Dynamics of the Distorted Diamond Chain
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p Dynamics of the Distorted Diamond Chain
H.-J. Mikeska and C. Luckmann
Institut f¨ur Theoretische Physik, Universit¨at Hannover, 30167 Hannover, Germany (Dated: October 22, 2018)
Abstract
We present results on the dynamics of the distorted diamond chain, S=1/2 dimers alternatingwith single spins 1/2 and exchange couplings J and J in between. The dynamics in the spin fluid(SF) and tetramer-dimer (TD) phases is investigated numerically by exact diagonalization for upto 24 spins. Representative excitation spectra are presented, both for zero magnetic field and inthe 1/3 plateau phase and the relevant parameters are determined across the phase diagram. Thebehaviour across the SF-TD phase transition line is discussed for the specific heat and for excitationspectra. The relevance of the distorted diamond chain model for the material Cu (CO ) (OH) (azurite) is discussed with particular emphasis on inelastic neutron scattering experiments, a recentsuggestion of one possibly ferromagnetic coupling constant is not confirmed. PACS numbers: 75.10.Jm, 75.10.Pq, 75.40.Gb, 78.70.Nx . INTRODUCTION The distorted diamond chain (DDC) is a one-dimensional (1D) quantum spin model withstructure as shown in Fig. 1(a) and hamiltonian H = N/ X n =1 n J S n +1 S n +2 + J ( S n S n +1 + S n +2 S n +3 ) + J ( S n S n +2 + S n +1 S n +3 ) o . (1)This model with spins 1/2 and all couplings antiferromagnetic may be strongly frustratedowing to the triangular building blocks and has receiced increasing interest in the last decadefor a number of reasons : It has a rich quantum phase diagram as shown in Fig. 1(b) (takenfrom Ref. 3). Here and in the following we choose a representation with J = 1 as energy unitand J , J as variables. This representation emphasizes the symmetry of the model underexchange of J and J . Three quantum phases have been discussed for the ground state ofthe model in zero magnetic field: For J , J ≪ J bonds and nearly free spins between these dimers.The low energy sector is governed by an effective antiferromagnetic Heisenberg chain with N/ J eff inthe following. This leads to the formation of a spin fluid (SF) phase with additional highenergy excitations. For intermediate J , J the ground state dimerizes, forming a twofolddegenerate sequence of alternating tetramers and dimers (TD phase). Finally, for both J , J ,sufficiently large, the ground state is ferrimagnetic with e.g. a ↑↑↓ structure of the unit cellof three spins (which satisfies J and J bonds and frustrates J ). These three phases can beclearly identified already in the symmetric model with J = J in the regimes J = J ≤ / / ≤ J = J ≤ J m (TD phase) and J m ≤ J = J (ferrimagnetic phase) with J m ≈ . .The generalization to the distorted diamond chain J = J leads to an even richer be-haviour including e.g. the trimer Heisenberg antiferromagnetic chain on the J = 0 axiswith the standard HAF for J = 1 as limiting case. The DDC model can be seen as gen-eralization of the HAF with NN and NNN interactions: The critical point of this model at J NNN /J NN ≈ . J NNN /J NN = 1 / / ≤ J = J ≤ J m with simple and exactly known dimerized ground states. 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J J (b) J2 = 1:0 J FRIDSF J1
FIG. 1: (Color online) Structure (a) and phase diagram (b) of the distorted diamond chain model((b) from Ref.3) between the SF and TD phases is a line of Kosterlitz-Thouless phase transitions. The point J = J = 1 / H c producesa magnetization plateau at 1/3 of the total saturation magnetization. The plateau statecorresponds to a fully saturated subsystem of spins 1/2 and all dimers ( J bonds) in theirsinglet state. Further increase of the moment requires breaking up at least one dimer withits large energy scale J = 1 and therefore a correspondingly higher field H ≥ H c (end ofthe plateau). Finally complete saturation is obtained at the field H sat , given by H sat J = 12 + 34 ( J + J ) + 12 (cid:26) J − J ) + 14 ( J + J ) + 1 − ( J + J ) (cid:27) . (2)The critical field H c1 (beginning of the 1/3 plateau) is determined by the level crossingbetween the saturated state of the effective HAF and its ’ferromagnon’ excitation with oneunit of magnetization less. If the mapping to the effective HAF applies, this gives the relation H c1 = 2 J eff . (3)Apart from the theoretical interest in investigating a model which allows to follow thevariation between different quantum phases, the DDC model is of