Half-Filled Lowest Landau Level on a Thin Torus

0.2 MB·English

by  Emil J. Bergholtz

Half-Filled Lowest Landau Level on a Thin Torus
∗ †
Emil J. Bergholtz and Anders Karlhede
Department of Physi
s, Sto
kholm University
AlbaNova University Center
5 SE-106 91 Sto
kholm, Sweden
0 (Dated: 2nd February2008)
0
We solve a model thatdes
ribes an intera
ting ele
tron gas in thehalf-(cid:28)lled lowest Landau level
2
on a thin torus, with radius of the order of the magneti
length. The low energy se
tor
onsists of

n non-intera
ting,one-dimensional,neutralfermions. Thegroundstate, whi
hishomogeneous,isthe
a Fermi sea obtained by (cid:28)lling the negative energy states and the ex
ited states are gapless neutral
J
ex
itations out of this one-dimensional sea. Although the limit
onsidered is exνtr=em1e/,2the solution
0 has a striking resemblan
e to the
omposite fermion des
ription of the bulk state(cid:22)the
2 ground state is homogeneous and the ex
itations are neutral and gapless. This suggests a one-
dimensionalLuttinger liquid des
ription, with possible observable e(cid:27)e
ts in transport experiments,

] ofthebulkstatewhereitdevelops
ontinuouslyfromthestateonathintorusastheradiusin
reases.
l
l
a
PACSnumbers: 73.43.Cd,71.10.Pm,75.10.Pq
h
-
s
e It was observed by Jiang et al in 1989 that the quan- fermion des
riptions at other (cid:28)lling fra
tions exists.
m
tum Hall (QH) system has a metalli
behavior at (cid:28)lling Herewestudytheintera
tingele
trongasinthelowest
ν =1/2 σ ν =1/2
xx
. fra
tion (cid:22) is(cid:28)nite and sampledependentas Landaulevelat onathintorus. Weobtainanex-
t T 0 σ
a → , whereas xy is unquantized and approximately a
tsolutionforaparti
ularshort-rangeele
tron-ele
tron
m
equal to its
lassi
al value [1℄. Early experiments also intera
tionthatisrelevantforatoruswith
ir
umferen
e,
L
1
- showed that there is a large density of low energy states , of the order of the magneti
length. The low-energy
d
[2℄, but no nonlo
al transport [3℄. se
tor
onsists of free neutral one-dimensional fermions.
n ν = 1/2
Expressedin terms of the originalele
trons, these"
om-
o The metalli
state was su

essfully des
ribed
c by Halperin, Lee and Read [4℄, who introdu
ed a mean positefermions"arenearestneighborele
tron-holepairs,
[ ex
itons,withahard-
ore
onstraint. Thisthusprovides
(cid:28)eldtheorywheretheexternalmagneti
(cid:28)eldis
an
elled ν =1/2
adipolepi
tureofthe state. Thegroundstateis
2 by a smeared out statisti
al (cid:28)eld, resulting in
omposite
a homogeneousFermi sea of the neutral fermions, whi
h
v fermions [5℄ moving in a zero (cid:28)eld(cid:22)ie in a two dimen-
4 sionalfreefermiongaswithaFermisurfa
e. Thispi
ture, supports gapless neutral ex
itations.
9
as well as Jain's general
on
ept of
omposite fermions,
3
wasspe
ta
ularly
on(cid:28)rmedbysurfa
ea
ousti
waveex-
6
perimentsperformedby Willett et al [6℄, and by ballisti
0
4 transport [7℄, whi
h showed that parti
les move in a re-
0 du
ed e(cid:27)e
tive magneti
(cid:28)eld (or, alternatively, have a
/ redu
ed
harge) whi
h approa
heszero as ν →1/2. ν = 1/2
at Figure 1: PhaseL1diagram for on a torus, where one
m Rezayianνd=Re1a/d2[8℄proposedami
ros
opi
wavefun
-
ir
umferen
Le1, ∼,5varies while the other isνin=(cid:28)n1i/te2. For a
tionforthe state,whi
hagreesverywellwithex- thin torus, magneti
lengths, the system
ν =1/2
- is that of nonintera
ting neutral one-dimensional fermions
a
tresultsforsmallsystems. Thetheoryforthe
d
n state was later further developed by several groups [9℄, (dipoles). The question mark indi
νat=es 1w/h2ether the state de-
velops
ontinuously into the bulk state or whether
o and a des
ription in terms of neutral dipoles was pro-
there is a phase transition. Based on the similarities of
c posed. L1
the state at short and the bulk state, su
h as a homoge-
:
v In spite of the impressive agreement between theory neousgroundstate andneutral gaplessex
itations, we
onje
-
Xi andexperimentthereare,inouropinion,importantques- ture that there is no phase transitionν.=T1h/is2suggests a one-
tions regarding the physi
s in the lowest Landau level dimensional des
ription of the bulk state as a Lut-
r tinger liquidrather than asafreetwo-dimensionalFermi gas.
a thatremaintobeanswered. Thereisnorealunderstand- L1
(For very short a
rystalline state determined by ele
tro-
ingof why the strongly
orrelatedele
tronsystem in the
stati
s alone is obtained.)
lowest Landau level, at various (cid:28)lling fra
tions, be
omes
a system of weakly intera
ting
omposite fermions(cid:22)no
The low-energy se
tor has many features in
ommon

