The role of density-dependent magnon hopping and magnon-magnon repulsion in ferrimagnetic spin-(1/2, S) chains in a magnetic field
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b The role of density-dependent magnon hopping and magnon-magnon repulsion inferrimagnetic spin-(1/2, S ) chains in a magnetic field W. M. da Silva
1, 2 and R. R. Montenegro-Filho Secretaria de Educa¸c˜ao da Para´ıba, 58015-900 Jo˜ao Pessoa-PB, Brasil Laborat´orio de F´ısica Te´orica e Computacional, Departamento de F´ısica,Universidade Federal de Pernambuco, 50760-901 Recife-PE, Brasil
We compare the ground-state features of alternating ferrimagnetic chains (1 / , S ) with S =1 , / , , / p s/S , with s = 1 /
2, considering the fully polarized magnetization as the boson vacuum. The single-particle Hamiltonian is a Rice-Mele model with uniform hopping and modified boundaries, whilethe interactions have a correlated (density-dependent) hopping term and magnon-magnon repul-sion. The magnon-magnon repulsion increases the many-magnon energy and the density-dependenthopping decreases the kinetic energy. We use density matrix renormalization group calculations toinvestigate the effects of these two interaction terms in the bosonic model, and display the quanti-tative agreement between the results from the spin model and the full bosonic approximation. Inparticular, we verify the good accordance in the behavior of the edge states, associated with theferrimagnetic plateau, from the spin and from the bosonic models. Furthermore, we show that theboundary magnon density strongly depends on the interactions and particle statistics..
I. INTRODUCTION
The gapped phases of magnetic insulators are respon-sible for magnetization ( m ) plateaus in the m vs. mag-netic field curves [1]. In one dimension, these incompress-ible phases satisfy the topological Oshikawa-Yamanaka-Affleck [2] criteria, and exhibit associated edge states inopen spin chains [3]. The gapped phases are separated bygapless phases that have a low-energy physics describedby the Tomonoga-Luttinger-liquid theory [4]. Thus, themagnetic field h induces quantum phase transitions in thespin chain, with quantum critical points at the plateauextremes [5, 6]. In the vicinity of the quantum criticalpoints, the magnons are in a high-dilute regime and canbe treated as a gas of hard-core bosons [7, 8] or fermions[9]. In this approximation, the energy is comprised onlyby a simple hopping term and the uniform Zeeman term,which plays the role of chemical potential in the effectivemodel. Following this approach, we can show that the m ( h ) curve (magnon density) presents a square-root sin-gularity in the gapless side of the transition. The firstcorrection to this law is linear and obtained by takinginto account magnon-magnon interaction through a phe-nomenological scattering length [10–14].Another kind of hopping term is known as corre-lated (or density-dependent ) hopping and is essentialin the modeling of a variety of quantum systems [15].One of these terms, the bond-charge interaction [16],is used to model electrons in strongly correlated ma-terials [17–21] and was particularly investigated in thecontext of high- T c superconductivity. Besides, the Hub-bard model with this term has an exact solution in aspecial point of the parameter space [22–25]. On theother hand, the extended boson Hubbard model with adensity-dependent hopping is an effective Hamiltonianfor bosonic molecules, typically polar species [26–28], inoptical lattices [29–36]. Further, general correlated hop- ping hard-core bosonic Hamiltonians are investigated tounderstand the physics of frustrated insulating magneticmaterials [29–31, 37].The ground-state of ferrimagnetic chains satisfy theLieb-Mattis theorem [38] and exhibit ferromagnetic andantiferromagnetic long-range orders [39]. and was inves-tigated in the isotropic [40–42] and anisotropic [41, 43, 44]cases. Ferrimagnetic systems ˙In particular, the behaviorof the edge states associated with the 1/3 magnetizationplateau of the AB anisotropic chain was recently inves-tigated [45]. Furthermore, rich phase diagrams are ob-served through doping [46–50] or adding geometric frus-tration [51–58] to the ferrimagnetic models. In partic-ular, ferrimagnetic spin-(1/2, S ) chains under an ap-plied magnetic field present magnetization plateaus at m = S − / m = S + 1 / m is the magnetization perunit cell [59–64]. On the experimental side, the one-dimensional magnetic phase of a variety of bimetalliccompounds was shown to be modeled by spin-(1/2, S )ferrimagnetic chains [65–70]. Recently, the full magne-tization curve of the charge-transfer salt (4-Br- o -MePy-V)FeCl was experimentally investigated [71] and shownto be a spin-(1/2, 5/2) chain above the three-dimensionalordering temperature.Since spin-( s , S ) ferrimagnetic chains have a long-ranged ordered ground state, linear and interacting spin-wave theory [72] from the classical ferrimagnetic state was used to characterize their low-energy magnetic ex-citations, mainly through the Holstein-Primakoff formal-ism [59, 60, 73–78]. Furthermore, linear spin-wave theoryfrom the fully polarized state [64] gives good results forthe gapped and gapless phases of spin-(1/2, S ) chains ina magnetic field.Here we show that the Holstein-Primakoff Hamiltonianup to order p s/S gives almost exactly results to theground-state phase diagram of ferrimagnetic spin-(1/2,S) chains in a magnetic field. We use the density matrixrenormalization group (DMRG) [79, 80] to obtain themagnetization curves, besides local properties, from theHolstein-Primakoff Hamiltonian and the spin Hamilto-nian. In addition to the simple hopping and the nearest-neighbor interacting terms, a correlated hopping term isessential to obtain a good accord between the numericalresults from the spin and bosonic models. In Sec. II wepresent the Holstein-Primakoff Hamiltonian and the an-alytical formulas for the hard-core boson approximation.In Sec. III we compare the DMRG results for the magne-tization and local properties from the Holstein-PrimakoffHamiltonian and the spin model. In Sec. IV we presenta summary of the main results of the manuscript. II. HOLSTEIN-PRIMAKOFF BOSONICHAMILTONIAN FROM THE FULLYPOLARIZED VACUUM
The alternating mixed-spin ( s = 1 / , S ) chain with L unit cells has the Hamiltonian H spin = J L X j =1 ( s j · S j + s j · S j − ) − BS ztot , (1)where S ztot = X j ( s zj + S zj ) (2)is the z -component of the total spin, and we considerthe magnetic field B in the z direction, with gµ B ≡ µ B is the Bohr magneton and g is the g -factor.The spin − / a sites, while spin − S to b sites along the chain, and we study chains for which S = 1 , / , , /
2, as schematically shown in Fig. 1(a).The ground-state total spin for B = 0 is S − s and hasa copy in each sector in the range − ( S − s ) ≤ S ztot ≤ ( S − s ), as expected from the Lieb-Mattis theorem [38],with a ferrimagnetic (FRI) long-range ordered state. Ifa little magnetic field is applied, the ground state with S ztot = ( S − s ) is chosen. Further, the ground state hasa finite gap to spin excitations carrying a spin ∆ S z =+1, which induces a magnetization plateau at m F RI =( S − s ). Also, a second magnetization plateau is the fullypolarized plateau at m F P = S + s .The fully polarized state is an exact ground state ofthe spin Hamiltonian, and we build the spin-wave the-ory considering it as the magnon vacuum. Making theHolstein-Primakoff mapping on a sites s zj = s − n aj ,s − j = √ sa † j (cid:16) − n aj s (cid:17) / ≈ √ sa † j (cid:16) − n aj s (cid:17) , (3) and on b sites: S zj = S − n bj ; S − j = √ Sb † j (cid:16) − n bj S (cid:17) / ≈ √ Sb † j (cid:16) − n bj S (cid:17) , (4) FIG. 1. (a) Schematic representation of the alternating spinmodel, with spin-1 / a sites, and spin- S at b sites, with S = 1 , / , , and 5 /
2. The z -direction is the direction of anapplied magnetic field B and the superexchange coupling is J .(b) Holstein-Primakoff bosonic Hamiltonian up to O ( S − / ),with a hopping term t = J √ sS , local potentials ε a = − SJ and ε b = − sJ , nearest-neighbor interaction V = J , anddensity dependent correlated hopping process X = J p s/S .The a sites have a hard-core constraint, n a ≤
