Quantum spin nematics, dimerization, and deconfined criticality in quasi-one dimensional spin-1 magnets

0.11 MB·English

by  Tarun Grover

Quantum spin nematics, dimerization, and deconfined criticality in quasi-one
dimensional spin-1 magnets
Tarun Grover1 and T. Senthil1,2
1Center for Condensed Matter Theory, Indian Institute of Science, Bangalore, India 560012 and
2Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
(Dated: February 6, 2008)
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Westudytheoreticallythedestructionofspinnematicorderduetoquantumfluctuationsinquasi-
0
one dimensional spin-1 magnets. If the nematic ordering is disordered by condensing disclinations
0
then quantum Berry phase effects induce dimerization in the resulting paramagnet. We develop
2
a theory for a Landau-forbidden second order transition between the spin nematic and dimerized

n states found in recent numerical calculations. Numerical tests of thetheory are suggested.
a
J PACSnumbers:

6
1
Studies of low dimensional quantum magnetism pro- situationsstudiedinRef. [3,4]forotherphasetransitions

] vide a good theoretical platform to develop intuition in quantum magnets.
l
e about the physics of strongly correlated many particle Inthispaperweprovideanunderstandingofthesephe-
- systems. In this paper we study various theoretical phe- nomena. Firstweprovidegeneralargumentsrelatingthe
r
t nomena in spin S = 1 quantum magnets with SU(2) spontaneous dimerization with one route to killing spin
s
invariant nearest neighbor interactions. Specifically we nematic order by quantum fluctuations. When applied
.
t
a focus on spin nematic order in such quantum magnets to one dimension our arguments explain the absence in
m and its destruction by quantum fluctuations. numerical calculations5 of the featureless quantum dis-
A general Hamiltonian describing such spin S = 1 ordered spin nematic proposed by Chubukov6 for spin-1
-
d quantum magnets takes the form chains. Furtherweshowthataputativedirectsecondor-
n der quantum phase transition between the spin nematic
o 2
c H = JijS~i.S~j −Kij S~i.S~j (1) and dimerized phases is described by a continuum field
[ <Xij> (cid:16) (cid:17) theory with the action

1
In real materials the ratio K/J is probably small; how- S = d3x|(∂ −iA )D~|2+r|D~|2+ (3)
v µ µ
ever it has been proposed that arbitrary values of K/J Z
1
6 can be engineered in ultra-cold atomic Bose gases with u |D~|2 2−v(D~)2(D~∗)2+ 1 (ǫ ∂ A )2
3 spin in optical lattices1. We will focus exclusively on a (cid:16) (cid:17) e2 µνλ ν λ
1 rectangular lattice where the couplings J,K on vertical
0 bondsareafactorofλsmallerthanthoseonthehorizon- HereD~ isacomplexthreecomponentvectorandAµ isa
7 tal bonds. This model was studied numerically recently non-compactU(1)gaugefield. Thenematicphaseoccurs
0 (for K > 0) in an interesting paper by Harada et al2. when D~ condenses while D~ is gapped in the dimerized
/
t In the isotropic limit λ = 1, they found that there is a paramagnet. This theory is ananisotropic versionof the
a
m firstorder phase transitionfroma collinearNeel state to non-compact CP2 model (NCCP2). The two compo-
a spin nematic state (along the line J = 0) with order nent version- the anisotropicNCCP1 model - describes
-
d parameter Neel-VBS transitions of easy plane spin-1/2 magnets on
n the square lattice3. A second order nematic-dimer tran-
o SαSβ +SβSα 2 sition on the rectangular lattice is possible if doubled in-
Q = − δ 6=0 (2)
c αβ (cid:28) 2 3 αβ(cid:29) stantonsinthegaugefieldAµareirrelevantatthecritical
:
v fixed point of this theory. These instantons are relevant
i eventhoughthereisnoorderedmoment<S~ >=0. This at the paramagnetic fixed point of Eqn. 3. This leads to
X
spin nematic state corresponds to the development of a confinement of the D~ fields and to dimer order. Indeed
r
a spontaneous hard axis anisotropy in the ground state. the dimer order parameteris simply the single instanton
When λ is decreased from 1 to make the lattice rect- operator. A direct second order nematic-dimer transi-
angular quantum fluctuations are enhanced. The Neel tion is thus accompanied by the dangerous irrelevance
and spin nematic phases then undergo quantum phase of doubled instantons and the associated two diverging
transitions to quantum paramagnets. Interestingly it is length/time scales.
