“Conventional” solid state materials Strongly correlated ...€¦ · Strongly correlated...
44
Eugene Demler Harvard University Strongly correlated many-body systems: from electronic materials to ultracold atoms to photons “Conventional” solid state materials Bloch theorem for non-interacting electrons in a periodic potential
Transcript of “Conventional” solid state materials Strongly correlated ...€¦ · Strongly correlated...
Eug
en
e D
em
ler
Harv
ard
Un
ive
rsity
Stro
ngly
corre
late
d m
any-b
ody s
yste
ms:
from
ele
ctro
nic
mate
rials
to u
ltracold
ato
ms
to p
hoto
ns
“Conventio
nal”
solid
sta
te m
ate
rials
Blo
ch
the
ore
m fo
r no
n-in
tera
ctin
g
ele
ctro
ns in
a p
erio
dic
po
ten
tial
BV
H
I
d
Firs
t sem
icond
ucto
r transis
tor
EF M
eta
ls
EF
Insula
tors
and
Sem
icon
du
cto
rs
Co
nsequences o
f the B
loch th
eore
m
“Conventio
nal”
solid
sta
te m
ate
rials
Ele
ctro
n-p
hon
on a
nd e
lectro
n-e
lectro
n in
tera
ctio
ns
are
irrele
vant a
t low
tem
pera
ture
s
kx
ky
kF
Land
au F
erm
i liquid
theory
: when fre
qu
ency a
nd
tem
pera
ture
are
sm
alle
r tha
n E
Fele
ctro
n s
yste
ms
are
equ
ivale
nt to
syste
ms o
f non-in
tera
ctin
g fe
rmio
ns
Ag
Ag
Ag
Stro
ng
ly c
orre
late
d e
lectro
n s
yste
ms
Quantu
m H
all s
yste
ms
kin
etic
energ
y s
uppre
ssed b
y m
agnetic
field
He
avy fe
rmio
nm
ate
rials
many p
uzzlin
g n
on-F
erm
i liq
uid
pro
pertie
s
Hig
h te
mpera
ture
superc
ond
ucto
rsU
nusual “n
orm
al”
sta
te,
Contro
vers
ial m
echanis
m o
f superc
onductiv
ity,
Severa
l com
petin
g o
rders
UC
u3.5
Pd
1.5
CeC
u2 S
i2
What is
the c
onnectio
n b
etw
een
stro
ngly
corre
late
d e
lectro
n s
yste
ms
and
ultra
cold
ato
ms?
Bose-E
inste
in c
ond
ensatio
n o
f
weakly
inte
ractin
g a
tom
s
Scatte
ring le
ng
th is
much s
malle
r than c
hara
cte
ristic
inte
rpa
rticle
dis
tances.
Inte
ractio
ns a
re w
eak
Density
10
13
cm
-1
Typic
al d
ista
nce b
etw
ee
n a
tom
s 3
00 n
m
Typic
al s
catte
ring le
ngth
10 n
m
New
Era
in C
old
Ato
ms R
esearc
h
Focus o
n S
yste
ms w
ith S
trong In
tera
ctio
ns
•A
tom
s in
op
tica
l lattic
es
•F
esh
ba
ch
reso
na
nce
s
•L
ow
dim
en
sio
na
l syste
ms
•S
yste
ms w
ith lo
ng
ran
ge
dip
ola
r inte
ractio
ns
•R
ota
ting
syste
ms
Fe
sh
ba
ch
reso
na
nce
an
d fe
rmio
nic
co
nd
en
sa
tes
Gre
iner e
t al., N
atu
re (2
003); K
ette
rlee
t al., (2
003)
Kette
rleet a
l.,
Natu
re 4
35, 1
047-1
051 (2
005
)
One d
imensio
nal s
yste
ms
Stro
ngly
inte
ractin
g
regim
e c
an b
e re
ache
d
for lo
w d
ensitie
s One d
imensio
nal s
yste
ms in
mic
rotra
ps.
