# Étude des modèles de Richardson-Gaudin de spin-½ avec champ magnétique arbitraire

Abstract : This thesis is divided into three parts and extended over seven chapters between the general introduction and conclusion. Indeed, the first part, consisting of four chapters, is mainly focused on a static study of Richardson-Gaudin models of spin 1/2 with arbitrary magnetic field including the possibility of a totally anisotropic XYZ coupling such that the terms $S^x_jS^x_k , S^y_j S^y_k mathrm{and} S^z_j S^z_k$ all have distinct coupling strengths. In this part we determine generically the energy spectrum, the central spin mean values, the wave functions and the two-point correlation functions of the considered models; which will be useful for an advanced dynamical analysis of these models. The calculation of the energy spectrum of the current system can exploit its integrability which is materialized by a set of equations constituting the integrability condition. This calculation is done independently of the symmetry of the model considered and involves the resolution of a system of quadratic equations that is relatively simple to solve numerically. The second part focuses on some examples of applications that could be implemented on the basis of the results obtained in part I. Indeed, it has recently been shown that the XX central spin model is integrable. It describes a particular spin coupled via XX spin-spin couplings to a bath of other spins while feeling a magnetic field along the $hat{z}$ axis, i.e. perpendicular to the plane in which the coupling exists. What interests us here is first to demonstrate that the XX central spin model remains integrable in the presence of an arbitrarily oriented magnetic field. In a second step we study whether the dark states remain in this case. The U(1) symmetry of the considered model is broken by the addition of an external field component in the XY plane, which could lead to a radically different physics. Nevertheless, we show that, for a sufficiently strong coupling, the structure of the dark states reappears even in the presence of a magnetic field in the plane. A simple understanding of this result and the demonstration of the universality of the results by a finite size scale study and for a variety of distinct distributions of the coupling constants at the different spins of the bath. Another example of application which closes this second part is that of the XXX dark states with arbitrary magnetic field. These states, although having a different physics from the one encountered in the previously studied models, exist and a generic law has been established to count them. This law has been verified up to 12 spins in the bath, so we can assume that it also works beyond 12 spins in the bath, we can conjecture that it is valid for any value of N. The third part of this project is dedicated to a dynamic study of non-integrable models similar to those studied in the first part of this manuscript. Indeed, these models are just the representation of Richardson-Gaudin models immersed in a time dependent magnetic field oriented along $hat{z}$. In a process of rotation around the $hat{z}$ axis of the magnetic field associated to the central spin, it is possible to define a time dependent basis in which the time dependent problem reduces to an independent problem. The resulting Hamiltonian can then be integrable and require the use of the results found in the first part to be solved. For these models, by imagining for example that the system is initially prepared in a dark state of the static Hamiltonian, we clearly see that its dynamics is not trivial and leads to perfect oscillations of this state. Finite size systems resulting from the central spin models treated in this thesis depend strongly on the distribution of couplings between the central spin and each nuclear spin constituting the bath.
Keywords :
Document type :
Theses
File URL :
https://docnum.univ-lorraine.fr/ulprive/DDOC_T_2021_0154_DIMO_PANJIO.pdf

https://hal.univ-lorraine.fr/tel-03418876
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Submitted on : Monday, November 8, 2021 - 10:43:08 AM
Last modification on : Tuesday, November 9, 2021 - 4:05:02 AM

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• HAL Id : tel-03418876, version 1

### Citation

Claude Dimo Panjio. Étude des modèles de Richardson-Gaudin de spin-½ avec champ magnétique arbitraire. Physique mathématique [math-ph]. Université de Lorraine, 2021. Français. ⟨NNT : 2021LORR0154⟩. ⟨tel-03418876⟩

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