AKLT

Also note the unpaired spins at the end of the chain, which are the fractionalized edge
modes. (Figure taken from the Wikipedia article: AKLT model.)there will be a phase factor ei2πSQ . It follows that for integer spins this factor
is always 1, and we conclude that the chain is gapped. For half-integer chains
the problem is much more complicated. The large fluctuations responsible forgenerating the mass gap typically have non-zero winding numbers and, because
Figure8: Graphicalthe AKLTcancellations.
ground state. Theblackdots denoteofthe sign,(−1)Q , representation
there may be ofimportantThus,althoughtheauxiliary
spin
1/2“particles”,
theovals
project
on
spin
1,
and
thelinesmean
thatargument based on the behaviour of the sigma model works for the integer
two spins
1/2itforma singletstate.
two of case.
the fourspin
1/2 on two together
adjacent
spin
chains,breaksdown inthe Since
half-integerThis
observation,sites forma singlet,
the maximal
total spin
is 1, soa theprojectionfor
onthe
spinHaldane
2 gives
withthe spin
1/2 chainbeing gapless,providesmotivation
zero.
Since
thisistruefor
allpairs,
thisstateisclearly
aneigenstateofHconjecture. Note that the most surprising result – that the integer spin AKLT
chain.
Also
note theat language
the end of ofthethechain,
which
are thefractionalizededge
isgapped– isunpaired
naturalspinsin thesigma
model,
while
it was hardermodes.
(Figure whattaken happens
from the Wikipedia
article: AKLT
tounderstandin the half-integer
case. model.)Only later was it proventhat the θ = π sigma model really is gapless [48].
We nowthe above
ratherabstract
argument,
whichtoalso
reliedprovidesthecomplement
extra dimension,and thecontrol
parameter
analogous
temperai2πSQ
there
will
be a phaseoffactor
eS,. It afollows
that forintegerspins
this factor
ontheassumption
large
with
description
of
a
very
instructive
andture is the ratio of two energy scales in the Hamiltonian [12, 49]. In this way,
isalways
1,andweconclude
that
the
chain
isgapped.
Forhalf-integer
chainsexactlysolvable model
fortheS =1, which
is a close cousin
the Heisenberg
the KT-transition
forms
basis
for understanding
how ofa one-dimensional
theproblem
ismuchmorecomplicated.
The
large
fluctuations
responsible
for
model
in Josephson
Eq. (1).
for
AKLT
chain,
namedafter
Figure9:
Thegraph TheintheHamiltonian
middleshowsundergoes
the thisenergy
a spinexcitation
intransitiona spinits
1
chain oftunneljunctionsaofzero-temperature
generating
themass
gap
typicallyhave
non-zerowinding
numbersand,becauseinventors
Aﬄeck,
Tom
Kennedy,
Elliott
LiebandHal
Tasaki,
is[1]
chain
a Ian
functionof
momentaclosetotheNéel
point
Q=1,whichcorresponds
from assuperconducting
toinsulating
behaviourastheJosephson
coupling
becof the sign, (−1)�Q , there may be important
cancellations.
Thus, although the� �� model�toa π phase
diﬀerence
betweenNisamespins
along�thethe Haldane
is2waschains;tween
junctionsis1 tuned[25]. theThe
later
realizedusinggap
ultra
1
argument
based onS
behaviourof theworks Pfor�ithe
�i+1
�i ·sigma
�i+1 model�the�integer
H=·S+cf
rac16SS+=
(S+S)
AKLT
2ii+1
clearlyvisible.
(FigurefromRef.
[31].)
cold atomic
gases2 trapped
to form discrete
lattices,and excitation
also hereinone
could
Figure9: Thethe middle
the energyof aThisa spin1
3spin
spin chains,itigraphbreaksindown
in theshows
half-integercase.
observation,
togetheriobserve
theKT-transition
[14].
chain
as aspinfunctionof momenta
close to the
Néel point
Qc = 1, whichcorrespondswith the1/2 chainbeing gapless,providesa motivation
for theHaldane(18)
Bothfermionicandbosonic
atoms
canbe
trappedin
opticallattices,
simtoaπphasediﬀerence
between
theNi
spins
along
thechains;theHaldane
gap
is�
conjecture.
