Lower lower-critical spin-glass dimension from quenched mixed-spatial-dimensional spin glasses

Abstract

By quenched-randomly mixing local units of different spatial dimensionalities, we have studied Ising spin-glass systems on hierarchical lattices continuously in dimensionalities 1 ≤ d ≤ 3. The global phase diagram in temperature, antiferromagnetic bond concentration, and spatial dimensionality is calculated. We find that, as dimension is lowered, the spin-glass phase disappears to zero temperature at the lower-critical dimension dc = 2.431. Our system being a physically realizable system, this sets an upper limit to the lower-critical dimension in general for the Ising spin-glass phase. As dimension is lowered towards dc, the spin-glass critical temperature continuously goes to zero, but the spin-glass chaos fully sustains to the brink of the disappearance of the spin-glass phase. The Lyapunov exponent, measuring the strength of chaos, is thus largely unaffected by the approach to dc and shows a discontinuity to zero at dc.

Phase Transitions of the Variety of Random-Field Potts Models

“…1 are identical, which ascertains that the Migdal-Kadanoff approximation is a physically realizable, robust approximation. Thus, this hierarchicalmodel/Migdal-Kadanoff approach correctly yields the lower-critical dimensions of Ising [25], Potts [19], XY models [25,26], the low-temperature critical phases of the antiferromagnetic Potts [27][28][29] and d = 2 XY models [25,26], the lower-critical dimensions of the spin-glass [30,31] and random-field Ising [18] models, chaos under rescaling in spin glasses [32], the experimental phase diagrams of surface systems [33], the phase diagrams of high-temperature superconductors [34], etc. Other physically realizable approximations have also been used in studies of polymers [35,36], disordered alloys [37], and turbulence [38].…”

Across Dimensions: Two- and Three-Dimensional Phase Transitions from the Iterative Renormalization-Group Theory of Chains

“…Furthermore, random local densities can be obtained for quenched random systems [9], in d = 1 using renormalization-group theory, and applied with our method to a variety of quenched random systems in d > 1. It would also be interesting to apply to systems which show chaos under direct renormalizationgroup theory, obtaining an alternate path to study such chaos [5,10,11].…”

“…III below. [4,5] Typical calculated renormalization-group flows of (J, H) are given in the lower panel of Fig. 3.…”

Renormalization-Group Theory of the Heisenberg Model in d Dimensions

“…Most recently, the changeover from first-to second-order phase transitions of q-state Potts models in d dimensions has been obtained by the Migdal-Kadanoff approximation. [14] In complex ordering systems with frozen microscopic disorder (quenched randomness), d c = 2 has been determined for the randomfield Ising [15,16] and XY models [17], and, yielding a non-integer value, d c = 2.46 for Ising spin-glass systems [18] (but reaching lower dimensions under spin-glass rewiring [9]). Study of the Migdal-Kadanoff approximation has led to the formulation of exactly soluble hierarchical models [19][20][21], yielding a plethora of exactly soluble models custom-fitted to the physical problems on hand.…”