Spinless Fermionic ladders in a magnetic field: Phase Diagram Sam T. Carr and Boris N. Narozhny The Abdus Salam ICTP, Strada Costiera 11, Trieste, 34100, Italy. Alexander A. Nersesyan The Abdus Salam ICTP, Strada Costiera 11, Trieste, 34100, Italy. and The Andronikashvili Institute of Physics, Tamarashvili 6, 0177, Tbilisi, Georgia 6 (Dated: February 6, 2008) 0 0 Thesystemofinteractingspinlessfermionshoppingonatwo-legladderexhibitsaseriesofquan- 2 tumphasetransitions whensubjected toanexternalmagneticfield. Athalffilling, theseareeither U(1) Gaussian phase transitions between two phases with distinct types of long-range order or n Berezinskii-Kosterlitz-Thouless transitions between ordered and gapless phases. a J PACSnumbers: 71.10.Pm;71.30.+h 9 ] I. INTRODUCTION behavior of many naturally found compounds7, includ- l e ing carbon nanotubes,8,9 as well as artificially manufac- - r tured structures.10 On the other hand, they provide a Thebehaviorofinteractingelectronsystemsunderthe t fertile ground for application of theoretical techniques s actionofanexternalmagneticfield,affectingorbitalmo- . developed for one-dimensional systems11 - i.e. non- t tionoftheparticles,isasubjectofintenseresearchofthe a perturbative approaches leading to asymptotically exact m last several decades. It has long been established that results. Furthermore, they are a first step for various the magnetic field has a dramatic effect on allproperties - attempts at generalization of the lore of physics in one d of the system. Even in the absence of interaction, the spatial dimension to higher-dimensional problems.12 n spectrum of the free Fermi gas is modified and exhibits o Landauquantization1inthecontinuumortheHofstadter As is often the case, situations where the number of c spectrum2 on a two-dimensional (2D) lattice. The most particles is commensurate with the lattice attract the [ most attention since only then can long range order spectacularconsequenceofthisphenomenonistheQuan- (LRO) develop. We have previously shown13 that at a 1 tum Hall Effect (QHE).3 Electron-electron interaction v compounds the complexity of the problem, giving rise quarterfilling,the presenceofthe flux resultsinexciting 4 to the Fractional Quantum Hall Effect (FQHE).4 effects which do not exist in the absence of the flux. In 6 particular the uniform external magnetic field can lead 1 The current theoretical understanding of the effect of to a staggered flux (or orbital antiferromagnet) phase, 1 the magnetic field on the properties of electron systems which furthermore has fractionally charged excitations. 0 was achieved by a combination of various methods and The purpose of the present paper is to describe the 6 techniques, each of which is strictly speaking only appli- fullphasediagramofinteractingspinlessfermionsonthe 0 cable in a certain parameter range. What is still lacking / two-leg ladder at 1/2-filling in the presence of an ex- t is a comrehensive approach that would unify all of the a ternal magnetic field. To drive a system to criticality different aspects of the problem into a single coherent m by applying the magnetic field is an intriguing possibil- picture. Perhapsatpresentsuch a goalis too ambitious. - ity which should be more accessible in experiment than d However, as a first small step in this direction, one can varying coupling constants. The phase diagram displays n ask whether such a comprehensive approach can be for- amultitudeofquantumphasetransitionsinducedbythe o mulated for a simpler model, that wouldon one hand be flux. There are two types of these phase transitions: (i) c a problem of interacting electrons in the magnetic field : Berezinskii-Kosterlitz-Thouless (BKT) transitions14 be- v and on the other hand would, at least in principle, allow tween ordered and disordered ground states, and (ii) i a generalization towards the original problem. X U(1) Gaussian phase transitions between different or- The simplest model of interacting fermions that in- dered ground states. Here we choose the fermions to r a corporates orbital effects of an external magnetic field be spinless in order to eliminate Zeeman splitting and is that of spinless fermions hopping on a two-leg ladder. focus on the orbital effects of the magnetic field. The Thismodelissimpleenoughnottoexhibitthemultitude case of spin-1/2 particles will be discussed in a separate of small gaps in the single-particle spectrum (character- publication. istic of the Hofstadter problem2). Yet the magnetic flux Traditionally15,16 ladder models have been treated in piercing the plaquettes of the ladder changes the ground two complimentary approaches. On one hand, one can state properties of the system giving rise to non-trivial start with the model of two decoupled chains, define the phases and inducing quantum phase transitions. low energy effective theory for each of them, and then Ladder models5,6 occupy a special place in the field of treatboththesingle-particletransversehoppingandtwo- strongly correlated electron systems. On one hand, they particle inter-chaincorrelationsperturbatively.17 On the describe (at least within some range of temperatures) otherhand,onecouldstartwiththeexactsingleparticle 2 basis (given by two bands) and then proceed with the models of spinless fermions and Hubbard-like chains of corresponding low energy limit.6 For the major part of spin-1/2 particles. In our case however, the SU(2) sym- this paper we will be using the latter approachwhich al- metry is explicitly broken in the Hamiltonian (1) by the lowsus to treatintra-andinter-chainprocessesonequal