Gaffnian holonomy through the coherent state method

0.53 MB·English

by  John Flavin

Gaffnian holonomy through the coherent state method
John Flavin,1 Ronny Thomale,2 and Alexander Seidel1
1Department of Physics and Center for Materials Innovation,
Washington University, St. Louis, MO 63130, USA
2Department of Physics, Stanford University, Stanford, CA 94305, USA
We analyze the effect of exchanging quasiholes described by Gaffnian quantum Hall trial state
wave functions. This exchange is carried out via adiabatic transport using the recently developed
coherentstateAnsatz. WearguethatourAnsatzisjustifiediftheGaffnianparentHamiltonianhas
a charge gap, even though it is gapless to neutral excitations, and may therefore properly describe
theadiabatictransportofGaffnianquasiholes. FornonunitarystatessuchastheGaffnian,theresult
2
of adiabatic transport cannot agree with the monodromies of the conformal block wave functions,
1
andmayormaynotleadtowell-definedanyonstatistics. UsingthecoherentstateAnsatz, wefind
0
two unitary solutions for the statistics, one of which agrees with the statistics of the non-Abelian
2
spin-singlet state by Ardonne and Schoutens.

n
a
J I. INTRODUCTION defines adiabatic transport.13,14 Indeed, in may cases of

6 interest, this remains a profound technical challenge.
1 In addition to wave functions arising from unitary

