Random quantum Ising chains with competing interactions

In this paper we discuss the criticality of a quantum Ising spin chain with competing random ferromagnetic and antiferromagnetic couplings. Quantum fluctuations are introduced via random local transverse fields. First we consider the chain with couplings between first and second neighbors only and then generalize the study to a quantum analog of the Viana-Bray model, defined on a small world random lattice. We use the Dasgupta-Ma decimation technique, both analytically and numerically, and focus on the scaling of the lattice topology, whose determination is necessary to define any infinite disorder transition beyond the chain. In the first case, at the transition the model renormalizes towards the chain, with the infinite disorder fixed point described by Fisher. This corresponds to the irrelevance of the competition induced by the second neighbors couplings. As opposed to this case, this infinite disorder transition is found to be unstable towards the introduction of an arbitrary small density of long range couplings in the small world models.

Figure 1

Ratio of the number of decimated fields over the number of decimated bonds as a function of the number of remaining clusters $N$ (see text). The scale invariance of this ratio at the transition is used to locate the critical point, which is evaluated as $h_{\max }=2.20\pm 0.05$ in this case. The initial size of this zig-zag ladder is $N_0=2^{14}=16384$ spins and the decimation was performed over 1000 samples.

Figure 2

Scaling behavior of the distribution of connectivity $P\left(c\right)$ for different number $N$ of remaining clusters (spins), for the zig-zag model $[J_{\max }^{\left(1\right)}=J_{\max }^{\left(2\right)}]$. The initial size is $N_0=2^{14}=16384$ spins, and the decimation was performed over 1000 samples. After a transient regime characterized by an algebraic distribution of connectivity, the distribution ultimately renormalizes towards a delta function $c=2$ corresponding to the topology of the chain.

Figure 3

Decimation ratio for the disordered quantum Ising model on a small world lattice. The initial size is $N_0=1024$, $q=0.1$, and 5000 samples were used.

Figure 4

The scaling behavior of the distribution of connectivity for $h_{\max }=1.5$, shows a broadening of this distribution up to some clear finite size topological effects inherent to small world models. The initial size is $N_0=1024$, $q=0.1$, and 5000 samples were used.