ontrollνed=mi1
/ro2s
opi
derivation of the mean (cid:28)eld the- ν = 1/2
ory at or, for that matter, of the
omposite with the bulk state, su
h as a homogeneous
groundstateandgaplessneutralex
itations,andwe
on-
je
ture that it develops
ontinuously into the bulk state
L
1
as → ∞, rather than being separated from this by
∗
†Ele
troni
address: ejbphysto.se a phase transition, see Fig. 1. Thνis=su1g/g2ests a one-
Ele
troni
address: akphysto.se dimensional des
ription of the bulk state as a
2
ν = 1/2
Luttinger liquid rather than as a two-dimensional Fermi However, at , the short range ele
trostati
and
gas [10℄. hoppingterms
ooperate, leading to a simple low-energy
We start by mapping the problem onto a one- se
tor for the thin torus.
dimensionallatti
emodel. FollowingHaldaneandRezayi We trun
ate the intera
tion in (2) and keep only the
we
onsider an ele
tron
on(cid:28)ned to the lowest Landau two shortest range ele
trostati
terms and the shortest
L ,L x y
1 2
level on a torus with lengths in the A~a=nd Bydxˆi- range hopping term
re
tions,respe
tively[11℄. InLandaugauge, − , H = [V nˆ nˆ +V nˆ nˆ
the magneti
translation operators be
ome Pn 10 n n+1 20 n n+2
V (c†c c c† +H.c.)] ,
T1 =e(L1/Ns)∂x , T2 =e(L2/Ns)(∂y+ix) , − 21 n n+1 n+2 n+3 (3)
(1)
nˆ =c†c
Ns where k k k. Thisprovidesagoodapproximationof
where is the number of (cid:29)ux quanta through the sur-
the intera
tion on a thin torus as dis
ussed below [13℄.
ℓfa
=e of t~hce/etBoru=s. 1(The magneti
length is set to one,
p .) These operators
ommute with Beforegivingthe detailsofouranalysisweoutline the
A~ logi
of the identi(cid:28)
ation of the low-energy se
tor of the
the Hamiltonian for the
harged parti
le
oupled to . ν =1/2
WTTα1NaTsv2Ψe=fu=en2
eπtiiφi/oαNnΨssT,aT2αrTe1=r.eqW1u,ii2tr,hedleψta0od=ibnψegPptenor=eiLoind1TLiL
2m2xuψe=p−t(2yo+TπanNψLps2h)aa2/sn=2ed,, hHtHooialpmhbpaeiivrlnttegosn
pthieaaarn
rmeg(e3bV)yo2a1nrt.ee.qWuAier
itndigne(cid:28)g.enaTwe
hihtaehps
auTriub2r
sogipafiavl
sepiestaerHastn′(in2oepqf(3u−t)ihv1ieas,lft2euhnple)tl
ewie2πmob/Ntasiψnmthem =1 e0i,g1e,n.s..tNatses 1m 2 0, ψ1m m grouping of the sites ′(2p,2p+1′) instead(cid:22)′ and a
orre-
, − . The states span sponding subspa
e HT [14℄. H (and HT) is the low-
the lowest Landau level and are lo
ated along the lines
y = 2πm/L energyse
torunderfairlygeneral
onditionssin
eit
on-
1 100110011001....
− . Thuswehaveobtainedthemappingonto
m tains the maximally hoppable state | i(cid:22)
a one-dimensional latti
e model, where numbers the
2π/L whi
hturnsouttobetheseedforthegroundstateandis
1 V
sites and the latti
e
onstant is . 20
alsothe lowestenergystateforthe -term(cid:22)andit has
H
Assuming translation invarian
e, the ele
tron-ele
tron
a low ele
trostat′i
energy by
onstru
tion. preserves
intera
tion Hamiltonian be
omes the subspa
e H and the hopping term
an be exa
tly
H = V c† c† c c , diagonalized in this spa
e giving non-intera
ting neutral
ee X X km n+m n+k n+m+k n (2)
n k>m fermions. Thegroundstateistheone-dimensionalFermi
line obtained by (cid:28)lling the negative energy states, and
where c†m
reates an ele
tron in state ψm and Vkm = the ex
itations are gapless ex
itations out of this Fermi
Vk,−m ≥0. To underHsteaend the physi
s ofVtph0e intera
tion sVe1a0.=T2hVe20elael
ltsrtoastteastii
ntHer′mhsavine t(h3e) laorweelsetsspo
srsui
bilael.el′eA
t-
it isusefulto divide intotwoparts: , the ele
tro-
p trostati
energy and we argue perturbatively that H is
stati
repulsionbetweentwoele
tronsseparated latti
e
Vm+p,m thelowenergyse
tor. However,weexpe
tthistobetrue