The study of fractional quantum Hall (FQH) liquids CFTs, there is also considerable interest in analytic trial
]
l has lead to the discovery of rich classes of topological states that are similarly related to nonunitary CFTs.4,15
l
a phases. Therapidprogressinthetheoreticalexploration The physical interpretation of such states remains much
h of this particular niche of correlated electron systems more subtle. Here, the field theoretic mapping employed
- was made possible by the development of a set of princi- in the unitary case does not lead to a topological quan-
s
e ples for the construction of “fixed point” or “prototype” tum field theory that can serve as the low energy ef-
m wave functions for possible FQH phases, as pioneered fective theory of the state in question. In particular,
. by Laughlin1 and then greatly expanded by other semi- the conformal block monodromies cannot be interpreted
t
a nal contributions.2–5 This progress was compounded by to describe adiabatic transport, as they do not result
m the discovery of a link between these special wave func- in unitary transformations on states. In contrast, adia-
- tions and conformal field theory (CFT).3 Via this link batic transport describes (a limit of) the time evolution
d the topological quantum field theory describing a state governed by a Hermitian Hamiltonian, and is therefore
n
can be inferred, which for some cases of interest is non- always described by a unitary transformation. For these
o
Abelian.3 The possibility of anyonic quasiparticles with reasons, it has been argued9,10,15–17 that states obtained
c
[ non-Abelianbraidingstatisticsmaybethemostspectac- from nonunitary CFTs describe gapless critical points
ular implication of this field theoretic mapping, and its within the phase diagram of quantum Hall states, es-
1
experimentalverificationremainsaprofoundchallengeto pecially in those cases where a local parent Hamiltonian
v
date, with topological quantum computing being a pro- for the state exists. Examples of the latter kind include
5
posedapplication.6,7 Evenintheory, adirectverification the Haldane-Rezayi (HR) state,4 and the state known as
5
3 of the implied field theoretic mapping is nontrivial, and Gaffnian, which has surfaced in the literature as early
3 involves the calculation of adiabatic transport of quasi- 1993 through a fixed many-body clustering property,18
. particles using full many-body wave functions. This is and which has been thoroughly discussed and was pro-
1
0 stilldifficultingeneral,buthasbeenpossibleforAbelian posed to be critical using a CFT construction.15
2 states8 and recently for p + ip wave superfluids and The question arises what hidden orders can be identi-
1 Moore-Read-type quantum Hall states.9–11 The general fiedinsuchnonunitarystates,whethertheyberemnants
: interpretationofanalytictrialwavefunctionsasdescrib- of topological orders or orders of a different kind. Un-
v
i ing topological phases is thus well accepted whenever fortunately,thereiscurrentlynoefficientanduniversally
X these wave functions are related to the conformal blocks applicable method to test for the topologically ordered19
r of a unitary rational CFT. In this case, the field theo- nature of a state directly through the study of ground
a
retic mapping described above implies that the statistics state properties. Much progress along these lines has re-
of quasihole-type excitations, as defined through adia- cently been made through the analysis of entanglement
batic transport, is directly encoded in the monodromies spectra,20,21whicharedirectlyrelatedtoedgespectra. It
of the associated conformal blocks.3,12 This means that hasbeenarguedthattheedgespectrumoftheGaffnianis
forsuchconformalblockwavefunctions,adiabatictrans- inconsistentwiththatofanyunitaryCFT,andthatthis
portisthesameasanalyticcontinuationinthequasihole contradictstheexistenceofagapinthebulkspectrum,16
coordinates, and Berry phase effects are trivial. This which is required for a topological phase. In principle,
thenallowsonetoobtainthequasiholestatisticsdirectly topological orders can be detected through nonlocal or-
from wave functions, but without explicitly calculating derparameters,22 thoughitremainsdifficulttoexplicitly
the Berry connection, or Wilczek-Zee connection, that construct such objects for general non-Abelian topolog-
2
ical orders in microscopic quantum Hall wave functions. ative result (no consistent anyon model) would further
The situation is similar in quantum magnetism. There, strengthen the case that the TT limit contains informa-
nonlocal operators detecting a topological phase can be tionaboutthegappedorgaplessnatureoftheunderlying
directly constructed for toy models23 defined on highly state. Thishasbeenexploredbyoneofusrecentlyforthe
constrainedHilbertspaceswhereagaugestructureisex- HR state,44 though not with regard to statistics. On the
plicit (see Ref. 24 for a general discussion). However, other hand, if a consistent anyon model is obtained, we
such order parameters generally remain elusive in mod- canarguethatthisisatleastaveryplausiblescenariofor
els where similar physics is emergent within the low en- the holonomies defined by the Gaffnian quasihole states,
ergy sector of a larger Hilbert space (e.g., Ref. 25). The as we will further elaborate below.
situationismuchsimplerinone-dimensionalsystemsex- Thecoherentstatemethodisbasedontheassumption
hibitingaHaldaneorLuttingerliquidphase. Thehidden of adiabatic continuity between states defined on a torus
orders of these phases can be probed through nonlocal with arbitrary aspect ratio and corresponding states in
objects measuring squeezed particle configurations,26,27 the thin torus limit. It further rests on general assump-
and their origin is quite manifest, e.g., in certain lim- tions about a coherent state Ansatz for localized quasi
its of Luttinger liquids where the wave function assumes holes in terms of adiabatically continued TT states. De-
a special factorized form.28–31 For topological orders, on tailed arguments in favor of this Ansatz have been given
theotherhand,themostdirectprobethatcan,inprinci- in Ref. 34. Some of these arguments, in particular the
ple,beimplementedatamicroscopicwavefunctionlevel justification for the factorized form of the Ansatz (see
isthestudyofthebraidingstatisticsoflocalizedelemen- Eq. (3) below), also rest on a notion of locality, which is
tary excitations. more subtle in a gapless state. We argue however, that
Inthispaper,weareinterestedinsomeformalproper- the necessary assumptions still apply, as long as there is
ties of the Gaffnian state. We ask the question whether a finite gap in the charge sector of the system, indepen-
the Gaffnian trial wave functions may define some uni- dent of the existence of gapless neutral excitations. The
taryanyonmodelthroughtheholonomycalculatedalong scaling of the charge gap of the Gaffnian state has been
exchange paths. Somewhat more physically speaking, discussed in some detail in Ref. 48, but at the moment,
thiscorrespondsthetheadiabatictransportoftrialstate the question whether it remains finite in the thermody-
quasiholes in the presence of a finite size gap. In- namiclimithasnotbeenconclusivelyresolvedtothebest
deed, this question is mathematically well defined. The of our knowledge.
quasihole trial states can be characterized as the unique There is a close connection between TT patterns and
zero-energy eigenstates (zero modes) of a local parent CFT fusion rules, which has been elaborated in Refs. 49
Hamiltonian.15,18 For given quasihole configuration, the and 50. In terms of the data used to construct an anyon
associatedconformalblockwavefunctionsdefineafinite- model, there is thus some similarity between the present
dimensionalsubspace,whichcanbeinterpretedasafiber method, and the procedure of constructing F-matrices
over a point in the quasihole configuration space. The consistent with given fusion rules, using the axioms of
question is thus whether the holomomy associated with modular tensor categories.51,52 Indeed, the two methods
exchange paths in this configuration space induces well- have so far given consistent, if not always identical, re-
definedstatistics. Itisclearfromtheoutsetthatifthisis sults in the unitary cases.34 Some differences between
so, the holonomies must be quite different from the con- these two approaches are worth noting. While the con-
formal block monodromies, since these holonomies give nectionwithtensorcategorytheorycanbephysicallyjus-
risetounitarytransformationsonfibers. Physically,this tified using the general framework of local quantum field
is clear from the fact that these holonomies describe the theory,53 the assumptions used in the coherent state ap-
adiabatic transport of quasiholes protected by a finite proach are not field theoretic in character. Indeed, the
size gap. Mathematically, it follows from the fact that F-matrix, although it could be ultimately constructed,
the connection on our vector bundle is a Wilczek-Zee does not directly appear in this approach, and none of
connection defined in terms of a physical scalar product. thedefiningconsistencyequationsofthisapproachmake
The question defined above can be rigorously ad- referencetoit. Moreover,thecoherentstatemethod,be-
dressed only by calculating the Wilczek-Zee connection ing ultimately based on adiabatic transport, could not
from the given analytic wave functions. Unfortunately, in principle lead to a nonunitary anyon model in its cur-
wedonotknowhowtodothisfortheGaffnianstate. In- rent formulation. A secondary, but non-negligible moti-
stead,wewillusethisquestionasatestbedforarecently vation for this work is thus to shine further light onto
developed coherent state method32–34 to calculate adia- the connection between these two different methods in
batic transport of quasiholes based on the “thin torus” the context of a nonunitary state.
(TT)32–44 or “dominance”18,45–47 patterns of the wave The paper is organized as follows. In Sec. II we will
