Lieb and hole-doped ferrimagnetism, spiral, resonating valence-bond states, and phase separation in large-U AB_{2} Hubbard chains
LLieb and hole-doped ferrimagnetism, spiral, resonating valence-bond states, and phase separationin large-U AB Hubbard chains
V. M. Martinez Alvarez and M. D. Coutinho-Filho Departamento de F´ısica, Laborat´orio de F´ısica Te´orica e Computacional,Universidade Federal de Pernambuco, Recife 50670-901, Pernambuco, Brazil
The ground state (GS) properties of the quasi-one-dimensional AB Hubbard model are investigated takingthe effects of charge and spin quantum fluctuations on equal footing. In the strong-coupling regime, we derivea low-energy Lagrangian suitable to describe the ferrimagnetic phase at half filling and the phases in the hole-doped regime. At half filling, a perturbative spin-wave analysis allows us to find the GS energy, sublatticemagnetizations, and Lieb total spin per unit cell of the effective quantum Heisenberg model, in very goodagreement with previous results. In the challenging hole doping regime away from half filling, we derive thecorresponding t - J Hamiltonian. Under the assumption that charge and spin quantum correlations are decoupled,the evolution of the second-order spin-wave modes in the doped regime unveils the occurrence of spatiallymodulated spin structures and the emergence of phase separation in the presence of resonating-valence-bondstates. We also calculate the doping-dependent GS energy and total spin per unit cell, in which case it is shownthat the spiral ferrimagnetic order collapses at a critical hole concentration. Notably, our analytical results inthe doped regime are in very good agreement with density matrix renormalization group studies, where ourassumption of spin-charge decoupling is numerically supported by the formation of charge-density waves inanti-phase with the modulation of the magnetic structure.
I. INTRODUCTION
Much attention has been given to quantum phase transi-tions [1, 2], which are phenomena characterized by the changeof the nature of the ground state (GS) driven by a non-thermalparameter: pressure, magnetic field, doping, Coulomb repul-sion, or competitive interactions. In this context, the studyof quasi-one-dimensional (quasi-1D) compounds with ferri-magnetic properties [3, 4] has attracted considerable theoreti-cal and experimental interest because of their unique physicalproperties and very rich phase diagrams. In particular, the GSof quasi-1D quantum ferrimagnets with AB or ABB (cid:48) unitcell topologies (diamond or trimer chains) described by theHeisenberg or Hubbard models [5] exhibit unsaturated spon-taneous magnetization, ferromagnetic and antiferromagneticspin-wave modes, effect of quantum fluctuations, and field-dependent magnetization plateaus, among several other fea-tures of interest.Of special interest is the topological origin of GS mag-netic long-range order associated with the unit cell structureof the lattice [5–12]. These studies have been motivatedand supported by exact solutions and rigorous results [13–18]; in particular, at half filling, the total spin per unit cellobeys Lieb-Mattis [13] (Heisenberg model) or Lieb’s theo-rem [15] (Hubbard model). On the other hand, it has beenverified that the ferrimagnetic GS of spin- / Heisenberg andHubbard /t - J AB chains, under the effect of frustration [19–23] or doping [7, 11, 24, 25], are strongly affected by quan-tum fluctuations that might cause its destruction and the occur-rence of new exotic phases: spiral incommensurate (IC) spinstructures, Nagaoka ( U → ∞ ) and resonating-valence-bond(RVB) states, phase separation (PS), and Luttinger-liquid be-havior. These features can enhance the phenomenology incomparison with a linear chain, which is dominated by thenontrivial Luttinger-liquid behavior that exhibits fractionalexcitations [26, 27], emergent fractionalized particles [28], and fractional-exclusion statistic properties [29] in the spin-incoherent regime [30]. In addition, investigations of transportproperties in AB chains, and related structures, have also un-veiled very interesting features [31].On the experimental side, studies [32–34] of the magneticproperties of homometallic phosphate compounds of the fam-ily A Cu (PO ) ( A = Ca , Sr , Pb ) suggest that in thesematerials the line of trimers formed by spin- / +2 ionsantiferromagnetically coupled do exhibit ferrimagnetism oftopological origin. Further, compounds Ca M (PO ) (M =Ni , Co) with a wave-like layer structure built by zigzag M-chains exhibit antiferromagnetic ordering (
M = Ni ) or param-agnetic behavior (
M = Co ) [35]. On the other hand, bimetal-lic compounds, such as
CuMn (S C O ) · . O [36], canbe modeled [36–38] by alternate spin- / - spin- / chainsand support interesting field-induced quantum critical pointsand Luttinger-liquid phase [37]. In addition, frustrated dia-mond ( AB topology) chains can properly model the com-pound azurite, Cu (CO ) (OH) , in which case the occur-rence of the / magnetization plateau is verified at highfields [39] in agreement with topological arguments [40] akinto those invoked in the quantum Hall effect. The spin-1/2trimer chain compound Cu (P O OH) , with antiferromag-netic interactions only, also display the 1/3 magnetizationplateau [41]. Interestingly, it has been established that in azu-rite the magnetization plateau is a dimer-monomer state [42],i.e., the chain is formed by pairs of S = 1 / monomers and S = 0 dimers, with a small local polarization of the diamondspins [43], in agreement with density functional theory [44].These dimer-monomer states have been found previously inthe context of modeling frustrated AB chains [45–47], andconfirmed through a modeling using quantum rotors [48]. Incontrast to azurite, whose dimers appear perpendicular to thechain direction, in the spin-1/2 inequilateral diamond-chaincompounds [49] A Cu AlO (SO ) (A = K , Rb , Cs) , themagnetic exchange interactions force the dimers to lie alongthe sides of the diamond cells and the monomers form a 1D a r X i v : . [ c ond - m a t . s t r- e l ] D ec Heisenberg chain. In fact, the low-energy excitations of thesenew compounds have been probed and a Tomonaga-Luttingerspin liquid behavior identified [50]. It is worth mentioningthat strongly frustrated AB chains can exhibit ladder-chaindecoupling [20], in which case the ladder is formed via thecoupling between dimer spins in neighboring AB unit cells.On the other hand, besides the above-mentioned quasi-1D compounds and related magnetic properties, consider-able efforts have been devoted to the study of supercon-ductivity and intriguing magnetic/charge ordered phases indoped materials [51, 52], in particular the formation of spin-gapped states in compounds such as the family of doped (La , Sr , Ca) Cu O . This compound is formed by one-dimensional CuO diamond chains, (Sr , Ca) layers, and two-leg Cu O ladders [53]. These results certainly stimulateexperimental and theoretical investigations of quasi-1D com-pounds in the hole-doped regime, which is the main focus ofour work, as described in the following.In this work, we shall employ an analytical approach suit-able to describe the strongly coupled Hubbard model on doped AB chains, which were the object of recent numerical studiesthrough density matrix renormalization group (DMRG) tech-niques [25]. Our functional integral approach, combined witha perturbative expansion in the strong-coupling regime, wasoriginally proposed to study the doped Hubbard chain [54],and later adapted to describe various doped-induced phasetransitions in the U = ∞ AB Hubbard chain [55]. In ad-dition, this approach was used to describe the doped stronglycoupled Hubbard model on the honeycomb lattice [56], whoseresults are very rewarding, particularly those for the GS en-ergy and magnetization in the doped regime, which com-pare very well with Grassmann tensor product numerical stud-ies [57].The paper is organized as follows: in Sec. II we review thefunctional integral representation of the Hubbard Hamiltonianin terms of Grassmann fields (charge degrees of freedom) andspin SU (2) gauge fields (spin degrees of freedom). In Sec. IIIwe diagonalize the Hamiltonian associated with the charge de-gree of freedom and obtain a perturbative low-energy theorysuitable to describe the ferrimagnetic phase at half filling andthe phases in the hole-doped regimes. In Sec. IV, we showthat the resultant Hamiltonian at half filling and large-U mapsonto the spin- / quantum Heisenberg model. In this regime,a perturbative series expansions in powers of /S of the spin-wave modes is presented, which allows us to calculate theGS energy, sublattice magnetizations, and Lieb GS total spinper unit cell in very good agreement with previous estimates.In Sec. V, we derive the low-energy effective t - J Hamilto-nian, which accounts for both charge and spin quantum fluc-tuations. We also present the evolution of the second-orderspin-wave modes, GS energy and total spin per unit cell un-der hole doping, thus identifying the occurrence of spatiallymodulated spin structures, with non-zero and zero GS totalspin, and phase separation involving the later spin structureand RVB states at hole concentration / . Remarkably, thesepredictions are in very good agreement with the DMRG datareported in Ref. [25]. Lastly, in Sec. VI, we present a sum-mary and concluding remarks concerning the reported results. II. FUNCTIONAL-INTEGRAL REPRESENTATION
