Magnetism and Thermodynamics of Spin-1/2 Heisenberg Diamond Chains in a Magnetic Field

TL;DR

This paper investigates the magnetic and thermodynamic properties of spin-1/2 Heisenberg diamond chains using advanced numerical methods, revealing coupling-independent peaks in structure factors and temperature-dependent double-peak features in susceptibility and heat capacity.

Contribution

It provides a comprehensive analysis of different coupling regimes in Heisenberg diamond chains and introduces new fitting equations for the static structure factor based on low-lying excitations.

Findings

Static structure factor peaks at specific wavevectors in zero field

Coupling-independent peaks at certain wavevectors in magnetization plateau

Double-peak structures in susceptibility and specific heat at finite temperatures

Abstract

The magnetic and thermodynamic properties of spin-1/2 Heisenberg diamond chains are investigated in three different cases: (a) J1, J2, J3>0 (frustrated); (b) J1, J3<0, J2>0 (frustrated); and (c) J1, J2>0, J3<0 (non-frustrated). The density matrix renormalization group (DMRG) technique is invoked to study the properties of the system in the ground state, while the transfer matrix renormalization group (TMRG) technique is applied to explore the thermodynamic properties. The local magnetic moments, spin correlation functions, and static structure factors are discussed in the ground state for the three cases. It is shown that the static structure factor S(q) shows peaks at wavevectors $q=a\pi /3$ (a=0,1,2,3,4,5) for different couplings in a zero magnetic field, which, however in the magnetic fields where the magnetization plateau with m=1/6 pertains, exhibits the peaks only at q=0, $2\pi /3$ and $4\pi /3$, which are found to be couplings-independent. The DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes, where two fitting equations are proposed. It is observed that the six modes are closely related to the low-lying excitations of the system. At finite temperatures, the double-peak structures of the susceptibility and specific heat against temperature are obtained, where the peak positions and heights are found to depend on the competition of the couplings. It is also uncovered that the XXZ anisotropy of F and AF couplings leads the system of case (c) to display quite different behaviors. In addition, the experimental data of the susceptibility, specific heat and magnetization for the compound Cu$_{3}$(CO$_{3}$)$_{2}$(OH)$_{2}$ are fairly compared with our TMRG results.

Figures

Equations

$$\mathcal{H}$$

$${\psi }_i$$

$$m(H)=m_p(\frac{H}{H_{c_1}})[1+{\alpha }_1(1-\frac{H}{H_{c_1}})-{\alpha }_2(1-\frac{H}{H_{c_1}}{)}^{2/3}],$$

$$m(H)$$

$$S(q)=\sum _j{e}^{iqj}⟨S_j^z{S}_0^z⟩,$$

$$S(q)=\frac{S(S+1)}{3}-\frac{\alpha (\cos q+e^{-\beta })}{\cos q+\cosh \beta },$$

$$S(q)$$

$$S(q)=\sum _{l=1}^6{\alpha }_l\sum _{m=0}^{\infty }2(6m+l{)}^{-\beta }\cos [(6m+1)q]+\frac{1}{4},$$

$${\psi }_i=\frac{1}{\sqrt{2}}(|{\uparrow }_{3i-2}{\uparrow }_{3i-1}{\downarrow }_{3i}⟩\pm |{\uparrow }_{3i-2}{\downarrow }_{3i-1}{\uparrow }_{3i}⟩).$$

$${\psi }_i$$

$$⟨{\mathbf{S}}_j\cdot {\mathbf{S}}_0⟩\propto (-1{)}^j\frac{1}{(2\pi {)}^{3/2}}\frac{\sqrt{\ln j}}{j}.$$

$$\mathcal{H}$$

$$S_j^{+}$$

$$S_j^z$$

$$a_{3i-2}$$

$$a_{3i-1}$$

$$a_{3i}$$

$$H$$

$$a_{1k}$$

$$a_{2k}$$

$$a_{3k}$$

$$H=E_g+\sum _k\sum _{i=1}^3{ϵ}_{ik}{\alpha }_{ik}^{+}{\alpha }_{ik}.$$

$$\left(\begin{array}[]{lcr}\omega_{1}&\gamma_{1}&\gamma_{3}\\ \gamma_{1}^{\ast}&\omega_{2}&\gamma_{2}\\ \gamma_{3}^{\ast}&\gamma_{2}^{\ast}&\omega_{3}\end{array}\right)\left(\begin{array}[]{c}u_{1i}\\ u_{2i}\\ u_{3i}\end{array}\right)=\epsilon_{ik}\left(\begin{array}[]{c}u_{1i}\\ u_{2i}\\ u_{3i}\end{array}\right).$$

Full text

I INTRODUCTION

Low-dimensional quantum spin systems with competing interactions have become an intriguing subject in the last decades. Among many achievements in this area, the phenomenon of the topological quantization of magnetization has attracted much attention both theoretically and experimentally. A general necessary condition for the appearance of the magnetization plateaus has been proposed by Oshikawa, Yamanaka and Affleck (OYA) OYA , stating that for the Heisenberg antiferromagnetic (AF) spin chain with a single-ion anisotropy, the magnetization curve may have plateaus at which the magnetization per site $m$ is topologically quantized by $n(S-m)=integer$, where $S$ is the spin, and $n$ is the period of the ground state determined by the explicit spatial structure of the Hamiltonian. As one of fascinating models which potentially possesses the magnetization plateaus, the Heisenberg diamond chain, consisting of diamond-shaped topological unit along the chain, as shown in Fig. 1, has also gained much attention both experimentally and theoretically (e.g. Refs. Drillon ; Sakurai ; Ishii ; Fujisawa ; CuCOOH1 ; CuCOOH2 ; TKS ; aaa1 ; aaa2 ; aaa3 ; aaa4 ; aaa5 ; faf1 ; faf2 ).

It has been observed that the compounds, A3Cu3(PO4)4 with A = Ca, SrDrillon , and Bi4Cu3V2O14Sakurai can be nicely modeled by the Heisenberg diamond chain. Another spin-$1/2$ compound Cu3Cl6(H2O)2$\cdot$2H8C4SO2 was initially regarded as a model substance for the spin-$1/2$ diamond chain Ishii , but a later experimental research reveals that this compound should be described by a double chain model with very weak bond alternations, and the lattice of the compound is found to be Cu2Cl4$\cdot$H8C4SO2 Fujisawa . Recently, Kikuchi et al.CuCOOH1 have reported the experimental results on a spin-$1/2$ compound Cu3(CO3)2(OH)2, where local Cu2+ ions with spin $S=1/2$ are arranged along the chain direction, and the diamond-shaped units consist of a one-dimensional (1D) lattice. The $1/3$ magnetization plateau and the double peaks in the magnetic susceptibility as well as the specific heat as functions of temperature have been observed experimentallyCuCOOH1 ; CuCOOH2 , which has been discussed in terms of the spin-$1/2$ Heisenberg diamond chain with AF couplings $J_{1}$, $J_{2}$ and $J_{3}>0$.

On the theoretical aspect, the frustrated diamond spin chain with AF interactions $J_{1}$, $J_{2}$ and $J_{3}>0$ was studied by a few groups. The first diamond spin chain was explored under a symmetrical condition $J_{1}=J_{3}$TKS . Owing to the competition of AF interactions, the phase diagram in the ground state of the spin-$1/2$ frustrated diamond chain was found to contain different phases, in which the magnetization plateaus at $m=1/6$ as well as $1/3$ are predictedaaa1 ; aaa2 ; aaa3 ; aaa4 ; aaa5 . Another frustrated diamond chain with ferromagnetic (F) interactions $J_{1}$, $J_{3}<0$ and AF interaction $J_{2}>0$ was also investigated theoretically, which can be experimentally realized if all angles of the exchange coupling bonds are arranged to be around $90^{\circ}$, a region where it is usually hard to determine safely the coupling constants and even about their signsfaf1 ; faf2 . Despite of these works, the investigations on the Heisenberg diamond spin chain with various competing interactions are still sparse.