Thyw
issen
et a
l., Eur. J
. Ph
ys. D
. (99);
Hanse
l et a
l., Natu
re (0
1);
Folm
an
et a
l., Adv. A
t. Mol. O
pt. P
hys. (0
2)
1D
confin
em
en
t in o
ptic
al p
ote
ntia
l
Weis
s e
t al., S
cie
nce (0
5);
Blo
ch e
t al.,
Esslin
ger
et a
l.,
Ato
ms in
optic
al la
ttices
Theory
: Jaksch
et a
l. PR
L (1
998)
Exp
erim
ent: K
asevic
het a
l., Scie
nce (2
001);
Gre
iner e
t al., N
atu
re (2
00
1);
Phillip
s e
t al., J
. Ph
ysic
s B
(200
2)
Esslin
ger
et a
l., PR
L (2
00
4);
and m
an
y m
ore
…
Stro
ng
ly c
orre
late
d s
yste
ms
Ato
ms in
optic
al la
ttices
Ele
ctro
ns in
Solid
s
Sim
ple
meta
ls
Pertu
rbatio
n th
eory
in C
oulo
mb in
tera
ctio
n a
pplie
s.
Band s
tructu
re m
eth
ods w
otk
Stro
ngly
Corre
late
d E
lectro
n S
yste
ms
Band s
tructu
re m
eth
ods fa
il.
Novel p
he
nom
ena in
stro
ngly
corre
late
d e
lectro
n s
yste
ms:
Quantu
m m
ag
netis
m, p
hase s
epara
tion, u
nconventio
na
l superc
onductiv
ity,
hig
h te
mp
era
ture
superc
ondu
ctiv
ity, fra
ctio
naliz
atio
n o
f ele
ctro
ns …
Stro
ngly
co
rrela
ted s
yste
ms o
f ultra
cold
ato
ms s
hould
als
o b
e u
sefu
l for a
pplic
atio
ns in
qua
ntu
m in
form
atio
n,
hig
h p
recis
ion s
pectro
scopy, m
etro
logy
By s
tudyin
g s
trongly
inte
ractin
g s
yste
ms o
f cold
ato
ms w
e
expect to
get in
sig
hts
into
the m
yste
rious p
ropertie
s o
f
nove
l qu
an
tum
mate
rials
: Quantu
m S
imula
tors
BU
TS
trongly
inte
ractin
g s
yste
ms o
f ultra
cold
ato
ms a
nd p
hoto
ns:
are
NO
T d
irect a
na
logu
es o
f cond
ensed m
atte
r syste
ms
These a
re in
de
pe
nd
en
t physic
al s
yste
ms w
ith th
eir o
wn
“pers
on
alitie
s”, p
hysic
al p
rop
ertie
s, a
nd th
eo
retic
al c
ha
lleng
es
New
Phenom
ena in
quantu
m
many-b
ody s
yste
ms o
f ultra
cold
ato
ms
Lon
g in
trinsic
time s
cale
s-
Inte
ractio
n e
nerg
y a
nd b
andw
idth
~ 1
kH
z-
Syste
m p
ara
mete
rs c
an b
e c
hanged o
ver th
is tim
e s
cale
Decou
plin
g fro
m e
xte
rna
l enviro
nm
ent
-Long c
ohere
nce tim
es
Ca
n a
ch
ieve
hig
hly
no
n e
qu
ilibriu
m q
ua
ntu
m m
an
y-b
od
y s
tate
s
New
dete
ctio
n m
eth
ods
Inte
rfere
nce, h
igher o
rder c
orre
latio
ns
Stro
ng
ly c
orre
late
d m
any-b
ody
syste
ms o
f photo
ns
Lin
ear g
eom
etric
al o
ptic
s
Stro
ng o
ptic
al n
onlin
earitie
s in
nanoscale
surfa
ce p
lasm
ons
Akim
ov
et a
l., Natu
re (2
007)
Stro
ngly
inte
ractin
g p
ola
ritons
in
couple
d a
rrays o
f cavitie
s
M. H
artm
ann e
t al., N
atu
re P
hysic
s (2
006)
Cry
sta
llizatio
n (fe
rmio
niz
atio
n) o
f photo
ns
in o
ne d
imensio
nal o
ptic
al w
aveguid
es
D. C
hang e
t al., N
atu
re P
hysic
s (2
008)
Stro
ngly
corre
late
d s
yste
ms o
f photo
ns
Outlin
e o
f these le
ctu
res
•In
troductio
n. S
yste
ms o
f ultra
cold
ato
ms.