The Hamiltonian of the one-band Hubbard model on chainswith AB unit cell topology is given by [7, 8, 10]: H = − (cid:88) (cid:104) iα,jβ (cid:105) σ { t αβij ˆ c † iασ ˆ c jβσ + H.c. } + U (cid:88) iα ˆ n iα ↑ ˆ n iα ↓ , (1)where i = 1 , . . . , N c (= N/ is the specific position of theunit cell, whose length is set to unity, N c ( N ) is the num-ber of cells (sites), α, β = A, B , B denote the type of sitewithin the unit cell, ˆ c † iασ ( ˆ c iασ ) is the creation (annihilation)operator of electrons with spin σ ( = ↑ , ↓ ) at site α of cell i ,and ˆ n iασ = ˆ c † iασ ˆ c iασ is the occupancy number operator. Thefirst term in Eq. (1) describes electron hopping, with energy t αβij ≡ t , allowed only between nearest neighbors A - B and A - B linked sites of sublattices A and B (bipartite lattice),and the second one is the on-site Coulombian repulsive inter-action U > , which contributes only in the case of doubleoccupancy of the site iα .At this point, it is instructive to digress on some fundamen-tal aspects of the formalism used in our work [54–56]. Withregard to the large-U doped Hubbard chain [54], U = ∞ AB Hubbard chain [55] and the Hubbard model on the honeycomblattice [56], it has been shown that the particle density productin Eq. (1) can be treated through the use of a decompositionprocedure, which consists in expressing ˆ n iα ↑ ˆ n iα ↓ in terms ofcharge and spin operators: ˆ n iα ↑ ˆ n iα ↓ = 12 ˆ ρ iα − S iα · n iα ) , (2)where ˆ S iα = 1 / (cid:88) σσ (cid:48) ˆ c † iασ (cid:48) σ σ (cid:48) σ ˆ c iασ , (3)and ˆ ρ iα = ˆ n iα ↑ + ˆ n iα ↓ , (4)are the spin-1/2 and charge-density operators, respectively, σ σ (cid:48) σ denotes the Pauli matrix elements ( (cid:126) ≡ , and n iα isan arbitrary unit vector. In fact, Eq. (2) follows from the iden-tity: ˆ ρ iα − ˆ n iα ↑ ˆ n iα ↓ = 2( ˆ S x,y,ziα ) = 2(ˆ S iα · n iα ) . Theconvenience of using the decomposition defined in Eq. (2),with explicit spin-rotational invariance for the large-U Hub-bard model, was discussed at length in Refs. [54–56].We start by using the Trotter-Suzuki formula [58, 59],which allows us to write the partition function, Z = Tr [exp( − β H )] , at a temperature k B T ≡ /β , as Z = Tr { ˆ T (cid:81) Mr =1 exp[ − δτ H ( τ r )] } , where ˆ T denotes thetime-ordering operator, the total imaginary time intervalis formally sliced into M discrete intervals of equal size δτ = τ r − τ r − , r = 1 , , ..., M, with τ = 0 , and τ M = β = M δτ , under the limits M → ∞ and δτ → . Weshall now introduce, between each discrete time interval, anovercomplete basis of fermionic coherent states [58, 59], ´ (cid:81) iασ dc † iασ dc iασ exp (cid:16) − (cid:80) iασ c † iασ c iασ (cid:17) |{ c iασ }(cid:105)(cid:104){ c iασ }| ,where { c † iασ , c iασ } denotes a set of Grassmann fields satisfy-ing anti-periodic boundary conditions: c † iασ (0) = − c † iασ ( β ) and c iασ (0) = − c iασ ( β ) ; while the set of unit vectorsdefines the vector field { n iα } , satisfying periodic ones: n iα (0) = n iα ( β ) , under a weight functional (see below).Thereby, following standard procedure [58, 59], the partitionfunction reads: Z = ˆ (cid:89) iασ D c † iασ D c iασ (cid:89) iα D n iα W ( { n iα } ) e − ´ β L ( τ ) dτ , (5)where the pertinent measures are defined by D c † iασ D c iασ ≡ lim M →∞ ,δτ → M − (cid:89) r =1 dc † iασ ( τ r ) dc iασ ( τ r ) , (6) D n iα ≡ lim M →∞ ,δτ → M − (cid:89) r =1 d n iα ( τ r ) , (7)the weight functional, W ( { n iα } ) , satisfies a normalizationcondition at each discrete imaginary time τ r : ˆ (cid:89) iα d n iα W ( { n iα ( τ r ) } ) = 1 , (8)and the Lagrangian density L ( τ ) is written in the form: L ( τ ) = (cid:88) iασ c † iασ ∂ τ c iασ − (cid:88) ijαβσ ( t αβij c † iασ c jβσ + H.c. )+ U (cid:88) iα [ ρ iα − S iα · n iα ) ] . (9)In order to fix W ( { n iα } ) one should notice that, in the op-erator formalism: ˆ ρ iα = ˆ ρ iα + 2ˆ n iα ↑ ˆ n iα ↓ . Therefore, usingEq. (2), the following identity holds [54]: ˆS iα · n iα ) = ˆ ρ iα (2 − ˆ ρ iα )2 , (10)which means that the square of the spin component opera-tor along the n iα direction has zero eigenvalues if the siteis vacant or doubly occupied, and a nonzero value only forsingly occupied sites, i.e., ( ˆS iα · n iα ) = 1 / . Now, takingadvantage of the choice of n iα , the local spin-polarization andspin-quantization axes are both chosen along the n iα direc-tion. Therefore, for singly occupied sites, we find S iα · n iα = p iα / , with p iα = ± , corresponding to the two possiblespin-1/2 states. Further, by incorporating vacancy and doubleoccupancy possibilities, corresponding to the four possible lo-cal states of the Hubbard model, one can write [54] p iα ˆS iα · n iα = ˆ ρ iα (2 − ˆ ρ iα )2 , (11)with p iα = ( ± . We stress that, due to fermion operatorproperties, the square of Eq. (11) reproduces Eq. (10), anda comparison between them implies, at arbitrary doping and U value, the formal equivalence between S iα · n iα ) and p iα (ˆ S iα · n iα ) . In this context, we remark that the origi-nal Coulomb repulsion term of the Hubbard Hamiltonian inEq. (1) is formally and energetically (eigenvalues) equivalentto both that in Eq. (9) or in its linear version through the fol-lowing replacement: S iα · n iα ) → p iα (ˆ S iα · n iα ) . In-deed, using the constraint in Eq. (11) we find, U (cid:80) iα [ ρ iα − p iα ( S iα · n iα )] = U (cid:80) iα [ ρ iα − ρ iα (2 − ρ iα )] , which is zerofor ρ iα = 0 , ; whereas, as expected, for double occupiedsites, ρ iα = 2 , the local energy is U . Therefore, Eq. (11) in itsGrassmann version, can be enforced by a proper choice of thenormalized weight functional: W ( { n iα } ) = lim M →∞ ,δτ → M (cid:89) r =1 W ( { n iα ( τ r ) } )= C exp (cid:110) − ˆ β dτ γ (cid:88) iα [ p iα S iα · n iα − ρ iα − ρ iα )] (cid:111) , (12)where γ → ∞ in the continuum limit ( M → ∞ , δτ → ),with delta-function peaks at the four local states of the Hub-bard model, and C is a normalization factor such that Eq. (8)holds. In fact, the product of W ( { n iα ( τ r ) } ) in Eq. (12) gener-ates a sum in r in the exponential of the suitable chosen Gaus-sian function, i.e., W ( { n iα } ) is such that in the continuumlimit, M → ∞ , δτ → , Eq. (12) obtains with a diverging γ ,as pointed out in Ref. [54]. In this way, using Eq. (12) for theweight functional in Eq. (5) for the partition function Z , andintegrating over { n iα } , the Lagrangian density L ( τ ) in Eq.(9) can thus be written in the following linearized form [54]: L ( τ ) = (cid:88) iασ c † iασ ∂ τ c iασ − (cid:88) ijαβσ ( t αβij c † iασ c jβσ + H.c. )+ U (cid:88) iα [ ρ iα − p iα ( S iα · n iα )] , (13)where the constraint in Eq. (11) was explicitly used.Now, since we are interested in studying the GS propertiesof the AB Hubbard chains, we choose the staggered factor p iα = +1 ( − at sites α = B , B ( A ) , consistent with thelong-range ferrimagnetic GS predicted by Lieb’s theorem athalf filling and for any U value [7, 8, 15], in which case weassume broken rotational symmetry along the z -axis. In thiscontext, by considering the symmetry exhibited by the ferri-magnetic order, let us define the SU (2) /U (1) unitary rotationmatrix [60] U iα = cos (cid:16) θ iα (cid:17) − sin (cid:16) θ iα (cid:17) e − iφ iα sin (cid:16) θ iα (cid:17) e iφ iα cos (cid:16) θ iα (cid:17) , (14)where θ iα is the polar angle between the z -axis and the unit lo-cal vector n iα and φ iα ∈ [0 , π ) is an arbitrary azimuth angledue to the U (1) gauge freedom of choice for U iα . Moreover,a new set of Grassmann fields, { a † iασ , a iασ } can be obtained,according to the transformation: c iασ = (cid:88) σ (cid:48) ( U iα ) σσ (cid:48) a iασ (cid:48) , (15)that locally rotates each unit vector n iα to the z -direction. Onthe other hand, if we express the product σ · n iα in matrixform: σ · n iα = (cid:20) cos ( θ iα ) sin ( θ iα ) e − iφ iα sin ( θ iα ) e iφ iα − cos ( θ iα ) (cid:21) , (16)we obtain, after using Eq. (14), U † iα ( σ · n iα ) U iα = σ z , (17)which explicitly manifest the broken rotational symmetryalong the z -axis. In this way, by substituting Eqs. (14) and(15) into Eq. (3), and using the above result, we find S iα · n iα = 12 (cid:88) σσ (cid:48) a † iασ [ U † iα ( σ · n iα ) U iα ] σσ (cid:48) a iασ (cid:48) = 12 (cid:88) σσ (cid:48) a † iασ ( σ z ) σσ (cid:48) a iασ (cid:48) ≡ S ziα ; (18)thereby, the constraint in Eq. (11) can be written in the form S iα · n iα = p iα ρ iα (2 − ρ iα )2 = 12 ( a † iα ↑ a iα ↑ − a † iα ↓ a iα ↓ ) , (19)where p iα = +1 ( − at sites α = B , B ( A ) . The choiceof p iα above implies Lieb’s ferrimagnetic ordering with the set { θ iA = θ iB = θ iB = 0 } , for all i , at half filling. However,in the hole doped regime away from half filling, the θ iα ’s canbe nonzero (e.g., θ iα = π for a spin flip, leading to a changein the sign of S ziα ); further, S ziα can be zero either by the pres-ence of holes or doubly occupied sites ( a † iα ↑ a iα ↑ = a † iα ↓ a iα ↓ ).Lastly, using Eqs. (15) and (19) into the Lagrangian, Eq. (13),we find, after suitable rearrangement of terms, L ( τ ) = L ( τ ) + L n ( τ ) , (20)where both Lagrangians are quadratic in the Grassmann fields: L ( τ ) = (cid:88) iασ a † iασ ∂ τ a iασ − (cid:88) iαjβσ ( t αβij a † iασ a jβσ + H.c. )+ U (cid:88) iασ (1 − p iα σ ) a † iασ a iασ , (21)and L n ( τ ) = (cid:88) iασσ (cid:48) a † iασ (cid:48) ( U † iα ∂ τ U iα ) σ (cid:48) σ a iασ − (cid:88) iαjβσσ (cid:48) t αβij [ a † iασ (cid:48) ( U † iα U jβ − σ (cid:48) σ a jβσ + H.c. ] , (22)with the first term in both Eqs. (21) and (22) being originatedfrom the first term in Eq. (13), the second ones come fromthe hopping term in Eq. (13), after a rearrangement of terms,while the last one in Eq. (21) (proportional to U ) is obtainedby using Eq. (19) in the last term of Eq. (13). It is worthmentioning that only charge degrees of freedom (Grassmannfields) appear in L ( τ ) , and spin degrees of freedom under the constraint in Eq. (19) [ SU (2) gauge fields { U † iα , U iα } , whichcarry all the information on the vector field { n iα } ] are nowrestricted to L n ( τ ) , which includes both spin and charge de-grees of freedom.In the large-U regime, double occupancy is energeticallyunfavorable and the factor − ρ iα is no longer needed inEq. (19), i.e., S iα · n iα = p iα ρ iα , with ρ iα = 0 or . Inthis case, a proper perturbative analysis will allow us to studyhole doping effects in Sec. V in a macroscopic fashion, so wedefine δ = 1 − N (cid:88) iα (cid:104) ρ iα (cid:105) , (23)which measures the thermodynamic average of hole dopingaway from half filling. In this context (strong-coupling limit),we take advantage of results derived from L ( τ ) (charge ef-fects in Sec. III), and at half filling (Sec. IV), in which casecharge degrees of freedom are frozen. III. CHARGE DEGREES OF FREEDOM AND THESTRONG-COUPLING LIMIT
In this section, we shall first diagonalize the Hamiltonianassociated with the Lagrangian L ( τ ) through the use of aspecial symmetry property of the AB chains and a canonicaltransformation in reciprocal space. Then, by introducing aperturbative expansion in the strong-coupling regime, a low-energy effective Lagrangian for the AB Hubbard chains athalf filling and in the doped regime will be obtained.
