Time-reversal-symmetry-breaking chiral spin liquids: a projective symmetry group approach of bosonic mean-field theories

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by  Laura Messio

Time-reversal-symmetry-breaking chiral spin liquids: a projective symmetry group
approach of bosonic mean-field theories
Laura Messio,1 Claire Lhuillier,2 and Gr´egoire Misguich1
1Institut de Physique Th´eorique, CNRS, URA 2306, CEA, IPhT, 91191 Gif-sur-Yvette, France
2Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, UMR 7600 CNRS,
Universit´e Pierre et Marie Curie, Paris VI, 75252 Paris Cedex 05, France
Projective symmetry groups (PSG) are the mathematical tools which allow to list and classify
mean-field spin liquids (SL’s) based on a parton construction. The seminal work of Wen and its
subsequent extension to bosons by Wang and Vishwanath concerned the so-called symmetric SL’s:
i.e. statesthatbreakneitherlatticesymmetriesnortimereversalinvariance. Herewegeneralizethis
approach to chiral (time reversal symmetry breaking) SL’s described in a Schwinger boson mean-
field approach. A special emphasis is put on frustrated lattices (triangular and kagome lattices),
wherethepossibilityofachiralSLgroundstatehasrecentlybeendiscussed. ThePSGapproachis
detailed for the triangular lattice case. Results for other lattices are given in the appendices. The
physicalsignificanceofgaugeinvariantquantitiescalledfluxesisdiscussedbothintheclassicallimit
and in the quantum SL and their expressions in terms of spin observables are given.
3
1 PACSnumbers: 75.50.Ee,71.10.Kt,75.10.Jm
0
2
n I. INTRODUCTION ponentiallydecayingcorrelationsforalllocalobservables
a (spins, dimers or spin nematic operators) and a spin gap
J to bulk excitations. They contrast to critical SL which
Symmetry breaking is an ubiquitous feature of the
0 have algebraic correlations and gapless excitations. It
low temperature behavior in condensed matter physics.
1 has been understood very early1,2 that the elementary
Solids or N´eel antiferromagnets are phases that break excitations of these resonating valence bond (RVB) SL
l] some essential symmetries of the physical laws: trans- carry a spin-1 contrarily to the spin-1 magnons of the
e lational symmetry or rotational spin symmetry. Under- 2
N´eel antiferromagnets. These emergent excitations are
- standing the nature of the broken symmetries, discrete
r called spinons. A natural framework to describe the SL
t or continuous, allows to understand the nature of the
s physics is the use of effective theories with the fractional
elementary excitations and to predict the low-energy be-
t. particlesaselementarybuildingblocks(partonconstruc-
a havior of the materials (Goldstone modes, Mermin Wag-
tion). Going from the original spins to these fractional-
m ner theorem, topological defects, ...). In some phases, at
ized spinons implies the introduction of gauge fields in
firstglance,thesymmetrycontentmaybehidden: asfor
- which the spinons are deconfined (SL) or glued (N´eel or-
d example in Helium liquids. The first obvious character
der). At first glance these approaches introduce via the
n is the absence of translation symmetry breaking and ab-
o gauge fields a considerable (infinite) amount of degrees
sence of a solid phase at zero temperature. It was very
c of freedom. In fact the number of possible distinct SL’s
early understood (F. London) that this absence of so-
[ is limited by the requirement that their physical observ-
lidification is due to the many-body quantum dynamics
ables do not break any lattice or spin symmetry and the
1 and the Helium phases have been named quantum liq-
v enumeration of the different classes of distinct SL’s can
uids to be contrasted to the more “classical liquids”. It
8 be done through group theory analysis.
was only decades after discovery of the 4He superfluidity
3 This was understood ten years ago by X.-G. Wen who
0 thatthenatureoftheorderparameterwasunveiled. The
developed a classification of symmetric SL using Projec-
2 understanding of the 3He order parameter has also been
tiveSymmetryGrouptechnique(PSG).3 Theanalysisof
. heavily dependent on group symmetry considerations.
1 Wen for fermionic spinons on the square lattice was ex-
0 Aparallelcanbedevelopedbetweenthisdistinctionof tendedbyWangandVishwanathtobosonicspinons.4 In
3 quantum liquids versus classical solids and that of spin these works, the definition of a SL is limited to spin sys-
1
liquids (SL’s) versus N´eel ordered phases. N´eel ordered temsthatdonotbreakanysymmetry,neitherSU(2)spin
:
v phases at least break translational symmetry of the lat- symmetry nor lattice symmetries nor time reversal sym-
Xi tice and rotational symmetry of the spins. They can metry. These SL’s have been dubbed by Wen symmetric
be described by a local order parameter and a Landau SL. This definition excludes chiral SL which break time-
r
a theory, whereas SL do not break any lattice symmetries reversal symmetry (and some minimal amount of lattice
nor spin rotation symmetry and cannot be described by symmetry) but which do not break SU(2) and do not
a local order parameter. Similarly to 4He, SL’s can be have long range order in spin-spin or dimer-dimer corre-
characterizedbyaninternalhidden,moreorlesscomplex lations.
order.
In the wake of Laughlin theory of FQHE, chiral
In this paper we are mainly concerned with topolog- SL’s have been very popular at the end of the eighties
ical SL. These SL’s are characterized at T = 0 by ex- (Kalmeyer and Laughlin5,6, Wen et al.7, Yang et al.8),
2
but, in the absence of indisputable candidates, this op- C. Choice of bond fields: A(cid:98)ij and B(cid:98)ij or A(cid:98)ij or
tionhasnearlydisappearedfrommanydiscussionsinthe B(cid:98)ij only. 4
last decade.