function. This method has been shown to be quite effi- summarizeourcalculation,whichleadstotwocloselyre-
cient for a number of interesting states based on unitary lated anyon models of Fibonacci type. All details are
CFTs, but has thus far not been applied to the nonuni- relegated to two Appendices. In Sec. III, we discuss
tary case. Our motivation to clarify the applicability of our results, and make contact with the non-Abelian spin
this method to a nonunitary state is twofold. A neg- singlet (NASS) of Ref. 54. In a future version of this
3
preprint soon to follow, we will also present new numer- α Thin torus pattern f1(α) f2(α)
ical studies on the zero mode spectrum of the Gaffnian 1 0020020011011011002002002 −s −2+s
state and its connection to root partitions. 2 1101101100200200200110110 −1+s 2−s
3 1101100101011011010110110 1 0
II. BRAIDING STATISTICS THROUGH THE TABLE I. c = 0 thin torus patterns for a two-quasi-hole
COHERENT STATE METHOD Gaffnian state, and the offset functions of the associated do-
main walls. The elementary domain wall strings are in bold,
and the orbital positions, 3n , are underlined. Patterns for
j
Our method to extract the statistics of a state from
c=1(−1)canbeobtainedbyshiftingeachoccupancynumber
thin torus patterns, based on the assumptions discussed oneorbitaltotheright(left),andtheshiftfunctionsobtained
in the Sec. I, has been documented in detail in a recent using f (c,α)=f (α)+c.
j j
paper.34 Here we will describe its setup for the Gaffnian
state,andfocusondifferencesthatarisecomparedtothe
discussion of the (k = 3) Read-Rezayi state5 as given in
Ref. 34.
The thin torus patterns of the bosonic ν = 2/3
are simple product states in a “rotated” or dual Lan-
Gaffnian, and their relation to the underlying mini-
daulevelbasis32–34, andmustbewelldistinguishedfrom
mal model CFT, have been thoroughly discussed by
Ardonne.49 These patterns can be identified in the usual the states |a1,...,an;c,α). Here, the ai label the or-
way32–44 when wave functions on the torus are consid- bital positions of the domain walls. For the topological
sector labels, we follow the convention of 34, where α
ered in the formal limit of an “extreme aspect ratio”,
distinguishes classes of sectors that are not related by
e.g. L (cid:29)1,L (cid:28)1. (Here and in the following, we set
x y
translation, and for given α, c = −1,0,1 distinguishes
themagneticlength(cid:96) equalto1.) Thepatternsemerg-
B
the three translationally related members of each class.
ing in this limit are occupancy numbers in a suitably
Below we will also use translational properties to define
chosenLandaulevelbasisonthetorus(see, e.g., 34), de-
a unique convention for how the c labels are to be as-
scribing the trivial non-interacting product state result-
signed. Asusual34,thedomainwallpositionscanchange
ing in this limit. For the six degenerate Gaffnian ground
only by multiples of a certain “stride” within each topo-
states, these patterns read 200200..., 020020..., and
logical sector (here, multiples of 3), and are thus of the
002002...,whichwewillcall“(200)-type”,or011011...,
generalforma =3n +f (c,α). f (c,α)isanoffsetfac-
101101..., and 110110..., which we refer to as “(011)- j j j j
tor that depends on the sequence of domain walls (i.e.,
type”.
As usual,32–44 we will assume that states defined on the topological sector and the domain wall in question).
For symmetric domain walls, its value is uniquely deter-
a “bulk torus”—L (cid:29) 1,L (cid:29) 1—can be evolved
x y
mined by symmetry. However, for asymmetric domain
adiabatically into the thin torus limit described above,
walls, acertainambiguityexistsaprioriinhowtodefine
where the evolution is governed by the Gaffnian par-
ent Hamiltonian15 with slowly varying aspect ratio. the domain wall position precisely with respect to the
adjacent orbitals. This is accounted for by the shift—or
Likewise, states with quasiholes on the bulk torus are
asymmetry—parameter s. See 34 and table I for details.
assumed to evolve into elementary domain walls be-
tween ground state patterns. There are three ele-
mentary domain wall strings: 100 and 001, which oc- By the assumption of adiabatic continuity, the “bare”
cur between (200)-type and (011)-type ground states, domainwallstates,|a ,...,a ;c,α)and|a ,...,a ;c,α),
1 n 1 n
and 010, which occurs between two different (011)- give rise to two different complete sets of zero energy
type ground states. These domain walls may link var- states (zero modes) of the Gaffnian parent Hamiltonian,
ious different combinations of ground state patterns, for any aspect ratio of the torus. These two mutually
thus forming charge 1/3 solitons. Representative ex- dualzero-modebasesareobtainedbyadiabaticallyevolv-
amples are given by the strings ...01101100200200... ing the respective bare states by means of a slow change
and...011011010110110.... Allcharge1/3domainwall in aspect ratio, and are denoted by |a ,...,a ;c,α(cid:105) and
1 n
patterns occurring in the TT limit are locally given by |a ,...,a ;c,α(cid:105)respectively(withthedependenceonas-
1 n
translated and/or inverted versions of these two types of pect ratio understood). These types of states generally
strings. This establishes a notion of “fusion rules” in the describe quasiholes that are localized in x or y, respec-
TT limit.49 tively, at the domain wall positions, and are delocalized
The thin torus states are naturally labeled by the po- intheperpendiculardirection(respectivelyyorx). Their
sitions of the domain walls together with a topological completeness within the zero-mode space implies that
sectorlabel—i.e., alabelidentifyingthesequenceofpat- states with localized quasiholes (in both x and y) can
terns between the domain walls. We use the notation be obtained by forming proper linear combinations. For
|a1,...,an;c,α) for the TT states appearing in the limit thesecoherentstates,withnquasiholeslocalizedatcom-
L (cid:29) 1,L (cid:28) 1, and |a ,...,a ;c,α) for the states plex positions h =h +ih , the following Ansatz has
x y 1 n j jx jy
appearing in the L (cid:28) 1,L (cid:29) 1 limit. The latter been motivated in previous works (see Ref. 34 and refer-
x y
4
ences therein):
n
(cid:88) (cid:89)
|ψ ({h})(cid:105)=N φ (h ,κa )|a ,...,a ;c,α(cid:105)
c,α α,j j j 1 n
a1<...<anj=1
(1)
n
|ψ ({h})(cid:105)=N(cid:48) (cid:88) (cid:89)φ¯ (h ,κa )|a ,...,a ;c,α(cid:105)
c,α α,j j j 1 n
a1<...<anj=1
(2)
FIG.1. Twoquasiholes,onestationaryandonemovingacross
the boundary, on a torus which is shown in a “repeated zone
with the Gaussian amplitude form factor
scheme”. The stationary quasihole is at the bottom of the
(cid:20) (cid:21) figure. Initially the moving quasihole (which starts at the
i
φ (h ,a )=exp (h +δ(α,j)/κ)κa −γ(h −κa )2 ,dotted-line circle) is ordered second, and the system is in
α,j j j 3 jy j jx j some configuration σ. After that quasihole moves across the
(3) boundary it becomes the first, and the system is in configu-
and its dual counterpart φ¯α,j(hj,aj) = ration gx(σ).
φ (−ih ,a )| . Here, κ = 2π/L , κ¯ = 2π/L ,
α,j j j κ→κ¯ y x
and δ(α,j) can be shown34 to be 0 or π, taking on
the same value for symmetry-related (translation or
inversion) domain walls. N and N(cid:48) are normalization
factors.
It must be emphasized that these coherent states are
valid34 onlyforwell-separatedquasiholes: Eq.(1)isvalid
when |h −h | (cid:29) 1 for all i,j, and Eq. (2) when
jx ix
|h −h | (cid:29) 1 for all i,j. When both conditions are
jy iy
satisfied, Eqs. (1) and (2) describe the same zero modes
for given quasihole positions, and must be related by a
linear transformation:
FIG.2. Theconfigurationσ istheoneinwhichtheleftmost
I
(cid:88) (first)quasiholeisbottommost,thenexttotheright(second)
|ψ ({h})(cid:105)= uσ ({h})|ψ ({h})(cid:105). (4)
c,α c,c(cid:48),α,α(cid:48) c(cid:48),α(cid:48) is the next above, and so on.
c(cid:48),α(cid:48)
Here, σ labels different “configurations” of the quasi-  
e2πi(L−1)/3 e−2πiL/3 e2πi/3
holes, i.e., components of the quasihole configuration 1
space that can be connected without violating the con- M = √  e−2πiL/3 1 e2πiL/3 . (7)
3
ditions |h −h |(cid:29)1, |h −h |(cid:29)1. We define these e2πi/3 e2πiL/3 e2πi(L−1)/3
jx ix jy iy
configuration-labels with respect to quasihole configura-
tions in the infinite plane. Toroidal periodicity leads to The h-independent product M ξσ may be identi-
c,c(cid:48) α,α(cid:48)
an equivalence between various triples (σ,(c,α),(c(cid:48),α(cid:48))) fied as the topological S-matrix of the problem. We will
that will be taken into account later. The transforma- now summarize how the missing information about ξσ
α,α(cid:48)
tion described by uσ ({h}) must also be unitary, canbeobtainedwithinthisformalism, andsubsequently
c,c(cid:48),α,α(cid:48)
since states in different topological sectors are generally this information can be used to gain knowledge about
orthogonal. braiding. For details we refer the reader to Ref. 34.
A lot is known about the transition functions Furtherconstraintsonthematrixξσ canbeobtained
α,α(cid:48)
uσ ({h}) from the properties of the zero-mode ba- bymoving(viaadiabatictransport)thepositionsofsome
c,c(cid:48),α,α(cid:48)
sis states—|a ,...,a ;c,α(cid:105) and |a ,...,a ;c,α(cid:105)—under of the n quasiholes across the boundaries of a rectan-
1 n 1 n
magnetic translations alone.34 Their dependence on h, gular coordinate chart on the torus, which we fix once
(c,c(cid:48)), and (α,α(cid:48)) separates into the following factorized and for all (see Fig. 1). It is important to note that
form: this can be done while maintaining |hjx−hix| (cid:29) 1 and
|h −h | (cid:29) 1, and this leads to new consistency con-
jy iy
uσ ({h})=u({h})M ξσ , (5) ditions on the ξσ matrix. We begin in a configuration
c,c(cid:48),α,α(cid:48) c,c(cid:48) α,α(cid:48) α,α(cid:48)
σ—e.g.,theoneshowninFig. 2—andcarryoutthispro-
with u({h}) and M fully determined by translational cess in two steps. First, we consider the adiabatic trans-
c,c(cid:48)
symmetry: port of the upper right particle along a path that moves
it further to the right. As soon as the particle crosses
 
