Properties and Detection of Spin Nematic Order in Strongly Correlated Electron Systems

0.31 MB·English

by  Daniel Podolsky

Properties and Detection of Spin Nematic Order in Strongly Correlated Electron
Systems
Daniel Podolsky and Eugene Demler
Department of Physics, Harvard University, Cambridge MA 02138
A spin nematic is a state which breaks spin SU(2) symmetry while preserving translational and
time reversal symmetries. Spin nematic order can arise naturally from charge fluctuations of a
5
spin stripe state. Focusing on the possible existence of such a state in strongly correlated electron
0
systems,webuildanematicwavefunctionstartingfromat J typemodel. Thenematicisaspin-
0
−
2 two operator, and therefore does not couple directly to neutrons. However, we show that neutron
scattering and Knight shift experiments can detect the spin anisotropy of electrons moving in a
n
nematic background. We find the mean field phase diagram for the nematic taking into account
a
spin-orbit effects.
J

4
1 I. INTRODUCTION and it consists of antiferromagnetically ordered domains

separated by anti-phase domain walls, across which the
]
el When ordered, classical spin systems can arrange direction of the staggered magnetization flips sign. The
order parameter is
- in a number of patterns, including (anti)ferromagnetic,
r
st canted,andhelicalstructures. Inadditiontothese,quan- S(r)=ΦeiKs·r+Φ∗e−iKs·r,
tum mechanics allows the formation of a wealth of mag-
.
at netic phases for quantum spins not available to their with Ks =(π,π)+~δ corresponding to antiferromagneti-
m classical counterparts. Due to its quantum numbers, cally ordered stripes with period 2π/δ . Here, the com-
| |
detection of such order is often difficult: For instance, plex vector Φ=eiθn takes its value within the manifold
-
d Nayak has considered a generalization of spin density ofground-statesS S /Z ;theZ quotientisnecessary
1 2 2 2
n wave (SDW) order, in which spin triplet particle-hole notto overcountph×ysicalconfigurations,asthe transfor-
o pairs of non-zero angular momentum condense with a mation eiθ eiθ, n n does not modify Φ. The
c → − → −
modulated density[1]. These states are characterized by real vector n gives the direction of the staggered mag-
[
spin currents rather than spin densities: thus, they do netization in the middle of a domain, while the phase

2 not couple at linear order to probes such as photons, factor eiθ specifies the location of the domain walls. A
v
neutrons,ornuclearspins. Onlyatsecondorderdothese shift in θ by 2π translates the system by the periodicity
9
phasescoupletoconventionalprobes,e.g. intwo-magnon of the SDW, and thus leaves the system invariant. As-
5
Raman scattering. Despite the challenges involved in sociated with collinear SDW order is charge order due
1
1 their detection, subtle forms of magnetic ordering such to the modulations in the amplitude of the local spin
1 asthese maybe necessaryto explainphenomenasuchas magnetization[12], which can be described by a general-
4 thespecificheatanomalyintheheavy-fermioncompound ized CDW order parameter
0 URu Si [2],andthepseudogapregimeinthecuprates[3].
t/ Pr2om2isingmaterialsfortheobservationofexoticmag- δρ(r)=ϕe2iKs·r+ϕ∗e−2iKs·r,
a
m netic phases include systems with strong antiferromag- forsomeSU(2)invariantobservableρ,notnecessarilythe
netic fluctuations suchasthe heavy-fermioncompounds, electrondensity. Inthe stripepicture, this CDWusually
-
d the organic superconductors, and the cuprates. For in- arisesfromtheaccumulationofholesatthedomainwalls.
n stance, the cuprates in the absence of carrier doping are Stripes were first observed in elastic neutron scat-
o antiferromagnetic Mott insulators at low temperatures. tering experiments on the spin-1 nickelate insula-
c As carriers are introduced through doping, the nature tor La Sr NiO , and coexistence of stripes with
v: of the magnetic order evolves until, for optimally doped superco2n−dxuctxivity4was first observed in underdoped
Xi and overdoped samples, the system becomes a metallic La1.6 xNd0.4SrxCuO4[13], where the unidirectional
paramagnet. In between these two limits, the under- chara−cter was demonstrated by transport[14] and
r
a doped cuprates have been argued to have spin glass[4] photoemission[15] measurements. In YBa2Cu3O6.35 and
and stripe phases[5]. The proximity between Mott insu- La1.6 xNd0.4SrxCuO4, it is observed that SDW order
lator and superconducting phases in the cuprates makes is firs−t destroyed by spin fluctuations[16]. In this case,
them ideal systems to study the hierarchy by which the memory of the charge modulation can remain, even af-
broken symmetry of Mott insulators is restored [6, 7]. ter averagingover the spin direction, resulting in a spin-
In this paper we will explore the possibility of detec- invariantCDWphase,whosepresencemayexplainrecent
tionofspinnematic order,adifferentquantummagnetic STM measurements in Bi Sr CaCu O [17, 18].
2 2 2 8+δ
phase, in strongly correlated electron systems[8, 9]. Ne- Forothermaterials,however,orinotherregionsofthe
maticorderhasbeenproposedasastateoriginatingfrom phase diagram, symmetry may be restored in a different
charge fluctuations of stripe order[10, 11]. A spin stripe order. In particular, Zaanen and Nussinov [10, 11] pro-
is a unidirectional collinear spin density wave (SDW), posedthatmanyexperimentalfeaturesofLaxSr1 xCuO4
−
2
magnons[10]. Hence, although spin nematic order can
haveimportantexperimentalconsequences,e.g. foranti-
ferromagnetic correlations in magnetic field experiments
on superconducting samples [10], its direct detection re-
mains a challenge.
The existence of other stripe liquid phases, different
from the spin nematic treated in this paper, as well as
proposalsfortheirdetection, arediscussedinRef.19. In
particular, the nematic state described there originates
from fluctuations of a unidirectional CDW that restore
translationalinvariancebut,bymaintainingamemoryof
theoriginalorientationoftheCDW,breaktherotational
symmetry (point group) of the lattice. Hence, unlike
the spin nematic, this “charge nematic” is SU(2) spin
invariant, and only breaks a discrete group.