1, while theconstraint n b ≤ S is imposed on b sites, with n a ( n b ) as thenumber of bosons in one a ( b ) site. The magnetic field B actsas a chemical potential µ in the bosonic model: µ = − B .(c) The bosonic approximations investigated: t , t − V , and t − X − V . In the t and t − V approximations, there is a hard-core constraint on all sites; while in the t − X − V model, the b sites can accommodate up to two magnons. where n aj = a † j a j and n bj = b † j b j , we arrive in the followingspin-wave Hamiltonian H SW = J X j ((cid:16) S − n bj (cid:17)(cid:16) s − n aj (cid:17) + √ sS "(cid:16) − n bj S (cid:17) b j a † j (cid:16) − n aj s (cid:17) ++ b † j (cid:16) − n bj S (cid:17)(cid:16) − n aj s (cid:17) a j + √ sS "(cid:16) − n aj s (cid:17) a j b † j +1 (cid:16) − n bj +1 S (cid:17) ++ a † j (cid:16) − n aj s (cid:17)(cid:16) − n bj +1 S (cid:17) b j +1 + (cid:16) s − n aj (cid:17)(cid:16) S − n bj +1 (cid:17)) + − B X j (cid:16) S + s − n aj − n bj (cid:17) + O ( S − ) . (5)Dropping the classical energy of the ferromagnetic state E class = 2 JLsS − B (cid:0) S + s (cid:1) L , the relevant magnon Hamil-tonian is: H SW = H t + H X + H V + O ( S − ) , (6)with H SW = H SW − E class .As sketched in Fig. 1(b), the H t term comprises ahopping process and distinct local potentials on a and b sites: H t = t X j (cid:16) b † j a j +1 + b † j +1 a j +1 + H. c. (cid:17) + X j [( ε a − µ ) n aj + ( ε b − µ ) n bj ] (7)with t = J √ sS , hopping parameter; ε a = − SJ , local potential on a sites; ε b = − sJ , local potential on b sites; µ = − B. (8)In an open chain, if a or b is a boundary site, the localpotential is half of the above value: ( ε (boundary) a = − SJε (boundary) b = − sJ. (9)Considering the local potentials, we see that the magnonhas a higher probability to be found on a sites, and thisprobability increases with S . However, since the hoppingparameter t increases with √ S , quantum fluctuations arerelevant for moderate values of S , and the magnons canovercome the potential barrier between a and b sites.We observe that the bulk Hamiltonian H t is a partic-ular case of the Rice-Mele model [81]: H Rice-Mele = X j (cid:16) t b † j a j +1 + t b † j +1 a j +1 + H. c. (cid:17) + X j ( ε a n aj + ε b n bj ) , (10)putting µ = 0 and considering the general case of al-ternating hopping: t ( t ) for intra-cell (inter-cell) hop-ping. The Rice-Mele Hamiltonian was originally pro-posed to model the physics of electrons in polymers, but is a paradigmatic model to the understanding of topolog-ical insulators, and can be realized by atoms in opticallattices [82]. The model presents the bulk-boundary cor-respondence [83], and an interacting version was recentlyinvestigated [83] to probe the connection between topol-ogy and particle-particle interactions [83]. The Rice-Melemodel recovers the bulk Hamiltonian H t for t = t .However, for an open chain, the mapping of the spinmodel requires local potentials on the boundary sites,Eq. (9).The second term in the bosonic Hamiltonian (6) is adensity-dependent or correlated hopping term given by H X = − X X j (cid:2) ( a † j + a † j +1 ) n bj b j + H. c. (cid:3) , (11)with X = J r sS . (12)Since s = 1 / n aj ≤
1, a hard-core constraint mustbe imposed on a sites for all models considered. Hence,we discard a term similar to the X term but with a and b variables exchanged, with X ′ = J p S/s as the correlatedhopping amplitude. As sketched in Fig. 1(b), the energyof the system is lowered by the hopping of a magnon to asite that is already occupied. In other words, the magnonprobability to overcome the potential barrier between a and b sites increases if there is one magnon on the b site.This term becomes relevant for higher magnon densities,since it is an interaction term, and X → S increases.The last term in the Hamiltonian (6), see Fig. 1(b),is a repulsive term between magnons in nearest neighborsites: H V = V X j n aj +1 ( n bj + n bj +1 ) , (13)with V = J , and increases the energy for higher magnondensities. For one magnon per unit cell, this term favorsthe magnon localization on alternating a sites, since thelocal potentials ( ε ) favors the occupation of a sites. Thus, ε (any density) and V (higher densities) favor magnon lo-calization on a sites, while quantum fluctuations (tunnel-ing between a and b sites) are favored by t (any density)and X (higher densities).In this work, we compare data from three approxima-tions of Hamiltonian (6), as summarized in Fig. 1(c),and from the spin Hamiltonian (1). The first, which weidentify as the t approximation, is an analytical solutionto the free hard-core model. In this approximation, allsites have a hard-core constraint and can be occupied byonly one magnon: n aj ≤ n bj ≤
1, for any j . Thisconstraint implies that there is not an energy contribu-tion from the X term, and we drop the nearest-neighborinteraction V . In the second approximation, t − V , wekeep the hard-core constraint but add the V contributionto H t . The last approximation, t − X − V , has the threeterms H t , H X , and H V . We consider a hard-core con-straint on a sites and a constraint n bj ≤ b sites. Aswe present below, the relaxation of the hard-core con-straint on b sites and the consequent activation of thecorrelated hopping term implies an excellent agreementbetween the results of this approximation and the spinmodel. A. Free hard-core magnons [ t -approximation]: H t and L → ∞ The first approximation to the magnetization curve, amany-magnon state, is to consider the magnons as freehard-core bosons or free fermions. This mapping is exactin the high-dilute regime of magnons, near the saturationfield. Here, we extend this approach for the full range ofthe magnetization, ( S − s ) ≤ m ≤ ( S + s ), and comparetheir results to more precise calculations considering theinteraction terms.The single-magnon energies are given by the term H t ,Eq. (7), in Hamiltonian (6). Using the following Fouriertransforms: ( a j = √ L P k e − ik/ e ikj a k ; b j = √ L P k e + ik/ e ikj b k , (14)where a phase e ± ik/ is included to ease the calculation, H t becomes: H t = X k (cid:0) a † k b † k (cid:1) (cid:18) ε a γ k γ k ε b (cid:19) (cid:18) a k b k (cid:19) , (15)where γ k = 2 t cos( k/ . (16)After diagonalization, the Hamiltonian is written as H t = X k ω ( − ) k n ( − ) k + ω (+) k n (+) k , (17)where the dispersion relations are ω ( ± ) k = ε a + ε b ± ω k = − J ( s + S ) ± ω k , (18) with ω k = s(cid:18) ε b − ε a (cid:19) + γ k = J p ( S − s ) + 4 sS cos ( k/ , (19)and n ( ± ) k = α ( ± ) † k α ( ± ) k , where (cid:18) α ( − ) k α (+) k (cid:19) = (cid:18) cos θ k − sin θ k sin θ k cos θ k (cid:19) (cid:18) a k b k (cid:19) , (20)with ( cos θ k = + S − s ω k ;sin θ k = − S − s ω k . (21)A magnetic field (or chemical potential) adds a + B en-ergy term to the two bands. -1 0 1k / π -2(S+s)-2S-2s = -10 E n e r gy / J inactive one - magnon states -1 -0.5 0 0.5 1k / π
02s = 1-1 -0.5 0 0.5 1k / π -2s = -10 B = ( s + S ) = − µ B=2S
FPFRI
B=0 vacuumone magnonper unit cell
FIG. 2. One-magnon states from the fully polarized (FP)vacuum and B = 0 as a function of lattice wave-vector k .In the t -approximation, the high energy band is inactive. For B = 2( s + S ) = − µ , the exact ground state is the FP vacuum;while there is one magnon per unit cell in the system for B = 2 S , and the ground state is ferrimagnetic (FRI). The magnon bands are shown in Fig. 2 for (i) B = 0;(ii) B = 2( s + S ), the saturation field, for which m = m FP = ( S + s ); and (iii) B = 2 S , the critical field of theferrimagnetic plateau, for which m = m FRI = ( S − s ), or1 magnon per unit cell: m FP − m FRI = ( S + 1 / − ( S − /
2) = 1. As expected from the Lieb-Mattis theorem,the magnetization in the null field is S − s . In the freehard-core boson or free fermion approximation, we fill thesingle-particle states following a Fermi distribution up tothe effective Fermi wave-vector k F . As shown in Fig. 2,if we follow this procedure the two bands should be filledfor B = 0 (two magnons per unit cell) and the magnetiza-tion curves of the spin model would not be reproduced.We have shown in Ref. [64] that this problem can beovercome, even for finite T , by introducing an effectivechemical potential µ to the upper band, in a way similarto Takahashi’s solution to the ferromagnetic linear chain[84]. In particular, µ → − J = − sJ as T →
0, suchthat the overall effect of µ at T = 0 is the suppressionof the upper band. Hence, in the free hard-core approx-imation, we must consider only the lower band in thecalculations, as schematically indicated in Fig. 2. Thus,for example, the energy per unit cell in the free hard-coreapproximation is written as EL = 1 L k F X k = − k F ω ( − ) k (22)for a magnon density per unit cell n , where k F = πn, (23)and n = m FP − m .
