Topological Paramagnetism in Frustrated Spin-One Mott Insulators

1.6 MB·English

by  Chong Wang

Topological Paramagnetism in Frustrated Spin-One Mott Insulators
Chong Wang, Adam Nahum, and T. Senthil
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
(Dated: January 7, 2015)
Timereversalprotectedthreedimensional(3D)topologicalparamagnetsaremagneticanalogsof
the celebrated 3D topological insulators. Such paramagnets have a bulk gap, no exotic bulk exci-
tations, but non-trivial surface states protected by symmetry. We propose that frustrated spin-1
quantummagnetsareanaturalsettingforrealisingsuchstatesin3D.Wedescribeaphysicalpicture
ofthegroundstatewavefunctionforsuchaspin-1topologicalparamagnetintermsofloopsoffluc-
tuatingHaldanechainswithnon-triviallinkingphases. Weillustratesomeaspectsofsuchloopgases
with simple exactly solvable models. We also show how 3D topological paramagnets can be very
naturally accessed within a slave particle description of a spin-1 magnet. Specifically we construct
5 slave particle mean field states which are naturally driven into the topological paramagnet upon
1
including fluctuations. We propose bulk projected wave functions for the topological paramagnet
0
based on this slave particle description. An alternate slave particle construction leads to a stable
2
U(1)quantumspinliquidfromwhichatopologicalparamagnetmaybeaccessedbycondensingthe

n emergent magnetic monopole excitation of the spin liquid.
a
J

6 Frustrated quantum magnets display a rich variety of In1DthefamiliarHaldane/AKLTchainistheonlytime
many–body phenomena. Some such magnets show long– reversal protected topological paramagnet while in 2D

] range magnetic order at low temperature, often selected there are no time reversal protected topological param-
l
e out of a manifold of degenerate classical ground states agnets. In3Dhowevertherearethreedistinctnon-trivial
-
by quantum fluctuations. A very interesting alternative phases [4, 10, 11] (corresponding to a classification by
r
t possibility — known as quantum paramagnetism — is the group Z2). Second, regarded as an electronic insu-
s 2
. the avoidance of such ordering even at zero tempera- lator, unlike the 1D Haldane chain [12], these 3D topo-
t
a ture. Quantum paramagnets may be of various types. logical paramagnets survive as distinct interacting SPT
m A fascinating and intensely–studied class is the quan- insulators [13]. The properties and experimental finger-
- tum spin liquids: these display many novel phenomena, prints of such topological paramagnets were described in
d
for instance fractionalization of quantum numbers and Refs. [4, 10, 11, 13]. However there is currently very lit-
n
topological order, or gapless excitations that are robust tleunderstandingofwheresuchphasesmightactuallybe
o
despite the absence of broken symmetries [1–3]. found. In this paper we propose that frustrated spin-1
c
[ Recently there has been much progress in understand- Mottinsulatorsmaybegoodplacestolookforanexam-

1 ing a different type of remarkable quantum paramagnet. ple of such phases.
v Thesearephaseswhichhaveabulkgapandnofractional Alreadyinthefamiliar1Dexampleitisthespin-1anti-
7 quantum numbers or topological order. Despite this, ferromagneticchain,ratherthanthespin-1/2chain,that
4 they have nontrivial surface states that are protected naturally becomes a topological paramagnet. In 3D for
0
by global symmetries. These properties are reminiscent one of the topological paramagnets we provide a phys-
1
of the celebrated electronic topological band insulators. ical picture and a parton construction which are both
0
Hencetheyhavebeencalledtopologicalparamagnets[4]. very natural for the spin-1 case. We hope that our ob-
.
1 Topologicalparamagnetsandtopologicalbandinsulators servations inspire experimental and numerical studies of
0
are both examples of what are known as Symmetry Pro- frustrated spin-1 quantum magnetism in the future. To-
5
tectedTopological(SPT)phases[5–7]. Aclassicexample wards the end of the paper we remark on materials that
1
: ofatopologicalparamagnetistheHaldane/AKLTspin-1 may form such interesting frustrated magnets.
v
chain: though this has a bulk gap and no bulk fraction-
The three 3D topological paramagnets that are pro-
i
X alization, it has dangling spin-1/2 moments at the edge tected by time reversal symmetry alone [4, 10, 11] all
which are protected by symmetry, for instance time re-
r allowforagappedsurfacewithZ topologicalorder(i.e.
2
a versal. In the last few years tremendous progress has
a gapped surface Z quantum spin liquid) even though
2
been made in understanding such SPT phases and their
the bulk itself is not topologically ordered. The prop-
physicalpropertiesindiversedimensions(forreviews,see
erties of this surface theory give a useful way to label
Refs. 8 and 9).
the bulk phases. The surface has gapped quasiparticle
The main focus of the present paper is on three- excitations — labelled ‘e’ and ‘m’ — which are mutual
dimensional topological paramagnets that are protected semions. These may be thought of as the electric charge
bytimereversal(wealsobrieflydiscusstopologicalpara- and magnetic flux of a deconfined Z gauge theory (like
2
magnets protected by other symmetries, notably conser- the vertex and plaquette defects of Kitaev’s toric code
vation of at least one spin component). These are inter- [14]). AttheSPTsurfacestheseparticleshaveproperties
esting for a number of reasons. First, time reversal is a — self-statistics or time reversal transformation proper-
robust symmetry of typical physical spin Hamiltonians. ties — that are impossible in a strictly 2D system, and
2
whichencodethetopologyofthebulkwavefunction. The spinliquidwithnon-trivialimplementationoftimerever-
three nontrivial bulk states are denoted: sal symmetry. Interestingly simply condensing the mag-
netic monopole of this U(1) spin liquid leads to an SPT
eTmT, efTmfT, efmf. state dubbed eCTmT in the presence of both spin rota-
tion and time reversal symmetries. If only time reversal
In the first and second, the surface e and m excitations
is present this becomes the eTmT state.
are each Kramers doublets under time reversal, denoted
by T. In the second and third they are fermions (f),
while in the first they are bosons. This paper focuses
I. LOOP GAS STATES
primarily on the ‘eTmT’ state.
