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Ground State of the S=1/2 Distorted Diamond Chain --- Model of Cu_3_Cl_6_(H_2_O)_2_・2H_8_C_4_SO_2_

Kiyomi Okamoto, Taskashi Tonegawa, Yutaka Takahashi, Makoto Kaburagi

Published Jan 1, 1999• 84 citations
OpenAlex

Abstract

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.

Related papers

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

Kiyomi Okamoto, Taskashi Tonegawa, Yutaka Takahashi +1 more

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.

arXiv:cond-mat/9907440•Journal of Physics Condensed Matter•Dec 9, 1999•66 citations

Ground State Properties of an S=1/2 Distorted Diamond Chain

Takashi Tonegawa, Kiyomi Okamoto, Hikihara, Toshiya +2 more

We investigate the ground state properties of an S=1/2 distorted diamond chain described by the Hamiltonian ${\cal H}=J_1\sum_\ell\bigl\{\bigl(\vecS_{3\ell-1}\cdot\vecS_{3\ell} +\vecS_{3\ell}\cdot\vecS_{3\ell+1}\bigr)+J_2\vecS_{3\ell-2}\cdot\vecS_{3\ell-1} +J_3\bigl(\vecS_{3\ell-2}\cdot\vecS_{3\ell}+\vecS_{3\ell}\cdot\vecS_{3\ ell+2}\bigr)-H S_{\ell}^z\bigr\}$ ($J_1,~J_2,~J_3\geq0$), which models well a trimerized S=1/2 spin chain system Cu$_3$Cl$_6$(H$_2$O)$_2$$\cdot$2H$_8$C$_4$SO$_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 $J_1=1.0$, $J_2=0.8$, and $J_3=0.5$, and the case (b) where $J_1=1.0$, $J_2=0.8$, and $J_3=0.3$. 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 $\cal H$ is spontaneously broken in the 2/3-plateau state as well as in the dimerized state.

arXiv:cond-mat/9912482•Dec 30, 1999•0 citations

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

Kazuhiro Sano, Ken'ichi Takano

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.

arXiv:cond-mat/0005281•Journal of the Physical Society of Japan•Aug 15, 2000•23 citations

Dynamics of a distorted diamond chain

H.-J. Mikeska, C. Luckmann

We present results on the dynamics of the distorted diamond chain, $S=1∕2$ dimers alternating with single spins $1∕2$ and exchange couplings ${J}_{1}$ and ${J}_{3}$ 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 $1∕3$ 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 ${\mathrm{Cu}}_{3}{(\mathrm{C}{\mathrm{O}}_{3})}_{2}{(\mathrm{O}\mathrm{H})}_{2}$ (azurite) material is discussed with particular emphasis on inelastic neutron scattering experiments; the recent suggestion of one possibly ferromagnetic coupling constant is not confirmed.

arXiv:0709.2863•Physical Review B•Feb 6, 2008•30 citations

Theoretical and experimental studies of melting of the 1/3 magnetization plateau in a frustrated $$S=1/2$$ antiferromagnetic trimerized quantum Heisenberg spin chain compound $$\hbox {Na}_2\hbox {Cu}_3\hbox {Ge}_4\hbox {O}_{12}$$

Sachin Kumar, Anup Kumar Bera, Amit Kumar +2 more

Abstract We report the magnetization process of $$S=1/2$$ S = 1 / 2 Heisenberg antiferromagnetic $$J_1{-}J_1{-}J_2$$ J 1 - J 1 - J 2 trimerized spin chain system (with frustration due to the next-nearest-neighbor interaction $$J_3$$ J 3 within the trimer) under applied magnetic field by means of density-matrix renormalization group method. Here, $$J'$$ J ′ s are the antiferromagnetic exchange interaction constants. Specifically, we explore the phenomenon of 1/3 magnetization plateau for the exchange coupling ratios $$ \alpha = 0.18, \beta =0.18 $$ α = 0.18 , β = 0.18 , and $$J_1=235$$ J 1 = 235 K, where $$\alpha =J_2/J_1$$ α = J 2 / J 1 , and $$\beta =J_3/J_1$$ β = J 3 / J 1 as found for the model compound $$\hbox {Na}_2\hbox {Cu}_3\hbox {Ge}_4\hbox {O}_{12}$$ Na 2 Cu 3 Ge 4 O 12 . It is found that a wide 1/3 quantum magnetization plateau state (observed in ground state magnetization) gradually disappears with increasing temperature beyond a crossover temperature $$T_{co}$$ T co due to thermal fluctuations. Based on numerical data, the width w(T) of the magnetization plateau is found to follow an exponential behave as $$\sim e^{-B/\sqrt{T_{co}-T}}$$ ∼ e - B / T co - T , where B is a constant and $$T_{co}$$ T co is estimated to be $$ \approx 60$$ ≈ 60

