Quantum and Topological Phase Transitions in Multi-QuDit Systems and 2D Materials

Abstract

This thesis is mainly focused on the study of quantum phase transitions of multiquDit systems (D-level many-body systems, extending standard 2-level qubit systems) and topological phase transitions in 2D materials. For this purpose, we have extended the concept of spin coherent states and its adaptation to parity symmetry from 2-level to D-level systems, using the representation theory of the unitary group U(D). Entanglement measures and phase space methods are also defined for the specific case of symmetric multi-quDits (bosons). The extension of quantum phase transitions from 2-level (qubits) to D-level (quDits) systems entails an enlarged variety of phases, which could be potentially exploited for quantum technological prospects. Parallelly, we have devoted our efforts to implement the topological phase transitions formalism in new 2D anisotropic materials such as phosphorene, which are a hot topic in material sciences and constitute the building blocks for future photonic and optoelectronic devices. This thesis is a compilation work of 7 publications [1–7] in scientific journals, which are indexed in the Journal Citation Report of the Science Citation Index, and are ranked in relevant positions, mostly in the first quartile (Q1) of the corresponding category. I have also published 4 international conference proceedings [8–11] derived from the main articles. The organization of this work begins with Chapter 1, an introduction to the state-of-the-art of quantum and topological phase transitions in the new quantum technological world, followed by the objectives and methodology. We find Chapter 5 at the end, a collection of the main results and conclusions derived from the publications, which compose the body of this thesis in Chapters 2, 3 and 4. A summary of these central chapters is as follows: In Chapter 2, we include 5 articles [1–5] arranged in 3 sections. In general, we study quantum phase transitions (QPT) in multi-quDit systems, using the 3-level Lipkin-Meshkov-Glick (LMG) model as paradigmatic example. The QPTs are characterized by the control parameter λ, measuring the interaction strength of the LMG model. The D-level or multi-quDit systems of N particles, will be modeled by collective spin operators generating a U(D) symmetry. Therefore, we have made a review of the construction of U(D) unitary irreducible representations and how to define coherent states (CS) with this symmetry, which will work as variational states modeling the lowest-energy eigenstates of our Hamiltonian in the thermodynamic limit N → ∞. In Section 2.1, the article [1] is presented. We extend the concept of quantum phase transitions, from totally symmetric to different U(3) permutation symmetry sectors of a system of identical particles, defining the so called mixed symmetry the role of a new control parameter, so that the phase space will have 4 phases and a "quadruple" point where all 4 phases coexist. In Section 2.2, the articles [2, 3] are presented. We compute entanglement and information measures for "symmetric" indistinguishable particles (bosons) in multi-quDit systems, restricting ourselves to the fully symmetric representations of U(D). In Section 2.3, the articles [4, 5] are presented. We define a generalized parity adaptation of U(D)-spin CS and make a phase space analysis of them and the LMG model eigenstates. In Chapter 3, the article [6] is presented. The Lieb-Mattis theorem is applied to U(N) quantum Hall ferromagnets at filling factor M for L Landau/lattice sites. The Hilbert space of the low energy sector in this model is identified with the carrier space of irreducible representations of U(N), described by rectangular Young tableaux of M rows and L columns, and associated with Grassmannian phase spaces GN M = U(N)/[U(M) × U(N − M)]. This chapter shed light on the manybody problems with mixed symmetry sectors, ranging from the LMG model in the previous chapter to the 2D materials in the next one. In Chapter 4, the article [7] is presented. We study how the transmittance and the Faraday angle are universal markers of topological phase transitions in a collection of 2D materials, including graphene and other Dirac materials, and HgTe quantum wells. We also show how these magnitudes become critical even for non-topological anisotropic materials such as phosphorene. For this purpose, we show how external electromagnetic fields affect these materials, and derive the current operators and the magneto-optical conductivities from the Kubo-Greenwood formula.