found that the spin nematic phase gives way to a dimer- We will provide two different arguments to justify our
ized quantumparamagnetwhere neighboringspin-1 mo- results. First we address the quantum disorderingof the
ments formstrongsingletsalongeveryotherbondinthe nematicbasedongeneraleffectivefieldtheoryconsidera-
horizontal direction. Further the quantum phase transi- tionsthatfocusonthepropertiesoftopologicaldefectsof
tionitselfappearstobesecondorderinviolationofnaive the nematic order parameter. Second we provide a more
expectations based on Landau theory but similar to the microscopic argument based on an exact slave ‘triplon’
2
representation7 of the S =1 operators. path. Thus there is a Berryphase of −1for closedpaths
The order parameter manifold for the spin nematic where dˆreturns to −dˆ.
state may be taken to be the possible orientations of Thephasefactorof−1fornontrivialclosedtimeevolu-
the spontaneous hard axis dˆ(the “director”) and thus tionsofdˆataspatialsitemay be naturallyincorporated
is S2/Z2. The Z2 simply reflects the fact that dˆand −dˆ intothe effective lattice modelofEqn. 4above. Firstwe
are the same state. The spin nematic state allows for notethataclosedloopintimewheredˆwindsbyπ corre-
Z2 pointvortexdefects (“disclinations”)intwospacedi- spondstoaconfigurationwithZ2 gaugeflux−1through
mensions. Thedirectordˆwindsbyπ onencirclingsucha the loop. The Berry phase is thus simply
disclination. AsdiscussedbyLammertetal8 forclassical
e−SB = σ (r) (7)
nematicsthefateofthedisclinationscruciallydetermines τ
thenatureofthe“isotropic”phaseobtainedwhenthene- Yr
matic is disordered by fluctuations. If the transition out Ateachspacepointtheproductoverthetime-likebonds
of the nematic occurs without condensing the disclina- measuresthefluxoftheZ2gaugefieldthroughtheclosed
tions then a novel topologically ordered phase - inter- timeloopatthatpoint. PreciselythisBerryphasefactor
preted in the present context as a quantum spin liquid arises in Z2 gauge theoretic formulations of a number of
- obtains. However the nematic may also be disordered different strong correlation problems9, and the theory is
in a more conventional way by condensing the disclina- knownasthe odd Z2 gaugetheory. Thus anappropriate
tions. In the present context we argue that non-trivial effective model for disordering the S = 1 spin nematic
quantum Berry phases associated with the disclinations state isa theoryofdˆcoupledto anoddZ2 gaugetheory.
lead to broken translational symmetry in this quantum
The spin nematic orderedphase correspondsto a con-
paramagnet. Furthermore this transition may be second densateofdˆ. InthisphasetheZ2disclinationsaresimply
order as described below (unlike the classical nematic-
associatedwith Z2 flux configurationsof the gauge field.
isotropic transition).
