Many-body spin Berry phases emerging from the $π$-flux state: antiferromagnetic/valence-bond-solid competition

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by  Akihiro Tanaka

Many-body spin Berry phases emerging from the π-flux state:
antiferromagnetic/valence-bond-solid competition
Akihiro Tanaka and Xiao Hu
Computational Materials Science Center, National Institute for Materials Science, Sengen 1-2-1 Tsukuba 305-0047, Japan
()
We uncover new topology-related features of the π-flux saddle-point solution of the D=2+1
5
Heisenbergantiferromagnet. Wenotethatsymmetriesofthespinonssustain abuilt-incompetition
0
0 between antiferromagnetic (AF) and valence-bond-solid (VBS) orders, the two tendencies central
2 to recent developments on quantum criticality. An effective theory containing an analogue of the
Wess-Zumino-Witten term is derived, which generates quantum phases related to AF monopoles
n
with VBS cores, and reproduces Haldane’s hedgehog Berry phases. The theory readily generalizes
a
J to π-fluxstates for all D.

9
Our understanding on quantum critical points parameter,the condensation of which would give way to
1
[1]/phases [2] in D=2+1 antiferromagets, and the issue aVBSstate. Indeedwewillseethatincorporationofthis

] ofdeconfinementthereinhaverecentlyundergonearapid feature is essential in recovering Haldane’s Berry phase
l
e sequence ofdevelopments. Competition betweenantifer- starting from the π-flux state.
r- romagnetic(AF)andvalence-bond-solid(VBS)-like fluc- Continuum fermion model The π-flux hopping Hamil-
st tuations constitute the basic premises for much of these tonian on a two dimensional square lattice is Hπ =
. activities. These theories have brought into wide recog- tc† T c ,whereT withµ=x,ygeneratestrans-
t i,µ,σ iσ µ iσ µ
a nitiontherelevanceofmonopoledefectsoftheAForder- Plation by one site. The π-flux condition imposes the
m
parameter and in particular the nontrivial Berry phase anticommutation relation {T ,T } = 0, which im-
x y
- factors [3,4] associated with such objects. Here, with mediately leads to the spinon’s dispersion E(k) =
d
n these new perspectives, we revisit the Berry phase ef- ±t cos2kx+cos2ky with Dirac nodes at k = (π2,±π2).
o fect [5] in states emerging from the π-flux saddle point It ips convenient to group together the four cites sharing
c solution of the Heisenberg antiferromagnet [6], a popu- a unit plaquette (Fig.1) into components (with spin in-
[
lar point of departure for studying undoped and lightly dices) of a Dirac spinor, Ψ=t (ψ1σ,ψ2σ,ψ3σ,ψ4σ) [8].

2 doped cuprate Mott insulators. We find that their topo-
v logicalpropertiesareratherrich. Amongourfindingsare
5 (1) a chiral symmetry of the π-flux Dirac fermion relat-
6 ing the AF and VBS orders, which lead us to a natural r+eˆ r+eˆ +eˆ
3 y x y
framework for studying their mutual competition, (2) a
1
0 lowenergyeffectivetheorywithanovelmany-spinBerry
5 phasetermforwhichthe contributionsfromacomposite
0 defect(seebelow)reproducethemonopoleBerryphases,
/ 4 3
t (3) a natural extension of such framework to arbitrary
a dimensionswithpossiblerelevancetohigherdimensional r r+eˆ
m
spin liquids. 1 2 x
-
d It is worth digressing on the second point before pro-
n ceeding to the more technical aspects. An important FIG.1. Lattice used to derivecontinuum Dirac theory
o feature of monopole excitations is the energy cost due
c
to the rapid modulation of the AF order near the sin- To fix the representation of the Dirac gamma matri-
:
v gular core. Meanwhile, in the discussions which follow, ces, we account for the π-flux condition by assigning
Xi thesystemtakesadvantageoftheinherentAF-VBScom- negative hopping integrals −t to links residing on ev-
petition and saves energy by escaping into a local VBS ery other horizontal rows; all other links have positive
r
a stateatthedefectcores. Suchphysicsshareinspiritwith hopping integrals, +t. Linearizing around the nodes,
work by Levin and Senthil [7], who study AF-VBS com- we arrive at the Dirac action (hereafter we employ Eu-
petition starting from the VBS side. In that work, the clidean spacetime conventions) L = iΨ¯6∂Ψ, where the
four-state clock ordering of the VBS state is disordered slashindicates the contractionwith the gammamatrices
throughtheintroductionofZ4 vortices. Closeinspection γ0 =σ0⊗σz,γ1 =−σ0⊗σx,γ2 =σy⊗σy. Here the first
ofthe lattice modelshowsthat these defects haveanAF (second) matrix within a direct product determines the
core,asopposedtoconventionalvorticeswithfeatureless block structure (the matrix elements within the cells).
