Lower-critical spin-glass dimension from 23 sequenced hierarchical models

The lower-critical dimension for the existence of the Ising spin-glass phase is calculated, numerically exactly, as $d_L=2.520$ for a family of hierarchical lattices, from an essentially exact (correlation coefficent $R^2=0.999999$) near-linear fit to 23 different diminishing fractional dimensions. To obtain this result, the phase transition temperature between the disordered and spin-glass phases, the corresponding critical exponent $y_T$, and the runaway exponent $y_R$ of the spin-glass phase are calculated for consecutive hierarchical lattices as dimension is lowered.

Figure 1

(a) The construction of the family of hierarchical lattices used in this study. Each lattice is constructed by repeatedly self-imbedding the graph. The graphs here are $n$ parallel series of $b=3$ bonds. The dimension $d=1+lnn/lnb$ of each lattice is given. The renormalization-group solution consists in implementing this process in the reverse direction for the derivation of the recursion relations of the local interactions. The lattices shown here and 19 other lattices with nearby fractional dimensions are used in our calculations. (b) The family of hierarchical lattices with $n_1$ parallel $b=3$ series of $n_2$ parallel bonds. The resulting hierarchical models are equivalent to the family in (a) with $n=n_1{n}_2$, with respect to identical critical exponents and phase diagram topology including the occurrence or nonoccurrence of a spin-glass phase.

Figure 2

Critical temperatures $1/J_C$ and critical exponents $y_T$ of the phase transitions between the spin-glass and paramagnetic phases as a function of dimension $d$, for the hierarchical models with antiferromagnetic bond concentration $p=0.5$. The runaway exponents $y_R$ of the spin-glass phase are also shown and give a perfect fit to $y_R=-1.30908+0.528513d-0.00354805d^2$, leading with a small extrapolation to the lower-critical dimension $d=2.520$ for $y_R=0$, with a very satisfactory correlation coefficient of $R^2=0.999999$.