Table Of Content

Exact SO(8) Symmetry in the Weakly-Interacting Two-Leg Ladder Hsiu-Hau Lin1, Leon Balents2 and Matthew P. A. Fisher2 1 Department of Physics, University of California, Santa Barbara, CA 93106-9530 2 Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030 (February 1, 2008) 8 We revisit the problem of interacting electrons hopping on a two-leg ladder. A perturbative 9 renormalization group analysis reveals that at half-filling the model scales onto an exactly soluble 9 Gross-Neveumodelforarbitraryfinite-rangedinteractions,providedtheyaresufficientlyweak. The 1 Gross-NeveumodelhasanenormousglobalSO(8)symmetry,manifestintermsofeightrealFermion n fields which, however, are highly non-local in terms of the electron operators. For generic repulsive a interactions, the two-leg ladder exhibits a Mott insulating phase at half-filling with d-wave pairing J correlations. Integrability of the Gross-Neveu model is employed to extract the exact energies, 8 degeneracies and quantum numbers of all the low energy excited states, which fall into degenerate 2 SO(8) multiplets. OneSO(8) vector includes two charged Cooper pair excitations, a neutral s=1 triplet of magnons, and three other neutral s = 0 particle-hole excitations. A triality symmetry ] l relates these eight two-particle excitations to two other degenerate octets which are comprised of e single-electronlikeexcitations. Inadditiontothese24degenerate“particle”statescostinganenergy - r (mass)mtocreate,thereisa28dimensionalantisymmetrictensormultipletof“bound”stateswith t s energy √3m. Doping away from half-filling liberates the Cooper pairs leading to quasi-long-range t. d-wavepairfieldcorrelations,butmaintainingagaptospinandsingle-electronexcitations. Forvery a low doping levels, integrability allows one to extract exact values for these energy gaps. Enlarging m thespace of interactions toinclude attractiveinteractions reveals that thereare fourrobust phases - possiblefortheweakcouplingtwo-legladder. Whileeachofthefourphaseshasa(different)SO(8) d symmetry, they are shown to all share a common SO(5) symmetry - the one recently proposed by n Zhang as a unifying feature of magnetism and superconductivityin thecuprates. o c [ I. INTRODUCTION the tendency forsingletbondformationacrossthe rungs of the ladder, spin-liquid behavior is expected.3,4,11,12 In 1 v Since the discovery of the cuprate superconductors1 the past several years there have been extensive analy- 5 therehasbeenrenewedinterestinthebehaviorofweakly ses of two-legladders, particularlythe Hubbard13,14 and 8 doped Mott insulators.2–4 There are two broad classes t-J models,3,15–17 both at half-filling and with doping. 2 of Mott insulators, distinguished by the presence or ab- Basedonnumericalmethods,includingMonteCarloand 1 sence of magnetic order. More commonly spin rota- density matrix renormalization group,3,4 as well as ana- 0 tionalinvarianceisspontaneouslybroken,andlong-range lytic approachesat weak coupling,18–25 the basic behav- 8 9 magnetic order, typically antiferromagnetic, is realized.5 ior is established. At half-filling there is a spin-liquid / There are then low energy spin excitations, the spin 1 phase with a spin-gap. Upon doping, the spin-gap sur- at magnons. Alternatively, in a spin-liquid Mott insulator vives, although smaller in magnitude, and the system m there are no broken symmetries, the magnetic order is exhibits quasi-long-rangesuperconducting pairing corre- - short-ranged and there is a gap to all spin excitations : lations,withapproximated-wavesymmetry. Thisbehav- d a spin-gap. ioris reminiscentofthatseenintheunderdopedcuprate n In the cuprates the Mott insulator is antiferromagnet- superconductors. o ically ordered, but upon doping with holes the antifer- Thereareanumberofexperimentalsystemswhichcan c : romagnetism is rapidly destroyed, and above a certain be described in terms of coupled two-leg ladders, which v level superconductivity occurs. Below optimal doping exhibit a spin-gap in the insulating compound.26–28 i X levels,thereareexperimentalsignsofaspingapopening These materials are often very difficult to dope. In one attemperatureswellabovethe transitionintothe super- case, doping has apparently been achieved, and under r a conductingphase.6–8Theapparentconnectionbetweena a pressure of 3GPa superconductivity is observed below spin-gapandsuperconductivityhasbeenasourceofmo- 12K.29,30 Carbon nanotubes31 constitute another novel tivation to search for Mott insulators of the spin-liquid material which can be modelled in terms of a two-leg variety. ladder.32–34 Specifically, the low energy electronic exci- Although spin-liquids are notoriously difficult to tations propagating down a single-walled nanotube can achieve in two-dimensions,9 it was realized that quasi- be mapped onto a two-leg ladder model with very weak one-dimensional ladders would be more promising. Par- interactions, inversely proportional to the tube radius. ticularattentionhasfocussedonthetwo-legladder.10 At An obvious advantage of such low-dimensional corre- half-fillingintheMottinsulator,thespinexcitationscan lated electron systems is (relative) theoretical simplic- bedescribedbyaHeisenbergantiferromagnet,anddueto ity. Indeed, in one-dimension many correlated elec- 1 tron models, including the Hubbard model, are exactly unifying the pair, magnon, and charge-density-wave ex- soluble.35 Unfortunately, the Mott insulating phases of citations. Indeed, the SO(5) subgroup rotating only the theseone-dimensionalmodelstypicallyhavegaplessspin- first five components of this vector is exactly the sym- excitations, and upon doping do not exhibit pairing. To metry proposed recently by Zhang41 to unify antiferro- date, we are unaware of any exactly soluble two-leg lad- magnetism and superconductivity in the cuprates. This der models which exhibit a gapped spin-liquid ground vectoroctet,referredtoas“fundamental”fermionsinthe state. field-theory literature, is related by a remarkabletriality symmetry42,43 (presentintheSO(N)GNmodelonlyfor In this paper, we revisit models of interacting elec- N =8)totwoothermassmoctets: spinorandisospinor tronhoppingonatwo-legladder,focusingonthe behav- multiplets, called the even and odd kinks. These sixteen ior near half-filling. For generic short-range potentials, particles have the quantum numbers of individual quasi- we derive a perturbative renormalization group valid for electrons and quasi-holes. The triality symmetry thus weakinteractions,muchsmallerthanthebandwidth.18,19 goes beyond the SO(8) algebra to relate single-particle Remarkably, at half-filling the renormalization group and two-particle properties in a fundamental way.42,43 transformationscalesthesystemtowardsaspecialmodel This relation also implies that pairing is present even in withenormoussymmetry-theSO(8)Gross-Neveu(GN) theMott-insulator: theminimumenergytoaddapairof model.36 Scaling onto the GN model occurs independent electrons (as a member of the SO(8) vector multiplet) is of the initial interaction parameters, provided they are m,reducedbyabindingenergyofmfromthecostof2m weak and predominantly repulsive. Thus, for weakly in- needed to add two quasi-electrons far apart. At energies teracting two-leg ladders at half-filling universal low en- abovethe 24mass m states,there exists anantisymmet- ergy properties are expected. Specifically, all properties rictensormultipletof28particleswithmass√3m. Each on energy scales of ordera characteristicGN mass (gap) can be viewed as bound states of two different funda- manddistancescaleslongerthanoroforderv/m(where mental fermions (or equivalently, two even or two odd v is the Fermi velocity) are universal and determined by kinks). Inthis waytheir quantumnumbers canbe easily the GN model. In terms of microscopic parameters, the deducedbysimpleaddition. Thetensorstatescontribute GNmass is oforderm te t/U,where t is the 1dband- ∼ − additional sharp (delta-function) peaks to various spec- widthandU isatypicalinteractionstrength,butismore tral functions, providing, for instance, the continuation profitably treated, along with v, as a phenomenological ofthemagnonbranchnearmomentum(0,0). Forconve- parameter. The universality predicted by the renormal- nience,thequantumnumbers(charge,spin,andmomen- ization group can be profitably exploited because the SO(8)GNmodelisintegrable,37–39sothatmanyofthese tum) of the vector and tensor excitations are tabulated in Tables 1 and 2. Finally, continuum scattering states universal properties can be computed exactly. To our enter the spectrum above the energy 2m. knowledge, this is the first integrable model for a Mott- insulating spin liquid. It describes a state we call the D- Combining the excitation spectrum of the GN model Mott phase, because the Mott-insulator has short-range with the non-interacting spectrum and some additional pairing correlationswith approximated-wavesymmetry. arguments,wehavealsoconstructedschematicformsfor We now summarize the results obtained from the SO(8) several correlation functions of interest. In particular, GN field theory. in Sec. V we give detailed predictions and plots of the single particle spectral function (measurable by photoe- The primary input from integrability is the complete excitation spectrum.37–40 The excitations of the GN mission), the spin spectral function (measurable by in- elastic neutron scattering), and the optical conductivity. model are comprised of “particles” (i.e. sharp excita- Integrability implies, for instance, sharp magnon peaks tions with a single-valued energy-momentum relation) in the spin structure factor at k = (π,π), (0,0), and organized into SO(8) multiplets, as well as continuum ( (k k ),π) with minimum energy m, √3m and scattering states of these particles. As expected for a ± F1 − F2 √3m respectively (here k and k are the Fermi mo- Mott-insulating spin-liquid with no broken symmetries, F1 F2 menta of the non-interacting system). Complete details each of these excitations is separated from the ground can be found in Sec. V. The optical conductivity has state by a non-zero gap. The lowest-lying particles three principal features: a Drude peak around zero fre- come in three octets, all with mass m, i.e. dispersing quency, with exponentially small weight ( e m/T) at as ǫ1(q) = m2+q2, where q is the deviation of the low temperature; an “exciton” peak aroun∼d ω−= √3m, particle’s momentum from its minimum energy value. p exponentially narrowat low temperatures; anda contin- One vector multiplet (conveniently denoted formally by uum for ω > 2m, due to unbound quasi-particle quasi- a vector of Majorana fermions η , A = 1...8) con- A hole pairs. S∼ee Sec. V for more details and a figure. sists entirely of collective two-particle excitations: two charge 2e “Cooper pairs” around zero momentum, a Our next calculations concern the relation of these re- ± triplet of spin-one “magnons” aroundmomentum (π,π), sultstoarecentstudyofmicroscopicallySO(5)invariant and three neutral spin-zero “charge-density-wave” (or ladder models by Scalapino, Zhang and Hanke (SZH).44 particle-hole pair) excitations. SO(8) transformations These authors consider the strong coupling limit of a rotate the components of the vector into one another, certainlocally-interactingtwo-legladder model designed 2 to exhibit exact SO(5) symmetry. Their model has an ter the system and effectively form a Luttinger liquid on-site interaction U t, an intra-rung interaction with a single gapless charge mode (with central charge | | ≫ V t, and a magnetic rung-exchange interaction J, c = 1). This phase (often denoted “C1S0”) still has a r|el|at≫ed to one another by the SO(5) symmetry. In the gap to spin excitations. Previous work18,20,21 has ap- U–V plane they derive astrong-couplingphase diagram, proached this phase via controlled perturbative calcu- including the case of attractive interactions with U and lations in the interaction strength, at fixed doping x V negative. We have analyzed general SO(5) invariant away from half-filling. Here, we are considering a dif- two-leg ladder models in the opposite limit of weak in- ferent order of limits, with fixed (albeit weak) interac- teractions, deriving as a