Non-planar structures are quite ubiquitous in classi- III. The search of SL 4
cal frustrated magnetism,9 and are associated to scalar A. Gauge invariance, fluxes and invariance gauge
chirality: S(cid:126) ·(S(cid:126) ×S(cid:126) ) (cid:54)= 0. In some cases where the goup (IGG) 5
1 2 3
ground state is non-planar this chirality can persists at B. The projective symmetry group (PSG) 5
finitetemperature10,11 althoughthemagneticorderitself C. The algebraic projective symmetry groups 5
is absent for T > 0 (Mermin-Wagner). A similar phe-
nomenon may take place in quantum systems at T = 0. IV. From chiral long range orders to chiral
There, the usual scenario is that of a gradual reduction SL’s 6
of the N´eel order parameter when the strength of the A. SU(2) symmetry breaking of symmetric
quantum fluctuations is increased. At some point the Ans¨azte 7
sublattice magnetization vanishes and the SU(2) sym- B. The chiral algebraic PSG’s: how to include
metry is restored (leading to a SL). Now, if the ordered weakly symmetric states 8
magnetic structure is chiral, the time-reversal symmetry C. Chiral algebraic PSG’s of lattices with a
T may still be broken at the point where the magnetic triangular Bravais lattice 8
order disappears, hence leading to a time-reversal sym-
metry breaking (TRSB) SL.36 Some TRSB SL have in- V. Strictly and Weakly symmetric Ans¨atze
deed been recently proposed on the kagome lattice12,13 on the triangular lattice with first
and there are probably other examples.14,15 neighbor interactions 9
ThegoalofthispaperistorevisitthePSGanalysisby A. Construction of WS Ans¨atze on the
relaxing the time-reversal symmetry constraint in order triangular lattice 9
to include chiral SL’s. The framework used here is the B. Condensation of the WS Ans¨atze: the
Schwinger-boson mean-field theory (SBMFT).37 But, as missing tetrahedral state 10
for the symmetric PSG, the symmetry considerations we
usehereshouldalsobevalidtoclassifySLinpresenceof VI. Fluxes 11
moderate fluctuations beyond mean-field. A. Definition and physical meaning in the
The paper is organized as follow. Sections II and III classical limit 12
arereviews,tokeepthisarticleself-contained. Section II B. Fluxes in quantum models 12
is a description of SBMFT to fix the notations and pre- 1. Spin-1/2 formulas 12
cise the present understanding of this approach. Sec. III 2. Fluxes in quantum spin S models 12
starts by recalling the gauge invariance of SBMFT and
3. Fluxes in SBMFT 13
then describes how the PSG is used to enforce the SL’s
C. Finite size calculations lattice symmetries
symmetries on mean-field theories.
and non local fluxes 13
In Sec. IV, the concept of PSG is extended to include
all chiral SL’s. In Sec. V all the chiral and non chiral VII. Conclusion 13
SL theories with explicit nearest neighbor gauge fields
on the triangular lattice are derived. As an example of A. The Bogoliubov transformation 14
application we propose a chiral SL as the ground state
of a ring-exchange model on the triangular lattice. The B. Bounds on self-consistent values of the MF
physical meaning of the fluxes and their expressions in parameters in SBMFT 14
termsofspinoperatorsisdevelopedinSec.VI,aswellas
the question of topological loops on finite size samples. C. The strange classical limit of the π flux
Sec. VII is the conclusion. Appendices contain proofs of Ansatz of Wang and Vishwanath4 15
some statements in the main text, technical details and
further applications to the square and kagome lattices. D. Weakly symmetric Ans¨atze on some usual
lattices 15
1. Lattices with a square Bravais lattices 15
2. Weakly symmetric Ans¨atze on the kagome
Contents
lattice 16
I. Introduction 1
E. Number of independent fluxes on a lattice17
II. Schwinger boson mean-field theory F. Example of non local fluxes breaking the
(SBMFT) 3 lattice symmetries 18
A. Bosonic operators and bond operators 3
B. The mean-field approximation 3 References 19
3
II. SCHWINGER BOSON MEAN-FIELD Hamiltonianrespectstherotationalinvariance. Onlylin-
THEORY (SBMFT) ear combinations of the two following operators and of
their hermitian conjugates obey this property:
We consider a spin Hamiltonian H(cid:98)0({S(cid:98)i}i=1...Ns) on 1
a periodic lattice with Ns spins, each of length S. H(cid:98)0 A(cid:98)ij = 2((cid:98)bi↑(cid:98)bj↓−(cid:98)bi↓(cid:98)bj↑), (3a)
can contain Heisenberg interaction or more complicated
1
terms such as cyclic exchange, all invariant under global B(cid:98)ij = 2((cid:98)b†i↑(cid:98)bj↑+(cid:98)b†i↓(cid:98)bj↓). (3b)
spin rotations (SU(2) symmetry) and by time-reversal
transformation T (H(cid:98)0({S(cid:98)i}) = H(cid:98)0({−S(cid:98)i})). We insist iandj arelatticesitesandtheseoperatorsarethusbond
on these symmetries since they are the basis of our con- operators. They are linked by the relation
struction.
1
lemFiinsdninotgotrhioeugsrlyoudnidffisctualttep(rGobSl)emofaanqduatnhteuSmBsMpFinTprporbo-- :B(cid:98)i†jB(cid:98)ij :+A(cid:98)†ijA(cid:98)ij = 4n(cid:98)i(n(cid:98)j −δij) (4)
videsanapproximatewaytotreattheproblem. Thisap-
where :.: means normal ordering.
proachcanbesummarizedbythefollowingsteps: i)The
AnyHamiltonianinvariantbyglobalspinrotationcan
spinoperators(hencetheHamiltonian)areexpressedus-
be expressed in terms of these operators only. For ex-
ingSchinwerbosons. ii)Asuitablerotationally-invariant
mean-field decoupling leads to a quadratic Hamiltonian ample, an Heisenberg term S(cid:98)i ·S(cid:98)j where i (cid:54)= j can be
decoupled as
H . iii) H is diagonalized using a Bogoliubov trans-
MF MF
formation and solved self-consistently.
S(cid:98)i·S(cid:98)j = :B(cid:98)i†jB(cid:98)ij :−A(cid:98)†ijA(cid:98)ij, (5a)
= 2:B(cid:98)i†jB(cid:98)ij :−S2, (5b)
A. Bosonic operators and bond operators
= S2−2A(cid:98)†ijA(cid:98)ij. (5c)
Let m to be the number of sites per unit cell in the
where the first line is true whatever the boson number,
lattice,andN numberofunit-cells,sothatN =N m
m s m but the last two lines use Eq. 4 and suppose that the
is the total number of sites. We define the two bosonic
constraint of Eq. 2 is strictly respected.
operators(cid:98)b† that create a spin σ =±1/2 (or σ =↑ or ↓)
iσ To make clear the physical significance of these two
on site i. The spin operators read:
bond operators in the case S = 1, we write them in
2
(cid:88)
S(cid:98)iz = σ(cid:98)b†iσ(cid:98)biσ, (1a) terms of projection operators P(cid:98)s on the singlet state and
σ P(cid:98)t on the triplet states:
S(cid:98)i+ =(cid:98)b†i↑(cid:98)bi↓, (1b) 1
S(cid:98)i− =(cid:98)b†i↓(cid:98)bi↑. (1c) A(cid:98)†ijA(cid:98)ij = 2P(cid:98)s (6a)
1
The Hamiltonian is thus a polynomial of bosonic opera- :B(cid:98)i†jB(cid:98)ij : = 4(P(cid:98)t−P(cid:98)s). (6b)
tors with only even degree terms. These relations imply
that the commutation relations [S(cid:98)iα,S(cid:98)iβ] = i(cid:15)αβδS(cid:98)iδ are WeseeinEq.6,that:B(cid:98)i†jB(cid:98)ij :representsaferromagnetic
verified. Asforthetotalspin,itreadsS(cid:98)(cid:126)i2 = n(cid:98)2i (cid:0)n(cid:98)2i +1(cid:1), contribution to Eq. 5a, whereas A(cid:98)†ijA(cid:98)ij gives the singlet
where n(cid:98)i =(cid:98)b†i↑(cid:98)bi↑+(cid:98)b†i↓(cid:98)bi↓ is the total number of bosons contribution.
