Magnetic field - temperature phase diagram of ferrimagnetic alternating chains: spin-wave theory from a fully polarized vacuum
aa r X i v : . [ c ond - m a t . s t r- e l ] A ug Magnetic field - temperature phase diagram of ferrimagnetic alternating chains:spin-wave theory from a fully polarized vacuum
W. M. da Silva and R. R. Montenegro-Filho
Laborat´orio de F´ısica Te´orica e Computacional, Departamento de F´ısica,Universidade Federal de Pernambuco, 50760-901 Recife-PE, Brasil (Dated: August 29, 2018)Quantum critical (QC) phenomena can be accessed by studying quantum magnets under an ap-plied magnetic field ( B ). The QC points are located at the endpoints of magnetization plateaus andseparate gapped and gapless phases. In one dimension, the low-energy excitations of the gaplessphase form a Luttinger liquid (LL), and crossover lines bound insulating (plateau) and LL regimes, aswell as the QC regime. Alternating ferrimagnetic chains have a spontaneous magnetization at T = 0and gapped excitations at zero field. Besides the plateau at the fully polarized (FP) magnetization;due to the gap, there is another magnetization plateau at the ferrimagnetic (FRI) magnetization.We develop spin-wave theories to study the thermal properties of these chains under an appliedmagnetic field: one from the FRI classical state, and other from the FP state, comparing theirresults with quantum Monte Carlo data. We deepen the theory from the FP state, obtaining thecrossover lines in the T vs. B low- T phase diagram. In particular, from local extreme points inthe susceptibility and magnetization curves, we identify the crossover between an LL regime formedby excitations from the FRI state to another built from excitations of the FP state. These two LLregimes are bounded by an asymmetric dome-like crossover line, as observed in the phase diagramof other quantum magnets under an applied magnetic field. I. INTRODUCTION
The theory of quantum phase transitions providesa framework from which the low-temperature behav-ior of many condensed-matter systems can be under-stood. The quantum critical point separates an insu-lating gapped phase and a gapless conducting phase.Of particular importance are magnetic insulators , forwhich the quantum critical regime can be experimentallyaccessed through an applied magnetic field. In these sys-tems, the gapped phases are associated to magnetizationplateaus in the magnetization curves.In one dimension, magnetization plateaus can be un-derstood as a topological effect through the Oshikawa,Yamanaka, and Affleck (OYA) argument , which gener-alizes the Lieb-Schultz-Mattis theorem . The OYA ar-gument asserts that a magnetization plateau is possibleonly if ( S u − m u ) = integer, where m u is the ground-statemagnetization and S u is the sum of the spins in a unitperiod of the ground state, respectively. If the groundstate does not present spontaneous translation symme-try breaking, S u is equal to the fully polarized magneti-zation per unit cell, while m u is the magnetization perunit cell of the system. The OYA argument was furtherextended to models in higher dimensions and to chargedegrees of freedom.Due to the gap closing a magnon excitation, the end-points of magnetization plateaus are quantum criticalpoints. In three-dimensional systems, this transitionis in the same universality class of the Bose-Einsteincondensation and was studied in a variety of magneticinsulators . In the magnetic system, the magnetiza-tion and the magnetic field play the role of the bosondensity and of the chemical potential, respectively, ofthe bosonic model. In one dimension the mapping to a hard-core boson model or a spinless fermion system implies a square-root singularity in the magnetizationcurve: m ∼ p | B − B c | as B → B c ; and, if three-dimensional couplings are present, the condensate canbe stabilized at temperatures below that of the three-dimensional ordering .Exactly at the quantum critical field, the magnonshave a classical dispersion relation, ω ∼ q , where q isthe lattice wave-vector. In one dimension, this quantumcritical field separates a gapped phase from a gapless Lut-tinger liquid (LL) phase , with excitations showinga linear dispersion relation, ω ∼ q . The predictions ofthe Luttinger liquid theory in magnetic insulators witha magnetic field, including the quantum critical regime,were investigated in many materials . For finite tem-peratures and B ≈ B c , the quantum critical regime isobserved, and the crossover line to the LL regime isgiven by T ( B ) ∼ a | B − B c | , with a universal, model-independent, coefficient a .One-dimensional ferrimagnets show spontaneousmagnetization at T = 0, as expected from the Lieb andMattis theorem , and a gap in the excitation spectrum isresponsible for a magnetization plateau in their magneti-zation curves at the ground-state magnetization value.In zero field, the critical properties in the vicinity ofthe thermal critical point at T = 0 were studied in theisotropic and anisotropic cases . Interesting physicsemerges through the introduction of destabilizing factorsof the ferrimagnetic state, such as doping or geo-metric frustration . The spin-wave