1.2

3)

We

hav

eau

gmen

ted

the

sum

ofkin

etic

ener

gies

of

each

site

wit

ha

cou

pli

ng,

J,

bet

wee

nro

tor

orie

nta

tion

son

nei

ghb

orin

gsi

tes.

Th

isco

up

lin

gen

ergy

is

min

imiz

edby

the

sim

ple

“mag

net

ical

lyor

der

ed”

state

inw

hic

hall

the

roto

rs

are

orie

nte

din

the

sam

ed

irec

tion

.In

contr

ast

,th

ero

tor

kin

etic

ener

gy

is

min

imiz

edw

hen

the

orie

nta

tion

ofth

ero

tor

ism

axim

all

yu

nce

rtain

(by

the

un

cert

ainty

pri

nci

ple

),an

dso

the

firs

tte

rminHR

pre

fers

aqu

antu

m

par

amag

net

icst

ate

inw

hic

hth

ero

tors

do

not

hav

ea

defi

nit

eori

enta

tion

(i.e

.,〈n〉

=0)

.T

hu

sth

ero

les

ofth

etw

ote

rms

inHR

close

lyp

ara

llel

those

1.4

Theo

reti

cal

Mod

els

15

ofth

ete

rms

inth

eIs

ing

mod

elHI.

As

inS

ecti

on

1.4

.1,

forg�

1,

wh

en

the

kin

etic

ener

gyd

omin

ates

,w

eex

pec

ta

qu

antu

mp

ara

magn

etin

wh

ich

,

foll

owin

g(1

.9),

〈0|ni·n

j|0〉∼

e−|xi−xj|/ξ.

(1.2

4)

Sim

ilar

ly,

forg�

1,w

hen

the

cou

plin

gte

rmd

om

inate

s,w

eex

pec

ta

mag-

net

ical

lyor

der

edst

ate

inw

hic

h,

asin

(1.1

2),

lim

|xi−xj|→∞〈0|ni·n

j|0〉=

N2 0.

(1.2

5)

Fin

ally

,w

eca

nan

tici

pat

ea

seco

nd

-ord

erqu

antu

mp

hase

tran

siti

on

bet

wee

n

the

two

ph

ases

atg

=gc,

and

the

beh

avio

rofN

0an

dξ

up

on

ap

pro

ach

ing

this

poi

nt

wil

lb

esi

mil

arto

that

inth

eIs

ing

case

.T

hes

eex

pec

tati

on

stu

rn

out

tob

eco

rrec

tfo

rd>

1,b

ut

we

wil

lse

eth

at

they

nee

dso

me

mod

ifica

tion

s

ford

=1.

Inon

ed

imen

sion

,w

ew

ill

show

thatgc

=0

forN≥

3,

an

dso

the

grou

nd

stat

eis

aqu

antu

mp

aram

agn

etic

state

for

all

non

zerog.

Th

eca

se

N=

2,d

=1

issp

ecia

l:T

her

eis

atr

ansi

tion

at

afi

nit

egc,b

ut

the

div

ergen

ce

ofth

eco

rrel

atio

nle

ngt

hdoes

not

obey

(1.2

)an

dth

elo

ng-d

ista

nce

beh

avio

r

ofth

eco

rrel

atio

nfu

nct

iong<gc

diff

ers

from

(1.2

5).

Th

isca

sew

ill

not

be

con

sid

ered

unti

lS

ecti

on20

.3in

Par

tII

I.

1.4

.3P

hysi

cal

reali

zati

on

sof

qu

an

tum

roto

rs

We

wil

lco

nsi

der

theN

=3

qu

antu

mro

tors

firs

t,an

dex

pose

asi

mp

lean

d

imp

orta

nt

con

nec

tion

bet

wee

nO

(3)

qu

antu

mro

tor

mod

els

an

da

cert

ain

clas

sof

“dim

eriz

ed”

anti

ferr

omag

net

s,of

wh

ich

TlC

uC

l 3is

the

exam

ple

we

hig

hli

ghte

din

Sec

tion

1.3.

Act

ual

lyth

eco

nn

ecti

on

bet

wee

nro

tor

mod

els

and

anti

ferr

omag

net

sis

far

mor

ege

ner

alth

an

the

pre

sent

dis

cuss

ion

may

sugg

est,

asw

ew

ill

see

late

rin

Ch

apte

r19

.H

owev

er,

this

dis

cuss

ion

shou

ld

enab

leth

ere

ader

toga

inan

intu

itiv

efe

elin

gfo

rth

ep

hysi

cal

inte

rpre

tati

on

ofth

ed

egre

esof

free

dom

ofth

ero

tor

mod

el.