Motivated by the recent experimental observation on the azurite compound Cu3(CO3)2(OH)2CuCOOH1 ; CuCOOH2 , we shall explore systematically the magnetic and thermodynamic properties of the spin-$1/2$ Heisenberg diamond chain with various competing interactions in a magnetic field, and attempt to fit into the experimental observation on the azurite in a consistent manner. The density matrix renormalization group (DMRG) as well as the transfer matrix renormalization group (TMRG) techniques will be invoked to study the ground-state properties and thermodynamics of the model under interest, respectively. The local magnetic moments, spin correlation functions, and static structure factors will be discussed for three cases at zero temperature. It is found that the static structure factor $S(q)$ shows peaks in zero magnetic field at wavevectors $q=a\pi/3$ $(a=0,1,2,3,4,5)$ for different couplings, while in the magnetic fields where the magnetization plateau with $m=1/6$ remains, the peaks appear only at wavevectors $q=0$, $2\pi/3$ and $4\pi/3$, which are found to be couplings-independent. These information could be useful for further neutron studies. The double-peak structures of the susceptibility and specific heat against temperature are obtained, where the peak positions and heights are found to depend on the competition of the couplings. It is uncovered that the XXZ anisotropy of F and AF couplings leads the system without frustration (see below) to display quite different behaviors. In addition, the experimental data of the susceptibility, specific heat and magnetization for the compound Cu3(CO3)2(OH)2 are fairly compared with our TMRG results.

The rest of this paper is outlined as follows. In Sec. II, we shall introduce the model Hamiltonian for the spin-$1/2$ Heisenberg diamond chain with three couplings $J_{1}$, $J_{2}$ and $J_{3}$, where three particular cases are identified. In Sec. III, the magnetic and thermodynamic properties of a frustrated diamond chain with AF interactions $J_{1}$, $J_{2}$ and $J_{3}>0$ will be discussed. In Sec. IV, the physical properties of another frustrated diamond chain with F interactions $J_{1}$, $J_{3}<0$ and AF interaction $J_{2}>0$ will be considered. In Sec. V, the magnetism and thermodynamics of a non-frustrated diamond chain with AF interactions $J_{1}$, $J_{2}>0$ and F interaction $J_{3}<0$ will be explored, and a comparison to the experimental data on the azurite compound will be made. Finally, a brief summary and discussion will be presented in Sec. VI.

II MODEL

The Hamiltonian of the spin-$1/2$ Heisenberg diamond chain reads

[TABLE]

where $\mathbf{S}_{j}$ is the spin operator at the $jth$ site, $L$ is the total number of spins in the diamond chain, $J_{i}$ ($i=1,2,3$) stands for exchange interactions, $\mathbf{H}$ is the external magnetic field, $g\mu_{B}=1$ and $k_{B}=1$. $J_{i}>0$ represents the AF coupling while $J_{i}<0$ the F interaction. There are three different cases particularly interesting, as displayed in Fig. 1, which will be considered in the present paper: (a) a frustrated diamond chain with $J_{1}$, $J_{2}$, $J_{3}>0$; (b) a frustrated diamond chain with competing interactions $J_{1}$, $J_{3}<0$, $J_{2}>0$; and (c) a diamond chain without frustration with $J_{1}$, $J_{2}>0$, $J_{3}<0$. It should be remarked that in the case (c) of this model, since the two end points of the $J_{2}$ bond represent the two different lattice sites, it is possible that $J_{1}$ and $J_{3}$ can be different, even their signs.

The magnetic properties and thermodynamics for the aforementioned three spin-$1/2$ Heisenberg diamond chains in the ground states and at finite temperatures will be investigated by means of the DMRG and TMRG methods, respectively. As the DMRG and TMRG techniques were detailed in two nice reviewsReview1 ; Review2 , we shall not repeat the technical details for concise. In the ground-state calculations, the total number of spins in the diamond chain is taken at least as $L=120$. At finite temperatures, the thermodynamic properties presented below are calculated down to temperature $T=0.05$ (in units of $|J_{1}|$) in the thermodynamic limit. In our calculations, the number of kept optimal states is taken as $81$; the width of the imaginary time slice is taken as $\varepsilon=0.1$; the Trotter-Suzuki error is less than $10^{-3}$; and the truncation error is smaller than $10^{-6}$.

III A Frustrated Heisenberg Diamond Chain ($J_{1},J_{2},J_{3}>0$)

III.1 Local Magnetic Moment and Spin Correlation Function

Figure 2(a) manifests the magnetization process of a frustrated spin-$1/2$ Heisenberg diamond chain with the couplings satisfying $J_{1}:J_{2}:J_{3}=1:2:2$ at zero temperature. The plateau of magnetization per site $m=1/6$ is observed. According to the OYA necessary condition OYA , the $m$ = $1/6$ plateau of a spin-$1/2$ Heisenberg chain corresponds to the period of the ground state $n=3$. Beyond the magnetization plateau region, the magnetic curve goes up quickly with increasing the magnetic field $H$. Above the upper critical field, the magnetic curve shows a s-like shape. To further look at how this magnetization plateau appears, the spatial dependence of the averaged local magnetic moment $\langle S_{j}^{z}\rangle$ in the ground states under different external fields is presented, as shown in Fig. 2(b). It is seen that in the absence of external field, the expectation value $\langle S_{j}^{z}\rangle$ changes its sign at every three sites within a very small range of $(-10^{-3},10^{-3})$ because of quantum fluctuations, resulting in the magnetization per site $m=\sum_{j=1}^{L}\langle S_{j}^{z}\rangle/L=0$. $\langle S_{j}^{z}\rangle$ increases with increasing the magnetic field, and oscillates with increasing $j$, whose unit of three spins is gradually divided into a pair and a single, as displayed in Fig. 2(c). At the field $H/J_{1}=1.5$, as demonstrated in Fig. 2(d), the behavior of $\langle S_{j}^{z}\rangle$ falls into a perfect sequence such as $\{...,(S_{a},S_{a},S_{b}),...\}$ with $S_{a}=0.345$ and $S_{b}=-0.190$, giving rise to the magnetization per site $m=1/6$. In addition, such a sequence remains with increasing the magnetic field till $H/J_{1}=2.5$, implying that the $m=1/6$ plateau appears in the range of $H/J_{1}=1.5\sim 2.5$, as manifested in Fig. 2(a). When the field is promoted further, the sequence changes into a waved succession with smaller swing of $(S_{a}-S_{b})$, as shown in Fig. 2(e), which corresponds to the fact that the plateau state of $m=1/6$ is destroyed, and gives rise to a s-like shape of $M(H)$. It is noting that when the plateau state of $m=1/6$ is destroyed, the increase of $m$ at first is mainly attributed to a rapid lift of $S_{b}$, and later, the double $S_{a}$ start to flimsily increase till $S_{a}=S_{b}=0.5$ at the saturated field.

The physical picture for the above results could be understood as follows. For the $m=1/6$ plateau state at $J_{1}:J_{2}:J_{3}=1:2:2$, we note that if an approximate wave function defined byaaa5

[TABLE]

where $\uparrow_{j}(\downarrow_{j})$ denotes spin up (down) on site $j$, is applied, one may obtain $\langle\psi_{i}|S_{3i-2}^{z}|\psi_{i}\rangle=1/3$, $\langle\psi_{i}|S_{3i-1}^{z}|\psi_{i}\rangle=1/3$, $\langle\psi_{i}|S_{3i}^{z}|\psi_{i}\rangle=-1/6$, giving rise to a sequence {…, ($\frac{1}{3}$, $\frac{1}{3}$, $-\frac{1}{6}$), …}, and $m=(\frac{1}{3}+\frac{1}{3}-\frac{1}{6})/3=1/6$, which is in agreement with our DMRG results {…, ($0.345$, $0.345$, $-0.190$), …}. This observation shows that the ground state of this plateau state might be described by trimerized states.

Let $H_{c_{1}}$ and $H_{c2}$ be the lower and upper critical magnetic field at which the magnetization plateau appears and is destructed, respectively. For $H_{c_{1}}\leq H\leq H_{c2}$, the magnetization $m=m_{p}=1/6$, namely, the system falls into the magnetization plateau state. For $0\leq H\leq H_{c_{1}}$ and $J_{1}:J_{2}:J_{3}=1:2:2$, the magnetization curve shows the following behavior

[TABLE]

where $H_{c_{1}}/J_{1}=1.44$, the parameters $\alpha_{1}=2/3$ and $\alpha_{2}=1$. Obviously, when $H=0$, $m=0$; $H=H_{c_{1}}$, $m=m_{p}$. A fair comparison of Eq. (3) to the DMRG results is presented in Fig. 3(a). For $H_{c2}\leq H\leq H_{s}$ and $J_{1}:J_{2}:J_{3}=1:2:2$, where $H_{s}$ is the saturated magnetic field, the magnetization curve has the form of

[TABLE]

where $k_{c}=(m_{s}-m_{p})/(H_{s}-H_{c_{2}})$ with $m_{s}$ the saturation magnetization, and $\beta_{1}$, $\beta_{2}$ the parameters. One may see that when $H=H_{c_{2}}$, $m=m_{p}$; $H=H_{s}$, $m=m_{s}$. A nice fitting to the DMRG result gives the parameters $H_{c_{2}}/J_{1}=3.15$, $m_{s}=1/2$, $H_{s}/J_{1}=4.55$, $\beta_{1}=0.143$, $\beta_{2}=0.178$ and $k_{c}=0.238$, as shown in Fig. 3(b). It should be remarked that from the phenomenological Eqs.(3) and (4), we find that, away from the plateau region, the magnetic field dependence of the magnetization of this model differs from those of Haldane-type spin chains where $m(H)\sim(H-H_{c1})^{1/2}$.