•B
og
oliu
bov
theory
. Spin
or
cond
en
sate
s.
•C
old
ato
ms in
optic
al la
ttices.
•B
ose H
ub
bard
mode
l and e
xte
nsio
ns
•B
ose m
ixtu
res in
optic
al la
ttices
Quantu
m m
agnetis
m o
f ultra
cold
ato
ms.
Curre
nt e
xperim
ents
: observ
atio
n o
f supere
xchange
•F
erm
ions in
optic
al la
ttices
Magnetis
m a
nd p
airin
g in
syste
ms w
ith re
puls
ive in
tera
ctio
ns.
Curre
nt e
xperim
ents
: Mott s
tate
•D
ete
ctio
n o
f many-b
od
y p
hases u
sin
g n
ois
e c
orre
latio
ns
•E
xperim
ents
with
low
dim
ensio
nal s
yste
ms
Inte
rfere
nce e
xperim
ents
. Analy
sis
of h
igh o
rder c
orre
latio
ns
•N
on-e
qu
ilibriu
m d
yna
mic
s
Em
phasis
of th
ese le
ctu
res:
•D
ete
ctio
n o
f many-b
ody p
hase
s
•D
ynam
ics
Ultra
cold
ato
ms
Most c
om
mon b
oso
nic
ato
ms: a
lka
li 87R
b a
nd 2
3N
a
Most c
om
mon fe
rmio
nic
ato
ms: a
lkali 4
0K
and 6
Li
Ultra
cold
ato
ms
Oth
er s
yste
ms:
BE
C o
f13
3Cs (e
.g. G
rimm
et a
l.)
BE
C o
f52C
r (Pfa
uet a
l.)
BE
C o
f84S
r (e.g
. Grim
m e
t al.),
87S
rand
88S
r(e
.g. Y
e e
t al.)
BE
C o
f 168Y
b, 1
70Y
b, 1
72Y
b, 1
74Y
b, 1
76Y
b
Quantu
m d
ege
nera
te fe
rmio
ns 1
71Y
b 1
73Y
b (T
akah
ash
i et a
l.)
Sin
gle
va
lence e
lectro
n in
the s
-orb
ital
and
Nucle
ar s
pin
Magnetic
pro
pertie
s o
f indiv
idual a
lkali a
tom
s
Zero
field
splittin
g b
etw
een a
nd
sta
tes
For 2
3Na A
HF
S =
1.8
GH
z a
nd fo
r 87R
b A
HF
S =
6.8
GH
z
Tota
l ang
ula
r mom
entu
m (h
yperfin
e s
pin
)
Hyp
erfin
e c
oup
ling m
ixes n
ucle
ar a
nd e
lectro
n s
pin
s
Magnetic
pro
pertie
s o
f indiv
idual a
lkali a
tom
s
Effe
ct o
f magn
etic
field
com
es fro
m e
lectro
n s
pin
gs =
2and µ
B=
1.4
MH
z/G
When fie
lds a
re n
ot to
o la
rge o
ne
can u
se (a
ssum
ing fie
ld a
long z
)
The la
st te
rm d
escrib
es q
uadra
tic Z
eem
an e
ffect q
=h 3
90 H
z/G
2
Magnetic
trappin
g o
f alk
ali a
tom
s
Magnetic
trapp
ing o
f neutra
l ato
ms is
due to
the Z
eem
an
effe
ct.