inter-chainhopping and the V interactionterm (the for- footing. However, it is well known18 (at least in the ab- mer is analogous to a Zeeman energy due to an external sence of the magnetic field) that for weak enough inter- magnetic field alng the x-axis,while the latter is a coun- chain tunneling there exists the phenomenon of Ander- terpart of an exchange anisotropy along the z-axis). son confinement, i.e. the suppression of the inter-chain The external magnetic field B is introduced by means single-particle tunneling by intra-chain two-particle cor- ofthePeierlssubstitution.19 IntheLandaugauge20 with relations. Thiseffect canbe seenineitherpicture, butit the vector potential A=B( y,0,0)the transverse hop- − is more intuitive to discuss it within the chainapproach. ping term is independent of the field, while the longitu- Bothwaysshouldproducethesamephysicalresults,but dinal hopping amplitude can be written as understanding the relation between the two approaches allows us to establish the limits of applicability of the t (σ)=t eiπσf, (2) 0 effective low energy theories that can be derived within k either picture. Moreover, we discuss how the Anderson wheref isthemagneticfluxthroughtheelementarypla- confinement regime is affected by the magnetic flux. quette in units of flux quantum φ =hc/e. Expressedin 0 The remainder of the paper is organized as follows. terms of the flux the model is explicitly gauge invariant. We start by defining the microscopic Hamiltonian of the model and proceed directly to the results, discussing the phase diagram and the other physical properties of the model. Then we outline the details of the calculations within the weak coupling (bosonization) approach. In a) Relative CDW c) BDW Section III we derive the effective low energy theory in the band picture. In Section IV we turn to the chain picture and discuss its relation to the band approach. Section V is devoted to the strong coupling limit of our b) CDW d) Relative BDW model and is followed by a brief summary of the results. Mathematical details are relegated to Appendices. e) OAF II. MODEL AND RESULTS FIG.1: Cartoon depictions ofpossible ordered phasesat 1/2 filling. Thedotsrepresentexcessfermiondimerizationonthe In this section we present our results. We start by sites. The ellipses represent excess occupation on the bonds. The arrows represent local currents. Note that the OAFand defining the microscopic Hamiltonian of our model and theBDW coexist - see Section III for an explanation. proceed to discuss the zero-temperature phase diagram. A. The Hamiltonian B. Phase Diagram Weconsideratight-bindingmodelofspinlessfermions on a two-leg ladder described by the Hamiltonian At half filling (one fermion per rung) the model (1) is characterizedbyaratherrichphasediagram. Depending 1 H = − 2 tk(σ)c†i,σci+1,σ+h.c. −t⊥ c†i,σci,−σ onthevaluesofmicroscopicparametersthegroundstate Xiσ h i Xiσ of the model may posess true long range order. Possi- + U n n +V n n . (1) bleorderedphasesareillustratedinFig.1. Thecartoons i+ i i,σ i+1,σ − showastrong-couplingpictureofthephases: chargeden- i iσ X X sity waves (CDW), where particles are localized on sites Here ci,σ is the electron annihilation operator on the oftheladder;bonddensitywaves(BDW)withdimerized chain σ = ± at the site i; niσ = c†i,σci,σ are the occupa- links along the chains;and the orbitalantiferromagnet17 tion number operators;t and t are the transverse and (OAF),sometimesreferredtoasthestaggeredfluxphase longitudinal hopping am⊥plitudesk, respectively. The last or a d-density wave, where the particle density remains twotermsinEq.(1)describenearestneighbourinter-and uniform,butthereexistnon-vanishinglocalcurrentsthat intra-chain interactions. The chosen form of short-range haveoppositedirectionsonalternatebonds. Noticethat, interaction is quite representative because it reflects the in the spin language, the Relative CDW is similar to a generic symmetry of the ladder and yields the most gen- spindensity wave(SDW)withspins polarizedalongz (a eral effective field theory in the low-energy limit. Our Neel state SDWz), whereas the OAF is equivalent to a notation reflects the well-known analogy between ladder SDWy. 3 In addition, the model (1) allows for various phases which varies along the horizontalaxis. Four different re- that do not posess long-range order. Using the afore- gions corresponding to various signs of the constants U mentionedanalogywiththe Hubbardchain,we maydis- andV areindicatedontheupperhorizontallineinFig.3. cuss the model (1) in terms of “spin” (or “relative”)and The analytic description of the phase boundaries, based “charge” (or “total”) sectors. In Section IIIB we show on the weak coupling theory, is given in Appendix A. that,inthelow-energylimit,the“charge”and“relative” The position of the boundaries depends on the applied sectors of the model asymptotically decouple. In all of field,theratioU/2V (i.e. θ)andtheratioofthehopping the orderedphases both sectors are gapped. However,it parameters, τ = t /t . We plot the phase diagram as a 0 is possible to have a phase where only one of the sectors function ofθ and fl⊥ux for τ =0.25,the value is arbitrary acquires a spectral gap. Phases where only the “charge” but representative as long as τ exceeds a possible gap sector is gapped, irrespective of the type of dominant in the “relative” sector. At other values of τ the topol- correlations, we will call the Mott Insulator (MI). The ogyofthe phasediagramandclassificationofthe phases cases where the gap exists in the “relative” sector only do not