The possibility of the existence of topological excitations in the anisotropic quantum Heisenberg model in one and two spatial dimensions is studied using the coherent state method. It is found that a part of the Wess-Zumino term contributes to the partition function, as a topological term for ferromagnets in the long-wavelength limit in both one and two dimensions. In particular, the XY limit of the two-dimensional anisotropic ferromagnet is shown to retain the topological excitations, as expected from the quantum Kosterlitz-Thouless scenario.
We show that the Wess-Zumino kinetic terms of individual spins combine to provide nontrivial topological terms for one-dimensional antiferromagnetic spin chains but do not produce a Hopf term for antiferromagnets on two-dimensional squares lattices. This result suggests that some other mechanism is needed to account for the neutral fermions which may be present in high-
Tc materials.
We examine spin vortices in ferromagnetic quantum Heisenberg models with planar anisotropy on two-dimensional lattices. The symmetry properties and the time evolution of vortices built up from spin-coherent states are studied in detail. Although these states show a dispersion typical for wave packets, important features of classical vortices are conserved. Moreover, the results on symmetry properties provide a construction scheme for vortex-like excitations from exact eigenstates, which have a well-controlled time evolution. Our approach works for arbitrary spin length both on triangular and square lattices.
These notes present a self-contained introduction to Wess-Zumino (WZ) terms in quantum systems with
SU(N) symmetry, emphasizing the interplay between geometry, topology, and condensed-matter applications. We begin with the
SU(2) spin coherent-state path integral, where the Berry phase appears as a WZ term encoding the symplectic structure of the Bloch sphere. This example is then used to introduce the geometric origin of topological terms, their relation to integral cohomology classes, and the role of Berry curvature as the first Chern class of the canonical
U(1) bundle. We next discuss physical realizations in which such geometric terms affect dynamics, including adiabatic Berry phases and geometric quantum noise in magnetic quantum dots. A substantial part of the notes is devoted to the condensed-matter motivation for higher
SU(N) symmetries, covering
SU(N) Heisenberg models,
SU(4) spin-orbital and spin-pseudospin systems, multipolar exchange interactions, and higher-spin multipolar orders. Finally, we develop the 0+1-dimensional
SU(N) superspin coherent-state construction, identify the phase space with
CPN−1, and derive explicit local WZ terms for
SU(3) and
SU(4). The appendices provide algebraic dictionaries connecting the abstract superspin language with concrete physical embeddings, including multipolar generator bases and several useful
SU(4) parametrizations.
We examine the effective field theory of the Bethe ansatz integrable Heisenberg antiferromagnetic spin chains. It shows that the quantum critical theories for the integer spin-S chains should be characterized by the SO(3)level-S Wess-Zumino-Witten model, and classified by the third cohomology group
H3(SO(3),Z)=Z. Depending on the parity of spin S, this integer classification is further divided into two distinct universality classes, which are associated with two completely different conformal field theories: the even-S chains have gapless bosonic excitations and the odd-S chains have both bosonic and fermionic excitations. We further show that these two classes of critical states correspond to the boundary states of two distinct topological phases in two dimension, which can be described by two-dimensional doubled SO(3) topological Chern-Simons theory and topological spin theory, respectively.