FIG. 1: Charge fluctuations of a spin stripe, provided that
the domain walls (curves) maintain their integrity. A spin II. SPIN NEMATIC ORDER PARAMETER
nematic state is a linear superposition of fluctuating do-
main configurations such as these. Within each domain, the
A spin nematic is a state that breaks spin SU(2) sym-
staggered magnetization (arrows) is well-defined, and it flips
metry without breaking time reversal invariance [8, 20].
sign across the anti-phase domain walls. Due to fluctua-
The presence of spin nematic order can be observed in
tions, translational symmetry is restored to the system, and
the magnetization has expectation value zero at every point. the equal time spin-spin correlator
However, SU(2) symmetry is not restored, as there is a pre- Sˆα(r )Sˆβ(r ) = C(r ,r )δαβ +ǫαβγAγ(r ,r )
ferred axis in which spinsalign, modulo a sign. [10, 11]. h 1 2 i 1 2 1 2
+Qαβ(r ,r ). (1)
1 2
This expressioncorresponds to the SU(2) decomposition
can be explained by assuming that the spin stripe order (1) (1) (0) (1) (2) . We consider the
sym asym sym
⊗ ∼ ⊕ ⊕
is destroyed by charge fluctuations. In this picture, dy- threetermsappearingineq. (1)inturn. Thescalarfunc-
namical oscillations in the anti-phase domain-walls of a tion C is explicitly spin invariant. It contains important
spin stripe lead to a restoration of translational symme- information regarding charge order, but it does not help
try and a loss of N´eel order. However, although both us in defining a nematic phase. On the other hand, the
charge and spin order seem to be destroyed in this pro- pseudovector function A = Sˆ Sˆ gives a measure
1 2
h × i
cess, δρ = 0 and S = 0, full spin symmetry need not of the non-collinearity of the spin vector field. However,
be restored: So long as neither dislocations nor topo- in the case at hand where the nematic state originates
logical excitations of the spin proliferate, the integrity from charge fluctuations of a collinear SDW phase, we
of the domain walls allows the staggered magnetization expect A to vanish. This is supported by the fact that,
on each oscillating domain to be well-defined. Thus, al- from the point of view of the Ginzburg-Landau free en-
though the local magnetization does not have an expec- ergy, the pseudovector A cannot couple linearly to any
tation value, the magnetization modulo an overall sign function of the SDW order parameter Φ. As an aside,
does. Thisisnematicorder,inwhichSand Sareiden- we note that for systems with spin exchange anisotropy
−
tified, see Fig. 1. The order parameter can be chosen to of the Dzyaloshinskii-Moriya (DM) form, the DM vec-
be SαSβ S2δαβ/3 nαnβ 1/3n2δαβ =0, which tor will couple linearly to A, as expected from the weak
h − i ∝h − i6
has a non-zero expectation value. Because translational non-collinearity (canting) in such systems. However,the
invariance is restored in this process, the nematic order expectation value of A in this case comes from explicit
parameter is spatially uniform, instead of being modu- breaking of the spin symmetry.
lated by some multiple of the SDW wave vector. In ad- Thus, all information of interest to us is contained
dition, we expect the nematic to be uniaxial, with a sin- in the symmetric spin-2 tensor Qαβ. We define a sym-
gle preferred axis n (mod Z ) inherited from the nearby metrized traceless spin correlator Qˆαβ,
2
collinear SDW.
1 δαβ
Direct observationof spin nematic order through con- Qˆαβ(r ,r )= (SˆαSˆβ +SˆβSˆα) Sˆ Sˆ (2)
1 2 2 1 2 1 2 − 3 1· 2
ventional probes is difficult. For instance, neutrons do
not couple to nematic order, which is a spin-two opera- whose expectation value yields Qαβ directly,
tor. Similarly, nematic order is translationally invariant, Qαβ(r ,r )= Qˆαβ(r ,r ) . (3)
and does not give Bragg peaks in X-ray experiments. In 1 2 h 1 2 i
principle, two-magnon Raman scattering can probe ne- Starting from a SDW state, as domain wall fluctua-
matic order, but in practice it is difficult to separate the tions grow to destroy charge order, translational invari-
contributiondue tothe nematic fromthe creationoftwo ance is restored to the system, insuring that Qαβ(r ,r )
1 2
3
is independent of the center-of-mass coordinate, i.e. Zero holes One hole Two holes
Qαβ(r1,r2)=Qαβ(r1+R,r2+R)foranydisplacement E S=3/2 S=0
R. This fact alone signals the breaking of spin sym- S=1/2 S=1
metry, since the choice of a non-trivial (i.e. not pro- S=1/2 S=0 S=1
portional to δab) tensor Qαβ has been made across the S=2, s S=3/2 S=0
system. Thus, while Qαβ(r1,r2) decays exponentially S=1, px S=1, py S=0, dx 2 + y 2 S=1/2 S=3/2 S=1
with distance r r due to the absence of long-range
N´eel order, the| o1n−set2o|f nematic order is reflected in the S=1, dxy ta Ω S=1/2 S=0 b Ω
translationally-invariantexpectationvalueinthematrix-
S=0, s Ω
valued function (3). Since the original SDW state has a
singlepreferredspindirectionnˆ,the ensuingnematic or-
FIG.2: Spectrumoft-J modelona2 2plaquette. Adapted
der will be uniaxial, ×
from [23].