The European Physical Journal B•Mar 1, 2025•1 citation

A dimerized and frustrated spin model with alternating ferromagnetic and antiferromagnetic exchanges: investigating quantum phases of Heisenberg spin chains with s = 1/2 and 1

Shaon Sahoo, V. M. L. Durga Prasad Goli, Diptiman Sen +1 more

We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic ($J_1^F$) and antiferromagnetic ($J_1^A$) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction ($J_2^F$). In this model frustration is induced by the non-zero $J_2^F$ and the effect of dimerization is realized by changing the strength of $J_1^A$ for a fixed value of $J_1^F$. The model with site spin $s$ behaves like a Haldane spin chain with site spin 2$s$ in the limit of vanishing $J_2^F$ and large $J_1^F/J_1^A$. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of $J_1^A-J_2^F$ is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.

•May 29, 2013•0 citations
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 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 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.
Theoretical and experimental studies of melting of the 1/3 magnetization plateau in a frustrated S=1/2S=1/2S=1/2 antiferromagnetic trimerized quantum Heisenberg spin chain compound Na2Cu3Ge4O12\hbox {Na}_2\hbox {Cu}_3\hbox {Ge}_4\hbox {O}_{12}Na2​Cu3​Ge4​O12​
Abstract We report the magnetization process of S=1/2S=1/2S=1/2 S = 1 / 2 Heisenberg antiferromagnetic J1−J1−J2J_1{-}J_1{-}J_2J1​−J1​−J2​ J 1 - J 1 - J 2 trimerized spin chain system (with frustration due to the next-nearest-neighbor interaction J3J_3J3​ J 3 within the trimer) under applied magnetic field by means of density-matrix renormalization group method. Here, J′J'J′ J ′ s are the antiferromagnetic exchange interaction constants. Specifically, we explore the phenomenon of 1/3 magnetization plateau for the exchange coupling ratios α=0.18,β=0.18 \alpha = 0.18, \beta =0.18 α=0.18,β=0.18 α = 0.18 , β = 0.18 , and J1=235J_1=235J1​=235 J 1 = 235 K, where α=J2/J1\alpha =J_2/J_1α=J2​/J1​ α = J 2 / J 1 , and β=J3/J1\beta =J_3/J_1β=J3​/J1​ β = J 3 / J 1 as found for the model compound Na2Cu3Ge4O12\hbox {Na}_2\hbox {Cu}_3\hbox {Ge}_4\hbox {O}_{12}Na2​Cu3​Ge4​O12​ Na 2 Cu 3 Ge 4 O 12 . It is found that a wide 1/3 quantum magnetization plateau state (observed in ground state magnetization) gradually disappears with increasing temperature beyond a crossover temperature TcoT_{co}Tco​ T co due to thermal fluctuations. Based on numerical data, the width w(T) of the magnetization plateau is found to follow an exponential behave as ∼e−B/Tco−T\sim e^{-B/\sqrt{T_{co}-T}}∼e−B/Tco​−T​ ∼ e - B / T co - T , where B is a constant and TcoT_{co}Tco​ T co is estimated to be ≈60 \approx 60≈60 ≈ 60
We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (J1FJ_1^FJ1F​) and antiferromagnetic (J1AJ_1^AJ1A​) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction (J2FJ_2^FJ2F​). In this model frustration is induced by the non-zero J2FJ_2^FJ2F​ and the effect of dimerization is realized by changing the strength of J1AJ_1^AJ1A​ for a fixed value of J1FJ_1^FJ1F​. The model with site spin sss behaves like a Haldane spin chain with site spin 2sss in the limit of vanishing J2FJ_2^FJ2F​ and large J1F/J1AJ_1^F/J_1^AJ1F​/J1A​. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of J1A−J2FJ_1^A-J_2^FJ1A​−J2F​ is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.