Density Wave States of Non-Zero Angular Momentum

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by  Chetan Nayak

Density Wave States of Non-Zero Angular Momentum
Chetan Nayak
Physics Department, University of California, Los Angeles, CA 90095–1547
(February 1, 2008)
We study the properties of states in which particle-hole pairs of non-zero angular momentum
condense. These states generalize charge- and spin-density-wave states, in which s-wave particle-
0 hole pairs condense. We show that the p-wave spin-singlet state of this type has Peierls ordering,
0
whilethed-wavespin-singletstateisthestaggeredfluxstate. WediscussmodelHamiltonianswhich
0
favor p-wave and d-wave density wave order. There are analogous orderings for pure spin models,
2
which generalize spin-Peierls order. The spin-triplet density wave states are accompanied by spin-

n 1 Goldstone bosons, but these excitations do not contribute to the spin-spin correlation function.
a Hence, they must be detected with NQR or Raman scattering experiments. Depending on the
J
geometry and topology of the Fermi surface, these states may admit gapless fermionic excitations.

1 AstheFermisurfacegeometryischanged,theseexcitationsdisappearatatransitionwhichisthird-
2
orderin mean-fieldtheory. Thesinglet d-waveandtriplet p-wavedensitywavestatesareseparated

fromthecorrespondingsuperconductingstatesbyzero-temperatureO(4)-symmetriccriticalpoints.
]
l
e PACS numbers:71.10.Hf, 75.10.Lp, 75.30.Fv, 71.27.+a,
-
r
t I. INTRODUCTION state – i.e. the CDW. However, they evade interactions
s
. which disfavor CDW order. Similarly, they are favored
t
a In recent years, a number of materials have been un- by superconductivity-favoring pair-hopping terms [4–7]
m coveredinwhichthecompetitionbetweenaneffectiveat- while evading interactions which disfavor superconduc-
- tractiveinteractionandshort-rangerepulsionappearsto tivity. Hence, they arerather naturalcandidatesfor sys-
d
leadto the formationof superconducting states inwhich tems with competing repulsive interactions.
n
o the Cooper pairs have non-zero relative angular momen- As in the case of higher angular momentum supercon-
c tum. Inthispaper,wesuggestthatsuchcompetitioncan ducting states, there is the possibility of gapless excita-
[ also lead to density-wave states formed by the conden- tions since the order parameter can have nodes on the

1 sation of particle-hole pairs of non-zero relative angular Fermi surface. To consider one consequence of this, sup-
v momentum. These states generalize the familiar charge- pose that the shape ofthe Fermi surface is such that the
3 and spin-density-wave states, in which s-wave particle- nodes of the order parameter do not lie on the Fermi
0 holepairscondense. Wediscussseveraldifferentpossible surface. Letusdistortthe shapeofthe Fermisurfaceby,
3
ordering schemes, the types of interactions which favor say,changingtheanisotropybetweenthehoppingparam-
1
0 them, their physical properties, and their possible rele- eters, which can be done by applying uniaxial pressure.
0 vance to experiments. At the mean-field level, a third-order phase transition
0 Several such states are already commonly known by canoccuratwhichgaplessexcitationsappear. Afterthis
t/ other names, as we will show below. The singlet l = 1 point,the systemremainscriticalasaresultofthenode.
a density-wave state is simply the Peierls state (or bond- Theanalogywithsupercondutivitycanbetakenastep
m
orderedwave),while the singlet l =2 density-wavestate further by combining density wave order with supercon-
d- isknownasthestaggeredfluxstateof[1–3]. However,the ducting order in a pseudospin SU(2) triplet following
n tripletanaloguesofthesestateshavenotbeendiscussed. Yang [8] and Zhang [9]. At a critical point between the
o Since the triplet analogues of these states break spin- two types of order, this pseudospin SU(2) could become
c rotational invariance, they have S = 1 Goldstone boson exact, giving – together with SU(2) spin symmetry – an
:
v excitations. However, the ground state does not have a O(4)-invariantcriticalpoint. Wediscussthepossiblerel-
i non-zero expectation value for the spin at any wavevec- evance of such a critical point to the pseudogap regime
X
tor. Hence,aswewillsee,theseGoldstonebosonscannot of the cuprate superconductors.
r
a be detected in experiments which couple simply to the Particle-hole condensates with non-zero angular mo-
spindensity,suchasneutronscatteringorNMR.Instead, mentum were considered in the context of excitonic in-
RamanscatteringorNQRarenecessaryto coupletothe sulators by Halperin and Rice [10]. They were redis-
Goldstone bosonsofthese moresubtle types ofordering. covered in the context of the mean-field instabilities of
More generaly, s-wave probes cannot couple directly to of extended Hubbard models by Schulz [11] and Ners-
the orders discussed here; instead, local probes or those esyan [12,13] and collaborators. At around the same
whichcoupletohigherpowersoftheorderparameterare time, Kotliar [1] and Marston and Affleck [2] found the
necessary. staggeredflux state as a mean-field solution of the Hub-
Thep-waveandd-wavedensitywavestatesarefavored bard model. However, it was apparently not recognized
by the same types ofinteractionswhich favorthes-wave that the singlet dx2−y2 density-wave state is the same as
1
thestaggeredfluxstate. Morerecently,thisstate[14–16] whilead superconductorhascosk a cosk areplaced
xy x y
−
and a related variant [17,18] have been discussed in the bysinkxa sinkya. Adx2−y2+idxysuperconductorbreaks
contextofthecupratesuperconductors. Aversionofthis T with the order parameter:
state (see the comments in the concluding section) has
appeared in mean field analyses of an SU(2) mean field ψα(k,t)ψβ( k,t) =
h − i
theory of the t J model [19,20]. The Nodal Liquid ∆0(coskxa coskya+isinkxa sinkya)ǫαβ (7)
− −
state of [21–23] also bears a family resemblance to the
Wecandefineanalogousordersfordensity-wavestates.
staggered flux state; we will return to the relationship
However,thespinstructureswillnolongerbedetermined
between these states in the concluding section.
byFermistatistics. Letusfirstconsiderthesingletorder-
ings. A singlet s-wave density wave is simply a charge-
II. ORDER PARAMETERS AND BROKEN density-wave∗:
SYMMETRIES
ψα†(k+Q,t)ψ (k,t) =Φ δα (8)
β Q β
Wedefinethedifferentpossibledensity-waveorderings (cid:10) (cid:11)
A singlet p density-wave state has ordering
x
by analogywiththe more familiarsuperconductingcase.