Qαβ(r ,r )=f(r r )Qαβ,
1 2 1− 2 0
(4) cluster. As is well known, it is impossible to describe
Qαβ =(nαnβ δαβ/3)S. nematic order in terms of a single spin-1/2 particle, as
0 −
the identification of “up” and “down” results in a trivial
Here,wehavedecomposedtheorderparameterintothree Hilbert space for the spin degree of freedom. Another
component objects: a function f describing the internal way to see this is that a spin-2 operator has a vanish-
structure ofthe nematic; the unit vectordirectorfieldn; ing expectationvalue with respectto anyspin-1/2state,
and the scalar magnitude S. As seen explicitly from (3), resulting in the identity Qˆαβ 0 whenever i=j.
the function f is parity-symmetric, f(r) = f( r). As ij ≡
This limitation can be overcome by coarse-graining a
−
will be shown below, f is dominated by the short range
group of spins and constructing a nematic wave func-
antiferromagnetic correlations between spins, and there-
tion out of them. In order to preserve the underlying
fore has a large contribution at the wave vector (π,π).
rotational symmetry of the system, we carry this out on
By definition, we choosethis contributionto be positive,
square 2 2 plaquettes of spins, and use energetic con-
f(π,π) > 0. With this convention, S > 0 corresponds ×
siderations to find the states most likely to contribute
to a N´eel vector that is locally aligned or anti-aligned
to magnetic order. We take the t-J model as a starting
with the director field n (sometimes referred to as the
point, analyzing the low energy Hilbert space in a man-
N+ phase in the literature ofclassicalliquid crystals,see ner similar to the projected SO(5) approachof Zhang et
e.g. Ref. 21), while S < 0 corresponds to a N´eel vec- al. [22] or the COREapproachofAltman andAuerbach
tor that is predominantly perpendicular to n (the N
− [23]. In these analyses, the lattice is first split into pla-
phase). We will show that, at low temperatures, S > 0
quettes. Oneachplaquette,theHamiltonianisdiagonal-
due to the local antiferromagnetic correlations whereas,
ized,andthemlowestenergystates ψ arekept,where
ν i
for anisotropic systems at high temperatures, a phase | i
ν 1,...,m and i labels the plaquette. For a lattice
with S < 0 is possible. Finally, we note that the uni- ∈ { }
composedofN plaquettes,thisallowsonetodefineapro-
directional SDW state (stripe phase) breaks the discrete jected subspace of the Hilbert space spanned by mN
rotational symmetry of a tetragonal lattice. This sym- M
factorizablestates of the form ψ ψ ψ .
metry may be restoredwhen charge fluctuations destroy | ν1i1⊗| ν2i2···| νNiN
Weassumethatthegroundstateiswell-containedin .
the SDW state to form the nematic. In this chase f(r) M
Figure 2 reproduces results in [23] for a t-J model on
will be symmetric under rotations on the plane by π/2,
a 2 2 plaquette. The lowest energy bosonic states are,
r R r. However, if a memory of the orientation ×
π/2 at half filling, the S =0 ground state Ω and the S =1
→
of the stripes survives the domain wall fluctuations, f magnontriplettˆ Ω ,andforaplaquet|teiwithtwoholes,
will not have such symmetry, yielding a “nematic spin- an S = 0 state†αˆb| Ωi . In addition, there are two low-
nematic”, i.e. a translationally invariant system that is †| i
lying S = 1/2 fermion doublets with one hole; however,
anisotropic in real space and in spin space.
unboundholesaredynamicallysuppressed,assupported
Sections III and IV will be devoted to understanding
by DMRG calculations on larger lattices, and we shall
thebehaviorofthethreecomponentfieldsofthenematic:
exclude the one-hole sector in what follows. Then, to
f,n, andS. Then,inSectionVwewillexplorehowthis
lowest order in the analysis, we only keep the low-lying
detailed knowledgecanbe usedin experimentalsearches
bosonic states, in terms of which a general low-energy
for nematic order.
plaquette state can be written as
III. NEMATIC WAVE FUNCTION |ψii = s+mαtˆ†α,i+cˆb†i |Ωii. (5)
(cid:16) (cid:17)
Inordertoexplorethepossiblesymmetrypropertiesof It is useful to get some intuition regarding the wave
the function f, we study its short wave length structure functions (5). Introducing the total spin and staggered
byexplicitconstructionofanematic operatoronasmall spinoperatorsonaplaquette,Sˆ =Sˆ +Sˆ +Sˆ +Sˆ and
1 2 3 4
4
Nˆ = Sˆ Sˆ +Sˆ Sˆ , as well as the DM-type vector, symmetry, with only singlet superconductivity and ne-
1 2 3 4
Dˆ =Sˆ −Sˆ Sˆ −Sˆ +Sˆ Sˆ Sˆ Sˆ , we find matic order surviving. We note that (8) ignores correla-
1 2 2 3 3 4 4 1
× − × × − × tions between spin degrees of freedom, controlled by tˆ ,
†α
Sˆα ∼iǫαβγtˆ†βtˆγ, and charge degrees of freedom, controlled by ˆb†. More
Nˆα ∼tˆα+tˆ†α, caocmcopulnetxanreemgaivtiecnwinavAepfupnecntdioixnsAt.hOatntathkeeoththisereffheacntdin,tino
Dˆα ∼i(tˆα−tˆ†α), practical calculations, one may consider a slightly sim-
pler wave function than (8) by replacing the θ by local
i
uptopositivemultiplicativefactorsanduptotermslying
Ising variables σ = 1 on each plaquette. This leads to
i
outside of the projected low-energy space. In particular, ±
the wave function
we find the expectation values on a single plaquette
hSˆii∼m∗×m, |ψ2i= " (s+σimαtˆ†α,i+cˆb†i)|Ωii#. (9)
Nˆi Re(s∗m), {σ1,σX2...}=±1 Yi
h i∼
Dˆ Im(s m), As before, this state has time-reversal and translational
i ∗
h i∼
symmetries restored, with only spin nematic and super-
ˆb s c,
h ii∼ ∗ conducting orders surviving. Finally, note that in order
Nˆiˆbi m∗c. (6) to produce a nematic state without any type of super-
h i∼
conductivity (singlet or triplet), it is necessary to intro-
Thelasttwoexpressionsdescribelocalsingletandtriplet duce another fluctuating Ising variable which multiplies
superconductivity, respectively. the term cˆb†i in (9), thus randomizing the relative phase
One can build a nematic on a cluster of plaquettes by
between all three components of the wave function.