1. Average local magnetizations
If the chain has a magnon density per unit cell n and considering the hard-core approximation, the aver-age magnetizations of a and b sites are given by ( h s z i = s − L P k F k = − k F h n ak i ( a sites); h S z i = S − L P k F k = − k F h n bk i ( b sites) . (24)Using Eqs. (20) and discarding terms involving the upperband, we obtain ( h n ak i = n ( − ) k cos θ k ; h n bk i = n ( − ) k sin θ k . (25)We, thus, have ((cid:10) s z (cid:11) = s − − S − s P k F k = − k F ω k ; (cid:10) S z (cid:11) = S − + S − s P k F k = − k F ω k , (26)after the substitution of Eqs. (21) in Eqs. (25) and theresults in Eqs. (24). III. DMRG RESULTS FOR THE SPIN MODELAND THE BOSONIC HAMILTONIANSA. Methodology
We use the density matrix renormalization group toobtain the magnetization curves and local properties ofthe spin, t − V , and t − X − V models; and also com-pare this data with the analytical results from the freehard-core model ( t -model) in the thermodynamic limit: L → ∞ . These approximations are summarized in Fig.1(c). All DMRG results (boson and spin models) are ob-tained through the Algorithms and Libraries for PhysicsSimulations (ALPS) project [85] for chains with L = 128unit cells, with one a site at one extreme and a b siteat the other. If the system has an a site at the leftextreme and a b in the right ( a - b boundaries), the renor-malization process for the magnetization step inside theferrimagnetic plateau becomes trapped in a metastable state for S = 3 / , , and 5 /
2. In these cases, the globalenergy minimum is reached by the algorithm if the chainshave a b site at the left extreme and an a site at the rightextreme ( b - a boundaries). In the other magnetizations,this is irrelevant, i. e., the same state is calculated forthe a - b or the b - a boundaries. We retain a maximum of243 states per block and the maximum discarded weightless than 10 − .For the spin model, the magnetization curves are cal-culated from the lowest energy state for a fixed S z and B = 0: E ( S z ); since for B = 0 we need only to addthe Zeeman term, such that E B ( S z ) = E ( S z ) − BS z . Ina gapless (non-plateau) phase, the magnetization curvesare made of finite steps in a finite-size system. Definingthe extreme points of these finite steps as B − and B + ,we have B ± = ± [ E ( S z ± − E ( S z )] (27)for the step at S z . In a gapless phase, B − → B + as L → ∞ , while in a thermodynamic-limit plateau state B − = B + for L → ∞ . In the last case, B ± are quantumcritical fields separating the plateau insulating state froma gapless critical Luttinger liquid phase. For the bosonicmodels, the magnon density per unit cell n as a functionof chemical potential µ is obtained with a similar pro-cedure. We calculate the lowest energy state for a fixednumber of bosons N , with N = nL and µ = 0. The valueof the chemical potential at the extremes ( µ − and µ + ) ofthe step at N are given by µ ± = ± [ E ( N ± − E ( N )] . (28)A gapless phase has µ + → µ − as L → ∞ , while in aplateau insulating phase µ + = µ − in the thermodynamiclimit. The transformation from the boson to the spinlanguage is performed through the following equations: ( n = m − m FP , and B = − µ. (29) B. Magnetization curves and local magnetizations
The magnetization curves from the spin model and theboson Hamiltonians are shown in Fig. 3(a). The modelspresent two magnetization plateaus, one at the ferrimag-netic magnetization m FRI = S − / m FP = S + 1 /
2. The saturation field B F P , endpoint of the FP plateau, is obtained through the closingof the single-particle magnon gap with B , as sketched inFig. 2, at B = B FP = 2( S + s ). The saturation field B FP from any bosonic approximation is rigorously equalto its exact value since the fully polarized state, an ex-act eigenstate of the spin model, is the vacuum for thebosonic models.The Oshikawa-Yamanaka-Affleck topological criteria[2] states that a plateau can appear in the magnetiza-tion curve of spin systems if S u − m = integer, (30) B / J m / m s (1/2, 5/2)(1/2, 2)(1/2, 3/2)(1/2, 1) Spin, DMRG (L=128) t , L →∞ t - X - V , DMRG (L=128)(a) FRI plateauFRI plateauFRI plateau
FRIplateau arrows edge states
B / J n (1/2, 5/2)(1/2, 2)(1/2, 3/2)(1/2, 1) Spin, DMRG (L=128) t , L →∞ t - X - V , DMRG (L=128) (b) t - V , DMRG (L=128) GaplessLuttingerliquidFRI plateau ( n = 1) FIG. 3. (a) Magnetization per unit cell ( m ) of (1 / , S ) chains,with S = 1 , / , , and 5 /
2, normalized by its saturationvalue ( m s ) and (b) magnon densities per unit cell n as func-tions of B in units of J . DMRG data for (full lines) the spinmodel and for ( • ) the t − X − V approximation. We presentin both figures the results for the t -model (dashed lines) and L → ∞ . In (b), we also show the magnon density as a func-tion of B units of J for ( • ) the t − V approximation, calculatedwith DMRG for a system with L = 128. In the thermody-namic limit, the ferrimagnetic plateau is observed for n = 1,while the gapless Luttinger liquid phase for 0 < n <
1. Onboth figures, we indicate (arrows) the magnetic field at whichthe edge states are occupied by one magnon. where S u is the maximum spin of a unit cell, unless theground state spontaneously break translation symmetry.This corresponds to a number of magnons per unit cell n = 0 , , , . . . , S u ( S u − /
2) for integer (half-integer) S u ,from the fully polarized state. Since S u = S + 1 / S ) chains, plateauscould appear at m = S + 1 / , ( S − / , . . . , / S . In the spin-(1/2, S ) chains, thedata shows that there are two magnetization plateaus,one at the fully polarized ( n = 0) magnetization and theother at one magnon per unit cell ( n = 1), the ferrimag-netic plateau. The other possible magnetization plateausare inhibited by the magnon-magnon interaction term V [Eq. (13)]. The magnetization curves of ferrimagneticspin chains with 1 / < s < S can exhibit other plateausbetween the ferrimagnetic and the fully polatized ones.This is observed, for example, in spin-(1,2) and spin-(1,3/2) chains [86]. For these chains, the ferrimagneticplateau state has two magnons per unit cell, n = 2, and the magnetization curves also exhibits a plateau at n = 1.We also observe the occupancy of the edge states of theferrimagnetic plateau by one magnon at the indicatedmagnetic fields. These edge states appears in finite-sizeopen systems associated with topological aspects [3, 45]of the ferrimagnetic state.The non-interacting Rice-Mele model (10) does notpresent edge states for uniform hopping. However, re-cently [83], it was shown that an interacting fermionicsystem , with a local Coulomb interaction, presents ef-fective edge states, and that a fraction of the boundarycharge is, in fact, related to the bulk properties. Ournon-interacting Hamiltonian (7) has the modified localpotentials in the boundaries (9), required from the spinmapping, and thus localized edge states. Furthermore,our interacting model is a bosonic system having the cor-related hopping and nearest-neighbor coulomb repulsion.In the Sec. IV, we study the boundary magnon den-sity and compare it from relevant interacting and non-interacting bosonic models.The gapless phase between the magnetization plateausis a Luttinger liquid phase with power-law decay of thetransverse spin correlation functions [4] and has a dy-namical exponent z = 1.We notice in Fig. 3(a) that as B decreases from B FP , the magnetization from the free hard-core model, t -approximation, departs from that of the spin modelat roughly half filling of the lower magnon band ω ( − ) ,Eq. (18). At this filling, the interaction effects start tobecome relevant. The critical field of the ferrimagneticplateau from the free hard-core model: 2 S , see Fig. 2,becomes more near its value for the spin model as S in-creases, as also confirmed in Table I. This effect can beattributed to the local energy of the a sites that becomesdeeper in comparison with the local energy of the b sites,as can be seen in the energy term (7) and the sketch inFig. 1. Thus, the magnons become more localized on a sites as S increases, and the X and V energy terms,Eqs. (11) and (13), respectively, become lesser relevant.The X term, due to the low occupancy probability ofone b site by two magnons, and also because X → S → ∞ , see Eq. (12). The V term, on the otherhand, because the probability of finding two magnonsin nearest neighbor sites is also very low. Further, the t − X − V -approximation, which has all energy terms inHamiltonian (6) active, is in excellent agreement with theresults for the spin model. Even the location of the edgestates is well reproduced by the t − X − V -approximation.In Fig. 3(b) we show the average magnon density perunit cell h n i . Besides the models presented in (a), we addthe t − V -approximation. The presence of the magnon-repulsion makes the accordance with the spin model goodfor h n i > /
2, while in the t -approximation this agree-ment is good up to a value of h n i less than 1/2. As h n i →
1, reaching the ferrimagnetic plateau, the repul-sion V increases much the energy of the system, and,thus, implies a lower value of the critical magnetic field B F RI , compared to the spin model. The location of the
TABLE I. Critical field of the ferrimagnetic plateau in unitsof J for the spin-(1/2, S ) chains in the free hard-core bosonapproximation t -approximation and the spin model. S t -approx.: 2 S spin model: β S − ββ a ( h s z i ) and b ( h S z i ) sites forthe spin-(1/2, S ) chains in the free hard-core boson approxi-mation t and the spin model: ( h s z i , h S z i ), at m = S − s , theferrimagnetic magnetization. S t -approx. spin model1 ( − . , .
77) ( − . , . − . , .
34) ( − . , . − . , .
88) ( − . , . − . , .
40) ( − . , . edge states in the finite-size system is also different be-tween the spin and the t − V -approximation. Further,the X -term, the density-dependent hopping term, lowersthe energy, and the full h n i -curve of the spin model iswell reproduced by the t − X − V approximation. Wenotice, however, that only for the finite size spin-(1/2, 1)chain studied, the lower critical field of the ferrimagneticplateau is B ≈ . J . In Sec. IV, we present the magnoncurve for N > a and b sitesare shown in Fig. 4. Notice that the average spin at b sites, Fig. 4(b), is normalized by S . Also shown arethe probability of occupancy of a and b sites. We usethe expressions (26) to obtain the average spins from the t -approximation, while we calculate the average magnondensity from the spin model with ( h n a i = − L P Ll =1 h s zl i , h n b i = S − L P Ll =1 h S zl i , (31)for a finite spin chain of size L and open boundaries.The value of the average spin at a sites from any of theconsidered bosonic approximations is in good agreementwith its value from the spin model. A relevant departurebetween the approximations and the spin model occursin the average spin on b sites as the ferrimagnetic mag-netization plateau is approached. However, even the t -approximation provides good values for the average spins,as shown in Table II for the ferrimagnetic magnetiza-tion. Also, the results become indistinguishable, even on b sites, as S increases or as the fully polarized plateauis approached. Furthermore, as in the magnetization re-sults, the t − X − V model is an excellent approximationto the spin Hamiltonian. The data in Figs. 4(c) and (d)confirm that due to the higher value of the local potentialon a sites, the probability of occupancy of a sites is higher m / m s -0.2500.250.5 < S z a > (1/2, 1) ( / , / ) ( / , ) (1/2, 5/2) m / m s < S z b > / S (1/2, 1)(1/2, 3/2)(1/2, 2)(1/2, 5/2)(a) (b) Spin, DMRG(L=128) t , L →∞ t - X - V , DMRG (L=128) m / m s < n a > (1/2, 1)(1/2, 3/2)(1/2, 2) (1/2, 5/2) m / m s < n b > (1/2, 1)(1/2, 3/2)(1/2, 2)(1/2, 5/2) (c) (d) FIG. 4. Average spin at (a) a , h S za i , and (b) b , h S zb i ,sites, in this case normalized by S , for (1/2, S ) chains with S = 1 , / , , and 5 /
2, as a function of the normalized mag-netization m/m s , where m s is the saturation magnetizationof each chain. The hard-core boson t approximation in thethermodynamic limit (dashed lines), DMRG results for thespin model (full lines) and the t − X − V approximation ( • ),both for systems with L = 128. Average magnon densities at(c) a , h n a i , and (d) b , h n b i , sites as a function of m/m s andthe same legend of (a) and (b). than that on b sites, as discussed in the context of themagnetization curves in Fig. 3. In particular, quantumfluctuations are reduced as S increases, since h n b i → t − X − V models. The magnon densitiesfrom the spin model are calculated through ( h n al i = − h s zl i ; h n bl i = S − h S zl i , (32)and h n celll i = h n al i + h n bl i . We show the data for the spin-(1/2, 1) and spin-(1/2,5/2) chains, for two low magnondensities ( n = 8 / n = 16 / n = 72 / n = 116 / L = 128 unitcells. For the lower magnon densities (hard-core limit)the results for t − X − V -approximation departs from thespin model data near the boundaries for the spin-(1/2,1). However, the accordance between the two models isexcellent in the case of the spin-(1/2, 5/2), even near theboundaries, for the two lower magnon densities shown. n = 8/128 = 0.0625 n = 16/128 = 0.12500 n = 72/128 = 0.56250spin: a sitespin: b sitespin: cell t - X - V A v e r a g e m a gnon d e n s iti e s a l ong t h e c h a i n l n = 116/128 = 0.90625 (a) (1/2, 1) chain n = 8/128 = 0.0625 n = 16/128 = 0.12500 n = 72/128 = 0.56250spin: b sitespin: cell t - X - V l n = 116/128 = 0.90625 (b) (1/2, 5/2) chain spin: a site (1/2, 1) (1/2, 3/2)(1/2, 2) l (1/2, 5/2) (c) edge states D e n s it y c h a ng e a l ong t h e c h a i n δ < n > F R I + → F R I cell: spin a site: spin b site: spin FIG. 5. Average magnon densities along (a) the (1 / S = 1) chain and (b) the (1 / S = 5 /