Considera systemof electronsona squarelattice of side ψα†(k+Q,t)ψ (k,t) =Φ sink a δα (9)
β Q x β
a. A superconductor is defined by a non-vanishing ex-
pectation value of The sin(cid:10)glet p +ip density-w(cid:11)ave states are defined by:
x y
hψα(k,t)ψβ(−k,t)i (1) ψα†(k+Q,t)ψβ(k,t) =ΦQ (sinkxa+isinkya) δβα
A triplet superconductor is characterized by the expec- (cid:10) (cid:11) (10)
tation value
Similarly, the singlet dx2−y2 density-wave states have
ψ (k,t)ψ ( k,t) =∆~(p) ~σ γǫ (2)
h α β − i · α γβ ψα†(k+Q,t)ψ (k,t) =Φ (cosk a cosk a) δα
β Q x − y β
Fermi statistics requires that ∆~(p) be odd in p~. p-wave
(cid:10) (cid:11) (11)
superconductorscanhavethecomponentsof∆~(p)chosen
from sinkxa, sinkya, or sinkxa±isinkya. For instance, while the singlet dx2−y2 +idxy density-wave states have
ap superconductorwithallspins polarizedalongthe 3-
x
directionwillhave∆1+i∆2 6=0and∆3 =∆1−i∆2 =0: ψα†(k+Q,t)ψβ(k,t) =
Φ (cosk a cosk a+isink a sink a) δα (12)
ψ (k,t)ψ ( k,t) =∆ (sink a) σ+ γǫ (3) (cid:10) Q x (cid:11)− y x y β
h α β − i 0 x α γβ
These states belong to a class of states of the form:
Spin-polarized p and p + ip superconductors have
y x y
sinkxa replaced, repectively, by sinkya and sinkxa + ψα†(k+Q,t)ψβ(k,t) =ΦQf(k) δβα (13)
isink a. The analog of the A′ phase of 3He has equal
y
(cid:10) (cid:11)
numbers of and pairs: f(k) is an element of some representation of the space
↑↑ ↓↓
groupofthevectorQ~ inthesquarelattice. Inthispaper,
ψ (k,t)ψ ( k,t) =
α β we will focus primarily on the cases f(k) = sink a and
h − ∆0i sinkxa σα1 γ +sinkya σα2 γ ǫγβ (4) f(k) = coskxa coskya, but f(k) could be an exlement
−
of some larger representation. The s-wave (or extended
Anunpolarizedp su(cid:0)perconductorof pairshas(cid:1)∆ =0
x ↑↓ 3 6 s-wave)cases,f(k)= f(k),aretheusualcharge-density
and ∆ =∆ =0: | |
1 2 wave states.
ψ (k,t)ψ ( k,t) =∆ (sink a) σ3 γǫ (5) Q is the wavevector at which the density-wave or-
h α β − i 0 x α γβ dering takes place. It may be commensurate or in-
As in the polarized case, unpolarized p and p + ip commensurate†. For commensurate ordering such that
y x y
superconductors have sink a replaced, repectively, by 2Q is a reciprocal lattice vector, e.g. Q = (π/a,0) or
x
sink a andsink a+isink a. Inprinciple, morecompli-
y x y
catedorderparametersarepossible,withallcomponents
of ∆~ taking non-vanishing values. If any component of
∆~(p) is not real, time-reversalsymmetry (T) is broken. ∗Extended s-waveis also possible.
Ad-wavesuperconductormustbeaspin-singletsuper- †In this paper, we will take commensurate to mean the
conductor. A dx2−y2 superconductor has situation in which 2Q~ is a reciprocal lattice vector. The
term ‘incommensurate’ will actually include higher-order
ψ (k,t)ψ ( k,t) =∆ (cosk a cosk a) ǫ (6) commensurability.
α β 0 x y αβ
h − i −
2
Q = (π/a,π/a), we can take the hermitian conjugate of ... i Φ eiQ~·~xδα (18)
− | Q| β
the order parameter:
As a result of the i, the Q =(0,π/a) singlet p density-
x
ψ†β(k,t)ψ (k+Q,t) =Φ∗ f∗(k)δα wavestatesbreakT. However,thecombinationofT and
α Q β
ψβ†(k+Q+Q,t)ψ (k+Q,t) =Φ∗ f∗(k)δα translationbyanoddnumberoflatticespacingsremains
(cid:10) α (cid:11) Q β
Φ f(k+Q)δα =Φ∗ f∗(k)δα (14) unbroken. The same is true ofthe commensuratesinglet
(cid:10) Q (cid:11)β Q β p +ip density-wavestates. Examplesofcommensurate
x y
Therefore, for Q commensurate andincommensuratesingletpx andpx+ipy density-wave
states are depicted in figure 1.
f(k+Q) Φ∗
= Q (15)
f∗(k) ΦQ (a) (b)
(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) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)
Hence,iff(k+Q)= f∗(k),Φ mustbeimaginary. For (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)(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)(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)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
Q
−
singlet px ordering, this will be the case if Q = (π/a,0) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(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:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(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:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(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: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:1)(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)
or Q=(π/a,π/a). For singlet dx2−y2 ordering, this will
be the case if Q = (π/a,π/a). If f(k+Q) = f∗(k), Φ (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) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)
Q (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)(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)(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)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
must be real. For singlet p ordering, this will be the
x
case if Q = (0,π/a). For singlet dxy ordering, this will (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(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:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(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:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(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: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:1)(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)
be the case if Q=(π/a,π/a).
For incommensurate ordering, Φ can have arbitrary
Q
phase: the phase of Φ is the Goldstone boson of bro-
Q
kentranslationalinvariance,i.e. theslidingdensity-wave
mode. Impurities will pinthis mode – atsecondorderin (c)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (d)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
the impurity potential, as in the case of a spin-density- (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
wave – so we will not consider it further. (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(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:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(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:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(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)
All of these states break translational and rotational (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)
invariance. To further analyze the symmetries of these (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(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:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(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:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
sstpaatcees., Tithiessiinnsgtlreutctpivedetnosiwtyr-itweavthesesheaovredneorinn-gvsaninishrienagl (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(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:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(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:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(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:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
x
expectation value:
FIG. 1. (a) Q = (π/a,0) px density-wave state. (b)
ψ†α(~x,t)ψ (~x+axˆ,t) ψ†α(~x,t)ψ (~x axˆ,t) = Q=(0,π/a)px density-wavestate. (c)Q=(π/a,0)px+ipy
β i− β − density-wave state. (d) Incommensurate px density-wave
(cid:10) ...− 2 ΦQeiQ~·~x+Φ−Qe−iQ~·~x(cid:11)δβα (16) sistaltaer.geAbrurtowthleesrse ilsinneos nareet cbuornrdenst.whLeinree tthhieckkninesestiicndeinceartgeys
(cid:16) (cid:17)
bond strength. Arrowed lines denote currents.