The e
nerg
y o
f an a
tom
ic s
tate
depends o
n th
e m
agnetic
field
.
In a
n in
ho
mogen
eo
us fie
ld a
n a
tom
exp
erie
nces a
spatia
lly
vary
ing p
ote
ntia
l.
Exam
ple
:
Pote
ntia
l:
Magnetic
trappin
g is
limite
d b
y th
e re
quire
ment th
at th
e tra
pped
ato
ms re
main
in w
eak fie
ld s
eekin
g s
tate
s. F
or 2
3Na a
nd
87R
bth
ere
are
thre
e s
tate
s
Optic
al tra
ppin
g o
f alk
ali a
tom
s
Base
d o
n A
C S
tark
effe
ct
-pola
rizab
ility
Typic
ally
optic
al fre
quencie
s.
Pote
ntia
l:
Dip
ola
r mom
ent in
duced b
y th
e e
lectric
field
Far-o
ff-resona
nt o
ptic
al tra
p c
onfin
es a
tom
s re
gard
less
of th
eir h
yp
erfin
e s
tate
Bog
oliu
bo
v th
eory
of
weakly
inte
ractin
g B
EC
. C
olle
ctiv
e m
od
es
BE
C o
f spin
less
bosons. B
ogoliu
bov
theory
We c
onsid
er a
unifo
rm s
yste
m firs
t
For n
on-in
tera
ctin
g a
tom
s a
t T=
0all a
tom
s a
re in
k=
0sta
te.
Mean fie
ld e
qu
atio
ns
Min
imiz
ing w
ith re
spect to
N0
we fin
d
-boson a
nnih
ilatio
n o
pera
tor a
t mom
entu
m p
,
-stre
ngth
of c
onta
ct s
-wave in
tera
ctio
n
-volu
me o
f the s
yste
m
-chem
ical p
ote
ntia
l
BE
C o
f spin
less
bosons. B
ogoliu
bov
theory
We n
ow
expan
d a
rou
nd th
e n
oin
tera
ctin
gso
lutio
n fo
r sm
all U
0.
Fro
m th
e d
efin
ition o
f Bose o
pera
tors
When N
0>
>1
we c
an tre
at b
0as a
c-n
um
ber a
nd re
pla
ce
by -
means th
at p
,-ppairs
should
be c
ounte
d o
nly
once
n0
= N
0/V
-density
and µ
= n
0 U
0
Mean-fie
ld H
am
iltonia
n
Bogoliu
bov
transfo
rmatio
n
Boso
nic
co
mm
uta
tion re
latio
ns
are
pre
serv
ed w
hen
Bogoliu
bov
transfo
rmatio
n
Mean-fie
ld H
am
iltonia
n b
ecom
es
Bogoliu
bov
transfo
rmatio
n
Ca
nce
llatio
n o
f non-d
iago
na
l term
s re
quire
s
To s
atis
fy ta
ke
So
lutio
n o
f these e
qu
atio
ns
Mean-fie
ld
Ham
iltonia
n
Bogoliu
bov
modes
Dis
pers
ion o
f colle
ctiv
e m
odes
Defin
e h
ea
ling le
ngth
from
Lon
g w
ave
length
limit, , s
ound d
ispers
ion
Short w
ave
length
limit, , fre
e p
artic
les
Sou
nd v
elo
city
Pro
bin
g th
e d
ispers
ion o
f BE
Cby o
ff-resonant lig
ht s
catte
ring
For d
eta
ils s
ee
cond-m
at/0
005
00
1
Tre
at o
ptic
al fie
ld a
s c
lassic
al
Excita
tion ra
te o
ut o
f the g
rou
nd s
tate
|g>
Dyn
am
ical s
tructu
re fa
cto
r
For s
mall q
we
find
PRL 83:2876 (1999)
Ω Ω
PRL 88:60402 (2002)
Rev. Mod. Phys. 77:187 (2005)
Gro
ss-P
itaevskii
eq
ua
tion
Gro
ss-P
itaevskii
equatio
n
Ham
iltonia
n o
f inte
ractin
g b
oso
ns
Com
muta
tion re
latio
ns
Equ
atio
ns o
f motio
n
This
is o
pe
rato
r equatio
n. W
e c
an ta
ke c
lassic
al lim
it by a
ssum
ing
that a
ll ato
ms c
onden
se in
to th
e s
am
e s
tate
.
