Duality and multicritical point of two-dimensional spin glasses

Abstract

Determination of the precise location of the multicritical point and phase boundary is a target of active current research in the theory of spin glasses. [1][2][3][4][5][6][7][8][9][10][11] In this short note we develop a duality argument to predict the location of the multicritical point and the shape of the phase boundary in models of spin glasses on the square lattice.The first system we treat is a random Z q model with gauge symmetry which includes the ±J Ising model and the Potts gauge glass. Following the notation of Wu and Wang, 12) the partition function isHere, ξ i (= 0, 1, · · · , q − 1) is the q-state spin variable, J ij (= 0, 1, · · · , q − 1) is the quenched randomness, and V (·) is a periodic function with period q. The sum in the exponent runs over neighbouring sites on the square lattice. It is straightforward to generalize the duality transformation in the WuWang formalism 12) to the random model, and the result iswhere η ij denotes ξ i − ξ j and λ ij is the bond variable on the dual lattice corresponding to η ij and eṼ is the Fourier transform of e V . The prime in the sum indicates that the bond variables η ij and λ ij are constrained such that their sums over a closed loop vanish. 12) Thus the duality transformation consists in replacing the exponential in (2) with that in (3):1

Error thresholds for Abelian quantum double models: Increasing the bit-flip stability of topological quantum memory

“…(4), as conjectured in Refs. [39][40][41], These findings are sum m arized in the inset o f Fig. 3, w hich relates the num erical estim ates for the error thresholds (data points) to the respective hashing bound value (line).…”

Recent Progress in Spin Glasses

“…for the two dimensional model and for the three dimensional model. In two dimensions, the exact location of the N-point was predicted by [232]. The authors studied the duality transformation formulated in [233], applying it to a random model with Z q symmetry.…”

Spin glass reflection of the decoding transition for quantum error correcting codes

“…The same is true for CSS codes where the two generator matrices G X , G Z can be mapped to each other, e.g., by column permutations, as is the case for the toric codes and, more generally, for the hypergraph-product (HP) codes [15]. For many such models, the transition point at the Nishimori line can be obtained to a high numerical accuracy using the strong-disorder self-duality conjecture [22][23][24][25][26][27][28][29]…”

“…Location of the multicritical point: In many types of local spin glasses on self-dual lattices the transition from the ordered phase on the Nishimori line happens at a multicritical point whose location to a very good accuracy has been predicted by the strong-disorder self-duality conjecture [22][23][24][25][26][27][28][29]. In case of the Ising spin glasses, the corresponding critical probability p c ≈ 0.110 satisfies Eq.…”