change qualitatively. Similarly, if we modify our willbe calledtheLuther-EmeryLiquid(LEL).21 Finally, model(1)toincludeothershort-rangeinteractionterms, when both sectors are gapless, the system represents a the onlyeffect onthe phasediagramwouldagainbe just Luttinger Liquid (LL). the shift of the phase boundaries. Let us now describe the phase transitions induced by U the applied field (i.e. the vertical direction in Fig. 3). We assume that the inter-chain hopping parameter τ is O not too small (see next subsection). Note, that since A F Relative CDW the model (1) is invariantunder the transformationf LEL → 1 f, σ σ,we only need to consider the flux within − → − V the range 0<f <1/2. Moreover,when the flux is large enough,sin2πf >1 τ2,thereisabandgapinthesingle − LL MI particlespectrumofthe model. Thatstate is largelyun- affected by interaction effects (at least within the weak- coupling limit), and thus we will restrict our discussion tosmallervaluesoff. Whiletheweak-couplingapproach cannotbetrustedatfieldstooclosetothebandgaplimit (sincetheFermivelocitybecomestoosmall),wecontinue the phase boundaries up to that point. All phase transi- FIG. 2: Phase diagram at B = 0 (after Ref. 16). Phase boundaries correspond to the lines V = 0, U/2V = 2+τ2 tions of interest happen sufficiently far from that region. and U/2V = τ2. − Apart from a brief discussion in Sec. V, we will not con- − sider the details of the transition to the band insulator in this paper. The phase diagram in the absense of the flux is known.15,16 ForthesakeofclarityweincludeitinFig.2. The most interesting features of the phase diagram in This phase diagram is valid for suffiently large values Fig. 3 are a sequence of U(1) phase transitions between of t where delocalization of the fermions across the different ordered states and reentrant transitions. Un- rung⊥s suppresses the CDW phase (which happens in derstandingofthesetransitionsisbasedonthefactthat, the absence of the inter-chain tunneling). There are as shown in Section IIIB, in the low-energy limit the two ordered phases: (i) for purely repulsive interactions “charge” and “relative” degrees of freedom of the model U,V >0, one hasa relativeCDW as to be expected (see decouple, and each sector is described by a sine-Gordon Section V); (ii) for repulsive interchain interaction and model(seeEq.(12)andAppendix A). PhaseswithLRO nottoostrongattractivein-chaininteraction,theground correspond to strong-coupling regimes in both sectors. state is the orbital antiferromagnet.17,22 The phase di- The phases whose order parameters are mapped onto agram in Fig. 2 was obtained within a weak-coupling eachotherunderasignchangeofthecorrespondingcou- bosonization approach. The phases do exist when the plingconstant(theamplitudeofthecosineterm)aremu- coupling becomes strong, however the exact location of tually dual. The associatedU(1) Gaussiancriticality oc- the phase boundaries might change. cursattheself-duallines,i.e. whentheoneofthosecou- Once the magnetic field is applied, the system may pling constants vanish. Such a duality is commonplace exhibit additional phase transitions. In Fig. 3 we plot inlow-energyeffectivetheories,indeedmorecomplicated the entire weak-coupling phase diagram for the model non-Abelian dualities were found recently in the SU(4) at half-filling and sufficiently large t (see next subsec- Hubbard model.23 However, it has also been recently tion). The magnetic flux varies alon⊥g the vertical axis, shown24 thatforcertainladdermodelsthe(Abelian)du- so that the diagramin Fig. 2 corresponds to the bottom ality between different phases turns out to be not only a axis of Fig. 3. The ratio of the microscopic interaction symmetry emerging in the low-energylimit but an exact parametersofthe Hamiltonian(1),U/2V, is represented property of the underlying microscopic model. in Fig. 3 through the angular variable θ = tan 1U/2V Gaussiantransitionsoccurintwodomainsofthephase − 4 τ = 0.25 U<0, V<0 U<0, V>0 U>0, V>0 U>0, V<0 U=0 V=0 U=0 V=0 U=0 0,5 Band Insulator (Hofstadter flux state) 0,4 OAF 0,3 x u fl LEL CDW MI LL 0,2 (SC) (SC) relative (BDW) LL CDW (SC) 0,1 relative BDW MI LEL (CDW) OAF (SC) 0 −π −π/2 θ 0 π/2 π FIG. 3: The weak-coupling phase diagram in the magnetic field. We plot the flux along the vertical axis and the angle θ (defined as θ = tan−1U/2V) along the horizontal axis. As the phases depend on the signs of the interaction parameters, they are indicated at the top of the diagram. Ordered phases are illustrated pictorially in Fig. 1. The corresponding order parameters are listed in Table I. Thedisordered phases arecharacterized bydominant correlations (indicated in parentheses). Forlargevaluesoftheflux(sin2πf >1 τ2),thereisabandgapinthenon-interactingpicture. Thethicksolidlines(blueand − green online) represent U(1) Gaussian transitions between mutually dual ground states with long-range order, and the thick dotted lines (black and red online) are Berezinski-Kosterlitz-Thouless phase transitions corresponding to opening of a gap in one of the sectors. diagram: (i) for repulsive intrachain interaction (V >0) BKT transition) and the system becomes a MI, but now and weak attractive interchain interaction (U < 0), and the dominant correlation is that of the 2k component F (ii)forU >0andweakV <0. Inthefirstcase(i),atzero of the transverse bond density, labeled by “(BDW)”. flux the system is a MI. The dominant (longest-range) Thislattertransitionturnsouttobereentrant. Asthe correlation turns out to be that of the 2kF component fluxisfurtherincreasedthesystemreturns(againviathe