theory of fer-rimagnetic chains was developed from the classicalferrimagnetic ground state, considering free and interact-ing magnons, with emphasis on zero-field properties. Themagnetization curves of these systems under an appliedmagnetic field were discussed mainly through numericalmethods .In this work, we investigate the spin-wave theory of fer-rimagnetic alternating chains at low temperatures andin the presence of a magnetic field. We compare someresults with quantum Monte Carlo (QMC) data, ob-tained using the stochastic series expansion method codefrom the Algorithms and Libraries for Physics Simula-tions (ALPS) project , with 1 × Monte Carlo steps.We consider spin-wave excitations from the ferrimagneticand fully polarized classical states. In the ferrimagneticcase, we consider interacting spin-waves, while in thefully polarized, only free spin-waves are discussed. Con-sidering the whole values of magnetization, from zero tosaturation, the two approaches present similar deviationsfrom the QMC data. We deepen the theory from the fer-romagnetic ground state and obtain the crossover linesbounding the plateau and LL regimes. In particular,we show that susceptibility and magnetization data canbe used to identify a crossover between two LL regimes,one built from excitations of the ferrimagnetic magneticstate, and the other from the fully polarized one.This paper is organized as follows. In Sec. II wepresent the Hamiltonian model and discuss the mag-netization curves from QMC calculations. In Sec. IIIthe spin-wave theories from the FRI and FP classicalstates are discussed, particularly the methodology usedto obtain the respective magnetization curves with a fi-nite temperature, and make a comparison between theirresults and QMC data. In Sec. IV, we study LL andplateau regimes at finite temperature through the freespin-wave (FSW) theory from the FP vacuum (FSW-FPv). Finally, in Sec. V we summarize our results andsketch the T - B phase diagram from the FSW-FPv theoryof the alternating (1/2,1) spin chain. II. MODEL HAMILTONIAN AND QMCMAGNETIZATION CURVES
An alternating spin ( s , S ) chain has two kinds of spin, S and s , alternating on a ring with antiferromagnetic su-perexchange coupling J between nearest neighbors, anddescribed by the Hamiltonian H = J N X j =1 (cid:16) s j · S j + s j · S j +1 (cid:17) − B N X j ( S zj + s zj ) , (1)where B is the magnetic field and N denotes the numberof unit cells. We assume S > s and consider equal g -factors for all spins, defining gµ B = 1, where µ B is theBohr magneton. The magnetization per unit cell is givenby m = N X j ( S zj + s zj ) . (2)In Fig. 1 we show QMC results for m ( B ) for the (1/2,1) chain in the low- T regime. At T = 0, m ( B ) presents FIG. 1. (color online). Magnetization plateaus at finite tem-perature, Luttinger liquid phase and crossovers: QuantumMonte Carlo (QMC) data. Magnetization per cell m and thesusceptibility χ = ∂m/∂B as a function of magnetic field B for an alternating ( s = 1 / S = 1) chain with N = 256 unitcells and the indicated values of temperature T . The criticalendpoint of the ferrimagnetic (FRI) and the fully polarized(FP) plateaus are B c,F RI = 1 . J and B c,F P = 3 J , respec-tively. The presence of the FRI and FP plateaus, and theregion dominated by Luttinger liquid (LL) regime is a com-mon feature for all values of s and S , with S > s . As T → χ → ∞ at the critical values of B ; for T &
0, local maximain the χ curves marks the crossover from the LL regime tothe quantum critical regime. The local minimum in the χ curve (dashed line) between B c,F RI and B c,F P separates theLL regime into two regions: one with excitations from theFRI state, LL ; the other with excitations from the FP state,LL . two magnetization plateaus: the ferrimagnetic (FRI), at m F RI = ( S − s ), and the fully polarized (FP) one, at m F P = s + S . In particular, at T = 0, m = m F RI for B = 0, with a gapless Goldstone mode. There are quan-tum phase transitions at the endpoint of the plateaus: B = B c,F RI and B = B c,F P , respectively; which havethe values B c,F RI = 1 . J and B c,F P = 3 . J for the(1 /
2, 1) chain. At the critical fields, there is a tran-sition from a gapped plateau phase to a gapless Lut-tinger liquid (LL) phase, as B → B c,F RI from magneticfields B < B c,F RI , or B → B c,F P from magnetic fields B > B c,F P . In the LL phase, the excitations have alinear dispersion relation, ω ∼ q , and present critical(power-law) transverse spin correlations. Exactly at thecritical fields, the excitations have a classical dispersionrelation ω ∼ q and in the high diluted limit can be repre-sented by a hard-core boson model or a spinless fermionmodel. Hence, the magnetization has a square-root be-havior m ∼ p | B − B c | and a diverging susceptibility χ = ∂m/∂B ∼ / p | B − B c | as B → B c .For finite- T , but T →
