Advances in Condensed Matter Physics

Volume 2015 (2015), Article ID 954524, 3 pages

http://dx.doi.org/10.1155/2015/954524

## New Topological Configurations in the Continuous Heisenberg Spin Chain: Lower Bound for the Energy

Laboratoire de Physique Théorique et Modélisation, Université de Cergy-Pontoise, 95302 Cergy-Pontoise, France

Received 19 April 2015; Accepted 12 August 2015

Academic Editor: Oleg Derzhko

Copyright © 2015 Rossen Dandoloff. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In order to study the spin configurations of the classical one-dimensional Heisenberg model, we map the normalized unit vector, representing the spin, on a space curve. We show that the total chirality of the configuration is a conserved quantity. If, for example, one end of the space curve is rotated by an angle of 2*π* relative to the other, the Frenet frame traces out a noncontractible loop in and this defines a new class of topological spin configurations for the Heisenberg model.

#### 1. Introduction

It is well known that the two-dimensional continuous Heisenberg model has very nice topological properties [1]. The order parameter is a normalized vector field (therefore, the order parameter manifold is ). If we impose homogenous boundary conditions on the vector field (constant vector field), we can compactify the plane into and therefore the possible field configurations are classified by . The energy in each class is bounded from below , where is the number of times, is wrapped around , and is the coupling constant in the Heisenberg spin Hamiltonian. These configurations are metastable because the Hamiltonian is symmetric under homothety: (where is a real parameter). The one-dimensional Heisenberg model does not have this nice topological property. Under homogeneous boundary conditions, the line may be compactified to , and now and there are no different classes of configurations based on homotopy. In order to find out if there is a new topological structure in the one-dimensional case, one has to analyze the Heisenberg Hamiltonian in more detail. The vector field is normalized and therefore we will use the following representation for . Shankar [2] has found that for a finite two-dimensional system (e.g., of radius ) the situation is even worse; that is, all configurations are point-like. Now, we will concentrate on the finite one-dimensional case.

#### 2. Materials and Methods

In and variables, the Hamiltonian has the form where the subscript stands for and denotes the coordinate along . This Hamiltonian is not symmetric under homothety transformation and therefore the spin configurations are not metastable like in the case. The equations of motion for this spin Hamiltonian have been established [3] in taking and to be the conjugated generalized coordinate and momentum so that the Poisson bracket gives . The generator of translations (momentum) is given by the following expression:

It verifies the Poisson brackets: and . These relationships have first been proven for an infinite chain by Tjon and Wright [3]. is a constant of the motion.

For our analysis of the possible spin configurations, it is useful to map the unit vector to the unit tangent of a space curve [4]. Now, different space curves will represent different spin configurations. We will impose homogeneous boundary conditions; that is, spins at will be parallel. Then, curves representing the different spin configurations will tend to the straight line as . Here, we will concentrate on the geometrical and topological quantities characterizing these space curves. Of special interest for us will be the writhe of a curve (which characterizes the chirality of the curve). For a closed curve, it is defined as follows (see, e.g., [5]):

The tip of the radius vector draws the curve, while is the unit tangent. The writhe is properly defined on a ribbon or on a thin rod. After taking the width of the ribbon to zero, one gets the writhe for a closed curve. For the proper definition of the writhe of an open curve (where at both ends of the curve the tangents are parallel to each other), one has to complete the open curve by a closure at infinity which makes the combined curve a closed curve (see, e.g., Figure 1). We have to distinguish two classes of homotopically equivalent curves: one where two ends of the curve are rotated to each other by and the other where the two ends are rotated by [6, 7]. This last class includes the straight line (i.e., the ground state for the spin configuration). The first class of curves cannot be transformed continually into the ground state without crossing the curve itself. For this reason, we have to keep a memory of the triad defined on the ribbon [6]. These two classes are due to the fact that group is nonsimply connected manifold and closed loops in fall into two classes: those that can be contracted to a point and those for which this is impossible. A triad evolving on the space curve from to traces out a closed curve on (see, e.g., [8]).