special case the corresponding tions in the small doping limit, x 0. In this limit, the → weak-couplingphasediagramfortheirmodel. Infact,al- Cooper-pairs being dilute behave as hard-core bosons or thoughwehavenotexploredthefull9-dimensionalspace free fermions. Although the spin-gap is preserved in the completely,forallbarecouplingswehaveconsidered,in- doped state, it is discontinuous as x 0+. The discon- → cludingattractiveinteractionsthatbreakSO(5)symme- tinuity can be understood as the binding of an inserted try explicitly, the RG scales the system into the SO(5) spin-one magnonto a Cooper pair in the system to form subspace. When the interactions are predominantly re- a mass √3m tensor particle, reduced by the binding en- pulsive, the SO(5) system falls into the basin of attrac- ergy (2 √3)m from its bare energy. The spin-gap thus − tion of the D-Mott phase, and the above results apply. jumps from ∆ (x=0)=m to ∆ (x=0+)=(√3 1)m s s − Asnegativeinteractionsareintroduced,fourotherphases upon doping. Such binding of a pair to a magnon has emerge: an S-Mott spin-liquid, with short-rangeapprox- been observednumerically in both Hubbard andt-J lad- imate s-wave pairing symmetry, a charge-density-wave ders by Scalapino and White .45 Similarly, the energy to (CDW) state with long-range positional order at (π,π), add an electron (for the hole-doped system) jumps from a spin-Peierls phase with kinetic energy modulated at ∆ (x = 0) = 3m/2 to ∆ (x = 0+) = m/2, the same 1 1 (π,π), and a Luttinger liquid (C2S2, in the nomencla- as t−he energyto add a singl−e hole. When many pairsare tureofRef.18)phasecontinuouslyconnectedtothenon- present, we have not succeeded in obtaining exact ex- interacting system. The first two of these also occur in pressionsfor the spin and single-particlegaps, but argue the strong-coupling limit, though their positions in the that the spin gap should decrease with increasing dop- phase diagram (Fig. 10) are modified. The phase dia- ing,sincetheaddedmagnonisattractedtoanincreasing grams at weak and strong coupling differ in non-trivial densityofCooperpairs. Itseemslikely,however,thatin- ways,implying a rather complex evolutionof the system tegrability could be exploited even in this case to obtain withincreasingU andV. Inweakcoupling,allfournon- exact results, and hope that some experts may explore trivial phases have distinct asymptotic SO(8) symme- this possibility in the future. tries, enhanced from the common bare SO(5). Further- Finally, we briefly address the behavior of the spin- more, critical points describing the transitions between spectal function for the doped ladder at energies above the various phases can also be identified. In particular, the spin-gap. In a recent preprint SZH44 have argued the D-Mott to S-Mott and CDW to spin-Peierls criti- thatinthisregimethe spin-spectralfunctionforamodel calpointsarec=1 conformalfieldtheories(singlemode with exact SO(5) symmetry should exhibit a sharp res- Luttingerliquids),whichinweak-couplingareacccompa- onance at energy 2µ and momentum (π,π), the so- nied by a decoupled massive SO(6) sector. The S-Mott called π resonance (introduced originally by Zhang to to CDWandD-Mottto spin-PeierlstransitionsareIsing − explainthe 42meVneutronscatteringpeak inthe super- criticaltheories(c=1/2),withdecoupledmassiveSO(7) conducting Cuprates). We show that a delta-function sectors in weak-coupling. There is also a multi-critical π resonance requires, in addition to SO(5) symmetry, point describing a direct transition from the D-Mott to − the existence of a non-zero condensate density in the CDWorfromtheS-Motttospin-Peierlsphases,whichis superconducting phase. Since condensation is not pos- simplyaproductofthec=1andc=1/2criticalpoints. sible in one-dimension, this precludes a delta-function Our final results concern the effects of doping a small π resonance. Following a recent suggestionby Zhang,46 densityofholes(orelectrons)intotheD-Mottspin-liquid − we address briefly the possibility of a weaker algebraic phase at half-filling. For very small hole concentrations, singularity in the spin spectral function. Regardless of the modifications of the Fermi velocities by band curva- the nature of the behavior in the vicinity of ω = 2µ, we ture effects can be ignored, and the doping incorporated expect spectral weight at energies below 2µ but above simply by including a chemical potential term coupled the spin-gap ∆ discussed above. to the total charge Q in the GN model; H = H µQ. s µ An analogous procedure is employed by Zhang41−in his The remainderof the paper is organizedas follows. In study ofthe SO(5)non-linearsigmamodel. Becausethe Sec. II we describe the model Hamiltonian for the inter- chargeQisaglobalSO(8)generator,integrabilityofthe acting ladder, reduce it to the continuum limit, bosonize GN model is preserved, and furthermore many of the the nine distinct interaction channels, and apply the SO(8) quantum numbers can still be employed to label renormalization group (RG) transformation. Sec. III the states. We find that doping occurs only for 2µ>m, details the simplifications that occur upon RG scaling, at which point Cooper pair “fundamental fermions” en- presents the bosonized form of the Hamiltonian in the 3 D-Mott phase, and for completeness demonstrates the ordinate running along the ladder and α = , is a spin ↑ ↓ short-ranged-wavecorrelationsfoundinRef.18Thebulk index. The parameters t and t are hopping amplitudes of the field-theoretic analysis is containedin Sec. IV. By along and between the leg’s of⊥the ladder. refermionizing the bosonized hamiltonian, we obtain the Being interested in weak interactions, we first diago- GN model exposing the exact SO(8) symmetry, and de- nalize the kinetic energy in terms of bonding and anti- scribe why this symmetry is hidden in the original vari- bonding operators: c = (a +( 1)ia )/√2, with i,α 1,α 2,α − ables. The triality symmetry is identified, and used to i = 1,2. The Hamiltonian is then diagonalized in mo- understand the degeneracy between the three mass m mentum space along the ladder, describing two