0, the magnetization m = 0 for B = 0, since the system is one-dimensional. Gappedmagnetic excitations are thermally activated and theplateau widths reduce. The susceptibility shows localmaxima, with distinct amplitudes, at B ≈ B c,F RI and B ≈ B c,F P marking the crossover between the LL regime,where the excitations have a linear behavior, ω ∼ q , tothe quantum critical regime, for which ω ∼ q . We candefine the local minimum in the χ curve, at B ≡ B i ,as a crossover between the region where the excitationsare predominantly from the FRI state, denoted by LL inFig. 1, and that where the excitations are predominantlyfrom the FP state, denoted by LL in Fig. 1. In partic-ular, for B ≈ B i , the magnetization curve has its morerobust value and behavior as the temperature increases,showing that the LL phase is more robust for B ≈ B i . III. SPIN-WAVE THEORY
The ferrimagnetic arrangement of classical spins is anatural choice of vacuum to study quantum ferrimagnetsthrough free spin-wave (FSW) theory , if we want tostudy excitations from the quantum ground state. Twotypes of magnon excitations are obtained, one ferromag-netic, which decreases the ground state spin by one unit,and the other antiferromagnetic, increasing the groundstate spin by one unit. In particular, the antiferromag-netic excitation has a finite gap ∆, which implies theexpected magnetization plateau at m = S − s and T = 0.However, at this linear approximation, quantum fluctua-tions are underestimated, giving poor results for the valueof antiferromagnetic gap, and other quantities, like theaverage spin per site.When one-dimensional ferromagnets are studiedthrough the linear spin-wave theory at finite tempera-tures, a diverging zero-field magnetization is obtained forany value of T . Takahashi modified the theoryby imposing a constraint on the zero-field magnetizationand an effective chemical potential in the thermal bosondistribution. This so-called modified spin-wave theorydescribes very well the low-temperature thermodynamicsof one-dimensional ferromagnets, and was further suc-cessfully adapted to other systems, including ferrimag-netic chains . In the case of ferrimagnets, the introduc-tion of the magnetization constraint in the bosonic dis-tribution, with the linear spin-wave dispersion relationsgives an excellent description of the low- T behavior. Thedescription of the intermediate- T regime can be improvedby changing the constraint .In this Section, we discuss interacting spin-wave the-ory using a ferrimagnetic vacuum (ISW-FRIv) for B = 0and T = 0, with the modified spin-wave approach (Taka-hashi’s constraint); and free spin-wave theory from a fullypolarized vacuum (FSW-FPv), also for B = 0 and T = 0. FIG. 2. (color online). Interacting spin-wave (ISW) magnonbranches from the classical ferrimagnetic vacuum (FRIv) - cal-culating the thermodynamic properties. (a) The classical fer-rimagnetic vacuum of the ( s , S ) chain. (b) Magnon dispersionrelations for the ( s = 1 / S = 1) chain with B = 0. There areferromagnetic and antiferromagnetic magnons, carrying spin∆ S z = − S z = 1, respectively. The values of the criti-cal fields are B ( ISW-FRIv ) c, FRI = 1 . J and B ( ISW-FRIv ) c, FP = 2 . J . Tocalculate the thermodynamic functions, the antiferromagnetic(ferromagnetic) magnons occupies their respective bands fol-lowing the Fermi (Bose) distribution function. An effectivechemical potential µ is introduced in the Bose distributionto prevent particle condensation at the k = 0 mode for B = 0 and T →
0. (c) For each value of T , we use a valueof µ such that m = 0 for B = 0. The inset shows that µ ( T → → T →
0. In this limit, both bands are emptyand m = ( S − s ) = 1 /
2, the FRI magnetization.
A. Spin-wave theory - ferrimagnetic vacuum
The Holstein-Primakoff spin-wave theory is developedfrom the classical ground state illustrated in Fig. 2(a),which has the energy E (FRIv) class = − JN sS − B (cid:0) S − s (cid:1) N .The bosonic operators a j ( a † j ) and b j ( b † j ), associated to A and B sites, respectively, have the following relation withthe spin operators (Holstein-Primakoff transformation): S + j = √ S (cid:16) − a † j a j S (cid:17) / a j , and S zj = S − a † j a j ; (3) s + j = b † j √ s (cid:16) − b † j b j s (cid:17) / , and s zj = b † j b j − s. (4)Putting the Hamiltonian (1) in terms of thesebosonic operators, expanding to quadratic order, Fouriertransforming and making the following Bogoliubovtransformation : a k = α k cosh θ k − β † k sinh θ k ,b k = β k cosh θ k − α † k sinh θ k , (5)tanh 2 θ k = 2 √ sSs + S cos (cid:16) k (cid:17) , (6)where k is the lattice wave-vector, the non-interactingspin-wave Hamiltonian is given by H (FSW-FRIv) = E + X k h ω (FRIv) k, − α † k α k + ω (FRIv) k, + β † k β k i . (7)The magnon branches obtained are: ω (FRIv) k,σ = σJ (cid:0) S − s (cid:1) − σB + Jω (FRIv) k , (8)with σ = ± , and ω (FRIv) k = r(cid:0) S − s (cid:1) + 4 sS sin (cid:16) k (cid:17) , (9)while the ground-state energy is E = J X k h ω (FRIv) k − (cid:0) S + s (cid:1)i . (10)The ω (FRIv) k, − modes carry a spin ∆ S z = −
1, having aferromagnetic spin-wave nature, and is gapless for B = 0;while ω (FRIv) k, + modes carry a spin ∆ S z = +1, having anantiferromagnetic spin-wave nature and has a gap ∆ =2 J ( S − s ) at B = 0. For the ( s = 1 / S = 1) chain ,for example, ∆ = 1, although the exact value is 1 . J ;while h S za i = 0 .
695 and h S zb i = − .
195 at T = 0, withthe exact values : h S za i = 0 .