To explore further the magnetic properties of the frustrated spin-$1/2$ Heisenberg diamond chain in the ground states with the couplings $J_{1}:J_{2}:J_{3}=1:2:2$ at different external fields, let us look at the static structure factor $S(q)$ which is defined as

[TABLE]

where $q$ is the wave vector, and $\langle S_{j}^{z}S_{0}^{z}\rangle$ is the spin correlation function in the ground state. As demonstrated in Fig. 4(a), in the absence of the external field, $S(q)$ shows three peaks: two at $q=\pi/3$, $5\pi/3$, and one at $q=\pi$, which is quite different from that of the spin $S=1/2$ Heisenberg AF chain, where $S(q)$ only diverges at $q=\pi$. As indicated by Eq. (5), the peaks of $S(q)$ reflect the periods of the spin correlation function $\langle S_{j}^{z}S_{0}^{z}\rangle$, i.e., the peaks at $q=\pi/3$ ($5\pi/3$) and $\pi$ reflect the periods of $6$ and $2$ for $\langle S_{j}^{z}S_{0}^{z}\rangle$, respectively. As shown in Fig. 4(b), in the absence of the external field, $\langle S_{j}^{z}S_{0}^{z}\rangle$ changes sign every three sites, which corresponds really to the periods of $6$ and $2$. With increasing the magnetic field, the small peak of $S(q)$ at $q=\pi$ becomes a round valley while the peak at $q=\pi/3$ ($5\pi/3$) continuously shifts towards $q=2\pi/3$ ($4\pi/3$) with the height enhanced, indicating the corruption of the periods of $6$ and $2$ but the emergence of the new period $\in$ $(3,6)$ for $\langle S_{j}^{z}S_{0}^{z}\rangle$, as shown in Fig. 4(c). At the field $H/J_{1}=1.5$, two peaks shift to $q=2\pi/3$ and $4\pi/3$ respectively, and merge into the peaks already existing there, showing the existence of the period 3 for $\langle S_{j}^{z}S_{0}^{z}\rangle$, as clearly displayed in Fig. 4(d). The valley and peaks of $S(q)$ keep intact in the plateau state at $m=1/6$. When the plateau state is destroyed at the field $H/J_{1}=4$, the peaks at $q=2\pi/3$ and $4\pi/3$ are depressed dramatically while the peaks at $q=\pi/3$, $5\pi/3$ and $\pi$ appear again with very small heights, revealing the absence of the period $3$ and the slight presence of the period $2$ and $6$, as shown in Fig. 4(e). At the field $H/J_{1}=4.8$, all peaks disappear and become zero, except for the peak at $q=0$, which is the saturated state. Therefore, the static structure factor $S(q)$ shows different characteristics in different magnetic fieldsnote1 . On the other hand, it is known that $S(q)$ also reflects the low-lying excitations of the system. It is thus reasonable to expect that the low-lying excitations of the frustrated diamond chain will behave differently in different magnetic fields.

To investigate the zero-field static structure factor $S(q)$ in the ground state for the frustrated spin-$1/2$ diamond chains with various AF couplings, the four cases with $J_{1}=1$, $J_{3}>0$, and $J_{2}=0.5$, $1$, $2$, and $4$ are shown in Figs. 5(a)-5(d), respectively. For $J_{2}=0.5$, as shown in Fig. 5(a), $S(q)$ displays a sharp peak at $q=\pi$ when $J_{3}<0.5$, and three peaks at $q=0$, $2\pi/3$ and $4\pi/3$ when $J_{3}>0.5$. It is shown from the ground state phase diagram aaa2 that the system is in the spin fluid (SF) phase when $J_{1}=1$, $J_{2}=0.5$, $J_{3}<0.5$, and enters into the ferrimagnetic (FRI) phase when $J_{1}=1$, $J_{2}=0.5$, $J_{3}>0.5$. For $J_{2}=1$, as indicated in Fig. 5(b), the incommensurate peaks exist, such as the case of $J_{3}=0.8$, where the system is in the dimerized (D) phase aaa2 . For $J_{2}=2$, as manifested in Fig. 5(c), $S(q)$ has a sharp peak at $q=\pi$ and two ignorable peaks at $\pi/3$ and $5\pi/3$ when $J_{3}$ $<1$; three sharp peaks at $q=0$, $2\pi/3$ and $4\pi/3$ at $J_{3}=1$; a round valley ($J_{3}=1.5$) or a small peak ($J_{3}=4$) at $q=\pi$ and two mediate peaks at $\pi/3$ and $5\pi/3$ when $J_{3}>1$. The system with $J_{1}=1$ and $J_{2}=2$ is in the D phase when $1<J_{3}<2.8$, and in the SF phase when $J_{3}<1$ or $J_{3}>2.8$ aaa2 . For $J_{2}=4$ revealed in Fig. 5(d), the situations are similar to that of Fig. 5(c), but here only the SF phase exists for the system with $J_{1}=1$, $J_{2}=4$ and $J_{3}>0$ aaa2 . It turns out that even in the same phase, such as the SF phase, the zero-field static structure factor $S(q)$ could display different characteristics for different AF couplings. In fact, we note that the exotic peak of $S(q)$ has been experimentally observed in the diamond-typed compound Sr3Cu3(PO4)4 SrCu .

If the spin correlation function for the spin-S chain can be expressed as $\langle S_{j}^{z}S_{0}^{z}\rangle=\alpha(-1)^{j}e^{-j\beta}$, where $\alpha$ and $\beta$ are two parameters, its static structure factor will take the form of

[TABLE]

which can recover exactly the $S(q)$ of the spin-S AKLT chain $S(q)=\frac{S+1}{3}\frac{1-\cos{q}}{1+\cos{q}+2/S(S+2)}$ SQAKLT with $\alpha=(S+1)^{2}/3$ and $\beta=\ln(1+2/S)$. Eq.(6) has a peak at wavevector $q=\pi$. By noting that the zero-field static structure factor $S(q)$ for the frustrated diamond chains displays peaks at wavevectors $q=a\pi/3$ $(a=0,1,2,3,4,5)$ for different AF couplings, the spin correlation function $\langle S_{j}^{z}S_{0}^{z}\rangle$ could be reasonably divided into six modes $\langle S_{6m+l}^{z}S_{0}^{z}\rangle=c_{l}+\alpha_{l}e^{-(6m+l)\beta}$ or $\langle S_{6m+l}^{z}S_{0}^{z}\rangle=\alpha_{l}(6m+l)^{-\beta}$ with $j=6m+l$ and $l=1,2,...,6$, whose contributions to the static structure factor should be considered separatelynote2 . Thus, the static structure factor for the present systems could be mimicked by a superposition of six modes, which leads to

[TABLE]

or

[TABLE]

respectively, depending on which phase the system falls into, where $\alpha_{l}$, $c_{l}$ and $\beta$ are couplings-dependent parameters.

As presented in Fig. 6, the DMRG results of the static structure factor as a function of wavevector are fitted by Eqs. (7) and (8) for the spin-$1/2$ frustrated Heisenberg diamond chains with various AF couplings in zero magnetic field. It can be found that the characteristic peaks can be well fitted by Eqs. (7) and (8), with only a slightly quantitative deviation, showing that the main features of the static structure factor for the present systems can be reproduced by a linear superposition of six modes. The fitting results give six different modes in general, as shown in Fig. 4(b).