at site i. To fix the “length” of the spins, the following
constraint must therefore be imposed on physical states:
B. The mean-field approximation
(cid:88)
n(cid:98)i = (cid:98)b†iσ(cid:98)biσ =2S. (2)
We now need two successive approximations to obtain
σ
a quadratic and solvable Hamiltonian. We first relax the
In traditional MF theories, the MF parameter is the
constraint on the boson number by imposing it only on
orderparameter(as forexamplethemagnetization(cid:104)S(cid:98)i(cid:105)) average:
and the MF Hamiltonian consequently breaks the initial
Hamiltoniansymmetries,exceptinthehightemperature (cid:104)n (cid:105)=κ. (7)
(cid:98)i
phase where the MF parameter is zero. Here, we would
like to describe SL’s that do not break any symmetry. whereκdoesnotneedtobeainteger. Toimplementthis
ThuswearegoingtoexpressH(cid:98)0 usingquadraticbosonic constraint, a Lagrange multiplier (or chemical potential)
(cid:80)
operators, requiring their invariance by global spin rota- λ is introduced at each site i and the term λ (κ−
i i i
tions. n ) is added to the Hamiltonian. κ can be continuously
(cid:98)i
The expectation value of these operators will then be varied to interpolate between the classical limit (κ=∞)
used as mean-field parameters, insuring that the MF and the extreme quantum limit (κ→0).
4
It should be reminded in general that fixing κ=2S to C. Choice of bond fields: A(cid:98)ij and B(cid:98)ij or A(cid:98)ij or B(cid:98)ij
studyaspin-S modelisnotnecessarilythebestchoiceas only.
intheSBMFT(cid:104)S(cid:98)2(cid:105)= 3κ(κ+2).16 Analternativechoice
i 8
couldbe tofix κ insucha way thatthe spinfluctuations AsintheexampleofEq.5,therelation4canbeusedto
and not the spin length have the correct value.38 eliminateAij orBij fromH(cid:98)MF. Ifwechoosetokeeponly
In a second step, bond operators fluctuations are ne- the B parameters, M is block diagonal with two blocks
ij
glectedandaMFHamiltonianH(cid:98)MFthatislinearinbond ofsizeNs andthevacuumofbosonsisaGS.Toobeythe
operators is obtained. For instance: constraintofEq.2,wehavetoadjustthebosondensities
by filling some zero-energy mode(s), therefore breaking
:B(cid:98)i†jB(cid:98)ij :(cid:39)(cid:104)B(cid:98)i†j(cid:105)B(cid:98)ij +B(cid:98)i†j(cid:104)B(cid:98)ij(cid:105)−|(cid:104)B(cid:98)ij(cid:105)|2. (8) the SO(3) symmetry. The GS is thus completely clas-
sical. On the contrary, we can keep the A only, but
Wereplace(cid:104)B(cid:98)ij(cid:105)and(cid:104)A(cid:98)ij(cid:105)bycomplexbondparameters then the singlet weight is overestimated, whijich can be
A and B . This MF approximation can be seen as the
ij ij introduce some bias on frustrated lattices where short-
first term of a large N expansion of a Sp(N) theory.17
The steps are explained in details in Ref. 16 in the very distance correlations are not collinear. Keeping A(cid:98)ij only
is a widespread practice in the litterature, but Trumper
similarcaseofanSU(N)theory. Thiszero’thorder1/N
et al.19 have explicitly shown that the bandwidth of the
expansion can be pursued to the first order.18 The MF
spectrum of excitations of the Heisenberg model on the
Hamiltonian is now a quadratic bosonic operator. It can
triangular lattice is twice too large when using A fields
be written in terms of a 2N ×2N complex matrix M ij
s s only. On the other hand, the use of both A and B
and of a real number (cid:15) depending on the A and B ij ij
0 ij ij restores the correct bandwidth and a improves quanti-
and on the Lagrange multipliers λ :
i tatively the excitation spectrum. Note that even on the
H(cid:98)MF =φ†Mφ+(cid:15)0. (9) stoqrusarime plartotvicees tthheegsirmouunltdansteaotueseunseergoyf.2b0oth bond opera-
where φ† = ((cid:98)b†1↑,(cid:98)b†2↑,...,(cid:98)b†Ns↑,(cid:98)b1↓,...,(cid:98)bNs↓).39 The ex- From a different point of view Flint and Coleman21
advise the use of both fields in order to have a large-
pression for M and (cid:15)0 depend on H(cid:98)0 and on the chosen
N limit where spin generators are odd under the time-
decoupling (for example using Eq. 5a, 5b or 5c).
reversal symmetry, as it is the case for SU(2).
The set of mean-field parameters {A ,B } appearing
ij ij
in H is called an Ansatz. Up to an equivalence rela-
MF
tionthatwillbedescribedinthenextsection,anAnsatz
defines a specific phase (ground state and excitations). III. THE SEARCH OF SL
Depending on the value of κ, this state can either have
N´eel long range order, or the bosons are gapped (several
Even when considering an Hamiltonian with nearest
types of SL are then possible).
neighbor interactions only, the dimension of the MF pa-
Inthefollowingwewillexplainandexploittherelation
rameters manifold is exponentially large.40 Moreover the
which exists between regular classical magnetic orders9
Lagrange multipliers λ make the search of the station-
and SL’s. i
ary points of the MF free energy difficult (constrained
To enforce self-consistency, the following conditions
optimization) as for each considered Ansatz, all λ must
should be obeyed: i
be adjusted to calculate the MF free energy. In Ref. 22
this optimization was carried out (without any simpli-
Aij =(cid:104)A(cid:98)ij(cid:105) and Bij =(cid:104)B(cid:98)ij(cid:105), (10)
fying/symmetry assumption) on square and triangular
which are equivalent to lattices with up to 36 sites. In almost all cases the
MF ground-state turned out to be highly symmetric, as
∂F ∂F
MF =0 and MF =0, (11) expected, but excited mean-field solutions are however
∂A ∂B
ij ij highly inhomogeneous (and often not understood yet).
