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arXiv:cond-mat/9901209 · Physical review. B, Condensed matter

Possible existence of topological excitations in quantum spin models in low dimensions

Ranjan Chaudhury, Samir K. Paul

Published Sep 1, 1999• 8 citations
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Abstract

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.

DOI
10.1103/physrevb.60.6234

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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.
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 importance of topological excitations in physics, mainly in the description of phase transitions, has a long history beginning with the superconductivity phenomenon in the rst years of the last century. However, the ideas behind this new concept has been more accurately described by Kosterlitz and Thouless, about 40 years ago. Conductor-superconductor, uidsuperuid, rough transitions and many others are characterized as being due the appearance of topological excitations in the system. Another concept of great importance is that the universality in phase transitions . Close to a phase transition the correlations of the system become innity, so that, the details of the model become irrelevant. In this situation, systems originaly so dierent as superconductors and ferromagnets, can be described by the same theory. The critical expoents, that caracterize the transition, depend only on the system dimension, the range of the potential and of their symmetries. This work is dedicated to study of the anisotropic Heisenberg model in two dimensions. This model has a non-usual phase transition, with quasi-long range order, characterized by a change in the behavior of the spin-spin correlation function C(r). In low temperature, T < TBKT, C(r) decays as a power law with distance, r. For temperatures greater than TBKT the correlation function falls o exponencially. The free energy has all the continuous derivatives, for this reason the transition is known as innity order transition. The BKT teory assume that the transition is driven by a vortice-antivortice unbiding mechanism in the system. In our study we did extensive numerical simulations with the tecnique known as Replica Exchange Wang-Landau. This method allow us to estimate the density of states g(E) of the model. With g(E) we can calculated the relevant termodynamic functions of the system (energy, magnetization, suceptibility and correlations for instance). Using Finite Size Scaling tecniques we have determinade the transition temperature TBKT and the behavior of the correlation function for a large range of temperatures. The calculations were done for various dilutions of non-magnetic sites, p, with p = 0,0;0,20;0,30 e 0,35, that allowed us to describe with great precision the critical behavior of the model in particular the correlation lentgh
We study quantum ferrimagnets in one, two, and three dimensions by using a variety of methods and approximations. These include: (i) a treatment based on the spin coherent state path-integral formulation of quantum ferrimagnets by taking into account the leading order quantum and thermal fluctuations (ii) a field-theoretical (non-linear σσσ-model type) formulation of the special case of one-dimensional quantum ferrimagnets at zero temperature (iii) an effective description in terms of dimers and quantum rotors, and (iv) a quantum renormalization group study of ferrimagnetic Heisenberg chains. Some of the formalism discussed here can be used for a unified treatment of both ferromagnets and antiferromagnets in the semiclassical limit. We show that the low (high) energy effective Hamiltonian of a (S_1, S_2) Heisenberg ferrimagnet is a ferromagnetic (antiferromagnetic) Heisenberg model. We also study the phase diagram of quantum ferrimagnets in the presence of an external magnetic field h (hc1<h<hc2h_{c1} < h < h_{c2}hc1​<h<hc2​) and show that the low- and the high-field phases correspond respectively to the classical Néel and the fully polarized ferromagnetic states. We also calculate the transition temperature for the Berezinskii-Kosterlitz-Thouless phase transition in the special case of two-dimensional quantum ferrimagnets.
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.
Quantum many-body edge and extended magnon excitations from the 1/3 plateau of the anisotropic Heisenberg model on an open AB2{AB}_{2}AB2​ chain in a magnetic field hhh are unveiled using the density matrix renormalization group and exact diagonalization. By tuning both the anisotropy and hhh in the rich phase diagram, the edge states penetrate in the bulk, whose gap closes in a symmetry-protected topological Kosterlitz-Thouless transition. Also, we witness the squeezed chain effect, the breaking of the edge state degeneracy, and a topological change of the excitations from gapped magnons with quadratic long-wavelength dispersion to a linear spinon dispersion in the Luttinger liquid gapless phase as the anisotropy $\ensuremath{\lambda}$ approaches the critical point from the $\ensuremath{\lambda}>0$ side of the phase diagram.