@incollection{statphys23_0310,
	title = {Ising model on the tree-like scale free network bounded by leaves},
	address = {Genova, Italy},
	author = {T. Hasegawa and K. Nemoto},
	booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics},
	editor = {Luciano Pietronero and Vittorio Loreto and Stefano Zapperi},
	month = {9-13 July},
	url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=310},
	year = {2007},
	biburl = {http://www.bibsonomy.org/bibtex/2919235adb6c6ea866861af2b51ce9d92/statphys23},
	abstract = {In recent years, the scale free networks (SFNs) have been studied as models to describe real-world systems topologically. A SFN has a power-law degree distribution $P(k) \propto k^{- \gamma}$, where degree $k$ is the number of edges connected to a vertex, and its network topology often affects various processes taking place there. Among them, the Ising models on the SFNs have been much investigated to show that their critical behaviors are far from those on periodic lattices and strongly depend on the exponent $\gamma$.

 We can classify a network with no cyclic paths into the following two tree-like structures: (i) the Bethe lattice-like structure (BLS) where the depth is infinite and there is no boundary, and (ii) the Cayley tree-like structure (CTS) where the depth is finite and the tree is bounded by leaves. It is already known that the ferromagnetic Ising model on the SFN with the BLS remains in the ferromagnetic phase at any finite temperature for $\gamma \le 3$, while the phase transition exists at a finite temperature $T_c$ for $\gamma>3$ and its critical exponents vary depending on the exponent $\gamma$. The transition temperature $T_c$ is given by $\tanh (J/T_c)=\langle k \rangle / \langle k (k-1) \rangle$, where $J (>0)$ is the ferromagnetic interaction and $\langle \cdots \rangle$ means the average with $P(k)$. On the other hand, the knowledge of the Ising model on the SFN with the CTS is still missing. The SFNs with the BLS enable us to perform detailed analysis for local properties of spins deep within the network, while the spin systems on the SFNs with the CTS shed lights on the influence of leaves for the system's behaviors. 

 In this paper, we derive the exact representations for a restricted magnetization and the zero-field susceptibility of the Ising model on the SFN with the CTS. We show that the system has no magnetization at any finite temperature, and (i) the one-spin susceptibility at the interior part of the tree diverges below the transition temperature $T_c$ of the SFN with the BLS for $\gamma >3$, while it diverges at any finite temperature for $\gamma \le 3$, and (ii) the susceptibility at the surface part diverges below a certain temperature $T_s$ given by $\tanh^2 (J/T_s)=\langle k \rangle / \langle k (k-1) \rangle$ for $\gamma >3$, while it diverges at any finite temperature for $\gamma \le 3$. Our result demonstrates that the ferromagnetic Ising model on the SFN with the CTS shows entirely different behaviors from those with the BLS.},
	keywords = {cayley complex critical ising model network phenomena statphys23 topic-11 tree }
}