choosing a factorizable state, in which s =0 and m is a
Despitethefactthatwehaveconsideredmanydifferent
constant real vector on every plaquette,
wavefunctions, depending onthe possible coexistence of
nematic order with different types of superconductivity,
|ψfacti= (mαtˆ†α,i+cˆb†i)|Ωii (7) the analysis above allows us to make strong predictions
i onthedominantshortwavelengthdependenceoff. This
Y
is because, of all the low energy plaquette states kept in
The constraint that m is real insures that no long range
theprojectedHilbertspace,onlythetriplettˆ Ω breaks
ferromagnetic or N´eel orders develop, whereas nematic †α| i
SU(2) symmetry. Hence, only this state can contribute
order does due to the SU(2)-symmetry breaking choice
directly to the nematic order parameter. There are six
of the vector m. Note from (6) that ψ is also
| facti differentpairsi=j ofsitesona2 2plaquette,andthe
a triplet superconducting state (except at half-filling, most general spi6n-2 operator Pˆαβ×on a plaquette can be
where c = 0). Order of this type is found, for instance,
writtenasalinearcombinationofthesix“link”operators
in the triplet superconducting state of quasi-one dimen-
Qˆ ,
sional Bechgaard salts, where the triplet superconduct- ij
ing order parameter is constant along the Fermi surface
Pˆαβ = α Qˆαβ. (10)
dueto thesplitting oftheFermisurfaceintotwodisjoint ij ij
Fermi sheets[24]. {i6=Xj}∈(cid:3)
On the other hand, in applications to materials such
It is useful to introduce an inner product for real func-
as the high T cuprates, we would like to introduce a
c tions α on the links of a plaquette, according to
ij
nematic state that is a singlet superconductor, instead
(α,β) = α β . With this, we can write a
of triplet. For this, the use of non-factorizable states is i=j (cid:3) ij ij
normalized b{a6sis}∈of plaquette operators Pˆ , correspond-
necessary. Forinstance,introducingalocalanglevariable P η
ing to 6 linearly independent functions αη which satisfy
θi [0,2π) on each plaquette, consider the state ij
∈ (αη,αν) = δην. We choose a basis of eigenoperators of
the symmetries of the plaquette, composed of operators
ψ = (dθ dθ ...) (8)
| 1i 1 2 with px, py, dx2+y2 and dxy symmetries, see Figure 3, in
Z
addition to two operators with s-wave symmetry,
" (s+mαcos(Q·ri−θi)tˆ†α,i+cˆb†i)|Ωii#, Pˆαβ = 1 Qˆαβ +Qˆαβ +Qˆαβ +Qˆαβ ,
Yi s1 2 12 23 34 41
(cid:16) (cid:17) (11)
where m is a constant real vector, and Q is the wave
vector of the underlying SDW. If the angle θi were held Pˆsα2β = √12 Qˆα13β +Qˆα24β .
constantacrossthe lattice, long rangeSDW orderwould (cid:16) (cid:17)
ensue. In contrast, by integrating independently over It is easy to see that among the six operators (10),
the θ at different sites, we introduce charge fluctuations only those with s-wave symmetry get a non-vanishing
i
that averageout the localmagnetizationto zero. Hence, expectation value with respect to the state (8). What’s
the state (8)has restoredtranslationalandtime-reversal more, the linear combination Pˆ = (Pˆ +√2Pˆ )/√3
sF s1 s2
5
1 2
β
s1 px dx 2 + y 2 para: S=0
γ
4 3
s2 py dxy N : S<0 N : S>0
FIG. 3: Basis of plaquette nematic operators with well-
defined symmetry properties. The solid lines denote links FIG. 4: Mean field phase diagram for a nematic without
with weight +1, dashed lines have weight 1. Thus, for in- anisotropy in the spin exchange interaction. The ordered
stance,Pˆs1 =(Qˆ12+Qˆ23+Qˆ34+Qˆ41)/2,wh−ichintermsofthe phases N± are uniaxial. All transitions are first order.
αij vectors reads αs1 =(eˆ12+eˆ23+eˆ34+eˆ41)/2. The other
basisvectorsareαs2 =(eˆ13+eˆ24)/√2,αpx =(eˆ23 eˆ41)/√2,
αpy = (eˆ12 eˆ34)/√2, αdx2+y2 = (eˆ12 eˆ23+eˆ34− eˆ41)/2, Therefore, at low temperatures, where the state of the
and αdxy =−(eˆ13 eˆ24)/√2. − − system is dominated by the low energy plaquette states,
−
S is positive.
has a vanishing expectation value, so that it is possible
to write anorthonormalbasis of operatorsin which only
IV. MEAN FIELD ANALYSIS
the basis operator,
Pˆ = 1 ( √2Pˆ +Pˆ ), (12) We now study the finite temperature phase diagram
sA √3 − s1 s2 of the spin nematic. We assume that superconductivity
is either present as a background phase throughout the
hasanon-zeroexpectationvalue. Thiscanbeunderstood
entire region of the phase diagram that we study, or not
as a consequence of local antiferromagnetic correlations,
present at all. Hence, we do not include explicitly the
since Pˆαβ can be expressed in terms of the plaquette
sA interaction between superconducting and nematic order
staggered magnetization N as
parameters. As is well known (see, e.g. Ref 25), symme-
1 δαβ tryallowstheinclusionofcubictermsintotheGinzburg-
Pˆαβ = NαNβ N2 . (13)
sA 2√6 − 3 Landau (GL) free energy of a nematic order parameter.
(cid:18) (cid:19) Thus,themostgeneralGLfreeenergyforaspin-isotropic
Using the relation system is, up to quartic order,
Qˆij = η αηijPˆη (14) FQ,iso =βTrQ20−γTrQ30+δTrQ40. (18)
X
whichholdsfori,jonthesameplaquette,weseethatthe By the identity (TrQ20)2 ≡ 2TrQ40, which holds for any
dominant short-range contribution to the nematic order 3 3 traceless symmetric matrix, we do not include the
parameter is te×rm (TrQ20)2 in (18). The phase diagram is shown in
Fig. 4. At high temperatures, β is large and positive,
Qαβ =αsA Pˆαβ . (15) andthesystemisdisordered,Q =0. Ontheotherhand,
ij ij h sA i 0
as temperature is reduced and the value of β decreases,
This is of the form (4) with
there is spontaneous breaking of spin SU(2) symmetry
f(k) = αsA(k) as the system undergoes a first order transition into the
nematic state. The nematic in this case is uniaxial as
4
= (coskxcosky coskx cosky), (16) desired(seesectionsIandII);inordertostabilizebiaxial
√6 − −
nematicorder,termsoforderQ5andQ6wouldhavetobe
0 0
Qαβ = Pˆαβ . added to the free energy[25]. We note that, as discussed
0 h sA i
insectionsIIandIII,S >0inthelowtemperaturephase.