792 and h S zb i = − . , shown in Fig.2(b), are: ˜ ω (FRIv) k,σ = ω (FRIv) k,σ − Jδω (FRIv) k,σ , (11)where δω (FRIv) k,σ = 2Γ ( S + s ) ω (FRIv) k sin ( k/ − Γ √ sS h ω (FRIv) k + σ ( S − s ) i , with Γ = 1 N X k sinh θ k , and (12)Γ = 1 N X k cos( k/
2) sinh θ k cosh θ k . (13)Up to O ( S ), the Hamiltonian is H (ISW-FRIv) = E g + X k (cid:0) ˜ ω (FRIv) k, − α † k α k + ˜ ω (FRIv) k, + β † k β k (cid:1) , (14)where E g = E class + E + E , (15)with E = − JN h Γ + Γ − (cid:16)p S/s + p s/S (cid:17) Γ Γ i . (16) At T = 0, the magnetization as a function of B , shownin Fig. 1 for the ( s = 1 / S = 1) chain, can be under-stood from these ferromagnetic (∆ S z = −
1) and anti-ferromagnetic (∆ S z = +1) magnon modes. For B = 0the two bands are empty and the magnetization is theferrimagnetic one. Increasing the magnetic field, the fer-romagnetic band acquires a gap which increases linearlywith B , while the gap to the antiferromagnetic banddecreases linearly with B . Notice, in particular, thatthe ferromagnetic band is empty for all values of B . At B = B ( ISW-FRIv ) c, FRI / /
2, the k = 0 mode of the an-tiferromagnetic band is the lower energy state, and at B = B ( ISW-FRIv ) c, FRI = ∆ the gap to this mode closes. Thevalue of B ( ISW-FRIv ) c, FRI is B ( ISW-FRIv ) c, FRI = ˜ ω (FRIv) , + = 2( S − s ) (cid:18) √ sS Γ (cid:19) J. (17)In particular, for the ( s = 1 / S = 1) chain, with Γ =0 .
305 and Γ = 0 . B ( ISW-FRIv ) c, FRI = 1 . J , which is veryclose to the exact value (1 . J ).The magnetization for B > ∆ is obtained by consider-ing the antiferromagnetic magnons as hard-core bosons ,or spinless fermions. The magnetization increases with B as the antiferromagnetic band is filled, and saturateswhen the Fermi level reaches the band limit, at k = π .The saturation field is B ( ISW-FRIv ) c, FP = ˜ ω (FRIv) π, + = 2 S − Γ + r Ss Γ ! J, (18)which for the ( s = 1 / S = 1) chain is B ( ISW-FRIv ) c, FP =2 .
74, departing from the exact value 3 J , but much betterthan the free spin wave result: 2 J .
1. Thermodynamics
For
T >
0, ferromagnetic and antiferromagnetic modesare occupied in accord to Bose-Einstein ( n (FRIv) k, − ) andFermi-Dirac ( n (FRIv) k, + ) distributions, respectively, as indi-cated in Fig. 2(a). The magnetization, for example, isgiven by m ( T, B ) = ( S − s ) + 1 N X k ( n (FRIv) k, + − n (FRIv) k, − ) . (19)We notice, however, that with T > B = 0the ferromagnetic band will be thermally activated and m → −∞ as T increases. This problem arises, also,in one-dimensional ferromagnetic chains, and was over-come by Takahashi , in the low- T regime, through theintroduction of an effective chemical potential µ in thebosonic distribution, and a constraint m ( B = 0 , T ) = 0.A similar strategy was applied to one-dimensional ferri-magnetic systems and good results were also obtainedin the low- T regime. The intermediate- T regime, wherethe minimum in the T χ curve of the ferrimagnets areobserved, can be more accurately described if other con-straints are used .Here, for B = 0, we use the simplest constraint m ( T, B = 0) = 0 , (20)since we are interested in the low- T regime, with n (FRIv) k, − = 1 e β [˜ ω (FRIv) k, − − µ ] − , (21) n (FRIv) k, + = 1 e β ˜ ω (FRIv) k, + + 1 . (22)In Fig. 2(b), we present m ( T, B = 0) for the indicatedvalues of T . As discussed, m → −∞ at µ = 0 and thevalue of µ for which the constraint m ( T, B = 0) = 0 issatisfied, monotonically decreases with T , in this low- T regime. A finite µ implies an effective gap for the ferro-magnetic band, with an exponential thermal activation oftheir magnons. In particular, notice that µ ( T →
0) = 0,as expected. To calculate the thermodynamic functionsfor B = 0, we consider the distributions in Eqs. (21) and(22) and use the same value of µ found in the case B = 0: µ ( B, T ) = µ ( B = 0 , T ), for any value of B .The magnetization as a function of B for T = 0,shown in Fig. 1, can be qualitatively understood fromthis theory. For B = 0, the magnetization m = 0,due to the constraint. As B increases, in the region0 < B < B c,F RI /
2, the gap to the ferromagnetic bandincreases, but this band is thermally activated and themagnetization decreases from the m = S − s value. Thiseffect can also be seen from Fig. 2(b). If we move theZeeman term, + B , from the ferromagnetic dispersion re-lation to the chemical potential, ˜ ω (FRIv) k, − → ˜ ω (FRIv) k, − − B and − µ → − ( µ − B ), in Eq. (21), the magnetization valueis the one shown in Fig. 2(b) for µ lower than that of B = 0, and m = 0. From Fig. 2(b), we see that increas-ing B (decreasing µ ) from B = 0 [from µ ( B = 0 , T )],the magnetization rises exponentially to the ferrimag-netic value. For B = B ( FSW, FPv ) c,F RI /
