Signatures of dissipative quantum chaos

Understanding the far-from-equilibrium dynamics of dissipative quantum systems, where dissipation and decoherence coexist with unitary dynamics, is an enormous challenge with immense rewards. Often, the only realistic approach is to forgo a detailed microscopic description and search for signatures of universal behavior shared by collections of many distinct, yet sufficiently similar, complex systems. Quantum chaos provides a powerful statistical framework for addressing this question, relying on symmetries to obtain information not accessible otherwise. This thesis examines how to reconcile chaos with dissipation, proceeding along two complementary lines. In Part I, we apply non-Hermitian random matrix theory to open quantum systems with Markovian dissipation and discuss the relaxation timescales and steady states of three representative examples of increasing physical relevance: single-particle Lindbladians and Kraus maps, open free fermions, and dissipative Sachdev-Ye-Kitaev (SYK) models. In Part II, we investigate the symmetries, correlations, and universality of many-body open quantum systems, classifying several models of dissipative quantum matter. From a theoretical viewpoint, this thesis lays out a generic framework for the study of the universal properties of realistic, chaotic, and dissipative quantum systems. From a practical viewpoint, it provides the concrete building blocks of dynamical dissipative evolution constrained by symmetry, with potential technological impact on the fabrication of complex quantum structures. (Full abstract in the thesis.)Comment: PhD Thesis, University of Lisbon (2023). 264 pages, 54 figures. Partial overlap with arXiv:1905.02155, arXiv:1910.12784, arXiv:2007.04326, arXiv:2011.06565, arXiv:2104.07647, arXiv:2110.03444, arXiv:2112.12109, arXiv:2210.07959, arXiv:2210.01695, arXiv:2211.01650, arXiv:2212.00474, and arXiv:2305.0966

Asymptotic safety in three-dimensional SU(2) Group Field Theory: evidence in the local potential approximation

We study the functional renormalization group of a three-dimensional tensorial Group Field Theory (GFT) with gauge group SU(2). This model generates (generalized) lattice gauge theory amplitudes, and is known to be perturbatively renormalizable up to order 6 melonic interactions. We consider a series of truncations of the exact Wetterich--Morris equation, which retain increasingly many perturbatively irrelevant melonic interactions. This tensorial analogue of the ordinary local potential approximation allows to investigate the existence of non-perturbative fixed points of the renormalization group flow. Our main finding is a candidate ultraviolet fixed point, whose qualitative features are reproduced in all the truncations we have checked (with up to order 12 interactions). This may be taken as evidence for an ultraviolet completion of this GFT in the sense of asymptotic safety. Moreover, this fixed point has a single relevant direction, which suggests the presence of two distinct infrared phases. Our results generally support the existence of GFT phases of the condensate type, which have recently been conjectured and applied to quantum cosmology and black holes.Comment: 43 pages, many figures; v2: minor correction

RG Flows and Fixed Points of O(N)rO(N)^r Models

By means of $\epsilon$ and large $N$ expansions, we study generalizations of the $O(N)$ model where the fundamental fields are tensors of rank $r$ rather than vectors, and where the global symmetry (up to additional discrete symmetries and quotients) is $O(N)^r$, focusing on the cases $r\leq 5$. Owing to the distinct ways of performing index contractions, these theories contain multiple quartic operators, which mix under the RG flow. At all large $N$ fixed points, melonic operators are absent and the leading Feynman diagrams are bubble diagrams, so that all perturbative fixed points can be readily matched to full large $N$ solutions obtained from Hubbard-Stratonovich transformations. The family of fixed points we uncover extend to arbitrary higher values of $r$, and as their number grows superexponentially with $r$, these theories offer a vast generalization of the critical $O(N)$ model. We also study sextic $O(N)^r$ theories, whose large $N$ limits are obscured by the fact that the dominant Feynman diagrams are not restricted to melonic or bubble diagrams. For these theories the large $N$ dynamics differ qualitatively across different values of $r$, and we demonstrate that the RG flows possess a numerous and diverse set of perturbative fixed points beginning at rank four.Comment: 60 pages + appendices and reference

Ward identities and combinatorics of rainbow tensor models

We discuss the notion of renormalization group (RG) completion of non-Gaussian Lagrangians and its treatment within the framework of Bogoliubov-Zimmermann theory in application to the matrix and tensor models. With the example of the simplest non-trivial RGB tensor theory (Aristotelian rainbow), we introduce a few methods, which allow one to connect calculations in the tensor models to those in the matrix models. As a byproduct, we obtain some new factorization formulas and sum rules for the Gaussian correlators in the Hermitian and complex matrix theories, square and rectangular. These sum rules describe correlators as solutions to finite linear systems, which are much simpler than the bilinear Hirota equations and the infinite Virasoro recursion. Search for such relations can be a way to solving the tensor models, where an explicit integrability is still obscure.Comment: 48 page

Enriching Majorana Zero Modes

My various projects in graduate school have centered around a common theme: harnessing relatively well-understood phases of matter and combining them to create exotic physics. They also involve Majoranas, or more accurately, defects that bind Majorana zero modes and are the centerpiece for topological quantum computation. We exploit and enrich this Majorana zero mode by employing topological superconductors, time crystals, and quantum dots and combining them together. Our first project involved joining Majorana nanowires and quantum dots to simulate the SYK model, a zero-dimensional strongly interacting phase with connections to black holes and holography. We follow by explaining how to combine spontaneous symmetry-breaking with topological superconductivity to recover parafermion physics in one dimension. We explain an exact mapping that relates fermions to parafermions, illustrating a deep connection between different one-dimensional phases of matter. We finally show that enhancing the topological superconductor with a time crystal, a phase of matter that spontaneously breaks time-translation symmetry, creates an anomalous zero mode that displays 4Tperiodicity in the Floquet drive. By combining these different phases in judicious ways we achieve exotic physics unattainable by the constituent parts. Our work thus illustrates profitable directions for harnessing Majorana zero modes to study the physics of exotic matter.</p