onstants, and , the amplitude for two parti
les
p under more general
onditions.
separatedadistan
e to hop symmetri
ally to asepara-
p+2m We now present our analysis for the trun
ated Hamil-
tion and vi
e versa. For a short-raVn(g~re)r=eal s2pδa(~
re) tonian (3) at ν = 1/2 and V10 = 2V20 ≡ 2α . The
ele
tron-elVek
mtro=n i(nkt2era
mtio2n)e−of2πt2h(ek2f+omrm2)/L21 ∇ enle
ntr.o.s.ntati
part, Hn|V=21=00,,1then 1has=tch†e0eigenstates
one (cid:28)nds − L [12℄. When | 1 2 Nsi, where i and | i | i, with en-
1
the torus be
omes thin, ie when de
reases, the dis-
2π/L ergies
1
tan
e between the single parti
le states2iδn
reases, N
hen
efewertermsin(2)
ontribute. Forthe∇ intera
- E =α s +n +n .
tion one (cid:28)nds that the range of the intera
tion is of the 0 (cid:16) 2 111 000(cid:17) (4)
order of six magneti
lengths.
ν =1/k n (n )
111 000
We
onsidertheele
trongasat(cid:28)llingfra
tion , Here, is the number of 3-strings, ie strings
k n n ...n
where is an integer, and assume that the number of
onsistingofthreenearbyele
trons(holes)in 1 2 Ns
νN k 3 k 2
s
ele
trons isaninteger. Themanyparti
lestates
an (a string of length ≥ is
ounted as − strings and
T κ N
1 s
be
hosen as eigenstates with momentum mod( ) periodi
boundary
onditions are assumed). Thus there
2π/N T y
s 2 0
(in units of ). translates the system in the - is a degenerate ground state manifold H
onsisting of
κ νN
s
dire
tion and
hTanges by (cid:22)ie by the number of all states where at most two ele
t′rons or two holes are
2 0
parti
les. Sin
e
omkmutes with the HamiltonTiakn, all next to ea
h other. Note that H ⊂ H . The ex
ita-
energy eigenstates are -fold degenerate [11℄. ( 2 pre- tions are 3-strings of either ele
trons or holes and ea
h
κ N α
s 0
serves mod( ) and hen
e
an be diagonalized along 3-string has energy . The statement about H follows
T N
1 s
with .) by indu
tion if the states without 3-strings for sites
At a (cid:28)xed (cid:28)lling fra
tion, the ele
trostati
repulsion are
onstru
ted by inserting an ele
tron and a hole in
N 2
s
strives to keep the parti
les apart, whereas the hopping − states without 3-strings. The energyof a3-string
termsfavourmaximallyhoppablestates. To(cid:28)nd thelow followsby
onsideringthe
hangeinenergyofoneorsev-
energy states is in general a very
ompli
ated problem. eral 3-strings when moving one
onstituent.
3
H ′ N =18
0 s
To diagonalize in H, we pro
eed as fol′lows. Th0e1re thisbdyex3a
tdiagonalizationinH ofupto sites.
aretwopossiblestatesforapairofsitesinH: |↓i≡| i The ≥ states
ontainmoredomainwallsanditiseasy
10
and | ↑i ≡ | i asn+d i=t isc†natucral to int=rodsu+
e the spin to see that the hopping be
omes more restri
ted(cid:22)thus
raising ope′rast+or, s−p=(s+2)p†−1 2p, | ↑i | ↓i. On we expe
t them to haNves high1e8r energy. We have veri(cid:28)ed
statesinH, des
ribehard
orebosons(cid:22) th′is nume′ri
ally for ≤ . Thus we
on
lude that
H andHT givethelow-energyse
torofthetheory(cid:22)the
they
ommute on di(cid:27)erent sites but anti
ommute on the
H XY 0
remaining states in H are separated from the ground
same site and is the nearest neighbor spin-1/2 V
d 21
state by a gap of order .