We have only written the modulated term; the ... refers
to the uniform contribution coming from the Fourier
transform of ψ†(k)ψ(k). The singlet dx2−y2 density-wave states have non-
vanishing expectation value:
Let us consider the commensurate and incommensu-
rate cases separately. The incommensurate singlet px ψ†α(~x,t)ψβ(~x+axˆ,t)+ψ†α(~x,t)ψβ(~x axˆ,t)
density-wave states completely break the translational ψ†α(~x,t)ψ (~x+ayˆ,t)+ψ†α(~x,t)ψ (~x a−yˆ,t) =
and rotational symmetries. If Φ = Φ∗ , T is pre- − (cid:10) β β − (cid:11)
served; otherwise, it is broken. QThe−sin−gQlet px + ipy (cid:10) ...+ 21 ΦQeiQ~·~x+Φ−Qe−iQ~·~x(cid:11)δβα (19)
density-wave states always break T. The commensurate (cid:16) (cid:17)
states,ontheotherhand,breaktranslationbyonelattice The incommensurate singlet dx2−y2 density-wave states
will preserve T if Φ = Φ∗ ; otherwise, they break
spacing; translation by two lattice spacings is preserved. Q −Q
T. The same is true of the incommensurate singlet d
From(15),acommensuratesingletp density-wavestate xy
x
density-wave states:
with Q=(π/a,0) must have imaginary Φ :
Q
ψ†α(~x,t)ψ (~x+axˆ+ayˆ,t) +
ψ†α(~x,t)ψ (~x+axˆ,t) ψ†α(~x,t)ψ (~x axˆ,t) = β
β − β − ψ†α(~x,t)ψ (~x axˆ ayˆ,t)
...+ Φ eiQ~·~xδα (17) (cid:10) β − − (cid:11)−
(cid:10) | Q| (cid:11) β (cid:10)ψ†α(~x,t)ψβ(~x−axˆ+ayˆ,t)(cid:11)−
Thesingletstateofthistypebreaksnoothersymmetries; ψ(cid:10)†α(~x,t)ψβ(~x+axˆ−ayˆ,t) (cid:11)=
1
iItf Qis u=su(0a,llπy/caa)l,leΦd thmeuPsteibeerlsresatla.te or bond order wave. (cid:10) ...− 4 ΦQeiQ~·~x(cid:11)+Φ−Qe−iQ~·~x δβα (20)
Q
(cid:16) (cid:17)
ψ†α(~x,t)ψ (~x+axˆ,t) ψ†α(~x,t)ψ (~x axˆ,t) = Incommensuratesingletdx2−y2+idxy density-wavestates
β β necessarily break T:
− −
(cid:10) (cid:11)
3
ψ†α(~x,t)ψβ(~x+axˆ,t)+ψ†α(~x,t)ψβ(~x axˆ,t) (a) (b)
−(cid:10)ψ†α(~x,t)ψβ(~x+ayˆ,t)+ψ†α(~x,t)ψβ(~x−−ayˆ,t)(cid:11) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(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:1)(cid:0)(cid:0)(cid:1)(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:1)(cid:0)(cid:0)(cid:1)(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:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
+i ψ†α(~x,t)ψ (~x+axˆ+ayˆ,t)
(cid:10) β (cid:11) (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)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)
+i ψ†α(~x,t)ψ (~x axˆ ayˆ,t) (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)(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) (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:10) β − − (cid:11)
−i(cid:10)ψ†α(~x,t)ψβ(~x−axˆ+ayˆ,t)(cid:11) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(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:1)(cid:0)(cid:0)(cid:1)(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:1)(cid:0)(cid:0)(cid:1)(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:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
i ψ†α(~x,t)ψ (~x+axˆ ayˆ,t) =
−(cid:10)(cid:10) ...+ β1 i −Φ eiQ~(cid:11)(cid:11)·~x+Φ e−iQ~·~x δα (21) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(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:1)(cid:0)(cid:0)(cid:1)(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:1)(cid:0)(cid:0)(cid:1)(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:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
2 − 4 Q −Q β
(cid:18) (cid:19)(cid:16) (cid:17)
The commensurate Q = (π/a,π/a) singlet dx2−y2
density-wave states must have imaginary Φ :
Q (c) (d)(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)(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)(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)
ψ†α(~x,t)ψ (~x+axˆ,t) + ψ†α(~x,t)ψ (~x axˆ,t) (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)(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)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
β β
− − (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)
ψ†α(~x,t)ψ (~x+ayˆ,t) + ψ†α(~x,t)ψ (~x ayˆ,t) = (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)(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) (cid:0)(cid:1)(cid:0)(cid:1)
(cid:10) β (cid:11) (cid:10) β − (cid:11) (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)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:10) (cid:11) (cid:10) ...+ 2i |ΦQ| eiQ~·(cid:11)~xδβα (22) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(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)
As a result of the i, the singlet dx2−y2 density-wave (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(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)
breaks T as well as translational and rotational invari-
ance. Thecombinationoftime-reversalandatranslation FIG. 2. (a) Commensurate dx2−y2 density-wave state.
by one lattice spacing is preservedby this ordering. The (b) Commensurate dxy density-wave state. (c) Commensu-
commensurate Q = (π/a,π/a) singlet dx2−y2 density- rate dx2−y2 +idxy density-wave state. (d) Incommensurate,
wave state is often called the staggered flux state. There T-preservingdx2−y2. Arrowless linesarebondswheretheki-
is also a contributionto this correlationfunction coming neticenergyislargebutthereisnonetcurrent. Linethickness
from ψ†(k)ψ(k) which is uniform in space (the ...); as indicates bond strength. Arrowed lines denotecurrents.
a result, the phase of the above bond correlation func-
where f(k) is chosen from the above set. Alternatively,
tion – and, therefore, the flux through each plaquette –
the particle-hole pairs can be unpolarized, e.g.
is alternating. The commensurate Q = (π/a,π/a) sin-
glet dxy must have real ΦQ; therefore, it does not break ψα†(k+Q,t)ψβ(k,t) =ΦQ sinkxaσβ1α+isinkyaσβ2α
T. On the other hand, the singlet dx2−y2 +idxy state
does break T. Note that the nodeless commensurate (cid:10) (cid:11) (cid:0) (25) (cid:1)
singlet dx2−y2 +idxy density-wave state does not break As in the superconducting case, more complicated order
more symmetries than the commensurate singlet dx2−y2 parametersarepossible,withallcomponentsof∆~ taking
density-wave state, in contrast to the superconducting
non-vanishing values.
case. Examples of singlet dx2−y2, dxy, and dx2−y2+idxy For commensurate ordering, we can follow the same
density-wave states are depicted in figure 2. logic as in (14). The phases of the components of Φ~ (p)
Q
We now consider the triplet density-wave states.
are constrained in the same way as the singlet order pa-
Triplet states all break spin-rotational invariance and,
rameters, as illustrated by the i in front of the second
therefore,haveGoldstonebosonexcitations. Wewilldis-
term in (25).
cuss the experimental consequences of these Goldstone
The orders discussed here can be generalized to other
bosons later. The triplet s-wave density wave state is
2Dlatticesandto3Dlattices. Theorbitalwavefunctions
simply a spin-density-wave. The triplet p-wave and d-
sink a, etc. will be replaced by representations of the
x
wave states are characterizedby:
point groups of these other lattices.
