Comments concerning the Ising model and two letters by N.H. March

Abstract

Two recent articles by Norman H. March that contain misleading statements concerning 3D Ising models, partly based on earlier erroneous work of Z.D. Zhang, are addressed.

A Method of Riemann–Hilbert Problem for Zhang’s Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure

“…(3) The Jordan-Wigner transform used in the very beginning of Zhang's original paper [3] is invalid. (4) The critical exponents α = 0 and γ = 5/4 for the solution to the 3D Ising model differ from perturbative, nonperturbative, and even experimental results, all of which are consistent with each other [16], with a special emphasis of new results of El-Showk et al [17] using convex optimization of the c-parameter within the conformal bootstrap approach to the four-point correlation functions. We do not want to repeat Zhang's detail responses published already [3,4,8,18,19].…”

“…Before processing the present work, we would like to compare the exact solution of the ferromagnetic 3D Ising model with other results in the literature. Since publication of Zhang's conjectured exact solution [3], there has been a series of criticisms from Wu, McCoy, Fisher and Chayes [11,12], Perk [13][14][15], and Fisher and Perk [16]. The main contents of their comments [11][12][13][14][15][16] are summarized briefly as follows: (1) The exact hightemperature series is not reproduced, which has been proved rigorously to be convergence at temperature greater than any positive number.…”

“…Since publication of Zhang's conjectured exact solution [3], there has been a series of criticisms from Wu, McCoy, Fisher and Chayes [11,12], Perk [13][14][15], and Fisher and Perk [16]. The main contents of their comments [11][12][13][14][15][16] are summarized briefly as follows: (1) The exact hightemperature series is not reproduced, which has been proved rigorously to be convergence at temperature greater than any positive number. (2) The exact low-temperature series is not reproduced, even in the first term.…”

Pushing the limits of Monte Carlo simulations for the three-dimensional Ising model

“…Similarly, by considering the critical behavior of |m| at K est c , we obtained β/ν = 0.518 01 (35), or β = 0.326 30 (22). (47)…”

“…New Monte Carlo simulation based, in part, on non-perturbative approaches [16][17][18], and tensor renormalization group the-ory with high-order singularity value decomposition [19] have also yielded very precise results. In clarifying work, Wu, McCoy, Fisher, Chayes and Perk [20][21][22] gave very convincing arguments that a supposed "exact" solution was simply wrong.…”