Magnetization steps and relevant cluster statistics for a diluted Heisenberg layer: Nearest-neighbor cluster model on the square lattice

A theory for magnetization steps (MST’s) from a strongly diluted Heisenberg antiferromagnet on the square lattice will be presented in several papers. In this first paper, general results for cluster models are reviewed, including an updated in-depth discussion of cluster types. A detailed equilibrium theory for the nearest-neighbor (NN) cluster model on the square lattice is then presented. The fraction $x$ of cations that are magnetic is assumed to be well below the site percolation concentration, $x_c=0.593$ for this model. Isotropic exchange interactions (Heisenberg exchange) are the only intracluster interactions. Neighbors are classified by symmetry, instead of the traditional classification by the distance $r$. Neighbors in the same symmetry class have the same isotropic exchange constant $J$, so that the $J$’s too can be classified by these symmetry classes. For the square lattice, twelve of the neighbors’ symmetry classes correspond to the twelve shortest $r$’s. The corresponding $J$’s are as follows: $J_1$ for NN’s, $J_2$ for second neighbors, and so on up to $J_{12}$. Clusters are divided into “types.” Each type, $c$, is specified by the cluster size $n_c$ (the number of spins), and by a “bond list” that specifies the $J$’s for all spin pairs in the cluster. The bond list determines the cluster’s exchange Hamiltonian. The relevant statistics is therefore the statistics of cluster types, not of cluster sizes. The main assumption in the statistics is that the magnetic ions are randomly distributed over all cation sites. For the NN cluster model on the square lattice, there are 3290 cluster types with sizes $n_c⩽12$. Perimeter polynomials (PP’s) for cluster types, analogous to the usual PP’s for cluster sizes, are given for all 3290 cluster types. The energy eigenvalues for the 10 cluster types with sizes $n_c⩽5$ were determined for magnetic ions with spin $S=5∕2$, such as ${\mathrm{Mn}}^{2+}$ and ${\mathrm{Fe}}^{3+}$. The average magnetic moment ${\mu }_c(T,B)$ per cluster, for these 10 cluster types, was then obtained at temperature $T$ and magnetic field $B$. At low $T$, all cluster types with sizes $n_c>1$ give rise to MST’s. The total magnetization $M$ is proportional to a statistically-weighted average of ${\mu }_c(T,B)$ over all cluster types. The contribution from cluster types with $n_c⩽5$ is dominant when $x⩽0.25$. This contribution is evaluated exactly. Two alternative approximations are used to evaluate the contribution from the larger clusters. Examples of calculated magnetization curves are given. In addition to the MST’s, the magnetization exhibits a fast rise at low $B$. This rise ends in a plateau (the plateau of “apparent saturation”). The apparent saturation value $M_s$ is calculated for $x$ up to 0.25. This paper is accompanied by electronically accessible tables (EPAPS) of numerical results for all 3290 cluster types.

Figure 1

Lattice sites on the square lattice. The central site is marked by “$\times$.” Other sites are divided into numbered symmetry classes.

Figure 2

Exchange bonds for the same triplet in three cluster models: $J_1$ model, $J_1-J_2$ model, and $J_1-J_3$ model. Solid and dashed lines represent $J_1$ and $J_2$ bonds, respectively.

Figure 3

Exchange bonds (shown as solid lines) connecting the spins (solid circles) in the 10 cluster types with sizes $n_c$ up to 5.

Figure 4

Upper row: The two configurations $3\alpha$ (straight), and $3\beta$ (L-shaped) of an open triplet (cluster of type 3). Lower row: The configurations $4A\alpha$ (straight), $4A\beta$ (L-shaped), and $4A\gamma$ (zig-zag) of the quartet type $4A$. The empty squares surrounding each configuration, both in the upper and lower rows, are cation sites that must remain vacant.

