Fixed-point distributions of short-range Ising spin glasses on hierarchical lattices

Fixed-point distributions for the couplings of Ising spin glasses with nearest-neighbor interactions on hierarchical lattices are investigated numerically. Hierarchical lattices within the Migdal-Kadanoff family with fractal dimensions in the range $2.58\le D\le 7$, as well as a lattice of the Wheatstone-Bridge family with fractal dimension $D\approx 3.58$ are considered. Three initial distributions for the couplings are analyzed, namely, the Gaussian, bimodal, and uniform ones. In all cases, after a few iterations of the renormalization-group procedure, the associated probability distributions approached universal fixed shapes. For hierarchical lattices of the Migdal-Kadanoff family, the fixed-point distributions were well fitted either by stretched exponentials, or by $q$-Gaussian distributions; both fittings recover the expected Gaussian limit as $D\to \infty$. In the case of the Wheatstone-Bridge lattice, the best fit was found by means of a stretched-exponential distribution.

Figure 1

Basic cells of the hierarchical lattices investigated herein, both with scaling factor $b=2$. (a) The diamond cell belongs to the Migdal-Kadanoff family [21, 22], is defined by $p$ parallel paths, and presents a fractal dimension $D=[ln(2p\left)\right]/ln2$. (b) The cell of the Wheatstone-Bridge family $\left(D\approx 3.58\right)$ whose hierarchical lattice approaches the cubic lattice [20]. The empty circles ($\mu$ and $\nu$) represent external sites of the cell, whereas the black circles are internal sites to be decimated in the RG procedure.

Figure 2

Flux diagrams of the probability distributions $P\left(K_{ij}\right)$, constructed from the coupling distributions of Eqs. (2, 3, 4), as well as from the $q$-Gaussian of Eq. (9) and stretched exponential of Eq. (11), represented in suitable variables [11, 29] for two hierarchical lattices of the Migdal-Kadanoff family [Fig. 1] with fractal dimensions: (a) $D=3$; (b) $D=6$. In each case an arrow indicates the point associated with the fixed-point (FP) distribution $P^{*}\left(K_{ij}\right)$; the region around this point is amplified in the inset of panel (a). In this representation one identifies clearly those initial distributions that differ most from the fixed-point distribution, as those that start the RG process with the “largest distance” from the latter (see, e.g., the evolution of the bimodal distribution). In the variables used, $\langle \rangle$ represent averages over the corresponding distributions.

Figure 3

Data of the fixed-point distribution for the Ising SG on a MK hierarchical lattice with $D=3$. For all data shown, the Gaussian distribution of Eq. (2) was considered as initial distribution, at its associated critical temperature (cf. Table 1). For a large range of RG iterations, from $n=4$ up to $n=16$ (denoted by different symbols), one gets typically a fixed-point distribution. (a) In the linear representation, both $q$-Gaussian [with $q=1.10\left(1\right)$] and stretched [with $\delta =1.76\left(5\right)$] fits are indiscernible. In the inset we represent the same data as ${ln}_{q}P\left(K_{ij}\right)$ vs $K_{ij}^2$, where the $q$-Gaussian fit (full green line) follows a straight line, being compared with the stretched-exponential one (dashed black line). (b) The same data of (a) is exhibited in a log-linear representation, where the $q$-Gaussian (full green line) and stretched exponential (dashed black line) fits are shown. In the inset we represent the same data by considering larger bins for the histograms (see text).

Figure 4

Data of the fixed-point distribution for the Ising SG on a WB hierarchical lattice with $D\approx 3.58$. The Gaussian distribution of Eq. (2) was considered as initial distribution, at its associated critical temperature (cf. Table 1). For a given range of RG iterations, from $n=4$ up to $n=10$ (denoted by different symbols), one gets typically a fixed-point distribution. (a) In the linear representation, the stretched exponential of Eq. (11) [with $\delta =1.57\left(3\right)$] is presented. (b) The same data of (a) is exhibited in a log-linear representation; in the inset we represent the same data by considering larger bins for the histograms (see text).

Figure 5

Fixed-point distributions for the hierarchical lattices of the MK family with several fractal dimensions in the range $2.58\le D\le 7$, following the $q$-Gaussian distribution fit of Eq. (9), are represented as full lines, whereas the one for the WB hierarchical lattice that follows the stretched-exponential fit [cf. Eq. (11)] is represented by a dashed black line. (a) In a log-linear representation, where one sees that the widths of these distributions decrease for increasing fractal dimensions (a similar behavior occurs if one considers the stretched-exponential fits from Table 1). (b) In the variables used, those $q$-Gaussian distributions with the same index $q$ do not depend on their width, and collapse into a single curve: consequently, the full red line (outer full line) applies to those hierarchical lattices of the MK family with fractal dimensions in the range $2.58\le D\le 5$, for which $q=1.10$ within the error bars (cf. Table 1); the full lines correspond to decreasing values of $q$ from outer to inner. The dashed blue line in panel (b) corresponds to the Gaussian distribution.

Figure 6

Data of fixed-point distributions are represented in a log-linear representation. (a) Ising SG on a MK hierarchical lattice with $D=3$; apart from the distributions used in Fig. 3 (stretched exponential and $q$-Gaussian), the Gaussian, $\alpha$-stable Lévy, and Student's $t$ distributions were also considered for fitting the data. (b) Ising SG on a WB hierarchical lattice with $D\approx 3.58$; apart from the stretched-exponential distribution used in Fig. 4, the Gaussian, $\alpha$-stable Lévy, Student's $t$, and stretched $q$-exponential [cf. Eq. (12)] distributions were also considered for fitting the data. The central region of the plots are amplified (linear-linear scale) in the respective insets.