2) with L = 128 unit cells. Magnondensities at ( • ) a sites, h n a,l i , at ( • ) b sites, h n b,l i , and ( • ) total magnon density, h n cell,l i , for unit cell l from the spin modeland ( N ) the t − X − V approximation, both calculated with DMRG for systems with L = 128. The following total averagemagnon densities are shown: n = 8 / , / , / , and 116 / m = 184 / , / , / , and 76 / m = 376 / , / , / , and 268 / δ h n i F RI +1 → F RI,l = h n i N = L,l − h n i N = L − ,l along the chain as a function of unit cell l shows the presence of the edge states for the four chains studied. (d) The average magnon probability density h n i along achain with L = 128 at (circles) a and (diamonds) b sites for (white symbols) N = L − N = L bosons fromthe t − X − V approximation of the (1/2,1) chain. (e) Sketch of the ground states for N = L − N = L bosons. Averagemagnon densities are shown as grey circles, fluctuations between a and b sites are indicated by a blue stripe, and we use a redfilled curve to represent the localized orbital in the boundary unit cell. For the two higher magnon densities, there is an excel-lent agreement between the spin model and the t − X − V approximation for the two chains.In Fig. 5(c), we show the magnon distribution inthe edge localized state occupied by one magnon for thefour chains studied. To calculate it, we notice that theedge state appears between the two magnetization steps: S z = L ( S − s ) and S z = L ( S − s ) + 1, which will joinin the thermodynamic limit, and make the S − s ferri-magnetic plateau, see Fig. 3(a) and (b). Thus, to visu- alize the spatial extent of the edge state, we consider themagnon distribution change, δ h n i F RI +1 → F RI , between atotal number of magnons N = L − S z = L ( S − s ) + 1]and N = L [ S z = L ( S − s )]: δ h n i F RI +1 → F RI,l = h n i N = L,l − h n i N = L − ,l . (33)We notice in the data shown in Fig. 5(c) that a tiny dis-crepancy is observed between the results for the t − X − V -approximation and the spin model in the case of the (1/2,1) chain, while for the other chains the agreement is excel-lent. The hole added to the many-magnon state at N = L becomes well localized in the boundary cell, mainly onthe boundary a site, thus implying the presence of thisempty orbital in the N = L − S z = L ( S − s )] state.The localization in the boundary increases with S as ex-pected from the increasing of the bulk gap. In fact, thefluctuations of the magnon density in the boundary be-come neglegible as S → ∞ , and the boundary hosts onemagnon in this limit. We can attibute this behavior tothe increasing potential barrier between bulk and edge,Eqs. (8) and (9), respectively, as S → ∞ .As an example of the data used to make the resultsshown in Fig. 5(d), we exhibit in Fig. 5(d) the averagemagnon density along the chain, h n i , for a and b sites,in the t − X − V approximation of the (1/2,1) chain.We notice that the a site at the left boundary is almostempty for N = L −
1, while its ≈ . N = L , implyingthe value shown in (c). We also notice a sizable changein the occupation of the b site at the left boundary forthe two fillings, while h n i does not change at the rightboundary. A sketch of the two ground states is shown inFig. 5(e). For N = L −
1, we have an insulating statewith the magnon average higher on a sites than on b sites.Quantum fluctuations between a and b sites in nearbyunit cells are expected, given the excellent accordancebetween the results from the t − X − V -approximationand the spin model in the high-density limit, m → m F RI .Adding one magnon to the ( L − m = S − s in the finite-size system, which is also an insulating state. The addedmagnon occupies the empty localized orbital state in theunit cell of the boundary a site. The penetration of thisedge state in the gapped bulk is very tiny and decreaseswith increasing S , see Fig. 5(c), as can be estimatedthrough the average magnon distribution. IV. THE RELEVANCE OF THE BOUNDARYPOTENTIALS AND INTERACTIONS IN THEBOUNDARY MAGNON DENSITY OF THEBOSONIC MODEL
With the help of Fig. 6, we discuss the relevance of thedifference between the local potential in the boundariesand the bulk sites on the boundary magnon densities,as well as the importance of interactions in it. We usethe parameters of the spin-(1/2, 1) chain, but focus onthe bosonic Hamiltonian, for three models in finite-sizechains: (1) the interacting model considering the map-ping to the spin system ( X = 0 , V = 0) - the t − X − V model; (2) the corresponding non-interacting model, i. e.,the t − X − V model with X = 0 and V = 0, which hasa hard-core constraint on a sites and the b sites can hostup to 2 bosons; and (3), the t approximation, which hasa hard-core constraint on a and b sites. One of the sets,Figs. 6(a) and (c), shows data for the boundary poten-tials distinct from the bulk potentials, with the values inEqs. (9) and (8), respectively. The other set, Figs. 6(b) N (b) Bound. pot. = Bulk pot. t - X - V (a) Bound. pot. ≠ Bulk pot. B = 0.06 J with X=0, V=0 t - X - V t - X - V t, finite Lt, finite L t - X - V with X=0, V=0 l l δ < n > N i → N f , l l l (d) Bound. = Bulk (c) Bound. ≠ Bulk
FIG. 6. DMRG results for finite-size modified Rice-Melemodels with L = 128. In (a) and (c), the boundary potentialshave the values in Eq. (9); while in (b) and (d), the boundarypotentials have the same values of the bulk local potentialson a and b sites, ε a and ε b in Eq. (8). Taking the parametersfrom the spin-(1/2, 1) chain, we show data for the full inter-acting model, t − X − V , the t − X − V model with X = 0 = V ,and the hard-core approximation t . In (a) and (b), we presentthe number of magnons N as a function of magnetic field (inunits of J ) B = − µ , where µ is the chemical potential inthe bosonic system. In (c) and (d), we show the change inmagnon density per unit cell δ h n i N i → N f ,l = h n i N f ,l − h n i N i ,l between states N = N i and N = N f as a function of unit cell l , where the arrows in (a) and (b) identify N i and N f for eachmodel. and (d), takes the value of the boundary potentials equalto the bulk ones, Eq. (8). The arrows in Figs. 6(a) and(b) indicate the edge states whose boson density change δ h n i N i → N f ,l = h n i N f ,l − h n i N i ,l (34)is shown in Figs. 6(c) and (d), respectively, where N f ( N i ) is the highest (lowest) value of N between the twomagnon density steps separated by the occupancy of theedge state.For the bulk different from the boundaries, Fig. 6(a),we see that the edge state is very robust, appearing evenin the t approximation. This does not occur for the t approximation if the boundary is equal to the bulk, Fig.6(b), which is an expected result