A Toy Model Approach to Fractal Nature: Thermodynamics on a Cantor-Lattice Ising Model

Abstract

Thermodynamics on a Cantor-lattice Ising model is studied to clarify effects of fractal structure. Exact solutions based on the transfer matrix are investigated for finite size systems, and it is found that there is non-trivial relationship between entropy and fractal structure. In order to understand the nature in the thermodynamic limit, the renormalization method is applied. The results suggest a possibility of residual entropy due to the competition between non-uniformity of the fractal structure and uniform external field. These pave a simple way to approach general behaviors of non-uniform systems including fractal structures, such as quasicrystals.

1. Introduction

Quasicrystals,^1–3) experimentally observed in binary or ternary alloys [e.g., Cd₆Yb and Al–Zn–Mg³⁾], have a self-similarity such as the Penrose lattice^4–9) with an inflation rule.¹⁰⁾ These kinds of quasicrystals often show fractal structures in their essentials. To clarify physical behaviors on fractal structures, we need to know how the structure affects such thermodynamic properties as entropy, internal energy, and so on. Traditionally, a lot of fractal structures have been investigated from theoretical points of view in statistical mechanics.^11–16) Actually, as is well known, fractal structures of local orders often appear even in uniform systems at critical temperatures, where the correlation functions diverge.^15,16) Furthermore, there are some previous theoretical approaches to physical properties such as magnetism on quasiperiodic structures.^17–21) However, effects of fractal structures, which are constitutionally given in quasicrystals, on thermodynamic properties have still remained unclear so far.

Here we introduce a Cantor-lattice Ising model^22–24) to study thermodynamic properties. This is a simple example of fractal structure where we can control its strength, called fractality. In a limiting case of this model, it corresponds to the one-dimensional Ising model, while it corresponds to an isolated Ising model in the other limiting case. We explain more details of the Cantor lattice and the controlling parameter λ in the following section. In Sec. 3, we show some exact results for finite size Cantor-lattice Ising model. The most interesting case is shown in thermodynamic limit. In the thermodynamic limit, the Cantor-lattice Ising models are analyzed by a renormalization treatment as discussed in Sec. 4. Summary and discussions are included in Sec. 5.

2. Cantor-Lattice Ising Model

The Cantor-lattice Ising model^22,23) is constructed by analogy with the Cantor set.²⁴⁾ Figure 1 shows the Cantor lattice. Introducing a concept of “generation” will help us to create the Cantor-lattice Ising model. The first generation has three spins {σ₁, σ₂, σ₃} and two interactions with the same magnitude J as shown in the top of Fig. 1. Once we obtain the first generation, the second generation is given by connecting three first generations by the interaction λJ. Here the parameter λ is assumed as |λ| < 1. And then, the third generation is given by a group of second generations with the interaction λ²J. In the same manner, the k-th generation is obtained as three (k − 1)-th generations connected by the interaction λ^k−1J. The model Hamiltonian of first generation is described as   

\begin{equation} \mathcal{H}_{1} = -J(\sigma_{1}\sigma_{2} + \sigma_{2}\sigma_{3}). \end{equation} (1)

Then, partial trace over the parameter σ₂ yields its partial partition function as   

\begin{align} \skew3\hat{Z}_{1}(\sigma_{1},\sigma_{3}) &\equiv \sum_{\sigma_{2} = \pm 1} \exp [K(\sigma_{1}\sigma_{2} + \sigma_{2}\sigma_{3})] \\ &= 2(\cosh^{2}K + \sigma_{1}\sigma_{3}\sinh^{2}K), \end{align} (2)

where the parameter K denotes βJ with the inverse temperature β = 1/k_BT. On the bases of eq. (2) and the above iteration rules, a partial partition function of the k-th generation is obtained by the (k − 1)-th generation as   

