Exact Thermodynamic Properties of (1,1/2) Mixed Diamond Chains with Strong Single-Site Anisotropy

TL;DR

This paper provides exact calculations of thermodynamic properties for mixed spin diamond chains with strong anisotropy, revealing detailed magnetic and thermal behaviors relevant to quantum spin systems.

Contribution

It introduces an exact analytical approach to determine thermodynamic properties of mixed spin chains with strong anisotropy, expanding understanding of quantum magnetic materials.

Findings

Exact magnetization curves obtained

Entropy and specific heat calculated precisely

Magnetic susceptibility characterized in detail

Abstract

The ground states and finite-temperature properties of mixed diamond chains with spins 1 and 1/2 are investigated in the limit of strong easy-axis anisotropy on the spin-1 sites. Magnetization curves, entropy, specific heat and magnetic susceptibility are exactly calculated using the method of \v{C}anov\`a et al. [J. Phys.: Condens. Matter 18 4967 (2006)].

Equations

$$\mathcal{H}=$$

$$+D\sum _{l=1}^L{S}_l^{z2}-H\sum _{l=1}^L\left({\tau }_l^{(1)z}+{\tau }_l^{(2)z}+S_l^z\right),$$

$$\mathcal{H}=$$

$$-H\sum _{l=1}^L\left({\tau }_l^{(1)z}+{\tau }_l^{(2)z}+S_l^z\right).$$

$$\Delta \to 1,J_{\mathrm{I}}\to 2,J_{\mathrm{H}}\to \lambda ,H_{\mathrm{I}}\to 2H,H_{\mathrm{H}}\to H.$$

$$W=\sum _{N_{\mathrm{d}}=0}^L{}_L{C}_{N_{\mathrm{d}}}{2}^{N_{\mathrm{d}}}=(2+1{)}^L=3^L.$$

$$W=\sum _{N_{\downarrow }=0}^L{}_L{C}_{N_{\downarrow }}{4}^{L-2N_{\downarrow }}={\left(\frac{17}{4}\right)}^L.$$

$$\chi T\propto \exp \left(-\frac{\left(2-\lambda \right)}{T}\right).$$

$$⟨N_{\mathrm{d}}⟩$$

Full text

We consider the mixed diamond chains with single-site anisotropy $D$ and magnetic field $H$ described by the following Hamiltonian:

[TABLE]

where $\mib S_{l}$ is a spin-1 operator, and $\mib\tau^{(\alpha)}_{l}(\alpha=1,2)$ are spin-1/2 operators in the $l$th unit cell. The total number of unit cells is denoted by $L$. The ground state and finite temperature properties of this model have been investigated by one of the authors and coworkers[1, 2]. Defining the composite spin operators $\mib T_{l}\equiv\mib\tau^{(1)}_{l}+\mib\tau^{(2)}_{l}$, it is evident that ${\forall l}\ [\mib T_{l}^{2},{\mathcal{H}}]=0$. Thus, we have $L$ conserved quantities $\mib T_{l}^{2}(\equiv T_{l}(T_{l}+1);\ T_{l}=0$ or 1). The total Hilbert space of the Hamiltonian (1) consists of separate subspaces, each of which is specified by a definite set of $\{T_{l}\}$.

A spin pair with $T_{l}=0$ is a singlet dimer that cuts off the correlation between $\mib S_{l}$ and $\mib S_{l+1}$. The segment including $n$ successive $\mib T_{l}$’s with $T_{l}=1$ and $n+1$ $\mib S_{l}$’s coupled with them is called a cluster-$n$. A cluster-$n$ is equivalent to a spin-1 antiferromagnetic Heisenberg chain of length $2n+1$ with alternating single-site anisotropy. A dimer-cluster-$n$ (DC$n$) phase consists of an alternating array of cluster-$n$’s and dimers.

Since all eigenstates are constructed as direct products of the eigenstates of cluster-$n$’s and dimers, the full thermodynamics of diamond chains can be formally formulated. However, the calculations of thermodynamic quantities include the contribution from cluster-$n$’s of arbitrary size. This is intractable in practice. In the region where the ground state is a DC$n$ phase with finite $n$, the contributions from large cluster-$n$’s are small. Hence, the thermodynamic properties can be estimated with enough accuracy[1, 2]. On the other hand, in the region where the ground state is a single infinite-size cluster (DC$\infty$ phase), this approximation is unreliable.