for the Rice-Mele model[83]. However, we notice in Fig. 6(a) that, in comparisonwith the spin model, the plateau width is very underesti-mated in the t approximation, since we are not discardingthe single-particle states in the upper band shown in Fig.2. We observe that this upper band was discarded for the L → ∞ results shown in Figs. 3 and 4, due to the in-troduction of the effective chemical potential connectedwith the Takhashi’s constraint [64]. Considering also the0spin model, the error in the plateau width for t − X − V interacting case is very tiny, with a lower critical field ≈ . J . Also in Fig. 6(a), bulk and boundaries dis-tinct, we note that the edge state in the non-interacting t − X − V model appears only for two bosons over the N = 127 state, Fig. 6(a), with one of the bosons oc-cupying an extended state and the other an edge state,as shown in Fig. 6(c). This shows that the distinctionbetween boundaries and bulk is not sufficient to a goodrepresentation of the edge state in the spin model, the interactions are essential for it. In the case of the t ap-proximation, the particle statistics provides a change inthe single particle quantum states that mimic the effectof the X and V interactions in the t − X − V model. InFig. 6 (b), where the boundary is equal to the bulk, weshow that edge states appear in the t − X − V model inthe interacting and non-interacting cases. As mentionedabove, the edge state of the non-interacting case doesnot appear in the t approximation. Further, the bound-ary density, Fig. 6 (d), in the interacting t − X − V caseappears in the right extreme and has a value approxi-mately equal to that in Fig. 6 (c). From these results,we can assert that the interactions X and V , and the par-ticle statistics, are essential to understand the boundarymagnon density. V. SUMMARY AND DISCUSSION
We have investigated the Holstein-Primakoff bosonicHamiltonian, up to order p s/S , of the alternating spin-( s = 1 / S ) chains in a magnetic field, and consideringthe fully polarized as the vacuum. Three bosonic Hamil-tonians were considered: the first has only a hoppingterm ( t -approximation) and distinct local potentials ( ε )on the spin-1 / a site) and the spin- S ( b site) sites , thisapproximation is a Rice-Mele Hamiltonian with bound-aries different from the bulk and uniform hopping term.The second one has the terms of the t -approximation plusa magnon-magnon repulsion V ( t - V -approximation); andthe last approximation shows the t and V terms, and adensity-dependent correlated hopping term X ( t − X − V -approximation). In the t − X − V approximation, b sitescan accommodate up to two magnons, while in the oth-ers a hard-core constraint is imposed on a and b sites.The local potentials, at any density, and V for higher densities favor magnon localization on the a sites. Onthe other hand, quantum fluctuations, magnon tunnelingbetween a and b sites, are favored by t , at any density,and X for higher densities. We use the density matrixrenormalization group to investigate the spin model andthe bosonic Hamiltonians t − V and t − X − V in finite-size open systems, while for the t -approximation we haveconsidered its analytical solution. We compare the mag-netization and magnon densities per unit cell as a func-tion of a magnetic field, average bulk density and localdensities along the chains from the spin model and thebosonic approximations. From the ferrimagnetic plateau(one magnon per unit cell) to saturation (empty chain),the t − X − V -approximation is in excellent agreementwith the spin model, while the t and t − V results de-part from that of the spin model as the magnon densityincreases. This, thus, shows the relevance of both interac-tion terms, magnon-magnon repulsion and the correlatedhopping term X , and its associated particle fluctuations,to describe the many-body system near the ferrimagneticplateau. The edge state associated with the insulatingferrimagnetic plateau is well reproduced by the bosonicmodel. In particular, we have shown that the magnonboundary densities are strongly dependent on the inter-actions and the particle statistics.The use of the fully polarized state as the vacuum en-ables a better understanding of the underlying quantumprocesses in the spin Hamiltonian, compared to a ferri-magnetic vacuum, which is not an exact eigenstate of thespin Hamiltonian. Our results also suggest that, beyondhopping and magnon-magnon interaction, the density-dependent hopping term increases the range of magne-tizations for which effective bosonic models can make agood description of the physical data from general spinsystems having ions with spins higher than 1/2. ACKNOWLEDGMENTS
We acknowledge support from Coordena¸c˜ao de Aper-fei¸coamento de Pessoal de N´ıvel Superior (CAPES), Con-selho Nacional de Desenvolvimento Cient´ıfico e Tec-nol´ogico (CNPq), and Funda¸c˜ao de Amparo `a Ciˆencia eTecnologia do Estado de Pernambuco (FACEPE), Brazil-ian agencies, including the PRONEX Program which isfunded by CNPq and FACEPE, APQ-0602-1.05/14. [1] T. Giamarchi, C. R¨uegg, and O. Tchernyshyov,Bose–Einstein condensation in magnetic insulators, Na-ture Physics , 198 (2008).[2] M. Oshikawa, M. Yamanaka, and I. Affleck, Magneti-zation plateaus in spin chains: “haldane gap” for half-integer spins, Physical Review Letters , 1984 (1997).[3] H. Hu, C. Cheng, Z. Xu, H.-G. Luo, and S. Chen,Topological nature of magnetization plateaus in period-ically modulated quantum spin chains, Physical Review B , 035150 (2014); H.-P. Hu, C. Cheng, H.-G. Luo,and S. Chen, Topological incommensurate magnetizationplateaus in quasi-periodic quantum spin chains, ScientificReports , 8433 (2015).[4] T. Giamarchi, Quantum Physics in One Dimension (Ox-ford University Press, 2004); R. Chitra and T. Gia-marchi, Critical properties of gapped spin-chains and lad-ders in a magnetic field, Physical Review B , 5816(1997). [5] S. Sachdev, Quantum Phase Transitions (CambridgeUniversity Press, 2001).[6] M. Vojta, Quantum phase transitions, Reports onProgress in Physics , 2069 (2003).[7] I. Affleck, Bose condensation in quasi-one-dimensionalantiferromagnets in strong fields, Physical Review B ,3215 (1991).[8] E. S. Sørensen and I. Affleck, Large-scale numerical ev-idence for Bose condensation in the S =1 antiferromag-netic chain in a strong field, Physical Review Letters ,1633 (1993).[9] A. M. Tsvelik, Field-theory treatment of the Heisenbergspin-1 chain, Physical Review B , 10499 (1990).[10] K. Okunishi, Y. Hieida, and Y. Akutsu, δ -function Bose-gas picture of S = 1 antiferromagnetic quantum spinchains near critical fields, Physical Review B , 6806(1999).[11] J. Lou, S. Qin, T.-K. Ng, Z. Su, and I. Affleck, Finite-sizespectrum, magnon interactions, and magnetization of S= 1 Heisenberg spin chains, Physical Review B , 3786(2000).