interest since it seems3o describe reasonably well the compound azurite, Cu (CO ) (OH) . Azurite has beeninvestigated in detail by static measurements (magnetization, susceptilibity, specific heat )as well as by high field ESR and the existence of the 1/3 magnetization plateau has beenclearly established. From these experiments, this compound appears to be in the SF phaseclose to the phase transition to the TD phase. Recently, however, the possibility of one of thecouplings J , J being ferromagnetic has been suggested from susceptibility and specific heatdata . Beyond the static properties investigated so far, the dynamics of the DDC and thematerial azurite in particular remain as a challenge to be understood both experimentallyand theoretically: the characteristic feature of the model, namely the presence of two degreesof freedom with different energy scales and their mutual influence will show up most clearly inthe energy spectra of the model. These are best investigated by inelastic neutron scattering(INS) experiments as clearly seen in recent work . For a more complete description bothof the DDC in the full phase diagram and of the results of INS experiments on azurite wepresent in the following results on the dynamics in the SF phase (section II) and in the TDphase (section III).A perturbative approach can be applied to obtain results in the regime J , J ≪ S z tot and wave vector k ) which is not sufficientto cover the excited dimer subspace with its higher energies. Since the elementary cell has3 spins, our system sizes are restricted to 4, 6 and 8 elementary cells. It turns out, however,that this for many aspects is sufficient to obtain reliable results for the infinite system whena finite size analysis is carried through. II. DYNAMICS IN THE SPIN FLUID PHASE
It is helpful to start the discussion from two well known limiting cases:(i) For J = J = 0 the system reduces to N/ N/ N/ fold degeneracy due to the4ree spins results. A magnetic field immediately saturates the free spin system leading to amagnetization of M sat / H c = J the dimers changeto their triplet states saturating the system. This behavior to a certain extent remains validon the symmetry line J = J where the total spins on all J − bonds are independentlyconserved: the ground state as well as the 1/3 magnetization plateau in low field remainunchanged whereas the transition to full saturation is determined by the effective interactionwhich develops between two neighboring dimers in their triplet state and finally leads to aneffective S = 1 chain. Qualitatively, for a large range of parameters the distorted diamondchain can be divided into two subsystems with clearly different energy scales, a low energypart of N/ N/ J = J , the spin 1/2 subsystemdevelops some coupling by polarizing intermediate dimers and the 2 N/ fold degeneracy islifted in favor of an effective Heisenberg chain with exchange J eff . In this regime, excitationsof the DDC remain well separated: they are in the low energy regime with energy scale J eff forming the spinon continuum of the HAF with N/ a , reciprocal lattice vector τ = 2 π/a ) or in the high energy regimewith energy scale J = 1 corresponding to the excitation of a dimer to its triplet state anddevelopping into a dispersive band with width ∆ dimer due to the coupling to the low energyspin subsystem. We will not consider in the following states with more than one exciteddimer.(ii) For J = 1 , J = 0 the system reduces to the Heisenberg antiferromagnet with N spins,forming a spinon continuum in the Brillouin cell with reciprocal lattice vector 3 τ , energyscale 1 and no additional high energy excitations. For 0 < J < a . This results in three excitationbranches (actually continua) which fill the energy range up to ǫ = π with small (for J slightly less then 1) gaps between them and an alternating sequence of minimum, maximumand minimum at wavevector k = π (in the following we will use exclusively the Brillouinzone with reciprocal lattice vector τ , corresponding to the full DDC). With increasing 1 − J ,these trimer bands develop increasingly larger gaps, finally the continuum of the effectiveHAF emerges from the lowest band and the two upper bands conspire to give the dimerexcitations decorated by continua of low energy spinon excitations.5sing this frame the lowest excitations of interest in the following are easily described:(i) the spinon continuum of the effective chain,(ii) the band with one excited dimer above the spinon continuum,(iii) the ’inverted ferromagnon’ i.e. the saturated effective HAF with one spin deviation( S z tot = N/ − S z tot = N/ ǫ (0)dimer ( k ) = 1 + δ (0)dimer + 12 ∆ (0)dimer cos k. (4)More precisely, this excitation is not a single band but a continuum due to the spinoncontinuum of initial states; however, we will only be able to discuss the lower edge of thisexcitation and therefore simplify the notation using Eq.