ToeachT-preservingsingletordering,wecanassociate
ψα†(k+Q,t)ψ (k,t) =Φ~ (k) ~σα (23)
β Q · β an ordering of a pure spin model, in the same way that
withthe(cid:10)componentsofΦ~ (k)(cid:11)chosenfrom,respectively, spin-Peierls ordering is related to Peierls ordering:
Q
sinkxa, sinkya, sinkxa isinkya; andcoskxa coskya, ψα†(k+Q,t)ψ (k,t) S~(k+Q,t) S~(k,t) (26)
± − α
sink a sink a, cosk a cosk a isink a sink a. → ·
x y x y x y
A state in which the−particle-h±ole pairs are polarized, T(cid:10)hese spin orderings ar(cid:11)e stDates in which the exEchange
which is the most direct analogue of a spin-density-wave energies are large along preferred directions. These pre-
has Φ~Q(p) of the form Φ3Q 6=0, Φ1Q =Φ2Q =0: ferred diractions oscillate from one lattice point to the
nextwithspatialfrequencyQ~. Thesimplestcaseisspin-
ψα†(k+Q,t)ψβ(k,t) =ΦQf(k)σβ3α (24) Peierlsordering,inwhichthespinsformdimers. Another
example is d orderingofa spin model, which takes the
(cid:10) (cid:11) xy
form:
4
S~(~x,t)S~(x+axˆ+ayˆ,t) + S~(~x,t)S~(x axˆ ayˆ,t) d2k (f(k))2
− − g =1
DS~(~x,t)S~(x axˆ+ayˆ,t)E DS~(~x,t)S~(x+axˆ ayˆ,t)E Z (2π)2 (ǫ(k) ǫ(k+Q))2+4g2 ΦQ 2(f(k))2
− − − − − | |
D =E 1D Φ eiQ~·~x+Φ e−iQ~·~xE q (32)
Q −Q
−4
(cid:16) (cid:17) The reduced Hamiltonian has long-ranged interactions,
in analogy with (20).
sothe variationalwavefunctionisessentiallycorrect. We
will now show that short-ranged Hamiltonians will in-
III. MODEL HAMILTONIANS clude terms of the form (27), and that the trial wave-
function(29)isreasonableforthese short-rangedHamil-
tonians.
Weareprimarilyconcernedinthispaperwiththeuni-
Consider,then, the followinglattice modelofinteract-
versalproperties of the states introduced above. We will
ing electrons:
notattempttoshowthatparticularrealisticmodelsofin-
teractingelectronshavep-ord-wavedensity-waveground H = t c† c +h.c. + U n n
states. Rather, we will content ourselves with discussing − iσ jσ i↑ i↓
the types of interactions which favor such orders and <Xi,j>(cid:16) (cid:17) Xi
+V n n
showing that they lead to energetically favorable trial i j
variational wavefunctions for some idealized Hamiltoni- <Xi,j>
t c† c c† c
ans. − c1 iσ jσ jσ i′σ
TheanalogoftheBCSreducedHamiltonianforsinglet <i,j>,<Xi′,j>,i6=i′
density-wave order is:
t c† c c† c
− c2 " i+xˆ,σ iσ− iσ i+xˆ,σ ×
H =Z (2dπ2k)2 ǫ(k)ψα†(k)ψα(k) − Xi (cid:16) c†i+xˆ+yˆ,σci+yˆ,σ−(cid:17)c†i+yˆ,σci+xˆ+yˆ,σ
d2k d2k′ (cid:16) (cid:17)
g f(k)f(k′)
+x y (33)
(2π)2 (2π)2 ×
Z (cid:20) → #
ψα†(k+Q)ψ (k)ψβ†(k′)ψ (k′+Q) (27)
α β
The first two terms are the usual hopping, t, and on-site
(cid:21)
repulsion, U of the Hubbard model. The third term is
In the triplet case, we replace the four-fermion operator
a nearest-neighbor repulsion, V. The third and fourth
of (27) by
terms lead to the correlatedmotion of pairs of electrons.
ψα†(k+Q)σaβψ (k)ψγ†(k′)σaδψ (k′+Q) (28) tc1 hops an electronfrom i′ to j when j is vacated by an
α β γ δ electron hopping to i. t hops nearest-neighbor pairs in
c2
thesamedirection. Termsofthisgeneralformhavebeen
We now introduce the variational wavefunction
discussed in [4–7] as a mechanism for superconductivity.
Ψ = u ψα†(k)+v ψα†(k+Q) 0 (29) As we will see below, they not only favor superconduc-
k,α k,α
tivity, but p- and d-wave density wave order as well.
(cid:12)(cid:12) E Yk,α(cid:0) (cid:1)(cid:12)(cid:12) E Fourier transforming the interaction terms into mo-
(cid:12) (cid:12)
Its energy can be minimized if we take mentum space, we see that terms of the form of the re-
duced interaction (27) are, indeed, present:
gΦ f(k)
Q
u v =
k,α k,α
(ǫ(k) ǫ(k+Q))2+4g2 Φ 2(f(k))2 d2k
q − | Q| Hint = (2π)2 "Uψ↑†(k1)ψ↑(k2)ψ↓†(k3)ψ↓(k4)
(30) Z
+ 2V cos(kx kx)a+cos(ky ky)a
in the singlet case and 3 − 4 3 − 4
(cid:16) (cid:17)
u ~σ βv = gΦ~Qf(k) −2tc1 cos(k1x−k4x)a+cos(k1y−k4y)a
k,α α k,β (ǫ(k) ǫ(k+Q))2+4g2 ΦQ 2(f(k))2 (cid:16)+2cosk1xacosk4xa+2cosk1yacosk4ya
− | |
q (cid:17)
(31) 2t sinkxasinkxa+sinkyasinkya
− c2 1 4 1 4 !
