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arXiv:cond-mat/0005281 · Journal of the Physical Society of Japan

Spin Gap ofS=1/2Heisenberg Model on Distorted Diamond Chain

Kazuhiro Sano, Ken'ichi Takano

Published Aug 15, 2000• 23 citations
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Abstract

We study the spin gap of the S=1/2 Heisenberg model on the distorted diamond chain, which is recently proposed to represent magnetic properties of Cu_3 Cl_6 (H_2 O)_2 2H_8 C_4 SO_2. This model is composed of stacked trimers and has three kinds of exchange interactions J_1, J_2 and J_3. Using the numerical diagonalization, we obtain a contour map of the spin gap in the J_2/J_1-J_3/J_1 plane. We argue possible values of the exchange constants based on the contour map and the observed value of the spin gap.

DOI
10.1143/jpsj.69.2710

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We study the spin gap of the S=1/2 Heisenberg model on the distorted diamond chain, which is recently proposed to represent magnetic properties of Cu_3 Cl_6 (H_2 O)_2 2H_8 C_4 SO_2. This model is composed of stacked trimers and has three kinds of exchange interactions J_1, J_2 and J_3. Using the numerical diagonalization, we obtain a contour map of the spin gap in the J_2/J_1-J_3/J_1 plane. We argue possible values of the exchange constants based on the contour map and the observed value of the spin gap.
Abstract. We study the ground state of the model Hamiltonian of the trimerized S = 1/2 quantum Heisenberg chain Cu3Cl6(H2O)2 · 2H8C4SO2 in which the non-magnetic ground state is observed recently. This model consists of stacked trimers and has three kinds of coupling constants between spins; the intra-trimer coupling constant J1 and the inter-trimer coupling constants J2 and J3. All of these constants are assumed to be antiferromagnetic. By use of the analytical method and physical considerations, we show that there are three phases on the ˜ J2 − ˜ J3 plane ( ˜ J2 ≡ J2/J1, ˜ J3 ≡ J3/J1), the dimer phase, the spin fluid phase and the ferrimagnetic phase. The dimer phase is caused by the frustration effect. In the dimer phase, there exists the excitation gap between the twofold degenerate ground state and the first excited state, which explains the non-magnetic ground state observed in Cu3Cl6(H2O)2 ·2H8C4SO2. We also obtain the phase diagram on the ˜ J2 − ˜ J3 plane from the numerical diagonalization data for finite systems. PACS numbers: 75.10.Jm, 75.40.Cx, 75.50.Ee, 75.50.GgGround state of an S = 1/2 distorted diamond chain 2 1.
We study the ground state of the model Hamiltonian of the trimerized S = 1/2 quantum Heisenberg chain Cu 3 Cl 6 (H 2 O) 2 2H 8 C 4 SO 2 in which a non-magnetic ground state was observed recently. This model consists of stacked trimers and has three kinds of coupling constant describing the couplings between spins: the intra-trimer coupling constant J 1 and the inter-trimer coupling constants J 2 and J 3 . All of these constants are assumed to be antiferromagnetic. By use of an analytical method and physical considerations, we show that there are three phases on the 2 - 3 plane ( 2 J 2 / J 1 , 3 J 3 / J 1 ): the dimer phase, the spin-fluid phase and the ferrimagnetic phase. The dimer phase is caused by the frustration effect. In the dimer phase, there exists an excitation gap between the twofold-degenerate ground state and the first excited state, which explains the non-magnetic ground state observed in Cu 3 Cl 6 (H 2 O) 2 ·2H 8 C 4 SO 2 . We also obtain the phase diagram on the 2 - 3 plane from the numerical diagonalization data for finite systems by use of the Lanczos algorithm.