\begin{align} \skew3\hat{Z}_{k}(\sigma_{1},\sigma_{N_{k}}) &= \sum_{\tau_{1},\tau_{2},\tau_{3},\tau_{4}}\skew3\hat{Z}_{k - 1} (\sigma_{1},\tau_{1})e^{\lambda^{k - 1}K\tau_{1}\tau_{2}}\\ &\quad \times\skew3\hat{Z}_{k - 1}(\tau_{2},\tau_{3})e^{\lambda^{k - 1}K\tau_{3}\tau_{4}}\skew3\hat{Z}_{k - 1}(\tau_{4},\sigma_{N_{k}}), \end{align} (3)

where the number of spins N_k = 3^k and the parameter τ_j (j = 1, 2, 3, 4) are the terminal Ising spins included in the (k − 1)-th generation. This procedure is nothing but the transfer matrix method.²⁵⁾ The above iterating relation (3) and the initial condition (2) can be easily extended in the case with including the magnetic field H. Here and after, we consider the case with magnetic fields. Using the partial partition function $\skew3\hat{Z}_{k}(\sigma _{1},\sigma _{N_{k}})$ given by eqs. (2) and (3), the exact partition function Z_k(T, H, λ) is in general obtained by   

\begin{equation} Z_{k}(T,H,\lambda) = \sum_{\sigma_{1},\sigma_{N_{k}}}\skew3\hat{Z}_{k}(\sigma_{1},\sigma_{N_{k}}). \end{equation} (4)

The above partition function Z_k(T, H, λ) is an exact solution of the Cantor-lattice Ising model. However, it is difficult to calculate it explicitly for sufficiently large systems though this distribution function can be treated rigorously for finite systems. Thus, to understand the thermodynamic limit, we need a renormalization method as discussed in Sec. 4.

Fig. 1

Schematic picture of the first, second, and k-th generation.

At the end of this section, we mention the meanings of the parameter λ. In the case of λ = 1 and k → ∞, the present model is equivalent to a one-dimensional Ising model. This dimension is of course one. On the other hand, in the case of λ = 0, the model corresponds to isolated three-spin clusters. The finite k cases also correspond to zero-dimensional even if λ = 1. Thus, in this case, the model can be regarded as zero-dimensional. According to these limiting cases, the Cantor lattice describes fractal systems in the case of 0 < λ < 1. We can treat the present Cantor-lattice Ising model as a toy model in order to observe the behavior of thermodynamic properties in fractal systems with controlling λ. Furthermore, as shown in the above, it is easy to obtain an exact solution of this model for finite k. This makes the model relatively easy to handle, and thus it is suitable for investigating how the fractal structure affects the thermodynamic properties of the system.

3. Thermodynamics on Finite Size Cantor-Lattice Ising Model

As discussed in the previous section, it is difficult to discuss the thermodynamic limit from eqs. (3) and (4). Therefore, discussing finite size systems cannot reveal the thermodynamics of the Cantor-lattice Ising model itself, but it does provide us with a clue as to how finite-size fractals approach to thermodynamic fractals. This may be similar to the relationship between approximants and quasicrystals.

In this section, we investigate finite size systems of the Cantor lattice. In this section, we treat the cases with k = 2 and 3. These cases give N₂ = 3² = 9 and N₃ = 3³ = 27 systems. The thermodynamic parameters are easily obtained using the partition function (4). As are well-known relations,²⁶⁾ the free energy f_k(T, H, λ) and the entropy s_k(T, H, λ) per one spin are obtained as   

\begin{equation} f_{k}(T,H,\lambda) = -\frac{k_{\text{B}}T}{N_{k}}\log Z_{k}(T,H,\lambda), \end{equation} (5)

and   

\begin{equation} s_{k}(T,H,\lambda) = -\frac{\partial}{\partial T}f_{k}(T,H,\lambda), \end{equation} (6)

respectively. Furthermore, we now introduce a new thermodynamic parameter “λ-conjugate” as   

\begin{equation} \Lambda_{k}(T,H,\lambda) \equiv - \frac{\partial}{\partial\lambda}f_{k}(T,H,\lambda). \end{equation} (7)

This parameter describes how the free energy changes with increasing λ.

