Table Of ContentTopological Quantum Liquids with Quaternion Non-Abelian Statistics
Cenke Xu1 and Andreas W. W. Ludwig1
1Department of Physics, University of California, Santa Barbara, CA 93106
(Dated: January 17, 2012)
Noncollinearmagneticorderistypicallycharacterizedbya“tetrad”groundstatemanifold(GSM)
of three perpendicular vectors or nematic-directors. We study three types of tetrad orders in two
spatial dimensions, whose GSMs are SO(3)= S3/Z2, S3/Z4, and S3/Q8, respectively. Q8 denotes
the non-Abelian quaternion group with eight elements. We demonstrate that after quantum dis-
ordering these three types of tetrad orders, the systems enter fully gapped liquid phases described
2 by Z2, Z4, and non-Abelian quaternion gauge field theories, respectively. The latter case realizes
1 Kitaev’s non-Abelian toric code in terms of a rather simple spin-1 SU(2) quantum magnet. This
0 non-Abeliantopologicalphasepossessesa22-foldgroundstatedegeneracyonthetorusarisingfrom
2 the22 representations of theDrinfeld double of Q8.
n
PACSnumbers:
a
J
4
The search for quantum liquid states has been one of
1
the main goals of condensed matter theory for decades.
] There are in general two different routes towards this
l
e goal,startingfromtwooppositelimits. Thefirstrouteis
- to startwith the quantum limit, say the large-N limit of
r
t the SU(N) antiferromagnet, and approach the physical
s
. system through an 1/N expansion. For instance, the
t
a 1/N expansionwithintheslavefermionformalismforthe FIG. 1: Three types of tetrad spin order. Type A phase has
m SU(N) antiferromagnet leads to the valence bond solid ground state manifold SO(3), which is equivalent to thecon-
- state with no classical counterpart [1]. In our current figurationsofthreeperpendicularvectors. TypeB phasehas
d work,wewilltakeasecondroutetowardstheliquidstate, GSMS3/Z4,twoofthethreeperpendicularvectorsarehead-
n less directors. In type C phase all three vectors are headless
which is by quantum disordering the semiclassical state.
o directors.
c Forinstance,itisunderstoodthatthevalencebondsolid
[ state (VBS) naturally emerges if quantum fluctuations
destroythesemiclassicalN´eelorderofaspin-1/2system. in turn three tetrad states, Type A, B, and C.
3
v ThisresultisbasedontheobservationthattheSkyrmion – Type A, with G = Z2: Let us first take G = Z2, and
1 of the N´eel order of a spin-1/2 antiferromagnet always thus the GSM is now SU(2)/Z2 = SO(3). SO(3) is pre-
7 carries lattice momentum [2, 3]. cisely the tetrad manifold, which corresponds to all the
6
In general, the ground state manifold (GSM) of spin configurations of three perpendicular vectors (Fig. 1A).
5
states can be written as One example of this case is the well understood non-
.
2 collinear spin density wave (SDW), for which the three
1 GSM=SU(2)/ G, (1) perpendicular vectors N~ , N~ and N~ that characterize
0 1 2 3
1 theGSMaredefinedasS~(~r)=N~2cos(2Q~·~r)+N~3sin(2Q~·
where G represents the unbroken subgroup of the SU(2)
: ~r); N~ = N~ ×N~ . Here Q~ is the spiral wave vector of
v spin symmetry in the ordered phase. G is at least Z 1 2 3
2 the SDW. It was pointed out in Ref. [6] that if quantum
i
X for spin-1/2 systems, because physical order parameters
fluctuations destroy the noncollinear spin density wave
should be invariant under spin rotation by 2π. In the
r (SDW), one interesting possibility is that the systemen-
a present paper, we will discuss the quantum disordered
tersaZ liquidstate. OnthetorusthisZ liquidground
phasesadjacenttosemiclassicalspinstateswhoseunbro- 2 2
state has a four-fold topological degeneracy [7]. Let us
ken symmetry G is a discrete subgroup of SU(2), either
briefly review why this is the case. The most convenient
Abelian or non-Abelian. All these states are “tetrad-
wayof parametrizingthe manifold SO(3)is by introduc-
like” states i.e. the GSM can be represented by three
ing CP1 spinor fields z =(z ,z )t as follows:
perpendicular vectors or nematic-directors (Fig. 1). We 1 2
will demonstrate that an exotic non-Abelian topologi- N~ ∼z†~σz, N~ ∼Re[ztiσx~σz], N~ ∼Im[ztiσy~σz]. (2)
1 2 3
cal liquid state can emerge after disordering a tetrad ne-
maticorderofafairlysimplespin-1system. Non-Abelian It is straightforwardto show that the vectors N~ are au-
a
statistics is a much sought-out phenomenon much dis- tomaticallyperpendiculartoeachotherafterintroducing
cussed in particular in fractional quantum Hall systems the spinor z . Since all the physical vectors N~ are bi-
α a
[4], and more recently also in certain topological insula- linears of z , the spinor z is effectively coupled to a Z
α α 2
tors (superconductors) [5]. In the sequel, we will discuss gauge field, which makes z equivalent to −z .