The la
st e
quatio
n b
ecom
es a
n e
quatio
n o
n th
e w
avefu
nctio
n
Gro
ss-P
itaevskii
equatio
n
Ana
lysis
of flu
ctu
atio
ns o
n to
p o
f the m
ean-fie
ldG
P e
quatio
ns le
ads to
the B
ogo
liubov
modes
Tw
o-c
om
po
ne
nt m
ixtu
res
Tw
o-c
om
ponent B
ose m
ixtu
re
Co
nsid
er m
ean-fie
ld (a
ll partic
les in
the c
ondensate
)
Re
pu
lsiv
e in
tera
ctio
ns. M
iscib
le a
nd im
mis
cib
le re
gim
es
Syste
m w
ith a
finite
density
of b
oth
specie
s
is u
nsta
ble
to p
hase
separa
tion
What h
app
ens if w
e p
repare
a m
ixtu
re o
f two
cond
ensate
s in
the im
mis
cib
le re
gim
e?
Tw
o c
om
pone
nt G
P e
quatio
n
Assum
e e
qua
l densitie
s
Ana
lyze flu
ctu
atio
ns
Bose m
ixtu
re: d
ynam
ics o
f phase s
epara
tion
Bose m
ixtu
re: d
ynam
ics o
f phase s
epara
tion
Defin
e
density
fluctu
atio
n
phase flu
ctu
atio
n
Equ
atio
ns o
f motio
n: c
harg
e c
on
serv
atio
n a
nd J
osep
hson re
latio
n
Equ
atio
n o
n c
olle
ctiv
e m
odes
Bose m
ixtu
re: d
ynam
ics o
f phase s
epara
tion
Here
When w
e g
et im
agin
ary
frequencie
s
Most u
nsta
ble
mode
Imagin
ary
frequencie
s in
dic
ate
exponentia
l gro
wth
of flu
ctu
atio
ns,
i.e. in
sta
bility
. The m
ost u
nsta
ble
mode s
ets
the le
ngth
for p
atte
rnfo
rmatio
n. N
ote
that
when . T
his
is re
quire
dby s
pin
conserv
atio
n.
PRL 82:2228 (1999)
Spin
or
co
nd
en
sate
s
F=
1
Spin
or
condensate
s. F
=1
Thre
e c
om
pon
ent o
rder p
ara
mete
r: mF=
-1,0
,+1
Co
nta
ct in
tera
ctio
n d
epen
ds o
n re
lativ
e s
pin
orie
nta
tion
When g
2>
0in
tera
ctio
n is
antife
rrom
agnetic
. Exam
ple
23N
a
When g
2<
0in
tera
ctio
n is
antife
rrom
agnetic
. Exam
ple
87R
b
F=
1 s
pin
or
condensate
s. H
am
iltonia
n
-spin
opera
tors
for F
=1
Tota
l Fz
is c
onserv
ed s
o lin
ear Z
eem
an te
rm s
hould
(usua
lly)
be u
nd
ers
tood a
s L
ag
range m
ultip
lier th
at c
ontro
ls F
z .
