We present results of ultrasonic measurements on a single crystal of the distorted diamond-chain compound azurite Cu$_3$(CO$_3$)$_2$(OH)$_2$. Pronounced elastic anomalies are observed in the temperature dependence of the longitudinal elastic mode $c_{22}$ which can be assigned to the relevant magnetic interactions in the system and their couplings to the lattice degrees of freedom. From a quantitative analysis of the magnetic contribution to $c_{22}$ the magneto-elastic coupling $G$ = $\partial J_2$/$\partial ε_b$ can be determined, where $J_2$ is the intra-dimer coupling constant and $ε_b$ the strain along the intra-chain $b$ axis. We find an exceptionally large coupling constant of $|G| \sim ($3650 $\pm$ 150) K highlighting an extraordinarily strong sensitivity of $J_2$ against changes of the $b$-axis lattice parameter. These results are complemented by measurements of the hydrostatic pressure dependence of $J_2$ by means of thermal expansion and magnetic susceptibility measurements performed both at ambient and finite hydrostatic pressure. We propose that a structural peculiarity of this compound, in which Cu$_2$O$_6$ dimer units are incorporated in an unusually stretched manner, is responsible for the anomalously large magneto-elastic coupling.
The natural mineral azurite Cu3(CO3)2(OH)2 is an interesting spin-1/2 quantum antiferromagnet. Recently, a generalised diamond chain model has been established as a good description of the magnetic properties of azurite with parameters placing it in a highly frustrated parameter regime. Here we explore further properties of this model for azurite. First, we determine the inelastic neutron scattering spectrum in the absence of a magnetic field and find good agreement with experiments, thus lending further support to the model. Furthermore, we present numerical data for the magnetocaloric effect…
In order to investigate the spin state of azurite, Cu3(CO3)2(OH)2, we have measured the thermal conductivity along the c-axis, $κ$c, perpendicular to the spin diamond-chains. It has been found that the temperature dependence of $κ$c shows a broad peak at ~ 100 K, which is explained as being due to the strong phonon-scattering by the strong spin-fluctuation owing to the spin frustration at low temperatures below ~ 100 K. Furthermore, it has been found that the temperature dependence of $κ$c shows another peak at low temperatures below 20 K and that $κ$c is suppressed by the application of magne…
High magnetic field $^{63,65}$Cu NMR spectra were used to determine the local spin polarization in the 1/3 magnetization plateau of azurite, Cu$_3$(CO$_3$)$_2$(OH)$_2$, which is a model system for the distorted diamond antiferromagnetic spin-1/2 chain. The spin part of the hyperfine field of the Cu2 (dimer) sites is found to be field independent, negative and strongly anisotropic, corresponding to $\approx$10 % of fully polarized spin in a $d$-orbital. This is close to the expected configuration of the "quantum" plateau, where a singlet state is stabilized on the dimer. However, the observed n…
The natural mineral azurite Cu$_3$(CO$_3$)$_2$(OH)$_2$ is a frustrated magnet displaying unusual and controversially discussed magnetic behavior. Motivated by the lack of a unified description for this system, we perform a theoretical study based on density functional theory as well as state-of-the-art numerical many-body calculations. We propose an effective generalized spin-1/2 diamond chain model which provides a consistent description of experiments: low-temperature magnetization, inelastic neutron scattering, nuclear magnetic resonance measurements, magnetic susceptibility as well as new …
Azurite, Cu3(CO3)2(OH)2, has been considered an ideal example of a one-dimensional (1D) diamond chain antiferromagnet. Early studies of this material imply the presence of an ordered antiferromagnetic phase below $T_N \sim 1.9$ K while magnetization measurements have revealed a 1/3 magnetization plateau. Until now, no corroborating neutron scattering results have been published to confirm the ordered magnetic moment structure. We present recent neutron diffraction results which reveal the presence of commensurate magnetic order in azurite which coexists with significant magnetoelastic strain. …
We consider the antiferromagnetic Heisenberg model on a distorted diamond chain and use the localized-magnon picture adapted to a distorted geometry to discuss some of its high-field low-temperature properties. More specifically, in our study we assume that the partition function for a slightly distorted geometry has the same form as for ideal geometry, though with slightly dispersive one-magnon energies. We also discuss the relevance of such a description to azurite.
