EUROPHYSICS LEERS OFFPRIN Vol. 64 Number 1 pp. 111 117 Magnetic susceptibility of an exactly solvable anisotropic spin ladder system A. P. onel, S. R. Dahmen, A. Foerster and A. L. Malvezzi Published under the scientific responsability of the EUROPEAN PHYSICAL SOCIEY Incorporating JOURNAL DE PHYSIQUE LERES LEERE AL NUOVO CIMENO
EUROPHYSICS LEERS
EUROPHYSICS LEERS 1 October 23 Europhys. Lett., 64 (1), pp. 111 117 (23) Magnetic susceptibility of an exactly solvable anisotropic spin ladder system A. P. onel 1,S.R.Dahmen 1,A.Foerster 1 and A. L. Malvezzi 2 1 Instituto de FísicadaUFRGS-Av.BentoGonçalves 9, Porto Alegre, RS, Brazil 2 Departamento de Física da UNESP - Av. Eng. Luiz E. C. Coube s/n Bauru, SP, Brazil (received 2 June 23; accepted 14 July 23) PACS. 7.1.Jm Quantized spin models. PACS. 71.1.Fd Lattice fermion models (Hubbard model, etc.). PACS. 3.6.Fd Algebraic methods. Abstract. We investigate the thermodynamics of an integrable spin ladder model which possesses a free parameter besides rung and leg couplings. he model is exactly solvable by means of the Bethe ansatz and exhibits a phase transition between a gapped and a gapless spin excitation spectrum. he magnetic susceptibility is obtained numerically and its dependence on the anisotropy parameter is determined. he spin gap obtained from the susceptibility curve and the one obtained from the Bethe ansatz equations are in very good agreement. Our results for the magnetic susceptibility fit well the experimental data for the organometallic compounds (IAP) 2CuBr 4 2H 2O(LandeeC.P.et al., Phys. Rev. B, 63 (21) 142(R)) Cu 2(C H 12N 2) 2Cl 4 (Hayward C. A., Poilblanc D. and Lèvy L. P., Phys. Rev. B, 4 (1996) R12649, Chaboussant G. et al., Phys. Rev. Lett., 19 (1997) 92; Phys. Rev. B, (1997) 346.) and (C H 12N) 2CuBr 4 (Watson B. C. et al., Phys. Rev. Lett., 86 (21) 168) in the strong-coupling regime. Introduction. he study of spin ladders, both from an experimental as well as from a theoretical point of view, has gained much attention in the last few years and has by now a literature of its own. Much of this interest was triggered by the discovery of high- c superconductivity in doped cuprate materials [1] and by the tremendous effort ensued while trying to understand the physics of 2d Cu-O lattices. However the lack of exactly integrable models in 2d as compared to the 1d case, where many such models have been known for decades, makes the task more difficult. hus spin ladders, being prototypes of quasi 1d systems, represent as such an excellent proving ground for studying the transition from the more amenable physics of one-dimensional quantum systems to two-dimensional behavior. Spin ladders exhibit antiferromagnetic (AF) ground states, where at each site of the lattice (in the undoped state) an electron occupies one of two spin states [2]. As such they can be reasonably well approximated by the archetypical AF Heisenberg model or some suitable generalization thereof. In one dimension the Heisenberg model is exactly solvable via the Bethe ansatz and from the solution it is known that in the AF regime, up to a given threshold of the spin exchange interaction, the elementary spin excitations are gapless. Beyond the threshold one has a Néel-like ground state with gapped excitations [3, 4]. However, the existence of a spin gap is critical for the occurrence of superconductivity under doping, whereby the holes introduced through doping undergo a Bose-Einstein condensation. By introducing the concept c EDP Sciences
112 EUROPHYSICS LEERS of a ladder model this apparent contradiction is resolved, since these systems allow for the formation of singlet states along the rungs which in turn are responsible for the appearance of a spin gap (singlet states can only form when there is an even number of legs [, 6]). he Heisenberg ladder, however, admits no exact solution and as a consequence many authors have considered generalized models which incorporate additional interaction terms without violating integrability [7 19], in the hope of gaining a deeper insight into the physics of ladder compounds. Notwithstanding the fact that these models exhibit realistic physical properties such as the existence of