Note that we can relate the real vector m in Eqs. (8)
Thus, the cubic coefficient γ must be positive.
and (9) to the director field n of the nematic, appearing
How is the above analysis modified by the presence of
in (4), through
spin-orbit effects? For definiteness, we concentrate on
Pˆαβ mαmβ m2δαβ/3, (17) the case of the cuprate LaxSr1 xCuO4 in the low tem-
h sA i∝ − perature orthorhombic (LTO)−phase. This compound
from which we conclude that n m. This can be used displays strong evidence for fluctuating stripe order in
∝
to constrain the sign of S, its underdoped regime[26], and the spin exchange con-
3 stants Jαβ are known in detail from neutron scattering
S = 2nαnβhPˆsαAβi>0. experiments in undoped La2CuO4[27, 28]. The analysis
6
presentedhere can be easily generalizedto other materi-
als. In LaxSr1 xCuO4, spin-orbit effects are small: The
anisotropic pa−rt of the spin exchange interaction Jαβ is
lessthan10 2 oftheisotropicpartinundopedLa CuO
− 2 4
[27, 28]. Yet, at low temperatures, this anisotropy leads
to a preferred direction for the N´eel order and to weak
ferromagnetism [29]. Thus, although the anisotropy is a
low energy effect, playing a weak role on the onset and
magnitude of the nematic order parameter, it may ulti-
mately fixthe preferredspinorientationforthe nematic.
This aspect of the interplay between J and Q will be
especially important when we try to separate their con-
tributionstotheanisotropyinthespinsusceptibility(see
cc
Eq. (26) below).
bb
Themostgeneralformofthespin-exchangeinteraction
for spins on nearest neighbor Cu sites is aa
H = JiαjβSiαSjβ (19) FIG. 5: The Dzyaloshinskii-Moriya vectors (dashed arrows)
Xhiji and ground state spin orientation (solid arrows) on a single
= J0,ijSi·Sj +Jsα,iβjSiαSjβ +Dij ·(Si×Sj) , CdeunOot2epClaunesiitnest.heTLhTeODMphavseectoofrsLaD2iCjupOo4in.tTahloenbgla±ckadaontds
Xhiji(cid:16) (cid:17) are staggered between adjacent bonds. This, combined with
easyplaneanisotropy,yieldaweaklycantedAForder,where
where Jαβ is a traceless symmetric tensor and D is the thespinsontheCusiteshavealargestaggeredcomponentin
s
Dzyaloshinskii-Moriya (DM) vector. Neither J nor J the c direction, and a small uniform component out of the
0 s ±
depend on the bond ij . It is convenient to work with plane (in theb direction, not shown).
h i
the principal axes of the system: the vectors a and c lie
on the CuO planes at 45 degrees from nearest neigh-
2
bor Cu-Cu bonds, and the vector b is normal to the DM vector induces a weak canting for staggered spins
CuO planes, see Fig. 5. In this basis, J is diagonal, that are perpendicular to it. This pushes N towards the
2 s
J 1+J =diag(Jaa,Jbb,Jcc)withJbb <Jaa Jcc [27]. plane normal to D, which requires λ′ > 0. Thus we see
0 s
For simplicity we take Jaa = Jcc in what fo≈llows, cor- that Js chooses an easy plane for the spins, and the DM
responding to easy plane antiferromagnetic interactions vector selects a preferred direction, c, within the easy
along the a-c plane. On the other hand, the DM vector plane [29], see Fig. 5. The ratio λ′/λ is unimportant in
D points along a, with the sign on each bond ij this case.
ij
given by the stagg±ered pattern shown in Fig. 5. Thuhs, i Wenowturnourattentiontothenematic. Inaddition
to the usual GL free energy for a nematic order param-
J +∆ 0 0 eter, eq. (18), the explicit symmetry breaking due to
0
J = 0 J 2∆ D , anisotropy in the spin exchange can be taken into ac-
ij 0
 0 −D J±+∆ count, to quadratic order in the spin-orbit interaction,
0
∓
by adding the terms,
 
where ∆ = (Jaa Jbb)/3 > 0. A useful measure of the
− F = F +F
relative importance of the anisotropies∆ and D is given Q Q,iso Q,anis
by the ratio x ≡ JD0∆2 , which is approximately equal to FQ,anis = αTr(JsQ0)+ α′DTQ0D. (21)
one in the LTO phase of La2CuO4[27, 28]. − J0
It is instructive to consider first the effects of spin
Notethat,unliketheN´eelcase(20),theorderparameter
anisotropy on an antiferromagnet. The GL free energy
Q couples linearly to the anisotropy. Thus, the symme-
F of a N´eel order parameter N is, to lowest order in
N
try is broken explicitly, and strictly speaking there is no
spin-orbit coupling,
disordered phase with Q = 0. This, however, does not
0
λ preclude the existence of crossovers in the order param-
F = λNTJ N+ ′(D N)2+µN2+νN4. (20)
N s eter, or even discontinuities at first order transitions, as
− J ·
0
we traverse the phase diagram. For weak anisotropy the
The term D N is a pseudoscalar and is forbidden by order parameter will be extremely small at high tem-
·
parityandtime-reversalsymmetries. Thefirsttwoterms peratures, and very sharp crossovers will be observed.
include the effects of anisotropy, and are both quadratic Another consequence of the linear coupling is that, un-
in the spin-orbit coupling. The coefficient λ is positive, likeantiferromagneticorderwhichalwaysliesontheeasy
as necessary to capture the tendency of spins to order plane of Jαβ, uniaxial nematic order can point either in
along the easy plane at low temperatures. Similarly, the thedirectionofmaximalcouplingofJαβ,orinthedirec-
7
β B 0.8
0.7
0.6
(cid:0)(cid:1)(cid:0)(cid:1) γ
(cid:0)(cid:1)C(cid:0)(cid:1)
0.5
A 0.4
S 0.3
S<0 S>0
N : N : 0.2
n c n=c
0.1
0
−0.1
FIG.6: Meanfieldphasediagramincludinganisotropyinthe
−0.2
spin exchange Jαβ, for the case Jbb < Jaa = Jcc, D = a. 0 0.1 0.2 0.3 0.4 0.5
All transitions are first order. Phase N+ is characterized±by β
S > 0 and n = c, phase N− by S < 0. The director n in
phase N− depends on the parameter w α′x/α: for w < 1, FIG. 7: Value of S along the dotted trajectory in Fig. 6.
