One of the most important quantum-mechanical properties is the property of exchange. The ideas behind this arise from the fact that elementary quanta are indistinguishable in the real word, but they are completely distinguishable in our theory, where we use different labels for the different particles. The procedure by which we make our theory indistinguishable as well leads to a new “force,” called quantum-mechanical exchange. In this chapter, we develop just what exchange is, we introduce the Heisenberg spin-exchange model and prove the Marshall sign rule, and we solve the one-dimensional version of the model using the so-called Bethe ansatz. You will find it amazing to learn the quantum origin of magnetism and details about how we work with quantum systems that have many particles. Always remember that permanent magnets are one of the best examples of macroscopic quantum mechanics.
To many physicists the study of magnetism begins with a Heisenberg model, or something similar, in which localized spins interact witb eacb other. Unfortunately, nature docs not present us witb sucb a starting point. In stead, at a reasonably (thougb not absolutely) funda mental level, we are presented witb the Scbriidinger equation for electrons wbose interaction is, to a very good approximation, spin independent. It is left to the Pauli principle to induce a spin dependence, and for tbis it is essential tbat the electrons are itinerant, i.e., rwt lo calized a priori. Given the subtlety of the phenomenon, it is crucial to elucidate the conditions which encourage or discourage the emergence of ferromagnetism. Some years ago Lieb and Mattis 1 showed that one never gets ferromagnetism in the ground states of itinerant electrons on a line. The results presented here include an extension of that earlier theorem (by a different method) to positive tempera tures, and an opposite statement for systems in any di mension with a certain parity t:onstraint (requiring hard -core repulsions)- for which we show that parallel spin alignments are favored at all temperatures. Exam ples of the second kind are one-dimensional systems of odd numbers of electrons on a ring (j.e., periodit: bound ary conditionS>, for which we extend the T-O result of Herring, 2 and a particular case of the Hubbard model whose ground state was shown to be ferromagnetic by Nagaoka J and Thouless. 45 We deal with the following general (spin-independent) Hamiltonian for N identical, itinerant particles of mass
We study the large-S limit of a one-dimensional quantum spin model in a transverse magnetic field. For S=1/2, this is dual to the quantum Hamiltonian limit of the two-dimensional axial next-nearest-neighbor Ising model. The phase diagram has three regions; paramagnetic, ferromagnetic, and a modulated phase of period 2. The Peschel-Emery curve is discovered to be a disorder line. The long-distance correlation functions behave differently on the two sides of the line, and the correlation length goes to zero upon approaching it from either side.
The Ising model provides a detailed mathematical description of ferromagnetism and is widely used in statistical physics and condensed matter physics. In this Student's Guide, the author demystifies the mathematical framework of the Ising model and provides students with a clear understanding of both its physical significance, and how to apply it successfully in their calculations. Key topics related to the Ising model are covered, including exact solutions of both finite and infinite systems, series expansions about high and low temperatures, mean-field approximation methods, and renormalization-group calculations. The book also incorporates plots, figures, and tables to highlight the significance of the results. Designed as a supplementary resource for undergraduate and graduate students, each chapter includes a selection of exercises intended to reinforce and extend important concepts, and solutions are also available for all exercises.
We consider the rectangular Ising model on a half-plane of infinite extent and study some of the consequences connected with the presence of the boundary. Only the spins on the boundary row are allowed to interact with a magnetic field
H. The method of Pfaffians is employed to obtain exact expressions for the partition function. It is found that the free energy is the sum of two terms, one of which is independent of
H and proportional to the total number of lattice sites, while the other depends on
H and is proportional to the number of lattice sites on the boundary. This separation makes it possible to define various thermodynamic quantities associated with the boundary. In particular, the boundary magnetization is shown to be discontinuous, in the ferromagnetic case, at zero magnetic field for temperatures below the bulk critical temperature
Tc. This discontinuity, which is the spontaneous boundary magnetization, goes to zero as ${(1\ensuremath{-}\frac{T}{{T}_{c}})}^{\frac{1}{2}}$ as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$. For
T=Tc, the discontinuity is of course absent, and the boundary magnetization behaves as $\ensuremath{-}\mathfrak{H}\mathrm{ln}\mathfrak{H}$ for small
H. The boundary susceptibility at zero magnetic field in the ferromagnetic case exhibits a logarithmic singularity at
T=Tc, both above and below transition. An interesting feature is that the ferromagnetic boundary magnetization, although discontinuous for
T<Tc, may be analytically continued beyond the point
H=0. We interpret this as a hystersis phenomenon which we study in detail by computing the probability distribution function for the average boundary spin. The correlation function for two spins, both on the boundary row, is also obtained exactly and its asymptotic behavior is given. Finally, we derive an expression for the magnetization in any row and explicitly evaluate it for the second row, i.e., the row next to the boundary.
We have provided a concise introduction to the Ising model as one of the most important models in statistical mechanics and in studying the phenomenon of phase transition. The required theoretical background and derivation of the Hamiltonian of the model have also been presented. We finally have discussed the computational method and details to numerically solve the two- and three-dimensional Ising problems using Monte Carlo simulations. The related computer codes in both Python and Fortran, as well as a simulation trick to visualize the spin lattice, have also been provided.