decou- octets. To help in developing an intuition for the GN pled (bonding and anti-bonding) bands. Focussing on model,severalapproximatepicturesarepresentedtoun- the case at half-filling with one electron per site, both derstand the excitations: a mean field theory which is bands intersect the Fermi energy (at zero energy) pro- asymptoticallyexactforN inageneralizedSO(N) vided t <2t. Moreover, due to a particle/hole symme- GN model, and a semi-cla→ssi∞cal theory based on the try pre⊥sent with near neighbor hopping only, the Fermi bosonized (sine-Gordon-like) form of the Hamiltonian. velocityv ineachbandisthesame,denotedhereafteras i We conclude Sec. IV by proving the uniqueness of the v. It is convenient to linearize the spectrum around the ground-state in the D-Mott phase and determining the Fermi points at k (see Fig. 1), which at half-filling ± Fi quantum numbers of the 24+28 = 52 particles. The satisfy k +k = π. Upon expanding the electron op- F1 F2 latter taskis complicatedby the necessityofintroducing erators as, Jordan-Wigner strings, which are required to preserve ginaturgoed-uincvedariiannbceosuonndizeartiaonn.unTphheyssticrainlggaoupgeerastyormsmmeotrdy- ciα ∼cRiαeikFix+cLiαe−ikFix, (2.2) ify the momenta of the certain excitations by a shift of the effective low energy expressionfor the kinetic energy (π,π) from their naive values determined from the GN takes the form, H = dx , with Hamiltonian density, 0 0 fermionoperators. Withthefield-theoreticanalysiscom- H plete,wegoontodiscusscorrelationfunctionsinSec.V, R givingdetailedpredictionsforthesingle-particlespectral H0 =v [c†Riαi∂xcRiα−c†Liαi∂xcLiα]. (2.3) function,spinspectralfunction,opticalconductivity,and Xi,α various equal-time spatial correlators. Sec. VI describes This Hamiltonian describes Dirac Fermions, with four the construction of general SO(5) invariant models in flavors labelled by band and spin indices. Since all weak-coupling, their phases, and the phase-diagram of flavors propagate both to the right and left with the the Scalapino-Zhang-Hankemodel in weak-coupling. Fi- same velocity, the model exhibits an enlarged symme- nally,Sec.VIIdescribesthebehavioroftheD-Mottphase try. Specifically,if the four right(andleft) movingDirac upondoping,includingthebehaviorofvariousgaps,and Fermions are decomposed into realand imaginary parts, a discussion of the status of the SO(5) “π resonance” in ψ =(ξ1 +iξ2 )/√2 where P =R/L and ξ1,ξ2 are one dimension. Various technical points and long equa- Piα Piα Piα Majorana fields, the eight right (and left) moving Ma- tionsareplacedintheappendices. AppendixAgivesthe jorana fields, denoted ξ with A = 1,2,...,8 form an full set of nine RG equations at half-filling, Appendix B PA eightcomponent vector. The Hamiltonian density, when discusses gauge redundancy and the multiplicity of the re-expressed in terms of these eight component vectors ground state in different phases, Appendix C constructs takes the simple form spinor and vector representationsof SO(5), Appendix D relatesSO(5)andSO(8)currents,andAppendixEgives 8 the five RG equations in the reduced SO(5) subspace. v = [ξ i∂ ξ ξ i∂ ξ ], (2.4) H0 2 RA x RA− LA x LA A=1 X II. MODEL which is invariant under independent global SO(8) rotations among either the right or left vector of Majorana We consider electrons hopping on a two-leg ladder as fields. This enlarged O(8) O(8) symmetry is only shown in Fig. 1. In the absence of interactions, the R × L present at half-filling with particle/hole symmetry. Hamiltonian consists of the kinetic energy, which we assume contains only near-neighbor hopping, e H0 = −ta†1α(x+1)a1α(x)+(1→2) x,α(cid:26) X t 1L 1R k −t⊥a†1α(x)a2α(x)+h.c. , (2.1) t 2L 2R x (cid:27) where aℓ(a†ℓ) is an electron annihilation (creation) operator on leg ℓ of the ladder (ℓ = 1,2), x is a discrete co- 4 FIG. 1: A two-leg ladder and its band structure. In the are also allowed, (here ˆ1 = 2,ˆ2 = 1). Because the low-energylimit, the energy dispersionis linearizednear currents (I ),I are (anti-)symmetric, one can always ij ij the Fermi points. The two resulting relativistic Dirac choose u = u for convenience. We also take uσ = 0 12 21 ii Fermionsaredistinguishedbypseudospinindicesi=1,2 since I = 0. With particle/hole symmetry there are ii for the anti-bonding and bonding bands, respectively. thus just three independent Umklapp vertices, uρ , uρ , 11 12 and uσ . Together with the six forward and backward 12 Electron-electron interactions scatter right-moving vertices, nine independent couplings are required to de- electrons into left-moving electrons and vice-versa, describe the most general set of marginal non-chiral four- stroying this large symmetry. For general spin- Fermion interactions for a two-leg ladder with parti- independent interactions the symmetry will be broken cle/hole symmetry at half-filling. downtoU(1) SU(2),correspondingtototalchargeand Since our analysis below makes heavy use of abelian spin conserva×tion. In the following we consider general Bosonization,47,35 it is convenient at this stage to con- finite-ranged spin-independent interactions between the sider the Bosonized form of the general interacting the- electrons hopping on the two-leg ladder. We assume the ory. To this end, the Dirac fermion fields are expressed typical interaction strength, U, is weak – much smaller in terms of Boson fields as thanthebandwidth. Wefocusontheeffectsoftheinter- actionsto leading non-vanishingorderinU. Inthis limit c =κ eiφPiα, (2.10) Piα iα it is legitimate to keep only those pieces of the interactions which scatter the low energy Dirac Fermions. Of whereP =R/L= . ToensurethattheFermionicoper- ± these, only those involving four-Fermions are marginal, ators anti-commute the Boson fields are taken to satisfy the rest scaling rapidly to zero under renormalization. Moreover,four-Fermioninteractionswhicharechiral,say [φPiα(x),φPjβ(x′)]=iPπδijδαβsgn(x−x′), (2.11) only scattering right movers, only renormalize Fermi ve- [φ (x),φ (x)]= iπδ δ . (2.12) Riα Ljβ ′ ij αβ locities and can be neglected at leading order in small