Con

sid

era

dim

eriz

edsy

stem

of“H

eise

nb

erg

spin

s”S

1i

an

dS

2i,

wh

erei

now

lab

els

ap

air

ofsp

ins

(a‘d

imer

’).

Th

eir

Ham

ilto

nia

nis

Hd

=K∑ i

S1i·S

2i+J∑ 〈ij〉

( S1i·S

1j

+S

2i·S

2j

) .(1

.26)

Th

eSni

(n=

1,2

lab

els

the

spin

sw

ith

ina

dim

er)

are

spin

op

erato

rsu

suall

y

rep

rese

nti

ng

the

tota

lsp

inof

ase

tof

elec

tron

sin

som

elo

cali

zed

ato

mic

stat

es.

See

Fig

1.3.

On

each

site

,th

esp

ins

Sni

ob

eyth

ean

gu

lar

mom

entu

m

16B

asi

cC

on

cepts

J

K

Fig

ure

1.3

Ad

imer

ized

qu

antu

msp

insy

stem

.S

pin

sw

ith

angu

lar

mom

en-

tumS

resi

de

onth

eci

rcle

s,w

ith

anti

ferr

omag

net

icex

chan

geco

up

lin

gsas

show

n.

com

mu

tati

onre

lati

ons

[ Sα,S

β

] =iεαβγSγ

(1.2

7)

(th

esi

tein

dex

has

bee

nd

rop

ped

abov

e),

wh

ile

spin

op

erato

rson

diff

eren

t

site

sco

mm

ute

.T

hes

eco

mm

uta

tion

rela

tion

sare

the

sam

eas

those

of

the

L

oper

ator

sin

(1.1

9).

How

ever

,th

ere

ison

ecr

uci

al

diff

eren

ceb

etw

een

Hil

ber

t

spac

eof

stat

eson

wh

ich

the

qu

antu

mro

tors

an

dH

eise

nb

erg

spin

sact

.F

or

the

roto

rm

od

els

we

allo

wed

stat

esw

ith

arb

itra

ryto

tal

an

gu

lar

mom

entu

m

`on

each

site

,as

in(1

.22)

,an

dso

ther

ew

ere

an

infi

nit

enu

mb

erof

state

s

onea

chsi

te.

For

the

pre

sent

Hei

senb

erg

spin

s,h

owev

er,

we

wil

lon

lyall

ow

stat

esw

ith

tota

lsp

inS

onea

chsi

te,

and

we

wil

lp

erm

itS

tob

ein

teger

or

hal

f-in

tege

r.T

hu

sth

ere

are

pre

cise

ly2S

+1

state

son

each

site

|S,m〉

wit

hm

=−S...S,

(1.2

8)

and

the

oper

ator

iden

tity

S2 ni

=S

(S+

1)(1

.29)

hol

ds

for

eachi

andn

.In

addit

ion

tod

escr

ibin

gT

lCu

Cl 3

,H

am

ilto

nia

ns

like

Hd

des

crib

esp

in-l

add

erco

mp

oun

ds

ind

=1

[39,

131]

an

d“d

ou

ble

laye

r”

anti

ferr

omag

net

sin

the

fam

ily

ofth

eh

igh

-tem

per

atu

resu

per

con

du

ctors

in

d=

2[5

81,

582,

387,

402,

517,

518,

159]

.

Let

us

exam

ine

the

pro

per

ties

ofHd

inth

eli

mitK�

J.

As

afi

rst

app

roxim

atio

n,

we

can

neg

lect

theJ

cou

pli

ngs

enti

rely

,an

dth

enHd

spli

ts

into

dec

oup

led

pai

rsof

site

s,ea

chw

ith

ast

ron

ganti

ferr

om

agn

etic

cou

pli

ng

Kb

etw

een

two

spin

s.T

he

Ham

ilto

nia

nfo

rea

chp

air

can

be

dia

gon

ali

zed

by

not

ing

that

S1i

and

S2i

cou

ple

into

stat

esw

ith

tota

lan

gu

lar

mom

entu

m

0≤`≤

2S

,an

dso

we

obta

inth

eei

gen

ener

gies

(K/2

)(`(`

+1)−

2S(S

+1)

),d

egen

eracy

2`

+1.