a spin gap [8 18] and magnetization plateaus at fractional values of the total magnetization [19], to our knowledge none of them has been used to predict physical quantities that could be compared directly to experimental data. he theory of spin ladders is now supported by a substantial body of experimental evidence thanks in part to the rapid advance of nano-engineered materials. he main goal of the present letter is to present an integrable spin ladder model whose magnetic susceptibility can be successfully compared to experimental data. We support our claim by calculating the magnetic susceptibility as a function of temperature and showing that it fits well the experimental data for the organometalliccompounds (IAP) 2 CuBr 4 2H 2 O [2], Cu 2 (C H 12 N 2 ) 2 Cl 4 [21, 22] and (C H 12 N) 2 CuBr 4 [23]. he model is solvable through the Bethe ansatz and incorporates an additional anisotropy parameter besides rung and leg couplings. he anisotropy-dependent critical line is determined through an explicit functional dependence of the excitation gap on the anisotropy t. he spin gap obtained from the susceptibility curve is compared to the one obtained from the Bethe ansatz equations and good agreement is found. he model. he Hamiltonian for the integrable anisotropicspin ladder system in the presence of an external magnetic field h reads [16] H = L [ J l h j,j+1 + ( σj τ j 1 ) h ( σ z 2 2 j + τj z ) ], (1) j=1 where h j,j+1 = 1 ( )( ) ( )( ) 1+σj z σj+1 z 1+τj z τj+1 z + σ + j 4 σ j+1 + σ j σ+ j+1 τ + j τ j+1 + τ j τ + j+1 + + 1 ( )( ) 1+σj z σj+1 z t 1 τ + j 2 τ j+1 +tτ j τ + j+1 + 1 ( )( ) t 1 σ + j 2 σ j+1 +tσ j σ+ j+1 1+τj z τj+1 z. L stands for the number of rungs and periodicboundary conditions are imposed along the legs. As usual, σ j ( τ j ) are Pauli matrices acting on site j of the upper (lower) leg, (J l ) is the strength of the rung (leg) coupling. he free parameter t introduces an anisotropy in the leg and interchain interactions. he cyclic 4-spin exchange terms have recently been discussed in the literature and shown to be essential in properly describing the properties of strong-coupling ladders [24]. he numerical estimates in our model confirm this assertion. By setting t 1 in eq. (1) one recovers the isotropic model of Wang [18] (which is moreover SU(4)-invariant with the proviso = ). he energy eigenvalues of (1) are given by E M 1 = J l j=1 ( 1 λ 2 j +1/4 2 + h ) ( + 1 2 J ) r L + h (M 2 + M 3 ), (2) J l J l J l J l where the λ j s satisfy a set of three-level nested Bethe ansatz equations [16].
A. P. onel et al.: Magnetic susceptibility of an exactly solvable etc. 113 6 gapped 4 3 Wang s point 2 1 gapless..2.4.6.8 1. t Fig. 1 he phase diagram of (1) in the ( t)-plane in units of J l. Wang s point (t = 1) is indicated by an arrow. he curve c =1+(t +1/t)/2 divides the gapped from the gapless phase. For the sake of convenience we take J l = 1 unless otherwise stated. he energy gap can be calculated using the exact Bethe ansatz solution and has the form ( =2 1 h 2 1 ( t + 1 )). (3) 2 t We verified that for h =and > 1+ 1 2 (t + 1 t ) there is an excitation gap in the energy spectrum and the ground state is given by a product of rung singlets, indicating a dimerization along the rungs. For = and no external field we find the critical value J c r =1+ 1 2 (t + 1 t ), indicating the critical line at which the quantum phase transition from the dimerized phase to the gapless phase occurs. his result is depicted in fig. 1. By choosing t real one has J c r 2. his corresponds to the strong-coupling limit where we believe the model to be a good candidate for describing organometallic ladder compounds. Comparison with experimental data. In order to calculate the magnetic susceptibility χ of our exactly solvable Hamiltonian and compare it to experimental data, we block-diagonalize eq. (1) for a sequence of finite chains (up to 6 rungs or 12 sites) and for different values of the coupling and anisotropy t. With these spectra we generate a sequence of finite-size.1.2 = = 8 = 12.1 = = 8 = 12.1 t = 1 t =.2 χ χ.1... 1 2 3 4. 1 2 3 4 Fig. 2 he magnetic susceptibility χ as a function of temperature for different values of the coupling and anisotropy t.