We investigate the ground state properties of an S=1/2 distorted diamond chain described by the Hamiltonian ${\cal H}=J_1\sum_\ell\bigl\{\bigl(\vecS_{3\ell-1}\cdot\vecS_{3\ell} +\vecS_{3\ell}\cdot\vecS_{3\ell+1}\bigr)+J_2\vecS_{3\ell-2}\cdot\vecS_{3\ell-1} +J_3\bigl(\vecS_{3\ell-2}\cdot\vecS_{3\ell}+\vecS_{3\ell}\cdot\vecS_{3\ ell+2}\bigr)-H S_{\ell}^z\bigr\}$ (J1, J2, J3≥0J_1,~J_2,~J_3\geq0J1​, J2​, J3​≥0), which models well a trimerized S=1/2 spin chain system Cu3_33​Cl6_66​(H2_22​O)2_22​$\cdot2H2H2H_8CCC_4SOSOSO_2$. Using an exact diagonalization method by means of the Lanczös technique, we determine the ground state phase diagram in the H=0 case, composed of the dimerized, spin fluid, and ferrimagnetic phases. Performing a degenerate perturbation calculation, we analyze the phase boundary line between the latter two phases in the $J_2,~J_3\llJ_1$ limit, the result of which is in good agreement with the numerical result. We calculate, by the use of the density matrix renormalization group method, the ground state magnetization curve for the case (a) where J1=1.0J_1=1.0J1​=1.0, J2=0.8J_2=0.8J2​=0.8, and J3=0.5J_3=0.5J3​=0.5, and the case (b) where J1=1.0J_1=1.0J1​=1.0, J2=0.8J_2=0.8J2​=0.8, and J3=0.3J_3=0.3J3​=0.3. We find that in the case (b) the 2/3-plateau appears in addition to the 1/3-plateau which also appears in the case (a). The translational symmetry of the Hamiltonian H\cal HH is spontaneously broken in the 2/3-plateau state as well as in the dimerized state.
We present results on the dynamics of the distorted diamond chain, S=1∕2S=1∕2S=1∕2 dimers alternating with single spins 1∕21∕21∕2 and exchange couplings J1{J}_{1}J1​ and J3{J}_{3}J3​ in between. The dynamics in the spin fluid (SF) and tetramer-dimer (TD) phases is investigated numerically by exact diagonalization for up to 24 spins. Representative excitation spectra are presented both for zero magnetic field and in the 1∕31∕31∕3 plateau phase and the relevant parameters are determined across the phase diagram. The behavior across the SF-TD phase transition line is discussed for the specific heat and for excitation spectra. The relevance of the distorted diamond chain model for the Cu3(CO3)2(OH)2{\mathrm{Cu}}_{3}{(\mathrm{C}{\mathrm{O}}_{3})}_{2}{(\mathrm{O}\mathrm{H})}_{2}Cu3​(CO3​)2​(OH)2​ (azurite) material is discussed with particular emphasis on inelastic neutron scattering experiments; the recent suggestion of one possibly ferromagnetic coupling constant is not confirmed.
The thermodynamic properties of the spin-1/2 diamond quantum Heisenberg chain model have been investigated by means of the transfer matrix renormalization group (TMRG) method. Considering different crystal structures, by changing the interactions among different spins and the external magnetic fields, we first investigate the magnetic susceptibility, magnetization, and specific heat of the distorted diamond chain as a model of ferrimagnetic spin systems. The susceptibility and the specific heat show different features for different ferromagnetic (F) and antiferromagnetic (AF) interactions and different magnetic fields. A 1/3 magnetization plateau is observed at low temperature in a magnetization curve. Then, we discuss the theoretical mechanism of the double-peak structure of the magnetic susceptibility and the three-peak structure of the specific heat of the compound Cu3(CO3)2(OH)2, on which an elegant measurement was performed by Kikuchi et al. [Phys. Rev. Lett. 94, 227201 (2005)]. Our computed results are consistent with the main characteristics of the experimental data. Meanwhile, we find that the double-peak structure of susceptibility can be found in several different kinds of spin interactions in the diamond chain. Moreover, a three-peak behavior is observed in the TMRG results of magnetic susceptibility. In addition, we perform calculations relevant for some experiments and explain the characteristics of these materials.
We consider the dimerized spin-1/2 Heisenberg chain with spin hexameric distortion of the exchange pattern and study the zero-temperature phase diagram in the parameter space (J1,J2,J3)(J_{1}, J_{2}, J_{3})(J1​,J2​,J3​) by continuum-limit bosonization approach and the exact diagonalization method. The phase diagram is rich and has two gaped dimer phases. We obtain an estimate of the critical line separating the different gapped dimer phases by the bosonization approach. The existence of the transition line and the difference between dimer phases is checked numerically. The behavior of the energy gap and the dimer order parameter supports the exact location of the gapless line.