A uniform approach to antiferromagnetic Heisenberg spins on low-dimensional lattices

Abstract

Using group theoretical methods we show for both the triangular and square lattices that in the continuum limit the antiferromagnetic order parameter lives on SO(3) without respect of the initial lattice. For the antiferromagnetic chain we recover the Haldane decomposition. This order parameter interacts with a local gauge field rather than with a global one as implicitly suggested in the literature which in our approach appears in a rather natural manner. In fact this merely corresponds to a novel extension of the spin group by a local gauge field. This analysis based on the real division algebras A d applies to low dimensional lattices (d = 1, 2, 3, 4). c 2000 Elsevier Science B. V. All rights reserved. 75.50.Ee, Since the discovery of the high temperature superconductivity bidimensional antiferromagnetic Heisenberg models and their continuum limit have attracted considerable interest [1][2][3][4][5]. The problem of the continuum limit is directly related to the presence or not of a Hopf term in the Lagrangian of the bidimensional antiferromagnetic Heisenberg model [6][7][8]. This problem is still open [9,10].The continuum limit of the bidimensional antiferromagnetic Heisenberg model is much more difficult to achieve than may be expected in particular for the triangular and hexagonal lattices. This difficulty has led the authors to outline the construction of novel local antiferromagnetic degrees of freedom for low dimensional antiferromagnetic lattices. This new construction naturally reproduces the real division algebra hierarchy [11][12][13] satisfied by the nonlinear σ-model [8,14,15] (which is accepted to correspond to the continuum limit of the Heisenberg Hamiltonian for classical isotropic antiferromagnets in two dimensions [16][17][18][19][20]). Real division algebras are known as powerful tools to describe regular tessellations in low dimension (the complex algebra C on plane and the quaternion algebra H in space [13,21]) and interactions in particle physics [12,14]. This powerfulness is employed to introduce effortlessly new degrees of freedom. This algebraic construction suggests an original microscopic mechanism and reveals the presence of a novel local gauge field which is expected to give rise to a Hopf term in the continuum limit [22]. In the present paper we will focus on the antiferromagnetic chain, the antiferromagnetic triangular and square lattices only.First, let us consider compact regular tessellations partitioning low dimensional Euclidian space (d = 1, 2, 3, 4) with identical regular polytopes [23]. Then each crystal will be characterized by a regular polytope: the crystal will have the dimension d of the regular polytope, the symmetries which leave the regular polytope invariant will leave the crystal invariant, the translations which reproduce the crystal with a single regular polytope will leave the crystal coinciding with itself, and the length of the regular polytope edge will correspond to the crystal step a. Implicitly we have associated to each crystal site (i) a spin operator...