Entanglement and symmetries in many-body quantum systems

As Schroedinger already recognised one century ago, entanglement is at the core of quantum mechanics. Nowadays it turns out to be the fundamental notion behind many quantum phenomena, from quantum algorithms to gravity, passing by critical phenomena and topological phases of matter, triggering unexpected connections between apparently far branches of physics. At the center of all these ideas, we find the (Rényi) entanglement entropies, which are powerful entanglement measures that provide fundamental insights into the investigated system or theory. This motivates the development of techniques for determining it, such as the replica trick: its implementation via the path-integral finds a wide place in this thesis. For example, we develop an efficient strategy to compute a generalised version of the Rényi entropies for all the eigenstates of a (1+1)-dimensional conformal field theory. This represents the starting point for a simulation scheme ideal to compute the entanglement in more generic (1+1)-dimensional quantum field theories (QFTs), e.g. after a quench in the sine-Gordon field theory. The study of entanglement also intertwines with another pillar of modern physics, i.e. symmetries and how their presence influences the properties of a system. Given the interest in this connection, this thesis addresses the question of how the entanglement splits into the different sectors of an internal symmetry. We approach the problem first in the QFT context, both for the free Dirac and complex scalar fields in two-dimensional spacetime, which have an abelian conserved charge, and systems having an internal Lie group symmetry to tackle the non-abelian case. Another typical framework in which we study the symmetry resolution of entanglement is lattice models, where different techniques can be exploited in order to derive exact results, ranging from the corner transfer matrix for gapped integrable systems to the connection between quadratic lattice Hamiltonians and their two-point correlation functions. The symmetry resolution also concerns other entanglement measures, namely, we analyse the behaviour of the operator entanglement, i.e. a key quantifier of the complexity of an operator, the symmetry-resolved mutual information, the effect of symmetries on entanglement negativity. The latter quantity is a genuine measure of quantum correlations in mixed states and a consistent part of the thesis is about this subject. For example, we study its time evolution after a quench and we provide an operatorial characterisation for entanglement in mixed states, which we dub negativity Hamiltonian

Complex Reaction Kinetics in Chemistry: A unified picture suggested by Mechanics in Physics

Complex biochemical pathways or regulatory enzyme kinetics can be reduced to chains of elementary reactions, which can be described in terms of chemical kinetics. This discipline provides a set of tools for quantifying and understanding the dialogue between reactants, whose framing into a solid and consistent mathematical description is of pivotal importance in the growing field of biotechnology. Among the elementary reactions so far extensively investigated, we recall the socalled Michaelis-Menten scheme and the Hill positive-cooperative kinetics, which apply to molecular binding and are characterized by the absence and the presence, respectively, of cooperative interactions between binding sites, giving rise to qualitative different phenomenologies. However, there is evidence of reactions displaying a more complex, and by far less understood, pattern: these follow the positive-cooperative scenario at small substrate concentration, yet negative-cooperative effects emerge and get stronger as the substrate concentration is increased. In this paper we analyze the structural analogy between the mathematical backbone of (classical) reaction kinetics in Chemistry and that of (classical) mechanics in Physics: techniques and results from the latter shall be used to infer properties on the former

Entanglement resolution of free Dirac fermions on a torus

Whenever a system possesses a conserved charge, the density matrix splits into eigenspaces associated to the each symmetry sector and we can access the entanglement entropy in a given subspace, known as symmetry resolved entanglement (SRE). Here, we first evaluate the SRE for massless Dirac fermions in a system at finite temperature and size, i.e. on a torus. Then we add a massive term to the Dirac action and we treat it as a perturbation of the massless theory. The charge-dependent entropies turn out to be equally distributed among all the symmetry sectors at leading order. However, we find subleading corrections which depend both on the mass and on the boundary conditions along the torus. We also study the resolution of the fermionic negativity in terms of the charge imbalance between two subsystems. We show that also for this quantity, the presence of the mass alters the equipartition among the different imbalance sectors at subleading order.Comment: 45 pages, 8 Figure