A. Charge degrees of freedom
We begin our discussion by considering the Lagrangian L in Eq. (21), and its corresponding Hamiltonian H , free ofthe SU (2) gauge fields. By performing the Legendre trans-formation: H = − (cid:80) iασ ∂ L ∂ ( ∂ τ a iασ ) ∂ τ a iασ + L , where ∂ L ∂ ( ∂ τ a iασ ) = a † iασ , the resulting H is given by H = − (cid:88) (cid:104) iα,jβ (cid:105) σ ( t αβij a † iασ a jβσ + H.c. )+ U (cid:88) iασ (1 − p iα σ ) a † iασ a iασ . (24)Further, since H ( L ) is quadratic in the Grassmann fields,the solution for the energy of the system is given by H in itsdiagonalized form [59].The AB unit cell topology exhibits a symmetry [9, 11, 24,25, 55] under the exchange of the labels of the B sites in agiven unit cell. Thus, we can construct a new set of Grass-mann fields possessing this symmetry, i.e., either symmet-ric or antisymmetric with respect to the exchange operation B ↔ B : ( d iσ , e iσ ) = 1 √ a iB σ ± a iB σ ) , b iσ = a iAσ . (25)In addition, as a signature of the quasi-1D structure of the AB chains, we notice that the B and B sites are located ata distance / (in units of length) ahead of the A site. There-fore, after Fourier transforming the above Grassmann fields,i.e., { d i,σ , e i,σ , b i,σ } = √ N c (cid:80) k e ikx i { d k,σ , e k,σ , b k,σ } , it isconvenient to introduce a phase factor e ik through the follow-ing transformation [55]: ( A kσ , B kσ ) = √ ( d kσ ± e ik b kσ ) , sothat H in Eq. (24) thus becomes H = (cid:88) kσ ε k [ A † kσ A kσ − B † kσ B kσ ] + U (cid:88) kσ (1 − σ ) e † kσ e kσ + U (cid:88) kσ [ A † kσ A kσ + B † kσ B kσ − σ ( A † kσ B kσ + B † kσ A kσ )] , (26)where ε k = − √ t cos( k/ , (27)with k = 2 πj ( N ) − π , and j = 1 , . . . , N/ . We can now ex-actly diagonalize H through the following Bogoliubov trans-formation: A kσ = u k α kσ − σv k β kσ , B kσ = σv k α kσ + u k β kσ , (28)with u k and v k satisfying the canonical constraint: ( u k ) +( v k ) = 1 , to maintain the anticommutation relations ofthe Grassmann fields. Due to the ferrimagnetic order of theGS, the above transformation is subject to a π periodicityof the Bogoliubov functions { u k , v k } and Grassmann fields { α kσ , β kσ } . The diagonalized H thus reads: H = − (cid:88) kσ ( E k − U α † kσ α kσ + (cid:88) kσ ( E k + U β † kσ β kσ + U (cid:88) kσ (1 − σ ) e † kσ e kσ , (29)where ( u k , v k ) = 1 √ (cid:18) ± | ε k | E k (cid:19) / , (30)and E k = (cid:113) ε k + U / . (31)As one can see from Eq. (29), the non-interacting tight bind-ing ( U = 0 ) spectrum of H present three electronic bands:a nondispersive flat band (related to the Grassmann fields { e † kσ , e kσ } , macroscopically degenerate), and two dispersiveones. In AB chains, flat bands are closely associated withferrimagnetism (unsaturated ferromagnetism) [5, 7, 8] at halffilling, in agreement with Lieb’s theorem [15, 16], or fullypolarized ferromagnetism [17] associated with the flat lowestband. We also stress that even at this level of approximationand in the weak coupling regime ( U = 2 t ), it was shown [7]that hole doping [parametrized by δ defined in Eq. (23)] can destroy the ferrimagnetic order and/or induce phase separa-tion in AB chains. As depicted in Fig. 1(a), the U = 0 spindegeneracy of the flat bands is removed by the Coulombianrepulsive interaction, in which case a gap U opens betweenthe e kσ modes: e k ↑ = 0 , where spins at sites B and B areup, and e k ↓ = U , where these spins are down. On the otherhand, the two dispersive bands are spin degenerated, and alsodisplay a Hubbard gap U separating the low ( α kσ ) -energy andhigh ( β kσ ) -energy modes [55]. B. Strong-coupling limit
In this subsection, we shall introduce a perturbative expan-sion in the strong-coupling regime ( U (cid:29) t ) in order to obtaina low-energy effective Lagrangian for the AB Hubbard chainat half filling and in the doped regime. First, we resume the re-sults of the previous section by writing the Grassmann fields d iσ and b iσ in terms of the Grassmann (Bogoliubov) fields α kσ and β kσ : ( d iσ , b iσ ) = 1 √ N c (cid:88) k ( e ikx i , e ik ( x i − ) ) × [( u k ± σv k ) α kσ ± ( u k ∓ σv k ) β kσ ] , (32)where the phase factor e − ik signalizes the quasi-1D AB structure, and the antisymmetric Grassmann field e i,σ remainsas defined in Eq. (25). In the strong-coupling limit, however,it will prove useful to define a set of auxiliary spinless Grass-mann fields [54, 55] in direct space associated with d iσ and b iσ : ( α i , β i ) = (cid:114) N c (cid:88) k,σ θ ( ± σ ) e ikx i ( α kσ , β kσ ) , (33)and a similar equation for ( α i , β i ) ↔ ( α kσ , β kσ ) is obtainedby the replacements: θ ( ± σ ) → θ ( ∓ σ ) and x i → x i − / ,where θ ( σ ) is the Heaviside function, while for the antisym-metric component, one has e i,σ = (cid:114) N c (cid:88) k e ikx i e k,σ . (34)Now, by expanding ( u k , v k ) in Eq. (30) in powers of t/U : ( u k , v k ) ≈ √ (cid:20) ± | ε k | U + O (cid:18) t U (cid:19)(cid:21) , (35)substituting these results into the Eq. (32), and using the in-verse transformation of Eq. (33), we can derive a perturbativeexpansion in powers of t/U for the Grassmann fields d iσ and b iσ in terms of the spinless Grassmann fields as follows: d iσ = θ ( σ ) α i + θ ( − σ ) β i + √ tU θ ( − σ )( α i + α i +1 )+ tU θ ( σ )[ √ β i + β i +1 ) − tU (2 α i + α i +1 + α i − )]+ O ( t /U ) , (36) b iσ = θ ( − σ ) α i − θ ( σ ) β i + √ tU θ ( σ )( α i + α i − ) − tU θ ( − σ )[ √ β i + β i − ) + tU (2 α i + α i +1 + α i − )]+ O ( t /U ) . (37)In the above derivation, we have used that θ ( σ ) θ ( σ (cid:48) ) = θ ( σ ) δ σ,σ (cid:48) . Notice that, since tU (cid:28) , in Eqs. (36) and (37)we can identify the fields α i ≈ a iA ↓ and α i ≈ ( a iB ↑ + a iB ↑ ) / √ , a result fully consistent with the low-energy spinconfiguration of the ferrimagnetic state discussed previously.Analogously, for the high-energy bands, the opposite spinconfiguration is observed, with spin up (down) present at sites A ( B , B ).Introducing Eqs. (36) and (37) into Eq. (24), with the aidof Eq. (25), we obtain a perturbative expression for H (low-energy sector) in terms of the spinless Grassmann fields up toorder J = 4 t /U : H = − J (cid:88) i [ α † i α i + α ( ) † i α i − β † i β i − β ( ) † i β i ] − J (cid:88) i [ α † i α i +1 + α ( ) † i α i +1 − β † i β i +1 − β ( ) † i β i +1 + H.c . ]+ U (cid:88) i [ β † i β i + β ( ) † i β i + e † i ↓ e i ↓ ] . (38)By applying Fourier transform to the above expression andrearranging the terms, we obtain H = − (cid:88) k J cos ( k/ α † k α k + α ( ) † k α k )+ (cid:88) k [2 J cos ( k/
2) + U ]( β † k β k + β ( ) † k β k )+ U (cid:88) kσ (1 − σ ) e † kσ e kσ . (39)In Fig. 1 we plot the electronic spectrum of the Hamilto-nian H , both in the weak and strong-coupling regime: (a)Eq. (29) for U = 2 t and (b) Eqs. (29) and (39) for U = 12 t ( J = 4 t /U = 1 / ), respectively, with t ≡ . We can noticethe presence of the shrinking phenomenon [7] as U increasesfrom t to t (strong-coupling regime) and that, for U = 12 t ,Eq. (39) is a very good approximation to Eq. (29). Notice-ably, the t (cid:28) U expansion of the fields allow us to identify α k ≈ a kA ↓ , α k ≈ ( a kB ↑ + a kB ↑ ) / √ (triplet state) and e k ↑ ≈ ( a kB ↑ − a kB ↑ ) / √ (singlet state), as the low-energyspin configuration of the ferrimagnetic state with single occu-pancy, where spins at sites A ( B , B ) are down (up), in agree-ment with Lieb’s theorem [7, 8, 15].In order to describe the most relevant low-energy pro-cesses that take place in this regime, one has to additionallyproject out the high-energy bands from H , that is, terms con-taining only fields related to the high-energy bands are ex-cluded. Therefore, after the Legendre transformation, H = − (cid:80) i,η i ∂ L ∂ ( ∂ τ η i ) ∂ τ η i + L , where η i = α i , α i , and e i ↑ (fields -1 0 1 k/π -2024 E ( a ) β kσ e k ↓ e k ↑ α kσ U = 2 t -1 0 1 k /π -2061214 ( b ) U = 12 t β k β k e k ↓ e k ↑ α k α k Figure 1. (Color online) Electronic spectrum of the Hamiltonian H :(a) Eq. (29) for U = 2 t and (b) Eqs. (29) and (39) for U = 12 t ( J =4 t /U = 1 / ), with t ≡ . Notice the band shrinking phenomenonas U increases from t to t (strong-coupling regime). The t (cid:28) U