We begin by explaining a physical picture of a suit-
In this section we describe a loop gas wavefunction
able ground state wave function for the eTmT topologi-
that is naturally adapted to spin one magnets and gives
cal paramagnet. This is most easily visualized on a dia-
an intuitive picture for the eTmT state. The wavefunc-
mond lattice. We first close–pack each interpenetrating
tion is a superposition of loop configurations, with each
fcc sublattice of the diamond lattice with closed loops.
loop representing an AKLT state [17] for the spins ly-
Oneachloopweplaceallthespin-1moments(locatedat
ing on it. A given configuration enters the superposition
the diamond sites) in the ground state of the 1D AKLT
with a sign factor determined by its topology: specifi-
chain. We then superpose all such loop configurations
cally, the loops come in two species, A and B (one as-
with a crucial (−1) sign factor whenever loops from the
sociated with each sublattice of the bipartite diamond
twodifferentfccsublatticeslink. Wearguethatthiscon-
lattice)andthesigndependsonthelinkingnumberofA
struction yields the topological paramagnet.
loops with B loops. This geometrical picture makes the
To understand the topological properties of such a
relationship between the bulk wavefunction and the sur-
wave function we describe a simple exactly solvable loop
face excitations particularly simple. The surface e and
gas Hamiltonian [15] — equivalent to two coupled Ising
gaugetheories—thatclarifiestheroleofthe‘(−1)linking’ m excitations are endpoints of the two species of AKLT
chains, and are Kramers doublets since an AKLT chain
sign structure. In this solvable model the loops do not
has dangling spin-1/2s at its ends.
have AKLT cores but there are two species of loops on
In Sec. II we describe a similar wavefunction for ‘pure
different sublattices with the mutual (−1) linking sign.
loops’, i.e. loops that do not carry an internal AKLT
Itdemonstratesverysimplyhowthissignleadstoastate
structure. Thismayberegardedasastateoftwocoupled
without intrinsic topological order. (This loop gas is not
Ising gauge theories. It is not in the eTmT phase, but it
in the eTmT state, because of the absence of AKLT
illustratesthebasicfeaturesoftheloopgasesinasimple
cores,butweshowittobenontrivialinadifferentsense.)
model with an exactly solvable Hamiltonian. This ‘pure
Next we use the two–orbital fermionic parton repre-
loop’modelisalsointerestinginitsownright: whenopen
sentation developed for spin-1 magnets [16] to construct
strands (as opposed to closed loops) are banished from
possiblegroundstates. Whenthefermionicpartonshave
the Hilbert space, i.e. when charge is absent, it is in a
the mean–field dispersion of a certain topological super-
nontrivialphasedespitetheabsenceoftopologicalorder.
conductor, we show that the gauge fluctuations associ-
Therefore it may be viewed as a ‘constraint–protected’
ated with the parton description convert the system into
state. It would be interesting to relate this to the recent
a topological paramagnet. In this construction the mean
ideas of Ref. [18]. We note that the models discussed
field state is unstable toward confinement by gauge fluc-
in Ref. [19] are also believed to be separated from the
tuations, as a result of a continuous nonabelian gauge
trivial phase by a phase transition, despite the absence
symmetry. Despite this the bulk gap survives, leaving
of topological order.
behind a non-trivial surface that we are able to iden-
tify as that of the eTmT topological paramagnet. As a Thewavefunctionsdiscussedhereareinasimilarspirit
warm up exercise to illustrate some of the ideas of this to the Walker Wang models, which are formulated in
3D construction, we also describe how to access the 1D terms of string nets with a nontrivial sign structure,
Haldane phase by confining a topological superconduc- and show bulk confinement and surface topological or-
tor of parton fermions. The 3D construction naturally der [11, 20, 21]. Constructions of SPTs using Walker
suggests alternative bulk wave functions for topological WangmodelsweregiveninRefs.[11,22]. 2D‘symmetry-
paramagnets, in the form of Gutzwiller–projected topo- enriched’ topological states [23–25] and SPT states [26]
logical superconductors. This may be fruitful for future havealsobeenconstructedbyattachingAKLTchainsto
numerical work on the energetics of microscopic models. loop-like degrees of freedom (see also [27]).
ThispartonconstructionalsogivesaccesstootherSPT
statesforquantummagnetsin3D.Forinstanceweshow
how to naturally obtain an SPT paramagnet (dubbed A. Fluctuating AKLT chains
eCmT in Ref. 10) protected by U(1)×ZT, where the
2
U(1) describes rotation about one spin axis, say S , and The diamond lattice is made up of two fcc sublattices,
z
ZT is time reversal. A and B. If C is a configuration of fully packed loops
2 A
Finally we show how to access a bulk U(1) quantum on A (with every A site visited by exactly one loop), we
3
define |C (cid:105) to be a product of AKLT states |L(cid:105) for each
A
of the loops L in C ,
A
(cid:89)
|C (cid:105)= |L(cid:105). (1)
A
L∈CA
Similarly|C (cid:105)isthestatecorrespondingtoaloopconfig-
B FIG.2. Forappropriateboundaryconditions,endpointsofA
urationC onB. TodefinetheAKLTstates|L(cid:105)fullywe
B and B chains (red and blue respectively) give surface excita-
must choose an orientation for the fcc links, as discussed tionswithmutualsemionicstatistics. Braidingtheanyonson
below (Sec. IB). thesurface(firstarrow)changesthesignofthewavefunction,
LetX(C ,C )bethemutuallinkingnumberofthetwo for consistency with the rule that configurations related by
A B
speciesofloops. Sincetheloopsareunoriented,thisisde- passing an A strand through a B strand in the bulk (second
fined modulo two: X(C ,C )=0,1. A schematic wave- arrow) appear in the wavefunction with opposite sign.
A B
function for the eTmT phase may be written in terms of
X(C ,C ):
A B
In contrast, |Φ(cid:105) is not expected to show topological
(cid:88)
|Φ(cid:105)= (−1)X(CA,CB)|CA(cid:105)|CB(cid:105). (2) order, despite the proliferation of long loops in Eq. 2.
CA,CA Instead it describes a phase in which the endpoints of
open chains are confined in the bulk. Furthermore there
For concreteness, we take periodic boundary conditions.
is no ground state degeneracy: states with odd winding
ThesumsoverC andC aretheneachrestrictedtoloop
A B numbers are not ground states (i.e. are not locally indis-
configurations with an even number of strands winding
tinguishable from |Φ(cid:105)).
around the 3D torus in each direction, for reasons dis-
Moredetaileddiscussionofthisisdeferredforthesolv-
cussed below. This global constraint, together with the
able model of Sec. II, but the basic idea is the following.
geometrical fact that the links of A never intersect those
While the amplitude (−1)X(CA,CB) depends on the global
of B, ensures that X(C ,C ) is well defined.
