Symmetry, complexity and multicritical point of the two-dimensional spin glass

Abstract

We analyze models of spin glasses on the two-dimensional square lattice by exploiting symmetry arguments. The replicated partition functions of the Ising and related spin glasses are shown to have many remarkable symmetry properties as functions of the edge Boltzmann factors. It is shown that the applications of homogeneous and Hadamard inverses to the edge Boltzmann matrix indicate reduced complexities when the elements of the matrix satisfy certain conditions, suggesting that the system has special simplicities under such conditions. Using these duality and symmetry arguments we present a conjecture on the exact location of the multicritical point in the phase diagram. Let us start by considering the Hamiltonian of the ±J Ising modelwhere J ij is ferromagnetic J(> 0) with probability p and antiferromagnetic −J with 1 − p. The sum extends over nearest neighbours on the square lattice. The randomness in J ij will be treated by the replica method. We will mainly consider the case of positive integer n, the number of replicas. Some results for the quenched limit n → 0 will be discussed in subsequent sections. The constraints of Ising spins and the ±J distribution of the interactions will be relaxed later in the present section.

Duality in Finite-Dimensional Spin Glasses

“…It is straightforward to apply the same type of argument to other lattices and other models. For example, models on the square lattice (such as the Gaussian Ising spin glass and the random chiral Potts model) can be treated very similarly: The differences lie only in the explicit expressions of x k as given in section 2.9 of reference [9] and section 4.1 of reference [11]. Also, the duality structure of the four-dimensional random plaquette gauge model is exactly the same as the ±J Ising model on the square lattice [10], and therefore the present analysis applies without change.…”

“…The discussions so far have already been given in references [9,10,11]. In those papers we went on to try to identify the multicritical point by the fixed-point condition of the principal…”

“…Conjecture 2 [9,8] The exact location of the multicritical point for the ±J Ising model on the square lattice with quenched randomness is given by the formula…”

“…More precisely, recent developments [8,9,10,11,12] to derive a conjecture on the exact location of the multicritical point in the phase diagram will be analyzed from a different view point.…”

Implications of ignorance for quantum-error-correction thresholds

“…We extend this to a noise model that contains two error ratesp X andp Z , and our assumptions about what these values are, p X and p Z . The critical region of the RBIM can be determined by an ansatz [7][8][9][10] and improved upon by a renormalisation style expansion [11]. The values resulting from this ansatz are numerically verified via explicit simulation of a correction algorithm, minimum weight perfect matching.…”