To further understand the above-mentioned behaviors of the zero-field static structure factor, $S(q)$, we have applied the Jordan-Wigner (JW) transformation to study the low-lying excitations of the spin-$1/2$ frustrated Heisenberg diamond chain with various AF couplings (see Appendix A for derivations). It can be seen that the zero-field low-lying fermionic excitation $\varepsilon(k)$ behaves differently for different AF couplings, as shown in Figs. 7(a)-(d). Obviously, these low-lying excitations are responsible for the DMRG calculated behaviors of S(q), where the positions of minimums of $\varepsilon(k)$ for different AF couplings, as indicated by arrows in Figs. 7(a)-(c), are exactly consistent with the locations of the peaks of zero-field static structure factor $S(q)$ shown in Figs. 6(a)-(c), respectively, although there is a somewhat deviation for Fig. 7(d) and Fig. 6(d). It also shows that the six modes suggested by Eqs. (7) and (8) is closely related to the low-lying excitations of the system.

III.2 Magnetization, Susceptibility and Specific Heat

The magnetization process for the spin-$1/2$ frustrated diamond chain with $J_{1}=1$, $J_{3}>0$, and $J_{2}=0.5$ and $2$ is shown in Fig. 8(a) and 8(b), respectively, where temperature is fixed as $T/J_{1}=0.05$. It is found that the magnetization exhibits different behaviors for different AF couplings: a plateau at $m=1/6$ is observed, in agreement with the ground state phase diagram aaa4 ; aaa5 ; for $J_{2}=0.5$, as shown in Fig. 8(a), the larger $J_{3}$ is, the larger the width of the plateau at $m=1/6$ becomes; for $J_{2}=2$, as presented in Fig. 8(b), the width of the plateau at $m=1/6$ becomes larger with increasing $J_{3}<1$, and then turns smaller with increasing $J_{3}>1$; for $J_{3}<1$, the larger $J_{2}$, the larger the width of the plateau at $m=1/6$; for $J_{3}=2$, the larger $J_{2}$, the smaller the width of the plateau at $m=1/6$. The saturated field is obviously promoted with increasing AF $J_{3}$ and $J_{2}$.

Figures 8(c) and 8(d) give the susceptibility $\chi$ as a function of temperature $T$ for the spin-$1/2$ frustrated diamond chain with $J_{1}=1$, $J_{3}>0$ and $J_{2}$ $=0.5$ and $2$, respectively, while the external field is taken as $H/J_{1}=0.01$. For $J_{2}=0.5$, as shown in Fig. 8(c), the low temperature part of $\chi(T)$ keeps finite when $J_{3}<0.5$, and becomes divergent when $J_{3}>0.5$. As clearly manifested in the inset of Fig. 8(c), $J_{3}=0.5$ is the critical value, which is consistent with the behaviors of static structure factor $S(q)$ in Fig. 5(a). For $J_{2}=2$, as shown in Fig. 8(d), an unobvious double-peak structure at low temperature is observed at small and large $J_{3}$ such as $0.2$ and $3$. The temperature dependence of the specific heat $C$ with $J_{1}=1$, $J_{3}>0$ and $J_{2}=0.5$ and $2$ is shown in Fig. 8(e) and 8(f), respectively, while the external field is fixed as $H/J_{1}=0.01$. For $J_{2}=0.5$, as given in Fig. 8(e), a double-peak structure of $C(T)$ is observed at low temperature for small and large $J_{3}$ such as $0.3$ and $1$. The case with $J_{2}=2$ shown in Fig. 8(f) exhibits the similar characteristics. Thus, the thermodynamics of the system demonstrate different behaviors for different AF couplings. As manifested in Fig. 5, the low-lying excitations behave differently for different AF couplings. The double-peak structure of the susceptibility as well as the specific heat could be attributed to the excited gaps in the low-lying excitation spectrumGuSu2 .

IV A Frustrated Diamond Chain with Competing Interactions ($J_{1},J_{3}<0,J_{2}>0$)

IV.1 Local Magnetic Moment and Spin Correlation Function

Figure 9(a) shows the magnetization process of a frustrated spin-$1/2$ Heisenberg diamond chain in the ground states with the couplings $J_{1}:J_{2}:J_{3}=-1:4:-0.5$. The plateau of magnetization per site $m=1/6$ is clearly obtained. To understand the occurrence of the magnetization plateau, the spatial dependence of the averaged local magnetic moment $\langle S_{j}^{z}\rangle$ in the ground states at different external fields is calculated. It is seen that in the absence of the magnetic field, as presented in Fig. 9(b), the expectation values of $\langle S_{j}^{z}\rangle$ change sign every three sites within a very small range of $(-2\times 10^{-4},2\times 10^{-4})$, resulting in the magnetization per site $m=0$. With increasing the field, the expectation values of $\langle S_{j}^{z}\rangle$ increase, whose unit of three spins is gradually divided into a pair and a single, as displayed in Fig. 9(c). At the field $H/|J_{1}|=0.05$, as illustrated in Fig. 9(d), the behavior of $\langle S_{j}^{z}\rangle$ shows a perfect sequence such as $\{...,(S_{a},S_{b},S_{b}),...\}$ with $S_{a}=0.496$ and $S_{b}=0.002$, giving rise to the magnetization per site $m=1/6$. In addition, the sequence is fixed with increasing the field until $H/|J_{1}|=3.2$, corresponding to the plateau of $m=1/6$. As the field is enhanced further, the double $S_{b}$ begin to rise, and the sequence becomes a waved series with smaller swing of $(S_{a}-S_{b})$ as revealed in Fig. 9(e), which corresponds to the plateau state at $m=1/6$ that is destroyed. It is noting that the increase of $m$ is mainly attributed to the promotion of double $S_{b}$, as $S_{a}$ is already saturated until $S_{a}=S_{b}=0.5$ at the saturated field.

As discussed above, the physical picture of the $m=1/6$ plateau state at $J_{1}:J_{2}:J_{3}=-1:4:-0.5$ can be understood by the following approximate wave function

[TABLE]

By use of this wave function, we have $\langle\psi_{i}|S_{3i-2}^{z}|\psi_{i}\rangle=1/2$, $\langle\psi_{i}|S_{3i-1}^{z}|\psi_{i}\rangle=0$, $\langle\psi_{i}|S_{3i}^{z}|\psi_{i}\rangle=0$, leading to a sequence of $\{...,(1/2,0,0),...\}$, and $m=(1/2+0+0)/3=1/6.$ This is in agreement with our DMRG results $\{...,(0.496,0.002,0.002),...\}$.

The static structure factor $S(q)$ of the frustrated spin-$1/2$ Heisenberg diamond chain with the competing couplings $J_{1}:J_{2}:J_{3}=-1:4:-0.5$ in the ground states is considered in different external fields. At zero field, shown in Fig. 10(a), $S(q)$ has three peaks at $q=\pi/3$, $5\pi/3$ and $\pi$ with mediate heights, which reflects the periods of $6$ and $2$ for $\langle S_{j}^{z}S_{0}^{z}\rangle$, respectively. As shown in Fig. 10(b), in the absence of the external field, $\langle S_{j}^{z}S_{0}^{z}\rangle$ changes sign every three sites, corresponding to the periods of $6$ and $2$. With increasing the field, the peak at $q=\pi$ becomes flat with height depressed forwardly, while the peak at $q=\pi/3$ ($5\pi/3$) is divided into two peaks shifting oppositely from $q=\pi/3$ ($5\pi/3$) with height decreased, indicating the corruption of the periods of $6$ and $2$ and the emergence of new periods for $\langle S_{j}^{z}S_{0}^{z}\rangle$, as shown in Fig. 10(c). At the field $H/|J_{1}|=0.05$, two shifting peaks have respectively reached $q=2\pi/3$ and $4\pi/3$, and are merged with the existing peaks, showing the occurrence of period $3$ for $\langle S_{j}^{z}S_{0}^{z}\rangle$, as clearly displayed in Fig. 10(d). The flat and peaks keep constant during the plateau state at $m=1/6$. When the plateau state is destroyed at field $H/J_{1}=3.27$, the peaks at $q=2\pi/3$ and $4\pi/3$ are depressed sharply, revealing the decay of the period $3$, as shown in Fig. 10(e). At the saturated field of $H/J_{1}=3.3$, all peaks disappear and become flat with the value zero, which is the saturated state. So, the static structure factor $S(q)$ shows different characteristics in different magnetic fields. Similar to the discussions in Fig. 4, the low-lying excitations of this frustrated diamond chain would also behave differently in different magnetic fields.