α α
2
Since the homotopy group π [SO(3)] = Z , the GSM
1 2
SO(3) supports vortex like topological defects with a Z
2
conservationlaw. This type oftopologicaldefectis often
called a vison. Pictorially, the vison can be viewed as
a configuration in which (for instance) N~ is uniform in
1
space,whileN~ andN~ haveavortex(Fig.2A). Afterwe
2 3
destroythe orderedstatewithquantumfluctuations,the
spinor z is gapped, but the Z conservation law of the
α 2
visonstillpersists. Thisimpliesthatthedisorderedphase
of the noncollinear SDW is equivalent to the deconfined
phase of Z gauge theory, where visons also have a Z
2 2
conservation law [6, 7]. In this phase the gapped spinor
z andthe visonhavemutualsemionicstatistics,i.e.the
α
wave function picks up a minus sign when z encircles
α
the vison adiabatically.
AswasdiscussedinRef.[6],thetransitionbetweenthe
ordered phase with GSM SO(3) and the Z2 deconfined FIG. 2: The configuration of vison defect in tetrad order A
liquidphaseiscontiuousandbelongstothe3DO(4)uni- and half-vison defect in tetrad orderB.
versality class. This is because the bosonic spinor field
z can also be viewed as a four component real vector,
α ThegaugefieldalwaysactsonZ byrightmultiplication.
whose order-disorder phase transition belongs to the 3D
Under the gauge transformation
O(4) universality class. The gapped Z gauge field does
2
notintroducesingularcorrectionsintheinfrared,i.e.the Z →Z(±iσz), (5)
3D O(4) universality class is unaffected by the the pres-
enceoftheZ gaugefield[6]. Becausethe physicalorder both N~ and N~ reverse direction, while N~ remains in-
2 2 3 1
parameters are bilinears of the spinor z , they acquire variant. Therefore the type B tetrad phase can be un-
α
a relatively large anomalous dimension as compared to derstoodasthecondensateoftheSU(2)rotorfieldZ (or
thestandardorderparametersattheWilson-Fisherfixed spinor z ) when it is coupled to the Z gauge field with
α 4
point of the O(4) Heisenberg model. Specifically, within gauge group Z from Eq. 4. Unlike the type A case, N~
4 2
a five-loopepsilonexpansionin d=4−ǫ dimensions the and N~ are no longer themselves physical order parame-
3
scaling dimension of these composite bilinears would be tersduetothepresenceoftheZ gaugefield;rather,the
4
estimated to be ηN~a ≈1.37 in (2+1) dimensions [8, 9] . physicalorderparameterQaib =NiaNib−31(N~i)2, i=2,3
– Type B, with G = Z4: Now let us move to the type- is of quadrupolar type.
B tetrad phase. The GSM can be characterized by Inadditionto the visondefectdiscussedin the type A
one vector and two directors, where again all three vec- phase, the type B phase also has a “half-vison” defect,
tor/directors are perpendicular to each other (Fig. 1B). i.e. the configuration in which N~ is uniform in space,
1
It is more convenientto describe this manifold using the while N~ and N~ have a half vortex (Fig. 2B). This de-
2 3
following slightly different representation of N~a: fecthasalogarithmicallydivergentinsteadofaconfining
energybecauseN~ andN~ arenematicdirectors. Wecan
N~1 ∼tr[Z†~σZσz],N~2 ∼tr[Z†~σZσx],N~3 ∼tr[Z†~σZσy], also describe this2half vor3tex as a Z gauge flux ±iσz in
4
the condensate ofZ. After encircling this flux, Z under-
Z =φ01+iφ1σx+iφ2σy +iφ3σz, goes a gauge transformationas in Eq. 5, and N~ and N~
2 3
reverse their directions.