2, the lower energyband is the antiferromagnetic (∆ S z = +1 magnons)fermionic band. This band is thermally activated for[ B ( FSW, FPv ) c,F RI / < B < B c,F RI , and the magnetization ishigher than S − s . The magnetization increases throughthe filling of this band, in accord to the Fermi distribu-tion, up to the saturation value m = s + S , which isexponentially reached. B. Spin-wave theory - fully polarized vacuum
In this section, we study the free spin wave theory froma fully polarized vacuum, illustrated in Fig. 3(a). Weshow that this theory provides a good description of thelow- T physics, and is quantitatively much better than thefree spin wave description from the ferrimagnetic vac-uum. The critical saturation field has an exact value, FIG. 3. (color online). Free spin-wave magnon branches fromthe classical ferromagnetic vacuum - calculating the thermo-dynamic properties. (a) The classical fully polarized vacuumof the ( s , S ) chain. (b) Free spin-wave (FSW) results for themagnon energies relative to the fully polarized vacuum (FPv)for T = 0 and B = 0 for the ( s = 1 / S = 1) chain. In thiscase, both branches are ferromagnetic with magnons carryinga spin ∆ S z = −
1. To calculate the thermodynamic functions,the lower (higher) magnon band is filled following the Fermi(Bose) distribution function. An effective chemical potential µ is introduced in the Bose distribution to prevent particlecondensation at the k = π mode for B = 0 and T →
0. Thecritical fields are B ( FSW, FPv ) c,F RI = 2 . J and B ( FSW, FPv ) c,F P = 3 . J .(c) The chemical potential µ is chosen such that m = 0 for B = 0. The inset shows that µ ( T → → − T →
0. Inthis limit only the lower energy band is occupied, implyingthat m → ( S − s ) = 1 /
2, the ferrimagnetic magnetization, as T → B → while the critical field at the end of the ferrimagneticplateau is B ( FSW, FPv ) c,F RI = 2 J .The Holstein-Primakoff transformation in this case is S + j = √ S (cid:16) − a † j a j S (cid:17) / a j , and S zj = S − a † j a j ;(23) s + j = √ s (cid:16) − b † j b j s (cid:17) / b j , and s zj = s − b † j b j , (24)with the two bosons lowering the site magnetization byone unit. To quadratic order in these bosonic operators,the Hamiltonian of the system, Eq. (1), is H (FSW-FPv) = E (FPv) class + J X j ( − s (cid:16) a † j a j + a † j +1 a j +1 (cid:17) − Sb † j b j + √ sS "(cid:16) a j + a j +1 (cid:17) b † j + (cid:16) a † j + a † j +1 (cid:17) b j + B X j (cid:16) a † j a j + b † j b j (cid:17)) , (25)with E (FPv) class = 2 JN sS − B (cid:0) S + s (cid:1) N . Fourier transformingthe bosonic operators and using the Bogoliubov transfor-mation a † k = α † k cos θ k − β † k sin θ k ; (26) b † k = β † k cos θ k + α † k sin θ k , (27)with tan 2 θ k = 2 √ sSS − s cos (cid:16) k (cid:17) , (28)the Hamiltonian in Eq. 25 is written as H (FSW-FPv) = E (FPv) class + X k h ω (FPv) k, α † k α k + ω (FPv) k, β † k β k i , (29)where the dispersion relations ω (FPv) k,η are ω (FPv) k,η = ( − η +1 r(cid:0) S − s (cid:1) + 4 sS cos (cid:16) k (cid:17) − (cid:0) S + s (cid:1) + B, (30)with η = 0 or 1.To discuss the T = 0 magnetization curve impliedby these spin-wave modes, we present in Fig. 3(b) thedispersion relations ω (FPv) k,η for the ( s = 1 / S = 1)chain and B = B ( FSW, FPv ) c,F P = 2 J ( s + S ) = 3 J . At B = B ( FSW, FPv ) c,F P = B c,F P , both bands are empty, and themagnetization is the fully polarized one. Decreasing B ,the η = 0 band is filled in accord to Fermi-Dirac statis-tics, and the magnetization decreases. The critical fieldat the end point of the ferrimagnetic plateau is obtainedmaking ω (FPv) π, = 0, which implies B ( FSW, FPv ) c,F RI = 2 SJ , equalto 2 J for the ( s = 1 / S = 1) chain. At this value of B ,the η = 0 band is totally filled and m = ( s + S ) −
1, giving1 / s = 1 / S = 1) chain. There is a gap of2( S − s ) J between the η = 0 and η = 1 bands, at k = π ;hence, the bosonic η = 1 band should start to be filledat B = B ( FSW, FPv ) c,F RI − S − s ) J , and the theory does notqualitatively reproduce the T → T , the magnetizationis given by m ( T, B ) = ( S + s ) − N X k [ n (FPv) k, + n (FPv) k, ] , (31)where n (FPv) k, = 1 e βω (FPv) k, + 1 , (32) n (FPv) k, = 1 e β [ ω (FPv) k, − µ ] − . (33)The constraint, which is applied at B = 0, is m ( T, B = 0) = 0 . (34)In Fig. 3(c) we present the magnetization as a function ofthe effective chemical µ for the indicated values of tem-perature. We note that m → −∞ as the temperatureincreases, similarly to the spin-wave theory with the fer-rimagnetic vacuum. However, in this case µ → − T →