Thus the Berry phase term associated with the gauge
Following Lammert et al8 we consider the quantum
field directly affects the dynamics of the disclinations.
phase transition out of the nematic using an effective Disorderedphases where dˆhas short rangedcorrelations
model in terms of the director dˆ. The dˆ,−dˆidentifica- may be discussed by integrating out the dˆfield. The re-
tion requires that the dˆvector is coupled to a Z2 gauge sultispureoddZ2 gaugetheoryonaspatiallatticewith
field. Thus we consider the following action on a three
rectangular symmetry. This theory is well understood.
dimensional spacetime lattice:
It is conveniently analysed by a duality transformation
to a stacked fully frustrated Ising model13 followed by a
S = Sd+SB (4) soft spin Landau-Ginzburg analysis10. This leads to a
Sd = − tdµσµ(r)dˆr.dˆr+µ (5) mapping to an XY model with four-fold anisotropy:
<rX,r+µ>
Sv =−tv cos(φR−φR′)−κ cos(4φR) (8)
Here r represent the sites of a cubic spacetime lattice, <RXR′> XR
µ = (x,y,τ), σµ(r) = ±1 is a Z2 gauge field on the link Here R,R′ are sites of the dual cubic lattice. The
between r and r+µ. The termS is the Berryphase to
B real and imaginary parts of the field eiφR correspond to
be elaborated below.
Fourier components of the Z2 vortex near two different
The Berry phases arise from the nontrivial quantum
wavevectors at which the quadratic part of the Landau-
dynamics of the dˆ vector and can be understood very
Ginzburg action has minima. The anisotropy is four-
simply by considering a single quantum spin S =1 with
fold on the rectangular spatial lattice as opposed to the
a time varying hard axis dˆ(τ) that represents the fluctu- 8-fold anisotropy that obtains with square symmetry10.
ating local director field: There is a disordered phase where the Z2 vortex has
short-ranged correlations: this corresponds to the topo-
2
H= dˆ(τ).S~ (6) logicallyorderedquantumspinliquidintheoriginalspin
(cid:16) (cid:17) model. In addition there are ordered phases associated
withcondensationoftheeiφR. Thesephasesbreaktrans-
For a time independent dˆ, the groundstate is simply the
lation symmetry. For the rectangular lattice of interest
state where the projection S~.dˆ= 0. The Berry phase is the natural symmetry breaking pattern is dimerization
obtained by considering a slow time varying closed path along the chain direction.
ofdˆintheadiabaticapproximation. Therearetwokinds We thus see that Berry phases associated with the
ofsuchclosedpathsthataretopologicallydistinct. First quantum dynamics of the director dˆ lead to dimeriza-
there are paths for which dˆreturns to itself. For such tionwhen the nematic orderis disorderedby condensing
paths it is easy to see that the Berry phase factor is 1. theZ2 disclinations. Thisanalysiscanbeeasilyrepeated
Then there are closed paths where dˆreturns to −dˆ. In in one spatial dimension. Then the Z2 disclinations are
the adiabatic approximation with S =1 it is easy to see point defects in spacetime. These are described by the
that the wavefunction acquires a phase of π for such a oddZ2 gaugetheoryin 1+1dimensions whichis always
3
confined and which has a dimerized ground state11. In withS givenineqn. 4. Thesumoverσ cannowbeper-
d
particular this argument shows that for S = 1 chains a formed. Theuniversalprpertiesarecorrectlycapturedin
featureless disordered spin nematic state will not exist. a model that keeps the lowest order cross term between
Returningtotwodimensionswemaynowwritedowna td and t . We therefore get
θ
field theory for the nematic-dimer transition. The Berry
phases on the disclinations are encapsulated in Eqn. 8. S =− tµcos(θr−θr′+Arr′)dˆr.dˆr′+U (∇~ ×A~)2
We now need to couple these back to the dˆvector. The <Xrr′> XP
main interaction between dˆand eiφR is the long ranged (20)
statisticalone: ongoingaroundaparticle createdbyeiφ withtµ ∼tdµtθ. Itisinstructivetointroducethecomplex
the vector dˆacquires a minus sign. vector D~ =eiθrdˆ that satisfies
r r
We proceedby firstignoringthe κ termin Eqn. 8 and
using a Villain representation to let |D~|2 = 1 (21)
D~ ×D~∗ = 0 (22)
Sv = UjR2R′ (9)
<RXR′> Thesecondconditionmaybeimposedsoftlybyincluding
a term
The jRR′ are integer valued currents of the eiφ that sat-
isfy the conditions v|D~ ×D~∗|2 =−v (D~)2(D~∗)2−1 (23)
(cid:16) (cid:17)
∇~.~j = 0 (10)
with v >0. Thus we arrive at the model
(−1)j = σ (11)
YP S = SD+SA (24)
The first equation expresses currentconservation(which SD = −t eiArr′D~r∗.D~r′ +c.c−v(D~)2(D~∗)2(25)
holdsatκ=0). Inthesecondthesymbol referstoa <Xrr′>
P
product over the four bonds of the directQlattice pierced
by <RR′ >. This term ensures that an eiφ particle acts with D~ satisfying |D~|2 = 1. At v = 0 this is the lattice
as π flux for the dˆfield. Solving Eqn. 10 by~j = ∇~ ×A~ CP2 model with a global SU(3) symmetry associated
(with A integer) we get with rotations of the D~ field. The v term breaks the
µ
SU(3) symmetry down to SO(3). Eqn. 3 is precisely a
(−1)∇~×A~ = σ (12) soft-spin continuum version of the lattice action above.