(4). Excitations (iii) and (iv) are therelevant excitations above the 1/3 plateau, we therefore use a notation giving their energiesin finite magnetic fields relative to the plateau ground state with S z tot = N/ ǫ ferrom ( k ) = 12 ∆ ferrom (1 + cos k ) + H − H c1 (5) ǫ (sat)dimer ( k ) = 1 + 12 ( J + J ) + δ (sat)dimer + 12 ∆ (sat)dimer (1 + cos k ) − ( H − H c1 ) . (6)The quantities ∆ and δ give the widths, resp. the nontrivial contributions to the minimumenergy (at k = π ) of the corresponding bands. The cosine dispersion of course is only validin lowest order and will change to a more complicated expression for real systems.In the model of an effective HAF for the low energy regime its exchange constant J eff determines the low energy spinon (i) and the inverted ferromagnon spectrum: ∆ ferrom = 2 J eff .Combined with ǫ (sat)dimer it is also sufficient to give the range of the plateau phase and tocharacterize its dynamics: In the presence of a finite field, spectra are identical to thosewithout field except for the shifts and splittings due to Zeeman energies. This establishesstates with an increasingly larger total spin S tot (in their maximum z − component) as groundstates. The plateau begins at the field H c1 when the S tot = N/ S tot = N/ − π ) by the external field, leading to H c = 2 J eff . The lowest excitation6or the plateau dynamics close to the field H c then is the ferromagnon of Eq.(5). When thefield is increased across the plateau regime, the S tot = N/ H c implying H c2 = H c1 + 1 + 12 ( J + J ) + δ satdimer . (7)The parameters determining the spectra can be calculated in perturbation theory in J , J and to lowest order are determined by the level spectrum of the general ( J = J ) tetramerwith 4 spins 0 . . . 3. This spectrum includes the lowest order information about J eff in thesinglet-triplet splitting of spins 0 and 3 and about ∆ (sat)dimer in the amplitude for the process | ↑ s i → | ↓ t + i ( s and t + are noninteracting dimer states) which determines the propagationof an excited dimer triplet. The results to lowest order in J , J are:2 J eff = ∆ ferrom = 2∆ satdimer = ( J − J ) δ satdimer = − ( J − J ) H c = 1 + 12 ( J + J ) . (8) J eff has been calculated in straightforward perturbation theory up to fifth order , basedon the splitting of the general tetramer into singlet and triplet states we have obtaineda result which accounts partly also for higher orders and allows reasonable estimates for | J − J | ≪
1, but finite J + J : J eff = 12 n(cid:8) ( J + J − + 3 ( J − J ) (cid:9) + J + J − (cid:8) J − J ) (cid:9) o . (9)In lowest order perturbation theory the relevant parameter, in addition to the energy scaleset by J and to J + J , is the exchange of the effective HAF determined by ( J − J ) andmany characteristic quantities of the DDC would be related by simple numerical factors if themapping were perfect. Whereas these perturbational results allow to discuss the dynamics inprinciple, J , J values of interest for the bulk of the phase diagram as well as for a materialsuch as azurite are beyond the validity of perturbation theory. We therefore present in thefollowing results from the numerical approaches described above. This will allow us to followthe essential aspects of the dynamics in the intermediate regime, i.e. through all of the SFphase. In Figs. 2 and 3 we show excitation spectra for three sets of exchange parameters:7et (a) represents the case of small couplings, set (b) is for a point in the phase diagramclose to the SF to TD transition thought to be qualitatively representative for azurite andset (c) shows results for the case of one coupling ferromagnetic. The data are obtained bydiagonalizing chains with 24 spins using the Lanczos algorithm.The low energy excitation spectra in zero external field are shown in Fig. 2. Here andin the following, energies are in units of J and wavevectors in units of π/ N = 24,resp. π/ N = 18). The spectra include all levels with S tot = 0 , S tot = 2 (forcompleteness) and S tot = 3. The latter band of excitations is the ’inverted ferromagnon’ anda cosine dispersion approximating the data points