In the present paper, we consider the easy-axis limit ($D\rightarrow-\infty$) of the Hamiltonian (1) where $S^{z}_{l}$ can only take the values $\pm 1$. Then, the Hamiltonian reduces to the form,

[TABLE]

This Hamiltonian is equivalent to Eq. (1) of Ref. \citenstr1 with

[TABLE]

Following Ref.\citenstr1, the Heisenberg spins $\mib\tau^{(1)}_{l}$ and $\mib\tau^{(2)}_{l}$ can be traced out[4] and the partition function reduces to that of the one-dimensional Ising model with $S_{l}^{z}=\pm 1$. The free energy ${\cal G}(T,H)$ at temperature $T$ can be obtained by the replacements (3) in the corresponding expressions for the spin-1/2 Ising-Heisenberg diamond chain in Ref. \citenstr1. Entropy, specific heat, magnetization, and magnetic susceptibility are obtained by appropriate differentiations.

The possible ground states are classified as follows:

1. i.

Néel phase (N) : $\forall l\ T_{l}=1,T_{l}^{z}=\pm 1,S_{l}^{z}=-T_{l}^{z}$. This phase corresponds to the ferrimagnetic phase of Hamiltonian (1) of Ref. \citenstr1 with $S=1/2$. However, in the present model, the total magnetization vanishes. This phase belongs to a DC$\infty$ phase. 2. ii.

Paramagnetic phase (P) : $\forall l\ T_{l}=0$. Ising spins $S_{l}^{z}$ are decoupled from each other and can take the values $\pm 1$ independently in the absence of the magnetic field $H$. For $H>0$, $\forall l\ S_{l}^{z}=1$. This phase corresponds to the DC0 phase. 3. iii.

Ferromagnetic phase (F) : $\forall l\ T_{l}^{z}=S_{l}^{z}=\pm 1$.

The corresponding ground-state energies per unit cell are given by E_{\rm N}=\frac{\lambda}{4}-2,\ \ E_{\rm P}=-\frac{3}{4}\lambda-H\ \ and $E_{\rm F}=\frac{\lambda}{4}+2-2H.$

The phase boundaries between these phases are shown in Fig. 1(a). The change of ground states with the magnetic field is clearly represented by the magnetization curves in Fig. 1(b) and (c). The stepwise magnetization curves at $T=0$ are smeared at finite temperatures as shown in these figures. It should be remarked that the P phase is induced not only by frustration ($\lambda$) but also by the magnetic field, even though all pairs $\mib T_{l}$ are nonmagnetic in the P phase and magnetic in the N phase realized for lower magnetic fields.

The temperature dependence of the entropy $\mathcal{S}$ in the absence of magnetic field is shown in Fig. 2(a) for various values of $\lambda$. For $\lambda<2$, the ground state is a Néel ordered state. Hence, the entropy $\mathcal{S}$ vanishes as $T\rightarrow 0$. On the other hand, for $\lambda>2$, the ground state is a paramagnetic state in which each $S_{l}^{z}$ can take two values. Hence, the entropy $\mathcal{S}$ tends to $L\ln 2$ as $T\rightarrow 0$. At $\lambda=2$, the creation energy of a dimer in the Néel background vanishes. Hence, an arbitrary number of dimers are present in the ground state. The number of configurations of $N_{\rm d}$ dimers is ${}_{L}C_{N_{\rm d}}$. Each segment between two dimers can take two states with total magnetization $\pm 1$. Hence, the number of states with $N_{\rm d}$ dimers is ${}_{L}C_{N_{\rm d}}\times 2^{N_{\rm d}}$. The total number of states $W$ is then given by

[TABLE]

This gives the residual entropy $\mathcal{S}=L\ln 3$ larger than the values for $\lambda\neq 2$. This is consistent with the data shown in Fig. 2(a).