U.18,19Alloftheremainingfour-Fermioninteractionscan Klein factors, satisfying beconvenientlyexpressedintermsofcurrents,definedas κ ,κ =2δ δ , (2.13) 1 { iα jβ} ij αβ Jij = c†iαcjα, Jij = 2 c†iασαβcjβ; (2.5) have been introduced so that the Fermionic operators in I =c ǫ c , I = 1 c (ǫσ) c , (2.6) different bands or with different spins anticommute with ij iα αβ jβ ij iα αβ jβ 2 one another. It will also be convenient to define a pair of conjugate where the R,L subscript has been suppressed. Both J non-chiral Boson fields for each flavor, and I are invariant under global SU(2) spin rotations, whereas J and I rotate as SU(2) vectors. Due to Fermi ϕ φ +φ , (2.14) statistics, some of the currents are (anti-)symmetrical iα ≡ Riα Liα θ φ φ , (2.15) iα ≡ Riα− Liα I =I I = I , (2.7) ij ji ij ji − which satisfy so that I =0 (no sum on i). ii [ϕ(x),θ(x )]= i4πΘ(x x). (2.16) The full set of marginal momentum-conserving four- ′ − ′− Fermion interactions can be written Here, and in the remainder of the paper, we denote by (1) =bρJ J bσJ J , Θ(x)theheavysidestepfunctiontoavoidconfusionwith HI ij Rij Lij − ij Rij · Lij the θ fields defined in Eq. 2.15 above. The field θ is a +fρJ J fσJ J . (2.8) iα ij Rii Ljj − ij Rii· Ljj displacement (or phonon) field and ϕiα is a phase field. The Bosonized form for the kinetic energy Eq. 2.3 is Here f and b denote the forward and backward ij ij (Cooper) scattering amplitudes, respectively, between v = [(∂ θ )2+(∂ ϕ )2], (2.17) bands i and j. Summation on i,j = 1,2 is implied. H0 8π x iα x iα To avoid double counting, we set fii = 0 (no sum on Xi,α i). Hermiticity implies b = b and parity symmetry 12 21 whichdescribesdensitywavespropagatinginbandiand (R L) gives f = f , so that there are generally eigh↔tindependent1c2ouplin2g1sbρ,σ, bρ,σ, bρ,σ,andfρ,σ. At with spin α. 11 22 12 12 This expression can be conveniently separated into half-filling with particle/holesymmetry b =b . Addi- 11 22 charge and spin modes, by defining tional momentum non-conserving Umklapp interactions of the form θ =(θ +θ )/√2 (2.18) iρ i i ↑ ↓ HI(2) =uρijIR†ijILîˆj −uσijI†Rij ·ILîˆj +h.c. (2.9) θiσ =(θi↑−θi↓)/√2, (2.19) 5 andsimilarlyforϕ. The√2ensuresthatthesenewfields totalelectricchargeandS thetotalz-componentofspin, z satisfy the same commutators,Eq. (2.16). It is alsocon- generate“translations”proportionaltoainthetwofields venient to combine the fields in the two bands into a ϕ and ϕ . To see this, we note that Q = dxρ(x) ρ+ σ+ ± combination, by defining with ρ(x) = ∂ θ /π the momentum conjugate to ϕ , x ρ+ ρ+ R whereas S can be expressed as an integral of the mo- z θµ± =(θ1µ±θ2µ)/√2, (2.20) mentumconjugatetoϕσ+. Since the totalchargeis conserved, [Q,H] = 0, the full Hamiltonian must therefore where µ=ρ,σ, and similarlyfor ϕ. It will sometimes be be invariantunderϕ ϕ +aforarbitraryconstant ρ+ ρ+ convenient to employ charge/spin and flavor decoupled → a, precluding a cosine term for this field. Similarly, con- chiral fields, defined as servationofS implies invarianceunderϕ ϕ +a. z σ+ σ+ → The conservation law responsible for the symmetry un- φ =(ϕ +Pθ )/2, (2.21) Pµ µ µ der shifts of the third field, θ , is present only in the ± ± ± σ weak coupling limit. To see t−his, consider the opera- with P =R/L= . ± tor, = k J +k J , with J = (N N ), The Hamiltonian density 0 can now be re-expressed P F1 1 F2 2 i α Riα − Liα H where N is the total number of electrons in band i in a charge/spinand flavor decoupled form, Piα P with spin α and chirality P. At weak coupling with v Fermi fields restricted to the vicinity of k , this oper- = [(∂ θ )2+(∂ ϕ )2]. (2.22) Fi H0 8π x µ± x µ± ator is essentially the total momentum. Since the total µ, X± momentum is conserved up to multiples of 2π, one has Thefieldsθρ+ andϕρ+ describethetotalchargeandcur- ∆P =±2πn=±2n(kF1+kF2) for integer n. Moreover, since the Fermi momenta k are in generalunequal and rent fluctuations, since under Bosonization, c†PiαcPiα = incommensurate,thisimplieFsithat∆J =∆J = 2n,or ∂xTθρh+e/πintaenrdacvtPiocn†PiHαacmPiαilt=on∂iaxnϕsρ+ca/nπ.also be readily ex- equivalently that J1−J2 is conserved1at wea2k co±upling. Since J J = dxj(x) with j(x) = ∂ ϕ /π the mo- pressed in terms of the Boson fields. The momentum mentum1−conj2ugate to θ , this conservaxtioρn−law implies conserving terms in Eq. 2.8 can be decomposed into two invariance underRθ ρ−θ +a. contributions, (1) = (1a)+ (1b), the first 2 involving ρ− → ρ− HI HI HI gradients of the Boson fields, The remaining 5 Boson fields, entering as arguments of various cosine terms, will tend to be pinned at the 1 HI(1a) = 16π2 Xµ± Aµ±[(∂xθµ±)2−(∂xϕµ±)2], (2.23) pamnriidnnicϕmipσa−leo,pfarrteehcedlusuedaelpstoopteionnntneiianalgsn.obtTohtewhrosfiooefltdhtsah.testSehine5cuefinetclhederstr,aeiθnaσtr−ye with coefficient A = 2(cρ fρ ) and A = (cσ fσ)/2, whereas thρe±second11co±ntr1i2bution inσv±olves−cos1i1ne±s various competing terms in the potential seen by these 12 5 fields, minimization for a given set of bare interaction of the Boson fields: strengths is generally complicated. For this reason we (1b) = 2Γbσ cosϕ cosθ employ the weak coupling perturbative renormalization HI − 12 ρ− σ+ grouptransformation, derived in earlier work.18,19 Upon +cosθσ+(2bσ11cosθσ +2Γf1σ2cosϕσ ) systematically integrating out high-energy modes away − − −cosϕρ−(Γb+12cosθσ−+b−12cosϕσ−), (2.24) forrodminathteeaFnedrmFierpmoiinfitesldans,datsheetnorfersecnaolirnmgatlhizeastpioantigarlocuop- with b±12 = bσ12 ±4bρ12. Similarly, the Umklapp interac- (RG) transformations can be derived for the interaction tions can be Bosonized as, strengths. Denoting the nine interaction strengths as g , i the leading order RG flow equations take the general HI(2) =−16Γuρ11cosθρ+cosϕρ−−4uσ12cosθρ+cosθσ+ form, ∂ℓgi = Aijkgjgk, valid up to order g3. For com- −cosθρ+(2u+12cosθσ−+2Γu−12cosϕσ−), (2.25) pAlpetpeennedsisxtAhe. ORuGrflaopwproeqacuhatiisontos ainrteeggrivaetentehxeplRicGitlyfloiwn with u =uσ 4uρ . Here Γ=κ κ κ κ is a prod- equations, numerically if necessary, to determine which uct of±Klein1f2ac±tors1.2Since Γ2 = 1,1↑we1↓ca2n↑ta2↓ke Γ = 1. of the nine coupling constants are