Figure 2(a) shows the temperature dependence of the entropy on the second generation k = 2, while Fig. 2(b) shows that on the third generation k = 3. The parameter λ is assumed as λ = 0.0, ±0.25, ±0.50, ±0.75, ±1.0. As shown in Fig. 2, the entropy per one spin has an asymptotic value k_B log 2 for higher temperatures. On the other hand, in lower temperatures, the temperature dependence of the entropy obviously changes with increasing the parameter λ. This is because the correlation becomes stronger with increasing λ. Note that the case of λ = 0 is different from the other cases. In this case, the system corresponds to isolated three spin clusters, and then, the entropy becomes larger than the other cases because of its extensivity. Figure 3 shows how the entropy depends on λ. Both of (a) and (b) are calculated on third generation, under zero magnetic field (a) and under a uniform finite magnetic field H/J = 0.3 (b). The temperature k_BT/J is assumed as k_BT/J = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 in both of (a) and (b). It may be interesting to note that the entropy is an even function for λ under zero magnetic field, namely s_k(T, 0, λ) = s_k(T, 0, −λ) owing to the entropy describing just fluctuations (non-directed) as shown in Fig. 3(a). On the other hand, this even symmetry is broken under a finite magnetic field as shown in Fig. 3(b).

Fig. 2

Temperature dependence of the entropy on the second generation (a) and the third generation (b). The external field H does not applied. The parameter λ is assumed as |λ| = 0.0, 0.25, 0.50, 0.75, 1.0.

Fig. 3

λ dependence of the entropy without external fields (a) and under a finite external field (b). The external field is assumed as H/J = 0.3 in the right panel (b). Both (a) and (b) were calculated at the temperatures k_BT/J = 0.1, 0.2, 0.3, 0.5, 0.6. These results were obtained on third generation models.

Figure 4 shows the temperature dependence of the λ-conjugate Λ_k(T, H, λ) defined in eq. (7) where the parameter λ is assumed as |λ| = 0.25, 0.5, 0.75, 1.0, and the magnetic field H does not applied. If we assume λ = 0, the λ-conjugate Λ_k(T, 0, λ) vanishes for all temperatures. It is monotonically increasing for all temperatures with increasing λ, and it vanishes in higher temperatures.

Fig. 4

Temperature dependence of the λ-conjugate on third generation models. Here we assumed |λ| = 0.25, 0.50, 0.75, 1.0.

4. Thermodynamic Limit of Cantor-Lattice Ising Model

In the previous section, behaviors of the entropy and λ-conjugate in finite generations are exactly obtained by the iterating relation (3). However, this method does not help us approach to the thermodynamic limit k → ∞. In this section, we treat the Cantor-lattice Ising model by a renormalization method^27,28) in order to approach to the thermodynamic limit.

The renormalization procedure is as follows. At first, we prepare a chain of three spins {σ₁, σ₂, σ₃} with interaction J, namely $\mathcal{H}_{0} = - J(\sigma _{1}\sigma _{2} + \sigma _{2}\sigma _{3}) - H(\sigma _{1} + \sigma _{2} + \sigma _{3})$. And then, we trace out the parameter σ₂ in the partition function $Z_{0} = \sum\nolimits_{\sigma _{2}}e^{ - \beta \mathcal{H}_{0}} $ and obtain an effective two spin system $\mathcal{H}_{1}(\sigma _{1},\sigma _{3}) = - k_{\text{B}}T\log Z_{0} =- J_{\text{eff}}^{(1)}\,\sigma _{1}\sigma _{3} - H_{\text{eff}}^{(1)}(\sigma _{1} + \sigma _{3})$. Secondly, three effective two spin systems are connected as   

\begin{align} \mathcal{H}^{0}{}_{2}(\sigma_{1},\sigma_{2}) &= \mathcal{H}_{1}(\sigma_{1},\tau_{1}) + \mathcal{H}_{1}(\tau_{2},\tau_{3}) + \mathcal{H}_{1}(\tau_{4},\sigma_{2}) \\ &\quad + \lambda J(\tau_{1}\tau_{2} + \tau_{3}\tau_{4}). \end{align} (8)