the right boundary of our coordinate chart, the particle
iπ (cid:88)
u({h})=exp 3 hjxhjy, (6) reappearsontheleftboundary. Viewedinthiscoordinate
j chart, thequasihole configurationhaschangedfrom σ to
5
a new configuration g (σ). At the same time, with the
x
particle that was formally rightmost now being leftmost,
thetopologicalsectorlabel(thesequenceofpatterns ap-
pearing in the L →0 limit) of the state has changed. If
y
αwasthelabelofthetopological(super-)sectorinitially,
the new label F(α) is easily found from the associated
thintoruspatterns.34 Thus,whilethetransitionfunction
expressing the duality relation (4) is defined in terms of
ξσ rightbeforecrossingtheboundaryofthecoordinate
α,α(cid:48)
chart, it is defined in terms of ξgx(σ) right thereafter.
F(α),α(cid:48)
Is is easy to work out a condition enforcing continuity FIG.3. Exchangepathfortwoquasiholesatpositionshi and
across the boundary, relating these two matrices,34 hi+1. First,thequasiholeathi+1 isdraggedtoha. Therethe
coherentstaterepresentationischangedfromtheoriginalba-
ξgx(σ) =B−1diag[e−2πiL/3−iLδ(α,n)/3−i/3(cid:80)kδ(α,k)]ξσ sistothedualbasisusingΞσ. Thatquasiholeisthendragged
toh ,andthestateischangedbacktotheoriginalbasisusing
b
×diag[e−2πifσn(α)/3], (8) Ξσ(cid:48). The quasihole at h , now the i-th, is moved to h , then
b c
bothquasiholesaremovedtotheirfinalpositions: Thequasi-
where the argument of diag[...] specifies the α-th diago-
hole at h goes to h and the quasihole at h goes to h .
i i+1 c i
nal entry of a diagonal matrix. The matrix B is defined Using the coherent state Ansatz, this gives rise to Eq. (12).
as B =δ . A similar process can be considered
α,α(cid:48) α,F(α(cid:48))
where the topmost quasihole of a configuration σ moves
up,crossingtheupperboundaryofthecoordinatechart, where F (α) is the (super-)sector resulting from an in-
τ
becomingthebottommostquasiholeintheresultingnew versionofthepatternsassociatedwiththe(super-)sector
configuration gy(σ). The analogous continuity condition α. Operating τx and τy in succession on a system in the
on the ξ matrices reads, configuration σ gives σ again, i.e., g (g (σ )) = σ .
I I τy τx I I
In this way Eqs. (10) and (11) allow us to further con-
ξgy(σ) =diag[e−2πifj(α)/3]ξσ strain the elements of ξσI.
×diag[e−2πiL/3−iLδ(α,n)/3−i/3(cid:80)kδ(α,k)]B. (9) To braid two quasiholes, we adiabatically transport
themalongthepathinFig.3. Tothisend,weconsidera
ThepurposeofEqs. (8)and(9)istwofold: Theyconnect configuration σ where both quasiholes are adjacent both
ξ matrices for different quasihole configurations, which in terms of their x and their y coordinates, i.e., they
will be essential in describing braiding processes, and can be exchanged without violating |h −h | (cid:29) 1 or
jx ix
they impose constraints on any given ξ matrix. To see |h −h |(cid:29)1 for any other pair of quasiholes. The re-
jy iy
the latter, focus on the quasihole configuration shown in sult of the adiabatic transport can then be worked out
Fig. 2, which we will now refer to as σ . It is easy to see directly from the coherent state Ansatz (1) given known
I
thatσ isinvariantunderthetwomovesdescribedabove transition functions, i.e., the matrices ξσ.32–34 It can be
I
performed in succession, i.e., g (g (σ )) = σ . Hence, expressed as follows:
y x I I
Eqs. (8) and (9) together constrain the matrix elements
   