Figure 5

Various probabilities $P$ for the $J_1$ model on the square lattice, plotted as a function of the fraction $x$ of cations that are magnetic. Solid curves are probabilities $P_c$ for individual cluster types. Dashed curves are sums of probabilities $P_c$. Some of these sums are over all cluster types of a given size. Others are sums over all cluster types with sizes larger than a specified size. Part (a) shows the probability $P_1$ for singles, and the sums $P_{>5}$ and $P_{>12}$ for sizes exceeding 5 and 12, respectively. Part (b) shows the probabilities $P_2$ and $P_3$ for pairs and triplets, respectively, and the sums $P_4$ and $P_5$ over all quartet types and over all quintet types, respectively.

Figure 6

(a) Probabilities $P_c$ for the three types of quartets: $4A$, $4B$, and $4C$. (b) Probabilities $P_c$ for the four types of quintets: $5A$, $5B$, $5C$, and $5D$.

Figure 7

Average zero-temperature magnetic moment ${\mu }_c$ for all cluster types of size $n_c⩽5$, plotted as a function of reduced magnetic field $b=g{\mu }_{\mathrm{B}}B∕\mid J_1\mid$. These results are for $S=5∕2$. The notation for the cluster types is as in Fig. 3. Note the position of the zero on the abscissa scale.

Figure 8

$x$ dependence of the parameters ${\eta }_{>5}$ and $b_s$ in Eq. (23) for the linear ramp. (a) ${\eta }_{>5}$, for any value of $S$; (b) $b_s$ for $S=5∕2$.

Figure 9

Calculated reduced magnetization $m=M∕M_0$ as a function of the reduced magnetic field $b=g{\mu }_{\mathrm{B}}B∕\mid J_1\mid$. These results are for $x=0.10$ at $T=0$, assuming $S=5∕2$. (a) Results for reduced fields up to magnetic saturation. (b) Expanded view of the results for $b<8.5$. The MST’s labeled as 2 and 3 are from pairs and triplets, respectively. The plateau corresponding to the apparent saturation of the magnetization is labeled as $m_s$.

Figure 10

(a) Calculated reduced magnetization $m$ at $T=0$ for $x=0.20$ and $S=5∕2$. The R&R approximation was used for the remainder. The solid and dashed curves at the bottom of the figure represent the remainder corrections obtained from the R&R approximation and from the CQUIN’s method, respectively. (b) The upper curve shows an expanded view of the low-field results in part (a). The plateau corresponding to the apparent saturation is labeled as $m_s$. The lower curve is the normalized susceptibility $dm∕db$, computed at $k_{\mathrm{B}}T∕\mid J\mid =0.01$ to avoid the infinities at $T=0$. The cluster types making the dominant contributions to the most conspicuous susceptibility peaks are indicated. Peaks labeled as 2 (i.e., from pairs) also contain small contributions from quartets of type $4C$. Small peaks that are visible but are not labeled are from various types of quintets.

Figure 11

The magnetization curve for $x=0.20$ for the same parameters as in Fig. 10b, except that the normalized temperature is $k_{\mathrm{B}}T∕\mid J_1\mid =0.1$, instead of zero.

Figure 12

The reduced apparent saturation value of the magnetization, $m_s=M_s∕M_0$, as a function of $x$, up to $x=0.25$. The solid curve is based on the present work. The dashed curve is based on an earlier, less accurate, treatment (see text).

Figure 13

Various properties of cluster types $c$ of size $n_c=6$: the bond list; an example (one configuration); the perimeter polynomial $D_c\left(q\right)$; the reduced field $b_s\left(c\right)$ at the last MST for $S=5∕2$; and the value of ${\eta }_c=S_c\left(0\right)∕n_{c}S$.

Figure 14

Various properties of cluster types $c$ of size $n_c=7$: the bond list; an example (one configuration); the perimeter polynomial $D_c\left(q\right)$; the reduced field $b_s\left(c\right)$ at the last MST for $S=5∕2$; and the value of ${\eta }_c=S_c\left(0\right)∕n_{c}S$.