expansion of the fields identifies α k ≈ a kA ↓ , α k ≈ ( a kB ↑ + a kB ↑ ) / √ and e k ↑ ≈ ( a kB ↑ − a kB ↑ ) / √ , where spins at sites A ( B , B ) are down (up), in agreement with Lieb’s theorem [15]. related to the low-energy bands), with ∂ L ∂ ( ∂ τ η i ) = η † i , the La-grangian associated with H (up to order J ) is given by L = (cid:88) i,η i η † i ∂ τ η i − J (cid:88) i ( α † i α i + α ( ) † i α i ) − J (cid:88) i ( α † i α i +1 + α ( ) † i α i +1 + H.c . ) . (40)We shall now focus on the U (cid:29) t perturbative expansion of L n , Eq. (22), which amounts to consider the most significantlow-energy processes, after the use of Eqs. (36) and (37) for d iσ and b iσ in terms of the spinless Grassmann fields. How-ever, terms allowing interband transitions between low- andhigh-energy bands do exist in L n . In this context, we apply asuitable second-order Rayleigh-Schr¨odinger perturbation the-ory [54, 55], consistent with the strong-coupling expansion,so that the modes associated with the high-energy bands areeliminated. Lastly, by adding L to the perturbative expansionof L n , which leads to the cancellation of the exchange termsin Eq. (40), the effective low-energy Lagrangian density of the AB Hubbard model in the strong-coupling limit (up to order J ) reads: L eff ( τ ) = L ( I ) + L ( II ) + L ( III ) + L ( IV ) , (41)where L ( I ) = (cid:88) i α † i ∂ τ α i + (cid:88) i α ( ) † i ∂ τ α ( ) i + (cid:88) i e † i ↑ ∂ τ e i ↑ , (42a) L ( II ) = (cid:88) iσ (cid:110) θ ( − σ )( U ( b ) † i ∂ τ U ( b ) i ) σ,σ α ( ) † i α ( ) i + θ ( σ ) 12 [( U ( d ) † i ∂ τ U ( d ) i ) σ,σ + ( U ( e ) † i ∂ τ U ( e ) i ) σ,σ ] × ( α † i α i + e † i ↑ e i ↑ ) + (cid:20) θ ( σ ) 12 [( U ( d ) † i ∂ τ U ( e ) i ) σ,σ +( U ( e ) † i ∂ τ U ( d ) i ) σ,σ ] α † i e i ↑ + H.c. (cid:105)(cid:111) , (42b) L ( III ) = − t (cid:88) iσ (cid:110) θ ( − σ )( U ( b ) † i U ( d ) i ) σ, − σ α ( ) † i α i + θ ( σ )( U ( d ) † i U ( b ) i +1 ) σ, − σ α † i α ( ) i +1 + θ ( − σ )( U ( b ) † i U ( e ) i ) σ, − σ α ( ) † i e i ↑ + θ ( σ ) (cid:16) U ( e ) † i U ( b ) i +1 (cid:17) σ, − σ e † i ↑ α ( ) i +1 + H.c. (cid:27) , (42c) L ( IV ) = − J (cid:88) i ; i (cid:48) = i,i +1; σ θ ( σ ) | ( U ( d ) † i U ( b ) i (cid:48) ) σ,σ | α † i α i − J (cid:88) i ; i (cid:48) = i,i +1; σ θ ( σ ) | ( U ( e ) † i U ( b ) i (cid:48) ) σ,σ | e † i ↑ e i ↑ − J (cid:88) i ; i (cid:48) = i,i − σ θ ( − σ )[ | ( U ( b ) † i U ( d ) i (cid:48) ) σ,σ | + ( U ( b ) † i U ( e ) i (cid:48) ) σ,σ | ] α ( ) † i α ( ) i , (42d)where U ( b ) i = U iA , U ( d,e ) i = 1 √ U iB ± U iB ) , (43)in which case we took advantage of the symmetry of the AB chain under the exchange operation B ↔ B , in correspon-dence with Eq. (25). From the above equations, we see that thekinetic term is represented by L ( I ) and is related to the chargedegrees of freedom only, whereas L ( II ) describes the dynam-ics of the spin degrees of freedom coupled to the charge fields.On the other hand, L ( III ) exhibit first-neighbor hopping con-tributions between charge degrees of freedom in the presenceof SU (2) gauge fields, while L ( IV ) is the spin exchange termin the presence of the charge Grassmann fields. IV. HALF-FILLING REGIME
Let us now discuss some basic aspects of the localized mag-netic properties related to the spin degrees of freedom. At halffilling, i.e., δ = 0 , we have (cid:104) α † i α i (cid:105) = 1 , (cid:104) α (1 / † i α (1 / i (cid:105) = 1 , (cid:104) e † i ↑ e i ↑ (cid:105) = 1 , and (cid:104) α † i e i ↑ (cid:105) = 0 (no band hybridization) as theelectrons tend to fill up the lower-energy bands, whereas thehigher-energy ones remain empty. As a consequence, a fer-rimagnetic configuration of localized spins emerges, i.e., thecharge degrees of freedom are completely frozen, such that (cid:104) α † i ∂ τ α i (cid:105) = (cid:104) α (1 / † i ∂ τ α (1 / i (cid:105) = (cid:104) e † i ↑ ∂ τ e i ↑ (cid:105) = 0 , with for-bidden hopping. Therefore, only terms from L II and L IV in Eqs. (42b) and (42d), respectively, give nonzero contributionsand the resulting effective strong-coupling Lagrangian at halffilling, defined in Eq. (41), reads: L Jeff = (cid:88) iασ θ ( p iα σ )( U † iα ∂ τ U iα ) σ,σ − J (cid:88) (cid:104) iα,jβ (cid:105) σ θ ( p iα σ ) (cid:12)(cid:12)(cid:12) ( U † iα U jβ ) σ,σ (cid:12)(cid:12)(cid:12) , (44)where the staggered factor p iα was defined in Eq. (11), and usewas made of the matrix transformations defined in Eq. (43) inorder to sum up the squares of the SU (2) gauge field prod-ucts in the exchange contribution from L IV in Eq. (42d).Now, using the following Legendre transform: H Jeff = − (cid:80) iασ ∂ L Jeff ∂ ( ∂ τ U iα ) σ,σ ( ∂ τ U iα ) σ,σ + L Jeff , where ∂ L Jeff ∂ ( ∂ τ U iα ) σ,σ = θ ( p iα σ )( U † iα ) σ,σ , we get the respective quantum HeisenbergHamiltonian written in terms of the SU (2) gauge fields at halffilling as H Jeff = − J (cid:88) (cid:104) iα,jβ (cid:105) σ θ ( p iα σ ) (cid:12)(cid:12)(cid:12) ( U † iα U jβ ) σ,σ (cid:12)(cid:12)(cid:12) . (45)Further, using the definition of the SU (2) /U (1) unitaryrotation matrix Eq. (14), it is possible to write [54–56] (cid:12)(cid:12)(cid:12) ( U † iα U jβ ) σ,σ (cid:12)(cid:12)(cid:12) = (1 + n iα · n jβ ) , where n iα =sin( θ iα ) [cos( φ iα )ˆ x + sin( φ iα )ˆ y ] + cos( θ iα )ˆ z is the unit vec-tor pointing along the local spin direction. Lastly, by using theconstraint as given in Eq. (19), we can identify the spin field { S iα } at the single occupied sites: S iα = p iα n iα / , (46)where p iα = +1 ( − at sites α = B , B ( A ) , in order toobtain H Jeff = J (cid:88) i (cid:104) ( S B i + S B i ) · ( S Ai + S Ai +1 ) (cid:105) − JN c . (47)The above expression is indeed that of the quantum antifer-romagnetic Heisenberg spin-1/2 model on the AB chain inzero-field, which takes into account the effects of zero-pointquantum spin fluctuations. In fact, to achieve this goal, we an-alyze the Hamiltonian, Eq. (47), by means of the spin-wavetheory, which has proved very successful in describing theproperties of the GS and low-lying excited states of spin mod-els. The predicted results provide a check of the consistencyof our approach and will be fully used in our description ofthe doped regime.We shall first introduce boson creation and annihilation op-erators via the Holstein-Primakoff [58] transformation: S A,zi = − S + a † i a i ,S A, + i = ( S A, − i ) † = √ Sa † i f A ( S ) , (48)for a down-spin on the A site, and S B l ,zi = S − b † li b li ,S B l , + i = ( S B l , − i ) † = √ Sf B ( S ) b li , (49)for an up-spin on the B l site, with l = 1 , , and f r ( S ) = (cid:16) − n r S (cid:17) / = 1 − n r S + . . . , (50)where S is the spin magnitude, and n r = a † i a i or b † li b li . Theoperators a † i and a i (or b † li , b li ) satisfy the boson commutationrules. Under the above transformation, the spin Hamiltonian,Eq. (47) is mapped onto the boson Hamiltonian: H Jeff = E − JN c + H + H + O ( S − ) , (51)where E = − S JN c , (52)is the classical GS energy and H and H are the quadraticand quartic (interacting) terms of the boson Hamiltonian, suit-able to describe the quantum AB Heisenberg model via aperturbative series expansion in powers of /S . By Fouriertransforming the boson operators, we find H = 2 JS (cid:88) k (2 a † k a k + (cid:88) l b † lk b lk )+ (cid:88) k,l =1 , JSγ k ( a † k b † lk + a k b lk ) , (53)where we have defined the lattice structure factor as γ k = 1 z (cid:88) ρ e ikρ = cos (cid:18) k (cid:19) , (54)with z denoting the coordination number ( z = 4 for the AB chain), while ρ = ± / connects the nearest neighbors A - B and A - B linked sites of sublattices A and B , and H = − J N (cid:88) ,l =1 , δ , (cid:110) γ − a † a b † l b l +( γ a † b † l b † l b l + γ − a † a † a b † l + H.c. ) (cid:111) . (55)For simplicity, we use the convention for k , for k , and soon. Also, the δ , = δ ( k + k − k − k ) is the Kronecker δ function, and expresses the conservation of momentum towithin a reciprocal-lattice vector G .We shall consider H first, which is the term leading tolinear spin-wave theory (LSWT). In fact, H is diagonalizedusing the following Bogoliubov transformation: a k = u k β k − v k α † k ,b lk = 1 √ u k α k − v k β † k + ( − l ξ k ] , with l = 1 , , (56) ( u k , v k ) = (3 + (cid:112) − γ k , √ γ k ) (cid:113) (3 + (cid:112) − γ k ) − γ k , (57)where u k and v k satisfy the constraint u k − v k = 1 . Thus, H = E + (cid:88) k ( (cid:15) α ) k α † k α k + (cid:15) β ) k β † k β k + (cid:15) ξ ) k ξ † k ξ k ); (58) E = JS (cid:88) k ( (cid:113) − γ k − , (59) (cid:15) α,β ) k = JS ( (cid:113) − γ k ∓ , (cid:15) ξ ) k = 2 JS, (60)where E is the O ( S ) quantum correction to the GS energy,and (cid:15) α,β ) k , (cid:15) ξ ) k are the three spin-wave branches providedby LSWT, both in agreement with previous results [19, 61].In fact, it is well known that systems with a ferrimagnetic GSnaturally have ferromagnetic and antiferromagnetic spin-wavemodes as their elementary magnetic excitations (magnons).For the AB chain, there are three spin-wave branches: anantiferromagnetic mode ( (cid:15) β ) k ) and two ferromagnetic ones( (cid:15) α ) k and (cid:15) ξ ) k ). The mode (cid:15) α ) k is gapless at k = 0 , i.e., theGoldstone mode, with a quadratic (ferromagnetic) dispersionrelation (cid:15) α ) k ∼ k . The other two modes are gapped. No-tice that the gapped ferromagnetic mode (cid:15) ξ ) k is flat, and isclosely associated with ferrimagnetic properties at half fill-ing [7, 17]. Since the dispersive modes preserve the localtriplet bond, they are identical to those found in the spin- / - spin-1 chains [62–65]. These chains also exhibit interestingfield-induced Luttinger liquid behavior [66].Now, our aim is to obtain the leading corrections to LSWT,i.e., second-order spin-wave theory to the GS energy, sublat-tice magnetizations and Lieb GS total spin per unit cell. In do-ing so, we develop a perturbative scheme for the descriptionof this quartic term. First, we decompose the two-body termsby means of the Wick theorem, via normal-ordering protocolfor boson operators. Conservation of momentum to within areciprocal-lattice vector, implies: k = k + q , k = p − q , k = k and k = p . Then, we need to look at the possiblepairings of the 4 operators, as for example, in the first term ofEq. (55): a † k + q a p b † l,k b l,p − q , a † k + q a p b † l,k b l,p − q , a † k + q a p b † l,k b l,p − q . Under this procedure, and by substituting the Bogoliubovtransformation, Eqs. (56)-(57), into Eq. (55), we find H = E + (cid:88) k ( δ(cid:15) ( α ) k α † k α k + δ(cid:15) ( β ) k β † k β k + δ(cid:15) ( ξ ) k ξ † k ξ k ) , (61)where E /N c = − J ( q + q − √ q q ) , (62)and the corresponding corrections for the spin-wave disper-sion relations read: δ(cid:15) ( α ) k = J [ u k ( √ q − q ) + 2 v k ( √ q − q )]+ 4 Jγ k u k v k (cid:20) √ q − q (cid:21) + O ( S − ) , (63) δ(cid:15) ( β ) k is obtained from δ(cid:15) ( α ) k through the exchange of u k ↔ v k , and δ(cid:15) ( ξ ) k = J ( √ q − q ) + O ( S − ) . (64)In Eqs. (62)-(64) above, the quantities q and q are definedby (thermodynamic limit) q = 12 π ˆ π − π dk ( v k ) , q = 12 π ˆ π − π dk ( γ k u k v k ) . (65)We remark that in deriving Eqs. (62)-(64), we have neglectedterms containing anomalous products, such as, α † k β † k and ver-tex corrections.Lastly, the above results of our perturbative /S series ex-pansion lead to the effective Hamiltonian: H Jeff = E JGS − JN c + (cid:88) k ( (cid:15) αk α † k α k + (cid:15) ( β ) k β † k β k + (cid:15) ( ξ ) k ξ † k ξ k ) , (66)where E JGS = E + E + E , (67)which can be read from Eqs. (52), (59), and (62), respectively,is the second-order result up to O (1 /S ) for the GS energy,and (cid:15) ( s ) k = (cid:15) s ) k + δ(cid:15) ( s ) k , with s = α, β, ξ, (68)are the corresponding second-order spin-wave modes, wherethe linear and the second-order correction terms are given byEq. (60) and Eqs. (63)-(65), respectively. A. Second-order spin-wave analysis
Our perturbative /S series expansion approach is able toimprove the LSWT result for the gap ∆ = J of the antiferro-magnetic mode, which should be compared with the second-order result derived from (cid:15) ( β ) k , Eqs. (60), (63) and (68), at k = 0 : ∆ = (1 + √ q ) J (cid:39) . J , in full agreementwith similar spin-wave calculations for AB [19] and spin- / -spin- [64, 65] chains, and in agreement with numeri-cal estimates using exact diagonalization, ∆ = 1 . J , forboth AB [5] and spin- / -spin- [63] chains. On the otherhand, the LSWT predicts a gap ∆ flat = J for the flat fer-romagnetic mode ( (cid:15) ( ξ ) k ) in AB chain, whereas our second-order spin-wave theory finds, using Eqs. (60), (64) and (68): ∆ flat = (1 − q + √ q ) J (cid:39) . J , in full agreementwith a similar spin-wave procedure [19]. Surprisingly, the es-timated value from Exact Diagonalization (ED) [5]: ∆ flat =1 . J , lies between these two theoretical values. In fact,analytical approaches are still unable to reproduce the ob-served level crossing found in numerical calculations [5, 19]for the two ferromagnetic modes. This is probably due to thefact that the different symmetries exhibited by the localizedexcitation (flat mode) and the ferromagnetic dispersive modeare not explicitly manifested in the analytical approaches, sothe levels avoid the crossing. B. Ground state energy
In the thermodynamic limit, the second-order result for theGS energy of the AB chain per unit cell reads: E JGS N c = − JS + JS π ˆ π − π dk (cid:18)(cid:113) − γ k − (cid:19) − J ( q + q − √ q q ) . (69)We remark that, at half filling, we shall not consider the con-stant term − JN c in Eq. (51), with the purpose of compari-son with preceding results. Performing the integration overthe first BZ and taking S = 1 / , we obtain that the GS en-ergy per site at zero-field is given by − . J . This resultagrees very well with values obtained using exact diagonal-ization [45] ( − . J ) and DMRG [67] ( − . J ) tech-niques. For the spin- / - spin-1 chain, the value obtainedusing DMRG [62] is − . J . To compare it with our find-ing, we need to multiply this value by / (ratio between thenumber of sites of the two chains), yielding − . J . C. Sublattice magnetizations and Lieb GS total spin per unitcell
In order to derive results beyond LSWT, we intro-duce staggered magnetic fields coupled to spins S A,zi and S B l ,zi , with l = 1 , , through the Zeeman terms: − h A (cid:80) i S A,zi and − h B l (cid:80) i S B l ,zi , which are added to H Jeff in Eq. (47). Thus, (cid:104) S A,z (cid:105) and (cid:104) S B l ,z (cid:105) correspondingto sublattices A and B l are obtained from (cid:104) S A,z (cid:105) = − (1 /N c ) (cid:80) i =1 , [ ∂E i ( h A ) /∂h A ] | h A =0 , and an analogousequation for (cid:104) S B l ,z (cid:105) using Eqs. (59) and (62): ( (cid:104) S A,z (cid:105) , (cid:104) S B l ,z (cid:105) ) = ∓ S ± (cid:18) , (cid:19) π ˆ π − π dkv k ∓ (cid:18) , (cid:19) q πS ˆ π − π dk γ k (9 − γ k ) / + O ( 1 S ) . (70)Carrying out the above integration, we obtain (cid:104) S A,z (cid:105) = − . and (cid:104) S B l ,z (cid:105) = 0 . . These results are ingood agreement with those obtained using DMRG [11] andED [5] techniques: (cid:104) S A,z (cid:105) = − . and (cid:104) S B l ,z (cid:105) =0 . , respectively, and with values for (cid:104) S A,z (cid:105) and (cid:104) S B l ,z (cid:105) for the spin- / - spin-1 chain [62–65]. Althoughat zero temperature, the sublattice magnetizations are stronglyreduced by quantum fluctuations, as compared with their clas-sical values, the unit cell magnetization remains S L ≡ / ,where S L is the Lieb GS total spin per unit cell, in full agree-ment with Lieb’s theorem [5, 15] for bipartite lattices: S L = 12 (cid:107) N A − N B (cid:107) , (71)with N A ( N B ) denoting the total number of spins in sublattice A ( B ) per unit cell.0 V. t - J HAMILTONIAN: DOPING-INDUCED PHASES,GROUND STATE ENERGY AND TOTAL SPIN