114 EUROPHYSICS LEERS -2 t = 1 = = 8 = 12-1 t =.2 = = 8 = 12 ln(χ 1/2 ) -4-6 ln(χ 1/2 ) -2-3 -8-4 -1 1 1 1/ - 1 1 1/ Fig. 3 A logarithmic plot of the susceptibility χ as a function of the inverse temperature 1 from which the spin gap can be obtained. susceptibilities χ (L) as a function of temperature and then apply a standard Bulirsch-Stoer acceleration technique [2] to obtain the thermodynamic limit lim L χ (L). Before presenting the results a few remarks on the behavior of χ as a function of and t are necessary, as well as the procedure used for choosing the values of our parameters. First we observe that for a fixed value of t there is a smoothing-out of the susceptibility curve for increasing, whereas for fixed the magnetic susceptibility increases with decreasing t, as can be seen in fig. 2. he second point is that in all cases the susceptibility presents an exponential decay for low temperature, where is the spin gap of the system χ e, (4) which is in agreement with the result of royer et al. [26] for the Heisenberg spin ladder model. By linearizing eq. (4) a numerical value for the spin gap ( ) can be found (some examples are depicted in fig. 3). When compared to the exact expression of the spin gap obtained from the Bethe ansatz equations (3) at =, an excellent agreement is found, as can be seen in table I. We remark that the behavior of the magnetic susceptibility for the critical value J c r is of the form χ 1 indicating a typical quantum critical behavior. For the isotropic case this was predicted in [17, 18]. By setting our parameters accordingly we were able to obtain the spin gap for the compounds (IAP) 2 CuBr 4 2H 2 O [2] Cu 2 (C H 12 N 2 ) 2 Cl 4 [21, 22], (C H 12 N) 2 CuBr 4 [23] which were previously studied by different authors using different methods. Our couplings and anisotropy were chosen so as to give the best fit of the experimental able I Spin gap obtained from the linearization of eq. (4) compared to the analytical result obtained from the Bethe ansatz for different values of and t (J l =1). t t t 1. 6..98 1. 12. 11.93 1. 2. 19.88.3 4.37 4.42 8.3 1.37 1.36 12.3 18.37 18.32.2 2.8 2.88.2 8.8 8.81.2 16.8 16.76
A. P. onel et al.: Magnetic susceptibility of an exactly solvable etc. 11 able II he values of parameters for different compounds as explained in the text. All data but the dimensionless t are in kelvin. Compound J l t exp (IAP) 2CuBr 4 2H 2O 1. 13..86 12. 12.3 11.9 [2] Cu 2(C H 12N 2) 2Cl 4 2.4..2344 1.8 13.41 1.8 [22] (C H 12N) 2CuBr 4 4. 4..31 9.8 9.9 9. [23] data. his choice as well as the experimental value exp [26] are given in table II. For the sake of clarity we present the results on three different compounds separately and refer the reader to the articles cited for more details on the experimental setup. (IAP) 2 CuBr 4 2H 2 O. Our results for this organometallic compound are depicted in fig. 4. his substance has been engineered through molecular-based magnetism in order to exhibit quasi two-dimensional behavior and exchange strengths small enough to render the critical point accessible through experiments [2]. he crystallographic structure is triclinic and the cuprate chains are isolated from each other by the aminopyridinium molecule. he exchange interaction between the Cu +2 ions takes place through the Br-Br contacts whose distances are 3.8 Å (rung) and 4.3 Å (along the leg). he next shortest distance between bromium atoms is along the diagonal of the ladder. With.74 Å it is too large and can thus be neglected. Cu 2 (C H 12 N 2 ) 2 Cl 4. he ladder structure of this compound arises from the stacking of Cu-Cu binuclear units (each Cu is to be seen as belonging to one leg). For magnetic fields below h = 7. the one-dimensional ground state is a valence-bond singlet structure on each rung and the system is gapped. Above this lower critical field, the first excited state (a triplet) becomes the ground state and the system has a non-zero magnetization. here are basically three different exchange paths between Cu ions: the intramolecular exchange constant between Cu-Cu ions takes place through two Cl ions, while the intermolecular J l involves inplane unpaired electron densities of Cu ions of different molecules. he third intermolecular χ [1-3 emu/mol] 3 2 2 1 1 J l =1 K = 13 K and t =.86 therm. lim. data (IAP) 2 CuBr 4.2H 2 O 2 2 1 1 1 1 2 1 2 3 4 6 7 8 9 1 [K] Fig. 4 Magnetic susceptibility vs. temperature. he full line is the result of exact diagonalization followed by extrapolation of finite-size data obtained from chains of up to 12 spins. he circles represent the experimental data obtained by Landee et al. [2]. he inset depicts the low-temperature regime.