≡
n=b,while for w>1, n=a. Theparameter w ismaterial- The parameters α∆ = 0.1, w < 1, and γ = δ = 1 are cho-
dependentandisunlikelytochangeacrossthephasediagram. sen for concreteness; β increases with temperature. As the
Thus, only a single type of N− phase is accessible within a anisotropy ∆ is decreased, or for trajectories that start deep
givenmaterial. Thedashedarrowshowsapossibletrajectory inside the N+ phase due to a larger value of γ, the value of
where a first order transition is crossed as temperature is in- S in the high temperature phase is reduced. Similarly, for
creased; alternatively, for smaller values of γ, the first order s|m|all values of γ, the first order transition may be avoided
transitionmaybeavoidedandstrongcrossoverbehaviormay and instead a strong crossover within the N− phase may be
be observed instead. observed.
V. DETECTION
In the presence of a spin nematic order parameter
Qαβ Qαβ(r ,r ) of the form (4), symmetry allows the
tion of minimal coupling of Jαβ, depending on the sign ij ≡ i j
term
of S. For the state with nematic order and S > 0, we
expectthedirectorfieldntolieontheeasyplaneandbe = g QαβSˆαSˆβ (22)
orthogonaltoD,i.e. beparalleltothedirectionthatthe Hint − ij i j
ij
X
SDW state would take in the absence of charge fluctua-
to enter the Hamiltonian, which can be thought of as a
tions. Thisconstrainsthelinearcoefficientsappearingin
local spin-spin interaction mediated by the nematic or-
(21) to be positive, α>0 and α >0.
′
der. Thecoefficientg ispositive,asnecessaryforthesta-
bility of the nematic order. When the system develops
In order to obtain the mean field phase diagram for a long-range nematic order, electrons move in a nematic
spinnematic,we comparethe minima ofF forthe direc- background which acts like an effective anisotropic spin
tor field n pointing along the various principal axes eˆ . exchange. This leads to anisotropy in the spin response
i
TheresultisshowninFig.6. Thelowtemperaturephase function. In the random phase approximation (RPA),
N+ is a uniaxial nematic characterized by S > 0 and when an external magnetic field B is applied to the sys-
n=c. The high temperature phase N− is also uniaxial, tem, H = H0 +Hint − iBi ·Sˆi, the electrons see an
butithasS <0. ThedirectorfieldinN dependsonthe effective field
− P
Trehlaistivqeuastnrteintygtihsomfathteertiawlodaenpiseontdreonpty,taenrdmsi,swun≡likαe′lxy/αto. Beαff(k)=Bα(k)+gQαβ(k)hSˆβ(k)i, (23)
change significantly over the phase diagram (except, of leading to the dynamic spin response
course,acrossastructuralphasetransition),sothatonly
one of the following scenarios should be observed within χαβ (k,ω)=χ (k,ω) ˆ1 +gQ(k)χ (k,ω) −1 αβ(24)
RPA 0 2 2 0
a given material: For w < 1, n = b, whereas for w > 1, ×
n=a. Thecoefficientβincreaseswithtemperature,pos- Spin nematic order thuhs(cid:0) induces anisotropy in(cid:1)thie spin
siblytuningafirstorderphasetransitionbetweenphases susceptibility. We introduce a tensor Ξαβ, defined by
N+ and N , or otherwise moving the system through a
−
sharp crossoverwithin the N− phase. In either case, the Ξαβ(k,ω) 1 (χ−1)αβ +(χ−1)βα δαβ (χ−1)cc.
temperature dependence of the order parameter leads to ≡ 2 − 3
c
strong experimental signatures discussed in Section V. (cid:2) (cid:3) X
Figure 7 shows the value of S as we move across the Experimental measurements of χαβ from polarized neu-
N+/N phase transition along the dashedline in Fig. 6. tronscatteringexperiments(forarbitrarywavevectork)
−
8
(0,π)
canbe used to compute Ξαβ, whichis the naturalobject
to consider when studying nematic order. In the current
ky
approximation, we find
Ξαβ (k,ω)=gQαβ(k). (25)
RPA
Similarly, Knight shift measurements give access to the (−π,0) (π,0)
local static spin susceptibility, i.e. the susceptibility in-
tegrated over all wave vectors k. To linear order in the
nematicorderparameterQ,theanisotropyintheKnight
shift over the various principal axes α is proportional to
d2k
[χ (k,ω =0)]2Ξαα(k,ω =0), (0,−π) kx
− (2π)2 0
Z
FIG.8: Wavevectordependenceoff(k)correspondingtothe
whEevreennoinsutmhemaabtisoenncoeveorfαneismiamtpiclieodr.dering, spin-orbit plaquette nematic operator PˆsA. This is expected to be the
dominant contribution to f, and enterstheanisotropy of the
couplingleads to anisotropyin the antiferromagneticex- spin susceptibility through Eq. 26. The “+” and “-” regions
change Jαβ, which enters the Hamiltonian in a term of show thesign of f(k) on theBrillouin zone.
ij
theform(22)withgQ J,seeeq.(19). Thecomplete
→−
expression for Ξαβ is therefore
whereas, if w <1, then n=b, and
Ξαβ (k,ω)=gQαβ(k) Jαβ(k). (26)
RPA − s Ξaa = Ξbb/2=Ξcc,
−
Note that only the traceless symmetric part of J con- Ξcc = g S f(k)/3 ∆η(k),
ij
| | −
tributes to Ξ. We face the challenge of untangling
the contributions to (26) due to the anisotropy in J Thus, if the system goes through a phase transition be-
from those coming from the presence of nematic or- tween phases N+ and N−, the discontinuity in S, shown
der. For this, the analysis of Section III is pivotal, inFig.7,togetherwiththechangeinthedirectorfieldn,
in particular eq. (16), which gives the dominant wave wouldgiveaveryclearexperimentalsignatureofnematic
vector dependence of the nematic contribution to the orderinKnightshiftandpolarizedneutronscatteringex-
anisotropy in the spin response function, see Figure 8. periments.