growing large. ± Hereafter, we will put Γ=1. In the absence of electron-electron interactions, the Under anumericalintegrationofthese nine flowequa- Hamiltonian is invariant under spatially constant shifts tionsitisfoundthatsomeofthecouplingsremainsmall, ofanyof the eightnon-chiralBosonfields, θ andϕ . while others tend to increase, sometimes after a sign µ µ With interactions five of the eight Boson ±fields ent±er change, and then eventually diverge. Quite surprisingly, as arguments of cosines, but for the remaining three – though, the ratios of the growing couplings tend to ap- ϕ ,ϕ and θ – this continuous shift symmetry is proach fixed constants, which are indepependent of the ρ+ σ+ ρ stillpresent. For−thefirsttwofields,theconservationlaw initial coupling strengths, at least over a wide range in responsibleforthissymmetryisreadilyapparent. Specif- the nine dimensional parameter space. These constants ically,the operatorsexp(iaQ)andexp(iaS ), withQthe can be determined by inserting the Ansatz, z 6 g g (ℓ)= i0 , (2.26) with a = 1,..,4, and P = R/L = as before. The i (ℓ ℓ) ± d first three of these chiral fields satisfy the commutators − Eq. (2.11) and (2.12). But for the fourth field, since intotheRGflowequations,toobtainninealgebraicequa- φ = Pφ , the second commutator is modified to tions quadratic in the constants gi0. There are various [φP4 ,φ ]=Pρ−iπ. distinct solutions of these algebraic equations, or rays in R4 L4 − Intermsofthesenewfields,thefullinteractingHamil- the nine-dimensional space, which correspond to differ- toniandensity alongthe specialraytakesanexceedingly entpossiblephases. Butforgenericrepulsiveinteractions simple form: = + , with between the electrons on the two-leg ladder, a numerical H H0 HI integration reveals that the flows are essentially always = v [(∂ θ )2+(∂ ϕ )2], (3.5) 0 x a x a attracted to one particular ray. In the next sections we H 8π a shallconsiderthepropertiesofthisphase,which,forrea- X sons which will become apparent, we denote by D-Mott. g = ∂ φ ∂ φ HI −2π2 x Ra x La a X III. D-MOTT PHASE 4g cosθ cosθ . (3.6) a b − a=b X6 In the phase of interest, two of the nine coupling constants, bρ and fσ, remain small, while the other seven Wenowbrieflydiscusssomeofthegeneralphysicalprop- 11 12 erties which follow from this Hamiltonian. In the next grow large with fixed ratios: sections we will explore in detail the symmetries present 1 1 in the model, and the resulting implications. bρ12 = 4bσ12 =f1ρ2 =−4bσ11 = (3.1) Groundstate properties ofthe aboveHamiltonian can be inferred by employing semi-classical considerations. 1 2uρ11 =2uρ12 = 2uσ12 =g >0. (3.2) Since the fields ϕa enter quadratically, they can be in- tegrated out, leaving an effective action in terms of the Oncetheratio’sarefixed,thereisasingleremainingcou- four fields θa. Since the single coupling constant g is pling contant, denoted g, which measures the distance marginally relevant and flowing off to strong coupling, from the origin along a very special direction (or “ray”) these fields will be pinned in the minima of the cosine intheninedimensionalspaceofcouplings. TheRGequa- potentials. Specifically,therearetwosetsofsemiclassical tions reveal that as the flows scale towards strong cou- ground states with all θa =2naπ or all θa =(2na+1)π, pling, they are attracted to this special direction. If the wherena areintegers. Excitationswillbeseparatedfrom initial bare interaction parameters are sufficiently weak, the ground state by a finite energy gap, since the fields the RG flows have sufficient “time” to renormalize onto areharmonicallyconfined,andinstantonexcitationscon- thisspecial“ray”,beforescalingoutoftheregimeofper- necting different minima are also costly in energy. turbativevalidity. Inthiscase,thelowenergyphysics,on Since both θσ fields are pinned, so are the spin-fields the scale of energy gaps which open in the spectrum, is in each band, θi±σ (i = 1,2). Since ∂xθiσ is proportional universal,dependingonlyonthepropertiesofthephysics to the z-component of spin in band i, a pinning of these alongthisspecialray,andindependentofthepreciseval- fields implies that the spin in each band vanishes, and ues of the bare interaction strengths. excitationswith non-zerospinareexpected to costfinite Toexposethisuniversalweakcouplingphysics,weuse energy: the spin gap. This can equivalently be inter- Eq. 3.2 to replace the nine independent coupling con- preted as singlet pairing of electron pairs in each band. stants in the most general Hamiltonian with the single Itisinstructivetoconsiderthepairfieldoperatorinband parameter g, measuring the distance along the special i: ray. Doing so reveals a remarkable symmetry, which is mostreadilyexposedintermsofanewsetofBosonfields, ∆i =cRi↑cLi↓ =κi↑κi↓e√i2(ϕiρ+θiσ). (3.7) defined by, With θ 0, ϕ can be interpreted as the phase of the iσ iρ ≈ pair field in band i. The relative phase of the pair field (θ,ϕ) = (θ,ϕ) , (θ,ϕ) = (θ,ϕ) , 1 ρ+ 2 σ+ in the two bands follows by considering the product (θ,ϕ) =(θ,ϕ) , (θ,ϕ) =(ϕ,θ) . (3.3) 3 σ 4 ρ − − ∆1∆†2 =−Γeiθσ−eiϕρ−, (3.8) The first three are simply the charge/spin and flavor fieldsdefinedearlier. However,inthefourthpairoffields, with Γ = κ1 κ1 κ2 κ2 = 1. Since θ4 = ϕρ the rel- θ and ϕ have been interchanged. It will also be useful to ative phase is↑al↓so p↑inn↓ed by the cosine poten−tial, with consider chiral boson fields for this new set, defined in a sign change in the relative pair field, ∆1∆†2 < 0, cor- the usual way, responding to a D-wave symmetry. Being at half-filling, the overall charge mode, θ is also pinned – there is ρ+ φ =(ϕ +Pθ )/2, (3.4) a charge gap – and the two-point pair field correlation Pa a a 7 function falls off exponentially with separation. We re- where ψ = (ψ ,ψ ), and τ is a vector of Pauli ma- a Ra La fer to this phase as a “D-Mott” phase, having D-wave trices acting in the R,L space. Here, summation over pairing correlations coincident with a charge gap. Upon repeated indices, a = 1,2,..,4 is implicit. It is remark- doping the D-Mott phase away from half-filling, gapless able that the Hamiltonian can be