is shown for qualitative comparison tothe effective model with its exact cosine dispersion. Fig. 3 shows the excitation bands in amagnetic field H c (beginning of the 1/3 plateau). In magnetic field two Zeeman componentsof the S tot = 4 , k = 0 level are relevant: The S z tot = 4 component turns into the plateauground state, whereas the S z tot = 3 component becomes the top of the inverted ferromagnonband, it is identical to that of Fig. 2 (apart from Zeeman shift) and now the lowest excitation.In addition, Fig. 3 shows the lowest excitation band with S tot = 5 which requires breakingone dimer ( J ) bond. Cosine dispersions as approximation to the data points are includedfor these two bands. For completeness we also show the first full band with S tot = 4 abovethe plateau ground state.Among the data shown, the excitations of interest from an experimental point of view(with large transition matrix elements for e.g. INS) are the spinon continuum in zero fieldand the inverted ferromagnon band as well as the excited dimer band in the plateau field.In addition, in zero field there will be transitions with energy of the order of J to an exciteddimer band with S tot = 1 resulting from breaking a J bond on top of the effective chaingroundstate. This is more difficult to deal with than the excited dimer excitation shown inFig. 3 which is on top of the less complex saturated effective chain state. We will discussthese excitations below, based on calculations of all eigenvalues of a N = 18 chain (see tableIII). States with S tot > S tot = 4 above the plateauground state will be only weakly excited in INS and analogous experiments: in particularstates in the S tot = 4 band are obtained from the saturated state by a virtual excitation | s ↑i → | t + ↓i . They have an excited dimer and an overturned spin (compared to thesaturated state) in the low energy subsystem and thus require two spin flips to be excited.8 e x c it a ti on e n e r gy S_tot = 0S_tot = 1S_tot = 2S_tot = 3S_tot = 4 (plateau) (a) J_2 = 1.0, J_1 = 0.3, J_3 = 0 e x c it a ti on e n e r gy S_tot = 0 S_tot = 1S_tot = 2S_tot = 3 S_tot = 4 (plateau) (b) J_2 = 1.0, J_1 = 0.7, J_3 = 0.3 0 1 2 3 4wave vector00.511.5 e x c it a ti on e n e r gy S_tot = 0 S_tot = 1S_tot = 2S_tot = 3 S_tot = 4 (plateau) (c) J_2 = 1.0, J_1 = -1.0, J_3 = 0.4
FIG. 2: (Color online) Low energy spectrum of the DDC for N = 24 , H = 0, J = 1 . J = 0 . , J = 0, (b) J = 0 . , J = 0 .
3, (c) J = − . , J = 0 . e x c it a ti on e n e r gy S_tot=3S_tot=4 (plateau)S_tot=5
J_2 = 1.0, J_1 = 0.3, J_3 = 0 (N = 24, lowest states for given S_tot) e x c it a ti on e n e r gy S_tot=3S_tot=4 (plateau)S_tot=5
J_2 = 1.0, J_1 = 0.7, J_3 = 0.3 (N = 24, lowest states for given S_tot) e x c it a ti on e n e r gy S_tot=3S_tot=4 (plateau)S_tot=5
J_2 = 1.0, J_1 = -1.0, J_3 = 0.4 (N = 24, lowest states for given S_tot)
FIG. 3: (Color online) Excitation spectrum of the DDC above the 1/3 plateau for N = 24 , H = H c , J = 1 . J = 0 . , J = 0, (b) J = 0 . , J = 0 .
3, (c) J = − . , J = 0 .
10e now discuss how the dynamics changes with varying exchange constants: Set (a)shows the behaviour typical for the weakly coupled DDC: the bands are well separated inenergy and the cosine dispersion is nearly perfect. Set (b) displays what is expected for amaterial such as azurite: in zero field a spinon continuum should be clearly visible whereasin the plateau regime two separate bands dominate the picture. The cosine approximationto the dispersion is less applicable, actually the spectrum of the inverted ferromagnon isclose to linear for smaller wave vectors. For the set (c) which serves as an example for thealternative suggesting one ferromagnetic coupling , the dynamics in zero field is seen to besurprisingly close to that of set (a). This may explain the emergence of the ferromagneticalternative from a discussion of static quantities. However, these two sets lead to stronglydiffering dynamics in finite field as seen by comparing Fig. 3 (b) and (c): The standardantiferromagnetc model (b) implies two well separated bands with rather small widths,whereas the partly ferromagnetic alternative (c) is characterized by an overlap of the twobands and a strong dispersion of the excited dimer band.For a semiquantitative discussion of the low energy dynamics of the DDC the mappingto the model of an effective HAF is rather useful. Therefore we discuss shortly the qualityof this mapping for H = 0 in Fig. 4 for the two parameter sets (a) J = 0 . , J = 0 and (b) J = 0 . , J = 0 .
