Fractional Fermions with Non-Abelian Statistics

0.38 MB·English

by  Jelena Klinovaja

Fractional Fermions with Non-Abelian Statistics
Jelena Klinovaja and Daniel Loss
Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
(Dated: January 25, 2013)
We introduce a novel class of low-dimensional topological tight-binding models that allow for
boundstatesthatarefractionallychargedfermionsandexhibitnon-Abelianbraidingstatistics. The
proposed model consists of a double (single) ladder of spinless (spinful) fermions in the presence
of magnetic fields. We study the system analytically in the continuum limit as well as numerically
3
in the tight-binding representation. We find a topological phase transition with a topological gap
1
thatclosesandreopensasafunctionofsystemparametersandchemicalpotential. Thetopological
0
phase is of the type BDI and carries two degenerate mid-gap bound states that are localized at
2
opposite ends of the ladders. We show numerically that these bound states are robust against a

n wide class of perturbations.
a
J
PACSnumbers: 71.10.Fd;05.30.Pr;71.10.Pm

4
2
Introduction. Topological properties of condensed

]
matter systems have attracted considerable attention in
l
al recent years. In particular, Majorana fermions [1], be-
h ing their own antiparticles, are expected to occur in a
- numberofsystems,e.g. fractionalquantumHallsystems
s
e [2,3],topologicalinsulators[4–6],opticallattices[7,8],p-
m wavesuperconductors[9], nanowireswith strong Rashba
. spin orbit interaction[10–15], and carbon-basedsystems
t
a [16–18]. Another class of topological systems is given by
m
bound states of Jackiw-Rebbi type [19] containing frac-
- tionalchargee/2[22–27]. Suchexoticquantumstatesare
d
interestingintheirownright,andduetotheirspecialro-
n
o bustness against many forms of perturbations they offer FIG. 1. Double-ladder tight-binding model, consisting of a
c thepossibilityforapplicationsinquantumcomputations, lower (σ = −1) and an upper (σ = 1) ladder, each lying
[ especially when they exhibit non-Abelian statistics such in the xy-plane and held at chemical potentials µσ. Here,
1 as Majorana fermions [20, 21]. ltaxdd(reerd,alnindktsz)(ibslutehelininktsr)at-hcehaininte,r-tlyad(dgreerehnoplipniknsg)atmhpeliitnutdrae-,
v In this Letter we introduce a surprisingly simple class andax,y,z arethecorrespondinglatticeconstants. Auniform
2 of models supporting a topological phase with bound magnetic field B=B1+B2 is applied in the yz-plane. The
2
8 statesthatpossessnotonlyfractionalchargebutalsoex- associated magnetic flux results in phases φ1 and φ2 in the
hoppingamplitudes between different chains, see Eqs. (2,3).
5 hibitnon-Abelianstatisticsunderbraiding. Thesebound
1. states behave in many ways similar to the well-studied
0 Majorana fermions in superconducting-semiconducting forthe boundstates, andconfirmthese findings by inde-
3 nanowires[20], butincontrasttothemthey arecomplex
pendentnumericsoftheunderlying tight-bindingmodel.
1
fermions, and quite surprisingly, emerge in the absence
We further test the stability of these states numerically
:
v of superconductivity and without BCS-like pairing. against a wide class of perturbations and show that the
i
X The two non-interacting tight-binding models we pro- bound states are robust against most of them, except of
r pose consist of a double ladder containing spinless parti- local charge fluctuations, against which they are partly
a cles in a uniform magnetic field and a single ladder con- protected by charge neutrality.
taining spinful particles in the presence of both uniform Tight-binding model. Weconsideradouble-laddersys-
and spatially periodic magnetic fields. We find a topo- tem consisting of four coupled chains aligned along x-
logical phase transition in these systems when varying direction, see Fig. 1. Two upper (lower) chains form
systemparametersorthechemicalpotential,withachar- the upper (lower) ladder. Each chain is labeled by two
acteristicclosingandre-openingofatopologicalgap. In- indices τ and σ, where τ = 1 refers to the left/right
±
side the topologicalphase we find two degenerate bound chains, and σ = 1 refers to the upper/lower ladders.