of the total charge density (hence the label “(CDW)” in BKT transition) back to the relative CDW state. There Fig. 3; see Section IIID for details on dominant corre- is another example of a reentranttransitionin the phase lations). As the flux is increased, the system becomes a diagram, for U < 0 and small V the zero-field ground relativeBDW(viatheBKTtransitionwhereagapopens state is a Luttinger Liquid, which once subjected to the in the “relative” sector of the effective theory). Further external field first becomes a LEL by opening a gap in increase of f drives the system through the U(1) transi- the“relative”sector,andthenathigherfieldcomesback tion to a CDW phase. At larger values of the flux (ap- to the LL state in which the most singular fluctuations proaching the band gap limit) values of flux, the system are those of the pairing operatorat momentum π 2k . F eventuallybecomesanOAF,againthroughtheGaussian Inthe LELphasethe “relative”sectorisgappeda−ndthe transition. “charge” sector is characterized by the dominant corre- Thesecondtransition(ii)occursatsmallvaluesofthe lation of the pairing operator at zero momentum. flux. The zero-field ground state is an OAF. As we turn Finallythereisalargepartofthephasediagramwhich on the flux, the system undergoes a U(1) transition to- is robust against the application of the external field. wardsa relativeCDWstate. Further increaseof the flux When both inter- and intra-chain interactions are at- resultsinclosingofthegapintherelativesector(viathe tractive, the LL (that is the zero-field ground state) is 5 mostly unaffected by the field. More interesting is the analogof the Meissner effect that can be seenin bosonic situationwhenbothinteractionsarerepulsive. Thezero- ladders.25 Details are presented in section IV. field ground state is the relative CDW. It turns out that this long-range order survives under the application of the field (except possibly for the transitionto the MI for weak V discussed above). III. LOW ENERGY EFFECTIVE THEORY: BAND BASIS C. Commensurate-Incommensurate transition InthisSectionwederivetheeffectivelowenergytheory for the model (1) taking the exact single particle spec- The above phase diagram breaks down if the param- trum as our starting point. As mentioned above, there eter h = [sin2(πf) + τ2]1/2, which in the noninteract- exists an alternative approach,which starts with discon- ing case determines the splitting between the Fermi mo- nected (but interacting) chains. The relation between menta of different bands, is too small (see Sections III the two will be discussed in the next Section. and IV). Then, the part of the phase diagram that cor- responds to attractive inter-chain interaction U < 0 ac- quires additional ordered phases. This is the result of additional inter-band scattering processes that at larger I II h violate momentum conservation in the low energy ef- (k) (k) fective theory (based on the two-band description). The latterissue revealsthe dichotomybetweenthe twostart- k k ing points already mentioned in the Introduction: chain basis versus band basis. If one starts with a solution (howevercomplete)fortwoindependentchainsandthen tries to take into account the inter-chain hopping (as well as the flux) in perturbation theory, then the pro- III IV (k) (k) cesses mentioned above are present in the theory from the beginning. In the case when these processes gen- k k erate a gap in the spectrum of relative degrees of free- dom, a finite splitting of the Fermi momenta would not take place unless the parameter h exceeds its critical value comparable with the gap. This is the well-known commensurate-incommensurate transition.16 As the pa- rameter h increases further, the “two-chain” approach FIG. 4: Possible types of the single-particle spectrum as a fails because renormalization of the parameters of the function of in-chain momentum. For any given τ, increasing theory becomes sizable at sufficiently large h. At that thefluxwilleventuallyopenaband-gapinthenon-interacting point one would be forced to start with the exact, two- spectrum. Close to this transition one of the two bands is band single-particle spectrum of the ladder. However, almost empty while the other is almost full. At this point, thiswouldseeminglyneglecttheprocessesinquestionas curvature effects of the spectrum become important; these they appear to violate momentum conservation. are beyond thescope of this paper. In Section IV we discuss the relation between the two approachesto ladder problems andshowthat if one uses either approach properly, then the final result is inde- pendent of the starting point, as should be expected. The new phases at U < 0 naturally emerge through the A. Single-particle spectrum commensurate-incommensurate transition.16 We shall also show that, regardless of the starting point, some properties of the system are not accessible The single-particle part of the Hamiltonian (1) can be within the effective low energy theory. The quantity in diagonalized by the unitary transformation questionisthediamagnetic(orpersistent)current,which turns out not to be an infra-red property. All electrons c (k) = u α +v β , 1 k k k k participateinthiscurrent. Inparticular,thecurvatureof c (k) = v α u β (3) thesingle-particlespectrumattheFermipointsbecomes 2 k k− k k important, so that linearization of the spectrum, being the usual prerequisite in the derivation of any effective where the “coherence factors” u ,v [which are positive k k low-energy theory, completely destroys this effect. Con- as the signs are written explicitly in Eq. (3)] are