Quadra
tic Z
eem
an e
ffect c
auses th
e e
nerg
y o
f mF=
0sta
te to
be lo
wer th
an th
e e
nerg
y o
f mF=
-1,+
1sta
tes.
The a
ntife
rrom
agn
etic
inte
ractio
n (g
2>
0) fa
vors
the n
em
atic
(pola
r) sta
te (m
F=
0a
nd its
rota
tions). T
he fe
rrom
agn
etic
in
tera
ctio
n (g
2<
0)
favors
spin
pola
rize
d s
tate
(mF=
+1)
and
its ro
tatio
ns).
Phase d
iagra
m o
f F=
1 s
pin
or
condensate
s
g2>
0A
ntife
rrom
agn
etic
g2<
0F
erro
magn
etic
Shaded re
gio
n: m
ixtu
re o
f all th
ree
sta
tes. T
here
is X
Y c
om
ponent
of th
e s
pin
.
Shaded re
gio
n: m
ixtu
re o
fm
F =-1
,+1
sta
tes. T
his
sta
telo
wers
inte
ractio
n e
nerg
y.
Natu
re 3
96:3
45 (1
998)
Usin
g m
agnetic
field
gra
die
nt
to e
xp
lore
the p
hase d
iagra
m
of F
=1 a
tom
s w
ith A
F in
tera
ctio
ns
23N
a
Natu
re 4
43:3
12 (2
006)
Ma
gn
etic
dip
ola
r inte
ractio
ns
in u
ltracold
ato
ms
Ma
gn
etic
dip
ola
r inte
ractio
ns in
sp
ino
rco
nd
en
sa
tes
Com
paris
on o
f conta
ct a
nd d
ipola
r inte
ractio
ns.
Typic
al v
alu
e a
=100a
B
θ
For 8
7Rb µ
=µB
and ε
=0.0
07 F
or 5
2Cr µ
=6µ
Band ε
=0.1
6
Bose c
ond
ensatio
n
of 5
2Cr.
T. P
fau
et a
l. (2005)
Revie
w:
Menotti e
t al.,
arX
iv071
1.3
42
2
Ma
gn
etic
dip
ola
r inte
ractio
ns in
sp
ino
rco
nd
en
sa
tes
Inte
ractio
n o
f F=
1 a
tom
s
Ferro
magn
etic
Inte
ractio
ns fo
r 87R
b
Sp
in-d
ep
enent
part o
f the in
tera
ctio
n is
sm
all.
Dip
ola
r inte
ractio
n m
ay b
e im
porta
nt
(D. S
tam
per-K
urn
)
a2-a
0=
-1.0
7 a
BA
. Wid
era
, I. Blo
ch e
t al.,
Ne
w J
. Ph
ys. 8
:152 (2
00
6)
Sp
on
tan
eo
usly
mo
du
late
d te
xtu
res in
sp
ino
rco
nd
en
sa
tes
Fourie
r spectru
m o
f the
fragm
ente
d c
ondensate
Ven
ga
latto
reet a
l.P
RL (2
008
)
Energ
y s
cale
s: im
porta
nce o
f Larm
or
pre
cessio
n
S-w
ave S
catte
ring
•Sp
in in
de
pend
ent (g
0 n =
kH
z)
•Sp
in d
epe
ndent (g
s n=
10 H
z)
Dip
ola
r Inte
ractio
n•A
nis
otro
pic
(gd n
=10 H
z)
•Lon
g-ra
ng
ed
Mag
netic
Fie
ld•L
arm
or P
recessio
n (1
00 k
Hz)
•Quadra
tic Z
eem
an (0
-20 H
z)
B
F
Red
uced
Dim
en
sio
nality
•Quasi-2
D g
eom
etry
spin
dξ
<
Sta
bility
of s
yste
ms w
ith s
tatic
dip
ola
r inte
ractio
ns
Ferro
magn
etic
config
ura
tion is
robust a
gain
st s
mall
pertu
rbatio
ns. A
ny ro
tatio
n o
f the s
pin
s c
onflic
ts w
ith
the “h
ead to
tail”
arra
ngem
ent
Larg
e flu
ctu
atio
n re
qu
ired to
reach a
low
er e
nerg
y c
onfig
ura
tion
XY
com
pon
ents
of th
e
spin
s c
an lo
wer th
e e
nerg
y
usin
g m
odula
tion a
long z
.