A B topology of the loop configurations, it amounts to the
TheentanglementbetweenthetwosublatticesinEq.2
simple local rule that the amplitude changes sign if an A
isentirelyduetothesignfactor. Firstconsiderwhathap-
strandispassedthroughaB strand. Itisusefultoimag-
pens in the absence of this sign factor. Each sublattice
ine a hypothetical parent Hamiltonian that imposes this
then hosts a superposition of loop configurations with
(cid:80) sign rule. But the sign rule cannot be consistently im-
positive amplitude, e.g. |C (cid:105). By analogy with the
CA A posedifthewavefunctionincludesopenstrandsorconfig-
usual picture of deconfined Z gauge theory as a super-
2 urations with odd winding numbers (see below). Similar
positionofelectricfluxloopconfigurations[28],wewould
phenomena occur in the confined Walker–Wang models
expect such a state to show Z topological order. (It is
2 [11, 20, 21].
a 3D version of the ‘resonating AKLT’ states studied in
However,openendpointsaredeconfinedatthebound-
2D [23–25].) The endpoint of an open AKLT chain is
ary, for appropriate boundary conditions. The minus
the deconfined Z charge in this state. Associated with
2
sign associated with passing an A strand through a B
the topological order is ground state degeneracy — dif-
strand in the bulk means that the endpoints are mutual
ferent ground states are distinguished by the parity of
semions [29] — see Fig. 2. They are also Kramers dou-
the winding number in each spatial direction.
blets. These surface properties are the defining features
oftheeTmT state. Thewavefunction|Φ(cid:105)hasmoresym-
metrythansimplytimereversal(e.g. separatespinrota-
tion symmetries for each sublattice) but if it is indeed in
the eTmT phase then these symmetries could be weakly
broken without leaving the phase.
B. Further details on fluctuating AKLT state
To write the AKLT-based state explicitly it is conve-
nient to represent the spin-one at each site i in terms of
auxiliary spin-1/2 bosons [17, 27]. If the boson creation
operators are b† (α =↑,↓), then S(cid:126) = 1b† (cid:126)σ b . The
iα i 2 iα αβ iβ
occupation number b† b is equal to two to ensure spin
αi iα
one at each site. The AKLT state |L(cid:105) is then created
FIG. 1. Two species of AKLT loops, one on each sublattice by acting on the boson vacuum with operators S† that
ij
of the diamond lattice. The loops live on the links of the fcc create singlet pairs on the links of the loop, which we
sublattices,i.e. onnext-nearest-neighbourbondsofdiamond. normalize as S† = √1 (b† b† − b† b† ). This operator
ij 3 i↑ j↓ i↓ j↑
4
link, and an up spin as an unoccupied one. The num-
ber of occupied links at each vertex is always even in the
stateweconsider,sotheconfigurationsofoccupiedlinks,
C and C , can be decomposed into closed loops.1 We
A B
refer to C and C as loop configurations. Other solv-
A B
able loop gas/string net models have been considered in
Refs. [11, 21], using the Walker Wang construction [20].
The ‘pure loop’ state analogous to |Φ(cid:105) above is (again
we sum only over loop configurations with even winding
numbers on each sublattice):
FIG. 3. Left: Loops on interpenetrating cubic lattices A and
(cid:88)
B. The state |Ψ(cid:105) is a superposition of such configurations |Ψ(cid:105)= (−1)X(CA,CB)|C (cid:105)|C (cid:105). (4)
A B
with signs determined by linking of A and B loops. Right:
the product of Pauli matrices defining the flip term F on a CA,CB
plaquette (see Eq. 6). We may view C and C as the electric flux line config-
A B
urations for a pair of coupled Z gauge fields, with one
2
Z gauge field living on each cubic lattice. Imposing the
2
is antisymmetric in (i,j), so to define |Φ(cid:105) we must fix
above sign structure for the two sets of electric flux lines
an orientation for the links of each fcc sublattice. (The
isequivalenttobindingtheelectricfluxlineofeachgauge
fcc lattice has four sublattices, a, b, c, d, so for example
fieldtothemagneticfluxlineoftheother,aswillbeclear
we could orient the links from a → b, a → c, a → d,
shortly.
b→c→d→b, with the orientations on each sublattice
It is straightforward to write down a gapped parent
relatedbyinversionsymmetry.) Thenforeachsublattice
Hamiltonian H for |Ψ(cid:105), using the fact that flipping
linking
(cid:89) the occupancy of all the links on the plaquette changes
|C(cid:105)= S† |vac(cid:105), (3)
ij thelinkingnumberX(C ,C )ifandonlyifthelinkpierc-
A B
(cid:104)ij(cid:105)∈C ing the plaquette is occupied. H is a sum of terms
linking
for the plaquettes p of each cubic lattice:
where i is the site at the tail of the oriented link (cid:104)ij(cid:105).
These states satisfy (cid:104)C|C(cid:105) = (cid:81) (1 + (−1)(cid:96)/3(cid:96)−1),  
loops
where (cid:96) is the length of a given loop [17]. (cid:88) (cid:88)
Hlinking =−J FAp+J FBp. (5)
It should be noted that that expectation values in the
p∈A p∈B
state |Φ(cid:105) are nontrivial, in particular because overlaps
(cid:104)C|C(cid:48)(cid:105)fordistinctC,C(cid:48) arenonzero. Sowhileitisplausi-
TheoperatorsF andF fliptheoccupancyofthelinks
A B
ble that |Φ(cid:105) is in the eTmT phase, this cannot be estab-
on a plaquette, with a sign that depends on whether the
lishedpurelyanalytically. Forexample,thestatecouldin
link piercing it is occupied. Allowing p to denote both a
principlebreakspatialorspinrotationsymmetrysponta-
plaquetteandthelinkpiercingit,anddenotingthePauli
neously. A cautionary example is given by the uniform-
operators on A and B by (cid:126)σ and (cid:126)τ respectively,
amplituderesonatingvalencebondstateforspin-1/2son
the cubic lattice: this has weak N´eel order [30], despite F =τz (cid:89)σx, F =σz (cid:89)τx. (6)
being a superposition of singlet configurations which in- Ap p l Bp p l
l∈p l∈p
dividually have trivial spin correlations. In the present
model, the entanglement between sublattices supresses Theseoperatorsallcommute,sotheHamiltonianistriv-
off-diagonalelementsofthereduceddensitymatrixwhen ially solvable. |Ψ(cid:105) is the unique ground state and min-
written in the AKLT-chain basis [31]. Together with the imises each term of H since F|Ψ(cid:105) = |Ψ(cid:105) for each
linking
non-bipartitenessofthefcclattice,thismakesspinorder plaquette operator.
seemlesslikely. Butsince|Φ(cid:105)isintendedtoillustratethe Thestate|Ψ(cid:105)containsonlyclosedloops,i.e. itsatisfies
topological structure of the phase, and not as a ground
state of a realistic Hamiltonian, it may not be crucial (cid:89)σz =1 for v ∈A, (cid:89)τz =1 for v ∈B (7)
l l
whether it is in the desired phase as written or whether
l∈v l∈v
further tuning of the amplitudes is required.
where v denotes a vertex and l ∈v the links touching v.