Then, tracing out the dummy parameters τ₁, τ₂, τ₃, τ₄ like above, we can obtain the effective Hamiltonian   

\begin{equation} \mathcal{H}_{2}(\sigma_{1},\sigma_{2}) = -J_{\text{eff}}^{(2)}\sigma_{1}\sigma_{2} - H_{\text{eff}}^{(2)}(\sigma_{1} + \sigma_{2}). \end{equation} (9)

Connecting three $\mathcal{H}_{2}(\sigma _{1},\sigma _{2})$ systems as   

\begin{align} \mathcal{H}^{0}{}_3(\sigma_{1},\sigma_{2}) &= \mathcal{H}_{2}(\sigma_{1},\tau_{1}) + \mathcal{H}_{2}(\tau_{2},\tau_{3}) + \mathcal{H}_{2}(\tau_{4},\sigma_{2}) \\ &\quad+ \lambda^{2}J(\tau_{1}\tau_{2} + \tau_{3}\tau_{4}), \end{align} (10)

we obtain again an effective Hamiltonian of the third generation as   

\begin{equation} \mathcal{H}_{3}(\sigma_{1},\sigma_{2}) = -J_{\text{eff}}^{(3)}\sigma_{1}\sigma_{2} - H_{\text{eff}}^{(3)}(\sigma_{1} + \sigma_{2}), \end{equation} (11)

by tracing out the dummy parameters τ₁, τ₂, τ₃, τ₄. By repeating this procedure, one can obtain an iterating relation, namely renormalization flow, of $J_{\text{eff}}^{(k)}$ and $H_{\text{eff}}^{(k)}$ for the effective model. Now we redefine two parameters $K_{k} = \beta J_{\text{eff}}^{(k)}$ and $h_{k} = \beta H_{\text{eff}}^{(k)}$ to describe the renormalization flow. And then, the relation is obtained as   

\begin{equation} K_{k + 1} = \frac{1}{4}\log\frac{f_{1}(K_{k},h_{k},\lambda^{k - 1})f_{3}(K_{k},h_{k},\lambda^{k - 1})}{f_{2}^{2}(K_{k},h_{k},\lambda^{k - 1})}, \end{equation} (12)

and   

\begin{equation} h_{k + 1} = \frac{1}{4}\log\frac{f_{1}(K_{k},h_{k},\lambda^{k - 1})}{f_{3}(K_{k},h_{k},\lambda^{k - 1})} \end{equation} (13)

under the initial condition   

\begin{equation} K_{1} = \frac{1}{2}\log\Big[e^{2h}\cosh 2K\sqrt {1 - \tanh^{2}2K\tanh^{2}h}\Big] \end{equation} (14)

  

\begin{equation} h_{1} = \frac{1}{4}\log\left[\frac{1 + \tanh 2K\tanh h}{1 - \tanh 2K\tanh h}\right]. \end{equation} (15)

Here the analytical function f₁, f₂, f₃ of K, h, λ are easily obtained from the iterating procedure though the explicit form is too complicated to be described. In the Cantor-lattice Ising model, the effective interaction $J_{\text{eff}}^{(k)}$ as well as K_k vanishes in the thermodynamic limit k → ∞. This is because the parameter λ is defined in the range of |λ| < 1. This is very important, because in the case of finite systems, there remain the spin interactions. This complicates the computation for directly estimating physical quantities. For finite systems of a certain small generation, it is better to use the exact solution of the distribution function as shown in Sec. 3. The reason why we call this renormalized Hamiltonian “effective model” here is due to the vanishment of the spin interactions at the renormalization fixed point.