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 diagonalisation 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 …
Spin-1/2 compounds A3Cu3AlO2(SO4)4 (A=K, Rb, and Cs) have one-dimensional (1D) inequilateral diamond chains. We analyze the temperature dependence of the magnetic susceptibility and determine the magnetic exchange interactions. In contrast to the azurite, a dimer is formed on one of the sides of the diamond. From numerical analyses of the proposed model, we find that the dimer together with a nearly isolated 1D Heisenberg chain characterizes magnetic properties including magnetization curve and magnetic excitations. This implies that a dimer-monomer composite chain without frustration is a goo…
We consider the antiferromagnetic spin-1/2 $XXZ$ Heisenberg model on a frustrated diamond-chain lattice in a $z$- or $x$-aligned external magnetic field. We use the strong-coupling approach to elaborate an effective description in the low-temperature strong-field regime. The obtained effective models are spin-1/2 $XY$ chains which are exactly solvable through the Jordan-Wigner fermionization. We perform exact-diagonalization studies of the magnetization curves to test the quality of the effective description. The results may have relevance for the description of the azurite spin-chain compound…
The magnetization process of the $S=1/2$ distorted diamond spin chain with the anisotropic ferromagnetic interaction is investigated using the numerical diagonalization of finite-size clusters. It is found that the spin nematic and SDW Tomonaga-Luttinger liquids can appear for sufficiently large easy axis anisotropy.
The frustrated quantum spin system on the distorted diamond chain lattice suitable for the alumoklyuchevskite is investigated using the numerical diagonalization of finite-size clusters and the level spectroscopy analysis. It is found that this model exhibits three quantum phases; the ferrimagnetic phase, the spin gap one, and the gapless Tomonaga-Luttinger liquid depending on the exchange coupling parameters. The ground state phase diagram is presented.
The $S=1/2$ distorted diamond spin chain with the anisotropic ferromagnetic interaction is investigated using the numerical diagonalization and the level spectroscopy analysis. It is known that the system exhibits a plateau of the magnetization curve at the 1/3 of the saturation. The present study indicates that as the anisotropy is varied the quantum phase transition occurs between two different mechanisms of the 1/3 magnetization plateau. The phase diagram with respect to the anisotropy and the ferromagnetic coupling is also presented.
The frustrated quantum spin system on the distorted diamond chain lattice is investigated using the numerical diagonalization of finite-size clusters and the level spectroscopy analysis. In the previous work this system was revealed to exhibit the 1/3 magnetization plateau due to two different mechanisms depending on the coupling parameters, and the phase diagram at the 1/3 magnetization was obtained. In the present work it is found that the 1/3 magnetization plateau vanishes for sufficiently large $XY$-like coupling anisotropy. The phase diagram based on the level spectroscopy analysis is als…
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.
By use of mainly the exact diagonalization and the level spectroscopy method, we investigate the ground-state phase diagrams of the $S=1/2$ distorted diamond type quantum spin chain with the monomer-monomer interactions and/or ferromagnetic interactions for the zero magnetic field case, as well as the $M=M_{\rm s}/3$ case and the $M=(2/3)M_{\rm s}$ case, where $M$ is the total magnetization and $M_{\rm s}$ is the saturation magnetization. The magnetization plateau at $M=M_{\rm s}/3$ vanishes in the region where the ferromagnetic interaction is rather strong. The monomer-monomer interaction rem…
We consider a spin-1/2 Ising-XYZ distorted diamond chain with the XYZ interaction between the interstitial Heisenberg dimers, the nearest-neighbor Ising coupling between the nodal and interstitial spins, respectively, and the second-neighbor Ising coupling between the nodal spins. The ground-state phase diagram of the spin-1/2 Ising-XYZ distorted diamond chain exhibits several intriguing phases due to the XY anisotropy and the second-neighbor interaction, whereas the model can be exactly solved using the transfer-matrix technique. The quantum entanglement within the Heisenberg spin dimers is s…