z =(z , z )t =(φ +iφ , −φ +iφ )t. (3)
1 2 0 3 2 1 When the rotor field Z that couples to the Z gauge
4
field is gapped out, the system is described by a pure
Z is a SU(2) matrix, sometimes called the SU(2) slave
Z -gauge theory – a ‘Z -liquid’ phase. If the system in
rotor field[23]. Z has an action by SU(2) (left 4 4
left
this phase is definedonthe torus,thenthere canbe four
multiplication) and by SU(2) (right multiplication).
right
different fluxes through each cycle of the torus: 0, π/2,
While SU(2) transformations correspond to the phys-
left
π, 3π/2. Each of these different flux combinations cor-
ical SU(2) spin rotation symmetry, SU(2) transfor-
right
responds to an independent topological sector. There
mations contain the gauge symmetry as a subgroup.
NowletustakeN~ avector,whileN~ andN~ areboth is thus a 16 fold topological degeneracy on the torus.
1 2 3
headlessdirectors. InordertomakeN~ andN~ headless, Recently, one of the authors of the present paper pro-
2 3
posed that the type B phase is an intermediate phase
wecancoupleZ toagaugefieldtakingvaluesinagroup
of the Hubbard model on the honeycomb lattice [10],
with group elements:
sandwichedbetweenafullygappedspinliquidphaseand
Z ={1, iσz, −1, −iσz}. (4) a pure N´eel order with ground state manifold S2. In
4
3
through Schwingber bosons [13]. This GSM is equiva-
lent to that of the biaxial nematic order [25] of a liquid
crystal[15, 16]. Similar “triatic” nematic spin order was
also found in numerical work on SU(2) spin-1/2 mod-
els with both two-spin and four-spin interactions on the
triangular lattice [17].
It is still most convenient to describe this phase with
the SU(2) rotor variable Z, but now Z is coupled to a
discretenon-Abeliangaugefieldtakingvaluesinthenon-
Abelian Quaternion group Q ,
8
FIG.3: (a),thetopologicaldegeneracycanbecountedasthe
number of inequivalent commuting gauge fluxes Φ1 and Φ2 Q ={±1, ±iσx, ±iσy, ±iσz}. (7)
8
throughbothcyclesofthetorus. (b),Theringexchangeterms
Eq. 10 are ring product of gauge field on four links of each Again, the gauge field acts on the rotor field Z by right
square,andintheringexchangetermsthelinksareconnected
multiplication. As a consequence of the action of this
in thesequenceof thearrows around each plaquette.
gaugegroup,N~ inEq.3becomeheadlessnematicdirec-
i
tors.
Ref. [10], the vector N~ is the N´eel order, whereas the Now we will describe this gauge theory based on the
1
directors N~ and N~ are spin nematic orders. The tran- non-Abelian quaterion group Q8 in more detail. Follow-
2 3
ing the general construction in Ref. [7], we define an 8-
sition between type B tetrad order and the Z liquid
4
dimensional Hilbert space H on each link (i,µ) of the
phase also belongs to the 3D O(4) universality class, for
lattice, whose basis elements we denote by |g i. Here i
the same reason as in the type A case. i,µ
denotes a lattice site, µ = xˆ,yˆ a unit vector in a (posi-
– Type C, with G = Q : Now let us move on to the
8 tive) lattice direction, and g denotes any of the eight
type C phase, whose ground state is characterized by i,µ
elements of the group Q . Now we define, for any group
threeperpendicularnematicdirectors. Thisphasecanbe 8
element h ∈ Q , and on every link i,±µ of the lattice,
obtained from a system of spin-1 SU(2) quantum spins 8
operators Th and Qh with the following action on
Sˆa possessing both two-spin and four-spin interactions i,±µ i,±µ
i the basis vector |g i residing on that link:
[11, 12]. For a spin-1 system it is often convenient [13] i,µ
tpoarianltlreoldwuictehShU~bi(,3)thSechSwOi(n3g)erspbionsosnysm~bmi.etrWyhiesnbhr~bo∗kienis, Tih,+µ|gi,µi=δh,gi,µ|gi,µi, Tih+µ,−µ|gi,µi=δh−1,gi,µ|gi,µi,
while there is no spin polarization on any site, thus the Qh |g i=|hg i, Qh |g i=|g h−1i,
i,+µ i,µ i,µ i+µ,−µ i,µ i,µ
system only has nematic order. Since h~b∗i k h~bi, the
Schwinger boson~b can be rewritten as h~b i = eiθN~ . N~ (Th )† =Th =Th−1 ,
i i i i,µ i,µ i+µ,−µ
is in fact a nematic director, because the transformation
N~ →−N~ canbecancelledbythetransformationθ →θ+ (Qh )−1 =(Qh )† =Q(h−1). (8)
i,µ i,µ i,µ
π, which is part of the U(1) gauge symmetry associated
withtheSchwingerboson~b. ThevectorsN~ areprecisely Th and Qf turn out to satisfy the following algebra:
i i,µ i,µ
the nematic directors in Fig. 1C.
