Mathematical structure of the three-dimensional (3D) Ising model

Abstract

An overview of the mathematical structure of the three-dimensional (3D) Ising model is given, from the viewpoints of topologic, algebraic and geometric aspects. By analyzing the relations among transfer matrices of the 3D Ising model, Reidemeister moves in the knot theory, Yang-Baxter and tetrahedron equations, the following facts are illustrated for the 3D Ising model: 1) The complexified quaternion basis constructed for the 3D Ising model represents naturally the rotation in a (3 + 1)dimensional space-time, as a relativistic quantum statistical mechanics model, which is consistent with the 4-fold integrand of the partition function by taking the time average. 2) A unitary transformation with a matrix being a spin representation in 2 n⋅l⋅o -space corresponds to a rotation in 2n⋅l⋅o-space, which serves to smooth all the crossings in the transfer matrices and contributes as the non-trivial topologic part of the partition function of the 3D Ising model. 3) A tetrahedron relation would ensure the commutativity of the transfer matrices and the integrability of the 3D Ising model, and its existence is guaranteed also by the Jordan algebra and the Jordan-von Neumann-Wigner procedures. 4) The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases φ x , φ y , and φ z . The relation with quantum field and gauge theories, physical significance of weight factors are discussed in details. The conjectured exact solution is compared with numerical results, and singularities at/near infinite temperature are inspected.The analyticity in β = 1/(k B T) of both the hard-core and Ising models has been proved for β > 0, not for β = 0. Thus the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D Ising model. 2 in the 4D space to the energy spectrum of the system.After publication of ref. [2] , two rounds of exchanges of Comments / Responses / Rejoinders appeared in 2008-2009. [3-8] The main objections of these Comments and Rejoinders [3,5,6,8] are summarized briefly as follows: The conjectured solution disagrees with low-temperature and high-temperature series, while the convergence of the high-temperature series has been rigorously proved. There are some problems with weight factors w y and w z (such as they are first defined to be real values, but actually can be complex, and the different weights are taken for infinite and finite temperature). There is a technical error in eq. (15) of ref. [2] for the application of the Jordan-Wigner transformation. The main rebuttals in my Responses [4,7] are also summarized briefly here: The objections in the Comments/Rejoinders [3,5,6,8] are limited to the outcome of the calculations and there were no comments on the topology-based approach underlying the derivation. All the well-known theorems for the convergence of the high-temperature series are proved only for β (= 1/k B T) > 0, not for infinite temperature (β = 0). Exactly infinite temperature has been never touched in these the...

Toward a final theory of critical exponents in terms of dimensionality d plus universality class n

“…Ising case [4,5], and it is now established to take that value to 'experimental' accuracy. For discrete values of d = 4, 3 and 2, we can therefore usefully write that…”

“…The critical behavior is one of the most interesting topics in condensed mater physics [1][2][3][4][5][6][7][8][9]. Six critical exponents at/near the critical point of a second-order phase transition show some universality behaviors in various materials.…”

“…Although there has been intensive work on this subject, it is still a big challenge to understand deeply the physics beyond critical phenomena in various fields. It is interesting to give a new view on the critical exponents obtained in several models, varying from Gaussian model [2] to string theory model [1], Ising model [3][4][5], XY model and to Heisenberg model [2]. It is important to understand in deep the relationship between these models.…”

Unified theory of critical exponents generated by the Ising Hamiltonian for discrete dimensionalities 2, 3 and 4 in terms of the critical exponentη

“…[9] These two values are very close to γ = 5/4 proposed by Zhang based on his two clearly stated conjectures. [6,7] In summary, the unified theory presented here in Equations(1)-(8) embrace the exactly known critical exponents generated by the d-dimensional Ising Hamiltonian solely for the three discrete values d = 2, 3 and 4, and the results of Table 1 of the unified theory in terms of d and η must not be used for other values of d, either integers or having non-integer character as in the 2 -expansion. Since for d = 3, the third row of Table 1 gives explicit formula for all the critical exponents in terms of η, it will be of considerable interest in the present context to have (i) experimental values and (ii) computer simulation for the exponent η in three dimensions.…”

“…For the specific value η = 1/8, Zhang's proposals based on his two conjectures are recovered. [6,7] Earlier, K. Wilson, however, had proposed γ = 1.247 from the low-order terms in the 2 -expansion. [8] Kardar's book contains the value γ = 1.238 obtained by Borel summation of the epsilon expansion.…”