Susceptibilities for the Müller-Hartmann-Zitartz countable infinity of phase transitions on a Cayley tree

We obtain explicit susceptibilities for the countable infinity of phase transition temperatures of Müller-Hartmann-Zitartz on a Cayley tree. The susceptibilities are a product of the zeroth spin with the sum of an appropriate set of averages of spins on the outermost layer of the tree. A clear physical understanding for these strange phase transitions emerges naturally. In the thermodynamic limit, the susceptibilities tend to zero above the transition and to infinity below it.

Figure 1

Schematic of the countable infinity of phase transitions of MHZ. The dots correspond to the countable infinity of phase transitions (as one traverses vertically by varying $H$ through 0 at a given temperature in the ordered phase) of integer order.

Figure 2

A Cayley tree of depth $n=3$ and coordination number $z=3$.

Figure 3

A plot of the magnetization $m=\frac{1}{N}{\sum }_i\langle {\sigma }_i\rangle$ versus magnetic field from Monte Carlo simulations of a finite-sized system of coordination number $z=3$ and depth $n=8$. Data are shown for different runs at the MHZ transition temperatures $T_2,T_{10},T_{1000},T_{\infty }\equiv T_{\mathit{BP}}$. At higher transition temperatures, the curve becomes smoother and smoother, indicating that in the thermodynamic limit the singularity would occur at a higher-order differentiation with respect to $H$. Since our system is finite, the data are practically indistinguishable for $T_{1000}$ and $T_{\infty }$.