±
states,onelocalizedattherightandoneattheleftendof The tight-binding Hamiltonian of a (τ,σ)-chain reads
the system. These bound states are fractionally charged
fermions and are shown to exhibit non-Abelian braiding Hτx,σ =tx (c†τ,σ,n+1cτ,σ,n+h.c.)+µσ c†τ,σ,ncτ,σ,n,
X X
statisticsoftheIsingtype. Westudythesystemsanalyt- n n
(1)
icallyinacontinuumapproach,finding explicitsolutions
2
where c (c ) is the annihilation (creation) opera-
τ,σ,n †τ,σ,n
tor on site n of the (τ,σ)-chain. The sum runs over N
sitescomposingthechain. Here,t isthehoppingmatrix
x
element in x-direction, and µ is the chemical potential
σ
of the (τ,σ)-chain.
The intra-ladder coupling is given by
Hσy =ty (einφ1c†1,σ,nc¯1,σ,n+h.c.), (2)
X
n
where the phase φ accompanies the hopping matrix el-
1
ement t in y-direction. This phase arisesfromthe mag-
y FIG. 2. The spectrum of the upper (orange) and lower (red)
netic flux through the unit cell produced by a magnetic chains in the first Brillouin zone. The chemical potentials
field B1 in z-direction. The inter-ladder coupling be- are chosen such that the system is at the charge-neutrality
tween two left or two right chains is given by point. Thefilled(empty)statesareindicatedbydark-colored
(light-colored)lines. ThetwoFermiwavevectorsaregivenby
Hτz =tz (einφ2c†τ,1,ncτ,¯1,n+h.c.). (3) kF,1 = π/4ax and kF,¯1 = 3π/4ax. The intra-ladder hopping
Xn ty (green dashed line) and the inter-ladder hopping tz (blue
dottedlines)leadtoopeningofgapsattheFermilevel(ǫ=0).
Similarly, the phase φ arises from the flux through
2
the unit cell produced by a magnetic field B in y-
2
direction. We note that in practice only one total field, The kinetic part of the Hamiltonian corresponding to
B = B1 + B2, needs to be applied in the yz-plane, Hx [see Eq. (1)] is rewritten as
τ,σ τ,σ
see Fig. 1. The total tight-binding Hamiltonian for the P
double-ladder model is given by
Hx = i~υ dx R ∂ R L ∂ L , (6)
− F Z τ†,σ x τ,σ− †τ,σ x τ,σ
H = [Hx +(Hy+Hz)/2]. (4) Xτ,σ (cid:0) (cid:1)
τ,σ σ τ
X
τ,σ
where we dropped the fast oscillatory terms, and where
Now we focus ona particularcaseof the abovemodel. υ = √2t a /~ is the Fermi velocity. The intra-chain
F x x
First, we fix the chemical potentials on the upper and couplings, given by Hz and Hy [see Eqs. (2) and (3)],
τ σ
lowerladders to be of opposite signs, µ1 =−µ¯1. Second, lead to mixing between Rτ,σ and Lτ,σ belonging to dif-
we assume the upper ladder to be at quarter-filling, i.e. ferent chains. The intra-ladder hopping yields
µ = √2t . The magnetic fields are chosen such that
1 x
Bφ1 == πΦ/2/4aanda φ2an=d Bπ, o=r iΦn /te2ramas ,ofwfiheelrdesΦtrenisgtthhse, Hy =Z dx ty(cid:16)R1†,1L¯1,1+R¯1†,¯1L1,¯1+h.c.(cid:17), (7)
1 0 x y 2 0 x z 0
flux quantum, and a are the corresponding lattice
x,y,z while the inter-ladder hopping yields
constants. Assuming thatt t ,t , we treatfromnow
x y z
≫
on inter-chain hoppings as small perturbations.