We investigate the ground-sate phase diagram of the XXZ version of the S=1/2 distorted diamond chain by use of the degenerate perturbation theory near the truncation point. In case of the XY-like interaction anisotropy, the phase diagram consists of the Neel phase and the spin-fluid phase. For the Ising-like interaction anisotropy case, it consists of three phases: the ferrimagnetic phase, the Neel phase and the spin-fluid phase. The magnetization in the ferrimagnetic phase is 1/3 of the saturation magnetization. The remarkable nature of the phase diagram is the existence of the Neel phase, al…
The ground states of distorted $S=1$ diamond chains are investigated for two types of distortion called type A and B [J. Phys. Soc. Jpn. 79 (2010) 114703]. For the type A distortion, Haldane phases with and without spontaneous translational symmetry breakdown are present for large values of parameter $λ$ that parametrize the strength of frustration. For small $λ$, the Haldane phase and two quantized ferrimagnetic phases in the undistorted chain remain stable even for strong distortion. In contrast, for the type B distortion, the quantized ferrimagnetic phases with and without spontaneous trans…
The ground states of two types of distorted mixed diamond chains with spins 1 and 1/2 are investigated using exact diagonalization, DMRG, and mapping onto low-energy effective models. In the undistorted case, the ground state consists of an array of independent spin-1 clusters separated by singlet dimers. The lattice distortion induces an effective interaction between cluster spins. When this effective interaction is antiferromagnetic, several Haldane phases appear with or without spontaneous translational symmetry breakdown (STSB). The transition between the Haldane phase without STSB and tha…
The magnetic Ln$^{3+}$ ions in the fergusonite and scheelite crystal structures form a distorted or stretched diamond lattice which is predicted to host exotic magnetic ground states. In this study, polycrystalline samples of the fergusonite orthotantalates $M$-LnTaO$_4$ (Ln = Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er) are synthesized and then characterized using powder diffraction and bulk magnetometry and heat capacity. TbTaO$_4$ orders antiferromagnetically at 2.25 K into a commensurate magnetic cell with $\vec{k}=0$, magnetic space group 14.77 ($P2_1$$'/c$) and Tb moments parallel to the $a$-axis. No…
We explore, at T=0, the magnetic properties of the $S=1/2$ antiferromagnetic distorted diamond chain described by the Hamiltonian ${\cal H} = \sum_{j=1}^{N/3}{J_1 ({\bi S}_{3j-1} \cdot {\bi S}_{3j} + {\bi S}_{3j} \cdot {\bi S}_{3j+1}) + J_2 {\bi S}_{3j+1} \cdot {\bi S}_{3j+2} + J_3 ({\bi S}_{3j-2} \cdot {\bi S}_{3j} + {\bi S}_{3j} \cdot {\bi S}_{3j+2})} \allowbreak - H \sum_{l=1}^{N} S_l^z $ with $J_1, J_2, J_3\ge0$, which well models ${\rm A_3 Cu_3 (PO_4)_4}$ with ${\rm A = Ca, Sr}$, ${\rm Bi_4 Cu_3 V_2 O_{14}}$ and azurite $\rm Cu_3(OH)_2(CO_3)_2$. We employ the physical consideration, the d…
Magnetization plateaus, spin-canted orders and field-induced transitions in a spin-1/2 Heisenberg antiferromagnet on a distorted diamond-decorated honeycomb lattice
Institute of Physics, Faculty of Science, P. J. Šafárik University（帕日·沙法里克大学理学院物理研究所）
We investigate the ground state properties of an S=1/2 distorted diamond chain described by the Hamiltonian ${\cal H}=J_1\sum_\ell\bigl\{\bigl(\vecS_{3\ell-1}\cdot\vecS_{3\ell} +\vecS_{3\ell}\cdot\vecS_{3\ell+1}\bigr)+J_2\vecS_{3\ell-2}\cdot\vecS_{3\ell-1} +J_3\bigl(\vecS_{3\ell-2}\cdot\vecS_{3\ell}+\vecS_{3\ell}\cdot\vecS_{3\ ell+2}\bigr)-H S_{\ell}^z\bigr\}$ ($J_1,~J_2,~J_3\geq0$), which models well a trimerized S=1/2 spin chain system Cu$_3$Cl$_6$(H$_2$O)$_2$$\cdot$2H$_8$C$_4$SO$_2$. Using an exact diagonalization method by means of the Lanczös technique, we determine the ground state phase…