in the triplet case, and require Φ to satisfy the gap (cid:16) (cid:17)
Q
equation: ψα†(k )ψ (k )ψβ†(k )ψ (k )
1 α 2 3 β 4
× #
(34)
5
Let us now consider various candidate orderings and neutron scattering‡. When another symmetry – in addi-
the terms which favor or penalize them. Antiferromag- tionto translationalinvariance–isbroken,this iseasier.
netic order is favoredby U but penalized by V. Charge- Let’s first consider broken time-reversal symmetry.
density-wave order is favored by V but penalized by U. The commensurate singlet dx2−y2 density wave state –
p and d-wave superconductivity are favored by t and orstaggeredfluxstate[1–3]–breaksT;thereisanalter-
c2
t respectively and penalized by V. p and d-density- natingpatternofcurrentscirculatingabouteachplaque-
c1
wave order are favored by t and t , respectively, and tte of the lattice. These currents produce an alternating
c2 c1
are both favored by V. The density-wave states can be magnetic field measurable by µSR and, in principle, by
favored over the others by taking V large. The p- or neutron scattering [3]. The magnitude of the current
d-wave states can be favored by taking t or t large. along a link of the lattice will be:
c2 c1
To be more precise, the mean-field equations for various
et
ordered states read: j = Φ 10−5Ampere Φ (39)
Q Q
¯h ∼ ×
d2k (f(k))2
λ =1 Now, ΦQ is related to the maximum of the gap ac-
Z (2π)2 (ǫ(k) ǫ(k+Q))2+4λ2 ΦQ 2(f(k))2 cording to ∆0 = gΦQ where g is the appropriate cou-
− | |
pling constant. Let us suppose, for the purposes of
q
(35)
illustration, that the formation of the ordered state
is driven by λ . Then, for λ small, Φ
where dDW dDW Q ∼
(t/λ )e−(const.)t/λdDW. Alternatively, we may take
dDW
the high-T context as a guideline: observed gaps are
λ =8V +96t c
dDW c1
100 300K, while interactions such are 1eV. In
λpDW =4V +16tc1+16tc2 t∼his cas−e, we expect Φ 10−2. This tran∼slates to a
λCDW =16V +96tc1+16tc2 2U Q ∼
− magnetic field at the center of each plaquette on the or-
λ =2U (36)
AF
der of 10G. The muons in a µSR experiment might see
Hence, the singlet dx2−y2 density-wave state will be the a lowwer field if they sit at points of high symmetry or
ground state if awayfromthe plane. The orbitalmagnetic moments are
likely to be dwarfed by local spin moments [3]. Incom-
8t <U 4V <48t mensurate ordering may or may not break T; if it does,
c1 c1
−
8t <2V +40t the above analysis applies.
c2 c1
(37) In the Φ3Q 6= 0, Φ1Q = Φ2Q = 0 triplet dx2−y2 density
wave state, there are counter-circulating currents of up-
whilethesingletpx density-wavestatewillbetheground anddown-spinelectrons. These currentscancel, sothere
state if is no net current circulating about each plaquette, but
thereisanalternatingpatternofspincurrentscirculating
6V +40t <U <2V +8t +8t
c1 c1 c2 about each plaquette. The checkerboard pattern of spin
2V +40t <8t
c1 c2 currents will generate, via the spin-orbit coupling,
(38)
d2k d2q
assumingthatthevanHovesingularitiesareattheantin- HSO = (2π)2 (2π)2 E~(q)· 2~k+~q ×
odes of the order parameters. Otherwise, the p- and d- Z ψ†(cid:16)α(k+q(cid:17))~σ βψ (k) (40)
wave density wave states will be favored over somewhat α β
smaller regions of parameter space. By including spin- a quadrupolar electric field which is, in principle, mea-
dependent interactionssuchasJS~i S~i, wecanfavorthe surable in NQR experiments. With the above estimate
·
tripletp-ord-wavestates. Hence,itappearsthattheor- ofthecurrent,anucleuswithanon-zeroquadrupolemo-
derings discussed in this paper are viable. The detailed ment would have an induced splitting of order 10Hz.
energeticsatlargecouplingstrengths–whichsurelyhold We now turn to broken spin-rotational invariance,
in physically interesting systems – are beyond the scope characteristic of the triplet states. Since it transforms
of this paper.
IV. EXPERIMENTAL SIGNATURES
‡One can expect, on general grounds, that incommensurate
singlet ortriplet p-ord-wavedensity-waveorderat wavevec-
Wenowturntothequestionoftheexperimentalsigna- torQ~ will induceCDW order at 2Q~ since a term of theform
tures of such states. Since the order parameter changes Φ2Qρ2Q orΦ~Q·Φ~Qρ2Q isallowedbysymmetryintheeffective
signas the Fermi surface is circled,there is no net CDW action. Nevertheless,wemaywishtodistinguishsuchastate
or SDW order which could be measured in, for instance, from one which has only CDW order.
6
non-trivially under the point groupof the square lattice, 1 (ǫ(k) ǫ(k+Q))2+4∆2(k) (46)
the triplet order parameter Φ~ will not couple to pho- 2 −
Q q
tons,neutrons,ornuclearspinsaccordingtoΦ~ F~,where
Q Let’s consider the situation in which there is a node,
·
F~ is,respectively,B~, S~N,orI~. Saidmorephysically,the i.e. when the argument of the square root vanishes (we
tripletorderedstatesdonothaveanomalousexpectation discuss below the conditions under which this occurs).
