Small Spin Clusters Mimicking a Temperature-Induced Phase Transition: Spins on Vertices of Regular Octahedron and Icositetrachoron

The octahedron bears the Schläfli symbol²⁸⁾ of \(\{3,4\}\) and satisfies both conditions of the triangles and even-membered paths around each spin. The ground state configuration of the “highest order” realizes by arranging three orientations (say, R, G, and B), which satisfy the optimum condition, on three diagonal pairs. This configuration is equivalent to the ground state of the antiferromagnetic 3-state Potts model.²⁷⁾ However, each spin can flip without energy penalty to the image reflected by the mirror plane containing four bonding spins. This mechanism is the same as the origin of the macroscopic degeneracy of the triangular lattice case.²²⁾ The regular (highest-order) configuration and an example of flipped ones are illustrated in Fig. 1. While fixing R and G, the two remaining spins can independently take either B or B′. Thus, the ground states are \(2^{2}\)-times degenerate for a fixed pair of R and G. Note that the spin configuration can continuously change orientation while maintaining optimum conditions among all spins.

Before proceeding into the simulation results of the present model, we see what happens in ordinary cases by looking at the behaviors of the 3-state Potts models²⁷⁾ on the same cluster geometry. The interaction is written as \begin{equation} V_{ij} = - J\boldsymbol{s}_{i}\cdot\boldsymbol{s}_{j}, \end{equation} (3) where the three states are expressed by mutually orthogonal unit vectors in dimension three. Note that an antiferromagnetic (neighbor-non-favoring) model (\(J<0\)) does not suffer from any frustration for the octahedral cluster, similar to the present model. Under this setting, we can use the nematic order parameter as an indicator of an average order between two spins, i and j. Irrespective of the sign of J, the local order \(\langle P_{2}(\boldsymbol{s}_{i}\cdot\boldsymbol{s}_{j})\rangle\) is 1 for the diagonals in the ground states.

We computed the properties analytically while considering all (\(= 3^{6}\)) spin configurations. Figure 2 shows the temperature dependences of \(\langle P_{2}\rangle_{\text{e}}\) (\(|\boldsymbol{d}|=1\)) and \(\langle P_{2}\rangle_{\text{d}}\) (\(|\boldsymbol{d}|=\sqrt{2}\)) for ferromagnetic and antiferromagnetic 3-state Potts models. Since the octahedral cluster is compatible with the optimal spin configurations without frustration in both cases, the limiting magnitudes at \(T=0\) are either \(-1/2\) or 1. With increasing temperature, they gradually and monotonously reach 0 (vanishing the spin order). The heat capacity exhibits a broad maximum around the temperature of the maximum where \(|d\langle P_{2}\rangle/dT|\) exhibits a maximum. An essential feature is that they retain the order of averaged magnitudes independently of temperature. The ferromagnetic model always has a larger average for the neighbors \(\langle P_{2}\rangle_{\text{e}} >\langle P_{2}\rangle_{\text{d}}\), while the antiferromagnetic one keeps the counter relation. We can translate the latter fact that the sublattice order is always in the antiferromagnetic case and only weakens with increasing temperature. Thus, irrespective of temperature, the point group assignable to the cluster while considering the sublattice order is \(D_{2h}\) of cuboids, of which three pairs of faces are distinct. In contrast, that of the ferromagnetic case is \(O_{h}\) of cubes with six equivalent faces.

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Figure 2. (Color online) Temperature dependence of two-point correlation functions, \(\langle P_{2}\rangle_{\text{e}}\) (\(|\boldsymbol{d}|=1\)) and \(\langle P_{2}\rangle_{\text{d}}\)) (\(|\boldsymbol{d}|=\sqrt{2}\)), for ferromagnetic (superscript F) and antiferromagnetic (AF) 3-state Potts models [defined by Eq. (3)] on an octahedral cluster consisting of 6 spins on vertices.