Continuum model. The most convenient way to an- Hz =Z dx tz(cid:16)R1†,1L1,¯1+R¯1†,1L¯1,¯1
alyze the tight-binding Hamiltonian in Eq. (4) is to
go to the continuum limit. For this we first derive +R¯1†,¯1L¯1,1+R1†,¯1L1,1+h.c.(cid:17). (8)
the spectrum via Fourier transformation along the x-
Next, we introduce a new basis φ =
awxhiesrefocr eachis(τth,σe)a-cnhnaiihni,lactiτo,σn,nop=era√t1NorPofktehikenaexleccτt,rσo,kn, (R1,1,L1,1,R1,¯1,L1,¯1,R¯1,1,L¯1,1,R¯1,¯1,L¯1,¯1) to rewrite
τ,σ,k the total Hamiltonian H = Hx + Hy + Hz as
with momentum k. The Hamiltonian Hx has the well-
τ,σ H = dx φ (x) φ(x) in terms of the Hamiltonian
knownspectrum, ǫk,σ,n =µσ+2txcos(kax). At quarter- densityR , † H
filling (µ 1 = √2tx) the Fermi momenta are given by H
± ±
kF,1 = π/4ax and kF,¯1 = 3π/4ax, see Fig. 2. We em- =~υ kˆη +t σ η + ty(τ η τ σ η )+δµσ , (9)
phasize here that the system is charge neutral, and the H F 3 z 1 1 2 1 1− 2 3 2 3
shiftedchemicalpotentialsjustredistributeelectronsbe-
where the Pauli matrix η acts on the right- and left-
tween chains. i
mover subspace, and the Pauli matrices σ and τ act
Next, welinearizethe spectrumaroundthe Fermimo- i i
on the chain subspaces. The momentum operator is de-
mentabyexpressingtheannihilationoperatorsΨ(x)that
finedaskˆ= i~∂ ,witheigenvaluekcountedhenceforth
actonthestatesclosetotheFermilevelintermsofslowly − x
from the corresponding Fermi points. Here, we assume
varying right (R ) and left (L ) moving fields as
τ,σ τ,σ
againthatthechemicalpotentialsoftheupperandlower
Ψ(x)= eikF,σxRτ,σ(x)+e−ikF,σxLτ,σ(x) . (5) chains are opposite in sign, however, small deviations
Xτ,σ (cid:2) (cid:3) from quarter-filling, δµ = δµ1 = −δµ¯1, are taken into
3
account. In addition, we neglect any constant shifts of
the spectrum. 0.05
TheHamiltoniandensity allowsustodeterminethe
0
H (a) (b)
topological class of the system [28]. The system is in-
variant under the time-reversal operation , defined by -0.05
T
UreT†laHti∗o(n−.kS)iUmTila=rlyH,(tkh)e.chInadrgeee-dc,onUjTug=atiτo1nη1sysmatmisefitersytohpis- 790 800 810 820 L L L L
erationC,definedbyUC†H∗(−k)UC =−H(k),canbesat-
isfied by U = τ σ η . Thus, the system belongs to the FIG. 3. (a) The relevant part of the spectrum for a double-
C 1 1 3
topologicalclass BDI [28]. The one-dimensional systems laddersystemoflengthL/ax =401foundbynumericaldiag-
onalization of thetight-bindingHamiltonian H [see Eq. (4)].
of this class are allowed to have an arbitrary number of
Theenergy ǫN corresponds totheNth energy level. Thepa-
bound states inside the energy bulk gap [28]. To deter-
rameters tz/tx =0.05, ty/tx =0.1, and δµ=0 are chosen to
mine if bound states are present in the system for some satisfythetopologicalcriterionEq. (12). Inthemiddleofthe
givensetof parameters,we follow the method developed gap, ǫ=0, thereare twodegenerate boundstates(red dots).
in Refs. 27 and 29. (b) The probability density |ψτ,σ|2 of the left state in the
The eight spectrum branches of are given by (τ,σ)-chain. Thedensitypatternsarethesameforallchains,
H in full agreement with thecontinuum solution Eq. (13).