given sequently, within the bosonization approach in the con- by (if not stated otherwise, in this section we will mea- textofthefermionladder,itisimpossibletodescribethe surethemomentumk inunitsoftheinverselongitudinal 6 lattice spacing, 1/a) absence of interaction) is given by 1 sinksinπf dk u2 = 1 j = 2t sinπf cosk[n (k)+n (k)] k 2" − sin2ksin2πf +τ2# h reli − 0 2π( α β Z 1 p sinksinπf sin2kcosπf[n (k) n (k)] v2 = 1+ . (4) α − β , (8) k 2" sin2ksin2πf +τ2# − sin2ksin2πf +τ2 ) p p The resulting single-particle Hamiltonian describes two bands where nα(β)(k) = hc†α(β)(k)cα(β)(k)i are the occupation numbers of the two bands. The current (8) is a periodic function of the flux with a period ∆f = 1. In Fig 5 we H0 = ǫαα†kαk+ǫββk†βk , plot the flux dependence of j within a single period. rel Xk h i Notice that the current chanhgesiits sign under transfor- mation f π f; at f = 1/2 j = 0 due to the with the spectrum rel → − h i recovery of time reversal symmetry at this point. ǫ = 2t coskcosπf sin2ksin2πf +τ2 . (5) α(β) 0 − ± (cid:20) q (cid:21) In the absense of the flux the coherence factors areinde- pendent of momentum (u2 = v2 = 1/2) and the Hamil- τ = 0.1 tmoentiraincbHa0ndcosnesaicshtswoifththteheucsuosailnseysmpemcetrtruimc,asnpdliatnbtyis2ytm-. nits) ττ == 00..23 u Inthepresenceofthefluxthespectrumcantakeone⊥of y r the four typical shapes depending on the value of τ and ra theflux. TheseareillustratedinFig.4. Forcompleteness bit r a we include a detailed discussion of the properties of the ( spectrum as a function of flux and interchainhopping in nt e r Appendix B. r u c The half-filled ladder is characterized by zero chem- ical potential. When the spectrum (5) takes the form 0 0,2 0,4 0,6 0,8 1 depicted in the two bottom graphs in Fig. 4, the system flux at µ = 0 is a band insulator. In that case interaction effects (as well as the external field) are not expected to FIG. 5: Diamagnetic current as a function of flux in the ab- drastically change the nature of the ground state of the sence of interaction. Only one period in f is shown. The non-interacting system. We will not discuss that case in cuspscorrespond tothe band gap opening. the present paper. The top two graphs in Fig. 4 describe the “metallic” Inthelimitτ 0,whentheladderdecouplesintotwo phase of the non-interacting system. In this case at half → completely disconnected chains, the appearance of the filling both bands are partially filled and each band is flux in the Hamiltonian (1) is a gauge artifact. Indeed, characterizedby its own Fermimomentum kα(β) satisfy- F a careful evaluation of the integral in Eq. (8) will show ingkα+kβ =π. Inwhatfollowswewillusethenotation that j = 0 at τ = 0. Expanding Eq. (8) for small F F rel k kα (so that kβ =π k ) with f,τ h 1iand recovering the dependence on the lattice F ≡ F F − F ≪ spacing a, one finds that coskF = sin2πf +τ2. (6) j = vFfτ2 1+O(f2,τ2) . (9) q h reli 3a In the presence of the magnetic flux there exists a fi- (cid:2) (cid:3) nite diamagnetic current in the ground state of the sys- Despite being small in this limit, the diamagnetic cur- tem. The current operator along the oriented link be- rentisnotaninfra-redphenomenon. Itsdependenceonτ tween sites n and n+1 of the chain σ is isapparentlytheeffectofafinitecurvatureofthesingle- particle spectrum. Notice, that as seen from Fig 5, j rel jn,σ =−it0 eiπfσc†n,σcn+1,σ−h.c. . (7) isnon-zeroevenintheinsulatingphase. Thusthecurhrenit is a non-universal quantity contributed by all electrons (cid:0) (cid:1) This current flows in opposite directions on the two legs and not only those in the vicinity of the Fermi points. of the ladder, so that the total currentj =j +j Consequently,effects relatedto sucha persistentcurrent tot n,+ n, will have zero expectation value, while the expectatio−n can not be addressed in terms of any Lorentz-invariant value of the relative current j = j j (in the effective low-energy theory (we will further comment on rel n,+ n, − − 7 this issue in Section IV). Thus, at present we are un- scribed in terms of smoothly varying chiral (right and able to calculate the effect of the interaction on the dia- left) fermionic fields, R (x) and L (x). This de- α(β) α(β) magnetic current. However, it is clear that even in the fines the continuum limit of the model in which the presence of interaction the current will still persist and non-interactingpartofthelatticeHamiltonian,including all the correlation-related phenomena discussed in this boththeinter-chainhoppingandthecouplingtotheflux, paper will coexist with it. transforms to the kinetic energy of the chiral particles: H = iv dx R ∂ R L ∂ L , B. Interaction Hamiltonian 0 − F ν† x ν − †ν x ν ν=α,βZ X (cid:0) (cid:1) Now we are going to apply the standard rules of where v = 2t asink /cosπf is the Fermi velocity F 0 F Abelian bosonization15,16 to derive the effective low- which at half filling is the same for both bands. energy theory. First, we will assume that the Fermi en- Specializing to the vicinity of the four Fermi points ergy is sufficiently far from the bottom of the β-band. in the coherence factors Eqs. (4) we find the low-energy Then we linearize the two-band spectrum Eq. (5) in correspondencebetweentheoriginallatticeoperatorsc i,σ the vicinity of the four Fermi points, kα and kβ. and the chiral fields R and L . Then, the interaction ± F ± F ν ν The associated low-energy degrees of freedom are de- terms in the model (1) become H a g (:J J :+:J J :)+g (:J J :+:J J :)+g (:J J :+:J J :) int 1 Rα Lα Rβ Lβ 2 Rα Rβ Lα Lβ 3 Rα Lβ Lα Rβ ≈ ( i X +g4 :Rα†LαRβ†Lβ :+:L†αRαL†βRβ : −g5 :Rα†L†αRβLβ :+:RαLαRβ†L†β : + (cid:16) (cid:17) (cid:16) (cid:17) +g6 :Rα†(xi)Rα†(xi+1)Lβ(xi)Lβ(xi+1):+:Rα(xi)Rα(xi+1)L†β(xi)L†β(xi+1): + (cid:16) (cid:17) +g6 :L†α(xi)L†α(xi+1)Rβ(xi)Rβ(xi+1):+:Lα(xi)Lα(xi+1)Rβ†(xi)Rβ†(xi+1): . (10) ) (cid:16) (cid:17) where J =:R R : and J =:L L : are the chiral densities of the right- and left-moving fermions with the band Rν ν† ν Lν †ν ν index ν (the symbol “::” stands for normal ordering). The first three terms in Eq. (10), characterized by coupling constants g ,g and g , describe the density-density 1 2 3 interaction, whereas terms with amplitudes g , g and g correspond to the interchain umklapp, interchain back- 4 5 6 scatteringandin-chainumklappterms,respectively. Explicitexpressionsfortheg intermsoftheoriginalmicroscopic i theory is givenin Appendix A. The coupling constants depend onthe interactionconstants ofthe microscopicmodel (1) and, through the coherence factors Eq. (4), on the external flux. The latter dependence plays an important role because it is responsible for the sequence of phase transitions, described in Section IIB), that are not accessible at f =0. The interaction Hamiltonian H in Eq. (10) is the most general form of four-fermion interaction in the band int representation,consistentwithmomentumconservation(modulothereciprocallatticevector). Allothertermscontain strongly oscillating exponentials and thus do not contribute to the low-energy theory. In particular, this argument applies to the term g7e2i(kFα−kFβ)xRα†LαL†βRβ +h.c. (11) Note, that if one starts building the low energy theory approximating the ladder by two uncoupled chains, then the Fermi momenta of the two bands are equal and the above term should be included in Eq. (10). We will discuss this term and the relation between the two approaches to bosonization in ladder models in Section IV. However it is immediately clear, that omitting Eq. (11) from Eq. (10) can only be valid at long distances x kα kβ 1 or, | | ≫ | F − F|− equivalently, at low energies ω v kα kβ . | |≪ F| F − F| Now we bosonize the theory in the standard manner (our conventions are outlined in Appendix C). As usual, the density-density terms (represented in Eq. (10) by g , g , and g ) renormalize the Fermi velocities and the scaling 1 2 3 dimensions of the vertex operators. Introducing symmetric and antisymmetric combinations of the bosonic fields, φ = (φ φ )/√2, we diagonalize the quadratic part of the effective bosonized Hamiltonian. The latter is then α β re±presented±by twosine-Gordonmodelsdefined inthe symmetric andantisymmetricsectorswhicharecoupledby the 8 in-chain umklapp term g : 6 v+ 1 g = F K (∂ θ )2+ (∂ φ )2 4 cos√8πφ H 2 + x + K x + − 2π2α2 + (cid:20) + (cid:21) 0 +vF− K (∂ θ )2+ 1 (∂ φ )2 + g5 cos√8πθ g6 cos√8πφ cos√8πθ . (12) 2 − x − K x − 2π2α2 −− π2α2 + − (cid:20) − (cid:21) 0 0 The cosine terms in Eq.(12), whenrelevant(these are cell doubling and is Z . The complexity of the formulae, 2 the casesK <1,K >1,respectively),areresponsible relating the four coupling constants in (12) to the two + for a dynamical gene−ration of a mass gap in the corre- interaction parameters in the original Hamiltonian (1), sponding sector and, therefore, for the U(1) phase tran- as well as the magnetic field, leads to a rich phase dia- sitions described in Section IIB. For weak interaction, gram,aswewillnowdemonstrateusingthejustoutlined g /πv 1, the “Luttinger liquid” parameters K are strategy. i F c|lo|se to u≪nity (see Appendix A). Consequently, th±e co- sine terms having scaling dimensions 2K and 2/K in + the symmetric and antisymmetric sectors, respectiv−ely, C. Ordered Phases are nearly marginal. The g term that couples the two 6 sectorsisthereforestronglyirrelevantbecauseofitsscal- As there are four distinct Fermi points in our model, ingdimension2K +2/K 4. Theonlysituationwhen + any local operator will contain four dominant Fourier the g term may become−im∼portant is the case when one 6 components of the sectors is gapped, while the amplitude of the co- sine term in the other sector vanishes, i.e. either g4 = 0 (x ) = (x)+( 1)n (x) n 0 π or g = 0. In this case the g term can generate the O O − O 5 6 missing cosine in the Gaussiansector andevenmake the latter massive. This mechanism was recently discussed + cos(2kFxn)[O2kF(x)+(−1)nOπ−2kF(x)]. inRef.26inthecontextoftheMottinstabilityofahalf- Here is the smooth part of the operator (x ) cor- filled fermionic ladder with U = 0. Since in our model 0 n O O responding to characteristic momentum q 0; is the the presence of the g and g terms is generic, and the π 4 5 ∼ O staggeredpartcontributedbymomentaq π,whichcan lines g = 0 and g = 0 characterize the phase bound- 4 5 ∼ originate from some inter-band pairing; the components aries, the only effect of the in-chain (g ) Umklapp scat- 6 and can be present due to in-band pair- tering would be to modify the equations that determine Oin2gk.FAt halOf fiπl−li2nkgF, it is only the staggeredpart that can the phase boundaries without changing the topology of acquire an expectation value and serve as an order pa- the phase diagram. Being interested in the description rameter;howeverinsomeofthegaplessphasesdominant of distinct phases rather than their precise location, we correlationsmay occur at 2k or π 2k rather than π. will ignore the g6 term in the remainder of this paper. F − F Localoperatorsofinterestinthecaseofhalf-filledlad- Thus the effective low energy theory for our model der are listed in Table I, which includes the microscopic Eq.