Note
that
the
mostsurprising
result–that
theinteger
spin
chain
where Si is a spin 1 operator at the lattice site i, and P2 projects on theilarly toelectrons
in thea crystal
lattice.
This also makes
it possible towith
engineerclearlyvisible.
(FigurefromRef.
[31].)
singletateachlink
in
chain
gives
aneigenstateof
theHamiltonianzero
is gapped corresponding– is natural in tothespinlanguageof thesigma model,
wasfindhardersubspace
2 on twoadjacentlattice whilesites. itTothetopologically
nontrivial
bands,isand
we mention
two recent
examples.
energy.
Since
theHamiltonian
asumofprojectors,
thegroundstateenergyto understandwhat
happens
the half-integer
case. Only
was it provengroundstate, weimaginethatineachlink in the chain
hosts later
two auxiliaryspinIn
abe2014experiment,
thewe
group
led bythat
ImmanuelBlochmanagedatogrounddesignhas
tonon-negative
andconclude
wehaveconstructedthattheθ=πsigma
model
reallyisgapless[48].
1/2 that are projected to a spin 1. As explained in Fig. 8, forming a spina latticewithtopologically
non-trivial
bands,
similarweto the ones
studiedin arethestatethefullinteracting
model.
From
thefigure
see
that
there
Weofatnow
complement
above
abstract
which
also
reliedsingleteach
linkin the thechain
givesrather
an eigenstate
ofargument,
the also
Hamiltonian
with
zero
famouspaper byThouless
et al.of[51].
These
bands
werethenpopulated
by atwo“unpaired”
spin
1/2
degrees
at thetwoendsof thechain,
which
on theassumptionof
a of
description
ofthea veryinstructiveandenergy.
Since theHamiltoniana freedomsum
projectors,groundstate energy
1787 large S,iswith
gasofexample
cold bosonic
Rb atoms,and using an intricate
measuringprocedure,
isantoof modelquantum
numbersincetheexactly
solvablefor
Swe= conclude
1, fractionalization,
which thatis a weclosecousinof original
the Heisenberghasbenon-negative
andhaveconstructeda degrees
groundthey
could wereexperimentally
determinetheChernunpaired
number forthegivelowestbandoffreedom
spin
1! Hamiltonian
Onecan showthatriseamodel(1).
Theforthisthe
AKLT
aftertoare
itsstate
ofintheEq.
fullinteractingmodel.
Fromthefigurewechain,alsospins
seenamed
that there
exp
toC=0.99(5)[3].
1 degeneracy
doublethegroundbutatthestriking
propertyof theinventors
Ian Aﬄeck,
Tom
Kennedy,
Elliott
Lieband ends
Hal Tasaki,is [1] which
two“unpaired”
spinof1/2
degreesofstate,freedomthemosttwoof thechain,
Also
in 2014,
theitgroup
ledby Tilman
Esslinger
madean experiment
withAKLT
chain
isthathasaHaldanegap,
aswasshown
analyticallyinalater��is an exampleof quantum
numberfractionalization,since
theoriginaldegrees
�
���
40
2
1 �authors
cold byK atoms
inan �opticallattice to� simulate
the1 precise model
proposedarticle
the same� the unpaired
�i +rise�i+1to) aH
Si ·1!S+[2].
cf+= spins
P2give(SSoffreedomOne
canrac16
show Sthat
AKLT =were spin
i+1
i · Si+1by The
Haldane
in
1988
[29].
This
showsthatrealitysometimes
surpasses
dreams.