To further investigate the zero-field static structure factor $S(q)$ in the ground state for the frustrated spin-$1/2$ diamond chains with various $J_{1}$, $J_{3}<0$ and $J_{2}>0$, two cases with $J_{1}=-1$, $J_{3}<0$ and $J_{2}=1$ and $4$ are illustrated in Fig. 11(a) and 11(b), respectively. For $J_{2}=1$, $S(q)$ shows a round peak at $q=\pi$, two sharp peaks at $\pi/3$ and $5\pi/3$ when $|J_{3}|<1$, and a very sharp peak at $q=0$ when $|J_{3}|>1$. For $J_{2}=4$, $S(q)$ shows three peaks at $q=\pi/3$, $\pi$ and $5\pi/3$ as $|J_{3}|<1$, and a very sharp peak at $q=\pi$ and nearly ignorable peaks at $\pi/3$ and $5\pi/3$ as $|J_{3}|>1$. In general, the zero-field static structure factor $S(q)$ shows different characteristics with different competing couplings, whose exotic characteristics could be experimentally observed in the related diamond-typed compounds.

As shown in Fig. 12, the DMRG results of the static structure factor as a function of wavevector are fitted by Eq.(7) for the spin-$1/2$ frustrated Heisenberg diamond chains with $J_{1}$, $J_{3}<0$ and $J_{2}>0$. It can be found that the characteristic behaviors can be nicely fitted by Eq.(7), with only a slightly quantitative deviation, showing that the main features of the static structure factor for the present systems can be captured by a superposition of six modes. It is consistent with the fact that the spin correlation function for the present systems has six different modes, as manifested in Fig. 10(b).

* *Similar to the case (a) with all AF couplings in the last section, the characteristics of zero-field $S(q)$ for the spin-$1/2$ frustrated Heisenberg diamond chain with $J_{1}$, $J_{3}<0$ and $J_{2}>0$ can be further undersood in terms of the low-lying excitations of the system (see Appendix A). By means of the JW transformation, the zero-field low-lying fermionic excitation $\varepsilon(k)$ of the present case is calculated, as shown in Figs. 13(a)-(b). It is found that the zero-field low-lying fermionic excitation $\varepsilon(k)$ differs for different couplings, but the positions of minimums of $\varepsilon(k)$ for different couplings, as indicated by arrows in Figs. 13(a) and (b), appear to be the same. One may see that these positions coincide exactly with the locations of peaks of the zero-field static structure factor $S(q)$ manifested in Figs. 12(a) and (b), respectively, showing that our fitting equation is qualitatively consistent with the low-lying excitations of the system.

IV.2 Magnetization, Susceptibility and Specific Heat

Figures 14(a) and 14(b) show the magnetization process for the spin-$1/2$ frustrated diamond chain at a finite temperature $T/|J_{1}|=0.05$ with $J_{1}=-1$, $J_{3}<0$, and $J_{2}=1$ and $4$, respectively. It is shown that the magnetization exhibits different behaviors for different $J_{1}$, $J_{3}<0$ and $J_{2}>0$. A plateau at $m=1/6$ is observed at small $|J_{3}|$; with a fixed $J_{2}$, the larger $|J_{3}|$, the smaller the width of the plateau at $m=1/6$, and the plateau disappears when $|J_{3}|$ exceeds the critical value; for a fixed $J_{3}$, the larger $J_{2}$, the wider the width of the plateau at $m=1/6$; the saturation field is obviously depressed with the increase of $|J_{3}|$ at a fixed $J_{2}$, and is enhanced with the increase of $J_{2}$ at a fixed $J_{3}$.

Figures 14(c) and 14(d) manifest the susceptibility $\chi$ as a function of temperature $T$ for the spin-$1/2$ frustrated diamond chain with $J_{1}=-1$, $J_{3}<0$ and $J_{2}=1$ and $4$, respectively, where the external field is taken as $H/|J_{1}|=0.01$. For $J_{2}=1$, the low temperature part of $\chi(T)$ keeps finite when $|J_{3}|<1$, and becomes divergent when $|J_{3}|>1$. As clearly revealed in the inset of Fig. 14(c), $J_{3}=-1$ is the critical value, which is in agreement with the behaviors of static structure factor $S(q)$ shown in Fig. 11(a). For $J_{2}=4$, a clear double-peak structure of $\chi(T)$ is obtained at $|J_{3}|=8$. The temperature dependence of the specific heat $C$ with $J_{1}=-1$, $J_{3}<$ [math] and $J_{2}=1$ and $4$ is shown in Figs. 14(e) and 14(f), respectively, where the external field is fixed as $H/|J_{1}|=0.01$. For $J_{2}=1$, a double-peak structure of $C(T)$ is observed for the case of $|J_{3}|=0.5$. The case with $J_{2}=4$ shown in Fig. 14(f) exhibits the similar characteristics. It is also found that, owing to the competitions among $J_{1}$, $J_{3}$ and $J_{2}$, the thermodynamics demonstrate rich behaviors at different couplings. As reflected in Fig. 11, the low-lying excitations behave differently with various F interactions $J_{1}$, $J_{3}$ and AF interaction $J_{2}$, while the excitation gaps could induce the double-peak structure in the susceptibility as well as in the specific heatGuSu2 .

V A Diamond Chain without Frustration ($J_{1},J_{2}>0,J_{3}<0$)

V.1 Local Magnetic Moment and Spin Correlation Function

Figure 15(a) shows the magnetization process of a non-frustrated spin-$1/2$ Heisenberg diamond chain in the ground states with the couplings satisfying $J_{1}:J_{2}:J_{3}=1:2:-0.5$. The plateau of magnetization per site $m=1/6$ is observed. The appearance of the magnetization plateau can be understood from the spatial dependence of the averaged local magnetic moment $\langle S_{j}^{z}\rangle$ in the ground states under different external fields. In absence of the external field, the expectation values of $\langle S_{j}^{z}\rangle$ change sign every one site with a waved swing within a very small range of $(-2\times 10^{-4},2\times 10^{-4})$, giving rise to the magnetization per site $m=0$. Under a finite field, every three successive spins have gradually cooperated into a pair and a single, as shown in Fig. 15(c). At $H/J_{1}=0.9$, as given in Fig. 15(d), $\langle S_{j}^{z}\rangle$ shows a perfect sequence such as $\{...,(S_{a},S_{b},S_{b}),...\}$ with $S_{a}=0.393$ and $S_{b}=0.053$, resulting in the magnetization per site $m=1/6$. Moreover, the sequence is fixed with the field increased until $H/J_{1}=1.8$, corresponding to the plateau state of $m=1/6$. As the field is increased further, double $S_{b}$ begin to increase, and the sequence becomes a waved succession with a smaller swing of $(S_{a}-S_{b})$, as manifested in Fig. 15(e), which corresponds to the fact that the plateau state at $m=1/6$ is destroyed. It is observed that the increase of $m$ at first is mainly attributed to the speedy boost of double $S_{b}$, and later, $S_{a}$ starts to increase weakly until $S_{a}=S_{b}=0.5$ at the saturated field.

For this non-frustrated case with couplings $J_{1}:J_{2}:J_{3}=1:2:-0.5$, the obtained perfect sequence of $\{...,(0.393,0.053,0.053),...\}$ for the $m=1/6$ plateau state could be understood by the following approximate trimerized wave function

[TABLE]

According to this function, we find $\langle\psi_{i}|S_{3i-2}^{z}|\psi_{i}\rangle=7/18$, $\langle\psi_{i}|S_{3i-1}^{z}|\psi_{i}\rangle=1/18$, $\langle\psi_{i}|S_{3i}^{z}|\psi_{i}\rangle=1/18$,* *giving rise to a sequence of $\{...,(7/18,1/18,1/18),...\}$. It turns out that $m=(7/18+1/18+1/18)/3=1/6$. This observation implies that the ground state of the plateau state can also be described by the trimerized states.

