Ising spin glasses on Wheatstone–Bridge hierarchical lattices

Abstract

Nearest-neighbor-interaction Ising spin glasses are studied on three different hierarchical lattices, all of them belonging to the Wheatstone-Bridge family. It is shown that the spin-glass lower critical dimension in these lattices should be greater than 2.32 . Finite-temperature spin-glass phases are found for a lattice of fractal dimension D ≈ 3.58 (whose unit cell is obtained from a simple construction of a part of the cubic lattice), as well as for a lattice of fractal dimension close to five. In the former case, the estimates of spin-glass critical temperatures associated with symmetric Gaussian and bimodal distributions are very close to recent results from extensive numerical simulations carried on a cubic lattice, suggesting that whole phase diagrams presented, obtained for couplings following non-centered distributions -not known up to the moment for Bravais lattices -should represent good approximations.

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

“…has a cell of the Wheatstone-Bridge family (D % 3.58), whose associated hierarchical lattice has been used in the literature to approximate models on the cubic lattice [20,35,36,38,54], In order to approach appropriately fixed-point distributions, good estimates of the paramagnetic-SG critical temperatures are necessary in each case; except for the MK lattice with fractal dimension D % 2.58, we have computed critical temperatures Tc within a three-decimal-digit certainty, with the associated error bars on the fourth decimal digits. The estimates of Tc were carried by considering, in all cases, single pools of size M = 106 real numbers.…”

“…(iv) Recently, the estimates of item (iii) were improved further through the hierarchical lattices defined by the cell of Fig. 1(b) [36], which when compared with the estimates of Ref. [14] yields a relative discrepancy of about 3% in the Gaussian case, whereas for the symmetric bimodal distribution the two estimates essentially coincide (leading to a relative discrepancy of about 0.3%).…”

“…Defining the dimensionless couplings, Kij = PJij IP = 1 /(JkT)], the corresponding RG equations may be written in the general form [35,36],…”

“…Operationally, for a given initial distribution, the associated critical temperature is estimated approximately, by following numerically P(Kij), within the standard narrowing RG proce dure (see, e.g., Refs. [7,35,36]). Herein, we have estimated Tc in most cases with a three-decimal-digit certainty (error bars on the fourth digit); therefore, we will deal with distributions that will be considered as "fixed-point" distributions for a certain number of RG iterations (typically around 10 iterations).…”

“…These lattices has been very useful for Ising SGs [7,10,11,16,[24][25][26][27][28][29][30][31][32][33][34][35][36][37][38], m ostly due to the possibility o f obtaining estim ates that are, in som e cases, very close to those o f more tim e-consum ing techniques, by perform ing relatively low -tim e-consum ing num erical com putations. Some o f these results concerning the low er critical dim ension d/, above w hich one finds a SG phase at finite tem peratures, as well as critical-tem perature estim ates, should be m entioned, (i) The bounds for the low er critical dim ension, 2 < dt < 3, w ere first obtained on M K lattices [7], a few years before their confirm ation in Refs.…”