Theproblemcanbeconsiderablysimplifiedifwerestrict
whereF istheMFfreeenergy,togetherwiththecon-
MF
our search to states respecting some (or all) the symme-
straint
∂F triesofH(cid:98)0. Suchsymmetriesaredividedintoglobalspin
(cid:104)n (cid:105)=κ ⇔ MF =0. (12) rotations, lattice symmetries and time reversal symme-
(cid:98)i ∂λ
i try. We have assumed from the beginning that H(cid:98)0 is
The next step is to calculate the mean values of the op- invariantbyglobalspinrotationsandchosentheMFap-
erators A(cid:98)ij and B(cid:98)ij either in the GS of H(cid:98)MF if the tem- proximation in such a way that it remains true for H(cid:98)MF,
perature is zero, or in the equilibrium state for non-zero butthechoiceofaspecificAnsatzmayormaynotbreak
temperatures. In both cases one needs to use a Bogoli- other discrete symmetries. The fore-coming section ex-
ubov transformation to diagonalize H . As this trans- plains howto find allAns¨atze such as thephysical quan-
MF
formation is often explained in the simple case of 2×2 titiesareinvariantbyallthelatticesymmetriesX,either
matrices (or for particular sparse matrices), we explain strictly(forsymmetricSL’s)oronlyuptoatime-reversal
the algorithm in a completely general case in App. A. transformation (Chiral SL’s).
5
We will now define some groups specific to an Ansatz: Eq. 13 with θ(i) = π for all lattice sites i. In the par-
the invariance gauge group in Sec. IIIA and the projec- ticular cases where we can divide the lattice in two sub-
tivesymmetrygroupinSec.IIIB.Then,inSec.IIIC,we lattices such as A = 0 whenever i and j are in the
ij
define the algebraic projective symmetry group, which is same sublattice (bipartite problem), the IGG is enlarged
associated to a lattice symmetry group and not specific to U(1). The later situation corresponds, for instance,
to a particular Ansatz on this lattice. to an Ansatz on a square lattice with only first-neighbor
A . The transformations of the IGG are then given by
ij
θ(i)=θ on a one sublattice and θ(i)=−θ on the other,
A. Gauge invariance, fluxes and invariance gauge with arbitrary θ ∈[0,2π[.
goup (IGG)
LetG (cid:119)U(1)Ns bethesetofgaugetransformations. A B. The projective symmetry group (PSG)
gauge transformation is characterized by an angle θ(i)∈
[0,2π[ateachsiteandtheoperatorG(cid:98) whichimplements Let X be the group of the lattice symmetries of the
the associated gauge transformation
Hamiltonian H(cid:98)0 (translations, rotations, reflections...).
From now on, for the sake of simplicity, we discard the
(cid:98)bjσ →(cid:98)bjσeiθ(j) =G(cid:98)†(cid:98)bjσG(cid:98) (13) hat on the gauge and symmetry operators. The effect of
an element X of X on the bosonic operators is
is given by
  X :(cid:98)bjσ →(cid:98)bX(j)σ. (16)
(cid:88)
G(cid:98) =expi (cid:98)b†jσ(cid:98)bjσθ(j). (14)
The effect of X on the Ansatz is:
j
(cid:26)
A →A ,
A wave function |φ(cid:105) respects a symmetry F(cid:98) if all the jk X(j)X(k) (17)
B →B .
jk X(j)X(k)
physicalobservablesmeasuredinthestateF(cid:98)|φ(cid:105)areiden-
tical to those measured in |φ(cid:105). It does not mean that We know that a gauge transformation does not change
|φ(cid:105)=F(cid:98)|φ(cid:105),butthatthetwowavefunctionsareequalup any physical quantities. What about the lattice sym-
to a gauge transformation: ∃G(cid:98) ∈G,|φ(cid:105)=G(cid:98)F(cid:98)|φ(cid:105). metries ? We know from Sec. IIIA that if the Ans¨atze
The action of G(cid:98) on the Ansatz is: before and after the action of X have the same physical
quantities, they are linked by a gauge transformation: it
(cid:26)Ajk →Ajkei(θ(j)+θ(k)), (15) thus exists at least one gauge transformation GX such
B →B ei(−θ(j)+θ(k)), as G X leaves the Ansatz unchanged. The set of such
jk jk X
transformations of G×X is called the projective symme-
such as H(cid:98)MF remains unaffected by G(cid:98). We note that try group (PSG) of this Ansatz. Note that this group
(cid:104)A(cid:98)jk(cid:105)and(cid:104)B(cid:98)jk(cid:105)aregaugedependent: theyarenotphys- only depends on the Ansatz and on X, but not on the
ical quantities as they do not preserve the on-site boson details of the Hamiltonian. Thus, an Ansatz is said to
number. As any such quantity, their mean values cal- respectalatticesymmetryX ifitexistsatransformation
culated using H(cid:98)0 is zero when the average is taken on GX ∈G such that the Ansatz is invariant by GXX.
all gauge choices. Using H(cid:98)MF, it can be non-zero as the The IGG of an Ansatz is the PSG subgroup formed
gauge symmetry is explicitly broken by the choice of the by the set of gauge transformations GI associated to the
Ansatz. identity transformation I of X. For each lattice symme-
We have seen that changing the gauge modifies the try X ∈ X respected by the Ansatz, the set of gauge
Ansatzbutnotthephysicalquantities. Conversely,iftwo transformations GX such as GXX is in the PSG is iso-
MF Hamiltonians give rise to the same physical quanti- morph to the IGG: for any GI in the IGG, (GIGX)X is
ties,thentheirAns¨atzearelinkedbyagaugetransforma- inthePSG.Thus,theconditionforanAnsatztorespect
tion. In fact two types of physical quantities are directly all the lattice symmetries is that its PSG is isomorph to
relatedtotheAnsatz: theMFparametermoduli(related IGG×X.
to the scalar product of two spins), and the fluxes. The
fluxes are defined as the arguments of Wilson loop oper-
atorssuchas(cid:104)B(cid:98)ijB(cid:98)jkB(cid:98)ki(cid:105)orof(cid:104)A(cid:98)†ijA(cid:98)jkA(cid:98)†klA(cid:98)li(cid:105). Bycon- C. The algebraic projective symmetry groups
struction these quantities are gauge invariant and define
the Ansatz up to gauge transformations. The physical An Ansatz is characterised (partially) by its IGG and
meaning of fluxes will be addressed in Sec. VI. its PSG. In turn, we know from these groups which lat-
The gauge transformations that do not modify a spe- tice symmetries it preserves. Reversely we now want to
cific Ansatz form a subgroup of G called the invariance impose lattice symmetries and find all Ans¨atze that pre-
gaugegroup(IGG).Italwayscontainstheminimalgroup servesthem. Toreachthisgoal,weproceedintwosteps.