Figure 3 shows the heat capacity C and susceptibility χ of the octahedral cluster together with additional data. Both quantities directly reflect fluctuation in, respectively, energy and nematic order parameter. Consequently, they serve as indicators of phase transitions.^1–3) They exhibit peaks at different temperatures in the present case, resulting in three temperature regions. The highest temperature region corresponds to the isotropically disordered state of spin alignment. The decreasing trend in heat capacity on heating and roughly constant magnitude of σ (Fig. 4) on the low-temperature side are consistent with this assignment. Although the crossover between the highly disordered and locally ordered states is widely seen in usual spin models, irrespective of ferromagnetic or antiferromagnetic interactions, only a broad peak appears in such cases. A distinction in the present case from such cases manifests in the peak at the lower temperature. Indeed, a high-temperature peak of the triangular lattice is due to the crossover, whereas a low-temperature one corresponds to the putative phase transition exhibiting exciting properties.^22,26)

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Figure 3. (Color online) Heat capacity C per spin (a) and susceptibility χ (b) of the clusters of spins on the vertices of the regular octahedron (cross, \(z=4\)) and icositetrachoron (circle, \(z=8\)) with error bars estimated from the statistics of separate simulations. The heat capacity of the triangular lattice²²⁾ (\(z=6\)) is shown for reference by a dotted curve in the upper panel.

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Figure 4. (Color online) Nematic order parameter σ (circle) and two-point correlation functions [plus-sign for edge \(\langle P_{2}\rangle_{\text{e}}\) (\(|\boldsymbol{d}|=1\)), cross for diagonal \(\langle P_{2}\rangle_{\text{d}}\) (\(|\boldsymbol{d}|=\sqrt{2}\))] for the octahedral cluster consisting of 6 spins on vertices. The horizontal line indicates the magnitude calculated from the optimal condition for the neighbors. In both, error bars indicate the standard deviation in 9 simulations.

Because of the smallness of the cluster, we can define only two local quantities: \(\langle P_{2}(\boldsymbol{s}_{i}\cdot\boldsymbol{s}_{j})\rangle\) averaged over bonding edges \(\langle P_{2}\rangle_{\text{e}}\) (\(|\boldsymbol{d}|=1\) with a vector \(\boldsymbol{d}\) between spins) and that over non-bonding diagonals \(\langle P_{2}\rangle_{\text{d}}\) (\(|\boldsymbol{d}|=\sqrt{2}\)). Figure 4 shows their temperature dependence. \(\langle P_{2}\rangle_{\text{e}}\) points 2/5 on cooling. This magnitude is expected for ground states with the optimal spin configuration for all pairs of spins connected by edges. On the other hand, as for the diagonal magnitude at low temperatures, although a naïve average over the degenerating ground states (ca. 0.78) is within error bars, we will need to consider some particular spin configurations from the whole set of ground states for complete understanding. The uniaxiality (nematic order with a finite σ) may choose some particular spin configurations from the whole set of ground-state configurations. Indeed, the simulation result is close to one (≈ 0.725), which is expected for the subset involving the disorder of one-third spins.²⁶⁾

Both quantities, \(\langle P_{2}\rangle_{\text{e}}\) and \(\langle P_{2}\rangle_{\text{d}}\), vary starting from the limiting magnitudes with increasing temperature (Fig. 4). The former is roughly parallel with the nematic order parameter. On the other hand, the latter starts to decrease appreciably around \(T\approx 0.1V_{0}/k_{\text{B}}\), which corresponds to the significant maximum in heat capacity Fig. 3(a). Since their temperature dependences differ, they cross around \(T_{\text{c}}\approx 0.285V_{0}/k_{\text{B}}\). Thus, the cluster changes its character from ferroic to antiferroic only upon temperature variation. Considering the case of the 3-state Potts models in the previous section, we can convincingly claim the distinct symmetries at sufficiently high and low temperatures. Although significant fluctuation originating in a small number of spins in a cluster makes it challenging to locate the transition temperature, it seems reasonable to assign it the temperature of the crossing. Note that this choice needs no outside references.

