Subir Sachdevqpt.physics.harvard.edu/talks/tasi_sachdev1.pdf · 2010-06-14 · Subir Sachdev TASI...
Transcript of Subir Sachdevqpt.physics.harvard.edu/talks/tasi_sachdev1.pdf · 2010-06-14 · Subir Sachdev TASI...
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
Monday, June 7, 2010
6. Square lattice: Fermi surfaces and spin density waves Fermi pockets and Quantum oscillations
7. Instabilities near the SDW critical point d-wave superconductivity and other orders
8. Global phase diagram of the cuprates Competition for the Fermi surface
Outline
Monday, June 14, 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
Monday, June 7, 2010
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
Monday, June 7, 2010
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
Monday, June 7, 2010
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
.
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
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
Monday, June 7, 2010
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.
Monday, June 7, 2010
Square lattice antiferromagnet
H =�
�ij�
Jij�Si · �Sj
J
J/λ
Weaken some bonds to induce spin entanglement in a new quantum phase
Monday, June 7, 2010
Square lattice antiferromagnet
H =�
�ij�
Jij�Si · �Sj
J
J/λ
Ground state is a “quantum paramagnet”with spins locked in valence bond singlets
=1√2
����↑↓�−
��� ↓↑��
Monday, June 7, 2010
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
Monday, June 7, 2010
λλc
Excitation spectrum in the paramagnetic phase
Monday, June 7, 2010
λλc
Excitation spectrum in the paramagnetic phase
Monday, June 7, 2010
λλc
Excitation spectrum in the paramagnetic phase
Monday, June 7, 2010
λλc
Excitation spectrum in the paramagnetic phase
Monday, June 7, 2010
λλc
Excitation spectrum in the paramagnetic phase
Sharp spin 1 particle excitation above an energy gap (spin gap)
Monday, June 7, 2010
λλc
Excitation spectrum in the Neel phase
Monday, June 7, 2010
λλc
Excitation spectrum in the Neel phase
Spin waves
Monday, June 7, 2010
λλc
Excitation spectrum in the Neel phase
Spin waves
Monday, June 7, 2010
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
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)
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
Monday, June 7, 2010
�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
Excitation spectrum in the paramagnetic phase
�ϕ
Spin S = 1“triplon”
V (�ϕ)V (�ϕ) = (λ− λc)�ϕ2 + u
��ϕ2
�2
λ >λ c
Monday, June 7, 2010
�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
Excitation spectrum in the paramagnetic phase
�ϕ
Spin S = 1“triplon”
V (�ϕ)V (�ϕ) = (λ− λc)�ϕ2 + u
��ϕ2
�2
λ >λ c
Monday, June 7, 2010
�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
Excitation spectrum in the paramagnetic phase
�ϕ
Spin S = 1“triplon”
V (�ϕ)V (�ϕ) = (λ− λc)�ϕ2 + u
��ϕ2
�2
λ >λ c
Monday, June 7, 2010
�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
Excitation spectrum in the paramagnetic phase
�ϕ
Spin S = 1“triplon”
V (�ϕ)V (�ϕ) = (λ− λc)�ϕ2 + u
��ϕ2
�2
λ >λ c
Monday, June 7, 2010
�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
Excitation spectrum in the paramagnetic phase
�ϕ
Spin S = 1“triplon”
V (�ϕ)V (�ϕ) = (λ− λc)�ϕ2 + u
��ϕ2
�2
λ >λ c
Monday, June 7, 2010
�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)
Excitation spectrum in the Neel phase
Spin waves (“Goldstone” modes)
and a longitudinal “Higgs” particle
V (�ϕ) = (λ− λc)�ϕ2 + u��ϕ2
�2
λ <λ c
Monday, June 7, 2010
λλc
Excitation spectrum in the Neel phase
�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)
Spin waves (“Goldstone” modes)
and a longitudinal “Higgs” particle
V (�ϕ) = (λ− λc)�ϕ2 + u��ϕ2
�2
λ <λ c
Monday, June 7, 2010
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)
Excitation spectrum in the Neel phase
Spin waves (“Goldstone” modes)
and a longitudinal “Higgs” particle
V (�ϕ) = (λ− λc)�ϕ2 + u��ϕ2
�2
λ <λ c
Monday, June 7, 2010
λλcCFT3
O(3) order parameter �ϕ
S =�
d2rdτ�(∂τϕ)2 + c2(∇r �ϕ)2 + s�ϕ2 + u
��ϕ2
�2�
Monday, June 7, 2010
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
Monday, June 7, 2010
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)
Monday, June 7, 2010
TlCuCl3
Monday, June 7, 2010
TlCuCl3
An insulator whose spin susceptibility vanishes exponentially as the temperature T tends to zero.
Monday, June 7, 2010
λλc
Pressure in TlCuCl3
Monday, June 7, 2010
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
Monday, June 7, 2010
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
Monday, June 7, 2010
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
Monday, June 7, 2010
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)
Monday, June 7, 2010
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)
Monday, June 7, 2010
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
Monday, June 7, 2010
Graphene
Semi-metal withmassless Dirac fermions
t
Brilluoin zone
Q1
−Q1
Monday, June 7, 2010
Graphene
U/t
t
Diracsemi-metal
Insulating antiferromagnetwith Neel order
t
Monday, June 7, 2010
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