Z formed by the identity and by the transformation The first one is to find the set of the so-called algebraic
2
6
PSG’s.3,4 They are subgroups of G ×X verifying alge- with p = 0 or 1. This constraint coming from the com-
braic conditions necessarily obeyed by a PSG. Contrary mutation relation between x and x must be obeyed by
1 2
tothePSGofanAnsatz, thealgebraicPSG’sexistinde- all algebraic PSG’s.
pendently of any Ansatz and only depend on the lattice
It is useless to list all algebraic PSG’s for the sim-
symmetry group X and on the choice of an IGG (chosen
ple reason that some of them are equivalent and give
asthemoregeneral). AnalgebraicPSGdoesnotdepend
Ans¨atze with the same physical observables. Two (al-
on the details of the lattice such as the positions of the
gebraic or not) PSG’s are equivalent if they are related
sites. However, depending on these details, an algebraic
by a gauge transformation G: for any gauge transforma-
PSG may have zero, one, or many compatible Ans¨atze.
tionG associatedtothelatticesymmetryX inthefirst
The second step consists, for a given lattice, in finding X
PSG, GG G−1 belongs to the set of gauge transforma-
all the Ans¨atze compatible with a given algebraic PSG. X
tions associated to X in the second PSG. We are only
Let us detail the algebraic conditions verified by the
interested in equivalence classes of PSG’s.
algebraic PSG’s. The group X is characterized by its
generatorsx ...x . Ageneratorx hasanordern ∈N∗ Taking algebraic PSG’s in different classes does not
1 p a a
such as xna is the identity (if no such integer exists, we imply that they have no common Ans¨atze: a trivial ex-
a
set n = ∞). For any transformation X ∈ X, there ampleistheAnsatzwithonlyzeroparameters,belonging
a
0ex≤istks a<unniqiufenoridsefirneditep,rkodu∈cZt Xif n=ot.xTk11h.e.r.uxlkpeps uwsietdh ttohaatnayrealignenboraoicthPeSrGcl’ass.sBauntdehaacvhecslpaescsifiinccpluhdyessicAalnpsr¨aotpze-
a a a a
erties.
to transform an unordered product into an ordered one
are the algebraic relations of the group. Each of these OnceallthealgebraicPSG’sclassesaredetermined,it
rules implies a constraint on the Gxa (chosen as one of remains to find the possible compatible Ans¨atze for one
the gauge transformation associated to xa). Basically, it representant of each class. As an example of compati-
statesthatifalatticesymmetryX canbewritteninsev- bility condition, let’s take the case where X belongs to
eralwaysusingthegenerators, thegaugetransformation the considered algebraic PSG (i.e. G = I). Then an
X
GX is independent of the writing (up to an IGG trans- Ansatz can be compatible with this algebraic PSG only
formation). The subgroups of G×X respecting all these if, for any couple of sites (i,j), A =A . If such
ij X(i)X(j)
constraints are the algebraic PSG’s. compatible Ans¨atze exist, they respect the lattice sym-
To illustrate the idea, let us consider a basic exam- metries by construction (in the sense that their physical
ple where X is generated by two translations x1 and quantitiesdoso). Wenowwanttoimposethetimerever-
x2. Both transformations have an infinite order n1 = sal symmetry: among the compatible Ans¨atze, we only
n2 = ∞. We have X ∈ X written as product of gen- keep those that are equivalent to a real Ansatz up to a
erators X = xm1xm2xm3xm4... and we would like o gauge transformation. We call them strictly symmetric
1 2 1 2
write it as X = xp1xp2. The need algebraic relation Ans¨atze (weakly symmetric ones are defined in the next
1 2
is simply the commutation between the two translations section).
: x x = x x . We then have p = m + m + ...
1 2 2 1 1 1 3 To completely define an Ansatz, it is sufficient to give
and p = m + m + .... We will now see that this
2 2 4 the algebraic PSG and the values of the MF parameters
implies a constraint on G and G . Suppose that we
x1 x2 on non symmetry-equivalent bonds. For example, on a
have an Ansatz unchanged by G x and G x . Then
x1 1 x2 2 square (or triangular or kagome) lattice with all usual
the inverses x−1G−1 or x−1G−1 too are in the PSG. So,
1 x1 2 x2 symmetries (see Fig. 2) and only first neighbor interac-
the product G x G x x−1G−1x−1G−1 ∈ PSG. This
product has bexe1n 1choxs2en2to1 maxk1e t2he axl2gebraic relation tions, the Aij and Bij of one bond are enough.
x x = x x (⇔ x x x−1x−1 = I) appear after the fol-
1 2 2 1 1 2 1 2
lowing manipulations:
G x G x x−1G−1x−1G−1 ∈PSG
x1 1 x2 2 1 x1 2 x2
⇔ G (x G x−1)x x x−1x−1(x G−1x−1)G−1 ∈PSG IV. FROM CHIRAL LONG RANGE ORDERS
x1 1 x2 1 1 2 1 2 2 x1 2 x2 TO CHIRAL SL’S
⇔ G (x G x−1)(x G−1x−1)G−1 ∈PSG.
x1 1 x2 1 2 x1 2 x2
The expressions in parenthesis in the last line are pure We will now show that the zoo of N´eel LRO obtained
gaugetransformationsandthefullresultingexpressionis from the strictly symmetric Ans¨atze misses the chiral
a product of gauge transformations. Thus, we can more states which are exact ground states of a large num-
precisely write: ber of frustrated classical models. This will lead us in
a straightforward manner to the construction of chiral
G (x G x−1)(x G−1x−1)G−1 ∈IGG. (18)
x1 1 x2 1 2 x1 2 x2 algebraic PSG’s in which time reversal and some lattice
If the IGG is Z , this constraint can be written in term symmetries can be broken (Sec. IVB). This generalised
2
framework will then be illustrated on the triangular lat-
of the phases θ (i) of the gauge transformation G as:
X X
tice in Sec. V and on the square and kagome lattice in
θ (i)+θ (x−1i)−θ (x−1i)−θ (i)=pπ, (19) App. D.
x1 x2 1 x1 2 x2
7
A. SU(2) symmetry breaking of symmetric Ans¨azte This extra constraint can make the classical limit prob-
lem unsolvable: no classical magnetization pattern is
To simplify, we suppose that all lattice sites are equiv- then compatible with the Ansatz. An example of such
alentbysymmetryandonlyconsiderAns¨atzesuchasthe a situation was studied by Wang and Vishwanath4 (see
λ areallequaltoasingleλ. EvenifanAnsatzisstrictly App. C).
i
symmetric, it does not always represent a SL phase. As Wecantaketheproblemoftheclassicallimitfromthe
iswellknowninSBMFT,aBosecondensationofzeroen- other side. We begin from a classical state, from which
ergy spinons can occur and leads to N´eel order. We will we calculate (cid:104)(cid:98)biσ(cid:105) and the Ansatz (using Eq. 20 and 21).