singularcores;hencetheircondensationleadstotheN´eel Hermele et al have recently shown that the algebraic
state. Likewise, it is natural to expect a VBS core to be spin liquid described by the π-flux Hamiltonian is sta-
present in a hedghog-like configuration of the AF order ble against monopoles, at least for large-N [2]. We now
1
moveawayfromthiscriticalphasebysupplementingthe which enables one to rewrite the variation of the
theory with mass terms so that the system can acquire fermionicdeterminantS =−lndetD[v ]intoaform
eff (2+1)
AF order. Indeed, earlier worksonthe π-flux state show suitable for generating a derivative expansion:
[9]thatasubstantialimprovementonthe variationalen-
1
ergy is acheived by adding on a spin density wave term δS =−tr D†δD . (3)
HSDW = iM(−1)ix+iyc†iσσzciσ, which appears to be eff (cid:20)−∂2+m2−m6∂V(2+1) (cid:21)
in accordPwith angular resolved photoemission experi-
It is easy to see that a nonlinear sigma (NLσ) model
ments on cuprate Mott insulators [10]. Being interested
S = 1 d3x(∂ v )2 arises,withg anonuniversal
in extracting the dependence of the effective action and NLσ 2g µ (2+1)
possible Berry phase terms on the N´eel unit director n, coupling coRnstant. Less trivial is an imaginary contribu-
we make in HSDW the generalization σz → n·σ, which tion to eq.(3), δSB2+P1, which we pick up at third order in
powers of 6∂V . As is usual with Wess-Zumino type
isphysicallyapotentialenergytermimposingthe gener- (2+1)
ationofAFspinmomentwiththe space-time-dependent terms, one recovers the action S2+1 from its variation
BP
orientation n(τ,x,y). In the continuum limit this be- δS2+1withtheaidofanauxiliaryvariablet∈[0,1]which
BP
comes the mass term LAF = imAFΨ¯(n·σ)Ψ. Next, we smoothly sweeps the extended function v(2+1)(t,xµ) be-
recall[8,11]thatincontrasttousualirreduciblerepresen- tween its two asymptotics, a fixed value at t = 0, say
tationsforDiracfermionsinD=2+1wherethenotionof (0,0,0,0,1), and the physical value at t = 1, v(2+1)(xµ).
chirality is absent (no “γ ”s), we have two generators The result is
5
of chiral transformations at our disposal, γ = σ ⊗σ
3 z y −2πiǫ 1
andγ5 =σx⊗σy whichanticommutemutuallyaswellas SB2+P1= Area(aSbc4d)eZ dtZd3xva∂tvb∂τvc∂xvd∂yve, (4)
withthespace-timecomponentsγ ,γ andγ . Onereads 0
0 1 2
off from the explicit matrix elements that the effects of
5
chiral mass terms proportional to Ψ¯γ3Ψ and Ψ¯γ5Ψ each where Area(S4) = Γ2π(52) = 38π2. Topologically, this is -
2
amounttobreakinglatticetranslationalsymmetrybyin- i2π times the winding number which counts the number
troducing bond alternationst→t+(−1)iµδt in the hor- of times the compactified “space-time” {(t,xµ)} isomor-
izontal (µ=x) and vertical (µ=y) directions. A crucial phic to S3 × S1 ∼ S4 wraps around the target space
observation here is that the SDW and the two VBS or- (also S4) for v (t,x ). It is important not to con-
(2+1) µ
dering potentials iΨ¯QΨ, Ψ¯γ Ψ and Ψ¯γ Ψ (Q ≡ n·σ) fusethisactionwithHopforChern-Simonsterms,which
3 5
all belong to the family of chirally rotated mass terms havebeenstudiedextensivelyinthecontextofchiralspin
Lchiral ∝iΨ¯Qei(αγ3+βγ5)⊗QΨ (α,β ∈R), i.e. they trans- systems [14] and are strongly tied to the dimensionality
formintooneanotherbysuitablechiraltransformations. D=2+1. Indeed, as we now show, the foregoing readily
Finally we mention that in the 4D irreducible represen- generalizes to theories of AF-VBS competition in arbi-
tation we have another possible mass term [11] Ψ¯γ γ Ψ trary space-time dimensions D=d+1, where for each d
3 5
which has relevance to chiral spin liquids (see footnote we find topological terms that are generalized versions
[21]). We do not retain this P- and T-violating term. of S2+1. We will return to the physical contents for the
BP
Bosonization The preceeding implies that the sponta- specific case of D=2+1 later.