(ǫ±1,2)2 =(~υFk±δµ)2+t2z, (10)
(ǫ±3,4)2 =(~υFk)2+t2z+t2y +δµ2 [30]. Here, we have introduced the notations,
±2qt2zt2y+δµ2[(~υFk)2+t2y]. (11) f(x)=eiθ/2(eikF,1xe−x/ξ1+iδµx/~υF e−ikF,1xe−x/ξ2),
−
The system is gapless only for one particular set of pa- eiθ = t2 δµ2+iδµ /t . (14)
rameters, t2 = t2 + δµ2. Otherwise, the spectrum is (cid:16)q y− (cid:17) y
y z
gapped, and there is a possibility for the existence of The localization length of the bound states is given
bound states inside this gap. Further, we are interested by ξ = max ξ , ξ , where ~υ ξ = t , and ~υ ξ =
1 2 F 1 z F 2
instatesexactlyinthemiddleofthegap,i.e. atzeroen- { }
t2 δµ2 t . Close to the phase transition point the
ergy. Inaddition,wefocusonsemi-infinitechains,sothe q y− − z
localizationlengthis determined by ξ , whereasdeep in-
boundary conditions areimposed only onthe left (right) 2
side the topologicalphase by ξ . If ξ is comparable with
end. This implies that the chain length L is much larger 1
L, the two bound states localized at opposite ends over-
thanthelocalizationlengthξoftheboundstateswefind.
lap. As a result, the energy levels are split from zero
In order to address the existence of bound states, we
energy, and the corresponding wavefunctions are given
first find four fundamental decaying solutions of the sys-
by the symmetric and antisymmetric combinations,
tem of linear differential equations, following from the
Schrodinger equation associated with [see Eq. (9)].
Second, the dimension of the null spaHce of the corre- ψs/a(x)=ψL(xL)±iψR(xR). (15)
sponding Wronskian leads us to a topological criterion
We note that these right or left localized bound states
[27, 29] that separates a topological phase (with bound
are fractionally charged fermions of charge e/2 [22–24].
states) from a trivial phase (without bound states). We
Theresultsobtainedaboveinthecontinuummodelare
find that bound states exist providedthe following topo-
ingoodagreementwiththenumericalresultsobtainedby
logical criterion is satisfied,
direct diagonalization of the tight-binding Hamiltonian
H given in Eq. (4), see Fig. 3.
t2 >t2+δµ2. (12)
y z Non-Abelian statistics. In the absence of overlap be-
tweentheboundstatesψ (x)andψ (x),thezero-energy
Working in the operator basis Ψ(x) = L R
levelis two-folddegenerate,so it canpotentially be used
(Ψ1,1,Ψ¯1,1,Ψ¯1,¯1,Ψ1,¯1), where Ψτ,σ(x) is the anni-
forbraidingandultimatelyfortopologicalquantumcom-
hilation operator on the (τ,σ)-chain, we find the
putation. Takingintoaccountthatthe systemischarge-
wavefunction of the state localized at the left end of the
neutral, we focus on the states that have equal density
double-ladder system explicitly,
distributions on both ends,
if(x)
ψ (x)=ψ (x) eiαψ (x), (16)
 f (x)  L R
ψ (x)= ∗ , (13) ± ±
L i( 1)nf (x)
− − ∗  where α is an arbitrary phase (see also below). The cre-
 ( 1)nf(x) 
 − −  ation operator F† corresponding to the state ψ (x) is
and the wavefunction of the state localized at the right given by F± = ±dx ψ±(x)·Ψ≡ √12(FL±eiαFR),±where
end,ψ (x)=ψ (L x),wherenlabelsthesite,x=na F (F )isthelReft(right)endstateannihilationoperator.
R L∗ − x L R
4
During the braiding process, which corresponds to ex-
changingtherightandleftendstates,theparity,defined
byP =1 F†F ,shouldbeconserved[21]. Asaresult,
theo±ldstat−est±ran±sformintonewonesasF e iαF
−
± →± ±
under braiding, or in terms of the left/right states,
F F , F e 2iαF . (17)
L R R − L
→ →
The corresponding unitary operator ULR that imple- FIG. 4. Single-ladder model. A uniform magnetic field B¯1
ments this braiding rule is found to be ULR = is applied perpendicular to the ladder. A spatially periodic
e−iα(n++n−)+iπn−, where n = F†F . Indeed, it is magneticfieldB¯2 with period 2ax isappliedin thexy-plane.