AnystatesatisfyingF|Ψ(cid:105)=|Ψ(cid:105)foralltheplaquetteop-
II. ‘PURE LOOP’ STATE eratorsmustalsosatisfythesevertexconditions,because
(cid:81) σz and (cid:81) τz can be written as products of Fs.
l∈v l l∈v l
Itisenlighteningtolookatthesimplestmodel[15]that
captures the (−1)linking sign structure. To this end we
take a system of spin-1/2s on the links of two interpen-
etrating cubic lattices A and B, as shown in Fig. 3. We 1 With a harmless ambiguity when the number of occupied links
think of a down spin (in the ‘z’ basis) as an occupied atavertexexceedstwo.
5
the corresponding state with τx =−1 on the occupied
plaquettes. Let ∂M be the loop configuration given by
the boundaries of the membranes in M. Then |Ψ(cid:105) can
be written (neglecting an overall constant)
(cid:88) (cid:88)
|Ψ(cid:105)= |C (cid:105)|M(cid:105). (8)
A
CA M
∂M=CA
Fig. 4 shows the geometrical interpretation of this state.
Itisasoupofτx membranes,withσz loopsgluedtotheir
boundaries.
FIG. 4. After a basis change, |Ψ(cid:105) is a superposition of mem- Confinement of string endpoints is easy to see in this
braneconfigurations(τx =−1onshadedplaquettes)withred basis. Apairofvertexexcitationsatwhich(cid:81) σz =−1
l∈v l
loops (where σz =−1) glued to membrane boundaries. (The are connected by an open string. Since the boundary of
red loops are σ–electric lines and the membrane boundaries M contains only closed loops, the open string makes it
are τ–magnetic lines.)
impossible to satisfy the gluing of strings to membrane
boundariesdemandedbytheF termsinH . Ifthe
B linking
separation of the vertex defects is D, there must be at
We may regard Eqs. 7 as the gauge constraints for a least D unsatisfied links, giving a linear confining poten-
pairofpureZ2 gaugetheories(theZ2 versionsof∇(cid:126).E(cid:126) = tial for such defects. For similar reasons, a configuration
0). The two electric fields are given by σz and τz and withanoddnumberofwindingσz strandsinsomedirec-
live on the links of A and B respectively. The magnetic tioncostsan energyproportionalto thespatial extent of
field of each gauge field lives on the links of the opposite the system in this direction. By symmetry, this applies
latticetoitselectricfield. Forexamplethemagneticfield equally to the τz strings that are present in the original
of σ is given by (cid:81) σx, where p is a plaquette of A, or basis.
l∈p
equivalently a link of B. Wecanalsounderstandtheconfinementofstringend-
In this language, H simply glues the electric flux points algebraically (Refs. [11, 21] give analogous argu-
linking
lineofeachspeciestothemagneticfluxlineoftheother. mentsforbulkconfinementandsurfacetopologicalorder
The σ–magnetic flux and the τ–electric flux are equal in the Walker Wang models). The Hamiltonian in Eq. 5
since F = 1, and the σ–electric and τ–magnetic fluxes is clearly exactly soluble not just for the ground state
A
are equal via F =1. but for all excited states. An ‘elementary’ excitation is
B
The state |Ψ(cid:105) is not topologically ordered. Neither is given by a ‘defect’ in some square plaquette, say on the
it a time-reversal protected SPT: it can be adiabatically B lattice, with
transformed to a product state without breaking time
F =−1 (9)
reversal symmetry. However it is protected if impose Bp
Eqs. 7 as constraints: i.e. if we forbid open strands, as
whileF =+1onallotherplaquettesofeithersublattice.
opposed to closed loops. In the gauge theory language,
Such a defect plaquette costs energy 2J. It leads to a
this means forbidding charge. With this constraint it is violation of the closed loop vertex constraint for σz on
impossible to reach a trivial state without going through
the two vertices of the A sublattice connected by the
aphasetransition, asfollowsfromtheself–dualityofthe
A-link that penetrates the defect plaquette. Thus the
state described in Sec. IIA.
excitation we have created has two string end-points on
We will explain these features from several points of nearestneighborA-sites. Tomovethesestringendpoints
view below. One convenient approach which leads to a apart by a distance D we must create o(D) such defect
geometricpictureistoswitchfromthe(σz,τz)basisused plaquettes. Consequently the energy cost is also o(D)
in Eq. 4 to the (σz,τx) basis. The σz configuration is a and we have linear confinement of string endpoints.
loop configuration on the A lattice, as above. We rep- In the gauge theory language, the reason for the ab-
resent the τx configuration by a configuration of mem- sence of deconfined excitations is that the tensionless
branes made up of plaquettes on the A lattice: τpx =−1 lines in this state are not lines of pure electric flux, but
representsanoccupiedplaquette,andτpx =1anunoccu- rather of electric flux together with magnetic flux of the
pied one. other species. If such lines could end, their endpoints
The FB terms in Hlinking act on a link of the A lattice would be deconfined excitations. But the Hilbert space
together with the four plaquettes touching it. FB = 1 does not allow for such excitations: a magnetic flux line
imposestherulethattheσz loopsaregluedtothebound- cannot terminate in the bulk (by virtue of its definition
ariesoftheτx membranes,i.e. tothelinkswhereanodd in terms of e.g. (cid:81)τx).
number of occupied plaquettes meet. This is the gluing Despite the lack of deconfined endpoints in the bulk,
of σ–electric flux lines (where σz = −1) to τ–magnetic A and B strings that terminate on a boundary can give
flux lines (where (cid:81)τx =−1) mentioned above. deconfinedeandmparticlesinasurfaceZ topologically
2
Let M denote a membrane configuration, and |M(cid:105) orderedstate. Toseethis, weterminatethesystemasin
6
Fig. 5, including in the Hamiltonian the natural plaque-
tte and vertex terms at the surface. The surface string
operators that create pairs of e or pairs of m excitations
can then be written explicitly (see Fig. 5). They sat-
isfy the same algebra as the string operators in the toric
code [14], confirming that e and m are mutual semions
FIG. 6. Under the mapping (11), a σz (or τz) operator on a
as expected from the heuristic argument of Fig. 2.
link is exchanged with a product of τx (resp. σx) operators
We can adiabatically transform |Ψ(cid:105) to a product state
on the surrounding links of the other lattice. (Links of one
so long as we allow the intermediate states to violate the
lattice can equally be thought of as plaquettes of the other.)
closed–loop constraints on at least one sublattice. The
membrane picture gives an obvious way to do this, by
giving the membranes in M a surface tension. If ‘Area’ to itself, but exchanges the trivial state with a topologi-
denotes the number of occupied plaquettes in M, the cally ordered one. Thus there is no adiabatic path from
interpolating state is |Ψ(cid:105)tothetrivialstate. Iftherewere,dualitywouldyield
an adiabatic path from |Ψ(cid:105) to the topologically ordered
|Ψ(cid:105) =(cid:88) (cid:88) e−γ×Area|C (cid:105)|M(cid:105). (10) state,andthisisimpossiblesince|Ψ(cid:105)isnottopologically
γ A
ordered.