3. We compare the numerical spectrum for N = 24 (dots) with the energiesof the N = 8 HAF with unity exchange constant (open circles). The energies of the DDChave been scaled by an effective exchange constant J eff , chosen to reproduce the ( N = 8)maximum spinon energy at k = π , S tot = 1. In (b) energy levels of the N = 8 HAF whichare beyond the range of the Lanczos calculation for the DDC model have been omitted fromthe plot to obtain a clearer picture. Whereas the mapping for the small parameter values inset (a) is nearly perfect throughout all of the spectrum, substantial deviations are seen forthe parameter set (b): the low energy spinon part is still reproduced well by the effectivemodel, but the high energy part, in particular the ferromagnon band with S tot = 3 is verydifferent both in energy and in dispersion. A cosine dispersion is only a rough approximationto the spectrum.Thus a quantitative experimental investigation of the dynamics will contribute substan-tially to locating the position of a specific compound in the phase diagram of Fig. 1(b). Fora quantitative overview (and possibly use in determination of coupling parameters from ex-periment) we reduce in the following the information in these spectra to a few characteristic11 e x c it a ti on e n e r gy S_tot = 0 S_tot = 1 S_tot = 2S_tot = 3S_tot = 4 (HAF for N=8) vs (J_1 = 0.3, J_3 = 0 for N = 24)low energy J_eff = 0.06 e x c it a ti on e n e r gy S_tot = 0 S_tot = 1 S_tot = 2S_tot = 3S_tot = 4 (HAF for N=8) vs (J_1=0.7, J_3 =0.3 for N = 24)low energy J_eff = 0.212
FIG. 4: (Color online) Spectrum of the effective HAF (N=8) vs spectrum of the DDC (N=24) for(a) J = 0 . , J = 0 and (b) J = 0 . , J = 0 . numbers to be presented in tables I and II below. From the numerical data we have calcu-lated values for the quantities determined by the effective exchange between the quasi-freespins. We give in table I numbers for J eff determined from the maximum spinon energy at k = π/ , S tot = 1 (when multiplied by 1.7964.., the corresponding energy in the N=8 HAFchain, these numbers lead back to the energy for the DDC model) and for the S tot = 1 spinon12t k = π (gapped due to discreteness). Further we give the width of the ferromagnon band(which would be 2 J eff if the mapping to the effective model were perfect) and the width ofthe dimer band above the plateau with S tot = N/ H c1 ) and end ( H c2 ) of the plateau aswell as the saturation field H sat . In the standard case (actually some exceptions exist closeto the phase transition line) H c1 is identical to the ferromagnon width from table I. Intable II we also give the energy scale which is relevant for an application of the numericalresults to azurite: Using the experimental number H sat = 33 T the value of the coupling J is calculated from Eq. (2) and the values given in table II (in both meV and T ) may serveto obtain energies and fields applying to azurite in standard units. In tables I and II threeregimes of the SF phase are covered: (a) values along the J = 0 axis (i.e. for Heisenbergtrimer model) (b) values along a diagonal path which appears as the most interesting one fordiscussing azurite and (c) tow examples for ferromagnetic coupling. We will discuss belowthe possibility of such an interaction from the point of view of inelastic excitations. Valuesalong the line J = 0 .
6, passing through the phase transition at J ≈ . J eff :(i) the maximum of the effective one spinon dispersion at wave vector π/ ǫ ( π/
2) = π/ J eff ,(ii) the critical field which determines the beginning of the plateau: H c1 = 2 J eff ,(iii) the width of the effective ’inverted ferromagnon’ as the lowest excitation at the be-ginning of the plateau (in fact, its minimum defines H c1 ): ǫ fm (0) − ǫ fm ( π ) = 2 J eff . Thisis automatically equal to (ii) from the definition of H c1 , but the cosine dispersion is anadditional independent property. Since the mapping is only approximate, these quantitiesdiffer as is seen in the numerical data and the differences characterize the quality of themapping. Actually there are more possibilities to extract J eff from the numerical data suchas the energy of the lowest spinon singlet at k = π and the ground state energy (suit-ably extracted from the energy of the saturated subsystem state), but numbers from theseapproaches essentially confirm the picture as it has emerged from the tables above. Theessential conclusion for the real DDC is that the effective coupling J eff depends on energy.The quantities in tables I and II have been calculated for N = 24, but a comparison withresults for N = 12 and N = 18 shows that finite size effects are very small, indeed negligible13 ouplings J , J J eff from J eff from ferromagnon dimer bandspinon maximum spinon at k = π width width0.02, 0.00 2.06 10 − − − − J , J H c H c H sat energy scale J in meV resp. T − . meV ≡ . T . meV ≡ . T . meV ≡ . T . meV ≡ . T . meV ≡ . T . meV ≡ . T . meV ≡ . T . meV ≡ . T . meV ≡ . T . meV ≡ . T TABLE II: DDC parameters related to the characteristic fields and unit of energy for azurite (seetext) w a v e v ec t o r w a v e v ec t o r J_1 = 0.7, J_3 = 0.3J_1 = 0.8, J_3 = 0.2 excited dimer dispersion for H = H_c1combine N=18 and N=24: S^z_tot = N/6 +1