written locally in the charge fluctuations are expected in the (ρ+) sector, and “fundamental” fermion variables, which are themselves power-law D-wave pairing correlations develop. highly non-locally related to the “bare” electron opera- It is worthnoting that the fully gapped D-Mott phase tors. has a very simple interpretation in the strong cou- AfurthersimplificationarisesuponchangingtoMajo- pling limit. Two electrons across each of the rungs of rana fields, the two-legged ladder form singlets, of the usual form , , , where the two states refer to electrons on 1 |le↑g↓1io−r|2↓, r↑eispectively. This two-electron state can be ψPa = √2(ηR2a+iηR2a−1). (4.5) re-written in the bonding anti-bonding basis, and takes the form, , , ,wherethe twostatesnowre- The Hamiltonian density then takes the manifestly in- ferto bond|i↑n↓g−anid−a|n−ti-↑b↓oindingorbitals. This resembles variant form a local Cooper pair, with a relative sign change between 1 1 bondingandanti-bondingpairs: anapproximateD-wave = η i∂ η η i∂ η +gGABGAB, (4.6) H 2 RA x RA− 2 LA x LA R L symmetry. where the currents are IV. SO(8) GROSS-NEVEU MODEL GAB =iη η , A=B, (4.7) P PA PB 6 As shown above, the bosonized effective Hamiltonian and A,B =1...8. on energy scales of order the gap is exceptionally simple in the D-Mott phase. In this section, we show that this simplicityisindicativeofahighersymmetry,andexplore B. SO(8) Symmetry its ramifications upon the spectrum. Eq. 4.6 is the standard form for the SO(8) Gross- Neveu model, which has been intensively studied in the A. Gross-Neveu Model literature.36–40,42,43 We firstdiscussits manifestsymme- try properties. An obvious symmetry of the bosonic action, Eqs. 3.5- The 28 currents GAB generate chiral SO(8) transfor- 3.6,ispermutationofthe fields θa →Pabθb, wherePab is mations. For g = 0,PEq. 4.6 has two independent sym- apermutationmatrix. Infact,thisisonlyasmallsubset metries under separate rotations of the left- and right- ofthetrueinvariancesofthemodel. Asisoftenthecase, moving fields. For g = 0, however, only simultaneous abelian bosonization masks the full symmetry group. It 6 rotations of both chiralities are allowed. More precisely, can be brought out, however, by a refermionization pro- the unitary operators cedure. We define “fundamental” (Dirac) fermion operators ψ with a=1,2,3,4 via Pa U(χ )=eiχAB dx(GARB+GALB), (4.8) AB ψPa =κaeiφPa, a=1...3 R ψ =Pκ eiφP4, (4.1) generate global orthogonaltransformations of the Majo- P4 4 rana fields, and P = R,L = 1, as before. The Klein factors are given by ± U†(χ)ηPAU(χ)=OAB(χ)ηPB, (4.9) κ =κ κ =κ , (4.2) where the orthogonalmatrix (χ) is given by 1 2↑ 2 1↑ O κ =κ κ =κ . (4.3) 3 1↓ 4 2↓ (χ)=eiχABTAB. (4.10) O In the re-Fermionization of the fourth field we have chosen to include a minus sign for the left mover. This is Here the TAB (A > B) are the 28 generators of SO(8) convenient, due to the modified commutators between inthe fundamentalrepresentation,with matrixelements tthhee “lesfttaannddarrdi”ghftorfimeldins:E[qφ.R42,.1φ2L.4]=−iπ, in contrast to [tThAeBη]CPAD =train(δsAfoCrδmBDas−SδOAD(δ8B)Cv)e/c2t.orEs.q.S4im.9ilianrdliyc,atthesetchuart- In these variables, the effective Hamiltonian density rents GAB are rank 2 SO(8) tensors. P becomes It is worth noting that despite the non-local relation betweenthefundamentalandbarefermionoperators,the =ψ iτz∂ ψ g ψ τyψ 2, (4.4) SO(8)symmetryremainslocalinthebareelectronbasis. H a† x a− a† a (cid:0) (cid:1) 8 This followsfromthe factthat the chiralSO(8) currents symmetry in the right and left moving sectors. The spin in the two bases are actually linearly related, i.e. currents J can then be shown to satisfy, P GAB =MABCDG˜CD, (4.11) [Ja(x),Jb(x′)]=δ(x x′)iǫabcJc(x)+ (4.19) P P P P P − P P where GÃPB =iξPAξPB, and the bare Majoranaoperators i2πkδabδ′(x−x′), (4.20) are defined by with a,b,c= x,y,z and k = 2. This is referred to as an 1 SU(2) current algebra at level (k) two. c = (ξ +iξ ), (4.12) P1↑ √2 P2 P1 We conclude this subsection by answering a question which may have occurred to the alert reader: why is 1 c = (ξ +iξ ), (4.13) the symmetry of the model SO(8) rather than O(8)? P1↓ √2 P4 P3 Based on Eq. 4.6, it would appear that any transfor- 1 mation of the form η η would leave the cP2↑ = √2(ξP6+iξP5), (4.14) Hamiltonianinvariant,PinAclu→dinOgAimBpPrBoperrotationswith 1 det = 1. The presence of such improper rotations c = (ξ +iξ ). (4.15) O − P2↓ √2 P8 P7 means O(8) = SO(8) × Z2, since any orthogonal matrix can be factored into a product of matrix with de- ThepreciseformsofthetensorsM arecomplicatedand terminant one and a particular (reflection) matrix, e.g. P notparticularlyenlightening. Nevertheless,theexistence r = δ 2δ δ . We have already shown above OAB AB − A1 B1 of the linear relation between currents implies that the that the SO(8) symmetry is physical – i.e. the symme- unitary operator U(χ) also generates local rotations of try generators act within the Hilbert space of the phys- the bare electron fields. In these variables, however, the ical electrons. It is straightforward to show that the 2 Z SO(8) symmetry is hidden, because M = M , which reflection is however, unphysical. To see this, imagine R 6 L implies different rotations must be performed amongst performingthe reflectioneffectedby r above,which 2 Z O right- and left-moving electron operators. takesη η . Usingthebosonizationrules,thiscor- P1 →− P1 Finally, it is instructive to see how the conserva- responds to θ θ and ϕ ϕ . Returning to the 1 1 1 1 →− →− tion of total charge and spin, corresponding to a global physicalfields, onefindsthatthe bareelectronoperators U(1) SU(2)symmetry,isembeddedinthelargerSO(8) transform much more non-trivially: × symmetry. Tothisend,considerthetotalelectroncharge operator, Q, which in terms of the low energy fields can cPiα −Z→2 cPiαψP†1. (4.21) be written, As weshallshowin Sec.IV.E.3,a singleGNfermionop- Q=2 dx ψP†1ψP1 =2 dx(G2R1+G2L1), (4.16) etarakteosra,spuhcyhsiacsalψeP†le1,ctisrounnoppheyrsaictaolr.inTthoeaZn2urnepflheycstiicoanltohnues, Z XP Z which implies that the symmetry cannot be effected by a unitary