Multifractality meets entanglement: relation for non-ergodic extended states

In this work we establish a relation between entanglement entropy and fractal dimension $D$ of generic many-body wave functions, by generalizing the result of Don N. Page [Phys. Rev. Lett. 71, 1291] to the case of {\it sparse} random pure states (S-RPS). These S-RPS living in a Hilbert space of size $N$ are defined as normalized vectors with only $N^D$ ($0 \le D \le 1$) random non-zero elements. For $D=1$ these states used by Page represent ergodic states at infinite temperature. However, for $0<D<1$ the S-RPS are non-ergodic and fractal as they are confined in a vanishing ratio $N^D/N$ of the full Hilbert space. Both analytically and numerically, we show that the mean entanglement entropy ${\mathcal{S}_1}(A)$ of a subsystem $A$, with Hilbert space dimension $N_A$, scales as $\overline{\mathcal{S}_1}(A)\sim D\ln N$ for small fractal dimensions $D$, $N^D< N_A$. Remarkably, $\overline{\mathcal{S}_1}(A)$ saturates at its thermal (Page) value at infinite temperature, $\overline{\mathcal{S}_1}(A)\sim \ln N_A$ at larger $D$. Consequently, we provide an example when the entanglement entropy takes an ergodic value even though the wave function is highly non-ergodic. Finally, we generalize our results to Renyi entropies $\mathcal{S}_q(A)$ with $q>1$ and to genuine multifractal states and also show that their fluctuations have ergodic behavior in narrower vicinity of the ergodic state, $D=1$.Comment: 7 pages, 4 figures, 92 references + 9 pages, 9 figures in appendice

Non-Hermitian Rosenzweig-Porter random-matrix ensemble: Obstruction to the fractal phase

We study the stability of non-ergodic but extended (NEE) phases in non-Hermitian systems. For this purpose, we generalize a so-called Rosenzweig-Porter random-matrix ensemble (RP), known to carry a NEE phase along with the Anderson localized and ergodic ones, to the non-Hermitian case. We analyze, both analytically and numerically, the spectral and multifractal properties of the non-Hermitian case. We show that the ergodic and the localized phases are stable against the non-Hermitian nature of matrix entries. However, the stability of the fractal phase depends on the choice of the diagonal elements. For purely real or imaginary diagonal potential the fractal phases is intact, while for a generic complex diagonal potential the fractal phase disappears, giving the way to a localized one.Comment: 10 pages, 6 figures, 66 reference

Non-Hermiticity induces localization: good and bad resonances in power-law random banded matrices

The power-law random banded matrix (PLRBM) is a paradigmatic ensemble to study the Anderson localization transition (AT). In $d$-dimension the PLRBM are random matrices with algebraic decaying off-diagonal elements $H_{\vec{n}\vec{m}}\sim 1/|\vec{n}-\vec{m}|^\alpha$, having AT at $\alpha=d$. In this work, we investigate the fate of the PLRBM to non-Hermiticity. We consider the case where the random on-site diagonal potential takes complex values, mimicking an open system, subject to random gain-loss terms. We provide an analytical understanding of the model by generalizing the Anderson-Levitov resonance counting technique to the non-Hermitian case. This generalization identifies two competing mechanisms due to non-Hermiticity: one favoring localization and the other delocalization. The competition between the two gives rise to AT at $d/2\le \alpha\le d$. The value of the critical $\alpha$ depends on the strength of the on-site potential, reminiscent of Hermitian disordered short-range models in $d>2$. Within the localized phase, the wave functions are algebraically localized with an exponent $\alpha$ even for $\alpha<d$. This result provides an example of non-Hermiticity-induced localization.Comment: 4.5 pages, 4 figures, 57 references + 5 pages, 4 figures in Appendice

Modificações na estrutura da proteção no direito do trabalho

Orientadora: Professora Dr.ª Aldacy Rachid CoutinhoInclui referências: p. 152-155Resumo: Esta dissertação tem o objetivo de demonstrar que a proteção, alicerçada no princípio da proteção do Direito do Trabalho, está sendo gradativamente reduzida, devido às alterações que estão ocorrendo na atualidade, e que sua defesa é uma necessidade. A demonstração da necessidade de se manter a proteção tem fundamento histórico e correlacionado aos momentos economicos. A principal razão da modificação da proteção, ou mesmo sua exclusão vêm do fato de que as mudanças na economia, com a internacionalização dos mercados, forçam a implementação da fragmentação e da flexibilização dos contratos trabalhistas, deixando trabalhadores à mercê dos interesses do capital. O próprio Direito do Trabalho está em risco, pois sua existência se fundamenta no diferencial entre os agentes de seu interesse, ou seja, a parte do pressuposto que haja diferenças entre empregados e empregadores no momento do contrato. A intervenção estatal é seu instrumento de realização dessas normas. A retirada do estado e a defesa a livre negociação causam e destruição do seu objetivo, qual seja, a proteção do trabalhador.Abstract: This dissertation aims at demonstrating that protection based on the principle of Right to Work protection has been reduced gradually due to changes that have been on the rise lately, and its defense is a necessity. The demonstration of the necessity of keeping such protection has a historic support, and it is correlated to economic moments. The main reason of protection change. Or even its exclusion arises from the fact that changes in economy together with the intemationalization of markets force the implementation of both fragmentation and flexibleness of employment contracts, which leave workers at the mercy of the interests of capital. Right to Work itself is at risk once its existence is based on the differential between agents of its own interest, that is; it comes from the presupposition that there are differences between workers and employers at the moment of hiring. The State intervention is an instrument by which such norms are performed. The withdrawal of the State and the defense for free negotiation cause the objective to be unstructured, whatever it may be, the workers" protection