of ξσI. |Ψ1(cid:105) |Ψ1(cid:105)
Yet another way in which the transition functions and  |Ψ (cid:105)   |Ψ (cid:105) 
hence the ξ-matrices are related for different quasihole  .2 →eiΦABΞσ(Ξσ(cid:48))† .2  . (12)
configurations is by mirror symmetry. In the presence of  ..   .. 
aconstantmagneticbackgroundfield,mirrorsymmetries |Ψ (cid:105) |Ψ (cid:105)
n n
exist only in conjunction with time-reversal symmetry,
which we will leave understood. The operators τx and Here, ΦAB is the Aharonov-Bohm phase, equal to the
τ associated with mirror reflections across the x and y charge of a quasihole, −1/3, times the area enclosed by
y
axis are therefore anti-linear operators. It is then sim- the braiding path. The quantities |Ψα(cid:105) are the three-
ple to find relations between the ξ-matrices using these component column vectors with entires |ψc,α(cid:105). Ξσ is the
operations,34 matrix ξσ⊗M (where we identify the states |ψc,α(cid:105) with
a formal tensor product basis |α(cid:105)⊗|c(cid:105)). σ is the initial
ξgτx(σ) =(Bτ)−1diag[eiL/3(cid:80)kδ(α,k)](ξσ)∗ configurationofthequasiholes, andσ(cid:48) theotherconfigu-
×diag[e2πi/3(cid:80)k(1+δ(α,k)/π)fk(α)], (10) ration that occurs during braiding (Fig. 3), i.e., the one
obtainedfromσ bycrossingthelineh =h ortheline
ξgτy(σ) =diag[e2πi/3(cid:80)k(1+δ(α,k)/π)fk(α)](ξσ)∗ h =h . jx ix
jy iy
×diag[eiL/3(cid:80)kδ(α,k)]Bτ. (11) diaIgtotnuarlnsinouthtethcaltatbheelsr,eis.ue.l,ttohfebrbariadiidngmiastarilxwaΞyσs(bΞlσo(cid:48)c)k†
iImnathgeeaobfoσveu,ngτdxe(rστ)i,stahnedcosinmfiiglaurrlaytigont(hσa).tiTsthheemmairtrroixr iξsσ(oξfσt(cid:48)h)†e ftohrem“χreid(nu)ce⊗d”Icmbarxa×idcmmaxa,twrihxeraesswoecicaatelldχwi(inth) =a
x τy
B appearing above is defined by (B ) = δ , counter-clockwise exchange of the i-th and i+1-st of n
τ τ α,α(cid:48) α,Fτ(α(cid:48))
6
quasiholes. This fact is a direct consequence of trans- α Thin torus pattern f (α) f (α) f (α)
1 2 3
lational invariance. Moreover, χi(n) is found to be in- 1 002001101101101011011011002002 −s 0 s
dependent of the initial configuration σ, as one would
2 110110101101100200200200110110 1 −2+s 1−s
expect.
3 110110020020020011011010110110 −1+s 2−s −1
The various constraint equations discussed so far still
4 110110101101101011011010110110 1 0 −1
admitmanysolutionsforthebraidmatrix. Thisischiefly
due to the fact that the asymmetry parameter s intro- TABLE II. c = 0 thin torus patterns for a three-quasihole
duced above is still undefined. A final set of constraint Gaffnian state, and the offset functions of the associated do-
equations comes from the imposition of certain locality main walls. The elementary domain wall strings are in bold,
constraintsonthebraidmatrixthatcanbemotivateddi- and the orbital positions, 3nj, are underlined. Patterns for
rectly from the thin torus patterns.33,34 Said succinctly, c=1(−1)canbeobtainedbyshiftingeachoccupancynumber
oneorbitaltotheright(left),andtheshiftfunctionsobtained
we mean by locality that the result of braiding should
using f (c,α)=f (α)+c.
only depend on the sequence of three ground-state pat- j j
terns forming the two domain walls associated with the
braidedquasiholes,andthatonlythepatterninthemid-
dle may change as a result of braiding. The constraint III. DISCUSSION
equations following from this, combined with the above
symmetries, then lead to a discrete set of (usually inti-
The two solutions obtained in the preceding Sec. II
mately related) solutions for the statistics. We will dis-
are related to one another simply by complex conjuga-
cuss the full set of constraint equations and their solu-
tion and an overall Abelian phase. They describe Fi-
tions in App. A for the two-quasihole case, and in App.
bonacci anyons and are thus closely related to those ob-
B for the three-quasihole case. Here we will summarize
tained for the k = 3 Read-Rezayi (RR) state using the
the results of this calculation by giving the (reduced)
same method.34 In essence, the solutions obtained from
braid matrices obtained from it for both two and three
the RR patterns and those obtained here are the same
particles:
uptoanAbelianphase. Thisstatementexcludes“global
exchange paths” on the torus involving processes such
χ1(2)=ξ(1,2)(ξ(2,1))† as the one depicted in Fig. 1, as we will further explain
e−iπa  below. This close correspondence between the solutions
found from RR and Gaffnian patterns is a manifestation
=e−iπ/3 eiπaϕ−1 e−2iπaϕ−1/2 ,
  of level-rank duality between the associated SU(2) and
e−2iπaϕ−1/2 ϕ−1 3
SU(3) fusion rules, respectively. This duality was also
2
(13) discussed by Ardonne49 in terms of domain walls. It is
manifest in the Bratteli diagrams of Fig. 4, which in the
present context represent the rules for domain wall for-
χ (3)=ξ(1,2,3)(ξ(2,1,3))† mation between ground state patterns for the respective
1
states. Note, however, that there is a three to two corre-
 