hain. Expressing the bosons in terms osf−fer=miKonsd
using tKhep =JoerdiπaPn-pjW=−11idg†jndejr transformation, p p p, obWtaeinnedowfo
roHnsidiner(3th)ewshtνeanb=ilVit11y0/2o=f t2hVe20soalnudtioinnvwesetihgaavtee
where , the Hamiltonian (3) is simply
tdh†1aKtNosf/2frdeNesf/e2r+miHon.cs.,],Hw=henαN2ress+triV
2t1e[dPtopN=sH/12′−[11d5℄†p.+T1dhpu+s, wfsohhroeratth-rreaarnniggteedoienfst
LeLrr1iab.
=etWsio2tenπh/(cid:28)eV√r(s2~rt)lnn=o2t∇e=2t5δhs.(at3~rat)t,feoVro1n0th=ae 2trheVai2nl0st=poar2u
αes
1
after a Fourier transformation, the ground state is ob-
orrespondsto V = 3α . Thehoppingterm
tainedby(cid:28)lElin=gaαlltheV2n1egativeenergNystates. Thisstate in
luded in (3) is then 21V =8 ,9wαhereasVthe =lead1inαg
has energy 2 − π per site (if s → ∞) and sup- ignored terms are small: 30 128 and 31 32 .
pco†rtcs ne=ut1rδal gapless ex
itations. One readily (cid:28)nds that This is
lose to the solvable point.
h m ni 2 mn, and hen
e the s′tate is homogeneouTs. We have performed density matrix renormalization
2
′This solves the problem in H and, by a
tion of , in group (DMRG) [16℄ stVu(d~ri)es=on a2δt(h~ri)n
ylinder with the
HT. Itremainsto
onsiderthestatesinthegroundstate Hamiltonian (2) and ∇ in
luding intera
-
0
manifold H that are in neither of these subspa
es. We tions that extend over up to six latti
e
onstants [17℄.
will now arguethat these areseparated from the ground We(cid:28)ndagroundstatethatishomogeneoustoveryhigh
V
21
state byHa gap of order generated by the′ hopping a