Experimentallyitwasobservedthatthetriangularlat- Qf Th Q(f−1) =Tfh,
i,µ i,µ i,µ i,µ
tice spin-1 material NiGa S has no global spin order
with time-reversal symmet2ry4breaking at low tempera- Qf Th Q(f−1) =Thf−1. (9)
i+µ,−µ i,µ i+µ,−µ i,µ
ture, but it still has gapless excitations with linear dis-
persion [14]. It has been proposed[11, 12] that the can- The dynamics of the discrete gauge field is given by the
didate ground state of this system is characterized by a following ‘ring exchange’ term [26]:
‘tetrad’ of three perpendicular nematic directors Na on
three different sublattices denoted by i = 1,2,3. iThis Hrqing =X−K Tih+i+xˆ,xˆyˆ,yˆTih+i+xˆ+xˆ+yˆyˆ,−,−xˆxˆTih+i+yˆ,yˆ−,−yˆyˆTih,xˆi,xˆ
proposed ground state thus has exactly the same GSM h
as that of Fig. 1C. The physical order parameter of this × tr[Ghi+xˆ,yˆGhi+xˆ+yˆ,−xˆGhi+yˆ,−yˆGhi,xˆ]+H.c. (10)
state is the quadrupolar spin order parameter[24]:
G is the two dimensional representation Eq. 7 of the
h
Qab ∼NaNb− 1(N~ )2 ∼hSˆaSˆb− 2i. (6) group element h∈Q8, and Ph denotes summation over
i i i 3 i i i 3 all group elements hi,µ ∈Q8 on each link.
The direction of µ in the ring exchange term on each
Here Sˆa is the spin-1 operator on sublattices i = 1,2,3. plaquette follows the arrows in Fig. 3. Here we always
i
This equation defines Na, and it is precisely the ne- assumeK >0,which favorsthe gaugeflux througheach
i
matic director Na in the previous paragraph introduced plaquette to be 1.
i
4
The SU(2) rotor field Z is defined on the vertices i of order, and we propose that it will drive the system into
i
thesquarelattice. Right-andleft-multiplicationofZ by the phase described by the pure quaternion nonabelian
SU(2) transformations SU(2) and SU(2) is gener- gauge theory. This gapped phase is a realization of the
right left
ated by the operators Ja and Ja, satisfying the commu- non-Abelian toric code phase built on a finite group G,
R L
tation relations (see also Ref. 18) proposedbyKitaev[7]. InthepresentcaseG=Q . Due
8
to the non-Abelian nature of the group Q , this gauge
8
[JRa,L,JRb,L] = iǫabcJRc,L, [JRa,JLb]=0. (11) theory is known to possess a rich set of gapped excita-
tions exhibiting non-Abelian statistics, which are char-
In particular, Ja and Ja act as follows:
L R acterizedby the representationsofthe so-calledDrinfeld
eiθ~·J~RZe−iθ~·J~R =Ze−i~θ2·~σ; eiθ~·J~LZe−iθ~·J~L =eiθ~2·~σZ(.12) double [19–21] of the group Q8. These excitations are
the following:
(i) magnetic excitations are located at the centers of
The quaternion gauge group is a subgroup of the
the plaquettes of the lattice (see Fig. 3), andarecharac-
SU(2) transformation.
right
terized by the product of group elements around a pla-
ThefullHamiltonianwithboth,rotorandgaugefields
quette. Since the product can be taken over different
reads
closed loops enclosing the same “magnetic flux”, a mag-
H = Hrot− X t tr[ZiTih,µi,µGhi,µZi†+µ]+Hrqing neticexcitationisnotcharacterizedbyagroupelementg
i,µ,hi,µ itself, but by its conjugacy class Cg = {h−1gh: h∈G}.