FIG. 5: (Color online) Excited dimer band of the DDC (N=24 and N=18) above the 1/3 plateauground state (in a magnetic field H c ) for J = 0 . , J = 0 . J = 0 . , J = 0 . to the accuracy given. This is due to the fact that e.g. J eff is determined by comparing twofinite systems, the HAF chain with N = 8 and the DDC with N = 24. This is illustrated inFig. 5 where the excited dimer bands for two different sets of couplings are shown combiningresults for N = 24 and N = 18 in the same graph. We conclude that the the effectiveparameters considered so far can be reliably determined for the infinite chain. The situationis different for the last quantity of interest, the dynamics of the one dimer excitations for H = 0. These excitations originate by exciting one dimer from singlet to triplet on top of thespinon continuum of the effective HAF chain (instead of on top of the saturated effectiveHAF chain for ∆ dimer as given above). In the following we give results on this zero fieldbranch as far as numerically possible:At zero field, the excited dimer states form a continuum as well and the infinite chainlimit can only be obtained by extrapolation in 1 /N . With N = 12 and N = 18 as theonly available numbers of spins the extrapolation can only be done for wavevectors k = 0and k = π . The lowest energy levels in our finite systems result from coupling of spinonsat wavevectors 0 (singlet only) and π (singlet and triplet) to the dimer triplet, i.e. weget a band of one singlet, three triplets and one quintuplet. However, the separation both15 ouplings J , J ǫ dimer ( k = 0) ǫ dimer ( k = π )0.30, 0.00 0.861 0.8930.60, 0.05 0.988 0.9780.60, 0.25 0.720 0.7110.65, 0.25 0.854 0.7930.70, 0.30 0.864 0.7760.40, -1.00 1.226 1.255TABLE III: Energies ǫ ( k = 0) and ǫ ( k = π ) in the DDC lowest excited dimer band (onset of thecontinuum). Values extrapolated from N = 12 and N = 18 to infinite N between these multiplets and to the higher levels is due to the finite size, whereas in thethermodynamic limit a continuum with energy of the order of J will result. This is to someextent reflected in the results of finite size extrapolation of the multiplet energies relatedto one excited dimer: There is a clear tendency for the energies to converge to the samevalue. Thus only one energy in this high energy subspace can be given reliably, no reliableinformation can be obtained in this approach about splitting into bands. In table III wegive the energy of the lowest state in the continuum of excited dimers obtained this way fora number of coupling constants. III. DYNAMICS IN THE DIMER-TETRAMER PHASE
The ground state in the tetramer-dimer phase is twofold degenerate and develops fromthe ground state on the symmetry line J = J . On this line the two ground states can bewritten down explicitly (even for finite systems with an arbitrary even number of cells of 3spins). They are given by the alternating sequence of the dimer singlet S and the lowesttetramersinglet T . This allows the two equivalent configurations . . . S T S T S T . . . and . . .
T S T S T S . . . (10)describing the two degenerate ground states.The lowest excitations above these ground states are obtained as solitons which are definedby gluing together the two degenerate ground states in a localized region on the chain. This16ives e.g. the state . . .
S T S ∗ S T . . . (11)where ∗ denotes a free spin. A soliton is possible only with two dimer singlets adjacentto each other and a free spin between them whereas a configuration . . . T T . . . obviouslydoes not exist. On the symmetry line J = J there are N/ J = J . Theproperties of a single soliton can suitably be investigated for chains with an odd numberof cells N/ T S ∗ → ∗
S T (12)with amplitude t leads to the propagation of solitons and the formation of a ground stateband with energy ǫ sol ( k ) = E + 2 t cos 2 k. (13)Thus, from numerical calculations for an odd number of cells the hopping amplitudeis easily determined even when only small systems are available. For infinite chains withperiodic boundary conditions and an even number of cells N/ k + k and k − k , i.e. ǫ ( k ) = 2 E + 4 t cos k cos 2 k . (14)Numerically obtained soliton spectra are shown in Fig. 6 for (a) N = 15 and (b) N = 18for the point J = 0 . , J = 0 .