Symmetry-resolved entanglement entropy in Wess-Zumino-Witten models

We consider the problem of the decomposition of the R\'enyi entanglement entropies in theories with a non-abelian symmetry by doing a thorough analysis of Wess-Zumino-Witten (WZW) models. We first consider $SU(2)_k$ as a case study and then generalise to an arbitrary non-abelian Lie group. We find that at leading order in the subsystem size $L$ the entanglement is equally distributed among the different sectors labelled by the irreducible representation of the associated algebra. We also identify the leading term that breaks this equipartition: it does not depend on $L$ but only on the dimension of the representation. Moreover, a $\log\log L$ contribution to the R\'enyi entropies exhibits a universal form related to the underlying symmetry group of the model, i.e. the dimension of the Lie group.Comment: 31 pages, v2: minor change

Full counting statistics and symmetry resolved entanglement for free conformal theories with interface defects

We consider the ground state of two species of one-dimensional critical free theories coupled together via a conformal interface. They have an internal $U(1)$ global symmetry and we investigate the quantum fluctuations of the charge across the impurity, giving analytical predictions for the full counting statistics, the charged moments of the reduced density matrix and the symmetry resolved R\'enyi entropies. Our approach is based on the relation between the geometry with the defect and the homogeneous one, and it provides a way to characterise the spectral properties of the correlation functions restricted to one of the two species. Our analytical predictions are tested numerically, finding a perfect agreement

Entanglement asymmetry as a probe of symmetry breaking

Symmetry and symmetry breaking are two pillars of modern quantum physics. Still, quantifying how much a symmetry is broken is an issue that has received little attention. In extended quantum systems, this problem is intrinsically bound to the subsystem of interest. Hence, in this work, we borrow methods from the theory of entanglement in many-body quantum systems to introduce a subsystem measure of symmetry breaking that we dub entanglement asymmetry. As a prototypical illustration, we study the entanglement asymmetry in a quantum quench of a spin chain in which an initially broken global $U(1)$ symmetry is restored dynamically. We adapt the quasiparticle picture for entanglement evolution to the analytic determination of the entanglement asymmetry. We find, expectedly, that larger is the subsystem, slower is the restoration, but also the counterintuitive result that more the symmetry is initially broken, faster it is restored, a sort of quantum Mpemba effect, a phenomenon that we show to occur in a large variety of systems.Comment: 7 pages, 5 figures. Text reorganized, new results for interacting integrable and non-integrable spin chains added. Final version published in Nature Communication

Symmetry decomposition of negativity of massless free fermions

We consider the problem of symmetry decomposition of the entanglement negativity in free fermionic systems. Rather than performing the standard partial transpose, we use the partial time-reversal transformation which naturally encodes the fermionic statistics. The negativity admits a resolution in terms of the charge imbalance between the two subsystems. We introduce a normalised version of the imbalance resolved negativity which has the advantage to be an entanglement proxy for each symmetry sector, but may diverge in the limit of pure states for some sectors. Our main focus is then the resolution of the negativity for a free Dirac field at finite temperature and size. We consider both bipartite and tripartite geometries and exploit conformal field theory to derive universal results for the charge imbalance resolved negativity. To this end, we use a geometrical construction in terms of an Aharonov-Bohm-like flux inserted in the Riemann surface defining the entanglement. We interestingly find that the entanglement negativity is always equally distributed among the different imbalance sectors at leading order. Our analytical findings are tested against exact numerical calculations for free fermions on a lattice.Comment: 48 pages, 7 figure

More on symmetry resolved operator entanglement

The `operator entanglement' of a quantum operator $O$ is a useful indicator of its complexity, and, in one-dimension, of its approximability by matrix product operators. Here we focus on spin chains with a global $U(1)$ conservation law, and on operators $O$ with a well-defined $U(1)$ charge, for which it is possible to resolve the operator entanglement of $O$ according to the $U(1)$ symmetry. We employ the notion of symmetry resolved operator entanglement (SROE) introduced in [PRX Quantum 4, 010318 (2023)] and extend the results of the latter paper in several directions. Using a combination of conformal field theory and of exact analytical and numerical calculations in critical free fermionic chains, we study the SROE of the thermal density matrix $\rho_\beta = e^{- \beta H}$ and of charged local operators evolving in Heisenberg picture $O = e^{i t H} O e^{-i t H}$. Our main results are: i) the SROE of $\rho_\beta$ obeys the operator area law; ii) for free fermions, local operators in Heisenberg picture can have a SROE that grows logarithmically in time or saturates to a constant value; iii) there is equipartition of the entanglement among all the charge sectors except for a pair of fermionic creation and annihilation operators.Comment: 26 pages, 6 figure

