Chaotic spin correlations in frustrated Ising hierarchical lattices

Spin-spin correlations are calculated in frustrated hierarchical Ising models that exhibit chaotic renormalization-group behavior. The spin-spin correlations, as a function of distance, behave chaotically. The far correlations, but not the near correlations, are sensitive to small changes in temperature or frustration, with temperature changes having a larger effect. On the other hand, the calculated free energy, internal energy, and entropy are smooth functions of temperature. The recursion-matrix calculation of thermodynamic densities in a chaotic band is demonstrated. The leading Lyapunov exponents are calculated as a function of frustration.

Figure 1

The family of hierarchical models from Ref. 1. In unit (a), there are $p$ cross bonds. In unit (b), two paths, consisting of $m_1$ and $m_2>m_1$ bonds in series, are in parallel. In (c), the final graph of the model is assembled with $p_a$ and $p_b$ of each unit in parallel. Each wiggly bond, representing an infinite antiferromagnetic coupling, has the effect of reversing the sign of an adjoining bond.

Figure 2

Renormalization-group flow topologies, Lyapunov exponents, and spin-spin correlations, as Hamiltonian parameters are scanned. In both examples, $p=5$, $p_b=2$, and $m_2=m_1+5$. In the upper panels, the lower curve, visible on the left, is a line of unstable fixed points, giving the second-order phase transition between the paramagnetic (below) and ordered (above) phases. In the ordered phase, only fixed points, limit cycles, and chaotic bands that are stable (attractive) are shown. In the lower panels, the spin-spin correlation function $⟨s_i{s}_j⟩$ for spins separated by a distance $2^n$ are given for consecutive $n$ at each value of the Hamiltonian parameters. It is shown in the text that the Lyapunov exponents, middle panels, apply to both upper and lower panels. Left panels: scanning $p_a$, which increases the disordering by ordering effect, at $m_1=5$. Right panels: scanning $m_1$, which increases the ground-state entropy of per first renormalized bond at $p_a=50$.

Figure 3

(a) Number of visits per interaction interval $\Delta t=0.005$ for 5000 chaotic iterations in the trajectory starting at $t^{\left(0\right)}=0.5$ and for $p=4$, $p_a=40$, $p_b=1$, $m_1=4.7$, and $m_2=m_1+1$. (b) Number of visits per correlation interval $\Delta ⟨s_i{s}_j⟩=0.005$ for the trajectory in (a).

Figure 4

Overlaps between consecutive groups of $N$ iterations for the trajectory in Fig. 3a. The relative difference $\frac{1}{200}{\Sigma }_{i=1}^{200}\left|\Delta n_i\right|/{\overline{n}}_i$, between two consecutive groups, in the number of visits $n_i$ to each of the 200 interaction intervals $i$ is shown as a function of group size $N$.

Figure 5

The spin-spin correlation function $⟨s_i{s}_j⟩$ for spins separated by a distance $2^n$ for $K=2.5$, $p=4$, $p_a=40$, $p_b=1$, $m_1=3.7,4.7,5.7$, and $m_2=m_1+1$.

Figure 6

Deviations, for small temperature or frustration change, in the spin-spin correlation function $⟨s_i{s}_j⟩$ for spins separated by a distance $2^n$ and for $K=0.8$, $p=4$, $p_a=40$, $p_b=1$, $m_1=8$, and $m_2=m_1+1$. In (a) and (b), between the two trajectories, $\Delta K=0.001$ and $\Delta p_a=0.001$, respectively.

Figure 7

For model parameters $p=4$, $p_a=10$, $p_b=1$, $m_1=7$, and $m_2=m_1+1$, (a) free energy per bond, $F=\text{ln}Z/N_{\text{bond}}$, (b) internal energy, $U=⟨s_i{s}_j⟩$, (c) entropy per bond, $S=\text{ln}Z/N_{\text{bond}}-K⟨s_i{s}_j⟩$, versus temperature $K^{-1}$.