CA M
∂M=CA The duality transformation makes sense for states
obeying the closed loop constraint. (To be precise, we
When γ = 0 this is the initial state, and when γ → ∞
must also impose the global constraint that the loop
only the term with zero area survives. This is the state
configurations have even winding in each direction.) As
with no loops and no membranes, i.e. the product state
shown in Fig. 6, its action is:
|σz =1(cid:105)|τx =1(cid:105). To get a gapped parent Hamiltonian
(cid:89) (cid:89)
for|Ψ(cid:105) ,wemodifytheplaquettefliptermF inH σz ←→ τx, τz ←→ σx (11)
γ A linking l p p l
(cid:104) (cid:105)
toF =(coshγ)−1 τz(cid:81) σx+(sinhγ)τx . Thispre- p∈l l∈p
Ap p l∈p l p
Herep∈ldenotesthefourplaquettespsurroundinglink
serves the simple algebraic properties of the plaquette
l. Wehavelabelledtheσsbylforlinkandtheτsbypfor
terms.
plaquette, butthedualityactsonthetwosetsofdegrees
offreedomsymmetrically. Itpreservesthelocalityofany
Hamiltonian acting in the constrained Hilbert space.
A. Self-duality of |Ψ(cid:105) and protection by constraints
For completeness, we write the action of the duality
on states explicitly. Return to the picture of loops +
When the interpolating state above is rewritten in the membranes on the A lattice, i.e. the (σz,τx) basis. One
original (σz,τz) basis, it includes configurations with may check that any state satisfying the constraints can
open strands, as well as closed loops, on the B lattice. be written as a sum over two loop configurations on the
Whatifweimposetheconstraintthatbothlatticeshave same lattice,
onlyclosedloops? Inthiscaseitisimpossibletogofrom
(cid:88)
|Ψ(cid:105)toatrivialstatewithoutaphasetransition. (Wewill |f(cid:105)= f(CA,CA(cid:48) )|CA(cid:105)σ|(cid:103)CA(cid:48) (cid:105)τ, (12)
take the reference trivial state to be that with no loops, CA,CA(cid:48)
|trivial(cid:105)=|σz =1(cid:105)|τz =1(cid:105).)
This follows from a simple duality transformation where|(cid:103)CA(cid:48) (cid:105)τ isdefinedastheuniformsuperpositionofall
whichexchangestheelectricfluxofeachspecieswiththe membrane configurations |M(cid:105)τ with boundary ∂M =
magneticfluxoftheotherspecies. Thedualitymaps|Ψ(cid:105) CA(cid:48) . We have added subscripts to the kets as a reminder
of the degrees of freedom involved. (C is the σ–electric
A
flux configuration, and C(cid:48) the τ–magnetic flux config-
A
uration; the fact that the wavefunction depends on M
only through ∂M is simply a statement of gauge invari-
ance.) The duality then simply exchanges the two kinds
of loops,
f(C ,C(cid:48) )←→f(C(cid:48) ,C ). (13)
A A A A
FIG. 5. String operators creating surface excitations. Left: The flip operators FA and FB (Eq. 6) are clearly in-
acting with a chain of σx operators on the links of the upper variant under the duality in Eq. 11 and therefore so is
layer(Alatticesurface)givesapairofeexcitations(i.e. end- H . (We can also see that |Ψ(cid:105) is invariant from
linking
points of bulk A strings). Right: a pair of m excitations (i.e. Eq.13andEq.8.) Ontheotherhand, thetrivialHamil-
endpointsofB strings)arecreatedbyachainofτx operators tonian
(thick green strand) on the lower layer (B surface), together (cid:32) (cid:33)
with σz operators on the corresponding links in the upper H =− J(cid:88)σz+J(cid:88)τz (14)
layer (thick purple links). trivial l l
l∈A l∈B
7
is exchanged with However, the above argument is incomplete as it does
not rule out the possibility of getting to a trivial state
 
by proliferating nearby pairs of open strands on opposite
(cid:88)(cid:89) (cid:88)(cid:89)
Hdeconfined =−J σlx−J τlx, (15) sublattices. Suchapairgivestwospin–1/2swhichcanbe
p∈Al∈p p∈Bl∈p boundintoasinglettoavoidagaplessdegreeoffreedom.
Inthegaugetheory,suchpairscorrespondtoboundpairs
which describes a pair of deconfined Z2 gauge theories. ofelectriccharges,onefromeachZ2gaugefield. Thesta-
This establishes the claim at the beginning of this sub- bilityoftheeTmT statesuggeststhatthepureloopstate
section: while the linking state is invariant, the trivial remainsprotectedevenwhensuchdoublechargesareal-
state is exchanged with a topologically ordered state. It lowed. We note that at the surface these double charges
follows that the linking state is in a distinct phase from correspondtotheboundstateoftheeandmparticle(in
the trivial state if we do not allow open endpoints in the the surface topological order). This is a Kramers singlet
Hilbert space. (We know from Eq. 10 that they are in spin-0 fermion (conventionally denoted (cid:15)). The surface
the same phase if we do allow endpoints.) Fermistatisticssuggestsapotentialobstructionto‘trivi-
alizing’ the bulk by proliferating the double charges. We
leave an explicit demonstration of this for the future.
B. Heuristic relation between symmetry protection
of eTmT and closed–loop constraint
III. PARTON CONSTRUCTIONS
The proposed wave function for the eTmT phase
has the two loop species ‘stuffed’ with Haldane/AKLT
ThoughthedescriptionoftheeTmT topologicalpara-
chains. The linking sign factor ensures that the ground
magnetintermsofaloopgaswavefunctionisphysically
state is not topologically ordered as required for a topo-
appealing it is desirable to have alternate descriptions
logical paramagnet. In particular the open end-points of
which enhance our understanding and which may help
theloops—whichnowharboraKramersdoublet—are
with evaluating the energetic stability of this phase in
confined. However as described in Sec. IA the surface
microscopic models. To that end, in this section we pro-
implements time-reversal ‘anamolously’ exactly charac-
pose explicit parton constructions for some topological
teristic of the eTmT state.
paramagnets in spin-1 systems.