Finally, in the thermodynamic limit, Cantor-lattice Ising model is reduced as an effective model   

\begin{equation} \mathcal{H}_{\text{eff}} = -H_{\text{eff}}(\sigma_{1} + \sigma_{2}). \end{equation} (16)

In this effective (reduced) model, H_eff depends on the temperature T, original external field H, and the parameter λ, that is, H_eff = H_eff(T, H, λ) = H_eff(K, h, λ) where K = βJ and h = βH. This is a Hamiltonian whose distribution function corresponds to the original system for a finite number of iterations, and the term of effective field H_eff includes finite temperature effects. From the principle of renormalization theory, in the limit k → ∞, as shown in (16), this is the limit for dropping higher-order correlations.

Let us investigate the thermodynamic parameters, entropy and λ-conjugate, on the effective model (16). Using the effective field H_eff, the entropy per one spin, s(K, h, λ) = S/Nk_B, and the normalized λ-conjugate Λ₀(K, h, λ) = Λ/NJ are described as   

\begin{align} s(K,h,\lambda) &= \log 2\cosh h_{\text{eff}}(K,h,\lambda) \\ &\quad - K\frac{\partial h_{\text{eff}}(K,h,\lambda)}{\partial K}\tanh h_{\text{eff}}(K,h,\lambda) \end{align} (17)

and   

\begin{equation} \Lambda_{0}(K,h,\lambda) = \frac{1}{K}\frac{\partial h_{\text{eff}}(K,h,\lambda)}{\partial \lambda}\tanh h_{\text{eff}}(K,h,\lambda), \end{equation} (18)

where N and h_eff denote the number of spins such as N = 2, and h_eff(K, h, λ) = βH_eff(K, h, λ), respectively. In order to clarify the behaviors of s and Λ₀, it is necessary to investigate behaviors of ∂h_eff/∂K and ∂h_eff/∂λ. For this purpose, we numerically calculated the iteration relations (12) and (13) many enough times until the value ceased to fluctuate. The result is shown in Figs. 5 and 6. These figures show the entropy s(K, h, λ) under the external fields h/J = 0.1, 0.5, and 0.8, respectively, where the parameter λ is assumed as λ = −1.0, −0.5, 0.0, 0.5, 1.0. The horizontal axes are defined as K = J/k_BT, and then the larger K corresponds to lower temperature. Figure 5 shows that the entropy decreases monotonically with temperature when h/J is small (h/J = 0.1), regardless of the value of λ. On the other hand, when the external field is large, as shown in Fig. 6, a non-monotonic behavior appears in the temperature dependence of the entropy, especially in the negative λ cases. This is due to the competition between interactions that attempt to flip the boundary spins of the clusters (or generations) and the contribution of external fields that attempt to distribute the spins uniformly. The similar non-monotonic behavior of entropy owing to frustration is known in spin glass systems.²⁹⁾ Thus, as suggested in Fig. 6, there will appear residual entropy (especially in negative λ models) with external fields at lower temperatures although we could calculate only in the region K < 1 in the present calculations. We consider the origin of this residual entropy is the competition between the external fields and non-uniform interactions. On the other hand, As shown in Fig. 7, λ-conjugate shows a kind of weak oscillation with increasing inverse temperature with external fields while λ-dependence is very weak. This behavior does not appear in finite-size systems as shown in Fig. 4. These results may be important feature of the present toy model. To make the above difference clear, Fig. 8 shows the exact solution of second generation (2nd gen.) and the third generation (3rd gen.) as well as the thermodynamic limit (T.L.) obtained by renormalization methods under positive and negative λ = ±0.3 with the same external field (h/J = 0.1). The absolute value of λ-conjugate, |Λ(K, h, λ)|, is increases with increasing the inverse temperature K in the cases of second and third generations. This increasing behavior is not seen in thermodynamic limit (T.L.). In the present stage, as shown in this example, it is not trivial how the finite size systems necessarily connect to thermodynamic limit.