The essential character of our interaction [Eq. (1)] is that it hates the complete uniaxial order but still likes it in part. Indeed, systems on the simple cubic lattice undergo a phase transition from the isotropic (i.e., disordered) state to the uniaxial (nematic) state on cooling with \(0\leq r\lesssim 0.38\).²⁹⁾ Even in the two-dimensional cases, they possess potential instability to the order.²²⁾ The lattice geometry, full of triangles with an even-membered circumstance, is essential to establish the order without frustration.

Despite a small number of spins, the octahedral cluster exhibits a qualitative change, i.e., symmetry breaking upon temperature variation, if the interaction between spins prefers mutual twist. The change accompanies enhanced fluctuation in energy and order, yet without singularity in thermodynamic quantities. Here, we proceed to the case of another cluster. Its positive result reinforces our claim of the crucial importance of lattice geometry.

A regular icositetrachoron is a regular polytope, i.e., a hyper polyhedron in dimension four.²⁸⁾ It bears the Schläfli symbol of \(\{3,4,3\}\). In the four-dimensional Cartesian coordinate \((x_{1},x_{2},x_{3},x_{4})\), 24 equivalent vertices are represented by all permutations of \((\pm\sqrt{2}/2,\pm\sqrt{2}/2,0,0)\) with any combination of double signs. Figure 5(a) illustrates the adjacency of vertices. Each vertex is connected to 8 adjacent vertices through equivalent edges (of a unit length), resulting in \(24\times 8\div 2 = 96\) edges. Figure 5(b) is projection into a three-dimensional space with constant \(x_{4}\). The icositetrachoron is projected on a cuboctahedron, of which the vertices are images from the \(x_{4}=0\) sector. Each square face carries an additional vertex, which comes from either of two sectors, \(x_{4}=\pm\sqrt{2}/2\). The additional vertices form a regular octahedron in each sector. Since the distance between two points, \(\boldsymbol{p}\) and \(\boldsymbol{q}\), is defined as \(d=[\sum_{i=1,4} (p_{i}-q_{i})^{2}]^{1/2}\), a variety of distances between vertices is limited to \(\{1,\sqrt{2},\sqrt{3}, 2\}\), corresponding to the kth nearest-neighbor vertices. We can find these distances in Fig. 5(b) while taking only large spheres (vertices in the \(x_{4}=0\) sector) into account in the projection.

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Figure 5. (Color online) Representations of a regular icositetrachoron. (a) A graph representing connections between adjacent vertices. (b) The projection into a three-dimensional space (\(x_{4}=\text{constant}\)). Large spheres on vertices of a cuboctahedron belong to the sector \(x_{4}=0\), while small spheres at centers of square faces represent vertices belonging to either of the sectors with \(x_{4}=\pm\sqrt{2}/2\). The latter is doubly degenerate in this projection. Only the edges (i.e., bonds) from small to large spheres are indicated by thin bonds. The bonds between small spheres are invisible in this projection because they are inside the cuboctahedron.

Although it is not easy to recognize from the projection, 12 regular triangles and 8 regular octahedrons gather around each vertex. The “lattice” geometry of the icositetrachoron is compatible with the ground states mimicking that of the antiferromagnetic 3-state Potts model, as confirmed by the successful coloring with R, G, and B in Fig. 5. Besides, the spin systems on these “lattices” have the degeneracy of the ground states without energetic frustration, similarly to the triangular lattice.²²⁾ For example, each spin can flip without an energy penalty, as in the octahedron case (Fig. 1). With fixed R and G, the eight (\(= 24/3\)) spins can independently take either B or B′, resulting in the degeneracy of \(2^{8}\) for a particular pair of R and G. However, some of R or G spins can flip distinctly from the octahedron, resulting in additional degeneracy. The characterization of the ground states is challenging and currently incomplete.