55 close to the symmetry line in the phase diagram. ǫ sol ( k ) inFig. 6(a) clearly shows the cos 2 k dispersion, whereas the dispersion of ǫ ( k ) in Fig. 6(b) issomewhat more complicated due to the small size of the system. Also shown are the three,resp. four-soliton bands demonstrating the clear division of the spectra into distinct solitonsbands for this nearly symmetric set of couplings (the zero of energy in Fig. 6(a) is takenfrom the noninteracting limit). The situation is analogous to the Ising chain with smalltransverse interactions, the system where the dynamics of magnetic solitons was discussedfirst . Slightly different, the soliton spin 1/2 here is a real spin 1/2 which can be attributed17 e x c it a ti on e n e r gy S_tot = 1/2S_tot = 3/2S_tot = 1/2 (a) J_2 = 1.0, J_1 = 0.6, J_3 = 0.55 : solitons in TD phase (N = 15) e x c it a ti on e n e r gy S_tot = 0 S_tot = 0S_tot = 1S_tot = 2 (b) J_2 = 1.0, J_1 = 0.6, J_3 = 0.55 : solitons in TD phase (N = 18)
FIG. 6: (Color online) Soliton spectrum of the DDC ((a) N=15, (b) N=18) in the TD phase for J = 0 . , J = 0 .
55. The N=15 spectrum is shown in the complete Brillouin zone 0 . . . π todemonstrate the cos 2 k dispersion of the single soliton. to the free electron of the Cu ion between the two dimers forming the domain wall. Forsmall numbers N/ N = 15 and J = 0 . , J = 0 .
55 the hopping amplitude isdeduced as t ≈ . N = 18 the corresponding calculation18as to include the possibility of two neighboring solitons as well as the resulting symmetryeffects and gives a somewhat higher value, t ≈ . N = 24 (when only the lowest 2 soliton band is accessible in Lanczoscalculations) we obtain t ≈ . ± . N = 15within the uncertainty resulting from matching the cosine dispersion for the different wavevectors. IV. CROSSING THE PHASE TRANSITION LINE
In this section we present some data for the specific heat and for the spectrum of low-lying excitations in order to approach the behavior of the system when its coupling constantschange between well defined end points, one in the SF phase, the other one in the TD phase,thus crossing the phase transition line. Evidently, owing to the small system sizes accessibleonly in our calculations, we cannot claim that these data describe correctly the most inter-esting aspect, namely the critical behavior; on the other hand our data for both the specificheat and the spectrum of low-lying excitations set a reasonable frame for the transitionregime, to be filled by more detailed calculations later. In the following we present resultsfor the DDC on the line J = 0 . J . As discussed above, the phase transitionalong this line is of Kosterlitz Thouless type and can be considered as a generalization ofthe phase transition in the HAF with both nearest and next nearest neighbor exchnage. Wetherefore have applied the procedure of Ref. 12 to determine the critical coupling and findthat the phase transition occurs at J ≈ . J for fixed J for three differentvalues of the magnetic field. The data are obtained from the full spectrum for the N = 18chain and therefore cover reliably the complete temperature regime although critical prop-erties near the critical coupling J ≈ .
364 will appear smeared out. In all diagrams weuse a logarithmic temperature scale adequate to the strongly different energy scales. Forall magnetic fields the specific heat exhibits a high temperature peak at T ≈ .
5, whereasthe low temperature properties reflect the structure of the system: For H = 0 and low tem-perature (Fig. 7a) the SF phase is characterized by a continuously increasing contributionto the specific heat, the remnant of the effective Luttinger liquid. With increasing J this19 .01 0.1 1temperature (in units of J_2 = 1)0246 s p ec i f i c h ea t J_3 = 0.05J_3 = 0.32J_3 = 0.44J_3 = 0.55 (a) Specific heat for H = 0, J_3 = 0.60.01 0.1 1temperature (in units of J_2 = 1.0)0246 s p ec i f i c h ea t J_3 = 0.05J_3 = 0.32J_3 = 0.44J_3 = 0.55(b) Specific heat for H = 0.6, J_1 = 0.6 s p ec i f i c h ea t J_3 = 0.05J_3 = 0.32J_3 = 0.44J_3 = 0.55(c) Specific heat for H = 1.3, J_1 = 0.6
FIG. 7: (Color online) Specific heat of the DDC (from all levels for N = 18) for H = 0 (a), H = 0 . H = 1 . J = 0 . J through the phase transition. e x c it a ti on e n e r gy ( o ff s e t by . f o r eac h s p ec t r u m ) J_3 = 0.05J_3 = 0.32J_3 = 0.44J_3 = 0.55
FIG. 8: (Color online) Low-lying spectra ( S tot = 0 (red/full) and S tot = 1 (blue/open)) of theDDC (N=24) for J = 0 . J = 0 . , . , . , .