U U (ii)electricchargesarelocatedattheverticesofthelat-
Hrot = XX 2RJRa2,i+ 2LJLa2,i (13) tice (seefig. 3). Anelectric chargerepresentsaviolation
i a
of the vertex constraintof Eq. 14 and corresponds to an
This Hamiltonian is subject to the following quaternion irreducible representation α of the group G. Transport-
gauge group constraint: ing an electric chargeα arounda magnetic flux Cg along
aclosedpathyieldstherepresentationmatrixD(α)(g)of
eiπJRa,i =Qiiσ,+axˆQiiσ,−axˆQii,σ+ayˆQii,σ−ayˆ. (14) the group element g.
(iii) the most general excitation contains both, mag-
a=x,y,z. TheunitaryoperatorTih,µi,µGhi,µ appearingin netic and electric charges (often called a “dyon”), and
Eq. 13 is the analogue of the conventional term eiA~i,µ·~σ is represented by a pair (Cg,a) as follows: when there
where A~i,µ is the gauge potential. The quaterniongroup is no magnetic charge, Cg = Cg=1, then a = α is an
gauge constraint Eq. 14 generates the following gauge electric charge, i.e. a representation of the group G.
transformations on both Z and Thi,µ: However when the magnetic charge associated with a
i,µ
“dyon” is not vanishing, i.e. when C 6= C , its elec-
g g=1
Thi,µ →Tfihi,µfi−+1µ, Z →Z G , (15) tric charge a is an irreducible representation a = αˆ of
i,µ i,µ i i fi the Normalizer N(g) = {h ∈ G : hg = gh} of g (con-
sisting of all those group elements commuting with g),
where f ∈ Q . The Hamiltonian Eq. 13 is invariant
i 8
which is in general not the entire group G, but only
under this gauge transformation. We have formulated
a subgroup thereof. – Let us count the total number
this model on the square lattice, but generalizations to
of excitations for the Drinfeld double of the quarternion
other lattices are straightforward. Again, the quantum
group Q . We use the following facts: there are 5 con-
phase transition between the ordered phase and quater- 8
jugacy classes {+1},{−1},{±iσa} where a=x,y,z; the
nion liquid phase belongs to the 3D O(4) universality
number of irreducible representationsof any finite group
class because the Q gauge field is always gapped.
8
equalsthenumberofconjugacyclasses;thecentralizerof
When U ,U ≫ t, the SU(2) rotor field Z is gapped
L R i
any ofthe three conjugacy classes{±iσa} is the Abelian
out, and the system is described by the pure quaternion
cyclicgroupZ offourelementsgeneratedbyiσa. Thus,
group gauge theory Eq. 10, plus the gauge constraints. 4
there are 5 excitations of the form (C ,α), 5 of the form
InthespinHamiltonian,therotorfieldZ canbegapped 1
i
byturningonthefollowingtermonthespinHamiltonian (C−1,α), and 4 of the form (Ciσa,αˆ), for eacha=x,y,z,
where αˆ labels the representations of Z . This amounts
considered in Ref. [11, 12]: 4
toatotalof5+5+3×4=22excitations. Sinceforagen-
H′ = X J′Qˆi·Qˆj, J′ >0. (16) eral 2D topological field theory the ground state degen-
eracyonthetorusequalsthe numberoftopologicalexci-
≪i,j≫
tations(“particles”),this degeneracyis 22inthe present
whereQˆ isthefive-componentquadrupoleorderparam- case. The fusion rules for these 22 particles can be ob-
i
eter introduced in Ref. [11–13]. Eq. 16 is an antiferro- tained fromthe modular S-matrix (throughthe Verlinde
quadrupole interaction between the 2nd neighbor sites Formula)which,inturn,canbeobtainedinthestandard
on the triangular lattice. This term energetically disfa- manner[20] from the Drinfeld double construction.
vorsthesystemtoformathreesublatticetetradnematic In general, for topological phases which are Drinfeld
5
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8
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