(1.3

0)

Not

eth

atth

ese

ener

gies

and

deg

ener

acie

sar

ein

on

e-to

-one

corr

esp

on

den

ce

wit

hth

ose

ofa

sin

gle

qu

antu

mro

tor

in(1

.22),

ap

art

from

the

diff

eren

ce

1.4

Theo

reti

cal

Mod

els

17

that

the

up

per

rest

rict

ion

on`

bei

ng

smal

ler

than

2Sis

ab

sent

inth

ero

tor

mod

elca

se.

Ifon

eis

inte

rest

edp

rim

aril

yin

low

ener

gy

pro

per

ties

,th

enit

app

ears

reas

onab

leto

rep

rese

nt

each

pai

rof

spin

sby

aqu

antu

mro

tor.

We

hav

ese

enth

atth

eK/J→∞

lim

itofHd

close

lyre

sem

ble

sth

eg→∞

lim

itofHR

.T

ofi

rst

ord

ering,

we

can

com

pare

the

matr

ixel

emen

tsof

the

term

pro

por

tion

altoJ

inHR

amon

gth

elo

w-l

yin

gst

ate

s,w

ith

those

of

the

Jte

rminHd;

itis

not

diffi

cult

tose

eth

atth

ese

matr

ixel

emen

tsb

ecom

e

equ

alto

each

oth

erfo

ran

app

rop

riat

ech

oice

of

cou

pli

ngs:

see

exer

cise

6.6

.1.

Th

eref

ore

we

may

con

clu

de

that

the

low

-en

ergy

pro

per

ties

of

the

two

mod

els

are

clos

ely

rela

ted

for

larg

eK/J

andg.S

omew

hat

diff

eren

tco

nsi

der

ati

on

sin

Ch

apte

r19

wil

lsh

owth

atth

eco

rres

pon

den

ceals

oapp

lies

toth

equantu

m

crit

ical

poi

nt

and

toth

em

agn

etic

ally

ord

ered

ph

ase

.

Th

em

ain

less

onof

the

abov

ean

alysi

sis

that

the

O(3

)qu

antu

mro

tor

mod

elre

pre

sents

the

low

ener

gyp

rop

erti

esof

qu

antu

manti

ferr

om

agn

ets

of

Hei

senb

erg

spin

s,w

ith

each

roto

rb

ein

gan

effec

tive

rep

rese

nta

tion

of

apa

ir

ofan

tife

rrom

agn

etic

ally

cou

ple

dsp

ins.

Th

est

rong-c

ou

pli

ng

spec

tra

clea

rly

ind

icat

eth

eop

erat

orco

rres

pon

den

ceLi

=S

1i+

S2i,

an

dso

the

roto

ran

gu

lar

mom

entu

mre

pre

sents

the

tota

lan

gula

rm

omen

tum

of

the

un

der

lyin

gsp

in

syst

em.E

xam

inat

ion

ofm

atri

xel

emen

tsin

the

larg

e-S

lim

itsh

ows

that

ni∝

S1i−

S2i:

the

roto

rco

ord

inat

eni

isth

ean

tife

rrom

agn

etic

ord

erp

ara

met

erof

the

spin

syst

em.

Mag

net

ical

lyor

der

edst

ates

of

the

roto

rm

od

elw

ith〈n

i〉6=

0,w

hic

hw

ew

ill

enco

unte

rb

elow

,ar

eth

eref

ore

spin

state

sw

ith

lon

g-r

an

ge

anti

ferr

omag

net

icor

der

and

hav

ea

van

ish

ing

tota

lfe

rrom

agn

etic

mom

ent.

Qu

antu

mH

eise

nb

erg

spin

syst

ems

wit

ha

net

ferr

om

agn

etic

mom

ent

are

not

mod

eled

by

the

qu

antu

mro

tor

mod

el(1

1.1

)–

thes

ew

ill

be

stu

die

din

Sec

tion

19.2

by

ad

iffer

ent

app

roac

h.

Let

us

now

consi

der

theN

=2

qu

antu

mro

tors

,an

din

trod

uce

thei

r

con

nec

tion

toth

esu

per

flu

id-i

nsu

lato

rtr

ansi

tion

of

boso

ns.