e2iπa spondence between the topological sectors in both cases,
 e2iπa  rather than one-to-one. This is so since each sector, i.e.,
=e−iπ/3 ,
 eiπaϕ−1 e−2iπaϕ−1/2  eachpathintheBrattelidiagram,isthreefolddegenerate
 
undertranslationsintheGaffniancase, butonlytwofold
e−2iπaϕ−1/2 ϕ−1
in the k = 3 RR case (at filling factor 3/2). The cor-
(14)
respondence between Gaffnian and RR sectors is perfect
if we limit ourselves to the n−1 generators of the braid
where√a = ±1/5, and ϕ is the golden ratio, ϕ = groupσi,i+1,i=1,...,n−1,whichexchangethei-thand
(1 + 5)/2. Here, the rows and columns refer to the i+1-st quasihole. These generate the full braid group in
α-supersectors of Tables I and II. The parameter s is theplane, butnotonthetorus. Onthetorus, thesegen-
found to have one of two values: s = 2−3a/2. Just as erators leave certain subspaces of topological sectors in-
in the case of the k = 3 Read-Rezayi state discussed in variant,whichallstartandendinthesamepatterninthe
Ref. 34, there are two solutions, one for each sign of a, topologicalsectorlabel. ThesesubspacesfortheGaffnian
which are related by an Abelian phase and complex con- arethenincorrespondencewithsimilarsubspacesforthe
jugation. Intheabove,wehavealsofixedagaugedegree RR state, in the sense that there are isomorphisms be-
of freedom associated with unitary transformations. tween them that commute with braiding, except for an
Together with the locality constraint described above, overall Abelian phase. This correspondence, however,
these two matrices imply the result of braiding any ad- gets spoiled by the inclusion of the remaining generators
jacent pair in a state of n quasiholes. A “tensor repre- on the torus, which mix the subspaces. This happens
sentation” of the statistics just as discussed in Ref. 55 differently for the Gaffnian and the RR case, since there
canthenimmediatelybeconstructedincompleteanalogy are six such subsectors in the former case, but only four
with Ref. 34. in the latter.
7
ing α degree of freedom is fully analogous in both cases.
This is how rank-level duality becomes manifest in the
present formalism. For similar reasons, our solutions for
the Gaffnian and RR states, which were both obtained
at the maximum (bosonic) filling factor, could be gener-
alized quite easily to lower filling factors.
As emphasized initially, the Fibonacci-type solutions
we obtained are distinct from the anyon model asso-
ciated with the conformal block monodromies of the
Gaffnian state. The latter describes so-called Yang-
Lee anyons, whose relation to Fibonacci anyons and
the associated Galois duality has enjoyed much inter-
est recently.56,57 Yang-Lee anyons are associated with
nonunitaryF-matricesconsistentwiththeSU(3) fusion
2
rules. It has been known for some time, however, that
the same fusion rules admit unitary solutions of the Fi-
bonacci type also; these are realized by the NASS state
of Ref. 54. Indeed, it is not difficult to perform checks
confirming that one of our solutions, that corresponding
FIG.4. Brattelidiagramsdepictingthec=0patternsinthe to s = 17/10, agrees exactly with the monodromies of
RR state (top) and the Gaffnian (bottom). Valid topological the NASS state (including the overall phase). The dom-
sectors on the torus are represented by paths which start at
inance patterns of the NASS state have been discussed
theleft,takeonestepright(eitherupordown)foreachquasi-
more recently,58 and it seems clear that the calculation
hole in the state, and end on the same type of ground state
presented here can be carried over to this state without
pattern—(200)-type or (011)-type—as they began, minding
essential changes. In all, this confirms once more for the
periodic boundary conditions. There is a one-to-one corre-
spondence between the paths in the lower diagram and the caseofSU(3)2 thatthecoherentstateAnsatzproducesa
patterns in Tables I and II, and also between the valid paths subset of all unitary anyon models consistent with given
in the upper and lower diagrams. This latter correspondence fusion rules.
is how the SU(2)3 and SU(3)2 rank-level duality manifests In the case of the unitary NASS state, our results
in terms of patterns. It should be noted that for each sector support the usual conjecture that the holonomies associ-
whichcorrespondstoapathinoneofthesediagrams,thereis atedwithadiabatictransportagreewithconformalblock
forRRanadditionalsectorrelatedtothefirstbytranslation,
monodromies. In the case of the Gaffnian, things are
and two additional translated sectors for the Gaffnian.
more subtle. We have argued that in this case, provided
thatthecoherentstateAnsatzisjustified,theholonomies
gives rise to a well defined Fibonacci anyon statistics,
The subtle differences between our solutions for the
possibly (given the twofold ambiguity of our solution)
Gaffnian and the RR case on the torus are of a piece
identical to those of the NASS state. In particular, we
with the difference in overall Abelian phase. It is well
believe that our Ansatz is indeed well justified if the gap
known that the overall Abelian phase could in principle
of the Gaffnian parent Hamiltonian does not close in the
assume any value in planar geometry, but on the torus,
chargesector. Ifso, thiswouldpresumablyremainafor-
it is constrained by the topological degeneracies charac-
mal property of the Gaffnian state that is not robust to
terizing the state. In the coherent state method, one
perturbations, even ones that do not open a gap in the
source of phase differences is the factor i/3 in the co-
neutral sector. However, it might shine new light on the
herent state Ansatz, Eq. (3), which is generally related
formal connections between the Gaffnian and the NASS
to the “stride” of the domain wall in a given topologic
state, which have been discussed previously.49 We leave
sector, which equals 3 in the present case and 2 in the
more rigorous investigations into this matter for future
RR case. This stride is of course identical to the center-
studies.
of-mass degeneracy. In particular, one may see that the
equationsobtainedfromglobalprocessessuchastheone
shown in Fig. 1 are quite sensitive to this stride and the
associated phase (see Eq. (A1) below). In view of the IV. CONCLUSION
importance of these processes in our method, and the
factthattheyspoilthecorrespondencebetweenGaffnian In this paper, we have analyzed the effect of exchang-
and RR topological sectors as explained above, it may ing quasiholes described by Gaffnian quantum Hall trial
not be clear a priori that the consistency equations we state wave functions via adiabatic transport, based on a
obtain in both cases admit closely related solutions, in coherentstateAnsatzintermsofadiabaticallycontinued
the sense discussed. That this is so can be traced back thin-torus zero modes. We have argued that this Ansatz
to the fact that the translational degree of freedom, c, maycorrectlydescribetheholonomiesofGaffnianquasi-
decouples early on (see below Eq. (5)), and the remain- holes as long as the charge sector of the state remains
8
gapped. In this case, we find that the statistics obtained respectively. Equation (A1) is satisfied when
are either closely related, or in fact identical, to those of
 