discuss how the Ans¨azte symmetry constraints the mag- What are the conditions on the classical state for the as-
netic order obtained after condensation, and establish a sociated Ansatz to be strictly symmetric ? As we look
relation with the regular states introduced in Ref. 9. for an Ansatz respecting all lattice symmetries, the ro-
The Bogoliubov bosons creation operators are linear tationally invariant quantities (as the spin-spin correla-
combinations of the (cid:98)biσ and (cid:98)b†iσ, such as their vacuum tions) must be invariant by all lattice symmetries, what
|˜0(cid:105) is a GS of H(cid:98)MF (see App. A). If the GS is unique, it severely limits the classical magnetization pattern. Such
a state is called a SO(3)-regular state. Mathematically,
must respect all the Hamiltonian symmetries and conse-
a state is said to be SO(3)-regular if for any lattice sym-
quently,cannotbreaktheglobalspinrotationinvariance.
metryX thereisaglobalspinrotationS ∈SO(3)such
Butwhenκincreases(wecontinuouslyadapttheAnsatz X
as the state is invariant by S X. Moreover, the time
to κ so that the self-consistency conditions remains ver- X
reversal symmetry (i.e. the Ansatz can be chosen to be
ified and the PSG remains the same), some eigen en-
real) imposes the co-planarity of the spins.41 The set of
ergie(s) decrease(s) to zero. The GS is then no more
coplanar SO(3)-regular states can be sent on the set of
unique as the zero mode(s) can be more or less popu-
condensed states of strictly symmetric Ans¨atze. In the
lated and the phases of each zero mode are free. It is
same way, we define the O(3)-regular states by including
then possible to develop a long range spin order.
global spin flips S →−S in the group of spin transfor-
This phenomena occurs when no λ verifies condition i i
mations. TheseO(3)-regularstatesarelistedinRef.9for
12. If λ increases the mean number of boson per site
severaltwo-dimensionallattices. TheO(3)-regularstates
increases up to a maximal number κ . At this point,
max
aredividedincoplanarSO(3)-regularstatesandinchiral
some eigen energies become zero. Increasing λ further is
states. In a chiral state, the global inversion S → −S
not possible as the Bogoliubov transformation becomes i i
cannot be “undone” by a global spin rotation. Equiv-
unrealizable (the M matrix of Eq. 9 has non-positive
alently, there exist three sites i, j, k such as the scalar
eigenvalues). To reach the required number of boson
per site, we have to fill the zero energy modes ˜b†, ˜b†, chiralitySi·(Sj∧Sk)isnonzero: thespinsarenotcopla-
1 2 nar. Then a strictly symmetric Ansatz, upon condensa-
...usingcoherentstateseα1˜b†1+α2˜b†2+...|˜0(cid:105)forexample. In tion, can only give coplanar SO(3)-regular states in the
thethermodynamicallimitthefractionofmissingbosons
classical limit, therefore missing all chiral O(3)-regular
is macroscopic and a Bose condensation occurs in each
states.
of the soft modes. The choice of the weight α of these
i This limitation can seem unimportant as most of the
modes fixes the direction of the on-site magnetization.
usual long-range ordered spin models have planar GS’s.
Detailedexamplesofmagnetizationcalculationsinacon-
But some new counter examples have recently been
densate are given by Sachdev.23
discovered. The first example is the cyclic exchange
In the classical limit (κ → ∞), all bosons are in the
model on the triangular lattice10 with a four sublattice
condensate and contribute to the on-site magnetization
tetrahedral chiral GS (see Fig. 1). More recently, two
m . The modulus |m | should be equal to κ/2 to satisfy
i i twelve sublattice chiral GS’s, with the spins oriented to-
Eq. 9. The(cid:98)biσ operators acquire a non-zero expectation wards the corners of a cuboctahedron, were discovered
value(cid:104)(cid:98)biσ(cid:105)andare(uptoagaugetransformation)linked on the kagome lattice with first and second neighbor
to mi by : exchanges11,24 (studiedinApp.D2). Asystematicstudy
(cid:18)(cid:104)(cid:104)(cid:98)(cid:98)bbii↑↓(cid:105)(cid:105)(cid:19)=(cid:18)(cid:112)|mi|(cid:112)−|mmziie|i+Armg(mzixi+imyi) (cid:19), (20) ohrafanstghieenscdoleafesdisnictreaerlvaGecatSlie’osdnovfthaslaiumtepst.lh9eemGodSe’slsaornedcihffierraelnftolratltaircgees
where Arg is the argument of the complex number and The theory of symmetric PSG is unable to encompass
mx,y,z are the magnetization components. These values such chiral states. In the following subsection, we will
i
are constrained by the Ansatz through: build TRSB SL Ans¨atze which include, upon condensa-
1 tion, all classical regular chiral states. This method was
Aij = 2((cid:104)(cid:98)bi↑(cid:105)(cid:104)(cid:98)bj↓(cid:105)−(cid:104)(cid:98)bj↑(cid:105)(cid:104)(cid:98)bi↓(cid:105)), (21a) already applied to the kagome lattice with up to third
1 neighbor interactions, leading to the surprising result of
Bij = 2((cid:104)(cid:98)b†i↑(cid:105)(cid:104)(cid:98)bj↑(cid:105)+(cid:104)(cid:98)b†i↓(cid:105)(cid:104)(cid:98)bj↓(cid:105)). (21b) a chiral state even in the purely first neighbor model.12
If this state is physically relevant or not is still an open
The supplementary constraint reads:
question, but independently, it shows that the omission
|m |∼κ/2 (22) of chiral Ans¨atze has prevented the discovery of more
i
8
σ
V σ
V2 R 2 R4
6
V V
FIG. 1: (Color online) Tetrahedral order on the triangular 1 1
lattice
FIG. 2: (Color online) Generators of the lattice symmetries
X on the triangular and square lattices. V is a translation,
i
σ is a reflection and R is a rotation of order i.
i
competitive MF solutions.
B. The chiral algebraic PSG’s: how to include consider Xe the subgroup of transformations of X that
weakly symmetric states can only be even. Mathematically, Xe is the subgroup
of X which elements are sent to the identity by all mor-
phismsfromX toZ . X containsatleastallthesquares
Thetime-reversaltransformationT actsonanAnsatz 2 e
oftheelementsofX as(cid:15) =(cid:15)2 =1. But,dependingon
by complex conjugation of the MF parameters.3 If an X2 X
the algebraic relations of X, it may contain more trans-
Ansatz respects this symmetry, it is sent to itself by T
formationsasweshowinthetriangularcaseinSec.IVC.