(1) consistsof two asymptoticallydecoupled sectors, lattice definitions and the bosonized form of the domi- each being a sine-Gordon model. In the case when a nant Fourier components. These are given up to mul- strong-couplingregimedevelopesineithersector,amass tiplicative factors; we preserve, however, prefactors pro- gapgetsgeneratedinthespectrum,andsemiclassicalso- portionaltothemagneticfluxtomakeclearwhichquan- lutionsofthe equationsofmotiondescribelockingofthe titiesdonotexistinzerofieldlimit. Wealsoindicatethe bosonic field in one of the infinitely degenerate minima LRO that appears when order parameters (first five op- of the cosine potential. Physical quantities evaluated on eratorsin the Table I – whichalsocorrespondto the five suchsolutionsmayeithervanishoracquireanonzeroex- “cartoons”in Fig 1) acquire nonzero expectation values. pectation value. The former would mean that the quan- An interesting observationthat can be made from Ta- tity in question is characterized by exponentially decay- bleIisthattheOAFandtheBDWarebothproportional ing correlations. In contrast, the latter corresponds to to the same low-energy operator, implying that the two long-rangecorrelations. Since local operatorsof the the- phases coexist. However, the OAF can exist already at oryhaveamultiplicativestructure,theycanindeedserve zero flux whereas the BDW order parameter is propor- as order parameters if gaps are generated in both sec- tional to the flux (at small f). This coexistence can be torssimultaneously. Themultiplicityoftheactualvalues understood by noticing that at f = 0 the BDW order 6 thatthe orderparameterwouldtakeonthesemiclassical parameter is defined in a gauge invariant way, i.e. with solutions, differing by a period of the cosine potential, flux-dependent phase factors explicitly included into its determines the degeneracy of the ordered ground state. definition [we remind that we have chosen the longitudi- The latter always appears to be associated with a unit nal Landau gauge, see Eq. (2)]. As a result, the BDW 9 TABLE I: Local operaors in the half-filled ladder Local Lattice Dominant Bosonized Ordered Operator Definition Component Form Phase J⊥ −it⊥ c†1c2−h.c. π cos√2πφ+cos√2πθ− OAF ρ− c†1hc1−c†2c2 i π cos√2πφ+sin√2πθ− Rel. CDW ρ ,+ eiπfc†1(xn)c1(xn+1)+e−iπfc†2(xn)c2(xn+1)+h.c. π sinπfcos√2πφ+cos√2πθ BDW k − ρk,− eiπfc†1(xn)c1(xn+1)−e−iπfc†2(xn)c2(xn+1)+h.c. π sin√2πφ+sin√2πθ− Rel. BDW ρ+ c†1c1+c†2c2 π tanπfsin√2πφ+cos√2πθ CDW − 2kF sin√2πφ+cos√2πφ − ρ c†1c2+c†2c1 2kF cos√2πφ+sin√2πφ O⊥sc c1c2 π 2kF iei√2πθ+cos[√2πφ (π −2kF)x] − −− − 0 tanπfei√2πθ+cos√2πθ − operator describing dimerization of the two chains with the couplings g (i = 1,3,4,5) are all of the same order, i zero relative phase acquires an admixture of the stag- there are important renormalizations of the parameters geredrelative current,proportionalto the flux atf 1. K emerging in the second-order16. This means that This admixture actually representsthe longitudinal≪part th±e exact positions of the phase boundaries depend also of the OAF order parameter (which is identical to J on g and g . These corrections, however, do not cause 4 5 by current conservation). The very appearance of suc⊥h qualitativechangesinthe overallphasediagramandcan anadmixture is a consequence ofthe explicit breakdown therefore be neglected in the leading order. For this rea- of time reversal symmetry, caused by the external flux, son,whendrawingconclusionsonrelevanceorirrelevance which is superimposed onthe spontaneous breakdownof of various perturbations, we will resort to an estimation this symmetry in the OAF phase. of their Gaussian scaling dimension. Another orderingin whichthe flux plays a crucialrole In the effective Hamiltonian (12) there are two cosine is the CDW phase. As already mentioned, the ground terms with amplitudes g and g . Both terms have the 4 5 stateofthe ladderatf =0andnottoosmallτ doesnot same periodin their respective variablesthatdefines the displaythis typeofLROasthe interchainhoppingtends values of the fields φ and θ for any semiclassicalsolu- + topreventdouble occupancyofthe rungs. Itisacurious tion. Depending on the sign−of g the field φ may take 4 + fact that under application of the flux this state can be one of the two possible sets of values, ϕ = n π/2, or 0 recovereddue to asimilar,althoughmoresubtle, admix- ϕ = π/8+n π/2(n=0, 1,...). Similarly,θ may ture with a staggered flux phase. Indeed, the bosonized taπkeoneofthe abovevaluesd±epending onthe spign−ofg . low-energy projection of the operator ρ+ has the form Consepquently thpere are four possible ordered phases. 5 fJ , where the operator diag (i)Ifbothg andg arenegative,thenthesemiclassical 4 5 solutionsareφ =ϕ andθ =ϕ . Ofallthe operators J i( 1)n σ c c h.c. , (13) + π 0 diag ∼ − †n,σ n+1,−σ− listedinTableIonlythestag−geredcomponentofthetotal σ X (cid:0) (cid:1) charge density ρ(s) has a non-zero expectation value on + represents an order parameterfor an OAF state with lo- the above solution. Therefore, the conditions g < 0 4(5) calcurrentseffectivelyflowingacrossthediagonalsofthe definethechargedensitywave(CDW).Thisphaseexists plaquettes.22 The scalar nature of the CDW under time onlyinthepresenceofthemagneticfield,whichmixesit reversalis not violated for the reasonalready mentioned up with an OAF phase, as explained previously. in the preceeding paragraph. (ii) For g > 0 and g < 0 we find the orbital anti- 4 5 Letusnowturntothederivationofthephasediagram. ferromagnet (OAF), since now φ = θ = ϕ and the + 0 For the cosine terms to become relevant and generate a staggeredcomponent of the inter-chain−current J gains gap in the spectrum, the “Luttinger liquid” parameters the expectation value. In contrast to the quarte⊥r-filled K+ andK shouldbe smallerandlargerthan1,respec- case13, the OAF phase exists even in the absence of the tively. Acc−ording to the definition of K , Eqs.