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arXiv:1211.5956 · Open Physics

Quantum corrections to ϕ 4 model solutions and applications to Heisenberg chain dynamics

Grzegorz Kwiatkowski, Sergey Leble

Published Jan 1, 2013• 0 citations
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Abstract

Abstract The Heisenberg spin chain is considered in ϕ 4 model approximation. Quantum corrections to classical solutions of the one-dimensional ϕ 4 model within the correspondent physics are evaluated with account of rest d-1 dimensions of a d-dimensional theory. A quantization of the model is considered in terms of spacetime functional integral. The generalized zeta-function formalism is used to renormalize and evaluate the functional integral and quantum corrections to energy in a quasiclassical approximation. The results are applied to appropriate conditions of the spin chain model and its dynamics, for which elementary solutions, energy and the quantum corrections are calculated.

DOI
10.2478/s11534-013-0267-4

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Abstract The Heisenberg spin chain is considered in ϕ 4 model approximation. Quantum corrections to classical solutions of the one-dimensional ϕ 4 model within the correspondent physics are evaluated with account of rest d-1 dimensions of a d-dimensional theory. A quantization of the model is considered in terms of spacetime functional integral. The generalized zeta-function formalism is used to renormalize and evaluate the functional integral and quantum corrections to energy in a quasiclassical approximation. The results are applied to appropriate conditions of the spin chain model and its dynamics, for which elementary solutions, energy and the quantum corrections are calculated.
The Wigner method is employed to develop explicit relationships among quantum-mechanical and classical equations of motion and quantum-mechanical and classical pair-correlation functions of spin operators. In this method, quantum corrections to the classical theory can be accurately estimated. The formalism is applied to a spin-\textonehalf{} Heisenberg model at high temperatures. The most attractive feature of this method is that many of the results are intuitively satisfying. Finally, we make a connection between our results and the classical calculations made by several other authors.
We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.
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.
Representations for the generating functionals of static correlators of zzz-components of spins in the XY and XXXXXX Heisenberg spin chains are obtained in the form of sums of the fermionic functional integrals. The peculiarity of the functional integrals in question is because of the fact that the integration variables depend on the imaginary time ``automorphically''. In other words, the integration variables are multiplied with a certain complex number when the imaginary time is shifted by a period. Therefore, the corresponding boundary conditions at the ends of the imaginary time segment are not of the form corresponding to fermionic, or bosonic, variables taken in the Matsubara representation at nonzero temperature. In fact, one part of sites of the models corresponds to the integration variables which are subjected to the unusual boundary conditions, while the variables on the other sites depend on the imaginary time conventionally, i.e., as fermions (or bosons). Thus a situation, when an ``automorphic'' boundary condition is the same for all sites of a chain spin model, is generalized. The results of the functional integration are obtained in the form of determinants of the matrix operators which are regularized by means of the generalized zeta-function approach. The partition functions of the models and certain correlation functions at nonzero temperature are obtained explicitly thus demonstrating correctness of the functional integral representations proposed.
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.