neous breaking of the chiral symmetry present in the AF-VBS competition for general D Our detour starts
π-flux state can lead to AF or VBS orders, depending by mentioning a generic property of Clifford algebras
onthe chiralanglesα and β. This motivates us to study [15] which lies behind this generalization. Property:
theAF-VBScompetitionintermsofthefollowingtheory Let n be the number of matrices spanning the algebra
with a generalized mass term (equivalent to Lchiral), {γi,γj} = 2δij(i,j = 1,...,n). Representations for this
algebra can be realized by a set of 2p ×2p γ -matrices
i
L2+1 =iΨ¯ 6∂+mV Ψ, (1) where either n = 2p or n = 2p +1. To see why this
F (2+1)
(cid:2) (cid:3) goes hand in hand with the construction of a fermionic
where V ≡ v1σ + v2σ + v3σ + iγ v4 + iγ v5, theory describing AF-VBS competition let us consider a
(2+1) x y z 3 5
and v ≡ (v1,...,v5) is a five component unit vec- π-flux state on a d-dimensional hypercubic lattice. The
(2+1)
tor. Thefirstthreecomponentscompriseavectorv ≡ latter, in analogy with the D=2+1 case, is defined by
AF
(v1,v2,v3)whichisparalleltonandincompetitionwith theanticommutivityamongthegeneratorsoftranslation
a VBS-like order parameter (v4,v5). We now show that {Tl,Tm} = 0 for l 6= m(l,m = 1,...,d). This gives rise
this theory can be “bosonized” in terms of v , to toDiracnodeswithintheBrillouinzone.(Ford=1,where
(2+1)
yieldaneffectiveactionwhichcontainsanewBerryphase therearenonotionsofflux-lineswhichpierceplaquettes,
term. Centraltothisfeatisthefollowingrelation[12,13] itsufficesto simply startwithafree tight-bindingmodel
satisfiedbytheDiracoperatorD[v ]=i6∂+imV which gives rise to massless Dirac fermions. The argu-
(2+1) (2+1)
and its hermitian conjugate, mentsbelowappliesforthiscaseaswell.) Ingoingtothe
continuumlanguage,Diracspinorsareconstructedbydi-
D†D=−∂2+m2−m6∂V , (2) vidingalllatticesitesintocellsconsistingof2d sites. The
(2+1)
2
Dirac matrices γ are therefore 2d×2d matrices. Mean- the mathematical property forementioned when we put
µ
while, what we wish to construct is a fermionic theory p = d. (It is also straightword to work out an explicit
of the form Ld+1 = iΨ¯[6∂ + V ]Ψ, where notations derivation of Ld+1 starting from the lattice theory.)