This can be combined with detailed knowledge of the Finally, we consider the prospect of observing direc-
form of J [27, 28], J (k) = η(k)∆diag(1,1, 2). As tor density waves (DDW), the Goldstone modes corre-
s s
before, ∆ = (Jaa Jbb)/3 > 0, and we assu−me that sponding to the nematic state, in neutron scattering ex-
anisotropyissmallb−eyondthenearest-neighborrangeto periments. To do this, we study the pole structure of
set η(k) = 2(cosk +cosk ). Thus, the contributions the RPA result (24). Expanding (24) about k = 0 and
x y
due to Q and J can be distinguished by the wave vec- ω = 0, we find two degenerate DDW modes, with speed
tor dependence of the signal, as one is proportional to vDDW g S ,
∝ | |
f(k) = 4 (cosk cosk cosk cosk ) and the other
√6 x y − x− y p k2
mtoeaηs(ukr)in=g2th(ceossuksxce+ptciobsilkityy).atFtohreepxoaimntpslek, =exp(πe,ri0m)eanntds χ⊥⊥(k,ω)∝ ω2−vD2DWk2, (27)
(0,π) are sensitive to Q, but not J, see Fig. 8.
with labelling either of the directions perpendicular
For completeness, we evaluate the expression (26) ⊥
to the director of the nematic. Equation (27) is con-
in the various phases shown in Fig. 6 for underdoped
sistent with an independent calculation of the DDW
LaxSr1 xCuO4. In the low temperature phase N+, we modesstartingfromaneffectivelowenergyquantumro-
expect,−
tormodel,seeSectionVI. WenotethattheDDWmodes
Ξaa = g S f(k)/3 ∆η(k), foundheremaybeoverdamped,dependingonthedetails
− | | − of the system. This would be the case, for instance, if
Ξbb = g S f(k)/3+2∆η(k),
v were smaller than the Fermi velocity. However,
− | | DDW
Ξcc = 2g S f(k)/3 ∆η(k), provided that the modes are underdamped, they can be
| | −
detected in neutron scattering experiments. This is sur-
InthehightemperaturephaseN−,wemustconsidertwo prisingat first,as localZ2 invarianceimplies that equal-
separate scenarios. If w >1, then n=a, and time correlators of the from SαSβ must decay expo-
h i ji
nentially at large distances x x . However, as (27)
Ξaa = 2g S f(k)/3 ∆η(k), | i − j|
− | | − indicates, the correlator at different times need not de-
Ξbb = g S f(k)/3+2∆η(k),
cayexponentially. Also note thatthe inelasticscattering
| |
Ξcc = g S f(k)/3 ∆η(k), peak(27)hasvanishingweightask 0,consistentwith
| | − →
9
theabsenceofanelasticBraggpeak. Unfortunately,this The product of nˆ operators then differs from the mean
feature makes DDW modes difficult to distinguish from field result by a correction,
the low energy collective modes of a quantum paramag-
net, computed in Appendix B. nˆαinˆβj −δijGαβ =δijραβ +λαijβ,
whichhasbeensplitintoalocaltermραβ,andanon-local
VI. QUANTUM ROTOR MODEL term, λαβ = 0 for i = j. Linearizing the equations of
ij
motion (31) with respect to these corrections, we obtain
Another approach to study Goldstone modes is to in-
iωLα = F(k)ǫαβγGβδργδ,
troduce a quantum rotor model which captures the low −
energy properties of the system. This can be done by iωραβ = (ǫβγδGαδ+ǫαγδGbd)(νLγ Bγ) (32)
− −
coarse-grainingspins in plaquettes, as carriedoutin sec-
where
tion III. There we found that the low energy plaquette
statesathalf-fillingcontainonesingletgroundstate,and d
one triplet state obtained by acting on the ground state F(k)=2g˜ (1 cosq ). (33)
i
by the staggered magnetization operator. This matches −
i=1
X
the low energy spectrum of a quantum rotor nˆ on the
i
plaquette i,whichcanbe writtenintermsofthe compo- Since we are interested in the linear response, we have
nent spins of the plaquette as excludedfromEq.(32)termsquadraticinB. Note that,
to this order in B, the non-local term λ drops out of
ij
Lˆ = Sˆ +Sˆ +Sˆ +Sˆ , the equations of motion.
i i1 i2 i3 i4
nˆ Sˆ Sˆ +Sˆ Sˆ . For the current case of interest, a uniaxial nematic,
i ∼ i1− i2 i3− i4 Gαβ is of the form
Here, the total spin Lˆ acts as the canonical conjugate of n2 S 0 0
nˆ, satisfying the commutation relations, G= 1 0− n2 S 0 .
3 − 
0 0 n2+2S
Lαi,Lβj = iδijǫαβγLγi,  
Solving (32) for L yields
h i
Lα,nβ = iδ ǫαβγnγ,
i j ij i
SF(k2)
hnαi,nβji = 0. (28) L⊥ = −ω2 νS2F(k)B⊥, (34)
−
h i Lnˆ = 0,
We introduce an effective Hamiltonian for these rotors,
consistent with local Z2 symmetry, where is either of the directions perpendicular to nˆ.