the NASS state, though this property will not be topo-
ξ ξ ξ
11 12 13
logicallyprotected. Thisobservationisofakindwiththe ξσI =ξ p2ξ −∆pξ . (A2)
statement that these states have the same fusion rules,  12 11 13 
ξ −∆pξ ξ
though their conformal block monodromies are different, 31 31 33
and are non-unitary in the Gaffnian case. Our results
appear to confirm in yet another case that the coher- Mirrorsymmetry,Eqs. (10)and(11),canalsoproducea
ent state Ansatz produces a set of unitary anyon models constraintequationbecausegτy(gτx(σI))=σI. However,
consistent with the fusion rules of the underlying state. inthecaseoftwoquasiholes,applyingEqs. (10)and(11)
in succession results in the trivial equation, ξσI =ξσI.
The process of solving for the reduced braid matrix is
similar to the solution given for n = 2 in Ref. 34. We
ACKNOWLEDGMENTS gain the following equations by demanding that ξσI is
unitary,
JFandASareindebtedtoS.Simonforhelpfuldiscus-
eiπ/3 =2∆η−Dpξ ξ +ηDpξ 2, (A3a)
sions on Gaffnian monodromies. RT thanks N. Regnault 11 12 13
for discussions as well as B. A. Bernevig and M. Greiter eiπ/3 =2ηDpξ 2−ξ 2, (A3b)
31 33
for collaborations on related topics. This work was sup-
0=∆η−Dp3ξ 2+∆η−Dpξ 2−∆ηDp2ξ 2,
ported by the National Science Foundation under NSF 11 12 13
(A3c)
Grant No. DMR-0907793. RT is supported by an SITP
fellowship by Stanford University. 0=η−Dξ31(cid:0)−p2ξ11+∆pξ12(cid:1)+ηDpξ13ξ33, (A3d)
whereη =exp(−2πi/3). Twoadditionalequationscome
fromtherequirementthatbraidingislocal;assaidabove,
Appendix A: Two-quasihole Solution thismeansthattheresultofbraidingshouldonlydepend
on the sequence of three ground-state patterns forming
the two domain walls associated with the braided quasi-
To work out the braid group for n quasiholes with
holes,andthatonlythepatterninthemiddlemaychange
general n, one needs to consider only braiding for pairs
as a result of braiding. Imposing these locality consider-
of quasiholes associated with all possible pairs of do-
ations tells us that χ (2) must be of the form
mainwalls,asgivenbyallpossiblecombinationsofthree 1
ground state patterns. Locality then implies that all the
 
other ground state patterns appearing in the topological · 0 0
sector label will not affect the result of braiding. To this χ (2)=0 · · , (A4)
1  
end, we will solve for the reduced braid matrix in the
0 · ·
simple cases of n=2 (here) and n=3 (in Appendix B).
Together, these results can be used to construct braid
where “·”s are unknown, potentially nonzero, matrix el-
matricesforn-quasiholestates, sincethesecasesexhaust
ements for which we will solve. By applying the form in
allpossiblesequencesofthreegroundstatepatternssep- Eq. (A4) to the matrix χ1(2) = ξσI(ξσI(cid:48))† derived from
arated by domain walls.
adiabatic transport, the zero elements give two more in-
Thesetofconstraintequationstosolveforthereduced dependent constraint equations,
braid matrix come from enforcing the unitarity of the
ξ matrices, the locality condition discussed in the main 0=η−D(1+p2)ξ ξ −∆ηDξ 2, (A5a)
11 12 13
text, and the global path relations Eqs. (8) and (9). We
0=η−Dξ (ξ −∆pξ )+ηDξ ξ . (A5b)
will begin by enforcing the latter. As discussed above, 31 11 12 13 33
we can apply Eqs. (8) and (9) in succession to constrain
ξσI because σI = gy(gx(σI)). Applying these two equa- Solving this system of six equations, (A3) and (A5),
tions,withthedatafromTableI,resultsintheconstraint is formally similar to the solution in the Appendices of
equation Ref. 34, so the details will not be repeated here. Just
as in that reference, there are two solutions: A special
    solution in which p=±i,
∆p−1 ∆p
ξσI =∆p 0 ξσI ∆p−1 0 , 1
    ξ 2 = ηDeiπ/3p−1, (A6a)
−1 −1 11 2
(A1) ξ =∆ξ , (A6b)
12 11
where p = −exp[−2πi(1+s)/3], ∆ = ξ =ξ =0, (A6c)
13 31
exp[−2πi(L/2+1)D/3], and D = 0 or 1 if the δ
ξ 2 =−eiπ/3, (A6d)
parameter for the 100-type domain walls is 0 or π, 33
9
which produces the reduced braid matrix can be applied in succession to ξσI to give
     
p 0 0 1 1
χ (2)=e2πi/3 0 p 0  (A7)  0 ηD   0 η−D 
1   ξσI = ξσI  . (B2)
0 0 e−iπ/3  η−D 0   ηD 0 
1 1
(but we will show in Appendix B that this solution is in-
consistentwiththeequationsfromthree-quasiholebraid- Equations (B1) and (B2) together constrain ξσI to be of
ing), and the consistent solution, the form
 
ξ112 = (η1D+eiπp/)32, (A8a) ξσI = ξξ1121 ηDξp122ξ11 −η−∆˜Dpξξ1122 −∆˜ξ−114pξ14 .
ξ12 =∆ξ11, (A8b) η−Dξ12 ∆˜pξ12 η−Dp2ξ11 −∆˜pξ14 
ξ −∆˜−1pξ −∆˜pξ ξ
ξ 2 =ηD(p+p−1)ξ 2, (A8c) 41 41 41 44
13 11 (B3)
ξ312 =ηD(p+p−1)ξ112, (A8d) The structure of this matrix is almost identical to the
ξ 2 =η−D(1−p)2ξ 2, (A8e) corresponding matrix ξ++ in Ref. 34, save that the
33 11 D-dependent phases ηD and ∆˜ are different. There,
∆˜ was defined such that ∆˜2 = eiπD, whereas here
which produces the reduced braid matrix
∆˜2 = e2iπD/3 = η−D. We might then find a braid ma-
χ (2)=e−iπ/3× trix with a non-trivial dependence on D. However, the
1
symmetry relations in Eqs. (8), (9), (10), and (11) also
 
p−1 0 0 have additional D-dependent phases compared to their
 0 p(p+p−1−1) ±eiπD/3(1−p)(cid:112)p+p−1corr.esponding forms in Ref. 34, and we will see that
 