(A4), this magnetic field.16,22 We will clarify this issue in Sec. V. translatesintothefollowingconditionso±ntheparameters At f = 0 the OAF coexists with BDW, as we already of the theory: 6 mentioned. (iii) When both interaction constants are positive the g +g >0; g g <0. (14) 1 3 1 3 − staggered component of the relative charge density ρ(s) Strictly speaking, these conditions are valid only to first hasanexpectationvalue (since inthis caseφ =ϕ an−d + 0 order in the Kosterlitz-Thouless RG equations. When θ =ϕ ). Wecallthecorrespondingorderedphasearel- π − 10 ativechargedensitywave(RelativeCDW).Themagnetic There are two other cases when only one of the condi- field has little effect on this phase except for the exact tions Eq. (14) is violated. Then only one of the sectors location of the phase boundary on the phase diagram, acquires a gap while the other remains gapless: which is beyond the scope of this paper. (i) If K < 1, K < 1, then the “charge” sector is + (iv) Finally, if g < 0 and g > 0 then φ = θ = ϕ gapped,butthe“rela−tive”sectorremainsgapless. Byfor- 4 5 + π andthe staggeredcomponentofthe relativelongi−tudinal mal analogy with the Hubbard model we call this phase bond density ρ(s) acquires a non-zero expectation value aMottInsulator. Insuchstate,incommensuratedensity , k− or bond-density correlations with characteristic momen- yieldingthe relativebonddensitywave(RelativeBDW). tum 2k = π are dominant. Indeed, depending on the Although the operator ρ(s) (and therefore its expecta- F 6 k,− sign of g4, either cos√2πφ+ or sin√2πφ+ acquire finite tion value) does not vanish in the absense of the mag- expectationvalues. Therefore,eitherthetransversebond netic field the relative BDW does not exist at f =0 (see densityρ orthe2k partofthetotalchargedensityρ F + Fig. 3) since the two conditions g4 < 0 and g5 > 0 can display sl⊥owestalgebraicdecay of the correspondingcor- be resolved only when f >0. relationfunctiondeterminedbythe“relative”sector(see All above long-range ordered states break sponta- Table I for bosonized expressions). So at g >0 4 neously translationalsymmetry of the underlying lattice cos(2k x/a) (perioddoubling)andthusaredoublydegenerate. Topo- F ρ (τ,x)ρ (0) . (17) logical excitations in these phases (Z -kinks) carry unit h ⊥ ⊥ i∼ v τ ixK− 2 | − − | chargeQ=1,asopposedtofractionalchargeQ=1/2in Ifg <0,thenEq.(17)appliestothecorrelationfunction the quarter-filled case. This follows from the definition 4 of ρ . of the fermionic number carried by a single kink, + (ii)IfK >1,K >1,thenthe“charge”sectorisgap- + less, but the relativ−e sector acquires a gap. By analogy ∞ Q = dx [J (x)+J (x)] with spin-gap systems, we call such a phase a Luther- Rν Lν νX=α,βZ−∞ Esemmeicrylaslisqicuaidl.v2a1luNesowdeipteinsdθin−gtohnatthteaksiegsnoonfego.f Itthetutrwnos 2 ∞ 5 = dx ∂ φ (x) (15) out,however,thatthisphasecanonlyoccurwheng <0, x + 5 rπ Z−∞ issotthhaattohfcothse√p2aπiθr−inig=6op0e,raatnodr athtezedroomminoamnetnctourmrel(awtiiothn andthefactthateachkinkinterpolatesbetweenthevac- the power law determined by the “charge” sector) uum values of the field φ at x that differ by a + → ±∞ period of the cosine potential, equal to π/2. 1 O (τ,x)O (0) . (18) h s†c sc i∼ v+τ ix1/K+ p | − | ThephaseboundariesasafunctionofU,V,f,τ canbe D. Non-ordered phases calculated by solving Eqs.(A1) for when g or g is zero, 4 5 or K or K is one. For completeness, these are writ- + In the previous subsection we have discussed the or- ten in Appen−dix A. The complete weak-coupling phase dered phases occuring under the conditions Eq.(14), i.e. diagramis plotted in Figure 3 and was discussed in Sec- when both sectors in the effective Hamiltonian (12) ac- tion IIB. quire gaps. In all other cases there exist gapless excita- tions. These are characterized by correlation functions that at large distances decay as a power law IV. LOW ENERGY EFFECTIVE THEORY: CHAIN BASIS (x) (0) 1/xd, hO O i ∼ In this section we briefly review the effective low en- where dis the scalingdimensionofthe operator . Cor- ergy theory that one can derive taking two independent O relations with slowest decay are usually referred to as chains as a starting point. Interchain hopping is then dominant. Inthe phasediagramFig.3wecategorizethe takeninto accountalreadyatthe bosonizationlevelsim- gapless phases according to their dominant correlations, ilarlytointeractionterms. Thisapproachisvalidaslong indicating the corresponding order parameter in paren- as t v (f)a, where v (f) is the renormalized veloc- F F theses. Inthissubsectionwebrieflydescribesuchphases. ity (⊥se≪e below). In the absence of the magnetic field the If the conditions Eq. (14) are reversed and K > 1, chain-basis description of the spinless ladder has been + K <1, then both sectors are gapless and the system is widelyusedinliterature.15,16Skippinginessentialdetails, a L−uttinger liquid. In this case the dominant correlation below we will give a brief review which will help to ana- functionisthatofthepairingoperatorO atwavevector lyze differences between the two approaches and further sc π 2k : clarify the role of the magnetic flux. F − In the chain-basis approach, one starts by lineariz- cos[(π 2kF)x/a] ing the fermion dispersion on each chain in the vicin- hOs†c(τ,x)Osc(0)i∼ |v+τ −ix|2/−K+|v−τ −ix|K−. (16) ity of the two Fermi points, ±kF = ±π/2, defines chiral