The static structure factor $S(q)$ of the non-frustrated spin-$1/2$ Heisenberg diamond chain in the ground states with the couplings $J_{1}:J_{2}:J_{3}=1:2:-0.5$ is probed in different external fields. As shown in Fig. 16(a), in absence of the external field, $S(q)$ shows a sharp peak at $q=\pi$, similar to the behaviors of the $S=1/2$ Heisenberg AF chain, which reflects the period of $2$ for $\langle S_{j}^{z}S_{0}^{z}\rangle$. As displayed in Fig. 16(b), at zero external field, $\langle S_{j}^{z}S_{0}^{z}\rangle$ changes sign every one lattice site, corresponding to the period of $2$. When the field is increased, the peak at $q=\pi$ becomes a flat with the height depressed greatly, while two new peaks with small heights at $q=2\pi/3$ and $4\pi/3$ appear, indicating the corruption of the period of $2$ and the emergence of the new period of $3$ for $\langle S_{j}^{z}S_{0}^{z}\rangle$, as demonstrated in Fig. 16(c). At the field $H/J_{1}=0.9$, the peaks at $q=0$, $2\pi/3$ and $4\pi/3$ become sharper. The flat and peaks of $S(q)$ keep unchanged during the plateau state at $m$ = $1/6$. When the plateau state is destroyed at the field $H/J_{1}=2.5$, the peaks at $q=2\pi/3$ and $4\pi/3$ are suppressed dramatically, revealing the decay of the period $3$, as shown in Fig. 16(e). At the field $H/J_{1}=2.8$, all peaks disappear, and become a flat with the value zero, except for the peak at $q$ $=0$, which is the saturated state. It can be stated that the static structure factor $S(q)$ shows various characteristics in different magnetic fields. Similar to what discussed in Figs. 4 and 10, the low-lying excitations of this non-frustrated diamond chain would also behave differently in different magnetic fields.

The zero-field static structure factor $S(q)$ in the ground state for the present system displays a peak at $q=\pi$ with different couplings, but whose static correlation function $\langle S_{j}^{z}S_{0}^{z}\rangle$ varies with the couplings. As illustrated in Fig. 17, only are the values of $\langle S_{j}^{z}S_{0}^{z}\rangle$ larger than zero presented for convenience. In order to gain deep insight into physics, for a comparison we also include the static correlation function for the $S=1/2$ Heisenberg antiferromagnetic (HAF) chain in Fig. 17(a), whose asymptotic behavior has the form of HAF1 ; HAF2

[TABLE]

Eq. (10) is depicted as solid lines in Figs. 17. To take the finite-size effect into account, the length of the diamond chain is taken as $L=90$, $120$ and $160$, respectively. As revealed in Fig. 17(a), the DMRG result of the $S=1/2$ HAF chain with an infinite length agrees well with the solid line. Fig. 17(b) shows that the static correlation function for the spin-$1/2$ diamond chain with frustrated couplings $J_{1}:J_{2}:J_{3}=1:1.2:0.5$ decays faster than that of the HAF chain. Compared with Figs. 17(c)-(f), it can be found that all the static correlation functions for the spin-$1/2$ non-frustrated diamond chain with AF interactions $J_{1}$, $J_{2}$ and F interaction $J_{3}$ fall more slowly than that of the $S=1/2$ HAF chain; for fixed AF interactions $J_{1}$ and $J_{2}$, the static correlation functions drop more leisurely with increasing the F interaction $|J_{3}|$; for fixed AF interaction $J_{1}$ and F interaction $J_{3}$, the static correlation functions decrease more rapidly with increasing the AF interaction $J_{2}$.

V.2 Magnetization, Susceptibility and Specific Heat

Figures 18(a) and 18(b) show the magnetization process for the spin-$1/2$ non-frustrated diamond chain at a finite temperature $T/J_{1}=0.05$ with $J_{1}=1$, $J_{3}<0$, and $J_{2}=0.5$ and $2$, respectively. It is found that the magnetization behaves differently with different AF interactions $J_{1}$, $J_{2}$ and F interaction $J_{3}$. A plateau at $m=1/6$ is obtained at small $|J_{3}|$; for fixed $J_{1}$ and $J_{2}$, the larger $|J_{3}|$, the narrower the width of the plateau at $m=1/6$, and after $|J_{3}|$ exceeds a critical value, the plateau at $m=1/6$ is eventually smeared out; for fixed $J_{1}$ and $J_{3}$, the larger $J_{2}$, the wider the width of the plateau at $m=1/6$; the saturated field is obviously unchanged with changing the F interaction $J_{3}$. The coupling-dependence of the spin-$1/2$ non-frustrated diamond chain with AF interactions $J_{1}$, $J_{2}$ and F interaction $J_{3}$ is similar to that of trimerized F-F-AF chainsGuSu1 .

Figures 18(c) and 18(d) present the susceptibility $\chi$ as a function of temperature $T$ for the spin-$1/2$ frustrated diamond chain with $J_{1}=1$, $J_{3}<0$ and $J_{2}=0.5$ and $2$, respectively, where the external field is taken as $H/J_{1}=0.01$. A double-peak structure of $\chi(T)$ is observed at small F interaction $|J_{3}|$ and disappears at large $|J_{3}|$. The temperature dependence of the specific heat $C$ with $J_{1}$ $=1$, $J_{3}<0$ and $J_{2}=0.5$ and $2$ is shown in Figs. 18(e) and 18(f), respectively, where $H/J_{1}=0.01$. It is seen that, when $J_{2}$ is small, $C(T)$ exhibits only a single peak; when $J_{2}$ is large, a double-peak structure of $C(T)$ is observed. In the latter case, the double-peak structure is more obvious for small F interaction $|J_{3}|$, and tend to disappear at large $|J_{3}|$. Therefore, the thermodynamics demonstrate various behaviors with different AF interactions $J_{1}$, $J_{2}$ and F interaction $J_{3}$.

V.3 Effect of Anisotropy of Bond Interactions

Some magnetic materials show different behaviors under longitudinal and transverse magnetic fields, showing that the anisotropy plays an important role in the physical properties of the system. First, let us investigate the XXZ anisotropy of the AF interaction $J_{2}$ on the properties of the spin-$1/2$ non-frustrated diamond chain with the couplings $J_{1}:J_{2z}:J_{3}=1:2:-0.5$ for various anisotropy parameter defined by $\gamma_{2}=J_{2x}/J_{2z}=J_{2y}/J_{2z}$, where the $z$ axis is presumed to be perpendicular to the chain direction. For $\gamma_{2}\geq 1$, the magnetization $m(H)$, susceptibility $\chi(T)$ and specific heat $C(T)$ are presented in Figs. 19(a), (b) and (c), respectively. With increasing $\gamma_{2}$, it is found that when the magnetic field $H$ is along the $z$ direction, the width of the magnetization plateau at $m=1/6$ as well as the saturation field are enlarged, while those are more increased for $H$ along the $x$ direction than along the $z$ direction; the peak of the susceptibility $\chi(T)$ for $H$ along the $z$ direction at lower temperature side is promoted, and the second round peak at high temperature side is depressed with a little shift, while $\chi(T)$ for $H$ along the $x$ direction shows the similar varying trend; the peak of the specific heat $C(T)$ for $H$ along the $z$ direction at lower temperature side leaves almost unchanged, and the second round peak at high temperature side moves towards the higher temperature side, while $C(T)$ for $H$ along the $x$ direction coincides with those for $H$ along the $z$ direction. For $0<\gamma_{2}<1$, the anisotropy just shows very reverse effect on the thermodynamic properties in comparison to what we discussed above.

Now let us discuss the effect of the XXZ anisotropy of $J_{3}<0$ on the magnetic and thermodynamic properties of the spin-$1/2$ non-frustrated diamond chain with the couplings $J_{1}:J_{2}:J_{3z}=1:2:-0.5$. Recall that as the $J_{2}$ bond connects two different lattice sites, as shown in Fig. 1, $J_{1}$ and $J_{3}$ can be different, even their signs. Define a parameter $\gamma_{3}$ to characterize the anisotropy as $\gamma_{3}=J_{3x}/J_{3z}=J_{3y}/J_{3z}$, where the $z$ axis is perpendicular to the chain direction. For $\gamma_{3}\geq 1$, the magnetization $m(H)$, susceptibility $\chi(T)$ and specific heat $C(T)$ are depicted in Figs. 19(d), (e) and (f), respectively. With increasing $\gamma_{3}$, it is seen that the width of the plateau at $m=1/6$ for $H$ along the $z$ direction becomes slightly wider, while it goes smaller for $H$ along the $x$ direction; the saturation field is not changed with $\gamma_{3}$ along both directions; the peak of $\chi(T)$ for $H$ along the $z$ direction at lower temperature side is promoted, and the second round peak at higher temperature side is slightly depressed, while the situations along the $x$ direction are just reverse, namely, the peak at lower temperature side is depressed, and the second peak at higher temperature side is slightly promoted; the peak of $C(T)$ for $H$ along the $z$ direction at lower temperature side leaves almost unchanged, and the second round peak at higher temperature side moves slightly to the higher temperature side, while $C(T)$ along the $x$ direction coincides with that along the $z$ direction. For $0<\gamma_{3}<1$, the situation just becomes reverse in comparison to what we discussed above.