YP Wenowconsidertheroleofthefourfoldanisotropyon
the disclinationfield eiφ (the κ termin Eqn8). Without
Now write A = 2a+s with a an integer and s = 0,1 so
this term the number conjugate to φ is conserved. In
that
the dual description this translates into conservation of
(−1)s = σ (13) the magnetic flux of the U(1) gauge field A~. Thus at
κ=0thegaugefieldisnoncompact. Theκtermhowever
YP YP
destroysthis conservationlaw - indeed four disclinations
which can be solved by choosing can be created or destroyed together. In the effective
model of Eqn. 3, a disclination in D~ corresponds to a
(−1)s =1−2s =σ (14)
configuration where the gauge flux is equal to π. Thus
the κ term may be interpreted as a doubled ‘instanton’
The integer constraint on A may be implemented softly
operator that changes the gauge flux by 4π.
by including a term
The nematic order parameter is simply related to the
−tθcos(2πa)=−tθσrr′cos(πArr′) (15) D~ fields:
WenowseparateoutthelongitudinalpartofAbyletting Q = Dα∗Dβ +c.c − δαβ (26)
αβ
(cid:18) 2 3 (cid:19)
1
A~ →A~+ ∇~θ (16)
π Thus when D~ condenses nematic order develops. The
After afurtherrescalingA→ A wefinallygetthe action paramagnetic phase occurs when D~ is gapped. In the
π absenceofinstantonsthelowenergytheoryofthisphase
S = S +S +S (17) hasafree propagatingmasslessphoton. Instantonshow-
d θ A
ever gap out the photon and confine the D~ fields. The
Sθ = −tθ σrr′cos(θr−θr′ +Arr′) (18) dimerorderparametereiφisthesingleinstantonoperator
<Xrr′>
andgetspinnedinthisphase. Adirectsecondordertran-
2
S = U ∇~ ×A~ (19) sitionbetweenthenematicanddimerizedstatescanthus
A
XP (cid:16) (cid:17) occur if doubled instantons are irrelevant at the critical
4
fixed point of the anisotropic NCCP2 action associated correlators. The symmetric part of this tensor is the ne-
with the condensation of D~. maticorderparameterandtheantisymmetricpartisthe
A different more microscopic argument can also be Neel vector.
usedtojustify Eqn. 3andprovidesfurtherinsight. Con-
IfnowasmallJ <0isturnedontheSU(3)symmetry
sider the following exact representation7 of a spin-1 op-
is explicitly brokendownto SO(3). This sign of J disfa-
erator at a site i in terms of a ‘slave’ triplon operator
vors Neel ordering so that nematic ordering wins in the
w~i: two dimensional limit. The NCCP2 field theory of the
S~ =−iw~†×w~ (27) transitiontothe dimer statemustthenbe supplemented
i i i with an anisotropy term v|N~|2 with v > 0 which due to
together with the constraint w~†.w~ = 1. The w~ satisfy Eqn. 31 is precisely the anisotropy term of Eqn. 3.