Antiferromagnetic insulators on the diamond lattice are candidate materials to host exotic magnetic phenomena ranging from spin-orbital entanglement to degenerate spiral ground-states and topological paramagnetism. Compared to other three-dimensional networks of magnetic ions, such as the geometrically frustrated pyrochlore lattice, the investigation of diamond-lattice magnetism in real materials is less mature. In this work, we characterize the magnetic properties of model A-site spinels CoRh2O4 (cobalt rhodite) and CuRh2O4 (copper rhodite) by means of thermo-magnetic and neutron scattering m…
We study the ground state of the model Hamiltonian of the trimerized $S=1/2$ quantum Heisenberg chain $\rm Cu_3 Cl_6 (H_2 O)_2 \cdot 2H_8 C_4 SO_2$ 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 $J_1$ and the inter-trimer coupling constants $J_2$ and $J_3$. All of these constants are assumed to be antiferromagnetic. By use of the analytical method and physical considerations, we show that there are three phases on the $\tilde J_2 - \tilde J_3$ plane (…
We consider the spin-1/2 antiferromagnetic Heisenberg model on three frustrated lattices (the diamond chain, the dimer-plaquette chain and the two-dimensional square-kagome lattice) with almost dispersionless lowest magnon band. Eliminating high-energy degrees of freedom at high magnetic fields, we construct low-energy effective Hamiltonians which are much simpler than the initial ones. These effective Hamiltonians allow a more extended analytical and numerical analysis. In addition to the standard strong-coupling perturbation theory we also use a localized-magnon based approach leading to a s…
We investigate the mixed diamond chain composed of spins 1 and 1/2 when the exchange interaction is alternatingly distorted. Depending on the strengths of frustration and distortion, this system has various ground states. Each ground state consists of an array of spin clusters separated by singlet dimers by virtue of an infinite number of local conservation laws. We determine the ground state phase diagram by numerically analyzing each spin cluster. In particular for strong distortion, we find an infinite series of quantum phase transitions by the cluster expansion method and conformal field t…
We investigate a mechanism to establish ground-state ferromagnetism in flat-band Hubbard systems based on a kind of {\it order-from-disorder} effect driven by dispersion. As a paradigm we consider a frustrated diamond chain, where for ideal diamond geometry the lowest one-electron band is flat, but the ground state remains paramagnetic for arbitrary on-site repulsion $U$. We focus on half filling of the flat band. By using numerical and analytical arguments we present the ground-state phase diagram for a distorted diamond chain, i.e., the former flat band becomes dispersive. Driven by the inte…
Magnetism and spin dynamics of Na$_5$Yb(MoO$_4$)$_4$: A weakly interacting rare-earth stretched diamond lattice
National Taiwan University（国立台湾大学）；Academia Sinica（中央研究院）；Sungkyunkwan University（成均馆大学）；KEK（高能加速器研究机构）；National Tsing Hua University（国立清华大学）；National Synchrotron Radiation Research Center（国家同步辐射研究中心）；University of Johannesburg（约翰内斯堡大学）
Thermal conductivity (κ) of a distorted spin diamond-chain system, Cu_3(CO_3)_2(OH)_2, is studied at low temperatures down to 0.3 K and in magnetic fields up to 14 T. In zero field, the κ(T) curve with heat current along the chain direction has very small magnitudes and shows a pronounced three-peak structure. The magnetic fields along and perpendicular to the chains change the κstrongly in a way having good correspondence to the changes of magnetic specific heat in fields. The data analysis based on the Debye model for phononic thermal conductivity indicates that the heat transport is due to …
We consider the $s=1/2$ transverse field $XY$ model on the frustrated diamond chain, considering anisotropic exchange parameters between nearest-neighbor spins. To this end, we employ three different methodologies: mean-field approximations, and state-of-the-art exact diagonalizations (ED), and density matrix renormalization group (DMRG) simulations. Within a mean-field theory, the Hamiltonian is fermionized by introducing the Jordan-Wigner transformation, and the interacting (many-body) terms are approximated to single-particle ones by a Hartree-Fock approach. We analyze the behavior of the i…