The simulation results of the icositetrachoron cluster are generally similar to the octahedron cluster, as seen in Fig. 3. However, the anomalies in heat capacity and susceptibility are more significant in the icositetrachoron than in the octahedron. The origin of more evident behavior may originate in the increased spin number and/or enhanced connectivity among spins. Remarkably, despite only 24 spins involved, the results for the icositetrachoron cluster enable similar analyses to the triangular lattice based on similar temperature dependence.²²⁾

Figure 3 compares the heat capacity C and susceptibility χ of the icositetrachoron cluster (the coordination number of a vertex being \(z=8\)) with others. The general trends are similar to, yet more significant than, the octahedron cluster (\(z=4\)). Namely, the heat capacity exhibits peaks around \(k_{\text{B}}T/zV_{0}\approx 0.03\) and 0.1, whereas the susceptibility only has a pronounced peak at the latter. The two peaks in heat capacity are similar to the triangular lattice, which undergoes a phase transition at a lower temperature and a crossover between the isotropic and sensitive (short-ranged nematic) states at a higher temperature. Its susceptibility shows a remarkable peak only at the higher temperature.

Figure 6 shows the nematic order parameter σ [Eq. (2)] and two-point correlations \(\langle P_{2}(\boldsymbol{s}_{i}\cdot\boldsymbol{s}_{j})\rangle\) at all possible distances as functions of temperature for the icositetrachoron cluster. Since the assumed interaction is partially ferroic (i.e., nematic), the σ is positive irrespective of temperature because of the finite-size correlation. However, upon decreasing temperature, it significantly grows around the temperature of the heat capacity peak at a higher temperature. Thus, the anomalies at the higher temperature originate in the crossover between the essentially free (well disordered) and the locally ordered nematic states. On the other hand, anomalous behavior is insignificant at the lower temperature. In contrast, two of the two-point correlations exhibit notable increases there. It is noteworthy that the two distances bear the same color as the starting (\(|\boldsymbol{d}|=0\)) spin in the coloring of Fig. 5. The contrasting temperature dependences followed by two distance groups resemble the triangular lattice.²²⁾

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Figure 6. (Color online) Nematic order parameter σ (circle) and orientational correlation \(\langle P_{2}[\boldsymbol{s}(0)\cdot\boldsymbol{s}(\boldsymbol{d})]\rangle\) (cross or plus-sign) at different distances \(|\boldsymbol{d}|\) in a cluster consisting of 24 spins on vertices of a regular icositetrachoron.

In the case of the triangular lattice, the analysis of the decay behavior of the two-point correlation offered a handy way to see the occurrence of a phase transition.²⁶⁾ It is doubtful whether this applies to the present case or whether the dependence is meaningful because we have only four distances. However, surprisingly, the distance dependence of the icositetrachoron cluster resembles the triangular lattice.²²⁾ Figure 7 illustrates those at selected temperatures. The decay at high temperatures is more substantial than a single power law, and the exponential law may apply. The power law describes the decay quite well at lower temperatures, as evident from the figure. Finally, the dependence exhibits a zigzag on further cooling, reflecting the splitting into the two distance groups, \(\{1,\sqrt{3}\}\) and \(\{\sqrt{2}, 2\}\). The former and the latter, respectively, bear different and same colors in the coloring in Fig. 5. Thus, the zigzag suggests the order of the antiferromagnetic 3-state Potts type consisting of three sublattices.

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Figure 7. (Color online) Two-point correlation of spin orientations in a cluster consisting of 24 spins on vertices of a regular icositetrachoron as a function of the logarithmic distance in dimension four at selected temperatures, \(k_{\text{B}}T/V_{0}=(0.81)^{j-10}\) (\(j=5{\text{–}}15\) from the bottom to the top).