i i i
usual boson commutation relations. The nematic order What may we say about the NCCP2 field theory
parameter is readily seen to simply be and the fate of doubled instantons? First as the criti-
cal theory is relativistic the dynamical critical exponent
Q = δαβ − wα†wβ +c.c (28) z = 1. Next we note that the instanton scaling dimen-
αβ (cid:28) 3 2 (cid:29) sionisexpectedtobebiggerforNCCP2 ascomparedto
NCCP1. In the isotropic case existing estimates4 give
Aswithotherslaveparticles,thisrepresentationleadsto 0.63 for the single instanton scaling dimension. In a
a U(1) gauge redundancy associated with letting naive RPA treatment of the gauge fluctuations the in-
stanton scaling dimension scales like m2N where m is
w~ →eiαiw~ (29)
i i the instantonchargeandN isthe numberofbosoncom-
ponents. Thus within this approximation we estimate
at each lattice site. It is convenient to first consider the
the doubled instanton scaling dimension in NCCP2 as
specialpoint Jij =0 where the Hamiltonianin Eqn. 1 is 3(2)2(.63) ≈ 3.78. This admittedly crude estimate nev-
known to have extra SU(3) symmetry. Then H may be 2
ertheless suggests that doubled instantons may be irrel-
rewritten (upto an overalladditive constant)
evant for NCCP2.
H =− K w~†.w~† (w~ .w~ ) (30) The possible irrelevance of the doubled instantons
ij i j i j
<Xij> (cid:16) (cid:17) has dramatic consequences for the phenomena at the
nematic-dimer transition. It implies that the critical
This is invariant under a global multiplication of w~ by fixed point has enlarged U(1) symmetry associated with
i
an SU(3) matrix U on one sublattice and by U∗ on the conservation of the gauge flux exactly like in Ref. 3.
other. SuchmagnetswerestudiedindetailinRef. 12and Thisenlargedsymmetryimpliesthatthe(π,0)columnar
wecantakeovermanyoftheirresults. Astandardmean dimer order parameter may be rotated into the (0,π)
field approximation with < w~ .w~ >=6 0 yields a param- columnar dimer or into plaquette order parameters at
i j
agnetic phase with gapped w~ particles in the d=1 limit (0,π), (π,0). Thus right at the critical point all these
whileintwodimensionsthew~ condensetherebybreaking different VBS orders will have the same power law cor-
the SU(3) symmetry. The theory of fluctuations beyond relations. It will be an interesting check of the theory of
mean field includes a compact U(1) gauge field. In the this paper to look for this in future numerical calcula-
paramagnetic phase instanton fluctuations of this gauge tions.
field confine the w~ particles and their Berry phases lead
In summary we have studied the destruction of spin
to dimerization on the rectangular lattice. The results
nematic order by quantum fluctuations in quasi-one di-
of Ref. 12 now imply that the transition associatedwith
mensionalspin-1magnets. WeshowedthatBerryphases
w~ condensation is described by an NCCP2 model with
associated with disclinations lead to dimerization if the
doubled instantons, i.e it is precisely of the form of Eqn.
nematic is disordered by their condensation. We pre-
3 but with v = 0. The triplon w~ on the A sublattice
i sented a continuum field theory for a Landau-forbidden
∼D~ while onthe other sublatticew~ ∼D~∗. Thus we see
i second order transition between nematic and dimerized
that the Neel vector N~ is simply related to D~ through
phases that generalizesearlier workonother transitions.
Future numerical work or cold atoms experiments may
N~ ∼−iD~∗×D~ (31)
be able to explore the physics described in this paper.
For the SU(3) symmetric Hamiltonian all eight compo- TS gratefully acknowledges support from the DAE-
nents of the tensor D∗D − |D|2δ /3 have the same SRC Outstanding Investigator Programin India.
α β αβ
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