operator within the Hilbert space of the elec- where ψ is a fundamental Gross-Neveu fermion. The P1 trons. For this reason, the true symmetry group of the U(1) charge symmetry is thus seen to be equivalent to ladder model is SO(8). the SO(2) symmetry of rotations in the 1 2 plane of − the eight-dimensional vector space. Similarly, the total GN Fermions spin operator (two-particle) y y S = dx[J (x)+J (x)], (4.17) R L La Ra Z with J (x) = J (x), can be re-expressed in terms of P Pii SO(8) generators by using, Ja(x)=ǫabcGbc, (4.18) P P y o y o y e y e witha,b,c=3,4,5=x,y,z. Thusweseetheequivalence La Ra La Ra betweentheSU(2)spinrotationsandSO(3)rotationsin Odd Kinks Even Kinks the 3-dimensionalsub-space3 4 5oftheeightdimen- (single-particle) (single-particle) − − sional vector space. Rotations in the five-dimensional FIG. 2: Triality between GN fermions, even kinks and subspace 1 2 3 4 5, correspond to global SO(5) odd kinks. The SO(8) GN Hamiltonian is identical in − − − − rotationswhichunifythechargeandspindegreesoffree- terms of these three sets of fermionic operators. Opera- dom. torsinthegrayareasarephysicalandgaugeindependent In the absence of interactions in the Gross-Neveu (see Sec. IV.E), while the other fermion operators must model, all ofthe excitations including spinremainmass- be “dressed” by an appropriate Jordan-Wignerstring to less. In this case there is an independent SU(2) spin remain in the physical Hilbert space. 9 C. Triality differs from those in the conventional GN model. This difference arises from the non-local relation between the Most of the above properties hold more generally for electronandGNfields. ExcitationswithintheGNmodel the SO(N) GN model, even for N = 8. However, the mustbeslightlymodifiedtosatisfygaugeinvariancewith case N =8 is extremely special, and6in fact possesses an respectto someunphyicaldegreesoffreedomintroduced additional triality symmetry not found for other N. A in the mapping. These modifications and the resulting useful reference is Ref. 42,43. spectrum in the D-Mott phase are described in the sub- To expose the additional symmetry, we return to the sequent subsection. sine-Gordon formulation. Essentially, the triality opera- Within the GN model, the excitations are of course tiontradestheoriginalbasis θ inthefour-dimensional organized into SO(N) multiplets, but are further con- a space of boson fields for eith{er o}ne of two other orthog- strainedforthecaseofinterest,N =8,bytriality. Inthis onal bases. Explicitly, the two alternate choices are the subsection,wediscussthelowest-lyingstates,theirmulti- even and odd fields θe/o, where pletstructuresandquantumnumbers,andgivesomeuse- a ful physical pictures to aid in understanding their prop- θe/o =(θ +θ +θ θ )/2, (4.22) erties. 1 1 2 3± 4 θe/o =(θ +θ θ θ )/2, (4.23) 2 1 2− 3∓ 4 θe/o =(θ θ +θ θ )/2, (4.24) 1. Results from integrability 3 1− 2 3∓ 4 e/o θ =(θ θ θ θ )/2. (4.25) 4 1− 2− 3± 4 The lowest-lying excitations are organized into three SO(8) vector multiplets, which are degenerate due to Heretheupperandlowersignsapplytotheevenandodd triality, for a total of 24 particles. Four of the 28 global fields, respectively, and identical definitions hold for the SO(8) generators may be chosen diagonal (to form the e/o e/o dual ϕ and chiral φ bosons. The bosonized Hamil- a Pa Cartan subalgebra). We will label the particles by the tonianinEqs.3.5,3.6isinvariantundereitherchangeof values of the four associated charges, denoted by the or- variables, i.e. dered quadruplet (N ,N ,N ,N ), and defined by 1 2 3 4 H[θ ]=H[θe]=H[θo]. (4.26) a a a N = dxψ ψ , (4.32) a a† a For eachof these bases, an inequivalent refermionization Z is possible, analogous to the introduction of the funda- (nosumona). Inthisnotation,oneSO(8)multipletcon- mental fermions in Eq. 4.1. In particular, the Hamilto- tains the states (known as fundamental fermions) with nian is unchanged in form when rewritten in terms of onlyoneofthefourN = 1,andallothersequaltozero. either the even or odd fermion operators, a ± The remaining 16 degenerate states have N = 1/2 for a ± a = 1,2,3,4, which are divided into those with an even ψPe/ao =κea/oeiφeP/ao. (4.27) numberofNa =+1/2(theevenkinks)andtheremainder withanoddnumber ofN =+1/2(the oddkinks). The It should be noted that the set of even and odd fermion a reasons for this terminology will become apparent later operators contains all the bare electron fields. In partic- inthissection. Eachparticlehasamassmanddisperses ular, (due to Lorentz invariance) as ǫ (q) = m2+q2, with 1 ψRe1 =cR1↑, ψLo1 =cL1↑, (4.28) mdeofimneedntruemlatqi.veSitnocethtehiereFleecrmtroinmboamnednotappekrat,ortshecPaiαctaurael ψe =c , ψo =c , (4.29) Fi R2 R2↑ L2 L2↑ momenta of each particle is offset from the GN model ψRe3 =cR2↓, ψLo3 =cL2↓, (4.30) momentum, q, by some amount. We will return to these ψe =c , ψo =c . (4.31) “base”momentalaterinthissubsection,aswellastothe R4 R1↓ L4 L1↓ other physical quantum numbers of the excitations. The other eight even and odd fields (ψLea and ψRoa) are At somewhat higher energies there is another multi- not simply related, however, to the electron fields. plet of 28 “particles”, which transform as an antisym- metric second rank SO(8) tensor. This multiplet can be viewed as two-particle bound states of the funda- D. Conventional Gross-Neveu Excitation spectrum mental Gross-Neveu fermions, or equivalently under triality as bound even-even or odd-odd kinks. Indeed, TheSO(N)GNmodelisintegrable,andtheexcitation of these 28 states, 24 have two zero charges and two spectrumisknownexactly. Toorganizethepresentation, N = 1. The other four are bound states of a fun- a ± we divide the discussion of the excitation spectrum into damental fermion with it’s anti-particle (an excition in two parts. In this subsection, we summarize known re- the semiconductor picture, below), so they do not carry sults for the conventionalGN model. The precise nature any of the four quantum numbers. Each of the 28 “par- of the excitations for the two-leg ladder model, however, ticle” states has a mass m =√3m. Finally, for energies 2 10