23
existence
ofofthephase buthasthebeen
confirmedby experii
i bothdouble degeneracy
the Haldaneground state,moststriking
propertyof theAt theandend byof his
paper Haldane
wrote:The“Whiletheparticular on
model
presented(18)ments
numericalsimulations.
firstexperiment
CsNiClwasAKLTchain
is that it
hasA.aT.
Haldaneas wasanalytically in a3 later[26] A.
F. Hebard
andFiory. gap,
Evidence
forshown
the Kosterlitz-Thoulesstranhere,
isunlikely
tobedirectly
physicallyrealizable,
itindicates.
.
.
”.
What�
donebyBuyersetal.
[13],andinFig.
9weshow
results
fromamore
recentwhereS
isaspin1operator
atthelatticesitei,andP
projects
onthei
2
article sitionby theinsame[2].
thin authors
superconducting
aluminum films. Physical Review Letters,
he could corresponding
notimagine wasthat 225onyears
later,
newlatticeexperimental
techniques
experiment[31].
subspace
to
spin
two
adjacentsites.
TofindtheJan
The44:291–294,existence of
the1980.
Haldane phase has been confirmed both by experiwouldmakeitpossibletocreateanartificial
stateofmatterthat
wouldindeed
groundstate,thateach linkinthehosts
two
auxiliary
spin
Later
workhasimagine
greatlysimulations.
deepened
ourTheunderstandingof thements
and
by we
numerical
firstchain
experimentonHaldane
CsNiClphase
3 wasprovide
that“unlikely”
realization.
[27]
Pavan
Hosurand
Xiaoliang
Qi.
Recentdevelopments
intransport
phe1/2thethat
are projected
to aandspinin 1.Fig.
As9 explainedin Fig.
8,is forming
spinofchain.
there
nolocalarecent
order
done
byHeisenberg
Buyerset antiferromagnetic
al. [13],
we Althoughshow results
froma morenomenaweyl semimetals.
Rendusparameter,it[31].
is insometimes
possible toComptes
characterise
it by Physique,a non-local14(9):857–870,
string orderexperiment
2013.
parameter[30] has
introducedin theof statistical[16].
17 context
Later work
greatly earlier
deepenedourunderstandingof the mechanics
Haldane phase
References
To thedefinea distinct phase of matter,chain.
it is importantthat thecharacteristicofHeisenberg
Although
is nolocal[28] B.
Jeanneret,antiferromagnetic
Ph. Flückiger, J. L. Gavilano,
Ch. there
Leemann,andP. order
Martiparameter,
itissometimes
possibletocharacterise
itbyanon-local
stringorder[1] IanAﬄeck,
TomKennedy,
Elliott
H
Lieb,andHal
Tasaki.
Rigorous
renoli. Critical phase fluctuations in superconducting wire networks. Phys18
parameter[30]valence-bondintroduced
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of statistical mechanics
[16].
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groundinstatesin1989.
antiferromagnets.
Physical Review
ical on
Review
B, 40:11374–11377,
Dec
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phaseofmatter,itisimportantthatthecharacteristicLetters, 59(7):799, 1987.
[29] Gregor Jotzu, Michael Messer, Rémi Desbuquois, Martin Lebrat, Thomas[2] IanAﬄeck, TomKennedy,
Elliottand Hal Tasaki.
Valencerealizabond
Uehlinger,
DanielGreif, andTilman
Esslinger.
Experimental
18H Lieb,groundstatesinisotropic
quantum
antiferromagnets.
In
Condensed
Mattion of the topological Haldane model with ultracold fermions. Nature,terPhysics and Exactly
Soluble Models, pages 253–304. Springer, 1988.
515(7526):237–240,
2014.
[3]
Aidelsburger,
MichaelLohse,C Schweizer,
Atala,
Julio Tin
[30]Monika
Tom Kennedy
and HalTasaki.
Hiddenz2 × z2 Marcos
symmetrybreaking
Barreiro,
Nascimbène,
NR Cooper,
Immanuel
and N Goldman.
Haldane Sgap
antiferromagnets.
PhysicalReviewBloch,B, 45(1):304,1992.