F (d+1) F
are obvious extensions from those used in the D=2+1
case. The number of Dirac matrices required for this TheDiracoperatorD[v(d+1)]obeyseqs.(2)and(3),in
purpose is 2d+ 1, there being in addition to the d+1 which the replacement v(2+1) → v(d+1) is to be made.
space-time components γ ,...,γ , a total of d chiral ma- Carrying out the derivative expansion as before we ob-
0 d
trices (“γ ”s), each standing for the directions available tain the low energyeffective theory which is an O(3+d)
5
for dimerization. We see that this fits in nicely with NLσ model suppplemented with the topological term
−2πi 1
SBd+P1 = Area(Sd+2)Z dtZ dd+1xǫα1···αd+3vα1∂tvα2∂τvα3∂x1vα4 ···∂xdvαd+3. (5)
0
For D=1+1 (d=1), the isotropic O(4) theory with the models within the context of D=2+1 quantum chromo-
partition function Z[v(1+1)] = Dv(1+1)e−(SNLσ+SB1+P1) dynamics [19], where current algebras similar to those
has been identified (e.g. [12,16]R) with the conformally which determine the conformal field theory contents of
invariant SU(2) Wess-Zumino-Witten (WZW) model, theWZWmodelwerederived. Weneedbeawarethough,
1
the fixed point theory for the S=1/2Heisenberg antifer- oftheprivellagedroleofdimensionalityD=1+1forwhich
romagnet [17]. It is instructive to analyze the effect of the coupling constantbecomes dimensionless. Hence the
introducingdifferenttypesofanisotropybetweentheAF intriguing possibility of a nontrivial infrared fixed point
and dimer sectors in this model [16], breaking down the is unresolved,andwarrantsfurther study fromthe view-
symmetry to O(3)×Z or lower (the appearance of Z point of exotic quantum spin systems.
2 2
is a remnant of the underlying lattice). First, the effec- We now let an anisotropic term favoring the AF sector,
tive theory for the AF limit reduces at the semiclassical e.g. of the form −αvAF2 with α>0, take us away from
level to the O(3) NLσ model at topological angle θ =π, theisotropicregime. Incontrasttosimilarmodelsinthe
as originally proposed by Haldane [3]. Taking the oppo- irreduciblerepresentation[20],heretherearenotopolog-
site limit with complete dimerization makes the kinetic icallyconservedfermioniccurrentswhichforbidschanges
and Berry phase terms of the O(3) AF sector vanish, in the Skyrmion number Qxy = 41π dxdyn·∂xn×∂yn
reflecting the quenching of the spin moment. An inter- [21]. Finding Berryphases accompaRnyingsuch processes
mediate situation arises when the anisotropy modulates requires us to extract the dependence of eq.(4) on n,
in space, physically corresponding to a distribution of which proceeds in two steps. We first integrate over the
nonmagnetic impurities. This induces S=1/2 moments auxiliary variable t. We use without loss of generality
in the background of a singlet state, whose spin Berry the parametrization v1 = sin(tϕ)π1, v2 = sin(tϕ)π2,
phases are responsible for novel power-law correlations. v3 = sin(tϕ)π3, v4 = sin(tϕ)π4, and v5 = −cos(tϕ),
Theisotropic(WZW)pointmaybeviewedinlightofthis where π = (π1,..,π4) is a four component unit vector.
picture as the case where the anisotropy acquires a tem- The resulting Lagrangiandensity is
poral dependence as well. The main insight gained from 1
L= sin2ϕ(∂ π)2+(∂ ϕ)2 +iθq +L , (6)
these examples is how the interplay between AF (spin 2g µ µ τxy anis
momentgenerating)anddimer(spinmomentquenching) (cid:2) (cid:3)
orderingtendenciesdeterminestheBerryphase,whichin where θ = π(1 − 89cosϕ + 81cos3ϕ), qτxy ≡
turn acts back on the ordering of the system. Turning 2π12ǫabcdπa∂τπb∂xπc∂yπd, and Lanis is the anisotropy
to higher dimensions, three dimensional spin liquid sys- term. Notice that the first and third terms vanish, as
tems which may be realized in frustrated magnets have they must, under the spin-moment quenching condition
lately received considerable interest, where again subtle ϕ = 0. When the phase field ϕ is locked at a con-
Berry phase effects due to monopole configurations can stantvalue-correspondingtofixingthe bondalternation
be present [18]. We believe our approach as applied to strength in the vertical direction relative to the mag-
the D=3+1 case with the novel Berry phase term S3+1 nitude of the remaining four components of v, this re-
BP
provides a new route to capture the topological proper- duces to the D=2+1 O(4) nonlinear sigma model with
ties of such systems. a θ-term [12]. (This intermediate model should thus
Monopoles and Berry phases Returning now to the be relevant to spin systems with anisotropic bond al-
D=2+1 case, the isotropic O(5) theory has the inter- ternation [22,23].) Going on to the second step, we
esting feauture ofbeing the higher dimensionalanalogue now parametrize the components of π as π1 = sinφn1,
of the WZW model in the sense detailed above. This π2 = sinφn2, π3 = sinφn3, π4 = −cosφ. The Berry
parallelism has been explored to some extent for related phase term can now be recasted in a way which explic-
itly depends on monopole-like configurations [24]. The
3
result, obtained by integrating by parts, is methods also offers a framework to explore exotic Berry
i 9 1 phases in D=3+1 spin systems as well.
