Low-temperature phase boundary of dilute-lattice spin glasses

Abstract

The thermal-to-percolative crossover exponent , well known for ferromagnetic systems, is studied extensively for Edwards-Anderson spin glasses. The scaling of defect energies are determined at the bond percolation threshold p c using an improved reduction algorithm. Simulations extend to system sizes above N =10 8 in dimensions d = 2 , . . . , 7. The results can be related to the behavior of the transition temperature T g ϳ͑p − p c ͒ between the paramagnetic and the glassy regime for p p c . In three dimensions, where our simulations predict = 1.127͑5͒, this scaling form for T g provides a rare experimental test of predictions arising from the equilibrium theory of low-temperature spin glasses. For dimensions near and above the upper critical dimension, the results provide a challenge to reconcile mean-field theory with finite-dimensional properties.The exploration of low-temperature properties of disordered systems remains an important and challenging problem. 1,2 The paradigmatic model for such phenomena is the Edwards-Anderson ͑EA͒ spin glass, 3Disorder effects arise via quenched random bonds, J i,j , mixing ferro-and antiferromagnetic couplings between nearestneighbor spins, which lead to conflicting constraints and frustrated variables. It is believed that an understanding of static and dynamic features of EA may aid a description of the unifying principles expressed in glassy materials. 3 Most insights into finite-dimensional systems have been gained through computational approaches that elucidate low-T properties. [4][5][6] Here, we extract the response induced through defect interfaces 7,8 at T = 0, created by fixing the spins along the two faces of the open boundary in one direction. Ground state energies E 0 and E 0 Ј of an instance of size N = L d are determined, which differ by reversing all spins on one of the faces. The distribution P͑⌬E͒ of interface energies ⌬E = E 0 Ј− E 0 created by this perturbation of scale L on the boundary is obtained. The typical energy scale, represented by the deviation ͑⌬E͒, grows as

Finite-size corrections for ground states of Edwards-Anderson spin glasses

“…-As a demonstration of our approach, we first treat the problem on strongly diluted EA-lattices exactly at p = p c , previously studied in ref. [22]. The fractal lattice at p c is too ramified to sustain order and y = y P < 0 for all d [42].…”

“…As a demonstration of our approach, we first treat the problem on strongly diluted EA-lattices exactly at p = p c , previously studied in Ref. [22]. The fractal lattice at p c is too ramified to sustain order and y = y P < 0 for all d. 42 Polynomial-time algorithms have been used to achieve system sizes up to N ≈ 10 8 , i.e., L ≤ 11 in d = 7, and the results for y P are recounted in Tab.…”

“…, 7 obtained numerically from domain wall excitations of ground states. The exponent yP refers to lattices at the bond percolation threshold pc [22], and y characterizes the T = 0 fixed point in the glassy phase (since Tc > 0 for d 3) for bond densities pc < p 1 [20]. The last column contains the measured values, denoted as ω, from the fit in fig.…”

“…Unfortunately, a direct study of this connection with an expansion in ǫ = d u − d is beset with technical difficulties 9 and results remain hard to interpret. Numerical approaches have renewed interest in this connection, based on simulations of 1d long-range models 12,[16][17][18] or of bond-diluted hypercubic lattices, [19][20][21][22] which we adapt for our study here. Prior to the development of this method, scaling studies on ground states beyond d = 4 (L ≤ 7) 23 were unheard of.…”

Critical behavior of three-dimensional Ising spin glass models

“…for p b > p SG . While investigations at T = 0 indicate that p SG should be identified with the bond-percolation point 60,61 (p perc = 0.2488126(5) on a simple cubic lattice 62 ), recent investigations of the critical behavior close to the percolation point suggest that p SG is larger than p perc . 63 The model can be extended by considering the distribution…”