Wenowbrieflyconsiderwhethertheresultsinthepre-
Historically the parton approach has provided varia-
vious subsection for the ‘pure loop’ state yield a heuris-
tional wave functions and effective field theories both
tic ‘bulk’ understanding of why the eTmT state is pro-
for spin liquids [3] and non-fractionalized symmetry-
tectedbytimereversal. Soletusimagineperturbingthe
breaking states [32]. The parton construction inevitably
schematic eTmT wavefunction of Sec. IA, and ask why
introduces a gauge symmetry. It describes a fraction-
wecannotreachatrivialstatewithoutaphasetransition.
alized spin liquid phase whenever it yields an emergent
We make use of the heuristic analogy between the
deconfined gauge field. To obtain a non-fractionalized
AKLT loops of the spin-1 system and the ‘pure loops’ of
phase such as conventional antiferromagnet or valence
the coupled gauge theory.2 The result for the pure loop
bond solid paramagnet, the gauge field should either be
state then indicates that if we only have closed AKLT
Higgsed or confined.
loops on each sublattice, we cannot get to a trivial state
Recentlythepartonconstructionhasbeenusedtocon-
without a phase transition. So, we must consider pro-
struct SPT states in two [33, 34] and three [35, 36] di-
liferating open strands on at least one sublattice. But
mensions. The general idea is to construct a gauge the-
in the spin-1 system, unlike the pure-loop system, open
ory (with matter fields) that is confined, but with cer-
strandsintroducebulkspin-1/2Kramersdoubletdegrees
tain non-trivial features surviving in the confined state
of freedom. (Binding these emergent spin–1/2s into sin-
that make it an SPT state. However, the currently
gletswithothersonthesamesublatticemerelyhealsthe
known constructions in three dimensions use either Z
AKLT chains, taking us back to the original situation 2
or U(1) gauge theories, which do not confine automati-
with separate closed loops on each sublattice.) When
cally: strong gauge coupling is needed to reach the con-
time reversal is broken, these spin–1/2s are innocuous
fined phase. Furthermore, the constructions using U(1)
— for example we can gap them out using a magnetic
gauge theories [35] require highly nontrivial dynamics of
field. But it is natural to expect that when time rever-
thegaugefieldstocondensecompositedyon-likeobjects.
sal is preserved they prevent us reaching a trivial state
In three dimensions, a continuous non-abelian gauge
without closing the gap.
symmetry is needed to guarantee confinement. We pro-
pose two parton constructions in three dimensions with
SU(2) gauge symmetry, which confine even if the bare
gaugecouplingissmall,givingrisetotopologicalparam-
2 Recallthatthesingletbasisallowsustorepresentanyspin-zero
state of the spin-1 system in terms of loops of spin-1/2 singlet agnets. A similar construction was used previously [34]
bonds; these may form minimal length ‘loops’ which backtrack in 2D to describe an SPT phase of a spin-1 magnet pro-
onasinglelink,i.e. spin-1singletbonds,orlongerAKLTloops. tected by spin SU(2) symmetry and time reversal. We
8
also propose a construction with U(1) gauge symmetry, counts the number of Majorana cones on the surface. It
which confines at sufficiently strong coupling. Crucially, was realized [38–40] that in the presence of interactions
this U(1) construction differs from previous ones in that the state with ν = 16 is trivial, while that with ν = 8
we only condense simple monopoles to confine the gauge is equivalent to a topological paramagnet. More specif-
theory,whichcanbeachievedatstrongcouplingwithout ically, for ν = 8 the surface state with four Dirac cones
exotic form of gauge field dynamics. (eight Majorana cones) can be gapped without breaking
The spin-1 operators are re-written using the two- any symmetry via strong interactions, and the result-
orbital fermionic parton representation proposed in ing gapped surface state must have intrinsic topologi-
Ref. 16, cal order. The simplest such topological order is a Z
2
gauge theory in which the e and m particles are bosons,
S(cid:126) = 1 (cid:88) f† (cid:126)σ f . (16) but transform under time-reversal as Kramers doublets
2 aα αβ aβ (T2 =−1). Thereforewecanputtheslavefermionsinto
a=1,2
a band with ν = 8, and let the gauge fields confine the
where a = 1,2 is the orbital index. As will be discussed fermions (either automatically through an SU(2) gauge
below,thetwo-orbitalstructureisnaturalfortopological field or at strong coupling through a U(1) gauge field).
bands corresponding to topological paramagnets. This Crucially, the topological quasi-particles (e and m) on
gives another reason for favoring spin-1 systems. the surface do not carry the gauge charge, and they sur-
The physical spin states are represented in the parton vive on the surface as deconfined objects. The resulting
description as phases are therefore confined paramagnets with nontriv-
(cid:16) (cid:17) ial surface states protected by time-reversal symmetry.
|Sz =0(cid:105)= √12 f1†↓f2†↑+f1†↑f2†↓ |vac(cid:105), Non-Kramers fermions, by contrast, cannot host non-
|S =+1(cid:105)=f† f† |vac(cid:105), |S =−1(cid:105)=f† f† |vac(cid:105). trivialbandstructurewithtime-reversalsymmetryalone.
z 1↑ 2↑ z 1↓ 2↓ However, if spin–S conservation is present, the band
z
where |vac(cid:105) is the state with no fermions. States in the structures can again be assigned an integer topological
physical spin Hilbert space thus have two fermions at invariant ν(cid:48) [37] which is the number of Dirac cones on
each site, (cid:80) f† f =2, and the two fermions form a the surface (or half the number of Majorana cones). It
aα aα aα
singlet in orbital space: denoting the Pauli matrices in is known [38–40] that with interactions the state with
orbital space by τx,y,z, this is (cid:80) f† (cid:126)τ f =0. ν(cid:48) = 8 is trivial, while that with ν(cid:48) = 4 is equivalent to
α aα ab bα
The representation in Eq. 16 actually has an Sp(4) a topological paramagnet. We can then put the slave-
gauge redundancy [16] which becomes apparent when fermions into a band with ν(cid:48) =4 and let the gauge fields
we represent the fermions using Majoranas, f = confine the fermions, which produces a topological para-
1(η −iη ). Here η are Hermitian operators satisfy- magnet with time-reversal and spin-S conservation.
2 1 2 1,2 z
ing {ηsI,ηs(cid:48)J} = 2δss(cid:48)δIJ, where s,s(cid:48) = 1,2 are the new In both cases we need to put the slave fermions into
indicesassociatedwiththeMajoranasandI,J represent band structures with four Dirac cones on the surface.
all other indices (site, spin, orbital). The Majorana rep- Band structures with two Dirac cones (ν = 4) have
resentation of the spin is been studied on the cubic [41] and diamond [42] lattices.
Therefore we can obtain the desired structure simply by
1
S(cid:126) = ηTΣ(cid:126) η, Σ(cid:126) =(ρyσx,σy,ρyσz), (17) putting the partons into two copies of the ν = 4 band.