05. The energy rangefor each spectrum is 0.5. contribution develops gradually into the gapped contribution of the TD phase with charac-teristic shoulders on both sides of the phase transition at J ≈ . H = 0 . H = 1 . J = 0 . J values. It would be interesting to see, using more powerful numerical methods, whether thedevelopment of this peak in the nearly saturated case shows critical properties.Fig. 8 shows a sequence of spectra of low-lying excitations ( S tot = 0 ,
1) in zero magneticfield obtained by the same procedure as the spectra shown in sections II and III. Qualita-tively, the transition from the SF phase with its gapless spinon continuum to the gappedsoliton spectra in the TD phase is clearly seen qualitatively, as far as possible for the limitedsize of the system: with increasing J the second degenerate ground state at k = π emerges,the spinon continuum is compressed into the soliton band and the gapless character dis-21ppears. Excitation spectra in finite magnetic field, in particular for fields in the plateauregime, on the other hand do not show specific variations but rather continuous changesacross the phase diagram without prominent features close to J = 0 . V. CONCLUSIONS
For the S = 1 / J eff . For parameters beyond the validity of a perturbativeapproach, this effective interaction has to be allowed to be energy dependent. The lowestexcitations in the plateau regime are the inverted ferromagnon and the propagating sin-gle dimer triplet excitation with, however, partly strong modifications of the correspondingcosine dispersions. The values of the characteristic parameters ( J eff , extent of the plateauregime, widths of the cosine bands) are given for typical paths crossing the SF phase. Thesedata should allow to decide whether a material such as Cu (CO ) (OH) (azurite) is suffi-ciently well described by the DDC model and, if so, to determine the corresponding cou-plings. The standard assumption for azurite is to take all couplings as antiferromagneticand we have shown that the spectra of low-lying excitations exhibit large and characteristicchanges when the possibility of one ferromagnetic coupling is introduced. We therefore ex-pect that our data will allow to interpret quantitatively experimentla data on azurite. Thisrefers in particular to the results of inelastic neutron scattering experiments . Considering22he present status of such investigations, our results do not confirm the conclusion of atleast one ferromagnetic coupling in azurite. Generally, our results lead us to describe thefollowing signatures when ferromagnetic couplings are present:(i) Whereas the dimer width is roughly 1/2 of the ferromagnon width for af couplings (assuggested by perturbation theory), for ferromagnetic couplings these widths tend to becomeequal.(ii) The sign of the couplings has a marked influence on the relative appearance of the ferro-magnon and the excited dimer band: For couplings J , J = ( − , .
4) these bands above the1/3 plateau overlap, whereas for ( − . , .
02) ferromagnon a swell as dimer width becomevery small and correspondingly H c becomes much smaller than in perturbation theory.The low-lying excitations in the TD phase with its twofold degenerate ground state areshown to be solitons. The width of the one soliton band as determined from the N = 15chain not too far from the symmetry line J = J reproduces well the soliton bands in the N = 24 chain and therefore gives reliably the tunneling amplitude for the soliton propagationin the thermodynamic limit.We have also shown spectra as well as the specific heat on a line across the SF-TD phasetransition. Although the small systems accessible to us do not allow to discuss criticalproperties of the DDC close to this Kosterlitz-Thouless transition, the variation of thedynamical properties through the transition become clear. In particular, only the low energyproperties, determining the behavior of the system at zero field, carry the signature of thephase transition.Generally, the DDC has many features in common with the antiferromagnetic Heisenbergchain with nearest and next-nearest exchange and the SF-TD phase transition is of the sametype as the KT transition at J NNN = 0 . ...J NN in this system. On the other hand, wehave shown that the additional degrees of freedom, resulting from the possibility to excitethe J − dimers to the triplet state, show up clearly in the dynamics. We leave to the futureto investigate the influence of these degrees of frededom on the phase transition using morepowerful anlytical and numerical methods. 23 cknowledgements We wish to thank H. Ohta, H. Kikuchi, K. Rule, S. S¨ullow and D. A. Tennant forstimulating discussions. We gratefully acknowledge that computational facilities for thenumerical calculations were generously provided by the John von Neumann-Institut forComputing at J¨ulich Research Center.
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