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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 expectation 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 incommensurate 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 invariance. 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) singlet 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 wavefunction(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 reduced 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 =√√22IRmenDψψγγ††((kk))ǫǫγγαασσααaaββψψββ††((−−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