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arXiv:cond-mat/9907440 · Journal of Physics Condensed Matter

The ground state of anS= 1/2 distorted diamond chain - a model of Cu3Cl6(H2O)2·2H8C4SO2

Kiyomi Okamoto, Taskashi Tonegawa, Yutaka Takahashi, Makoto Kaburagi

Published Dec 9, 1999• 66 citations
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Abstract

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.

DOI
10.1088/0953-8984/11/50/336

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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.
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 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.
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.
Quantum spin-1/2 antiferromagnetic Heisenberg trimerized chain with strong intradimer and weak monomer-dimer coupling constants is studied using the many-body perturbation expansion developed about the exact solution of the spin-1/2 Ising-Heisenberg diamond chain. The Heisenberg trimerized chain shows the intermediate one-third plateau, which was also observed in the magnetization curve of the polymeric compound Cu3_33​(P2_22​O6_66​OH)2_22​ affording its experimental realization. Within the strong-coupling method we have obtained the effective Hamiltonians for the magnetic-field range from zero to one-third plateau, and from one-third plateau to the saturation magnetization. The second-order perturbation theory shows reliable results even for the moderate ratio between weaker dimer-monomer and stronger intradimer coupling constants. We have also examined thermodynamic properties and recovered the low-temperature peak in the specific heat. The accuracy of the developed method is tested through a comparison with numerical density-matrix renormalization group and quantum Monte Carlo simulations. Using the results for the effective Hamiltonian we suggest straightforward procedure for finding the microscopic parameters of one-dimensional trimerized magnetic compounds with strong intradimer and weak monomer-dimer couplings. We found the refined values for the coupling constants of Cu3_33​(P2_22​O6_66​OH)2_22​ by matching the theoretical results with the available experimental data for the magnetization and magnetic susceptibility in a wide range of temperatures and magnetic fields.
We explore, at T = 0, the magnetic properties of the S = 1/2 antiferromagnetic distorted diamond chain described by the Hamiltonian = ∑ j = 1 N /3 { J 1 ( S 3 j −1 · S 3 j + S 3 j · S 3 j +1 )+ J 2 S 3 j +1 · S 3 j +2 + J 3 ( S 3 j −2 · S 3 j + S 3 j · S 3 j +2 )}− H ∑ l = 1 N S l z with J 1 , J 2 , J 3 ≥ 0, which models well A 3 Cu 3 (PO 4 ) 4 with A = Ca, Sr, Bi 4 Cu 3 V 2 O 14 and azurite Cu 3 (OH) 2 (CO 3 ) 2 . We employ the physical consideration, degenerate perturbation theory, level spectroscopy analysis of the numerical diagonalization data obtained by the Lanczos method and also the density matrix renormalization group (DMRG) method. We investigate the mechanisms of the magnetization plateaux at M = M s /3 and (2/3) M s , and also show the precise phase diagrams of the ( J 2 / J 1 , J 3 / J 1 ) plane concerned with these magnetization plateaux, where M = ∑ l = 1 N S l z and M s is the saturation magnetization. We also calculate the magnetization curves and the magnetization phase diagrams by means of the DMRG method.