Some attempts to analyze the decay behaviors indicated that a fit assuming two separate power-laws, \(\sigma_{1j}d^{-\nu}\) (\(j=0,1\)), with a shared power ν and different coefficients \(\sigma_{1j}\) reasonably describe the decay at a single temperature for the two distance groups. We choose \(j=0\) to the \(\{\sqrt{2}, 2\}\) group based on the same color as the central spin (\(d=0\)) in the coloring in Fig. 5. In the spirit of the analyses for the triangular lattice under external fields,²⁶⁾ the difference in fit parameters, such as \(\Delta\sigma_{1}=\sigma_{10}-\sigma_{11}\), may serve as an effective order parameter. Figure 8 indicates its temperature dependence. It is effectively null through high-temperature anomalies but becomes finite below the low-temperature anomaly in heat capacity.

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Figure 8. (Color online) \(\Delta\sigma_{1}\) (left axis, circle with error bar) and locally-defined apparent order parameters, \(A_{2\phi}^{k}\) defined by Eq. (4) (right axis, cross for \(k=1\) and plus-sign for \(k=2\)), for a cluster consisting of 24 spins on vertices of a regular icositetrachoron.

Alternately, we may define the quantity that senses the formation of the Potts-like order. In the case of the triangular lattice,²²⁾ the quantities were defined for hexagons around an arbitrarily chosen site, and they successfully served as locally-defined order parameters. Because the minimum path around a site is a square, we can take the following form in the present case: \begin{equation} A_{2\phi}^{k} = \frac{1}{4}[1-2P^{*}(d_{k})+P^{*}(\sqrt{2}d_{k})], \end{equation} (4) where \begin{equation*} P^{*}(d) = \frac{\langle P_{2}[\boldsymbol{s}(0)\cdot\boldsymbol{s}(d)]\rangle}{g^{\circ}(d)} \end{equation*} with \(g^{\circ} (d)\) being an “averaged” decay function of the correlation function necessary to correct the distance dependence. The index k represents the square consisting of the kth nearest neighbors from the central vertex.

Since the variety of d is limited to \(\{1,\sqrt{2},\sqrt{3}, 2\}\), only \(k=1, 2\), corresponding to \(d=1,\sqrt{2}\), are useful: \(k=4\) (\(d=2\)) points to a single vertex for each vertex and the \(k=3\), which means to be the nearest to the \(k=4\) vertex, results in a square with \(d=1\), i.e., \(A_{2\phi}^{3}=A_{2\phi}^{1}\). By definition, \(A_{2\phi}^{k}\) vanishes if the spin alignment on the square has the fourfold symmetry while assuming the adequate correction about the distance dependence of the correlation function.

The definition of \(g^{\circ} (d)\) depends on k of \(A_{2\phi}^{k}\). For \(k=1\), \(g^{\circ} (d)\) is a weighted average of two decay functions, i.e., \(g^{\circ} (d)=\frac{1}{3}(\sigma_{10}+2\sigma_{11})d^{-\nu}\). On the other hand, \(g^{\circ} (d)\) is given by \(\sigma_{10}d^{-\nu}\) for \(k=2\) because both the d's belong to the same distance group. Thus, definite \(A_{2\phi}^{1}\) indicates the violation of the fourfold symmetry, while \(A_{2\phi}^{2}\) should be null irrespective of such a phenomenon. Conversely, the deviation of the latter from zero is an indicator of the manipulation of the former.

Figure 8 compares the temperature dependence of \(A_{2\phi}^{k}\) with \(\Delta\sigma_{1}\). As expected, \(A_{2\phi}^{2}\) remains effectively null at all temperatures while \(A_{2\phi}^{1}\) deviates from zero around \(k_{\text{B}}T\approx 0.3V_{0}\). Thus, the temperature dependence indicates that the fourfold symmetry of the square formed by the nearest-neighbor spins is violated at low temperatures. Definite yet gradual temperature dependences of \(A_{2\phi}^{1}\) with \(\Delta\sigma_{1}\) remind us of the behavior of quantities similar to the former at short distances for the triangular lattice.²²⁾ Note that their growth becomes sharper at longer distances (around several to tens lattice spacings) and the blurred dependences originate in the uncertainty in the averaged decay function employed.