0, as shown in Fig. 3(b). Hence, a finite chemical potential µ = − η = 1 bandmust be considered in the T = 0 theory. With this chem-ical potential, the η = 1 band stays empty at T = 0 forany value of B .The thermodynamic functions are calculated using Eq.33, with µ ( T, B ) = µ ( T, B = 0). For finite T , thefermionic η = 0 band is completely filled and the oc-cupation of the η = 1 band is such that m = 0. Con-sidering the low- T regime, as B increases, the energyof the two bands raises, lowering the total occupationof the η = 1 band, since ω (FPv) k, − µ linearly increaseswith B for any k , and m increases. The magnetiza-tion exponentially reaches its value at the ferrimagneticplateau, m = S − s , as B increases, since n (FPv) k, → k and the η = 0 band is completely filled. For[ B ( FSW, FPv ) c,F RI / < B < B ( FSW, FPv ) c,F RI , with [ B ( FSW, FPv ) c,F RI /
2] re-lated to the point B = B c,F RI / η = 0 band decreases from the T = 0 case: n (FPv) k, = 1 for any k , and the magnetization is higher than S − s . The magnetization increases with B , and exponen-tially reaches the fully polarized value at B > B ( FSW, FPv ) c,F P ,since magnons at the η = 0 band are thermally excited. B(J) χ / 4 m FIG. 4. (color online). Comparison between results fromquantum Monte Carlo (QMC) method, N = 256 unit cells,and the two spin-wave approaches for the magnetization percell m and the susceptibility χ : ( s = 1 / S = 1) chainat temperature T = 0 . J/k B ). Results from the inter-acting spin-wave theory from a ferrimagnetic vacuum (ISW-FRIv) and free spin-wave theory from a ferromagnetic vacuum(FSW-FPv) compare well with QMC for B . B c,F RI and B & B c,F P . The maximum in χ related to B c,F RI ( B c,F P ) isbetter localized, compared to QMC, through the ISW-FRIv(FSW-FPv) approach. C. Comparison between QMC data and the twospin-wave approaches
In Fig. 4 we present magnetization and susceptibil-ity χ = ∂m/∂B as a function of B from ISW-FRIv andFSW-FPv theories along with QMC data, at T = 0 . J .Since the ISW-FRIv gives a better result for B c,F RI , thistheory is better in the vicinity of this critical field. Oth-erwise, the FSW-FPv approach is better in the vicinityof B c,F P . Further, the amplitudes of the two peaks in χ ( B ), which marks the crossover to the LL regime, havevalues lower than the ones given by QMC. The differencebetween the amplitudes of the spin-wave approaches andQMC data is related to limitations in the spin-wave theo-ries. Despite it, the description from both spin-wave the-ories are qualitatively excellent, and quantitatively veryacceptable in the low- T regime.Below we calculate the T vs B phase diagram in thelow- T regime from the FSW-FPv theory. We study thecrossover lines between the LL regimes and the quantumcritical regimes; as well as the crossovers lines betweenthe plateau regimes and the quantum critical regimes.We use the FSW-FPv approach since it has essentiallythe same precision of the ISW-FRIv theory, if we considera range of B from 0 to the saturation field; also, thecritical point B c,F P is exact in the FSW-FPv theory. IV. LUTTINGER LIQUID REGIME
In the LL phase, the dispersion relation can be approx-imated by ± v F | k − k F | , where v F is the Fermi velocity.Further, in this regime the magnetization has the form : m = m ( T = 0) − π v F ∂v F ∂B ( k B T ) + O ( T ) . (35)In our case, the Fermi velocity along the η = 0 bandis v F = [ ∂ω (FPv) k, /∂k ] k = k F , with k F calculated from ω (FPv) k, | k = k F = 0.In Fig. 5(a) we present v F as a function of B for the(1/2,1) chain. Near the critical fields, | ∂v F /∂B | is largeand v F little. For a fixed B & B ( FSW, FPv ) c,F RI , as shown inFig. 5(b), the magnetization presents a fast decay fromthe T = 0 value as T increases. Also, for B . B ( FSW, FPv ) c,F P ,as shown in Figs. 5(c), m increases from m (0). In bothcases, the curvature of the m ( T → B get closer to the critical fields. The crossover tem-perature T ( B ) of the LL regime at a fixed B is definedas the point at which m ( T ) departs from the quadraticbehavior in Eq. (35). So, T ( B ) is taken to be at theminima ( B & B ( FSW, FPv ) c,F RI ) and maxima ( B . B ( FSW, FPv ) c,F P )of the m ( T ) curve . In particular, as B → B c thecrossover line separates the LL regime and the quan-tum critical regime, for which the excitations have aquadratic dispersion relation. In this case, a universal,model independent, straight line k B T ( B ) = a | B − B c | ,with a = 0 . .In the inset of Fig. 5(a), we show that the minimumin the χ ( B ) = ∂m/∂B curve is found at B = B i , avalue of B at which | ∂v F /∂B | = 0. This value of B marks a crossover from the regime where excitations are B(J) v F ( J ) J χ ∂ v F __ ∂ B > 0 ∂ v F __ ∂ B < 0 B = B i ≈ (a) B = B i T = 0.01 J_k B T(J/k B ) (b) (c) (d) T(J/k B ) T(J/k B ) m B(J)01 J χ T = 0.085 (J/k B ) B = B i B = 2.05 JB = 2.10 JB = 2.20 J B = 2.366 JB = 2.95 JB = 2.90 JB = 2.80 J T = 0.085 (J/k B ) FIG. 5. (color online). Results from the free spin-wave ap-proach with the fully polarized vacuum (FSW-FPv). (a)Fermi velocity v F as a function of the magnetic field B and[(b), (c) and (d)] magnetization curves m ( T ). (a) ∂v F /∂B → + ∞ and v F → B → B ( FSW, FPv ) c,F RI = 2 . J , while ∂v F /∂B → −∞ and v F → B → B ( FSW, FPv ) c,F P = 3 . J .As shown in the inset, for B = B i ≈ . J , ∂v F /∂B = 0and the susceptibility χ ( B ) has a minimum at this value of B . (b) m ( T ) for the indicated values of B in the vicinity ofthe critical field B ( FSW, FPv ) c,F RI . (c) m ( T ) for values of B in thevicinity of the critical field B ( FSW, FPv ) c,F P . (d) m ( T ) for B = B i .The m ( T ) curves to order O ( T ), Eq. (35), are shown asdashed lines in (b) and (c) for the corresponding values of B ,arrows indicate local extreme points in m ( T ), which are usedas a criterium to identify the LL regime. The inset in (d)shows that the minimum in m ( T ) is associated to the localminimum in χ ( B ), which is found between the two criticalfields. predominantly from the FRI critical state to the regimewhere they come from the FP critical state. At B = B i ,the Fermi wave-vector is at the inflection point of thedispersion curve ( d ω (FPv) k, /dk = 0), since ∂v F ∂B = " d ω (FPv) k, dk k = k F (cid:18) ∂k F ∂B (cid:19) , (36)and k F increases monotonically with B between the crit-ical fields. If the value of k at the inflection point is k i ,we can calculate B i from the equation ω (FPv) k i , = 0. For the(1/2,1) chain, for example, B i = 2 . J and is indicatedin Fig. 5(a).At B = B i , ∂v F /∂B = 0 and the quadratic term in T(J/k B ) m T(J/k B ) m T(J/k B ) m B = 1.80 JB = 1.85 J
B(J) T ( J / k B ) B = 1.95 JB = 2.25 J B = 2.95 JB = 2.90 JB = 2.80 J (a) (b)(c) (d)
FSW-FPv minima(shifted)QMC minimaQMC maximaFSW-FPvmaxima a | B - B c , F R I | a | B - B c , FP | B > ~ B c,FRI B < ~ B c,FP FIG. 6. (color online). Magnetization per cell m ( T ) with fixed B : calculating the crossover lines bounding the Luttinger liq-uid regime. Quantum Monte Carlo (QMC) results for themagnetization curves m ( T ) and the crossover lines for a sys-tem with N = 128. (a) m ( T ) for values of B in the vicinity ofthe critical field B c,F RI = 1 . J . (b) m ( T ) for values of B inthe vicinity of the critical field B c,F P = 3 . J . (c) m ( T ) for avalue of B such that ∂χ/∂B ≈ T = 0 and inside the Lut-tinger liquid phase, dashed line in Fig. 1. (d) Local extremepoints of m ( T ) curves from QMC and free spin-wave from thefully polarized vacuum (FSW-FPv). In the case of the FSW-FPv local minima, we shift B by B c,F RI − B ( FSW, FPv ) c,F RI ≈ . J .The exact crossover straight lines as T →
0, extended in thefigure for better visualization: a | B − B c,F RI | and a | B − B c,F P | ,with a = 0 . T = 0 . m ( T ). Eq. (35) is absent. So, the more stable, against T , LLregion is found for B ≈ B i . Since the crossover temper-atures T ( B ) → T ( B ) linehas an asymmetric dome-like profile, which is a conse-quence of the v F curve, shown in Fig. 5(a) for the caseof the (1/2,1) chain, and is also observed in other quan-tum magnets .A minimum in the m ( T ) curve is also observed for B = B i , due to the O ( T ) in Eq. (35), as shown in Fig. 5(d).In this case, however, this extreme point is associatedwith the minimum in the χ ( B ) curve, at B = B i , asshown in the inset of Fig. 5(d).In Fig. 6 we show m ( T ) curves for the (1/2,1) chaincalculated with QMC method to discuss the qualitativelyagreement between these almost exact results and theconclusions from the spin-wave theory. In Figs. 6(a) and(b), we show the minimum (maximum) in the m ( T ) curvefor B & B c,F RI = 1 . J ( B . B c,F P = 3 J ). In Fig. 6(c),we calculate m ( T ) for a value of B in the vicinity of theminimum in the χ ( B ) curve, B = B i . Using the datain Fig. 1, it is located at B i = (2 . ± . J , and isindicated as a dashed line in that figure. As shown inFig. 6(c), the m ( T →
0) curve is also flat, as in Fig.5(d), for B = 2 . J . The minimum in the m ( T ) curve appears at T ≈ . J . As can be observed in the T = 0 . J susceptibility curve in Fig. 1, it is also associated withthe minimum in the χ ( B ) curve, at B ≈ B i .In Fig. 6(d), we compare the position of the localextreme points in the m ( T ) curves from QMC and FSW-FPv methods. The values of B at the minima of m ( T )were translated by B c,F RI − B ( FSW, FPv ) c,F RI ≈ . J . The linesfor the maxima in m ( T ) from both methods are in verygood agreement since the FSW-FPv is almost exact for T →
0, due to the low density of excited magnons in thistemperature regime. Otherwise, the minima from bothmethods do not compare well, except for T →
0, whichis dominated by the critical point.