In this section, we shall derive the corresponding t - J Hamiltonian suitable to describe the strongly correlated AB Hubbard chain in the doped regime, in which case both charge(Grassmann fields) and spin [ SU (2) gauge fields] quantumfluctuations are considered on an equal footing. Indeed, the t - J Hamiltonian can be derived by means of the followingLegendre transformation to Eq. (41): H t - Jeff = − (cid:88) i,µ = b,d,e ∂ L eff ∂ ( ∂ τ U ( µ ) i ) σ,σ ( ∂ τ U ( µ ) i ) σ,σ − (cid:88) i,ν i ∂ L eff ∂ ( ∂ τ ν i ) ∂ τ ν i + L eff , (72)where ∂ L eff ∂ ( ∂ τ ν i ) = ν † i with ν i = α i , α i , e i ↑ ; ∂ L eff ∂ ( ∂ τ U ( b ) i ) σ,σ = θ ( − σ )( U ( b ) † i ) σ,σ α ( ) † i α ( ) i , and ∂ L eff ∂ ( ∂ τ U ( d,e ) i ) σ,σ = θ ( σ ) [( U ( d,e ) † i ) σ,σ ( α † i α i + e † i ↑ e i ↑ ) + ( U ( e,d ) † i ) σ,σ ( α † i e i ↑ + e † i ↑ α i )] , from which we can write the effective t - J Hamilto-nian as H t - Jeff = H t + H J , (73)where H t = − t (cid:88) iσ { θ ( − σ )( U ( b ) † i U ( d ) i ) σ, − σ α (1 / † i α i + θ ( σ )( U ( d ) † i U ( b ) i +1 ) σ, − σ α † i α (1 / i +1 + θ ( − σ )( U ( b ) † i U ( e ) i ) σ, − σ α (1 / † i e i ↑ + θ ( σ )( U ( e ) † i U ( b ) i +1 ) σ, − σ e † i ↑ α (1 / i +1 + H.c. } , (74)and H J = − J (cid:88) i ; i (cid:48) = i,i +1; σ θ ( σ ) | ( U ( d ) † i U ( b ) i (cid:48) ) σ,σ | α † i α i − J (cid:88) i ; i (cid:48) = i,i +1; σ θ ( σ ) | ( U ( e ) † i U ( b ) i (cid:48) ) σ,σ | e † i ↑ e i ↑ − J (cid:88) i ; i (cid:48) = i,i − σ θ ( − σ )[ | ( U ( b ) † i U ( d ) i (cid:48) ) σ,σ | + | ( U ( b ) † i U ( e ) i (cid:48) ) σ,σ | ] α ( ) † i α ( ) i . (75)Notice that Eqs. (74) and (75) are identical to Eqs. (42c) and(42d), since Eqs. (42a) and (42b) were eliminated through theLegendre transformation.Some digression on H t - Jeff is in order. One of the key prop-erties of quasi-1D interacting quantum systems is the phe-nomenon of spin-charge separation, leading to the formationof spin and charge-density waves, which move independentlyand with different velocities. It has been demonstrated [24]that for δ > / the low-energy physics of the doped AB Hubbard chain in the U = ∞ coupling limit is described in terms of the Luttinger-liquid model, with the spin and chargedegrees of freedom decoupled. Most importantly, it has beenshown that for the AB t - J Hubbard chains [25] charge andspin quantum fluctuations are practically decoupled, as sug-gested by the emergence of charge-density waves in anti-phase with the modulation of the ferrimagnetic order. Onecan make use of this feature to formally split each term of the t - J Hamiltonian, Eq. (73)-(75) , into a product of two inde-pendent terms acting on different Hilbert spaces, i.e., we canenforce spin-charge separation and calculate the charge andspin correlation functions in a decoupled fashion.Therefore, from the above discussion, we shall considerthat the charge correlation functions are well described by aneffective spinless tight-binding model [24, 55, 68], since thehole (charge) density waves develop along the x -axis and inanti-phase with the modulation of the ferrimagnetic structure,as numerically observed in Fig. 2(b) of Ref. [25]. So, usingEqs. (33), with a/ → a (effective lattice spacing of the lin-ear chain: distance between A and B sites, see Fig. 2(a) ofRef. [25]), we find (cid:104) α (1 / † i α i (cid:105) = 1 N c (cid:88) kk (cid:48) e − ik ( x i − e ik (cid:48) x i (cid:104) Ψ | α † k α k (cid:48) | Ψ (cid:105) = 1 π ˆ k F ( δ ) − k F ( δ ) e ik dk = 2 π sin[ k F ( δ )] , (76)with | Ψ (cid:105) being the hole-doped ferrimagnetic GS,where k F ( δ ) = π N h N ≡ πδ is the Fermi wave vec-tor of the spinless tight-binding holes. In thesame fashion: (cid:104) α † i α (1 / i +1 (cid:105) = π sin[ k F ( δ )] and (cid:104) α (1 / † i e i ↑ (cid:105) = (cid:104) e † i ↑ α (1 / i +1 (cid:105) = 0 ; while (cid:104) α † i α i (cid:105) = (cid:104) α (1 / † i α (1 / i (cid:105) = (cid:104) e † i ↑ e i ↑ (cid:105) = (1 − π ´ k F ( δ ) − k F ( δ ) dk ) =(1 − δ ) . Here, we remark that the itinerant holes away fromhalf filling are associated with the lower-energy dispersive α k and α (1 / k bands [see Fig. 1(b) in Sec. (II)], thus contributingto the kinetic Hamiltonian in Eq. (74). On the other hand,the local correlations related to the lower-energy bands α k , α (1 / k , and e k ↑ , contribute equally to the exchangeHamiltonian in Eq. (75). Thereby, using the above tight-binding results for the charge correlation functions, H t - Jeff in Eqs. (73)-(75) gives rise to the δ -dependent Hamiltonian, H t - Jeff ( δ ) = H teff ( δ ) + H Jeff ( δ ) , written below: H t - Jeff ( δ ) = − t π sin[ k F ( δ )] (cid:88) i [( U ( b ) † i U ( d ) i ) ↓↑ + ( U ( d ) † i U ( b ) i +1 ) ↑↓ + H.c. ] − J (1 − δ )4 (cid:88) (cid:104) iα,jβ (cid:105) σ θ ( p iα σ ) | ( U † iα U jβ ) σ,σ | , (77)where the sum over σ was evaluated in Eq. (74) and the squareof the SU (2) gauge field products in the exchange contribu-tion have been summed up in Eq. (75), so that this contribu-tion is just (1 − δ ) times H Jeff at half filling, Eq. (45), oralternatively, in terms of spin fields, Eq. (47), or spin-waves,Eqs. (66)-(68). On the other hand, the SU (2) gauge fields ma-1trix elements: (cid:16) U ( b ) † i U ( d ) i (cid:17) ↓↑ and (cid:16) U ( d ) † i U ( b ) i +1 (cid:17) ↑↓ , that ap- pear in the kinetic contribution of Eq. (77), can be written interms of the spin fields [55, 56] as ( U ( b ) † i U ( d ) i ) ↓↑ + H.c. = (cid:88) l √ (cid:18)(cid:113) − S B l ,zi − S A,zi + 4 S A,zi S B l ,zi + (cid:113) S A,zi + 2 S B l ,zi + 4 S A,zi S B l ,zi (cid:19) , (78)and (cid:16) U ( d ) † i U ( b ) i +1 (cid:17) ↑↓ is obtained from Eq. (78) through the replacement S A,zi → S A,zi +1 , in which case we took advantage of the U (1) gauge freedom and Eq. (46). Notice that these square-root matrix elements depend on z -spin components only.At this stage, it will prove useful, in the calculation of the GS total spin in the doped regime, to consider H t - Jeff ( δ, h ) whichdescribes the system in the presence of a homogeneous magnetic field h = h ˆz = ( − h A + h B + h B ) ˆz , where the staggered fieldspoint along the local corresponding magnetizations in the ferrimagnetic phase have the same magnitude h . The magnetic fieldcouple with the spin fields through the Zeeman term (see Sec. IV C) and with the charge degrees of freedom through the magneticorbital coupling in the Landau gauge: A = hx ˆy . Since our aim is to study doping effect on the magnetization, we shall assumevanishingly small magnetic field in the context of linear response theory and perturbative expansion in the strong-couplingregime. Additionally, the magnetic orbital coupling can be considered through the so-called Peierls substitution [27, 69]: t → te i ´ jβiα A · d l , where iα and jβ are first-neighbor sites, and the flux quantum φ = hc/e ≡ . If one consider that the carrier isat the site iA , we have four hopping possibilities: iA → iB , and iA → ( i + 1) B , , so the total phase φ acquired by thecarrier in this prescription satisfies Stokes’ theorem: φ = (cid:23) unit cell A · d l = (cid:115) S h · d S = ha ( a ≡ . We also remark that,in order to obtain real values for the zero-field staggered magnetizations, we have considered, for convenience, an imaginarygauge transformation [56, 70]: A → i A . Therefore, by placing Eq. (78) and the similar matrix element into the kinetic term inEq. (77), making the above Peierls substitution, and using the Holstein-Primakoff and Bogoliubov transformations introduced inEqs. (48)-(50) and Eqs. (56)-(57), respectively, up to order O ( S − ) , we arrive at the following diagonalized kinetic Hamiltonian H teff ( δ, h ) : H teff ( δ, h ) = − √ π te − ( − h A + h B + h B ) sin[ k F ( δ )] (cid:88) k [4 S − v k − ( u k + 2 v k ) α † k α k − (2 u k + v k ) β † k β k − ξ † k ξ k ] , (79)where the doped-induced contributions for the spin dispersion relations are evidenced in the last three terms. On the other hand,by adding the Zeeman terms (see Sec. IV C) to the exchange contribution H Jeff ( δ ) , given in Eq. (77), we obtain H Jeff ( δ, h ) .Lastly, by adding the kinetic and the exchange contributions, we arrive at the effective t - J Hamiltonian in the presence of amagnetic field: H t - Jeff ( δ, h ) = − √ π te − ( − h A + h B + h B ) sin[ k F ( δ )] (cid:88) k (4 S − v k ) + J (1 − δ )( E JGS − JN c )+ (cid:88) k [ (cid:15) ( α ) k ( δ ) α † k α k + (cid:15) ( β ) k ( δ ) β † k β k + (cid:15) ( ξ ) k ( δ ) ξ † k ξ k ] − h A (cid:88) i S A,zi − h B (cid:88) i S B ,zi − h B (cid:88) i S B ,zi , (80)where E JGS is given by Eq. (67) and (69), and the corresponding spin-wave modes [see Eqs. (79), (60), (63)-(65), and (68)] ofthe doped AB t - J chain read: (cid:15) ( α ) k ( δ ) = 4 √ π t sin( πδ )[ u k + 2 v k ] + (1 − δ )( (cid:15) α ) k + δ(cid:15) ( α ) k ) , (81) (cid:15) ( β ) k ( δ ) is obtained from (cid:15) ( α ) k ( δ ) through the exchange u k ↔ v k and the replacement α → β , while (cid:15) ( ξ ) k ( δ ) = 4 √ π t sin( πδ ) + (1 − δ )( (cid:15) ξ ) k + δ(cid:15) ( ξ ) k ) . (82)We find it instructive to comment on the analytical structure of the above equations. Firstly, we mention the presence of theBogoliubov parameters [see Eqs. (57)] in a symmetric form in the kinetic terms of Eq. (81) and its analogous for (cid:15) ( β ) k ( δ ) ; besides,although the flat mode is strongly affected by the presence of holes, it remains dispersionless. In addition, using Eqs. (80) and(65), the total GS energy (no spin-wave excitations) per unit cell in the thermodynamic limit is readily obtained: E t - JGS ( δ, h ) /N c = − √ π te − ( − h A + h B + h B ) sin( πδ )(4 S − q )+ (1 − δ )( E JGS /N c − J ) − (cid:104) S A,z (cid:105) h A − (cid:88) l =1 , (cid:104) S B l ,z (cid:105) h B l , (83)2where (cid:104) S A,z (cid:105) and (cid:104) S B l ,z (cid:105) are the calculated sublattice magnetizations, at half filling and zero-field, given by Eqs. (70).In subsections V A, V B, and V C, we will show that theunderlying competing physical mechanisms: the magneticorbital response and the Zeeman contribution embedded inEqs. (80)-(83) will dramatically affect the behavior of the sys-tem under hole doping and, in particular, will lead to spiralIC spin structures, the breakdown of the spiral ferrimagneticGS at a critical value of the hole doping, a region of phaseseparation, and RVB states at δ ≈ / . A. Doped regime: Spin-wave modes
Before we go one step further to discuss relevant macro-scopic quantities, i.e., the GS energy and total spin in thedoped regime, we shall first undertake a detailed study, at amicroscopic level, of the hole-doping effect on the calculatedspin-wave branches given by Eqs. (81)-(82).Fig. 2 depicts the second-order spin-wave dispersion rela-tions at
J/t = 0 . and for the indicated values of δ . Withoutloss of generality, we set t = 1 in our numerical computations.At half filling, the antiferromagnetic mode (cid:15) ( β ) k , together withthe two ferromagnetic modes: the dispersive (cid:15) ( α ) k and the flatone (cid:15) ( ξ ) k , are shown in Fig. 2(a), which are defined in Eq. (68),and can be plotted using Eqs. (60) and (63)-(65).As the hole doping increases slightly, the abrupt decreaseof the peaks at k = 0 and k = π of the numerical DRMGstructure factor (see Fig. 3 of Ref. [25]), associated with theferrimagnetic order, manifests itself here through the openingof a gap in the ferromagnetic Goldstone mode (cid:15) ( α ) k , as seen inFig. 2(b), thus indicating that the system loses its long-rangeorder. Note that the antiferromagnetic mode (cid:15) ( β ) k is also sim-ilarly shifted. On the other hand, although the dispersion re-lation is modified for small values of the wave vector k , theminimum value of (cid:15) ( α ) k still remains at k = 0 up to the onset of the formation of spiral IC spin structures at δ c (IC) = 0 . (a value that should be compared with the numerical DMRGestimate of δ ≈ . ± . ), characterized by the flatten-ing of the dispersive spin-wave branches around zero. Uponfurther increase of δ , two minima form (around k = 0 ) andmove away from each other as one enhances the hole doping.This behavior is the signature of the occurrence of spiral ICspin structures (see Fig. 3 of Ref. [25]).Fig. 2(c) shows the onset of phase separation (PS) at δ (PS) = 0 . for J = 0 . , which is characterized by theoverlap of the two ferromagnetic modes at k = 0 . The sig-nature of this regime is the spatial coexistence of two phases:spiral IC spin structures at δ (PS) = 0 . and RVB statesat δ ≈ / , in very good agreement with the numerical es-timate of δ IC − PS ≈ . [25]. At δ ≈ / , the flat modehas the lowest energy, as illustrated in Fig. 2(d). This be-havior indicates that the RVB state is the stable phase at δ ≈ / and J = 0 . [25], and also in agreement with thenumerical DMRG studies [11, 24] and analytical prediction at U = ∞ [55]. Onset of ICOnset of PS (b)(a)(c) (d)
Figure 2. (Color online) Evolution of the zero-field second-orderspin-wave dispersion relations of the AB t - J chain as a function ofhole doping ( δ ): dispersive ferromagnetic (cid:15) ( α ) k and antiferromagnetic (cid:15) ( β ) k modes and the flat ferromagnetic one (cid:15) ( ξ ) k , at (a) half filling; (b)the onset of the spiral IC spin structures at δ c (IC) = 0 . , in whichcase the flattening of the gap of (cid:15) ( α ) k around k = 0 is observed; (c)the onset of PS at δ (PS) = 0 . , characterized by the overlap ofthe two ferromagnetic modes at k = 0 and by the spatial coexistenceof two phases: spiral IC spin structures, with modulation fixed at δ (PS) , and RVB states at δ ≈ / . (d) At δ = 1 / the flat modepresents the lowest energy, thus indicating that the short-range RVBstate is the stable phase. In order to better understand the rich variety of doping-induced phases in the system, in Fig. 3 we plot the evolution ofthe wave vector k min corresponding to the local minimum of (cid:15) ( α ) k ( δ ) , upon increasing the hole doping δ from to / . Thewave vector k min remains zero until it hits the onset dopingvalue δ c (IC) = 0 . , beyond which a square-root growth be-havior takes place [71]: [ δ − δ c (IC)] / (blue line), for δ closeto δ c (IC) . The square-root growth behavior is the signatureof the occurrence of a second-order quantum phase transitionfrom the doped ferrimagnetic phase to the IC spiral ferrimag-netic state with a non-zero value of the total GS spin, S GS .This result is supported by the behavior of ∆ k ≡ k max − π at which the local maximum of the numeric DMRG structurefactor S ( k ) near k = π is observed, as shown in the inset ofFig. 3 (taken from the inset of Fig. 3(b) of Ref. [25]). Forfurther increase of hole doping our result deviates from thesquare-root growth behavior and some very interesting fea-tures are to be noticed. The value of δ c = 0 . indicatesthe breakdown of the total S GS in the IC phase, as will beconfirmed by the explicit calculation of S GS , a macroscopicquantity, in Section V C. Thus, for . < δ < . the sys-tem displays an IC phase with zero S GS , in agreement withthe DMRG data (see Fig. 1(c) of Ref. [25]). At δ (PS) = 0 . k m i n / π Doped Ferri IC S GS = 0 IC S GS = 0 PS δ c (IC) δ c δ δ (PS) ∼ RVB
Analytic [ δ − δ c (IC)] / δ ∆ k / π Figure 3. (Color online) Evolution of k min (value of k at the localminimum of (cid:15) ( α ) k ( δ ) near k = 0 ) as a function of δ : doped ferri-magnetism for < δ < δ c (IC) ≈ . ; spiral IC spin structureswith non-zero (zero) S GS for δ c (IC) < δ < δ c ≈ . ( δ c < δ <δ (PS) ≈ . ), with a second-order quantum phase transition at δ c (IC) characterized by a square-root behavior [ δ − δ c (IC)] / (blueline), and a first-order transition at δ (PS) involving the IC spin struc-ture, with modulation fixed at δ (PS) , and short-range RVB states athole concentration 1/3. The inset shows DMRG data from Ref. [25]for ∆ k ≡ k max − π as a function of δ , where k max is the value of k at the local maximum of the structure factor S ( k ) near k = π , inqualitative agreement with the second-order transition at δ c (IC) . the system exhibits a first-order transition accompanied by thespatial phase separation regime: the IC phase with zero S GS and modulation fixed by δ (PS) in coexistence with the short-range RVB states at δ ≈ / , also consistent with the DMRGdata plotted in Fig. 4 of Ref. [25].Lastly, we emphasize that, despite the occurrence of severaldoping-induced phases in the DMRG studies [25]: Lieb fer-rimagnetism, spiral IC spin structures, RVB states with finitespin gap, phase separation, and Luttinger-liquid behavior, itis surprising and very interesting that the second-order spin-wave modes remain stable up to δ ≈ / , with predictionsin very good agreement with the DMRG studies [25]. In thiscontext, it is worth mentioning the long time studied case ofrare earth metals [72], where an external magnetic field can in-duce non-trivial phase transitions involving spiral spin struc-tures, well described by spin-wave theory. B. Doped regime: Ground state energy