V.4 Comparison to Experimental Results

Recently, Kikuchi et al. CuCOOH1 have performed a nice measurement on a spin-$1/2$ diamond-chain compound Cu3(CO3)2(OH)2, i.e., azurite. They have observed the $1/3$ magnetization plateau, unambiguously confirming the previous theoretical prediction. The two broad peaks both in the magnetic susceptibility and the specific heat are observed. We note that in Ref. CuCOOH1 , the experimental data at finite temperatures are fitted by the zero-temperature theoretical results obtained by the exact diagonalization and DMRG methods, while the result of the high temperature series expansion fails to fit the low-temperature behavior of the susceptibility. In accordance with our preceding discussions, by using the TMRG method, we have attempted to re-analyse the experimental data presented in Ref. CuCOOH1 to fit the experiments for the whole available temperature region.

Our fitting results for the temperature dependence of the susceptibility $\chi$ of the compound Cu3(CO3)2(OH)2 are presented in Fig. 20(a). For a comparison, we have also included the TMRG result calculated by using the parameters given in Ref. CuCOOH1 . Obviously, our TMRG results with $J_{1}:J_{2}:J_{3z}=1:1.9:-0.3$ and $J_{3x}/J_{3z}=J_{3y}/J_{3z}=1.7$ fit very well the experimental data of $\chi$, and the two round peaks at low temperatures are nicely reproduced, while the result with $J_{1}:J_{2}:J_{3}=1:1.25:0.45$ obtained in Ref. CuCOOH1 cannot fit the low-temperature behavior of $\chi$ GuSu3 . On the other hand, the fitting results for the temperature dependence of the specific heat $C(T)$ of the compound Cu3(CO3)2(OH)2 are shown in Fig. 20(b). The lattice contribution, which is included in the raw experimental data in Ref.CuCOOH1 , is subtracted according to $C(T)=C_{Exp}(T)-\alpha T^{3}$, where $\alpha$ is a parameter. Obviously, our TMRG result with the same set of parameters $J_{1}:J_{2}:J_{3z}=1:1.9:-0.3$ and $J_{3x}/J_{3z}=J_{3y}/J_{3z}=1.7$ fits also remarkably well the experimental data of $C(T)$, and the two round peaks at low temperatures are nicely reproduced, while the result with $J_{1}:J_{2}:J_{3}=1:1.25:0.45$ given in Ref. CuCOOH1 cannot fit the low-temperature behavior of $C(T)$, even qualitatively. In addition, the sharp peak of $C(T)$ experimentally observed at temperature around $2K$ cannot be reproduced by both sets of the coupling parameters, which might be a three-dimensional long-range ordering due to interchain interactions. The fitting results for the magnetization $m(H)$ of the compound Cu3(CO3)2(OH)2 are shown in Fig. 20(c). We would like to point out that the quantitative fitting by our above parameters to the width of the plateau is not so good, but the qualitative behavior is quite consistent with the experiments both in the transverse and longitudinal magnetic fields, say, $H_{c1}^{\parallel}>H_{c1}^{\perp}$, $H_{c2}^{\parallel}<H_{c2}^{\perp}$, and the saturation field is fixed along both directions, suggesting that our fitting parameters capture the main characteristics. It is worth pointing out that if the anisotropy ratio is increased up to $J_{3x}/J_{3z}=J_{3y}/J_{3z}=2.5$ with the same couplings $J_{1}:J_{2}:J_{3z}=1:1.9:-0.3$, the width of the $1/3$ plateau for $H\parallel b$ will be decreased to about one-half of that for $H\perp b$.

Therefore, our calculations show that (i) the best couplings obtained by fitting the experimental data of the susceptibility for the azurite could be $J_{1}:J_{2}:J_{3z}=1:1.9:-0.3$ with the anisotropic ratio for the ferromagnetic interaction $J_{3x}/J_{3z}=J_{3y}/J_{3z}=1.7$, where $z\perp b$; (ii) the compound may not be a spin frustrated magnet; (iii) the double peaks of the susceptibility and the specific heat are not caused by the spin frustration effect, but by the two kind of gapless and gapful excitations owing to the competition of the AF and F interactions.

One might argue that for this diamond chain compound, from the point of the lattice distance it is unlikely that $J_{1}$ is AF without XXZ anisotropy while $J_{3}$ is F with strong XXZ anisotropy. We may offer another possibility to support our findings, namely, the case of $J_{1}$ and $J_{3}$ with opposite signs is not excluded from the lattice structure of the compound. A linear relationship exists between the exchange energy and the metal-ligand-metal bridge angle: the coupling energy, positive (ferromagnetic) at angles near $90^{o}$, becomes increasingly smaller (more antiferromagnetic) as the angle increasesAngle . As the ferromagnetic coupling $J_{3}$ is determined by fitting the experimental low-temperature behaviors of $\chi(T)$ and $C(T)$, this fitting coupling parameters should not be impossible if one considers the angle of $J_{1}$ bridge to keep the antiferromagnetic coupling while the angle of $J_{3}$ bridge to induce the ferromagnetic coupling. On the other hand, we note that there is another compound with Cu ions, Cu2(abpt)(SO4)2(H2O)$\cdot$H2O, whose $g$ factors in XY plane are different from that in $z$ direction LargeXY . Besides, someone might argue that the condition $J_{2}\gg J_{1},|J_{3}|$ is necessary to explain the double peak behavior of the diamond chain. In fact, such an argument is not necessarily true, as manifested in Fig. 18(c), where the double peaks of $\chi(T)$ at low temperatures can also be produced with the parameters $J_{1}:J_{2}:J_{3}=1:0.5:-0.1$. In other words, the double-peak behavior of the diamond chain may not depend on whether $J_{2}\gg J_{1}$, $J_{3}$ or not, but may be strongly dependent on the competition of AF and F interactions, as discussed above.

VI Summary and discussion

In this paper, we have numerically studied the magnetic and thermodynamic properties of spin-$1/2$ Heisenberg diamond chains with three different cases (a) $J_{1}$, $J_{2}$, $J_{3}>0$ (frustrated), (b) $J_{1}$, $J_{3}<0$, $J_{2}>0$ (frustrated), and (c) $J_{1}$, $J_{2}>0$, $J_{3}<0$ (non-frustrated) by means of the DMRG and TMRG methods. In the ground states, the local magnetic moment, spin correlation function, and static structure factor are explored. The static structure factor $S(q)$ at zero field shows peaks at wave vector $q=0$, $\pi/3$, $2\pi/3$, $\pi$, $4\pi/3$ and $5\pi/3$ for different couplings, in which the peaks at $q=0$, $2\pi/3$ and $4\pi/3$ in the magnetization plateau state with $m=1/6$ are observed to be couplings independent. The DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes, where two fitting equations are proposed. It is seen that the six modes are closely related to the low-lying excitations of the system. At finite temperatures, the magnetization, susceptibility and specific heat are calculated, which show various behaviors for different couplings. The double-peak structure of the susceptibility and specific heat can be procured, whose positions and heights are found to be dependent on competing couplings. It has been shown that the XXZ anisotropy of F and AF couplings can have remarkable effect on the physical behaviors of the system. In addition, the experimental susceptibility, specific heat and magnetization of the diamond chain compound Cu3(CO3)2(OH)2CuCOOH1 can be nicely fitted by our TMRG results.

For the spin-$1/2$ frustrated Heisenberg diamond chains with AF couplings $J_{1}$, $J_{2}$ and $J_{3}$, the magnetization plateau at $m=1/6$ in the ground state coincides with a perfect fixed sequence of the averaged local magnetic moment such as $\{...,(S_{a},S_{a},S_{b}),...\}$ with $2S_{a}+S_{b}=1/2$, which might be described by trimerized states. On the other hand, the static structure factor $S(q)$ shows peaks at wave vectors $q=0$, $\pi/3$ ($5\pi/3$), and $2\pi/3$ ($4\pi/3$) for different external fields and different AF couplings. We note that the similar behavior of $S(q)$ has been experimentally observed in diamond-typed compound Sr3Cu3(PO4)4 SrCu . In addition, the DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes. It is observed that the six modes are closely related to the low-lying excitations of the present case. At finite temperatures, the magnetization $m(H)$, susceptibility $\chi(T)$ and specific heat $C(T)$ demonstrate different behaviors at different AF couplings, say, the magnetization plateau at $m=1/6$ is observed whose width is found to be dependent on the couplings; the double peak structure is observed for the susceptibility $\chi(T)$ and specific heat $C(T)$ as a function of temperature, and the heights and positions of the peaks are found to be dependent on the AF couplings.