B(J) C ( k B ) / T ( J / k B ) B = B i T(J/k B ) T → ∝ T LL regime B = B i LL phase (T = 0) B )00.050.1 C ( k B ) FIG. 7. (color online). Specific heat from the free spin-wavetheory from a fully polarized vacuum (FSW-FPv) for T → C ∼ T as T →
0, and
C/T is approximately constant for B ≈ B i = 2 . J . The in-set shows this linear behavior of C at B = B i . The crossoverfrom the T = 0 insulating plateau regime to the gapless quan-tum critical regime, at local maxima, are indicated by arrows. We determine the crossover lines between the LL andplateau regimes through specific heat data, C ( B ). InFig. 7 we present FSW-FPv results for C ( B ) in the low- T regime. In the LL phase, at T = 0, the specific heat C ∼ T as T →
0, and
C/T is approximately constantin the LL regime, as shown in Fig. 7. The range of B near B = B i is the more robust for this regime, andwe present in the inset of Fig. 7 the linear behaviorof C as a function of T . For B . B ( FSW, FPv ) c,F RI or B & B ( FSW, FPv ) c,F P , the excitations are exponentially activated andthe crossover to the quantum critical regime is marked bya local maximum in C ( B ). The points of these crossoverlines, T plateau ( B ) ∼ | B − B c | , are indicated by arrows inFig. 7. The quantum critical regime is bounded by thiscrossover line and that of the LL regime, which pointsappears as a second local maximum near B ( FSW, FPv ) c,F RI and B ( FSW, FPv ) c,F P in Fig. 7. V. SUMMARY AND DISCUSSIONS
FIG. 8. (color online). Spin-wave T − B phase diagram ofthe ( s = 1 / S = 1) chain from the FPv. The quantumcritical points B ( FSW, FPv ) c,F RI = 2 . J and B ( FSW, FPv ) c,F P = 3 . J bound the FRI and FP plateau regions, respectively. Increas-ing temperature, the plateau width decreases and the lines k B T = | B − B ( FSW, FPv ) c,F RI | and k B T = | B − B ( FSW, FPv ) c,F P | limitthe plateau regions for B . B c [ferrimagnetic (FRI) plateau]and B & B c [fully polarized (FP) plateau]. The LL regime hascrossover lines given by a | B − B ( FSW, FPv ) c,F RI | and a | B − B ( FSW, FPv ) c,F P | ,with a = 0 . B → B c , as indicated by local maximaof the susceptibility χ ( B ) = ∂m∂B , χ ( B ) max . Between theselocal maxima, there is a local minimum [ χ ( B ) min ] separatingthe regions under the influence of the B ( FSW, FPv ) c,F RI critical pointand that of the B ( FSW, FPv ) c,F P one. We have calculated the critical properties of alternat-ing ferrimagnetic chains in the presence of a magneticfield from two spin-wave theories. We determine the bet-ter low-energy description of the excitations, consideringthe level of approximation, comparing the results withquantum Monte Carlo data. These ferrimagnetic chainspresent two magnetization ( m ) plateaus, the ferrimag-netic (FRI) plateau, for which m = S − s and the fullypolarized (FP) one, at m = s + S . The first spin-wave the-ory, is an interacting spin-wave (ISW) approach with theFRI classical vacuum, ISW-FRIv. The second method-ology, is a free spin-wave (FSW) calculation from the FPstate, FSW-FPv. In both cases, two bands are obtained. To calculate the finite temperature ( T ) properties of thesystem, one of the bands is considered as a bosonic band,with an effective chemical potential to prevent boson con-densation at B = 0; while the other is considered as ahard-core boson band, with a fermionic one-particle ther-mal distribution. Near the endpoint of the FRI plateau,the ISW-FRIv theory is a better option; while the FSW-FPv is exact for T → T vs. B phase diagram of the system, we deepen thestudy on the FSW-FPv, calculating the finite T crossoverlines bounding the plateau and the Luttinger liquid (LL)regimes.In Fig. 8 we summarize our results in a T vs. B phasediagram, and show specific heat data C/T as a functionof B and T . In the FRI and FP plateau regions the exci-tations are gapped, and ( C/T ) → T →
0. The gapsclose at the quantum critical (QC) fields B ( FSW, FPv ) c,F RI = 2 J and B ( FSW, FPv ) c,F P = 3 J , and local maxima appears in the val-ues of C/T for a fixed T . These local maxima indicatethe crossover between the plateau and the QC regimes,and between the QC and LL regimes. As T →
0, thecrossover line between the plateau and the QC regimes(P-QC line) is a straight line k B T ( B ) = | B − B c | , for B c = B ( FSW, FPv ) c,F RI and B c = B ( FSW, FPv ) c,F P ; while a straightline a | B − B c | , with a model-independent constant a =0 . B = B ( FSW, FPv ) c,F RI [ B = B ( FSW, FPv ) c,F P ] was also cal-culated from local minima (local maxima) in the m ( T )curves: m ( T ) min [ m ( T ) max ]. The LL-QC lines were alsocalculated from local maxima in the susceptibility curve χ ( B ) at fixed T : χ max ( B ).The Luttinger liquid regime can be divided into tworegions, separated by the minimum in the χ ( B ) curvewith a fixed temperature, χ min ( B ). The value of themagnetic field at which this minimum occurs at T = 0, B i , is at the inflection point of the magnon band andchanges little with T . The line m ( T ) min as a functionof B meets the line χ min ( B ) for B ≈ B i . Finally, theLL regime has an asymmetric dome-like profile which isassociated with the Fermi velocity profile as a functionof B at the relevant magnon band, as observed in otherquantum magnets .We acknowledge financial support from Coordena¸c˜aode Aperfei¸coamento de Pessoal de N´ıvel Superior(CAPES), Conselho Nacional de Desenvolvimento Ci-entifico e Tecnol´ogico (CNPq), and Funda¸c˜ao de Am-paro `a Ciˆencia e Tecnologia de Pernambuco (FACEPE),Brazilian agencies, including the PRONEX Program ofFACEPE/CNPq. S. Sachdev,
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