(up to a gauge transformation). So, in an appropriate
OnceX isknown,wedefinethechiralalgebraicPSG’sof
gauge, all parameters can then be chosen real. In most e
X as the algebraic PSG’s of X . The method described
previousSBMFTstudies,thehypothesisoftimereversal e
previously to find all algebraic PSG’s applies the same
invariance of the GS was implicit, as only real Ans¨atze
way. We define X as the set of transformations which
were considered. In contrast to SU(2) global spin sym- o
may be odd (X − X ). It contains transformations of
metrythatcaneasilybebrokenthroughtheBoseconden- e
undetermined parities.
sationprocess,notransitionisknowntoproduceachiral
To filter the weakly symmetric Ans¨atze from those
ordered state out of a T-symmetric Ansatz. Indeed, chi-
compatible with the chiral algebraic PSG’s, we have to
ralAns¨atzehaveloopswithcomplex-valuedfluxeswhich
take care of the transformations of X . This gives two
evolve continuously with κ. We do not expect any sin- o
typesofextraconstraints. First,sametype(AorB)MF
gular behavior of these (local) fluxes when crossing the
parametersonbondslinkedbysuchtransformationmust
condensationpoint,sothegenericsituationisthatachi-
ralLROphasewillgiverisetoaTRSBSL12,13 whende- have the same modulus. The second constraint concerns
their phases, through the fluxes. The phases are gauge
creasingκ. Itisofcoursepossiblethatthelowest-energy
dependent,butthefluxesaregaugeindependent. Fluxes
Ansatz changes with κ but such a first-order transition
are sent to their opposite by T and as well as by the
has not reason to coincide with the onset of magnetic
odd transformations of X. Then are unchanged by even
LRO.
transformations. To find all WS Ans¨atze we then have
To obtain all chiral SL’s we have to explicitly break
to determine a maximal set of independent elementary
time-reversal symmetry at the MF level, in the Ansatz.
fluxes and distinguish all possible cases of parities for
For SO(3) classical regular states, a lattice transforma-
the transformations of X ((cid:15) =±1).
tion from X is compensated by a global spin rotation o X
We can now apply these theoretical considerations to
(that leaves the Ansatz unchanged). For O(3) classical
find all WS Ans¨atze on some usual lattices as the tri-
regular states, a lattice transformation X ∈ X is com-
angular, honeycomb, kagome and square lattice. The
pensated by a global spin rotation possibly followed by
calculations are detailed for the triangular lattice in the
an inversion S → −S . This defines a parity (cid:15) to be
i i X
following subsections and some results for the kagome
+1 if no spin inversion is needed, and −1 otherwise. In
and square lattice are given in App. D.
a chiral SL, the parity will be deduced from the effect
of X on the fluxes: (cid:15) = 1 if they are unchanged, −1
X
if they are reversed. With this distinction in mind we
willcallweakly symmetric Ans¨atze(WS)theAns¨atzere- C. Chiral algebraic PSG’s of lattices with a
spectingthelatticesymmetriesuptoT, whereasthethe triangular Bravais lattice
Ans¨atze respecting strictly all lattice symmetries and T
havealreadybeencalledstrictly symmetric (SS)Ans¨atze The first step is to find all chiral algebraic PSG’s. As
(all lattice symmetries are even). already mentioned, they only depend on the symmetries
The distinction between even and odd lattice symme- of Xe and on the IGG. We choose the most general case
tries (as defined by (cid:15)X) is the basis of the construction of IGG∼ Z2 and suppose that H(cid:98)0 respects all the lat-
of all WS Ans¨atze via the chiral algebraic PSG’s. Let us tice symmetries with the generators described in Fig. 2.
9
These symmetries are those of a triangular lattice, but and the algebraic PSG is transformed in an other ele-
the actual (spin) lattice of H(cid:98)0 can be any lattice with a ment of its equivalence class. Using the following gauge
triangularBravaislatticesuchasahoneycomb,akagome transformations:
or more complex lattices. The coordinates (x,y) of a
G :(x,y) → πx,
point are given in the basis of the translation vectors V , 3
1
V2 andtheeffectofthegeneratorsonthecoordinatesare G4 :(x,y) → πy,
weseethatachangeofp orp isagaugetransformation,
V :(x,y) → (x+1,y), (23a) 3 4
1
sowecansetthemtozero. Solvingthesetofequations26
V :(x,y) → (x,y+1), (23b)
2 leads to:
R :(x,y) → (x−y,x), (23c)
6
θ (x,y) = 0 (30a)
σ :(x,y) → (y,x). (23d) V1
θ (x,y) = p πx (30b)
V2 1
The algebraic relations in X are: (cid:18) (cid:19)
x+1
θ (x,y) = p πx y− +g (x∗,y∗),(30c)
V V = V V , (24a) R3 1 2 R3
1 2 2 1
σ2 = I (24b) withasupplementaryconstraintthatcanonlybetreated
R6 = I, (24c) when the spin lattice is defined:
6
V1R6 = R6V2−1 (24d) gR3(x∗,y∗)+gR3((−y)∗,(x−y)∗)
V2R6 = R6V1V2 (24e) +g ((y−x)∗,(−x)∗) = p π. (31)
R3 2
V σ = σV (24f)
1 2
This constraint only depends on the coordinates of the
R σR = σ. (24g)
6 6 sites in a unit cell (x∗ and y∗).
Eqs. 30 and 31 define the chiral algebraic PSG on the
Let us now determine the subgroup X of transforma-
e
triangular Bravais lattice. The full determination of the
tions which are necessarily even. It evidently includes
V2, V2 and R2 (noted R ). But there are more even WSAnts¨atzerequiresprecisedefinitionofthespinlattice
1 2 6 3 (triangular,honeycomb(m=2)orkagome(m=3))and
transformations in this subgroup. Using Eq. 24e we find
onthenumberofinteractionsincludedintheMFHamil-
(cid:15) (cid:15) = (cid:15) (cid:15) (cid:15) , so (cid:15) = 1. In the same way, using
V2 R6 R6 V1 V2 V1 tonian (first neighbor only or first and second neighbor;
Eq. 24d, we get (cid:15) =1. Thus X is generated by V , V
V2 e 1 2 AandB parameters,orAonly...). Thecaseofthetrian-
and R . The algebraic relations in X are
3 e
gular lattice (m = 1) with nearest neighbor interactions
V V = V V , (25a) and A and B MF parameters is described in the next
1 2 2 1
R3 = I, (25b) subsection.