± ± ±
easy to show that UL†RF±ULR = ±e−iαF±, and thus
UL†RFLULR =FR, and UL†RFRULR =e−2iαFL. In terms
of the F operators we have
R/L
ically that local fluctuations that preserve the symme-
ULR =e−i(α−π/2)(nR+nL)−iπ(eiαFL†FR+h.c.)/2, (18) try between the upper and lower ladders are not harm-
ful. This includes correlatedfluctuations ofchemical po-
wwohrekreofnsLu/cRh=douFbL†l/eRlFaLd/dRer.sNsimexitla,rlettouthseaossnuemperoaponseetd- tmenattriaixlse(l∆emµe1nt=s.−If∆tµh¯1e),symmamgnetertyicisflunxoets,praensdervheodp,ptinhge
for Majoranafermions [31]. The braiding operationscan perturbations act like a level-detuning, and the bound
be performed by exchanging bound states localized at states separate in energy independent of wave function
different double ladders. Using ULR given explicitly by overlap but proportional to their occupation probabil-
Eq. (18), one can show that two braiding operations do ity at the site of fluctuation. We emphasize that the
not commute, [Uij,Ujk] = 0, where the labels i,j,k de- single-ladder model is protected againstsuch symmetry-
6
note three states. Thus, we see that our bound states breaking terms, except against local chemical potential
obey non-Abelian braiding statistics. In particular, it fluctuations that break charge neutrality, i.e. ∆µ =
1
is interesting to consider the case of α = ±π/2, since ∆µ¯1. However, we note that chains without charge im-
these states, ψs/a(x) [see Eq. (15)], can be easily pre- purities are stable againstsuch fluctuations, as local dif-
pared by lifting the degeneracy temporarily (via tuning ferences in µ would lead to charge redistribution, restor-
the wave function overlap), resulting in filling either the ing a uniform chemical potential in the chain. Another
symmetric or the antisymmetric energylevels after some problem can arise from flux fluctuations. They can de-
dephasingtime. Moreover,wecandeterminethephaseα crease the Fourier components at k of the backscat-
F,σ
[see Eq. (16)] by projective measurements on the states terigtermsandtherebyreducethegaps. Asaresult,the
ψs/a. For example, the probability to measure the sym- system can move out of the topological phase. However,
metric state is given by ψs ψ+ 2 = sin2(α/2+π/4). these flux fluctuations become irrelevant deep inside the
|h | i|
All this taken together opens up the possibility to use topological phase.
these boundstatesfortopologicalquantumcomputation
along the lines proposed for Majorana fermions [31]. Conclusions. We have uncovered model systems of
Single-ladder model. An alternative representation strikingsimplicitythatallowforatopologicalphasewith
of the double-ladder model with spinless particles is a degenerate bound states that are fractionally charged
single-ladder model with spinful particles, see Fig. 4. In and obey non-Abelian braiding statistics. We have
thismodeltheσ = 1(lower)andtheσ =1(upper)lad- shownthattheseexoticquantumstatesareratherrobust
−
ders are identified with the spin-down and spin-up com- against a large class of perturbations. Quite remark-
ponents, resp. A uniform magnetic field B¯ is applied ably, our models demonstrate that non-Abelian states
1
perpendicular to the ladder, leading to both orbital and can exist in single-particle systems, without any correla-
spin effects. The Zeeman energy acts as a chemical po- tionsandinthecompleteabsenceofsuperconductivityor
tential with opposite signs for opposite spin directions, BCS-likepairing. Thisshouldopenthepathfornovelim-
µ = σgµ B¯ , where g is the g-factor, and µ is the plementations of topological matter in realistic systems.
σ B 1 B
Bohr magneton. A spatially periodic field B¯ of period One promising candidate system that suggests itself for
2
2a couples opposite spins, so the effective hopping is implementations of such tight-binding ladders are opti-
x
given by t = gµ B¯ . Such a periodic field can be pro- cal lattices [34], because they allow for a high degree of
z B 2
duced by nanomagnets [32] or, equivalently, by Rashba control and, in particular, possess the charge stability of
spin orbit interaction and a uniform magnetic field [33]. the type invoked here.
Stability against perturbations. We next address the
stability of the topological phase and the bound states This work is supported by the Swiss NSF, NCCR
against local perturbations. In general, we find numer- Nanoscience, and NCCR QSIT.
5
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