⊥
Notethat,unlikethecaseofantiferromagneticorder,χαβ
ν
H = Lˆ2 g˜ (nˆ nˆ )2. (29) has no off-diagonal contributions. This is attributed to
rotor 2 i − i· j
Xi Xhiji Ftim(ke)reveg˜rks2a,lainnvdarpioalnecsei.nFtohrelsounsgcewptaivbeililteynginthdsickate→th0e,
→
The dynamic spin susceptibility can be obtained, in the presence of two degenerate DDW modes, with velocity
long wave length limit k 0, by the linear response of v = νS2g˜. This is consistent with the RPA result
L to an external field Bi =→Be−i(ωt−k·ri), foDuDnWd above, Eq. (27).
p
Finally, we compute the dynamic spin susceptibility
HB0 = Bi Lˆi. (30) near k = (π,π) by replacing the term HB0 by a source
− ·
i that couples directly to the staggered magnetization,
X
The Heisenberg equations of motion for the rotors be- Hπ = B nˆ . (35)
B i i
come − ·
i
X
dLˆ Proceeding as above, we find the linear response
i = g˜ (nˆ nˆ )(nˆ nˆ )+B Lˆ
i j i j i i
dt · × ×
j∈XNN(i) n = −νGαβ Bβ, (36)
dnˆi = ν(Lˆi nˆi nˆi Lˆi) Bi nˆi (31) i ω2−zg˜n2 i
dt 2 × − × − ×
where z = 4 is the coordination number of the lattice.
IntheabsenceofanexternalfieldB,therotorcorrelator Note that, in the current approximation, the collective
is assumed to be highly local, modes near (π,π) are non-dispersing and gapped. The
spingapω = zg˜n2indicatesthatthereisnolongrange
nˆαnˆβ =δ (δαβn2/3+Qαβ) δ Gαβ. N´eel order in the system.
h i ji0 ij ≡ ij p
10
Jn =0 Jθ =0 Jn =8 Jθ =8
Jθ Jn Jθ Jn
CDW Nematic
XY SDW SDW
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)XY (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)Heis XY(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) Heis (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
cond unbound
Q=1/2 Q=1 defects defects Nematic CDW
(a) (b)
Ising K Ising K
(a) (b) K K
FIG. 9: (a) Phase diagram for eq. (37) for the case Jn =0, K=8
corresponding toa Z2 lattice gauge theory with a U(1) field. (cid:0)(cid:0)(cid:1)(cid:1)
TheIsingtransition as K is increased corresponds toabind- Jn (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)
ingofQ=1/2topological excitationsintoQ=1excitations. Nem (cid:0)(cid:0)(cid:1)(cid:1)SDW
(b) Case Jθ = 0, corresponding to a Z2 lattice gauge theory (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)
nweitthtraannsSitOio(n3)cafinelbde. spTlhiteinfitrosttwoordseerconnedmaotridcertotrpanarsiatmioangs-, Hei(cid:0)(cid:1)s (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)T(cid:0)(cid:1)P (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)
passing through a topologically ordered phase. Topo (cid:0)(cid:0)(cid:1)(cid:1)CDW
(cid:0)(cid:1)
(c) (cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:1)
(cid:0)(cid:1)
VII. LATTICE GAUGE THEORY XY Jθ
FIG. 10: Phase diagrams when any of the couplings in eq.
Upto now,we haveignoredthe possibilityoffraction-
(37)isinfinite. Inallofthesecases,topologicaldefectsinthe
alization and the richer phase structure that it allows.
gauge field are energetically forbidden, and we can work in
As an example, by introducing a disclination core en-
minimal gauge σij = 1. (a) Case Jn = , insuring that at
ergy, it is possible to split the phase transition for the least nematic order is present. K is irre∞levant in this situa-
onset of nematic order, which is strongly first order in tion, as the field σij is frozen. As Jθ is increased, thesystem
Landau theory, into two second order transitions [30]. undergoesanXYtransitionfromnematictoSDWorder. The
For a theorywith localZ2 gaugeredundancy,it is useful situationissimilartothecaseJθ =∞,shownin(b),butnow
todiscretizethemagneticdegreesoffreedomonalattice aHeisenberg transition is seen from CDW to SDWorder. In
andto introduce anauxiliarygaugefield σ = 1 living (c),K = ,leadingtoadynamicaldecouplingoftheθandn
on the bonds of the lattice. Under a gaugiej tra±nsforma- terms in t∞he action (37). For small Jθ and Jn, a paramagnet
with only topological ordersurvives.
tion at site i, the site variables pick up a minus sign,
ni ni and eiθi eiθi, while simultaneously the
→ − → −
bond variables surrounding i change sign, σ σ .
ij → − ij that the Z redundancy in n (θ) is removed everywhere.
The simplest gauge invariant action that can be written 2
Thus, the universality class of these transitions is the
under these conditions is
same as that of the corresponding ungauged theories.
S = J n σ n J σ cos(θ θ ) For instance, in Fig. 10(c), starting with the topolog-
n i ij j θ ij i j
− − −
ically ordered paramagnet at small J and J , we can
ij ij n θ
Xh i Xh i increase J until the onset of CDW order through an
θ
K σ . (37)
ij XY phase transition. Similarly, by increasing J we get
− n
X(cid:3) Y(cid:3) the onset of nematic order through a Heisenberg transi-
Unlike section IV, here we ignore the anisotropy in the tion. In these regions of phase space, the simultaneous
tensor Jαβ, which is weak and only affects the very low presence of both CDW and nematic orders imply SDW
energy physics. Figures 9 and 10 display the phase di- order. To see this, suppose that n m and eiθ m are si-
h i h i
agrams for extreme values of the couplings J , J and multaneously non-zero, where the subscript m indicates
n θ
K. Figure 9 shows the phase diagrams for the cases that we are working in minimal gauge. Then eiθn m is
h i
J = 0 and J = 0. These correspond to Z lattice non-zero, but this quantity is in fact independent of the
n θ 2
gauge theories with U(1) and SO(3) Higgs fields, re- gauge used.
spectively, which have been studied extensively in the Let us now consider the phase diagram when K = 0.
literature[30, 31, 32, 33]. In this case the auxiliary gauge field fluctuates strongly,
Figure 10 shows the situation when any of the cou- anditisusefultosum(37)overσij configurationstoget
plings in (37) is infinite. In all of these cases, the aux-
iliary field σ is completely ordered, and we can work J2 J2
ij S = n (n n )2 θ cos2(θ θ )
in “minimal gauge”, where σij = 1 for all bonds. This eff − 2 i− j − 2 i− j
correspondstoalackoftopologicaldefects,whichareen- Xhiji Xhiji
ergetically forbidden. Once a value of n (or θ) is chosen J n n cos(θ θ )+... (38)
′ i j i j
− · −
at a given lattice point, smoothness of the fields insures
Xhiji