Fig. 5

Temperature dependence of the entropy on renormalized effective models under the magnetic field H/J = 0.1. Here we assumed λ = −1.0, −0.5, 0.0, 0.5, 1.0. Note that the horizonal axis is defined by the normalized inverse temperature K = βJ = J/k_BT, and the range of entropy is different from Fig. 6.

Fig. 6

Temperature dependence of the entropy on renormalized effective models under the magnetic field H/J = 0.5. Here we assumed λ = −1.0, −0.5, 0.0, 0.5, 1.0.

Fig. 7

Temperature dependence of the λ-conjugate on renormalized models under the magnetic field H/J = 0.1 (a), and 0.5 (b). Here we assumed λ = −1.0, −0.5, 0.0, 0.5, 1.0. There was no significant difference in λ.

Fig. 8

Generation comparison in λ-conjugates. Here we assumed the external field h/J = 0.1. The parameter λ is assumed as 0.3 in (a) while it is assumed as −0.3 in (b).

5. Summary and Discussions

In the present study, we have introduced a simple toy model to investigate how the fractal structure affects such thermodynamic parameters as entropy. To achieve this purpose, we have focused on the Cantor-lattice Ising model. In this toy model, the parameter λ plays an important role to describe its fractality. And then, we introduced a thermodynamic parameter conjugate to the parameter λ, namely the λ-conjugate Λ = −∂F/∂λ. On the Cantor-lattice Ising model, entropy and λ-conjugate have been obtained by two methods. One is exact approach based on the transfer matrix method. The other is the renormalization method discussed in Sec. 4. Using both methods, we have confirmed behaviors of the thermodynamic parameters, which will give a simple picture on fractal systems. However, unfortunately, the results obtained by these two methods look different from each other. One possible reason is due to the difference of appropriate parameter for each method. Especially, present renormalization method does not work well without external fields. According to this difficulty, it is still unclear how the behaviors of thermodynamic parameters on finite-size systems are associated with the thermodynamic limit. This issue is left for future works. Finally, these results may be reproduced in optical lattice,³⁰⁾ where we can estimate the λ-conjugate from the λ dependence of its heat capacity.

Acknowledgments

This work was mainly supported by JSPS KAKENHI Grant Number JP19H05821. And one of the authors (YH) is also supported by JSPS KAKENHI Grant Number JP19K22117.

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Appendix

In this appendix, we show an interesting relation between the entropy and the λ-conjugate. Figure A1 shows this relation of the third generation. The entropy s_k(T, 0, λ) and the λ-conjugate Λ_k(T, 0, λ) are related to each other through two parameters T and λ. Then, a range of the values that they can take in the parameter space (Λ_k, s_k) is limited even though T and λ are independently defined. As shown in Fig. A1, the entropy cannot take smaller values when the λ-conjugate takes appropriate regions (|Λ_k(T, 0, λ)| $ \lesssim $ 0.25) in a finite-size Cantor-lattice Ising model. We consider the reason for this as follows. The λ-conjugate is defined as eq. (7) and it means the correlation between adjacent (k − 1)-th generation clusters in the k-th generation clusters. Thus, the small λ-conjugate corresponds to the less correlated models. Especially, the completely isolated system corresponds to Λ₃(T, 0, λ) = 0. In such a small Λ₃(T, 0, λ) case, the minimum entropy is larger than the other cases. In other words, the minimum entropy is a decreasing function of λ-conjugate. Furthermore, it is suggested that a discontinuous change will appear on the minimum entropy at Λ₃(T, 0, λ) = ±0.25.

Fig. A1

Parametric plot of the entropy vs. λ-conjugate on third generation models.