ForN

=2,

itis

use

ful

toin

trodu

cean

angu

lar

vari

ableθ i

onea

chsi

te,

soth

at

ni

=(c

osθ i,s

inθ i

).(1

.31)

Th

ero

tor

angu

lar

mom

entu

mh

ason

lyon

eco

mp

on

ent,

wh

ich

can

be

rep

-

rese

nte

din

the

Sch

rod

inge

rp

ictu

reas

the

diff

eren

tial

op

erato

r

Li

=1 i

∂ ∂θ i

(1.3

2)

acti

ng

ona

wav

efu

nct

ion

wh

ich

dep

end

son

allth

eθ i

.T

he

roto

rH

am

ilto

nia

n

isth

eref

ore

HR

=−Jg 2

∑ i

∂2

∂θ2 i−J∑ 〈ij〉

cos(θ i−θ j

),(1

.33)

subir

Rectangle

λλcCFT3

O(3) order parameter �ϕ

S =�

d2rdτ�(∂τϕ)2 + c2(∇r �ϕ)2 + (λ− λc)�ϕ2 + u

��ϕ2

�2�

Description using Landau-Ginzburg field theory


�1.0

�0.5

0.0

0.5

1.0

�1.0

�0.5

0.0

0.5

1.0

0.0

0.5

1.0

1.5

2.0

λλc


�ϕ

Spin S = 1“triplon”

V (�ϕ)V (�ϕ) = (λ− λc)�ϕ2 + u

��ϕ2

�2

λ >λ c


�1.0

�0.5

0.0

0.5

1.0�1.0

�0.5

0.0

0.5

1.0

�0.3

�0.2

�0.1

0.0

λλc

�ϕ

V (�ϕ)


Spin waves (“Goldstone” modes)

and a longitudinal “Higgs” particle

V (�ϕ) = (λ− λc)�ϕ2 + u��ϕ2

�2

λ <λ c


λλc


�1.0

�0.5

0.0

0.5

1.0�1.0

�0.5

0.0

0.5

1.0

�0.3

�0.2

�0.1

0.0

�ϕ

V (�ϕ)



V (�ϕ) = (λ− λc)�ϕ2 + u��ϕ2

�2

λ <λ c


λλc

�1.0

�0.5

0.0

0.5

1.0�1.0

�0.5

0.0

0.5

1.0

�0.3

�0.2

�0.1

0.0

�ϕ

V (�ϕ)




V (�ϕ) = (λ− λc)�ϕ2 + u��ϕ2

�2

λ <λ c


λλcCFT3

O(3) order parameter �ϕ

S =�

d2rdτ�(∂τϕ)2 + c2(∇r �ϕ)2 + s�ϕ2 + u

��ϕ2

�2�


S. Wenzel and W. Janke, Phys. Rev. B 79, 014410 (2009)M. Troyer, M. Imada, and K. Ueda, J. Phys. Soc. Japan (1997)

Quantum Monte Carlo - critical exponents


Quantum Monte Carlo - critical exponents

Field-theoretic RG of CFT3E. Vicari et al.

S. Wenzel and W. Janke, Phys. Rev. B 79, 014410 (2009)M. Troyer, M. Imada, and K. Ueda, J. Phys. Soc. Japan (1997)


TlCuCl3


TlCuCl3

An insulator whose spin susceptibility vanishes exponentially as the temperature T tends to zero.


λλc

Pressure in TlCuCl3


N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B 63 172414 (2001).

TlCuCl3 at ambient pressure


N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B 63 172414 (2001).

Sharp spin 1 particle excitation above an energy gap (spin gap)

TlCuCl3 at ambient pressure


Christian Ruegg, Bruce Normand, Masashige Matsumoto, Albert Furrer, Desmond McMorrow, Karl Kramer, Hans–Ulrich Gudel, Severian Gvasaliya,

Hannu Mutka, and Martin Boehm, Phys. Rev. Lett. 100, 205701 (2008)

TlCuCl3 with varying pressure

Observation of 3 → 2 low energy modes, emergence of new longi-tudinal mode (the “Higgs boson”) in Neel phase, and vanishing ofNeel temperature at quantum critical point


Prediction of quantum field theoryPotential for �ϕ fluctuations: V (�ϕ) = (λ− λc)�ϕ2 + u

��ϕ2

�2

Paramagnetic phase, λ >λ c

Expand about �ϕ = 0:

V (�ϕ) ≈ (λ− λc)�ϕ2

Yields 3 particles with energy gap ∼�

(λ− λc)