8
This can be easily done by taking advantage of the two
where ρx,y,z are Pauli matrices acting on the Majorana orbitals in Eq. 16, making the topological paramagnets
index. The generators of the gauge symmetry are ten very natural in spin-1 systems.
anti-symmetric imaginary matrices that commute with In the next section we outline a similar construction
the physical spin operators: fortheone–dimensionalHaldanechain,byconfiningslave
fermionswhichformfourcopiesoftheKitaevchain. This
Γ={ρy,ρyτx,z,ρx,zσy,ρx,zσyτx,z,τy}, (18)
illustratestheessentialideaofourconstructionsinasim-
pler and more familiar context.
where τ are Pauli matrices acting on the orbital index.
i
The spin in Eq. 17 is invariant under the Sp(4) gauge
transformation η →eiaiΓiη.
The effective field theory associated with the parton A. Parton construction for Haldane/AKLT chain
construction is a gauge theory. The gauge symmetry is
determined by the mean field band structure of the par- The Haldane phase is an SPT phase with gapless
tons, andisingeneralasubgroupofthefullSp(4)group boundary degrees of freedom that are protected by time
due to some generators being Higgsed. The gauge struc- reversal. As a warm–up exercise, we outline how this
tureallowssymmetrytoactprojectivelyontheηfermion phase can be constructed from a topological supercon-
[3]. In particular, time-reversal could be either Kramers ductor of slave fermions. This illustrates some features
(T2 =−1) or non-Kramers (T2 =1). wewillmeetagainin3D.Adifferentpartonconstruction
In 3D, band structures of Kramers fermions with T for the Haldane phase was considered in Ref. 43.
symmetry are classified by an integer index [37] ν which The fermions are taken to be non-Kramers (T2 = 1).
9
In 1D, superconducting band structures for free non- since M =tρy(1+ρzσyτy)=(2tρy)P , etc. So we may
+
Kramers fermions are labelled by a Z–valued index [37], rewrite H as
MF
ν,whichisthenumberofprotectedMajoranazeromodes
attheboundary. Thestatewithagivenν canbeviewed H =−1(cid:88)η(−)TMη(+). (25)
MF 2 i i+1
as ν copies of Kitaev’s p–wave superconducting chain
i
[44]. Interactionsreducethisclassificationto Z , i.e. the
8 Takingopenboundaryconditions, anddenotingtheleft-
ν = 8 state becomes trivial [6]. Further, the state with
most site of the chain by L, we see that the four modes
ν = 4 is topologically equivalent to the Haldane chain,
in η(+) do not appear in the Hamiltonian.
modulothepresenceofgappedfermionsinatrivialband. L
These four Majoranas correspond to two complex
Here we therefore put the slave fermions into four
fermion modes that can be occupied or unoccupied, i.e
copies of the Kitaev bandstructure, in an SU(2)–
toadegeneratefour-dimensionalboundaryHilbertspace.
symmetric manner. Gauge fluctuations (or Gutzwiller
Attheleveloffreefermions,thisdegeneracyisprotected
projection) will then remove the unwanted degrees of
freedom, leaving a topological paramagnet in the Hal- bytimereversalsymmetryT,underwhichη(+) isinvari-
L
dane phase. ant (since by definition Xη(+) =η(+)).3
Starting with an antiferromagnetic spin–1 chain, Once we go beyond mean field theory, the fermions
are coupled to confining gauge fluctuations. We will see
H=J(cid:88)S(cid:126)i.S(cid:126)i+1+..., (19) belowthattwoofthefourboundarystatesarenotgauge
i invariant — i.e. they can be thought of as having an
unscreened gauge charge sitting at the end of the chain.
we represent the spins with slave fermions as in Eq. 16
Confinement removes these states from the low energy
or equivalently Eq. 17. The valence bond picture of the
Hilbert space, leaving a single boundary spin–1/2 whose
AKLT state suggests using a mean-field Hamiltonian for
gaplessness is protected by time reversal.
the partons with hopping t and spin–singlet, orbital–
H treats spin and orbital degrees of freedom sym-
singlet pairing ∆, MF
metrically, and preserves SU(2) × SU(2) sym-
spin orbital
H =−(cid:88)(cid:104)t(cid:104)f†f +h.c.(cid:105)+∆(cid:104)f†σyτyf†T +h.c.(cid:105)(cid:105). metry. The four boundary states can be labeled by the
MF i i+1 i i+1 occupation numbers of two complex fermions c . Since
1,2
i
the partons transform as doublets under each SU(2), the
In terms of the Majoranas, this is fermionsc shouldalsoformdoubletsundereachSU(2).
1,2
In an appropriate basis the transformations are
1(cid:88)
H =− ηTMη , M =tρy+i∆ρxσyτy. (20)
MF 2 i i+1 SU(2) : (c ,c )T −→ U (c ,c )T,
spin 1 2 s 1 2
i
SU(2) : (c ,c†)T −→ U (c ,c†)T. (26)
We first consider this as a free fermion problem, then orbital 1 2 o 1 2
include the gauge fluctuations.
where U are SU(2) matrices. It follows that states
s,o
For simplicity take ∆ = t, which makes the terms in
which are singlets under SU(2) are doublets under
spin
H for different links commute. The Hamiltonian is
MF SU(2) and vice versa. We denote the spin doublet
orbital
simply four copies of the Kitaev chain, as can be seen
|↑(cid:105),|↓(cid:105)andtheorbitaldoublet|1(cid:105),|2(cid:105). Thespinoperator
immediately by going to a basis where σyτy is diagonal.
for the boundary spin-1 can be split into contributions
To be more explicit, it is useful to define the matrix from the dangling boundary modes η(+) and from η(−):
L L
X =ρzσyτy. (21) S(cid:126) =S(cid:126)(+)+S(cid:126)(−), with
L L L
Firstly, we use this to define the action of time reversal S(cid:126)(±) = 1η(±)TΣ(cid:126) η(±), Σ(cid:126) =(ρyσx,σy,ρyσz). (27)
T on the fermions: 8
T : η −→Xη. (22) We can make a similar splitting for the orbital spin T(cid:126),
which is related to S(cid:126) by swapping the σs for τs. We
This definition ensures that the spin changes sign under denote the matrices appearing in T(cid:126) by Ω(cid:126):
T and that H is invariant. The fermions are non-
MF
Kramers (T2 =1 on η). T(cid:126)(±) = 1η(±)TΩ(cid:126) η(±), Ω(cid:126) =(ρyτx,τy,ρyτz). (28)
Secondly let us define matrices that project onto a 8
given value of X, and corresponding fermion modes:
1
P = (1±X), η(±) =P η. (23)
± 2 ± 3 Any quadratic term iη(+)TAη(+) (where A is real antisymmet-
ric) is forbidden as it is odd under T. However in the presence
In an appropriate basis, η(+) has four nonzero compo- ofinteractionsafourfermiontermγ1γ2γ3γ4—whereγiarethe
nents. Next, note that componentsofη(+) insomebasis—isallowedbytimereversal,
and lifts the boundary degeneracy to a single doublet as in the
M =P MP , (24) Haldanechain[45].