For the spin-$1/2$ frustrated Heisenberg diamond chains with F couplings $J_{1}$, $J_{3}$ and AF coupling $J_{2}$, the magnetization plateau at $m=1/6$ in the ground state corresponds to a perfect fixed sequence of the averaged local magnetic moment such as $\{...,(S_{a},S_{b},S_{b}),...\}$ with $S_{a}+2S_{b}=1/2$, which could be understood by trimerized states. The static structure factor $S(q)$ shows peaks also at wave vectors $q=0$, $\pi/3$ ($5\pi/3$), and $2\pi/3$ ($4\pi/3$) for different external fields and different F couplings $J_{1}$, $J_{3}$ and AF coupling $J_{2}$, which is expected to be experimentally observed in the related diamond-type compound. In addition, the DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes with the fitting equations mentioned above. The six modes are closely related to the low-lying excitations of the system. At finite temperatures, the magnetization $m(H)$, susceptibility $\chi(T)$ and specific heat $C(T)$ demonstrate various behaviors for different couplings, namely, the magnetization plateau at $m$ = $1/6$ is observed whose width is found to depend on the couplings; the double-peak structure is also observed for the susceptibility $\chi(T)$ and specific heat $C(T)$, and the heights and positions of the peaks are found dependent on F couplings $J_{1}$, $J_{3}$ and AF coupling $J_{2}$.

For the spin-$1/2$ non-frustrated Heisenberg diamond chains with AF couplings $J_{1}$, $J_{2}$ and F coupling $J_{3}$, the magnetization plateau at $m=1/6$ in the ground state coincides with a perfect fixed sequence of the averaged local magnetic moment such as $\{...,(S_{a},S_{b},S_{b}),...\}$ with $S_{a}+2S_{b}=1/2$, which could be understood by trimerized states. The static structure factor $S(q)$ is observed to exhibit the peaks at wave vectors $q=0$ and $2\pi/3$ ($4\pi/3$) for different external fields and different AF couplings $J_{1}$, $J_{2}$ and F coupling $J_{3}$, which could be experimentally detected in the related diamond-type compound. In addition, it is found that the zero-field spin correlation function $\langle S_{j}^{z}S_{0}^{z}\rangle$ is similar to that of the $S=1/2$ Heisenberg AF chain. At finite temperatures, the magnetization $m(H)$, susceptibility $\chi(T)$ and specific heat $C(T)$ are found to reveal different behaviors for different couplings, i.e., the magnetization plateau at $m=1/6$ is obtained, whose width is found to depend on the couplings; the double-peak structure is observed for the temperature dependence of the susceptibility $\chi(T)$ and specific heat $C(T)$, where the heights and positions of the peaks depend on different AF couplings $J_{1}$, $J_{2}$ and F coupling $J_{3}$.

The effect of the anisotropy of the AF and F interactions on the physical properties of the non-frustrated Heisenberg diamond chain is also investigated. For the case of the couplings satisfying $J_{1}:J_{2z}:J_{3}=1:2:-0.5$, when the anisotropic ratio $\gamma_{2}=J_{2x}/J_{2z}=J_{2y}/J_{2z}\neq 1$, it is found that the width of the plateau at $m=1/6$, the saturation field, and the susceptibility $\chi(T)$ show the same tendency, but quantitatively different, under the external field $H$ along the $z$ and $x$ directions, while the specific heat $C(T)$ for $H$ along the $z$ direction coincides with that along the $x$ direction. For the case of the couplings satisfying $J_{1}:J_{2}:J_{3z}=1:2:-0.5$, when the anisotropic ratio $\gamma_{3}=J_{3x}/J_{3z}=J_{3y}/J_{3z}\neq 1$, it is seen that the width of the plateau at $m=1/6$, the saturation field, and the susceptibility $\chi(T)$ exhibit the opposite trends for $H$ along the $z$ and $x$ directions, while the specific heat $C(T)$ for $H$ along the $z$ direction also coincides with that along the $x$ direction.

For all the three cases, plateau states of $m=1/6$ are observed during the magnetization, whose static structure factor $S(q)$ shows peaks at wavevectors $q=0$, $2\pi/3$ and $4\pi/3$. But in absence of the magnetic field, the static structure factor $S(q)$ in the ground state displays peaks at $q=0$, $\pi/3$, $2\pi/3$, $\pi$, $4\pi/3$, and $5\pi/3$ for the frustrated case with $J_{1}$, $J_{2}$, $J_{3}>0$; peaks at $q=0$, $\pi/3$, $\pi$, and $5\pi/3$ for the frustrated case with $J_{1}$, $J_{3}<0$, $J_{2}>0$; and a peak at $q=\pi$ for the non-frustrated case with $J_{1}$, $J_{2}>0$, $J_{3}<0$. In addition, the DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes, where the fitting equation is proposed. At finite temperatures, the double-peak structure of the susceptibility and specific heat against temperature can be obtained for all the three cases. It is found that the susceptibility shows ferrimagnetic characteristics for the two frustrated cases with some couplings, while no ferrimagnetic behaviors are observed for the non-frustrated case.

The compound Cu3(CO3)2(OH)2 is regarded as a model substance for the spin-$1/2$ Heisenberg diamond chain. The $1/3$ magnetization plateau and the two broad peaks both in the magnetic susceptibility and the specific heat have been observed experimentallyCuCOOH1 . Our TMRG calculations with $J_{1}:J_{2}:J_{3z}=1:1.9:-0.3$ and $J_{3x}/J_{3z}=J_{3y}/J_{3z}=1.7$ capture well the main characteristics of the experimental susceptibility, specific heat and magnetization, indicating that the compound Cu3(CO3)2(OH)2 may not be a spin frustrated magnetGuSu3 .

Appendix A LOW-LYING EXCITATIONS OF SPIN-1/2 Frustrated

HEISENBERG DIAMOND CHAINS

In this Appendix, the low-lying excitations of the spin-1/2 frustrated Heisenberg diamond chain are investigated by means of the Jordan-Wigner (JW) transformation. The Hamiltonian of the system reads

[TABLE]

where $3N$ is the total number of spins in the diamond chain, $J_{i}>0$ ($i=1,2,3$) represent the AF coupling while $J_{i}<0$ the F interaction, and $H$ is the external magnetic field along the $z$ direction. In accordance with the spin configuration of the diamond chain, we start from the Jordan-Wigner (JW) transformation with spinless fermions

[TABLE]

where $j=1,\cdots,3N$. Because the period of the present system is 3, three kinds of fermions in moment space can be introduced through the Fourier transformations

[TABLE]

Ignoring the interactions between fermions, the Hamiltonian takes the form of

[TABLE]

where $E_{0}=\frac{N}{4}(2J_{1}+J_{2}+2J_{3}-6H)$, $\omega_{1}$=$-(J_{1}+J_{3})-H$, $\omega_{2}$=$-\frac{1}{2}(J_{1}+J_{2}+J_{3})-H$, $\omega_{3}$=$\omega_{2}$, $\gamma_{1}$=$(J_{1}e^{ik}+J_{3}e^{-i2k})/2$, $\gamma_{2}$=$(J_{2}e^{ik})/2$, and $\gamma_{3}$=$(J_{3}e^{ik}+J_{1}e^{-ik})/2$.

Via the Bogoliubov transformation

[TABLE]

the Hamiltonian can be diagonalized as

[TABLE]

The coefficients of the Bogoliubov transformation can be found through equations of motion $i\hbar\dot{a}_{ik}=[a_{ik},H]$:

[TABLE]

For a given $k$, the eigenvalues $\varepsilon_{ik}$ and eigenvectors $(u_{1i},u_{2i},u_{3i})$ can be numerically calculated by the driver $ZGEEV.f$ of the $LAPACK$, which is available on the websiteLAPACK . Figs.7 show the zero-field low-lying fermionic excitation $\varepsilon(k)$ for the frustrated diamond chain with different AF coupling, while Figs.13 present the zero-field low-lying fermionic excitation $\varepsilon(k)$ for the frustrated diamond chain with $J_{1}$, $J_{3}<0$ and $J_{2}>0$.

Acknowledgements.

We are grateful to Prof. D. P. Arovas for useful communication. This work is supported in part by the National Science Fund for Distinguished Young Scholars of China (Grant No. 10625419), the National Science Foundation of China (Grant Nos. 90403036, 20490210, 10247002), and by the MOST of China (Grant No. 2006CB601102).