�1.0

�0.5

0.0

0.5

1.0

�1.0

�0.5

0.0

0.5

1.0

0.0

0.5

1.0

1.5

2.0

�ϕ

V (�ϕ)


Neel phase, λ < λc

Expand �ϕ =�0, 0,

�(λc − λ)/(2u)

�+ �ϕ1:

V (�ϕ) ≈ 2(λc − λ)ϕ21z

Yields 2 gapless spin waves and one Higgs particle with

energy gap ∼�

2(λc − λ)

Prediction of quantum field theoryPotential for �ϕ fluctuations: V (�ϕ) = (λ− λc)�ϕ2 + u

��ϕ2

�2

Paramagnetic phase, λ >λ c

Expand about �ϕ = 0:

V (�ϕ) ≈ (λ− λc)�ϕ2

Yields 3 particles with energy gap ∼�

(λ− λc)

�1.0

�0.5

0.0

0.5

1.0

�1.0

�0.5

0.0

0.5

1.0

0.0

0.5

1.0

1.5

2.0

�ϕ

V (�ϕ)

�1.0

�0.5

0.0

0.5

1.0�1.0

�0.5

0.0

0.5

1.0

�0.3

�0.2

�0.1

0.0

�ϕ

V (�ϕ)


0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

L (p < pc)

L (p > pc)

Q=(0 4 0)

L,T1 (p < pc)

L (p > pc)

Q=(0 0 1)

E(p < pc)unscaledEn

ergy

√2*

E(p

< p c),

E(p

> p c) [

meV

]

Pressure |(p − pc)| [kbar]

TlCuCl3pc = 1.07 kbarT = 1.85 K

Energy of Higgs particle

Energy of triplon=

√2

Prediction of quantum field theory

V (�ϕ) = (λ− λc)�ϕ2 + u��ϕ2

�2

S. Sachdev, arXiv:0901.4103Monday, June 7, 2010

t

Graphene

e1

e2

e3


Graphene

Semi-metal withmassless Dirac fermions

t

Brilluoin zone

Q1

−Q1


Graphene

U/t

t

Diracsemi-metal

Insulating antiferromagnetwith Neel order

t


I.HONEYCOM

BLATTIC

EAT

HALF

FIL

LIN

G

. We

defi

ne

the

unit

lengt

hve

ctor

s

e1

=(1,0

),

e2

=(−

1/2,√

3/2)

,e3

=(−

1/2,−√

3/2).

(1)

Not

eth

atei·e

j=−

1/2

fori6=j,

ande1

+e2

+e3

=0.

We

take

the

orig

inof

co-o

rdin

ates

ofth

ehon

eyco

mb

latt

ice

atth

ece

nte

rof

anem

pty

hexagon

.T

he

Asu

bla

ttic

esi

tes

clos

est

toth

eor

igin

are

ate1,e2,

ande3,

while

the

B

subla

ttic

esi

tes

clos

etto

the

orig

inar

eat−e1,−e2,

and−e3.

The

reci

pro

cal

latt

ice

isge

ner

ated

by

the

wav

evec

tors

G1

=4π 3

e1

,G

2=

4π 3e2

,G

3=

4π 3e3

(2)

The

firs

tB

rillou

inzo

ne

isa

hex

agon

whos

eve

rtic

esar

egi

ven

by

Q1

=1 3

(G2−

G3)

,Q

2=

1 3(G

3−G

1)

,Q

3=

1 3(G

1−G

2),

(3)

and−Q

1,−Q

2,

and−Q

3.

We

defi

ne

the

Fou

rier

tran

sfor

mof

the

ferm

ions

by

c A(k

)=∑ r

c A(r

)e−ik·r

(4)

and

sim

ilar

lyfo

rc B

.

Now

we

defi

ne

the

conti

nuum

lim

itby

the

fiel

ds

ΨA1

=c A

(Q1)

,ΨB1

=c B

(Q1)

,ΨA2

=c A

(−Q

1)

,ΨB2

=c B

(−Q

1)

(5)

The

hop

pin

gH

amilto

nia

nis

H0

=−t∑ 〈ij〉

( c† Aiαc B

jα+c† B

jαc A

iα

)(6

)

wher

eα

isa

spin

index

.If

we

intr

oduce

Pau

lim

atri

cesτa

insu

bla

ttic

esp

ace,

this

Ham

il-

tonia

nca

nb

ew

ritt

enas

H0

=

∫ d2k

4π2c†

(k)[ −t(

cos(k·e

1)

+co

s(k·e

2)

+co

s(k·e

3)) τ

x

+t( si

n(k·e

1)

+si

n(k·e

2)

+si

n(k·e

3)) τ

y(7

)

1

The

low

ener

gyex

cita

tion

sof

this

Ham

ilto

nia

nar

enea

rk≈±Q

1.