3
R V = V R , (25c)
3 1 2 3
R3 = V1V2R3V2. (25d) V. STRICTLY AND WEAKLY SYMMETRIC
ANSA¨TZE ON THE TRIANGULAR LATTICE
As explained in Sec. IIIC, each of these relations gives WITH FIRST NEIGHBOR INTERACTIONS
a constraint on the gauge transformations associated to
these generators. The Eqs.25 imply that for any site i:
A. Construction of WS Ans¨atze on the triangular
lattice
θ (V−1i)−θ (i) = p π, (26a)
V2 1 V2 1
θ (i)+θ (R i)+θ (R2i) = p π, (26b)
R3 R3 3 R3 3 2 Thetriangularlatticehasasinglesiteperunitcelland
θR3(i)−θR3(V2−1i)−θV2(i) = p3π, (26c) the values of x∗ and y∗ are the coordinates of this site in
θ (V−1i)+θ (V−1V−1i) a unit cell, say (0,0). Eq. 31 simplifies into:
V2 1 R3 2 1
+θ (V R2i)−θ (i) = p π, (26d)
V2 2 3 R3 4 6gR3(0,0)=0. (32)
wtiohnerseapr1etworpit4tecnanmtaokdeuleoith2eπr).thWevealnuoete0o[xr]1t(hteheinetqeugear- TcahuesesotlhuetioIGnsGariseZgR,3(o0n,l0y)t=hektπh/r3e,ewvaitlhueks iknt=eg−er1.,B0,e1-
2
part of x and x∗ = x−[x] (0 ≤ x∗ < 1). By partially lead to physically different Ans¨atze.
fixing the gauge, we can impose Finally, we have 6 distinct algebraic PSG’s for the re-
duced set of symmetries X . They are characterised by
θ (x ,y ) = 0 e
V1 i i two integers p =0,1 and k =−1,0,1 and defined by:
1
θ (x∗,y ) = p πx∗.
V2 i i 1 i
θ (x,y) = 0 (33a)
V1
Through a gauge transformation G of argument θ , the
G θ (x,y) = p πx (33b)
θ of a lattice transformation X becomes: V2 1
X (cid:18) (cid:19)
x+1 kπ
θ (x,y) = p πx y− + (33c)
θX(i)→θG(i)+θX(i)−θG(X−1i). (28) R3 1 2 3
10
For each set ((cid:15) ,(cid:15) ), the compatible Ans¨atze are thus
R6 σ
limited to:
i) ((cid:15) ,(cid:15) )=(1,1): k =0, p =0 and φ =0 or π,
R6 σ 1 B1
ii) ((cid:15) ,(cid:15) )=(−1,−1): k =0, p =0 and φ =0 or π,
R6 σ 1 B1
iii) ((cid:15) ,(cid:15) )=(1,−1): φ =p π/2 or π+p π/2,
R6 σ B1 1 1
iv) ((cid:15) ,(cid:15) ) = (−1,1): k = 0, p = 0 and no constraint
R6 σ 1
on φ .
B1
A couple ((cid:15) ,(cid:15) ) does not characterize an Ansatz. A
R6 σ
givenAnsatz,canbefoundforseveralcouplesofparities.
For example, the Ans¨atze obtained for ((cid:15) ,(cid:15) ) = (1,1)
R6 σ
are also present for all other ((cid:15) ,(cid:15) ). Indeed as their
R6 σ
MF parameters are real, they are not sensitive to time
FIG.3: (Coloronline)Ans¨atzerespectingtheX symmetries reversal and any (cid:15) , (cid:15) can be chosen. From the clas-
e R6 σ
on the triangular lattice. All arrows carry Bij parameters sical point of view, these Ans¨atze describe coplanar spin
of modulus B1 and of argument φB1 and Aij parameters of configurations, which are invariant under a global spin
modulus A1 and of argument 0 on red arrows (choice of the flip followed by a π rotation around an axis perpendicu-
gauge), 2kπ/3 on blue ones and 4kπ/3 on green ones. On
lar to the spin plan.
dashed arrows A and B take an extra p π phase.
ij ij 1 Finally, there are nine different WS Ans¨atze families,
giveninTableI.Wenowconcludethissectionbyaseries
of remarks concerning the solutions we have obtained:
Now, we have to find all the Ans¨atze compatible with i) The number of WS Ans¨atze families is larger than
these PSG’s.42 The first useful insight is to count the the number of algebraic PSG of X , because the
e
number of independent bonds. Here, one can obtain any operators in X can act in different ways on the
o
bond from any other by a series of rotations and trans- Ans¨atze.
lations (i.e., elements of χ ). Thus, if we fix the value
e
of A and B on a bond ij, we can deduce all other ii) Amongstthese9Ans¨atzefamilies,onlythetwofirst
ij ij
bond parameters from the PSG. Note that A can be are non chiral, and the 6 others are TRSB Ans¨atze
ij
chosen real by using the gauge freedom. The value of all (by applying T, k = 1 is changed to k = −1 and
bond parameters are represented on Fig. 3 as a function φ to −φ ). The 6 families obtained by aplying
B1 B1
of their value on the reference bond. The unit cell of T are not listed here.
the Ansatz contains up to two sites because p may be
1
iii) These solutions are called families as the moduli
non-zero.
A and B can vary continuously without modi-
From now on we can forget about the PSG construc- 1 1
fying the symmetries. The third Ansatz has no
tionandonlyretainthedefinitionoftheAnsatzgivenby
fixed value for φ and includes the first and sec-
Fig.3anditsminimalsetofparameters: twointegersp1 B1
ond Ans¨atze families (they are kept as distinct as
and k, two modulus A and B , and one argument φ .
1 1 B1 they are non chiral).
Until now, we have only considered the subgroup X
e
and we have looked for Ans¨atze strictly respecting these
iv) The fluxes of these Ans¨atze are easily calculated
symmetries. We now want to consider all symmetries
using Fig. 3.
in X, but the symmetries in X will be obeyed mod-
o
ulo an eventual time-reversal symmetry. This requires v) ThedetailedlistofcompatibleAns¨atzedependson
supplementary conditions on the Ans¨atze of Fig. 3. As the choice of the mean-field parameters (here, non
explained in Sec. IVB, the transformations of X im- zero A and B on first neighbor bonds) as we
o ij ij
ply relations between the modulus and the arguments of explain in App: C by contrasting these results to
the Ansatz. Since we are in a very simple case, where those of Wang et al. on the same lattice.4
all bonds are equivalent in X , no extra relation on the
e
modulus can be extracted from X . However, some con-
o
ditions can be found by examining how the the fluxes
Arg(A A∗ A A∗) on an elementary rhomboedron and B. Condensation of the WS Ans¨atze: the missing
ij jk kl li
tetrahedral state
Arg(A B A∗)onanelementarytriangletransformwith
ij jk ki
R and σ. Assuming that neither A nor B are zero we
6 1 1
find: The SBMFT has already been used to study the anti-
ferromagnetic Heisenberg first-neighbor Hamiltonian on
2kπ(1−(cid:15)R6)/3 = 0 (34a) (tEheq.t5rica)nogrulwarithlabttoitche AwithantdheBA1ij9-o(Enlqy. 5dae)c.ouTphliengcl4a,2s3-
2kπ(1+(cid:15) )/3 = 0 (34b) ij ij
σ sical limit of this model gives the well known three sub-
(1+(cid:15)R6)φB1 = p1π (34c) lattice N´eel order with coplanar spins at angles of 120◦.
(1−(cid:15) )φ = p π (34d) The bond parameters obtained from this classical order
σ B1 1