values for the spin density but, rather, for spin currents; For simplicity, we will consider the commensurate Q~ =
spincurrentsdonotcouplesimplytotheseprobes. How- (π/a,π/a) singlet p density-wave state in a model with
x
ever, the order parameter Φ~Q will couple to such probes anisotropic nearest-neighbor hopping:
atsecondordersinceitssquaretransformstriviallyunder
the point group. Such a coupling will be of the form: ǫ(k)= 2t(rcosk a+cosk a) (47)
x y
−
2
H = d2x 2 F~ Φ~ F~ Φ~∗ Φ~ F2 with r <1. The mean-field quasiparticle energies are:
probe · Q · Q − Q
Z (cid:20) (cid:16) (cid:17)(cid:16) (cid:17) (cid:12) (cid:12) (cid:21)
(cid:12)(cid:12) (cid:12)(cid:12) (41) E(k)=± 4t2(rcoskxa+coskya)2+∆02sin2kxa (48)
q
Inthecaseofphotons,thiswillleadto2-magnonRaman Thereisanodeatk =0,k a=arccos( r). Expanding
x y
−
scattering. Thecouplingtonuclearspinscouplesdirectly about this node,
to the nuclear quadrupole moment, and will lead to a
shift in the nuclear quadrupole resonance frequencies. E(q)= v2q2+v2q2 (49)
In the presence of disorder, rotational symmetry will ± x x y y
q
bebroken. Hence,therewillbeasmallcoupling,propor- with momenta ~q now measured from the node and
tional to the disorder strength, of the Goldstone bosons
to s-wave probes such as NMR and neutron scattering. v =∆ a, v =2ta 1 r2 (50)
x 0 y
−
p
The effective Lagrangian for the quasiparticles near the
V. GAPLESS FERMIONIC EXCITATIONS
nodes can be written:
The mean-field Hamiltonian is: eff =χα†(∂τ τzvyi∂y τxvxi∂x)χα (51)
L − −
d2k TermswhichbreakthenestingoftheFermisurface,such
H = ǫ(k)ψα†(k)ψ (k)+
(2π)2 α asthechemicalpotentialornext-neighborhopping,open
ZB.Z. "
hole pockets at the nodes:
gΦ f(k)ψα†(k+Q)ψ (k) (42)
Q α # µ = µχα†χα (52)
L −
If we define the four componentobjectχAα accordingto We now turn to the question of when a p- or d-wave
density wave will havenodal excitations. Let’s againbe-
χ1α = ψα(k) (43) gin with the commensurate Q~ = (π/a,π/a) singlet px
χ ψ (k+Q)
(cid:18) 2α (cid:19) (cid:18) α (cid:19) density-wave state:
then the mean-field Hamiltonian can be written in the
ψα†(k+Q,t)ψ (k,t) =i Φ sink a δα (53)
form: β | Q| x β
(cid:10) (cid:11)
d2k 1 in a system in which the Fermi surface is nested at Q~.
H = χα†(k) (ǫ(k) ǫ(k+Q))τ +
ZR.B.Z.(2π)2 2 − z Tkhi=s s0tacrteoswseisllthhaevFeegrmapilseussrfeaxccei.taFtoironasniofptehneFneordmailsluinre-
x
1
face,thisneednotbethecase. Inananisotropicnearest-
∆(k)τ + (ǫ(k)+ǫ(k+Q)) χ (44)
x α
2 ! neighbor tight-binding model, Eq. (47), with r > 1,
the Fermi surface at half-filling is an open Fermi sur-
The integral is over the reduced Brillouin zone. The τ’s
face which does not cross the line k =0. Consequently,
x
are Pauli matrices; the ‘flavor’ index A = 1,2 on which
there are no gapless excitations. For r <1, however,the
they act has been suppressed. ∆(k) is defined by.
Fermi surface does cross the line k = 0, and there are
x
gapless excitations.
∆(k) ∆ f(k) gΦ f(k) (45)
≡ 0 ≡ Q Arethereanythermodynamicsingularitiesatthetran-
The single-quasiparticle energies are: sitionatwhichgaplessexcitationsoccur? Toanswerthis
question, let us consider the mean-field ground state en-
1 ergy:
E (k)= (ǫ(k)+ǫ(k+Q))
±
2 ±
7
d2k chemical potential. In such a case, the system is in the
E = E(k)
0 (2π)2 criticalphase. Regardlessofthedetailsofthebandstruc-
ZR.B.Z.
d2k ture,the curveǫ(k)=ǫ(k+Q)is determined by symme-
= ǫ2(k)+∆2(k) (54)
(2π)2 try for commensurate Q~: it is the set of points for which
ZR.B.Z. p ~k and ~k + Q~ are related by a symmetry of the square
The first and second derivatives of E0 are continuous. lattice. ForQ~ =(π/a,0),ǫ(k)=ǫ(k+Q)ifk = π/2a.
x
However,the third derivative of the ground state energy For Q~ =(π/a,π/a), ǫ(k)= ǫ(k+Q) if k k =± π/a.
x y
with respect to r contains a term of the form: ± ±
Ad-wavedensity-wavewillalwayshavenodallineswhich
∂3E d2k 8ǫ(k)t3cos3k a cross the curve ǫ(k) = ǫ(k+Q); a p-wave density wave
0 x
= +... (55) mayormaynot. Ifthereisnocrossingpoint,orthecross-
∂r3 ZR.B.Z.(2π)2 (ǫ2(k)+∆2(k))3/2 ingpointis notbelow the chemicalpotential,the system
isinthenon-criticalphase. Again,thenon-criticalphase
ThistermdivergesifthereisanodeontheFermisurface,
can have gapless excitations.
but is finite otherwise. Hence, the phase with a node on
In the case of incommensurate ordering, similar con-
the Fermi surface is a critical line with a singular third
siderations hold. Let us suppose that the Fermi surface
derivative of the ground state energy. We will call this
is nested at incommensurate Q~, i.e. if ~k is on the Fermi
phase the ‘critical phase’ of the p density wave. Note
x surface, then~k+Q~ is as well, and ǫ(k)= ǫ(k+Q)= µ.
that the second derivative of the ground state energy
If the Fermi surface intersects the nodal lines of ∆(k),
is everywhere continuous but nowhere differentiable in
thentherewillbe gaplessnodalexcitations. Ifthe chem-
the critical phase. It is separated by a third-order phase
ical potential is now lowered or the hopping parameters
transitionfromthephasewithnogaplessexcitations,the
are changed, so that the Fermi surface is no longer per-
non-critical phase of the p density wave.
x
fectly nested,then the nodeswillopenintohole pockets.
How does this observation generalize (a) away from
Again, as the nesting condition is approached, a third-
half-filling and to non-nested Fermi surfaces; and (b) to
orderphasetransitionwilloccur,asinthecommensurate
d-waveand/orincommensurateordering? Toanswer(a),
case.
let’schangethechemicalpotentialinordertomoveaway
In summary, the system will be in a ‘critical’ state if
from half-filling. Now,
nodal points are at or below the Fermi surface. Oth-
∂3E d2k 8ǫ(k)t3cos3k a erwise, the system will be ‘non-critical’, whether or not
0 x
= +... (56)
∂r3 ZE(k)<µ(2π)2 (ǫ2(k)+∆2(k))3/2 tthheerseetowthoesrtagtaepslaenssdetxhceiteanttiiornesc.riTtihcaeltprhaansseitiisonchbaertawcteeern-
Below half-filling, the denominator never diverges. ized, in mean-field-theory, by a diverent third derivative
Hence, the system is always in the non-critical phase, ofthegroundstateenergy. Thereisnoreasontomistrust
despite the fact that there are gapless excitations. As mean-fieldtheorysincetherearen’tstrongorderparame-
the chemical potential is increased, the system crosses a terfluctuations whichmightdestabilizeoutcalculations.
third-orderphasetransitionandentersthecriticalphase.