[12] I. Affleck, W. Hofstetter, D. R. Nelson, andU. Schollw¨ock, Non-Hermitian Luttinger liquids andflux line pinning in planar superconductors, Journal ofStatistical Mechanics: Theory and Experiment ,P10003 (2004).[13] I. Affleck, Luttinger liquid parameter for the spin-1Heisenberg chain in a magnetic field, Physical ReviewB , 132414 (2005).[14] L. Vanderstraeten, F. Verstraete, and J. Haegeman, Scat-tering particles in quantum spin chains, Physical ReviewB , 125136 (2015).[15] A. Dobry and A. Aligia, Quantum phase diagram of thehalf filled Hubbard model with bond-charge interaction,Nuclear Physics B , 767 (2011).[16] S. Kivelson, W.-P. Su, J. R. Schrieffer, and A. J. Heeger,Missing bond-charge repulsion in the extended Hubbardmodel: Effects in polyacetylene, Physical Review Letters , 1899 (1987).[17] J. Hirsch, Bond-charge repulsion and hole superconduc-tivity, Physica C: Superconductivity and its Applications , 326 (1989).[18] J. E. Hirsch and F. Marsiglio, Superconducting statein an oxygen hole metal, Physical Review B , 11515(1989).[19] G. I. Japaridze and A. P. Kampf, Weak-coupling phasediagram of the extended Hubbard model with correlated-hopping interaction, Physical Review B , 12822 (1999).[20] J. Vidal and B. Dou¸cot, Strongly correlated hopping andmany-body bound states, Physical Review B , 045102(2001).[21] A. Anfossi, P. Giorda, and A. Montorsi, Entanglementin extended Hubbard models and quantum phase transi-tions, Physical Review B , 165106 (2007).[22] L. Arrachea and A. A. Aligia, Exact Solution of a Hub-bard Chain with Bond-Charge Interaction, Physical Re-view Letters , 2240 (1994).[23] A. Schadschneider, Superconductivity in an exactly solv-able Hubbard model with bond-charge interaction, Phys-ical Review B , 10386 (1995).[24] F. Dolcini and A. Montorsi, Finite-temperature proper-ties of the Hubbard chain with bond-charge interaction,Physical Review B , 075112 (2002). [25] C. Vitoriano and M. D. Coutinho-Filho, FractionalStatistics and Quantum Scaling Properties of the Hub-bard Chain with Bond-Charge Interaction, Physical Re-view Letters , 146404 (2009).[26] M. Maik, P. Hauke, O. Dutta, M. Lewenstein, and J. Za-krzewski, Density-dependent tunneling in the extendedBose–Hubbard model, New Journal of Physics , 113041(2013).[27] S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz,Z. Cai, M. Baranov, P. Zoller, and F. Ferlaino, ExtendedBose-Hubbard models with ultracold magnetic atoms,Science , 201 (2016).[28] S. Fazzini, A. Montorsi, M. Roncaglia, and L. Barbi-ero, Hidden magnetism in periodically modulated one di-mensional dipolar fermions, New Journal of Physics ,123008 (2017).[29] R. Bendjama, B. Kumar, and F. Mila, Absence of single-particle bose-einstein condensation at low densities forbosons with correlated hopping, Physical Review Letters , 1 (2005).[30] K. P. Schmidt, J. Dorier, A. L¨auchli, and F. Mila, Single-particle versus pair condensation of hard-core bosonswith correlated hopping, Physical Review B , 174508(2006).[31] K. P. Schmidt, J. Dorier, A. M. L¨auchli, and F. Mila, Su-persolid Phase Induced by Correlated Hopping in Spin-1/2 Frustrated Quantum Magnets, Physical Review Let-ters , 090401 (2008).[32] M. Eckholt and J. J. Garc´ıa-Ripoll, Correlated hoppingof bosonic atoms induced by optical lattices, New Journalof Physics , 093028 (2009).[33] H. C. Jiang, L. Fu, and C. Xu, Pair superfluid and su-persolid of correlated hard-core bosons on a triangularlattice, Physical Review B , 045129 (2012).[34] O. J¨urgensen, F. Meinert, M. J. Mark, H.-C. N¨agerl, andD.-S. L¨uhmann, Observation of Density-Induced Tunnel-ing, Physical Review Letters , 193003 (2014).[35] K. Biedro´n, M. L¸acki, and J. Zakrzewski, Extended Bose-Hubbard model with dipolar and contact interactions,Physical Review B , 245102 (2018).[36] D. Johnstone, N. Westerberg, C. W. Duncan, andP. ¨Ohberg, Staggered ground states in an optical lattice,Physical Review A , 043614 (2019).[37] J. Stasi´nska, R. W. Chhajlany, O. Dutta, and M. Lewen-stein, Effects of extended correlated hopping in a bose-bose mixture, arXiv:1912.13359 (2019).[38] E. Lieb and D. Mattis, Ordering energy levels of inter-acting spin systems, Journal of Mathematical Physics , 749 (1962).[39] G.-S. Tian, Coexistence of the ferromagnetic and antifer-romagnetic long-range orders in the generalized antiferro-magnetic heisenberg model on a bipartite lattice, Journalof Physics A: Mathematical and General , 2305 (1994).[40] E. P. Raposo and M. D. Coutinho-Filho, Quantum crit-ical properties of ferrimagnetic hubbard chains, PhysicalReview Letters , 4853 (1997); Field theory of ferri-magnetic hubbard chains, Physical Review B , 14384(1999).[41] F. C. Alcaraz and A. L. Malvezzi, Critical behaviour ofmixed heisenberg chains, Journal of Physics A: Mathe-matical and General , 767 (1997).[42] B. Gu, G. Su, and S. Gao, Thermodynamics ofspin- 1 2 antiferromagnet-antiferromagnet-ferromagnet and ferromagnet-ferromagnet-antiferromagnet trimer-ized quantum Heisenberg chains, Physical Review B ,134427 (2006); S.-S. Gong, S. Gao, and G. Su, Ther-modynamics of spin-1/2 tetrameric Heisenberg antiferro-magnetic chain, , 14413 (2009); S.-S. Gong, W. Li,Y. Zhao, and G. Su, Magnetism and thermodynamicsof spin-(1/2, 1) decorated Heisenberg chain with spin-1pendants, , 214431 (2010).[43] T. Sakai and S. Yamamoto, Critical behavior ofanisotropic Heisenberg mixed-spin chains in a field, Phys-ical Review B , 4053 (1999).[44] L. M. Ver´ıssimo, M. S. S. Pereira, J. Streˇcka, and M. L.Lyra, Kosterlitz-Thouless and Gaussian criticalities in amixed spin-(1/2, 5/2, 1/2) branched chain with exchangeanisotropy, Physical Review B , 134408 (2019).[45] R. R. Montenegro-Filho, F. S. Matias, and M. D.Coutinho-Filho, Topology of many-body edge and ex-tended quantum states in an open spin chain: 1/3plateau, Kosterlitz-Thouless transition, and Luttingerliquid, Physical Review B , 035137 (2020).[46] A. M. S. Macˆedo, M. C. dos Santos, M. D. Coutinho-Filho, and C. A. Macˆedo, Magnetism and phase separa-tion in polymeric hubbard chains, Physical Review Let-ters , 1851 (1995).[47] R. R. Montenegro-Filho and M. D. Coutinho-Filho,Doped ab hubbard chain: Spiral, nagaoka andresonating-valence-bond states, phase separation, andluttinger-liquid behavior, Physical Review B , 125117(2006).[48] G. Sierra, M. A. Mart´ın-Delgado, S. R. White, D. J.Scalapino, and J. Dukelsky, Diagonal ladders: A classof models for strongly coupled electron systems, Phys.Rev. B , 7973 (1999).[49] R. R. Montenegro-Filho and M. D. Coutinho-Filho, Mag-netic and nonmagnetic phases in doped ab t-j hubbardchains, Phys. Rev. B , 115123 (2014).[50] K. Kobayashi, M. Okumura, S. Yamada, M. Machida,and H. Aoki, Superconductivity in repulsively interactingfermions on a diamond chain: Flat-band-induced pairing,Phys. Rev. B , 214501 (2016).