Rare thermal bubbles at the many-body localization transition from the Fock space point of view

In this work we study the many-body localization (MBL) transition and relate it to the eigenstate structure in the Fock space. Besides the standard entanglement and multifractal probes, we introduce the radial probability distribution of eigenstate coefficients with respect to the Hamming distance in the Fock space from the wave function maximum and relate the cumulants of this distribution to the properties of the quasi-local integrals of motion in the MBL phase. We demonstrate non-self-averaging property of the many-body fractal dimension $D_q$ and directly relate it to the jump of $D_q$ as well as of the localization length of the integrals of motion at the MBL transition. We provide an example of the continuous many-body transition confirming the above relation via the self-averaging of $D_q$ in the whole range of parameters. Introducing a simple toy-model, which hosts ergodic thermal bubbles, we give analytical evidences both in standard probes and in terms of newly introduced radial probability distribution that the MBL transition in the Fock space is consistent with the avalanche mechanism for delocalization, i.e., the Kosterlitz-Thouless scenario. Thus, we show that the MBL transition can been seen as a transition between ergodic states to non-ergodic extended states and put the upper bound for the disorder scaling for the genuine Anderson localization transition with respect to the non-interacting case.Comment: 13 pages, 5 figures, 94 reference

Return probability for the Anderson model on the random regular graph

We study the return probability for the Anderson model on the random regular graph and give evidence of the existence of two distinct phases: a fully ergodic and nonergodic one. In the ergodic phase, the return probability decays polynomially with time with oscillations, being the attribute of the Wigner-Dyson-like behavior, while in the nonergodic phase the decay follows a stretched exponential decay.We give a phenomenological interpretation of the stretched exponential decay in terms of a classical random walker. Furthermore, comparing typical and mean values of the return probability, we show how to differentiate an ergodic phase from a nonergodic one. We benchmark this method first in two random matrix models, the power-law random banded matrices, and the Rosenzweig-Porter matrices, which host both phases. Second, we apply this method to the Anderson model on the random regular graph to give further evidence of the existence of the two phases.Comment: 10 pages, 4 figures, 88 references (main text) + 5 pages, 9 figures (appendices

Light-harvesting efficiency cannot depend on optical coherence in the absence of orientational order

The coherence of light has been proposed as a quantum-mechanical control to enhance light-harvesting efficiency. In particular, optical coherence can be manipulated by changing either the polarization state or spectral phase of the illuminating light. Here, we show that, in weak light, controlling the light-harvesting efficiency using any form of optical coherence is impossible in all molecular light-harvesting systems and, more broadly, those composed of weakly interacting sub-units which lack fixed orientational order and operate on longer-than-ultrafast timescales. Under those conditions, optical coherence does not affect light-harvesting efficiency, meaning that it cannot be used as a form of control. In particular, control through the polarization state is lost in disordered samples or when the molecules reorient on the timescales of the light-harvesting, and control through the spectral phase is lost when the efficiency is time-averaged for longer than the coherence time of the light. In practice, efficiency is always averaged over long times, meaning that coherent optical control is only possible through polarisation in systems with orientational order.Comment: 8 + 5 pages, 5 + 1 figure

The Isotropic Fractionator as a Tool for Quantitative Analysis in Central Nervous System Diseases

One major aim in quantitative and translational neuroscience is to achieve a precise and fast neuronal counting method to work on high throughput scale to obtain reliable results.Here we tested the Isotropic Fractionator (IF) method for evaluating neuronal and non-neuronal cell loss in different models of central nervous system (CNS) pathologies.Sprague-Dawley rats underwent: (i) ischemic brain damage; (ii) intraperitoneal injection with kainic acid (KA) to induce epileptic seizures; and (iii) monolateral striatal injection with quinolinic acid (QA) mimicking human Hungtington’s disease.All specimens were processed for IF method and cell loss assessed.Hippocampus from KA-treated rats and striatum from QA-treated rats were carefully dissected using a dissection microscope and a rat brain matrix. Ischemic rat brains slices were first processed for TTC staining and then for IF.In the ischemic group the cell loss corresponded to the neuronal loss suggesting that hypoxia primarily affects neurons. Combining IF with TTC staining we could correlate the volume of lesion to the neuronal loss; by IF, we could assess that neuronal loss also occurs contralaterally to the ischemic side.In the epileptic group we observed a reduction of neuronal cells in treated rats, but also evaluated the changes in the number of non-neuronal cells in response to the hippocampal damage