Influencia de los abonos de liberación lenta y de sustratos con textiles industriales en el cultivo de Pelargonium zonale y Osteospermum ecklonis

[EN] This research evaluates the influence of textile waste from recycled tires as culture medium at different doses in order to reduce the costs. Additionally, it was studied the application of slow release fertilizer (0 and 2 g plant-1) in Pelargonium zonale and Osteospermum ecklonis. Five different substractes were used: Textile (TEX), TEX mixed with polyethylene aluminum (TEX + PE Al), TEX mixed with polyethylene copper (TEX + PE Cu), TEX mixed with PVC (TEX + PVC) using peat blonde and coconut fiber as a control (TU + CO). In a second experiment, the influence of textiles’ proportion in growing media based on peat and coconut fiber in Pelargonium zonale plants was studied. Analyses conducted at the end of the growing cycle, showed that plants grown with CO+TU had 100% of commercial plants, while those grown with textile reached 0%. Plants fertilized in a complementary manner with slow release fertilizer, showed greater vegetative development and commercial quality. Results from the second experiment showed that the lower proportion of textile and the higher proportion of fiber content in coconut based substrates peat, the greater development of the plant and the higher increase in commercial quality.[ES] La producción de planta ornamental es un importante sector que consume diversidad de sustratos, principalmente turbas rubia y negra, perlita, fibra de coco y poliestireno expandido, pero la preocupación por el medio ambiente lleva a la búsqueda de nuevos sustratos. En estos cultivos, prácticamente se ha abandonado el uso de abonos tradicionales, empleándose casi de forma exclusiva los fertilizantes de liberación lenta, la fertirrigación o la combinación de ambas técnicas. Con el fin de reducir los costes de cultivo, en este trabajo se evaluó la influencia de los residuos textiles procedentes del reciclado de neumáticos como medio de cultivo, y se estudió a su vez, la aplicación de abonos de liberación lenta a distintas dosis (0 y 2 g planta-1) en Pelargonium zonale y Osteospermum ecklonis. Se utilizaron 5 tipos de sustratos: Textil (TEX), TEX mezclado con polietileno de aluminio (TEX+PE Al), TEX mezclado con polietileno de cobre (TEX+PE Cu), TEX mezclado con PVC (TEX+PVC), utilizando turba rubia y fibra de coco como testigo (TU+CO). En un segundo experimento, se estudió la influencia de la proporción de textil en sustratos de cultivo a base de turba rubia y fibra de coco en plantas de Pelargonium zonale. Los análisis realizados al final del ciclo de cultivo, mostraron que, las plantas cultivadas con TU+CO presentaron el 100% de plantas comerciales, mientras que las cultivadas con textil alcanzaron un 0%. Las plantas fertilizadas, de manera complementaria, con abono de liberación lenta, presentaron un mayor desarrollo vegetativo y calidad comercial. Del segundo experimento, se obtuvo que, a menor proporción de textil y mayor contenido en fibra de coco en los sustratos a base de turba rubia, mayor fue el desarrollo de las planta y mayor calidad comercial.Murciano Silla, S. (2013). Influencia de los abonos de liberación lenta y de sustratos con textiles industriales en el cultivo de Pelargonium zonale y Osteospermum ecklonis. http://hdl.handle.net/10251/52527Archivo delegad

Lack of symmetry restoration after a quantum quench: an entanglement asymmetry study

We consider the quantum quench in the XX spin chain starting from a tilted N\'eel state which explicitly breaks the $U(1)$ symmetry of the post-quench Hamiltonian. Very surprisingly, the $U(1)$ symmetry is not restored at large time because of the activation of a non-abelian set of charges which all break it. The breaking of the symmetry can be effectively and quantitatively characterised by the recently introduced entanglement asymmetry. By a combination of exact calculations and quasi-particle picture arguments, we are able to exactly describe the behaviour of the asymmetry at any time after the quench. Furthermore we show that the stationary behaviour is completely captured by a non-abelian generalised Gibbs ensemble. While our computations have been performed for a non-interacting spin chain, we expect similar results to hold for the integrable interacting case as well because of the presence of non-abelian charges also in that case.Comment: 25 pages, 5 figures. Typos corrected, references adde