Performing the integration over the first BZ in Eqs. (65) and(69) and setting S = 1 / in Eq. (83), we find that the AB t - J ground state energy per unit cell as a function of hole dopingin zero-field reads: E t - JGS ( δ ) /JN c = − . tJ sin ( πδ ) − . − δ ) . (84)We shall now examine the case of small hole doping awayfrom half filling, i.e., with hole concentration ranging from δ = 0 up to δ = 0 . for two values of J : . and . . InFig. 4, we show the evolution of the GS energy per unit cell δ E t - J G S ( δ ) / J N c Analytic J = 0 . Numeric J = 0 . Analytic J = 0 . Numeric J = 0 . Figure 4. (Color online) Analytical prediction for the GS energy perunit cell of the AB t - J chain as a function of doping, and compar-ison with numerical data from DMRG technique for J/t = 0 . and J/t = 0 . [25]. At half filling ( δ = 0 ), both results meet at theexpected prediction [25]: ≈ − . . Note that we have added theterm − JN c with the intention of comparison with numerical calcu-lation. of the AB t - J model as a function of hole doping for bothmentioned values of J , and the comparison was made with thenumerical DMRG data [25]. From the two results at J = 0 . ,the only quantitative difference induced by the increase of thehole concentration is a crossing feature around δ ≈ . , whereour analytical result slightly change its behavior by loweringthe energy with respect to the numerical data [25]. In fact, be-cause our model assumes a ferrimagnetic state as the startingpoint, this change of behavior suggests that we have entered ina region of strong magnetic instabilities, and possibly indicat-ing a smooth transition to an incommensurate phase with zeroGS total spin beyond δ ≈ . , as confirmed by the numericaldata in Ref. [25] and illustrated in Fig. 3. On the other hand,at J = 0 . , although our results reproduce the numerical datawith an acceptable agreement, we observe a discrepancy thatincreases with δ . The cause of such discrepancy will be dis-cussed in the next subsection.With the purpose of determining the interplay between thecontribution of magnetic exchange and the itinerant kineticenergy to the zero-field GS energy Eq. (83), we take J = 0 . and show its evolution with doping in Fig. 5. We can seein the insets, Fig. 5(a) and Fig. 5(b), the competitive behav-ior of the two energetic contributions, i.e., the contribution ofthe exchange energy increases linearly with δ , while a practi-cally linear decrease of the hopping term is observed as oneenhances the hole doping. This competition indicates that aphase transition to a paramagnetic phase should occur at somecritical concentration value. C. Doped regime: Ground state total spin
The existence of a transition from an IC spiral ferrimag-netic phase to an IC paramagnetic one is a most interest-4 δ E t - J G S ( δ ) / J N c J = 0 . ExchangeHopping δ -2-2.2-2.4 E J ( δ ) / J N c δ E t ( δ ) / J N c Figure 5. (Color online) Ground-state energy per unit cell for the AB t - J chain as a function of δ for J = 0 . . In the insets, weillustrate the two energetic contribution due to (a) exchange and (b)hopping terms. ing feature observed numerically in doped AB t - J Hubbardchains [25]. In order to firmly corroborate the mentionedtransition, we have calculated the GS total spin per unit cellas a function of hole doping, S GS ( δ ) = (cid:80) α (cid:104) S α,z (cid:105) ( δ ) ,with α = A, B , B , by means of the zero-field derivativeof Eq. (83): (cid:104) S α,z (cid:105) ( δ ) = − (1 /N c )[ ∂E t - JGS ( δ, h ) /∂h α ] | h α =0 . (85)We thus find (cid:104) S α,z (cid:105) ( δ ) = (cid:104) S α,z (cid:105) ± √ π t sin( πδ )(4 S − q ) , (86)where + ( − ) corresponds to sublattice α = A ( α = B , ),and (cid:104) S A,z (cid:105) and (cid:104) S B l ,z (cid:105) are given by Eqs. (70). Therefore,by performing the integration over the first BZ of the threecontributions in Eq. (86), we finally obtain: S GS ( δ ) S L = 1 − . πδ ) , (87)where S L = (cid:80) α (cid:104) S α,z (cid:105) = 1 / is Lieb’s reference value forthe GS total spin per unit cell at half filling and zero-field (seeSection IV C).In Fig. 6 we plot the evolution of S GS , normalized by S L ,as a function of δ , and compare it with the numerical datafrom DMRG and Lanczos techniques [25], for J = 0 . (redsquares) and J = 0 . (blue circles). In the latter (former) case,the system undergoes a transition from the modulated itin-erant ferrimagnetic phase to an incommensurate phase withzero (nonzero) S GS . Notice that, in both cases, the transitionis characterized by a decrease of S GS from S L to or to aresidual value, regardless of the value that S GS takes after thetransition. Indeed, at J = 0 . and δ > . , the formationof magnetic polarons (onset of the Nagaoka phenomena thatsets in as U → ∞ ) with charge-density waves in phase withthe modulation of the ferrimagnetic structure, as indicated by Figure 6. (Color online) Ground-state total spin S GS per unit cell(solid magenta line), normalized by its value in the undoped regime: S L = , as a function of hole doping δ for the indicated values of J . In the figure, δ c ≈ . indicates the critical value of dopingat which the magnetic order is suppressed and a second-order phasetransition takes place. the DMRG data [25], leads to an incommensurate phase withnonzero S GS .Most importantly, we can observe in Fig. 6 that the value of S GS decreases practically linearly with δ until the magneticorder is completely suppressed at δ c ≈ . . This behavior issupported by numerical results [25], particularly in the regimewhere the Nagaoka phenomenon is not manifested, that is, at J = 0 . , as indicated in Fig. 3. In this regime, spin and chargequantum fluctuations destabilize the ferrimagnetic structureand trigger a transition to an incommensurate paramagneticphase at δ c , with S GS ∼ ( δ − δ c ) → . VI. CONCLUSIONS
In summary, we have presented a detailed analytical studyof the large-U Hubbard model on the quasi-one-dimensional AB chain. We used a functional integral approach combinedwith a perturbative expansion in the strong-coupling regimethat allowed us to properly analyze the referred system at andaway from half filling.At half filling, our model was mapped onto the quantumHeisenberg model, and analyzed through a spin-wave pertur-bative series expansion in powers of /S . We have demon-strated that the GS energy, spin-wave modes, and sublat-tice magnetizations are in very good agreement with previ-ous results. In the challenging hole doping regime awayfrom half filling, the corresponding t - J (= 4 t /U ) Hamilto-nian was derived. Further, under the assumption that chargeand spin quantum correlations are decoupled, the evolutionof the second-order spin-wave modes in the doped regimehas unveiled the occurrence of spatially modulated spin struc-tures and the emergence of phase separation (first-order tran-sition) in the presence of resonating-valence-bond states. Thedoping-dependent GS energy and total spin per unit cell are5also calculated, in which case the collapse of the spiral mag-netic order at a critical hole concentration was observed.Remarkably, our above-mentioned analytical results in thedoped regime are in very good agreement with density matrixrenormalization group studies, where our assumption of spin-charge decoupling is numerically supported by the formationof charge-density waves in anti-phase with the modulation ofthe ferrimagnetic structure.Finally, we stress that our reported results evidenced thatthe present approach, also used in a study on the compatibil-ity between numerical and analytical outcomes of the large-UHubbard model on the honeycomb lattice, was proved suitable for the AB chain (a quasi-1D system), where the impact ofcharge and spin quantum fluctuations are expected to manifestin a stronger way. We thus conclude that our approach offers aquite powerful analytical description of hole-doping inducedphases away from half filling in low-dimensional strongly-correlated electron systems. ACKNOWLEDGMENTS
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