[51] K. Hida, Magnetic Properties of the Spin-1/2Ferromagnetic-Ferromagnetic-Antiferromagnetic Trimer-ized Heisenberg Chain, J. Phys. Soc. Jpn. , 2359(1994).[52] K. Takano, K. Kubo, and H. Sakamoto, Ground stateswith cluster structures in a frustrated Heisenberg chain,Journal of Physics: Condensed Matter , 6405 (1996).[53] R. R. Montenegro-Filho and M. D. Coutinho-Filho,Frustration-induced quantum phase transitions in aquasi-one-dimensional ferrimagnet: Hard-core bosonmap and the tonks-girardeau limit, Physical Review B , 014418 (2008).[54] N. B. Ivanov, Spin models of quasi-1d quantum ferrimag-nets with competing interactions, Condens. Matter Phys. , 435 (2009).[55] T. Shimokawa and H. Nakano, Frustration-Induced Fer-rimagnetism in S = 1/2 Heisenberg Spin Chain, J. Phys.Soc. Jpn. , 043703 (2011).[56] S. C. Furuya and T. Giamarchi, Spontaneously magne-tized Tomonaga-Luttinger liquid in frustrated quantumantiferromagnets, Physical Review B , 1 (2014).[57] F. Amiri, G. Sun, H. J. Mikeska, and T. Vekua, Ground-state phases of a rung-alternated spin- 12 Heisenberg lad-der, Physical Review B , 1 (2015). [58] K. Sekiguchi and K. Hida, Partial Ferrimagnetism in S= 1/2 Heisenberg Ladders with a Ferromagnetic Leg, anAntiferromagnetic Leg, and Antiferromagnetic Rungs, J.Phys. Soc. Jpn. , 084706 (2017).[59] S. K. Pati, S. Ramasesha, and D. Sen, A density ma-trix renormalization group study of low-energy exci-tations and low-temperature properties of alternatingspin systems, Journal of Physics: Condensed Matter , 8707 (1997); Low-lying excited states and low-temperature properties of an alternating spin-1–spin-1/2chain: A density-matrix renormalization-group study,Physical Review B , 8894 (1997).[60] K. Maisinger, U. Schollw¨ock, S. Brehmer, H. J. Mikeska,and S. Yamamoto, Thermodynamics of the (1, 1/2) fer-rimagnet in finite magnetic fields, Physical Review B ,R5908 (1998).[61] A. S. F. Ten´orio, R. R. Montenegro-Filho, and M. D.Coutinho-Filho, Quantum phase transitions in alter-nating spin-1/2–spin-5/2 heisenberg chains, Journal ofPhysics: Condensed Matter , 506003 (2011).[62] J. Streˇcka and T. Verkholyak, Magnetic Signatures ofQuantum Critical Points of the Ferrimagnetic MixedSpin-(1/2, S) Heisenberg Chains at Finite Temperatures,Journal of Low Temperature Physics , 712 (2017).[63] J. Streˇcka, Breakdown of a Magnetization Plateau in Fer-rimagnetic Mixed Spin-(1/2,S) Heisenberg Chains due toa Quantum Phase Transition towards the Luttinger SpinLiquid, Acta Physica Polonica A , 624 (2017).[64] W. M. da Silva and R. R. Montenegro-Filho, Magnetic-field–temperature phase diagram of alternating ferrimag-netic chains: Spin-wave theory from a fully polarized vac-uum, Physical Review B , 214419 (2017).[65] M. Hagiwara, K. Minami, Y. Narumi, K. Tatani, andK. Kindo, Magnetic Properties of a Quantum Ferrimag-net: NiCu(pba)( D 2 O ) 3 · , 2209 (1998).[66] M. Hagiwara, Y. Narumi, K. Minami, K. Tatani, andK. Kindo, Magnetization Process of the S = 1/2 and 1Ferrimagnetic Chain and Dimer, Journal of the PhysicalSociety of Japan , 2214 (1999).[67] O. Kahn, Y. Pei, M. Verdaguer, J. P. Renard, andJ. Sletten, Magnetic ordering of manganese(II) cop-per(II) bimetallic chains; design of a molecular basedferromagnet, Journal of the American Chemical Society , 782 (1988).[68] A. Gleizes and M. Verdaguer, Ordered magnetic bimetal-lic chains: a novel class of one-dimensional compounds,Journal of the American Chemical Society , 7373(1981).[69] M. Verdaguer, A. Gleizes, J. P. Renard, andJ. Seiden, Susceptibility and magnetization ofCuMn(S C O ) · O, Physical Review B ,5144 (1984).[70] P. J. Van Koningsbruggen, O. Kahn, K. Nakatani, Y. Pei,J. P. Renard, M. Drillon, and P. Legoll, Magnetism of A-copper(II) bimetallic chain compounds (A = iron, cobalt,nickel): one- and three-dimensional behaviors, InorganicChemistry , 3325 (1990).[71] H. Yamaguchi, T. Okita, Y. Iwasaki, Y. Kono, N. Ue-moto, Y. Hosokoshi, T. Kida, T. Kawakami, A. Mat-suo, and M. Hagiwara, Experimental realization of Lieb-Mattis plateau in a quantum spin chain, Scientific Re-ports , 9193 (2020); H. Yamaguchi, Y. Iwasaki,Y. Kono, T. Okita, A. Matsuo, M. Akaki, M. Hagiwara, and Y. Hosokoshi, Low-energy magnetic excitations inthe mixed spin-(1/2, 5/2) chain, Physical Review B ,10.1103/PhysRevB.102.060408 (2020).[72] Y. Noriki and S. Yamamoto, Modified Spin-Wave The-ory on Low-Dimensional Heisenberg Ferrimagnets: ANew Robust Formulation, J. Phys. Soc. Jpn. , 034714(2017).[73] S. Brehmer, H.-J. Mikeska, and S. Yamamoto, Low-temperature properties of quantum antiferromagneticchains with alternating spins and, Journal of Physics:Condensed Matter , 3921 (1997).[74] S. Yamamoto and T. Fukui, Thermodynamic propertiesof heisenberg ferrimagnetic spin chains: Ferromagnetic-antiferromagnetic crossover, Physical Review B ,14008 (1998).[75] S. Yamamoto, S. Brehmer, and H.-J. Mikeska, Elemen-tary excitations of heisenberg ferrimagnetic spin chains,Physical Review B , 13610 (1998).[76] S. Yamamoto, T. Fukui, K. Maisinger, andU. Schollw¨ock, Combination of ferromagnetic andantiferromagnetic features in heisenberg ferrimagnets,Journal of Physics: Condensed Matter , 11033 (1998).[77] N. B. Ivanov, Magnon dispersions in quantum Heisen-berg ferrimagnetic chains at zero temperature, PhysicalReview B , 3271 (2000).[78] S. Yamamoto, Bosonic representation of one-dimensionalheisenberg ferrimagnets, Physical Review B , 064426(2004).[79] S. R. White, Density-matrix algorithms for quantumrenormalization groups, Physical Review B , 10345(1993). [80] S. R. White, Density matrix formulation for quantumrenormalization groups, Physical Review Letters ,2863 (1992).[81] M. J. Rice and E. J. Mele, Elementary Excitations of aLinearly Conjugated Diatomic Polymer, Physical ReviewLetters , 1455 (1982).[82] N. R. Cooper, J. Dalibard, and I. B. Spielman, Topologi-cal bands for ultracold atoms, Reviews of Modern Physics , 015005 (2019).[83] Y.-T. Lin, D. M. Kennes, M. Pletyukhov, C. S. Weber,H. Schoeller, and V. Meden, Interacting Rice-Mele model:Bulk and boundaries, Physical Review B , 085122(2020).[84] M. Takahashi, Few-dimensional heisenberg ferromagnetsat low temperature, Physical Review Letters , 168(1987).[85] B. Bauer, L. D. Carr, H. G. Evertz, A. Feiguin,J. Freire, S. Fuchs, L. Gamper, J. Gukelberger, E. Gull,S. Guertler, A. Hehn, R. Igarashi, S. V. Isakov,D. Koop, P. N. Ma, P. Mates, H. Matsuo, O. Parcol-let, G. Paw lowski, J. D. Picon, L. Pollet, E. Santos,V. W. Scarola, U. Schollw¨ock, C. Silva, B. Surer, S. Todo,S. Trebst, M. Troyer, M. L. Wall, P. Werner, and S. Wes-sel, The ALPS project release 2.0: open source softwarefor strongly correlated systems, J. Stat. Mech.: TheoryExp. (05), P05001.[86] T. Sakai and S. Yamamoto, Coexistent quantumand classical aspects of magnetization plateaux inalternating-spin chains, Journal of Physics CondensedMatter12