Above half-filling, it is always in the critical phase.
VI. TRANSITIONS TO SUPERCONDUCTING
Suppose, now, that we allow next-nearest neighbor
STATES
hopping t′, thereby spoiling nesting. The ground state
energy is given by
As Zhang [9] has recently emphasized, enhanced sym-
d2k metry can be dynamically generated at a critical point
E = E(k)
0 (2π)2 between two different ordered electronic states. The fo-
ZE(k)<µ
cus of that work was a critical point between an antifer-
d2k 1
=ZE(k)<µ(2π)2 "2(ǫ(k)+ǫ(k+Q))− rYoamngagnideetnatnifideda daxn2−SyU2 (s2u)pesrycmonmdeutcrtyor.(wInhiceha,rliteorgwetohrekr,
1 (ǫ(k) ǫ(k+Q))2+4∆2(k) (57) with SU(2) spin-rotational symmetry, trivially forms an
2 − O(4)=SU(2) SU(2) Z )whichisanexactsymmetry
q (cid:21) × × 2
of the Hubbard model at half-filling with µ=U/2. This
and
symmetry would be dynamically generated at a critical
∂3E d2k 4ǫ(k)t3cos3k a point between a CDW and an s-wave superconductor.
0 x
= We now consider the modification of this idea to p- and
∂r3 (2π)2 3/2
ZE(k)<µ (ǫ(k) ǫ(k+Q))2+4∆2(k) d-wave ordering.
−
+... (cid:16) (cid:17) We first consider a transition at half-filling between a
singlet commensurate dx2−y2 density-wave and a dx2−y2
This diverges if the nodal line of ∆(k) crosses the curve superconductor. Wegroupthetwoorderparametersinto
ǫ(k) = ǫ(k +Q) and this crossing point lies below the
8
a vector§: if H is given by∗∗:
int
√2Re ψ†(k+ q)ψ†( k+ q) d2q
↑ 2 ↓ − 2 H = u(0,0)λ(0,0)(q)λ(0,0)(q)+
Φi(q)f(k)=√2iImψαnn†DD(kψ↑†+(kQ++q22q))ψψ↓†α(−(kk−+2q2q))EEoo  (58) int uuZ((01,(,102))πλλ)((i201,,(cid:20)10))((qq))λλ((i01,,10))((qq))++uu((01,,10))λλ((i01Q,,10))((qq))λλ(i(01Q,,10))((qq))++
If, following Yang [8(cid:10)], we introduce the followi(cid:11)ng SU(2) a a Q aQ aQ
generators which we will call pseudospin SU(2) u(1,1)λ(1,1)(q)λ(1,1)(q)+u(1,1)λ(1,1)(q)λ(1,1)(q)
ia ia Q iaQ iaQ
(cid:21)
d2k
O3 = (2π)2 ψα†(k)ψα(k) + where
ZR.B.Z. (cid:18)
ψα†(k+Q)ψα(k+Q) λ(0,0) = (2dπ2k)2 f(k)ΨAα† k+ 2q ΨAα k− 2q
O+ =ZR.B.Z.(2ddπ22kk)2 iψ↑†(k)ψ↓†(−k+Q) (cid:19) λ(i1,0) =ZZ (2dπ2k)2 f(k)ΨAα†(cid:16)(cid:16)k+ 2q(cid:17)(cid:17) × (cid:16) q (cid:17)
O− = (2π)2 iψ↑(k)ψ↓(−k+Q) (59) τiBAΨBα k− 2
ZR.B.Z. d2k q (cid:16) (cid:17)
λ(1,0) = f(k)ǫαβΨ k+ ǫCA
then the order parameters form a triplet under this iQ (2π)2 Cα 2 ×
SU(2), Z (cid:16) (cid:17) q
τBΨ k+
iA Bβ − 2
Φ (q)f(k) ψ†(k+ q)ψ†( k+ q) d2k q (cid:16) (cid:17)
+ − ↑ 2 ↓ − 2 λ(0,1) = f(k)ΨAα† k+ σβ
ΦΦ−0((qq))ff((kk)) = i(cid:10)ψDψα†↑((kk++Qq2)+ψ↓2q()−ψkα+(kq2−)2qE)(cid:11) (60) a Z (2π)2 (cid:16) ΨA2β(cid:17) kiα−×2q
There is a small but imp(cid:10)ortant difference betw(cid:11)eenour λ(0,1) = d2k f(k)ǫABΨ k+(cid:16)q ǫγβ(cid:17)
pseudospin SU(2) and Yang’s [8]: the factors of i in the aQ (2π)2 Aγ 2 ×
definitions of O±. These are necessary since a commen- Z (cid:16)σβ Ψ (cid:17) k+ q
surate dx2−y2 density-wave breaks T, while a dx2−y2 su- d2k iαq Bβ(cid:16)− 2(cid:17)
perconductor does not. Consequently, our pseudospin λ(1,1) = f(k)ΨAα† k+ τB
ia (2π)2 2 iA×
SU(2) does not commute with T, which is an inversion Z (cid:16) (cid:17) q
followed by a rotation by π about the 3-axis. σiβαΨBβ k− 2
The electron fields transform as a doublet under the d2k q (cid:16) (cid:17)
pseudospin SU(2) as well as the spin SU(2). We will λ(ia1,Q1) = (2π)2 f(k)ΨCγ k+ 2 ǫCAτiBA×
group them into 4-component objects ΨAα, where A is Z (cid:16) ǫγβ(cid:17)σβ Ψ k+ q (64)
the pseudospin index, A = 1,2, and α is the spin index, iα Bβ − 2
α= , : (cid:16) (cid:17)
↑ ↓ These ‘microscopic’ Hamiltonians describe electrons
Ψ1α = ψα(k) (61) at half-filling with a nested Fermi surface and interac-
Ψ2α iǫαβψβ†( k+Q) tions which favor density-wave and superconducting or-
(cid:18) (cid:19) (cid:18) − (cid:19)
derequally. Inotherwords,theydescribeacriticalpoint
A ‘microscopic’ Hamiltonian which is O(4) invariant
can be written down: athalf-fillingbetweenadx2−y2 density-waveandadx2−y2
superconductor. Near the critical point, we can focus on
H =H +H (62) the low-energydegreesof freedom: the Goldstonemodes
0 int
and the nodal fermionic excitations. We can write down
an O(4) invariant action for this:
d2k
H = ǫ(k)ΨAα†Ψ (63)
0 ZR.B.Z.(2π)2 Aα S = dτ d2k ΨAα†(k)(∂ ǫ(k))Ψ (k)+
eff (2π)2 τ − Aα
Z
d2k d2q
ig dτ Φ (q)f(k)
§We will use underlined lowercase Roman letters such as Z (2π)2 (2π)2 i ×
i=1,2,3 to denotepseudospin triplet indices and uppercase
Roman letters to denote peudospin doublet indices A=1,2.