116 EUROPHYSICS LEERS 1, J l =2.4 K =13.2 K and t =.2344,8 Cu 2 (C H 12 N 2 ) 2 Cl 4 χ [emu/mol],6,4 1,,8,6,4,2,,2, 1 1 2 therm. lim. data 1 2 3 4 [K] Fig. Magnetic susceptibility vs. temperature. he full line is the result of exact diagonalization followed by extrapolation of finite-size data, as explained in the text. he circles were obtained from the experimental data of Chaboussant et al. [22]. he inset depicts the low-temperature regime. exchange introduces a frustration via next-nearest neighbor interaction and occurs through hydrogen bonds between the Cl and N ions. his interaction is however significantly smaller than and is taken altogether equal to zero in our model. he low-temperature gap has been determined through susceptibility measurements and NMR techniques to be = 1.8 ±.6 K with J l =2.4K and =13.2 K [22] (cf. table II). he agreement between the lowtemperature experimental data and our results is excellent, as can be seen in fig., and remains good for the whole temperature range. (C H 12 N) 2 CuBr 4. As in the previous compound, here we also have a binuclear Cu- Cu stacked structure. he intramolecular exchange coupling is mediated through an orbital overlap of Br ions adjacent to the Cu sites. he intermolecular exchange is also mediated by Br ions a bit further apart (8.97 Å in comparison to intramolecular distances of 6.934 Å). A diagonal frustration exchange should be present but it is expected to be weak [23]. Again the χ[1-3 emu/mol] 2 2 1 1 J l =4 K = 16 K and t =.31 therm. lim. data 2 2 1 1 1 2 3 (C H 12 N) 2 CuBr 4 1 2 3 4 6 7 8 9 1 [K] Fig. 6 Magnetic susceptibility vs. temperature. Circles represent experimental data obtained by Watson et al. [23]. he solid line is the result obtained as in the previous figures. he low-temperature regime is depicted in the inset.
A. P. onel et al.: Magnetic susceptibility of an exactly solvable etc. 117 fit is better for low temperatures, with an energy gap of 9.8 K (9. K, experimental [23]) and an exponential drop in the magnetic susceptibility, as depicted in fig. 6. Summary. In this letter we investigated an exactly solvable anisotropic spin ladder model which has an extra free anisotropy parameter besides the rung and leg couplings. he spin gap is anisotropy-dependent. he magneticsusceptibility was investigated and good agreement was found between the gap obtained from the exact Bethe ansatz solution and the one obtained from the magnetic susceptibility. Connections with strong-coupling compounds were discussed and our results reproduce well experimental data. he anisotropy parameter t has no obvious physical interpretation but, as shown in this letter, it affects the excitation spectrum and allows us to tune our model to different physical systems of interest, thus presenting a unified scenario for strong-coupling ladders. he authors would like to thank A. R. Pohlmann for clarifying some questions on the chemistry of organometallic compounds. his work has been financially supported by the Brazilian agencies CNPq (AP, AF and ALM) and FAPESP (ALM). REFERENCES [1] Bednorz J. B. and Müller K. A., Z. Phys., 64 (1986) 189. [2] Dagotto E., Rep. Prog. Phys., 62 (1999) 12. [3] Bethe H., Z. Phys., 71 (1931) 2. [4] Dagotto E. and Rice. M., Science, 271 (1996) 618. [] Rice. M., Gopalan S. and Sigrist M., Europhys. Lett., 23 (1993) 44. [6] White S. R., Noack R. M. and Scalapino D. J., Phys. Rev. Lett., 73 (1994) 886. [7] Albeverio S., Fei S.-M. and Wang Y., Europhys. Lett., 47 (1999) 364. [8] Batchelor M.. and Maslen M., J. Phys. A, 32 (1999) L377; 33 (2) 443; de Gier J. and Batchelor M.., Phys. Rev. B, 62 (2) R384. [9] Batchelor M.., de Gier J., Links J. and Maslen M., J. Phys. A, 33 (2) L97. [1] de Gier J., Batchelor M.. and Maslen M., Phys. Rev. B, 62 (2) 1196. [11] Links J. and Foerster A., Phys. Rev. B, 62 (2) 6. [12] Kundu A., J. Math. Phys., 41 (2) 721. [13] Foerster A., Hibberd K. A., Links J. R. and Roditi I., J. Phys. A, 34 (21) L2. [14] Ambjorn J. et al., J. Phys. A, 34 (21) 887. [1] Frahm H. and Kundu A., J. Phys. C, 11 (1999) L7. [16] onel A. P., Foerster A., Links J. and Malvezzi A. L., Phys. Rev. B, 64 (21) 442-1. [17] Nersesyan A. A. and svelik A. M., Phys. Rev. Lett., 78 (1997) 3939. [18] Wang Y., Phys. Rev. B, 6 (1999) 9236. [19] de Gier J. and Batchelor M.., Phys. Rev. B, 62 (2) R384. [2] Landee C. P. et al., Phys. Rev. B, 63 (21) 142(R). [21] Hayward C. A., Poilblanc D. and Lèvy L. P., Phys. Rev. B, 4 (1996) R12649. [22] Chaboussant G. et al., Phys. Rev. Lett., 19 (1997) 92; Phys. Rev. B, (1997) 346. [23] Watson B. C. et al., Phys. Rev. Lett., 86 (21) 168. [24] Nunner. S. et al., Phys. Rev. B, 66 (22) 1844(R); Haga N. and Suga S., Phys. Rev. B, 66 (22) 13241. [2] Bulirsh R. and Stoer J., Num. Math., 6 (1994) 413. [26] royer M., sunetsugu H. and Würtz D., Phys. Rev. B, (1994) 131.