L =− (2φ−sin2φ)(1− cosϕ+ cos3ϕ)ρ , (7)
BP 2 8 8 m We acknowledgehelpful discussionsand/orcorrespon-
where ρm = ∂τ(ǫ4aπbcna∂xnb∂cnc)+c.p. is the monopole dI.eHnceersbuwti,thC.PMroufsd.ryA,.SA.bMaunroavk,aLm.iBaanldenLts.,YGu..Baskaran,
charge density. Integrating ρ over a space-time re-
m
gion surrounding the center of the monopole event gives
the change in the Skyrmion number between the two
time-slices before and after occurence of the event, i.e.
dτdxdyρ =∆Q .
m xy
R
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a b
b a M. P. A.Fisher, Science 303 1490 (2004).
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2
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[9] T.C.Hsu,Phys.Rev.B4111379(1990);C.Gros,Phys.
At each lattice site there is a competition between spin
Rev. B 38 931 (1988).
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4 [10] F. Ronninget al, Science 282 2067 (1998)
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enceofL ,thelatteremergeslocallywhenamonopole
anis pelquist,D.NashandL.C.R.Wijewardhana,Phys.Rev.
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2
whichimplies,accordingtoeq.(7)thatS1 =1,withthe [13] ItisinterestingtonotethedifferenceofourV withthat
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superscript standing for the sublattice index. We then given in ref [12] (see eq.(18) of this paper), which stems
go around the plaquette counterclockwise as depicted in from ouruseof the4D representation whilethe2D rep-
resentation is employed in ref [12]. Indeed our fermionic
Fig. 2, which simultaneously rotates the orientation of
actioncanberecastedintoamodelinthelatterrepresen-
the VBS order parameter by 90 degrees increment. Not-
tationwithaninternalSO(5)symmetry.Weareindebted
ing that the orientation of the “VBS clock” is specified
to A. Abanov for kindly clarifying this point.
by the angle φ−ϕ, we must correct for this by also in-
[14] E.Fradkin,FieldTheoriesofCondensedMatterSystems,
crementing φ by −π (while keeping ϕ fixed)or φ by π
2 2 Addison-Wesley,Redwood City CA 1991.
(keeping ϕ fixed). Either way the Berry phase shifts by [15] L.Frappat,A.SciarrinoandP.Sorba,DictionaryonLie
π2∆Qxy. In this way, we find that in order to have AF Algebras and Superalgebras, Academic Press, San Diego
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for all four sites sharing the plaquette, we must have [16] A.TanakaandX.Hu,Phys.Rev.Lett.88127004(2002).
SB2P =eiπ2∆Qxy, SB3P =eiπ∆Qxy, SB4P =ei32π∆Qxy. [17] I. Affleck and F. D. M. Haldane, Phys, Rev. B 36 5291
This methods also applies to the case where the VBS (1987).
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[19] G. Ferretti and S. G. Rajeev, Phys. Rev. Lett. 69 2033
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(1992).
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the spin. In addition, the framework can be extended to
[21] Instead, one is lead to an interesting relation <
make it applicable for studying staggered flux states. In Ψ¯γ3γ5γ0Ψ >∝ qxy where qxy is the Skyrmion number
this sense, our method is capable of ’generating’ a rich density (i.e. thelocal spin chirality).
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[25] T. K.Ng, Phys. Rev.Lett. 82 3504 (1999).
ferromagnets. We have alsodemonstratedthat the same
4