Hamiltonien de Heisenberg

Résumé

Dans la théorie du magnétisme quantique, l'hamiltonien de Heisenberg décrit un ensemble de moments magnétiques localisés en interaction. Cet hamiltonien s'écrit : : H=\frac 1 2 \sum_{i,j} J_{ij} \vec{S}_i \cdot \vec{S}_j - \sum_i g_i \mu_B \vec{h}\cdot \vec{S}_i où \mu_B est le magnéton de Bohr, g_i est le rapport gyromagnétique du i-ème moment localisé, \vec{S}i est un opérateur de spin, \vec{h} est le champ magnétique externe, et J{ij} est la constante d'échange. Pour J_{ij}>0 l'interaction est antiferromagnétique et pour J_{ij}

Source officielle

https://fr.wikipedia.org/wiki/Hamiltonien_de_Heisenberg

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Publications associées (61)

Publications associées

Chargement

Personnes associées (9)

Personnes associées

Chargement

Unités associées (3)

Unités associées

Chargement

Concepts associés (13)

Concepts associés

Chargement

Cours associés (4)

Cours associés

Chargement

Séances de cours associées (13)

Séances de cours associées

Chargement

Asymptotic freedom, Haldane gap and edge states of SU(N) spin chains

Samuel Gozel

In this thesis we study various one-dimensional quantum spin systems with SU(2) and SU(N) symmetry. We investigate the short-distance behavior of the SU(2) Heisenberg model in the limit of large spin and show that there exists an extended regime where perturbation theory, in the form of spin-wave theory, can be successfully applied. The reason for this perturbative regime stems from the asymptotic freedom of the nonlinear sigma model onto which the Heisenberg model can be mapped. When considering observables which respect the rotational invariance of the model, we observe a cancellation of infrared divergences in the perturbative expansions, leading to a meaningful description of correlation functions. We then turn to the study of SU(N) models. Building on the representation theory of the SU(N) group and on the matrix product state (MPS) formalism, we introduce a generic method to construct Affleck-Kennedy-Lieb-Tasaki (AKLT) models having edge states described by any self-conjugate irreducible representation (irrep) of SU(N). A simple example is given by a spin-1 AKLT model having spin-1 edge states. The phase transition between this model and the original AKLT model is shown to be continuous and to correspond to a topological phase transition described by the SU(2) level-1 Wess-Zumino-Witten conformal field theory universality class. In addition we study an SU(3) AKLT model for the 3-box symmetric irrep at each site. We demonstrate that the edge states are adjoint edge irreps, we extract its correlation length and provide a useful construction as an optimal MPS. Finally, we develop a density matrix renormalization group algorithm based on standard Young tableaus and subduction coefficients to make full use of the non-abelian symmetry and to investigate the SU(3) Heisenberg model with 3-box symmetric irrep at each site. We show that the model has a finite gap above the singlet ground state, in agreement with an extension of the Haldane conjecture to SU(3) chains in the fully symmetric irreps. We also argue that there are five branches of elementary excitations living in four different irreps, each of which is gapped.

EPFL2020

Chargement

Linear Flavour-Wave Theory of SU(N) Lattice Models with Arbitrary Irreducible Representations

Francisco Hyunkyu Kim

This thesis explores various approaches of studying the long-range colour order of antiferromagnetic SU(N) Heisenberg models with the linear flavour-wave theory (LFWT). The LFWT is an extension of the well-known SU(2) spin-wave theory to SU(N), and this semi-classical method has been used to study SU(N) models with colour-ordered ground states in the fundamental irreducible representation (irrep). The aim of this thesis is to study various SU(N) Heisenberg models in different irreps and in various colour configurations, in part by extending the LFWT method to different irreps of SU(N). This will be achieved by using three different methods all using different bosonic representations. First, the LFWT willbe performed using the multiboson method that introduces a boson for each state of a given irrep. Then, a different SU(3) bosonic representation first introduced by Mathur & Sen will be presented and used to perform the LFWT calculations. Finally, another way of applying the LFWT will be shown using the bosons first used by Read & Sachdev for SU(N) irreps with rectangular Young tableaux. The specific models that will be treated by these methods are the fully antisymmetric SU(N) models on the square, honeycomb, and triangular lattices with m>1 particles per site, as well as the bipartite SU(3) Heisenberg chain in the adjoint irrep to show how the LFWT can be used for mixed symmetries. The last chapter of the thesis will be dedicated to another problem related to the accidental line of zero modes in the harmonic spectrum of the antiferromagnetic SU(3) Heisenberg model on the square lattice with one particle per site in a three-sublattice order. The objective will be to try to lift the accidental zero modes in the spectrum.

EPFL2019

Chargement

The spin-1/2 Ising-Heisenberg model on two geometrically related triangles-in-triangles lattices is exactly solved through the generalized star-triangle transformation, which establishes a rigorous mapping correspondence with the effective spin-1/2 Ising model on a triangular lattice. Basic thermodynamic quantities were exactly calculated within this rigorous mapping method along with the ground-state and finite-temperature phase diagrams. Apart from the classical ferromagnetic phase, both investigated models exhibit several unconventional quantum ordered and disordered ground states. A mutual competition between two ferromagnetic interactions of basically different character generically leads to the emergence of a quantum ferromagnetic phase, in which a symmetric quantum superposition of three up-up-down states of the Heisenberg trimers accompanies a perfect alignment of all Ising spins. In the highly frustrated regime, we have either found the disordered quantum paramagnetic phase with a substantial residual entropy or a similar but spontaneously ordered phase with a nontrivial criticality at finite temperatures. The latter quantum ground state exhibits a striking coexistence of imperfect spontaneous order with partial disorder, which is evidenced by a quantum reduction of the spontaneous magnetization of Heisenberg spins that indirectly causes a small reduction of the spontaneous magnetization of otherwise classical Ising spins. DOI: 10.1103/PhysRevB.87.024421

Amer Physical Soc2013

Chargement

Asymptotic freedom, Haldane gap and edge states of SU(N) spin chains

Samuel Gozel

EPFL2020