Z c
om
ponents
of th
e
spin
s c
an lo
we
r the e
nerg
y
usin
g m
odula
tion a
long x
X
Dip
ola
r inte
ractio
n a
ve
rag
ed
afte
r pre
ce
ssio
n
“Head to
tail”
ord
er o
f the tra
nsvers
e s
pin
com
ponents
is v
iola
ted
by p
recessio
n. O
nly
need to
check w
heth
er s
pin
s a
re p
ara
llel
Stro
ng in
sta
bilitie
s o
f syste
ms w
ith d
ipola
r inte
ractio
ns
afte
r avera
gin
g o
ver p
recessio
n
X
Insta
bilitie
s o
f F=
1 R
b(fe
rrom
agnetic
) condensate
due to
dip
ola
r inte
ractio
ns
Theory
: unsta
ble
modes in
the
regim
e
corre
spo
ndin
g to
Berk
ele
y e
xperim
ents
.
Chern
g, D
em
ler, P
RL (2
009)
Exp
erim
ents
.
Venga
latto
re e
t al. P
RL (2
008)
Berk
ele
y E
xperim
ents
: checkerb
oard
phase
Insta
bilitie
s o
f magnetic
dip
ola
r inte
ractio
ns:
genera
l analy
sis
α–
angle
betw
een m
agn
etic
field
and n
orm
al to
the p
lane
dn
–la
yer th
ickness
qm
easure
s
the s
trength
of q
uadra
tic
Zeem
an e
ffect
Insta
bilitie
s o
f colle
ctiv
e m
odes
Magneto
roto
nsofte
nin
g
Qm
easure
s
the s
trength
of q
uadra
tic
Zeem
an e
ffect
87R
b c
on
de
nsate
: m
ag
netic
su
pers
olid
?
Phase d
iag
ram
of 4
He
Possib
le s
upers
olid
phase in
4He
A.F
. Andre
ev a
nd I.M
. Lifs
hits
(1969):
Meltin
g o
f vacancie
s in
a c
rysta
l due
to s
trong q
uantu
m flu
ctu
atio
ns.
Als
o
G. C
heste
r (1970); A
.J. L
egg
ett (1
97
0)
T. S
chne
ider a
nd C
.P. E
nz
(1971
).
Fo
rma
tion
of th
e s
up
ers
olid
ph
ase
du
e to
so
ften
ing
of ro
ton
excita
tion
s
Reson
ant p
erio
d a
s a
functio
n o
f T
Inte
rlayer c
ohere
nce in
bila
yer
quantu
m H
all s
yste
ms a
t ν=
1
Hartre
e-F
ock
pre
dic
ts ro
ton
softe
nin
gand tra
nsitio
n in
to a
sta
te w
ith b
oth
in
terla
yer c
ohere
nce a
nd s
tripe o
rder.
Tra
nsport e
xperim
ents
suggest firs
t ord
er tra
nsitio
n in
to a
com
pre
ssib
le
sta
te.
L. B
rey
an
d H
. Fertig
(2000)
Eis
enste
in, B
oebin
ger
et a
l. (1994)
Low
energ
y e
ffectiv
e m
odel fo
r fe
rrom
agnetic
condensate
s
Mass s
upe
rflow
curre
nt
cause
d b
y s
pin
textu
re
Topo
log
ica
l spin
win
din
g(P
ontry
agin
ind
ex)
Magnetic
dip
ola
r inte
ractio
n.
