Jean-Marie Stéphan

On a compris ces dernières années que certaines mesures d'intrications sont un outil efficace pour la compréhension et la caractérisation de phases nouvelles et exotiques de la matière, en particulier lorsque les méthodes traditionnelles basées sur l'identification d'un paramètre d'ordre sont insuffisantes. Cette thèse porte sur l'étude de quelques systèmes quantiques à basse dimension où un telle approche s'avère fructueuse. Parmi ces mesures, l'entropie d'intrication, définie via une bipartition du système quantique, est probablement la plus populaire, surtout à une dimension. Celle-ci est habituellement très difficile à calculer en dimension supérieure, mais nous montrons ici que le calcul se simplifie drastiquement pour une classe particulière de fonctions d'ondes, nommées d'après Rokhsar et Kivelson. L'entropie d'intrication peut en effet s'exprimer comme une entropie de Shannon relative à la distribution de probabilité générée par les composantes de la fonction d'onde du fondamental d'un autre système quantique, cette fois-ci unidimensionnel. Cette réduction dimensionnelle nous permet d'étudier l'entropie aussi bien par des méthodes numériques (fermions libres, diagonalisations exactes, ...) qu'analytiques (théories conformes). Nous argumentons aussi que cette approche permet d'accéder facilement à certaines caractéristiques subtiles et universelles d'une fonction d'onde donnée en général.Une autre partie de cette thèse est consacrée aux trempes quantiques locales dans des systèmes critiques unidimensionnels. Nous insisterons particulièrement sur une quantité appelée écho de Loschmidt, qui est le recouvrement entre la fonction d'onde avant la trempe et la fonction d'onde à temps t après la trempe. En exploitant la commensurabilité du spectre de la théorie conforme, nous montrons que l'évolution temporelle doit être périodique, et peut même être souvent obtenue analytiquement. Inspiré par ces résultats, nous étudions aussi la contribution de fréquence nulle à l'écho de Loschmidt après la trempe. Celle-ci s'exprime comme un simple produit scalaire -- que nous nommons fidélité bipartie -- et est une quantité intéressante en elle-même. Malgré sa simplicité, son comportement se trouve être très similaire à celui de l'entropie d'intrication. Pour un système critique unidimensionnel en particulier, notre fidélité décroît algébriquement avec la taille du système, un comportement rappelant la célèbre catastrophe d'Anderson. L'exposant est universel et relié à la charge centrale de la théorie conforme sous-jacente.

Fluctuations of observables provide unique insights into the nature of physical systems, and their study stands as a cornerstone of both theoretical and experimental science. Generalized fluctuations, or cumulants, provide information beyond the mean and variance of an observable. In this paper, we develop a systematic method to determine the asymptotic behavior of cumulants of local observables as the region becomes large. Our analysis reveals that the expansion is closely tied to the geometric characteristics of the region and its boundary, with coefficients given by convex moments of the connected correlation function: the latter is integrated against intrinsic volumes of convex polytopes built from the coordinates, which can be interpreted as average shadows. A particular application of our method shows that, in two dimensions, the leading behavior of odd cumulants of conserved quantities is topological, specifically depending on the Euler characteristic of the region. We illustrate these results with the paradigmatic strongly-interacting system of two-dimensional quantum Hall state at filling fraction 1/2 1 / 2 , by performing Monte-Carlo calculations of the skewness (third cumulant) of particle number in the Laughlin state.

We investigate a one-dimensional free fermion model with nearest and next-nearest neighbor hopping, evolving in imaginary time from a product state with N consecutive fermions, and conditioned to go back to the same state after a given time. Such types of models are quantum reformulations of well-studied two-dimensional classical lattice models, which are known to give rise to limit shapes where expectation values of simple local observables, such as density, depend on position in an appropriate scaling limit. In the case of only nearest neighbor hopping, this model is known to have two fluctuating regions which can be tuned to merge depending on ratio between time and N . Correlations near the merger are governed by a so-called tacnode kernel. Here we show that another universal higher-order tacnode process can appear upon including the next-nearest neighbor term. We also discuss the limit shapes, and compute analytically the corresponding density profile.

We present Monte Carlo simulations of the two-dimensional one-component plasma (2D OCP) confined to a cylindrical geometry, focusing on density profiles, fluctuations, and their connection to bulk correlation functions. The cylindrical geometry eliminates geometric frustration, allowing for a precise study of boundary density oscillations, the dependence on boundary conditions, and their relationship to the melting transition and triangular lattice structure. By triangulating particle configurations, we quantify the exponential suppression of topological defects in the crystalline phase. Furthermore, we propose an oriented correlation function that better links boundary density profiles with bulk correlation functions, motivating anisotropic generalizations of the phase-field crystal model. These results provide new insights into the interplay between boundary effects, bulk correlations, and phase transitions in the 2D OCP.

