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arXiv:cond-mat/0101466 · Modern Physics Letters B

EFFECTIVE THEORY FOR A QUANTUM SPIN SYSTEM IN LOW DIMENSIONS – BEYOND THE LONG WAVELENGTH LIMIT

RANJAN CHAUDHURY, SAMIR K. PAUL

Published Apr 10, 2002• 6 citations
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Abstract

An effective theory for a quantum spin system in low dimensions is constructed in the finite-q regime. It is shown that there are field configurations for which Wess–Zumino terms contribute to the partition functions as topological terms for ferromagnets as well as antiferromagnets in both one- and two-dimensional lattices. This is in sharp contrast to the absence of topological excitations in two-dimensional quantum antiferromagnets in the long wavelength limit.

DOI
10.1142/s0217984902003713

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An effective theory for a quantum spin system in low dimensions is constructed in the finite-q regime. It is shown that there are field configurations for which Wess–Zumino terms contribute to the partition functions as topological terms for ferromagnets as well as antiferromagnets in both one- and two-dimensional lattices. This is in sharp contrast to the absence of topological excitations in two-dimensional quantum antiferromagnets in the long wavelength limit.
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{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).
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.
Microscopic models of quantum antiferromagnets are investigated on the basis of a mapping onto effective low-energy Hamiltonians. Lattice effects are carefully taken into account and their role is discussed. We show that the presence of an external staggered magnetic field modifies in a nontrivial way the usual mapping onto the nonlinear sigma model, leading to the appearance of terms neglected in previous works. Our analysis is compared with Lanczos diagonalizations of S=1S=1S=1 Heisenberg chains in a staggered field, confirming the validity of the single-mode approximation for the evaluation of the dynamical structure factor. The results are relevant for the interpretation of experiments in quasi-one-dimensional compounds. Microscopic realizations of SU(4) spin chains are also discussed in the framework of spin-orbital lattice systems. The low-energy physics is shown to be described by sigma models with topological angle $\ensuremath{\theta}$ in one dimension. This mapping strongly suggests that the one-dimensional CP3{\mathrm{CP}}^{3}CP3 model (with $\ensuremath{\theta}=\ensuremath{\pi})$ undergoes a second-order phase transition as a function of the coupling.
Low-temperature magnetic properties of both classical and quantum dimerized ferromagnetic spin chains are studied. It is shown that at low temperatures the classical dimerized model reduces to the classical uniform model with the effective exchange integral ${J}_{0}=J(1\ensuremath{-}{\ensuremath{\delta}}^{2})$, where $\ensuremath{\delta}$ is the dimerization parameter. The partition function and spin correlation function are calculated by means of mapping to the continuum limit, which is justified at low temperatures. The quantum model is studied using the Dyson-Maleev representation of the spin operators. It is shown that in the long-wavelength limit the Hamiltonian of the quantum dimerized chain reduces to that of the uniform ferromagnetic chain with the effective exchange integral ${J}_{0}=J(1\ensuremath{-}{\ensuremath{\delta}}^{2})$. This fact implies that the known equivalence of the low-temperature magnetic properties of classical and quantum ferromagnetic chains remains for the dimerized chains. The considered model is generalized to include the next-neighbor antiferromagnetic interaction.