Lowercase Roman indices a = 1,2,3 will be vector indices
(i.e. real spin triplet indices) and Greek letters α = 1,2 will ∗∗Wehaveonlywrittendownthequarticterms;higher-order
be used for real spin SU(2) spinor indices. Pauli matrices τi O(4) invariants also exist, but they are irrelevant at weak
willbeusedforpseudospin,whileσawillbereservedforspin. coupling.
9
q q
ǫαβΨ k+ ǫCAτiBΨ k+ + spin-triplets. Hence, the effective field theory for such a
Cα 2 A Bβ − 2
q q transition takes the form:
hǫ ΨAα†(cid:16) k+ (cid:17) τiBǫBCΨBβ(cid:16)† k+(cid:17)
αβ 2 A − 2
+ dτd2x (cid:16)∂µΦi 2(cid:17)+ 12rΦiΦi+ 41(cid:16)!u ΦiΦi(cid:17)2i (65S)eff = dτ d2k ΨAα†(∂ ǫ(k))Ψ +
Z (cid:18)(cid:0) (cid:1) (cid:0) (cid:1) (cid:19) (2π)2 τ − Aα
In this Lagrangian, we have rescaled all of the veloci- Z d2k d2q
ties and stiffnesses to 1. In general, these quantities will ig dτ Φa(q)f(k)
(2π)2 (2π)2 i ×
be different – breaking the O(4) symmetry – and this Z q q
ǫγασaβΨ k+ ǫCAτaBΨ k+ +
cannot be done. Symmety-breaking terms will be briefly α Cγ 2 A Bβ − 2
addressed below. hσaβǫ ΨAα†(cid:16) k+(cid:17)q τaBǫBCΨB(cid:16)γ† k+(cid:17) q
The transition between the dx2−y2 density-wave and α βγ − 2 A − 2
sthyme mdxe2t−ryy2-bsrueapkerincognfideuldct,owrhiischdrwiveewnilblycaallpus.eudospin-2 + dτd2x (cid:16)∂µΦi 2+(cid:17)12rΦaiΦai + 41!(cid:16)u ΦaiΦai(cid:17)2i
Z (cid:18)(cid:0) (cid:1) (cid:16) (cid:17) (cid:19)
=u Φ2+Φ Φ
Lu 0 + − where
=u (cid:0)Φ23−Φ21−Φ(cid:1)22 (66)
dFeonrsuity<-wa0v,etshteat3e-aisx(cid:16)ifsaviosreadn; feoarsyua>(cid:17)xi0s,atnhde 1th−e2d-xp2l−any2e ΦΦΦa2aa1 =√√22IRmenDψψγㆆ((kk))ǫǫγγαασσααaaββψψβↆ((−−kk))Eo  (68)
is an easy plane and the dx2−y2 superconducting state is  3   i ψnαD†(k+Q)σαaβψβ(k) Eo
favored.  
(cid:10) (cid:11)
WecanmoveawayfromanestedFermisurfacebytun-
ingthechemicalpotentialoranext-neighborhoppingpa-
VII. DISCUSSION
rameter. Sucheffectsareencapsulatedbyapseudospin-1
symmetry-breaking term:
In this paper, we have discussed the properties of or-
S = µO3
µ dered states in which particle-hole pairs with non-zero
= dτd2x ǫ Φ ∂ Φ +Ψ†τ3Ψ (67) angular momentum condense. These states generalize
ij i τ j
Z (cid:16) (cid:17) charge- and spin-density wave states in the same way
where O3 is the pseudospin SU(2) generator defined thatp-andd-wavesuperconductorsgeneralizes-wavesu-
above. If u=0, µ will immediately force the pseudospin perconductivity. However,unlikeinthe superconducting
into the 1 2 plane – i.e. the supercondcutor will be fa- case – where the Meissner effect follows directly from
−
vored. If u<0, the dx2−y2 density-wavestate will be fa- the brokensymmetry,irrespective of the pairing channel
voreduntilµ (√ u). Atthispoint,afirst-orderphase – the angular variation of the condensate makes p- and
c
∝ −
transition–thepseudospin-floptransition–willoccurat d-wave density-wave ordering difficult to detect. Exper-
which the pseudospin switches from an easy-axis phase iments seeking to uncover such order must (a) be sen-
toaneasy-planephase. IfweallowΦ tohaveadifferent sitive to spatial variations of kinetic energy or currents
0
velocity than Φ±, then this first order phase transition or (b) measure higher-order correlations of the charge
canbecometwosecondorderphasetransitions. Depend- or spin density. We explained how µSR, neutron scat-
ingonthevaluesoftheseparametersandthestrengthof tering, NQR, and Raman scattering can be used in this
quantum fluctuations, the intervening phase can either regard. Impurities, which break rotational invariance,
have both types of order or neither. would cause admixture of p- or d-wave ordering with s-
The criticalpoint occurs when the jump in Φ is tuned wave ordering. It is natural to wonder whether experi-
to zero. Hence, it is a tricritical point. At such a critical ments which appear to detect SDW order should be re-
point, O(4)-breaking terms can scale to zero. The criti- examined to see if they have actually uncovered p- or
cal point and the quantum critical region [24,25] are de- d-wave order which, as a result of impurities, is mas-
scribedbythephysicsofthecriticalfluctuationscoupled querading as s-wave order.
to nodal fermionic excitations. By arguments similar to Asinthesuperconductingcase,thenon-trivialpairing
thoseof[21],thenodalfermionsareneutral,spin-1/2ob- symmetry can lead to the existence of nodal excitations.
jects. A more detailed analysis will be given elsewhere As parameters such as the chemical potential or next-
[26]. neighborhoppingarevaried,nodalexcitationsappearat
Similarconclusionscanbe drawnfordxy anddx2−y2+ a transition which is third-order in mean field theory.
id transitions; the latter case is particularly simple The ‘phase’ with nodal excitations is always critical.
xy
since there are no fermions. In the case of transi- The analogiesbetween p- and d-wave density-waveor-
tions between p -wave density-wave and superconduct- dering and p- and d-wave superconductivity begs the
x
ingstates,theorderparametersarebothpseudospinand
10