We demonstrate that the widely used plasma analogy is unreliable at predicting edge properties of quantum Hall states. This discrepancy arises from a fundamental difference between quantum Hall droplets and plasmas (Coulomb gases): The former are incompressible liquids subject to area-preserving deformations, while the latter are governed by electrostatics and thus involve conformal transformations. Consequently, the plasma analogy fails at the edge, except in fine-tuned geometries, as it does not account for the emergent local edge velocity. We quantitatively show how the analogy's failure affects physical quantities, such as fluctuations of local observables and absorption rates in microwave spectroscopy, measurable in both solid-state experiments and quantum simulators. Published by the American Physical Society 2025

We study two-dimensional (2D) droplets of noninteracting electrons in a strong magnetic field, placed in a confining potential with arbitrary shape. Using semiclassical methods adapted to the lowest Landau level, we obtain near-Gaussian energy eigenstates that are localized on level curves of the potential and have a position-dependent height. This one-particle insight allows us to deduce explicit formulas for expectation values of local many-body observables, such as density and current, in the thermodynamic limit. In particular, correlations along the edge are long-ranged and inhomogeneous. As we show, this is consistent with the system’s universal low-energy description as a free 1D chiral conformal field theory of edge modes, known from earlier works in simple geometries. A delicate interplay between radial and angular dependencies of eigenfunctions ultimately ensures that the theory is homogeneous in terms of the canonical angle variable of the potential, despite its apparent inhomogeneity in terms of more naïve angular coordinates. Finally, we propose a scheme to measure the anisotropy by subjecting the droplet to microwave radiation; we compute the corresponding absorption rate and show that it depends on the droplet’s shape and the waves’ polarization. These results, both local and global, are likely to be observable in solid-state systems or quantum simulators of 2D electron gases with a high degree of control on the confining potential. Published by the American Physical Society 2024

The outcomes of measurements are characterized by an infinite family of generalized uncertainties, or cumulants, which provide information beyond the mean and variance of the observable. Here, we investigate the cumulants of a conserved charge in a subregion with corners. We derive nonperturbative relations for the area law, and more interestingly, the angle dependence, showing how it is determined by geometric moments of the correlation function. These hold for translation invariant systems under great generality, including strongly interacting ones. We test our findings by using two-dimensional topological quantum Hall states of bosons and fermions at both integer and fractional fillings. We find that the odd cumulants' shape dependence differs from the even ones. For instance, the third cumulant shows nearly universal behavior for integer and fractional Laughlin Hall states in the lowest Landau level. Furthermore, we examine the relation between even cumulants and the Rényi entanglement entropy, where we use our results for the fractional state at filling 1/3 to compare these quantities in the strongly interacting regime. We discuss the implications of these findings for other systems, including gapless Dirac fermions, and more general conformal field theories.

We study the time evolution of the spin-1/2 XXZ chain initialized in a domain wall state, where all spins to the left of the origin are up, all spins to its right are down. The focus is on exact formulae, which hold for arbitrary finite (real or imaginary) time. In particular, we compute the amplitudes corresponding to the process where all but k spins come back to their initial orientation, as a k -fold contour integral. These results are obtained using a correspondence with the six vertex model, and taking a somewhat complicated Hamiltonian/Trotter-type limit. Several simple applications are studied and also discussed in a broader context.

Understanding the fluctuations of observables is one of the main goals in science, be it theoretical or experimental, quantum or classical. We investigate such fluctuations in a subregion of the full system, focusing on geometries with sharp corners. We report that the angle dependence is super-universal: up to a numerical prefactor, this function does not depend on anything, provided the system under study is uniform, isotropic, and correlations do not decay too slowly. The prefactor contains important physical information: we show in particular that it gives access to the long-wavelength limit of the structure factor. We exemplify our findings with fractional quantum Hall states, topological insulators, scale invariant quantum critical theories, and metals. We suggest experimental tests, and anticipate that our findings can be generalized to other spatial dimensions or geometries. In addition, we highlight the similarities of the fluctuation shape dependence with findings relating to quantum entanglement measures.

The gapless modes on the edge of four-dimensional (4D) quantum Hall droplets are known to be anisotropic: they only propagate in one direction, foliating the 3D boundary into independent 1D conduction channels. This foliation is extremely sensitive to the confining potential and generically yields chaotic flows. Here we study the quantum correlations and entanglement of such edge modes in 4D droplets confined by harmonic traps, whose boundary is a squashed three-sphere. Commensurable trapping frequencies lead to periodic trajectories of electronic guiding centers; the corresponding edge modes propagate independently along S^1 S 1 fibers, forming a bundle of 1D conformal field theories over a 2D base space. By contrast, incommensurable frequencies produce quasi-periodic, ergodic trajectories, each of which covers its invariant torus densely; the corresponding correlation function of edge modes has fractal features. This wealth of behaviors highlights the sharp differences between 4D Hall droplets and their 2D peers; it also exhibits the dependence of 4D edge modes on the choice of trap, suggesting the existence of observable bifurcations due to droplet deformations.