0 ±e−iπD/3(1−p)(cid:112)p+p−1 p+p−1−1 by following the same steps as in that reference to find
the braid matrix solution, all the D-dependent phases
(A9)
will conspire to cancel save for those on the off-diagonal
elements which can be removed via a unitary transfor-
This two-quasihole reduced braid matrix is the same as
mation.
thatinRef. 34exceptfortwofeatures: TheD-dependent
ByrequiringthatξσI beunitary, wefindthefollowing
phaseontheoff-diagonalelements(thoughthiswilllater
constraint equations,
beremovedwithaunitarytransformation),andtheover-
all Abelian phase, which here is e−iπ/3 and in Ref. 34
1=|ξ |2+2|ξ |2+|ξ |2, (B4a)
was eiπ/2. 11 12 14
0=∆˜ξ ξ ∗+∆˜−1p2ξ ξ ∗−p|ξ |2−p|ξ |2,
12 11 11 12 12 14
(B4b)
(cid:16) (cid:17)
Appendix B: Three-quasihole Solution 0=ξ ξ ∗−2∆˜−1pξ ∗ +ξ ∗ξ , (B4c)
41 11 12 14 44
The solution for the reduced braid matrix of three 1=3|ξ41|2+|ξ44|2. (B4d)
quasiholes begins similarly to that for two quasiholes.
We first constrain ξσI using global path relations, Eqs. The locality of braiding tells us not only that some el-
(8) and (9), and mirror symmetry, Eqs. (10) and (11). ements of χ1(3) must be zero, as was the case for two
Applying the former two in succession and filling in the quasiholes,butalsothatthe2×2blockwithoff-diagonal
data from Table II gives the constraint elements must be equal to the equivalent 2×2 block in
χ (2). Thisisbecausethesequencesofgroundstatepat-
1
 ∆˜p−1   ∆˜p terns of the domain walls associated with the quasiholes
to be braided are the same for those two supersectors in
 0 −ηD  ∆˜p−1 0 
ξσI =∆˜p ξσI  −ηD mth,uesttwboe-aonfdthtehrfoeerm-quasiholecases. Inotherwords,χ1(3)
−1 −1
(B1) χ (3)=e−iπ/3×
1
where p is defined as in Appendix A, and ∆˜ =  
·
exp[−2πi(L/2+1)D/3]. This definition for ∆˜ is seem-
 · 
icnagsleyotfhtewsoamqueaasishtohleastLfor=∆1 imnoAdpuploen3d,iaxnAd,hbeuret Lin=th0e  p(p+p−1+1) ±eiπD/3(1−p)(cid:112)p+p−1 ,
(cid:112)
modulo 3. ±e−iπD/3(1−p) p+p−1 p+p−1+1
The two mirror symmetry equations, (10) and (11), (B5)
10
if Eq. (A9) is the correct reduced braid matrix for two- additional D-dependent phase and the overall Abelian
quasiholes, and of the form phase is different.
  We have yet to enforce consistency between the 2×2
·
blocks in the two- and three-quasihole braid matrices;
χ (3)=e2πi/3 · , (B6) to do so we equate Eq. (B9) to Eqs. (B5) and (B6)
1  
p 0 in turn. The latter produces a contradiction, because
 
0 e−iπ/3 the off-diagonal elements of Eq. (B9) are not zero for
p=±i. ThusEq.(A6)isnotaconsistentsolutionfortwo
with p = ±i, if Eq. (A7) is the correct matrix (which quasiholes, and Eqs. (A7) and (B6) are not consistent
we will show is not the case). We find our constraint braidmatrices. EnforcingconsistencybetweenEqs. (B5)
equations by equating the product χ1(3) = ξσI(ξσI(cid:48))†, and (B9) implies that
obtainedfromadiabatictransport,withtheformsabove.
We can perform the two solutions in parallel by using
p+p−1 =ϕ, (B10)
√
only the elements of Eqs. (B5) and (B6) that are zero in
where ϕ is the golden ratio, ϕ = (1+ 5)/2. In other
both. This produces the constraint equations
words,
0=p3ξ 2−∆˜2(p−1)pξ 2−p2ξ 2, (B7a)
11 12 14
(cid:18) (cid:19)
0=∆˜(p−1)p2ξ ξ +∆˜2pξ 2−p2ξ 2, (B7b) iπ
11 12 12 14 p=exp ± , (B11)
(cid:104) (cid:105) 5
0=−ξ pξ +∆˜(p−1)ξ +ξ ξ . (B7c)
41 11 12 14 44
or, if we define a = ±1/5, p = exp(iπa). This is all the
The set of constraint equations, (B4) and (B7), is solved
same as in Ref. 34. However, here the s parameter is
when
defined differently in terms of p than in Ref. 34; we have
eiθ1 defined p=−exp[−2πi(1+s)/3], so the s parameter is
ξ 2 = , (B8a)
11 (1+p)2 also constrained to be
ξ =∆˜−1ξ , (B8b)
12 11
3a
ξ 2 =(p+p−1−1)ξ 2, (B8c) s=2− . (B12)
14 11 2
ξ 2 =e2iθ2(p+p−1−1)ξ 2, (B8d)
41 11
(B8e) Consistency also implies
which produces the reduced braid matrix
eiθ1 =p2, (B13)
χ1(3)=e−iπ/3+iθ1× e2iθ2 =1. (B14)
 
1
 1 
 p(1−p) ±eiθ2+iπD/3p(cid:112)p+p−1−T(1h14e),e.xrpesrpesescitoinveslyfo,rhχa1v(e2)baenend χsi1m(3p)lifiinedEqwsi.th(1r3e)spaencdt
(cid:112)
±eiθ2−iπD/3p p+p−1−1 e2iθ2(1−p) to Eqs. (A9) and (B9) and have undergone a uni-
(B9) tary transformation in which each state is multiplied by
(−eiπD/3)#200, where #200 is the number of 200200...
Just as in Appendix A, Eq. (B9) is the same reduced strings in the thin torus pattern associated with that
braid matrix as was found in Ref. 34 for three quasi- state. This unitary transformation removes the depen-
holes, except thathere the off-diagonalelements havean dence on the unkonwn parameter D.
1 R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). 8 D. Arovas, J. R. Schrieffer, and F. Wilczek, Phys. Rev.
2 B. I. Halperin, Helv. Phys. Acta. 56, 75 (1983). Lett. 53, 722 (1984).
3 G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991). 9 N. Read, Phys. Rev. B 79, 045308 (2009).
4 F. D. M. Haldane and E. H. Rezayi, Phys. Rev. Lett. 60, 10 N. Read, arXiv:0807.3107 (2008).
956 (1988). 11 P. Bonderson, V. Gurarie, and C. Nayak, Phys. Rev. B
5 N. Read and E. H. Rezayi, Phys. Rev. B 59, 8084 (1999). 83, 075303 (2011).
6 A. Y. Kitaev, Ann. Phys. 303, 2 (2003). 12 C.NayakandF.Wilczek,Nucl.Phys.B.479,529(1996).
7 S. Das Sarma, M. Freedman, and C. Nayak, Phys. Rev. 13 M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).
Lett. 94, 166802 (2005). 14 F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984).