The temperature dependence of the entropy $\mathcal{S}$ at the triple point $(\lambda=0,H=2.0)$ is shown in Fig. 2(b). To estimate the residual entropy, let us consider the ground state of a diamond $\left|{S_{l}^{z}\tau^{(1)z}_{l}\tau^{(2)z}_{l}S_{l+1}^{z}}\right\rangle$. If $S_{l}^{z}=S_{l+1}^{z}=1$, the energy is equal to $-2$ irrespective of the values of $\tau^{(1)z}_{l}(=\pm 1/2)$ and $\tau^{(2)z}_{l}(=\pm 1/2)$. On the other hand, for $S_{l}^{z}=-S_{l+1}^{z}$ only the state with $\tau^{(1)z}_{l}=\tau^{(2)z}_{l}=1/2$ has the lowest energy $-2$. Namely, the spins $\tau^{(1)z}_{l}$, $\tau^{(2)z}_{l}$, $\tau^{(1)z}_{l-1}$, and $\tau^{(2)z}_{l-1}$ on both sides of ${N_{\downarrow}}$ sites with $S_{l}^{z}=-1$ are fixed, while those on other $N-2{N_{\downarrow}}$ sites can take four possible states freely. The number of configurations of the sites with $S_{l}^{z}=-1$ is ${}_{L}C_{{N_{\downarrow}}}$. Hence, the number of states with fixed ${N_{\downarrow}}$ is ${}_{L}C_{{N_{\downarrow}}}\times 4^{L-2{N_{\downarrow}}}$. The total number of states $W$ is then given by

[TABLE]

This gives the residual entropy $\mathcal{S}=L\ln(17/4)$ that is consistent with Fig. 2(b).

The temperature dependence of the specific heat $\mathcal{C}$ in the absence of magnetic field is shown in Fig. 3.

The low temperature peak moves to lower temperature as $\lambda$ approaches 2.0 from both sides. This peak disappears at $\lambda=2$. We can interpret that the entropy released under the peak turns into the increase in the residual entropy at $\lambda=2$. The similar phenomena are observed for $D=0$[1] at the phase boundary between DC$n$ phases with different $n$. The corresponding behavior of entropy is observed also for finite $D$.[2] However, exact thermodynamics is not available in the case of Néel ground state for finite $D$. In the present strongly anisotropic limit, it is shown explicitly that this phenomenon occurs also at the DC[math]-Néel phase boundary.

The temperature dependence of the magnetic susceptibility is shown in Fig. 4(a). For $\lambda>2$, the ground state is paramagnetic and the Curie law behavior $\chi T\rightarrow L$ is observed. This is the contribution of the spins ${S}^{z}_{l}(=\pm 1)$. On the other hand, the susceptibility shows an exponential behavior for $\lambda<2$ as shown in Fig. 4(b). However, this excitation energy is not the energy required to flip the spin $S^{z}_{l}$ or $T^{z}_{l}$ as in conventional antiferromagnets. The latter should be equal to $2$ irrespective of the value of $\lambda$, while the excitation energies estimated from the slopes of Fig. 4(b) clearly depend on $\lambda$. This can be understood in the following way: The excitation energy $E_{\rm d}$ of a dimer in the Néel background is given by $E_{\rm d}=\left(2-\lambda\right)$. This implies that the number of dimers $N_{\rm d}$ is proportional to $\exp\left(-\frac{\left(2-\lambda\right)}{T}\right)$. The segment between two dimers carries an Ising spin $\pm 1$ irrespective of its length. Each segment contributes to $\chi$ by $1/T$. Under the periodic boundary condition, the number of these segments is also equal to $N_{\rm d}$. Hence, we find

[TABLE]

This is consistent with the slopes of Fig. 4(b). Thus, the increase of the magnetic susceptibility with temperature in the case of Néel ground state results from the thermal excitation of nonmagnetic dimers.

At $\lambda=2$, the expectation value of $N_{\rm d}$ in the ground state is given by

[TABLE]

Hence, $\chi T/L$ tends to ${2}/{3}$ as $T\rightarrow 0$ as shown in Fig. 4(a) .

We thank S. Hoshino and H. Shinaoka for fruitful discussion and comments. This work is supported by JSPS KAKENHI Grant Number JP25400389.