ura
y and stronLg in=di
2aπt/io√n2slonf2gapless ex
itations in
1
termin . Intuitively,thismakessensesin
eH
ontains the region around that we have inves-
01100110011....110 4 . L . 8 L
1 1
themaximallyh′oppablestate| H i. Note tigated ( ). (When ...i.s even smaller, the
thatwhereasH isinvariantunder , otherstates inH0 ground state is a
rystal | ↑↑↑↑↑↑ ↑i(cid:22)the lowest en-
V
10
may mix with states not in the ground state manifold. ergy state for the shortest range ele
trostati
term .)
H H
Our pro
edure will be to simply diagonalize in the The low energy se
tor at the solvable point ( in (3)
V = 2V
0 10 20
ground state manifold H . with ′ ) is
ontained′in the spin-1/2 Hilbert
To des
ribe a general state in H0, we introdu
e the spa
e H. The states not in H are separated from the
a 00 b 11
notation | i ≡ | i and | i ≡ | i, along with | ↓i and low energy states by a gap. Small perturbations of the
(2p 1,2p) H
| ↑i, for the states on sites a − aba. b
ontainbasbthe Hamiltonianaroundthesolvablepointlead,inperturba′-
hopping terms: ↑↓↔↓↑, ↓ ↑↔ , ↑ ↓↔ , ↑ tion theory, to an e(cid:27)e
tive spin-1/2 Hamiltonian in H.
ba ba ab ab V
↔ ↑ and ↓ ↔ ↓(cid:22)all with strength 21. The generatedterms arespin operatorsofquadrati
and
A general state is uniquely des
ribed by a string of ↑ higher order. They all hav′e small
oe(cid:30)
ients sin
e there
, ,a b d
↓ and an′d
an b′e
hara
terized by the number of isagaptostatesnot′inH (thematrixelementsfortran-
alternatingH andHT′domainsiatisbbuiltupof. Any′pair sitionstostatesinHT vanish). Alltermsareirrelevsaznstzin
↑↓ or ↓↑ belongs to H and any or belongs to HT(cid:22)a thesenseoftherenormalizationgroup,ex
eptfor i i+n
.... ....
polarized string ↑↑↑↑ or ↓↓↓↓
an however belong whi
h makes the nonintera
ting fermion theory dKeve=lop1
to either domain. This implies tahat thbere is a domain intoaLuttingerliquidKwi=th1intera
tionparameter 6
wall in between ↑↓ (or ↓↑) and (or )(cid:22)
ounting the (at the solvable point ), see eg [18℄.
d
number of domain walls gives for a general state. Toobtainthegenerale(cid:27)e
tiveHamiltonianexpli
itlyis
ν =1/2 Vm0
oaf′sAthasnetdnatebea′sriesasantdnebieglhobnogrss
ttaootmeHbi0ifniiatfthiitoansdsoa(enasaen)qo,uta(
blobn)nu,tma(iabne↓ar)n,oyf sVneo2rnnv+-et1r,H0iv)′isazials.nzi+dHnos].wimeTvphelyre,bhthoe
peopemilneeg
tPtreoris,mtna[st(,i2
VV2tmennr,0m, −ws,ilVl,2ni−n,1g,p0erne−--
(b ), ( a) ( b)
d ↑ ↑ or ↓ H. It is straightforw′ard to s′how that eral,
ontribute in se
ond order perturbation theory.
is preserved by . The states in H and HT are the Based on the renormalization group argument, and
d = 1 d = 2
states. In the se
tor, we
onsider (cid:28)rst the supported by the DMRG
al
ulations, we
on
lude that
a b ab ν =1/2
sbtaates with one and one ndex=t t2o ea
h other, or the systemLon a thin torus is a Luttingerslziqszuid
, in a string of spins. These states are mapped fora(cid:28)nite rangeof 1 andthat thegenerationof i i+n
H ab
0
into ea
h other under . To be in H , must enter as terms indi
ates that the intera
tion parameter that de-
X ab H X X
≡↓ ↓. Under , this hops just like ↑: ↓↔↓ terminesthede
ayof
orrelationfun
tionsisshiftedfrom
V
21
withmat′rixelement .NThus6theproblemisequivalent its valueLat the solvable point.
s 1
to the H problem with − sites and one (cid:28)nds that′ When in
reasesfurther,thereiseitheraphasetran-
these states are separated from the ground state in H sition or the state develops
ontinuously into the bulk
V ν =1/2
21
by a gap of order . By
onsidering how hoppable the state. We
onje
ture that the latter is the
ase.
states are we expe
t the states just
onsidered to be the The main support for this
omes from the striking sim-
d=2
lowestenergystatesinthe se
tor. Wehaveveri(cid:28)ed ilarities of the low-energy se
tor on the thin torus and
4
ν = 1/3
the
ompositefermiondes
riptionofthebulkstate,most presumably generalize into a mapping of onto
s = 1
notably the homogeneous ground state and the gapless an
hain. In passing, we note that this suggests
neutralex
itations. Furthermo′re,wenotethattheredu
- that the Haldane
onje
ture for the gaps in spin
hains
tion of the Hilbert spa
e to H by itself implies that the [21℄ might apply to the two-dimensional ele
tron gas in

