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Orientador: Bin Kang Cheng / Dissertação (mestrado) - Universidade Federal do Parana / Resumo: No formalismo de Feynman na mecânica estatística quântica, a função partição pode ser representada como uma integral de caminho. O método variacional proposto recentemente por Feynman e Kleinert permite transformar a integral de caminho numa integral no espaço de fase. no qual as flutuações quânticas são consideradas introduzindo potencial clássico efetivo. Este método foi testado com sucesso para potenciais suaves e para o potencial com a singularidade delta de Dirac. Nesta dissertação, nos aplicamos o método para potenciais com singularidade forte :(a) um potencial quadrático e (b) um potencial linear, ambos com uma parede rígida na origem. Para que a densidade de probabilidade seja zero na origem, introduzimos o método de Feynman-Kleinert adaptado. Primeiro obtivemos o potencial clássico efetivo analiticamente e avaliamos os potenciais clássicos efetivos, energias livres { ou funções partições ) e densidades de probabilidades numericamente. Por ultimo comparamos nossos resultados com os exatos, os de Feynman-Kleinert e os semi-clássicos. Nossos resultados são bons para osciladores com baixas frequências angulares e potenciais lineares fracos, mesmo para baixas temperaturas. Para osciladores com frequências angulares altas e potenciais lineares fortes, os resultados são válidos somente a temperaturas mais altas ( beta menor ou igual 1). / Abstract: In Feynman's approach to quantum statistical mechanics, the partition function can be represented as a path integral. A recently proposed variational method of Feynmam-Kleinert is able to transformed the path integral into an integral in phase space, in which the quantum fluctuations have been taken care of by introducing the effective classical potentential. This method has been tested with succeed for the smooth potentials and for the singular potential of delta. In this dissertation, we apply the method to the strong singular potentials: (a) a quadratic potential and (b) a linear potential both with a rigid wall at the origin. By satisfying the density of the particle be vanish at the origin, we introduce an adaptated method of Feynman-Kleinert in order to improve the method. We first obtain the effective classical potential analytictly and then evaluate effective classical potentials, free energies (or partition functions) and the densities of particle numerically. Finally we compare our results with those of exact, Feynman- Kleinert and semi- classical. Our results are good for lower angular frequency of oscillator and for weak linear potential even at lower temperatures. For higher angular frequency of oscillator and for strong linear potential, our results are valid only in higher temperature (up to beta greather than or equal to 1).

Στην συγκεκριμένη εργασία ασχολούμαστε με την μελέτη θερμικά σύμπλεκτων (entangled) καταστάσεων πολλών κβαντικών bit (qubit) σε διάφορα μοντέλα Heisenberg με την μέθοδο Monte Carlo (MC). Αρχικά χρησιμοποιώντας τον μετασχηματισμό Suzuki-Trotter μετατρέπουμε την κβαντική μονοδιάστατη αλυσίδα των spin (μοντέλα Ising, Heisenberg με και χωρίς μαγνητικό πεδίο στις διευθύνσεις x,y,z) σε κλασικό δισδιάστατο πλέγμα. Εξετάζουμε την συμπεριφορά του συγκεκριμένου μετασχηματισμού για το αντισιδηρομαγνητικό Heisenberg ΧΧΧ μοντέλο, για το σιδηρομαγνητικό Heisenberg μοντέλο (ΧΧΧ και ΧΥΖ) χωρίς και με μαγνητικό πεδίο στις διευθύνσεις x,y,z για διάφορα μήκη της αλυσίδας, διαφορετικές διαστάσεις Trotter και διαφορετικό αριθμό Monte Carlo βημάτων (MCΒήματα). Μελετάμε την συμπεριφορά της θερμοχωρητικότητας, της ενέργειας, της μαγνητικής επιδεκτικότητας και της μαγνήτισης στις διευθύνσεις x,y,z. Επιβεβαιώνουμε την σωστή συμπεριφορά τους με βάση τα αναλυτικά αποτελέσματα. Τέλος γνωρίζοντας, ότι η κλασική συσχέτιση είναι το κάτω όριο της ποσότητας Localizable Entanglement, και ότι η ποσότητα Entanglement of Assistance είναι το πάνω όριο, εκτιμούμε για τα ίδια μοντέλα τη συμπεριφορά των ορίων και προσπαθούμε να εκτιμήσουμε το μήκος σύμπλεξης (Entanglement Length) για διάφορες θερμοκρασίες. / In the present Master Thesis we study the thermal entangled states of many qubits in a variety of Heisenberg models with the deployment of the Monte Carlo(MC) method. Initially we are using the Suzuki-Trotter decomposition in order to convert the one dimensional spin chain(models Ising, Heisenberg with and without magnetic field in the x,y,z axis) into a classical two dimensional lattice. We examine the behavior of the latter decomposition for the antiferromagnetic Heisenberg XXX model, the ferromagnetic Heisenberg model (XXX and XYZ) with or without magnetic field in the axis x,y,z for different chain lengths, Trotter dimensions and number of Monte Carlo Steps (MCSteps). We investigate the behavior of the following quantities: specific heat, energy, susceptibility and magnetization in the axis x,y,z. We confirm their proper behavior comparing to analytical and arithmetic results. Finally knowing that the maximum classical correlation function is the lower limit of the quantity Localizable Entanglement (LE) and that the quantity Entanglement of Assistance is the upper limit, we evaluate for the same models the behavior of the limits and we try to evaluate the Entanglement Length for a variety of temperatures.