Inte

rms

ofth

efiel

ds

nea

rQ

1an

d−Q

1,

we

defi

ne

ΨA1α(k

)=c A

α(Q

+k

)

ΨA2α(k

)=c A

α(−

Q+k

)

ΨB1α(k

)=c B

α(Q

+k

)

ΨB2α(k

)=c B

α(−

Q+k

)(8

)

We

consi

der

Ψto

be

a8

com

pon

ent

vect

or,

and

intr

oduce

Pau

lim

atri

cesρa

whic

hac

tin

the

1,2

valley

spac

e.T

hen

the

Ham

ilto

nia

nis

H0

=

∫ d2k

4π2Ψ† (k

)( vτykx

+vτxρzky

) Ψ(k

),(9

)

wher

ev

=3t/2

;b

elow

we

setv

=1.

Now

defi

ne

Ψ=

Ψ† ρzτz.

Then

we

can

wri

teth

e

imag

inar

yti

me

Lag

rangi

anas

L0

=−iΨ

(ωγ0

+kxγ1

+kyγ2)

Ψ(1

0)

wher

e

γ0

=−ρzτz

γ1

=ρzτx

γ2

=−τy

(11)

So

the

low

ener

gyth

eory

has

42-

com

pon

ent

Dir

acfe

rmio

ns.

A.

Antiferromagnetism

We

use

the

oper

ator

equat

ion

(val

idon

each

sitei)

:

U

( n↑−

1 2

)( n↓−

1 2

) =−

2U 3Sa2

+U 4

(12)

Then

we

dec

ouple

the

inte

ract

ion

via

exp

( 2U 3

∑ i

∫ dτSa2i

) =

∫ DJa i(τ

)ex

p

( −∑ i

∫ dτ

[ 3U 8Ja2

i−Ja iSa i

])(1

3)

We

now

inte

grat

eou

tth

efe

rmio

ns,

and

look

for

the

saddle

poi

nt

ofth

ere

sult

ing

effec

tive

acti

onfo

rJa i.

At

the

saddle

-poi

nt

we

find

that

the

low

est

ener

gyis

achie

ved

when

the

vect

orhas

opp

osit

eor

ienta

tion

son

the

Aan

dB

subla

ttic

es.

Anti

cipat

ing

this

,w

elo

okfo

r

aco

nti

nuum

lim

itin

term

sof

afiel

dϕa

wher

e

Ja A

=ϕa

,Ja B

=−ϕa

(14)

2

The

coupling

bet

wee

nth

efiel

dϕa

and

the

Ψfe

rmio

ns

isgi

ven

by

∑ i

Ja ic† iασa αβc iβ

=ϕa( c† A

ασa αβc A

β−c† B

ασa αβc B

β

) =ϕaΨ† τzσaΨ

=−ϕaΨρzσaΨ

(15)

Fro

mth

isw

em

otiv

ate

the

low

ener

gyth

eory

L=

Ψγµ∂µΨ

+1 2

[ (∂µϕa)2

+ϕa2] +

u 24

( ϕa2) 2 −

λϕaΨρzσaΨ

(16)

Not

eth

atth

em

atri

xρzσa

com

mute

sw

ith

all

theγµ;

hen

ceρzσa

isa

mat

rix

in“fl

avor

”

spac

e.T

his

isth

eG

ross

-Nev

eum

odel

,an

dit

des

crib

esth

equan

tum

phas

etr

ansi

tion

from

the

Dir

acse

mi-

met

alto

anin

sula

ting

Nee

lst

ate.

Inth

eN

eel

stat

e,w

ehav

eϕa

=N

0δ az

(say

),an

dso

the

dis

per

sion

ofth

eel

ectr

ons

is

ωk

=±√ k

2+λ2N

2 0(1

7)

nea

rth

ep

oints±Q

1.

Thes

efo

rmth

eco

nduct

ion

and

vale

nce

ban

ds

ofth

ein

sula

tor.

3

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