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Physical review. B, Condensed matter

Topological terms in one- and two-dimensional quantum Heisenberg antiferromagnets

Eduardo Fradkin, Michael Stone

Published Oct 1, 1988• 182 citations
Publisher

Abstract

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-${T}_{c}$ materials.

DOI
10.1103/physrevb.38.7215

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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{T}_{c}Tc​ materials.
Physical configurations corresponding to topological excitations present in the XY limit of a quantum spin 1/2 Heisenberg ferromagnet, are investigated on a two dimensional square lattice. Quantum vortices(anti-vortices) are constructed in terms of terms of coherent spin field components and the crucial role of the associated Wess-Zumino term is highlighted. It is shown that this term can identify a large class of vortices(anti-vortices). In particular the excitations with odd topological charge belonging to this class, are found to exhibit a self-similar pattern regarding the internal charge distribution. Our formalism is distinctly different from the coventional approach for the construction of quantum vortices(anti-vortices).
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 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)=ZH^{3}(SO(3),Z)=ZH3(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.
The dimensional crossover in a spin-S nearest-neighbor Heisenberg antiferromagnet is discussed as it is tuned from a two-dimensional square lattice, of lattice spacing a, towards a spin chain by varying the width Ly{L}_{y}Ly​ of a semi-infinite strip ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$. For integer spins and arbitrary Ly{L}_{y}Ly​, and for half integer spins with Ly/a{L}_{y}/aLy​/a an arbitrary even integer, explicit analytical expressions for the zero temperature correlation length and the spin gap are given. For half integer spins and Ly/a{L}_{y}/aLy​/a an odd integer, it is argued that the c=1c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1c=1 behavior of the SU(2)1{)}_{1})1​ Wess-Zumino-Witten fixed point is squeezed out as the width ${L}_{y}\ensuremath{\rightarrow}\ensuremath{\infty}$; here c is the conformal charge. The results specialized to S=12S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}S=21​ are applied to spin-ladder systems.
Bulk magnetism in solids is fundamentally quantum mechanical in nature. Yet in many situations, including our everyday encounters with magnetic materials, quantum effects are masked, and it often suffices to think of magnetism in terms of the interaction between classical dipole moments. Whereas this intuition generally holds for ferromagnets, even as the size of the magnetic moment is reduced to that of a single electron spin (the quantum limit), it breaks down spectacularly for antiferromagnets, particularly in low dimensions. Considerable theoretical and experimental progress has been made in understanding quantum effects in one-dimensional quantum antiferromagnets, but a complete experimental description of even simple two-dimensional antiferromagnets is lacking. Here we describe a comprehensive set of neutron scattering measurements that reveal a non-spin-wave continuum and strong quantum effects, suggesting entanglement of spins at short distances in the simplest of all two-dimensional quantum antiferromagnets, the square lattice Heisenberg system.