Favors
spin
skyrm
ion
s
Re
qu
ires n
et z
ero
spin
win
din
g
Magnetic
cry
sta
l phases in
F=
1 8
7Rb
ferro
magnetic
condensate
sE
xperim
enta
l constra
ints
:
1.
Dih
edra
l sym
metry
of th
e m
agn
etic
cry
sta
l
2.
Recta
ngu
lar la
ttice
3.
No
n-p
lanar s
pin
orie
nta
tion
Magnetic
cry
sta
l lattic
es
optim
ized w
ithin
each
cla
ss
Optim
al s
pin
config
ura
tion.
Chern
g, D
em
ler, u
npublis
hed
Sp
in c
orre
latio
n fu
nctio
ns
for s
pin
com
pone
nts
para
llelan
d p
erp
end
icula
r
to th
e m
agnetic
field
Magnetic
cry
sta
l phases in
F=
1 8
7Rb
ferro
magnetic
condensate
s
Ultra
cold
ato
ms in
optic
al la
ttices.
Band s
tructu
re. S
em
icla
sic
aldyna
mic
s.
Optic
al la
ttice
The s
imp
lest p
ossib
le p
erio
dic
op
tical p
ote
ntia
l is fo
rmed
by o
verla
ppin
g tw
o c
ounte
r-pro
pagatin
g b
eam
s. T
his
resu
lts
in a
sta
ndin
g w
ave
Avera
gin
g o
ver fa
st o
ptic
al o
scilla
tions (A
C S
tark
effe
ct) g
ives
Com
bin
ing th
ree p
erp
end
icula
r sets
of s
tandin
g w
aves w
e
get a
sim
ple
cubic
lattic
e
This
pote
ntia
l allo
ws s
epara
tion o
f varia
ble
s
Optic
al la
ttice
For e
ach c
oord
inate
we h
ave M
atth
ieu
eq
uatio
n
Eig
enva
lue
sand e
ige
nfu
nctio
ns
are
know
n
In th
e re
gim
e o
f deep
lattic
e w
e g
et th
e tig
ht-b
ind
ing m
ode
l
-bandg
ap
-re
coil e
ne
rgy
Lo
west b
and
Ban
d s
tructu
re
Optic
al la
ttice
Effe
ctiv
e H
am
ilton
ian fo
r non-in
tera
ctin
g
ato
ms in
the lo
west B
loch b
an
d
neare
st
neig
hbors
Sem
icla
ssic
aldynam
ics in
the la
ttice
1.
Ban
d in
de
x is
consta
nt
2.
3.
Blo
ch o
scilla
tions
Co
nsid
er a
unifo
rm a
nd c
onsta
nt fo
rce
PR
L 7
6:4
508 (1
996)
Sta
te d
ependent o
ptic
al la
ttices
Ho
w to
use s
ele
ctio
n ru
les fo
r optic
al tra
nsitio
ns to
make
diffe
rent la
ttice p
ote
ntia
ls fo
r diffe
rent in
tern
al s
tate
s.
Fin
e s
tructu
re fo
r 23N
a a
nd 8
7Rb
The rig
ht c
ircula
rly p
ola
rized lig
ht c
ouple
s
to tw
o
excite
d le
vels
P1/2
and P
3/2 . A
C S
tark
effe
cts
have o
pposite
sig
ns a
nd c
ancel
each o
ther fo
r the a
ppro
pria
te fre
quency. A
t this
frequency A
C S
tark
effe
ct
for th
e s
tate
com
es o
nly
from
pola
rized lig
ht a
nd
giv
es th
e p
ote
ntia
l .
Analo
gously
sta
te w
ill only
be a
ffecte
d b
y , w
hic
h
giv
es th
e p
ote
ntia
l .
Decom
posin
g
hyperfin
e
sta
tes w
e fin
d
PR
L 9
1:1
0407 (2
003)
Sta
te d
ep
en
de
nt la
ttice