hargeonaverageishomogeneous(cid:22)this,orsomesuitable a strong magneti
(cid:28)eld.
generalization thereof, is likely to be a good approxima- Thethintorus,or
ylinderforthatmatter,withamag-
L
1
tionalsowhen in
reasesandlongerrangeintera
tions neti
(cid:28)eld perpendi
ular to its surfa
e is probably not

ome into play. experimentally a

essible. Thus, the experimental
on-
Furthersupportforour
onje
ture
omes from
onsid- sequen
esoftheresultsinthisLetterpresumablydepend
ν = 1/3
ering the Laughlin (cid:28)lling fra
tion [19℄. Rezayi on whether the results are appli
able, mutatis mutandis,
and Haldane have shown that the Laughlin state is the tothebulk
aseaswe
onje
ture. Our
onje
tureimplies
ν =1/3 ν = 1/2
ground state also on a thin
ylinder and that it that the state is a one-dimensional Luttinger
develops
ontinuously from a
harge density wave state liquid rather than a two-dimensional Fermi theory. We
L
1
into the homogeneous Laughlin state as → ∞ [20℄. predi
t that this leads to observable e(cid:27)e
ts in the bulk
ν =1/2 I V
Our DMRG
al
ulations agree with this(cid:22)we (cid:28)nd, us- system, su
h as non-linear −
hara
teristi
s
L
1
ing (2), for a range of a
harge density wave state determined by the Luttinger liquid intera
tion parame-
in quantitative agreement with that of Rezayi and Hal- ter.
ν = 1/3
dane. Thus the ground state of the short range Ifour
onje
tureis
orre
t,thenitshouldbepossibleto
Hamiltoniandevelops
ontinuouslyintothehomogeneous understandtheexperimentalresultsthataresu

essfully
L
1
Laughlin state as →∞. This lends some support for explained by the standard
omposite fermion theory [4℄,
ν = 1/2
our
onje
ture that the state also develops adi- su
h as the ballisti
transport and the surfa
e a
ousti
abati
ally. However, in this
ase there is no gap and the waveresults. Wenotethattheappearan
eoflow-energy
issue is moredeli
ate. The argumentwould be strength- ex
itations that are neutral, and hen
e do not
ouple to
ν =1/2
enedifthepi
tureofthe stategivenabove
ould the magneti
(cid:28)eld, is
onsistent with the ballisti
trans-
ν = 1/3
be shown to generalize to the state on the thin port results.
torus,inwhi
h
aseitshouldberelevantalsoforthebulk WethankThorsHansHanssonandJonMagneLeinaas
ν =1/3
state. The mapping of the low energy se
tor at forvaluabledis
ussions. AndersKarlhedewassupported
ν = 1/2 s = 1/2 XY
onto an -spin
hain, would then by the Swedish Resear
h Coun
il.
[1℄ H.W.Jiang,H.L.Störmer,D.C.Tsui,L.N.Pfei(cid:27)er,andK.W. also been
onsidered byD.-H.Lee and J.M.Leinaas (private
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[10℄ Thepossibilitythatthe stateisaLuttingerliquidhas