− +
10
The pairs (|↑(cid:105),|↓(cid:105)) and (|1(cid:105),|2(cid:105)) are both Kramers dou- It is interesting to consider inversion symmetry here.
blets, sincethespinandorbitaloperatorsforthebound- In the free fermion problem, ν → −ν under inversion,
arymodes,S(cid:126)(+)andT(cid:126)(+),changesignunderT. Thiscan so that a nonzero value of ν can only be realised with
L L
also be checked explicitly by considering the transforma- a Hamiltonian which breaks inversion symmetry. With
tion of the boundary states (labeled by fermion occupa- interactions, ν (cid:39) ν +8, suggesting that ν = 4 can be
tion numbers) under T, with the fermions transforming realised in inversion–symmetric interacting system [47].
as T :c →c† . The present example is a nice realisation of this. The
1,2 1,2 mean field Hamiltonian H appears to break inversion
Now we consider the effect of gauge fluctuations or MF
symmetry. However, the symmetry can be restored by
Gutzwiller projection. We have listed the generators for
combining it with a gauge rotation. So the projected
the Sp(4) gauge group in Eq. 18. However, some gauge
wavefunction is actually inversion symmetric.
generators are Higgsed in the above mean field state. In
We now move on to 3D states.
general,todeterminetheunbrokengaugegroupwemust
examine Wilson loops of the form W = uˆ uˆ ..uˆ ,
whereH =(cid:80) ηTuˆ η [3]. Theunbroki1ein2 gia2iu3geginein1-
MF ij i ij j B. Cubic lattice
erators are those that commute with the Wilson loops.
Here, the only nontrivial Wilson loop is the matrix X
defined in Eq. 21. This leaves a subset of six unbroken Making use of the cubic band structure studied in
generators, which may be written in terms of the matri- Ref. 41, we construct an SU(2) gauge theory which con-
ces Ω appearing in the orbital spin (Eq. 28): fines to a topological paramagnet. We choose the mean
field Hamiltonian
Γ1D ={Ω(cid:126),XΩ(cid:126)}. (29) H =(cid:88)t ηTρyη + (cid:88) iχ(cid:48) ηTρxσyτyη (31)
MF ij i j ij i j
Taking linear combinations, we can use instead4 (cid:104)ij(cid:105) (cid:104)(cid:104)ij(cid:105)(cid:105)
(cid:88)
+ χ ηTρxσyη ,
ij i j
Γ ={P Ω(cid:126)P ,P Ω(cid:126)P }. (30)
1D + + − − (cid:104)(cid:104)(cid:104)ij(cid:105)(cid:105)(cid:105)
We denote the unbroken gauge group SU(2)(+) × where the nearest-neighbor hopping tij gives a π-flux on
orbital everysquareplaquette,thebody-diagonalpairingχ fol-
SU(2)(−) . ij
orbital lowsthepatternstudiedinRef.41,andthenext-nearest-
To make the Hamiltonian in Eq. (20) a reasonable neighbor pairing χ(cid:48) is a small perturbation introduced
ansatz, we must check that the Sp(4) gauge charges are ij
toreducethegaugegrouptoSU(2)andisnotresponsible
all zero on average: (cid:104)Γ (cid:105) = 0 for all i. Fortunately the
i for the gap or the band topology.
unbroken gauge symmetry Γ guarantees this.
1D To determine the unbroken gauge group, we examine
Theboundarymodesinvolveonlyη(+),soareinvariant
the Wilson loops as above. The fundamental nontrivial
underSU(2)(−) . However,|1(cid:105)and|2(cid:105)arenotinvariant onesareproportionaltoρzσy andρxσyτy. Theunbroken
orbital
under SU(2)(+) . Therefore after confinement only the gaugegroupisgeneratedbythoseoftheSp(4)generators
orbital
doublet |↑(cid:105), |↓(cid:105) survives, with corresponding spin S(cid:126)(+). thatcommutewiththeWilsonloops. Itisthenstraight-
L forwardtoseethattheunbrokengaugegroupisanSU(2)
This is the boundary spin-1/2 of the Haldane phase.
generated by
In this 1D example we can confirm explicitly that
Gutzwiller–projecting the mean-field wavefunction gives
Γ ={ρzσyτx,τy,ρzσyτz}. (32)
the desired SPT phase. In fact the Gutzwiller–projected cubic
state for ∆ = t, denoted |Ψspin(cid:105), is precisely the AKLT One can choose to implement time-reversal T as T :
state. To see this we adopt a trick from Ref. 43. Using η → iρzσyη, and it is straightforward to see that T :
the fact that the terms in HMF commute, we can check H →H ,S(cid:126) →−S(cid:126),and(cid:126)Γ →−(cid:126)Γ . Theband
MF MF cubic cubic
that|Ψ (cid:105)haszeroamplitudeforapairofadjacentsites
spin structure in Eq. (31) preserves time-reversal symmetry,
to be in a spin–two state. |Ψ (cid:105) is therefore the ground
spin and the SU(2) gauge rotation commutes with T. Notice
state of the AKLT Hamiltonian, since this can be writ- also that T2 =−1 on the η fermions.
tenasasumofprojectorsontothespin-twosubspacefor
each link.5
we simply Gutzwiller–project the BCS ground state of HMF.
The ground state of the Kitaev chain involves a long-range
4 To be more precise, the two types of generators in Eq. 29 cor- Cooper pair(cid:12)wavefu(cid:11)nction, C(r) = (L−2r)/L [46], so in the
respond to elements of the invariant gauge group (IGG) [3] at present case (cid:12)Ψspin is obtained by acting on the vacuum with
an exponented sum of long-range singlet creation operators,
different momenta, k = 0 and k = π, so when we take linear
combinationsthetwotypesofgeneratorsinEq.30alternateon exp((cid:80)i(cid:80)r>0C(r)fi†σyτyfi†+Tr), and projecting. The AKLT
evenandoddsites. Thisisnotcrucialhere. Anothersubtletyis state may of course be written using only short–range singlet
thattheIGGisenlargedatthespecialpoint∆=t. creation operators, |AKLT(cid:105)∝P(cid:81)i(fi†σyτyfi†+T1)|vac(cid:105). By the
5 This correspondence with the AKLT state is less obvious if previousargument,thetwostatesmustbeequivalent.