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  1. We present microscopic, multiple Landau level, (frustration-free and positive semi-definite) parent Hamiltonians whose ground states, realizing different quantum Hall fluids, are parton-like and whose excitations display either Abelian or non-Abelian braiding statistics. We prove ground state energy monotonicity theorems for systems with different particle numbers in multiple Landau levels, demonstrate S-duality in the case of toroidal geometry, and establish complete sets of zero modes of special Hamiltonians stabilizing parton-like states, specifically at filling factor\nu=2/3ν=2/3. The emergent Entangled Pauli Principle (EPP), introduced in [Phys. Rev. B 98, 161118(R) (2018)] and which defines the “DNA” of the quantum Hall fluid, is behind the exact determination of the topological characteristics of the fluid, including charge and braiding statistics of excitations, and effective edge theory descriptions. When the closed-shell condition is satisfied, the densest (i.e., the highest density and lowest total angular momentum) zero-energy mode is a unique parton state. We conjecture that parton-like states generally span the subspace of many-body wave functions with the two-bodyMM-clustering property within any given number of Landau levels, that is, wave functions withMMth-order coincidence plane zeroes and both holomorphic and anti-holomorphic dependence on variables. General arguments are supplemented by rigorous considerations for theM=3M=3case of fermions in four Landau levels. For this case, we establish that the zero mode counting can be done by enumerating certain patterns consistent with an underlying EPP. We apply the coherent state approach of [Phys. Rev. X 1, 021015 (2011)] to show that the elementary (localized) bulk excitations are Fibonacci anyons. This demonstrates that the DNA associated with fractional quantum Hall states encodes all universal properties. Specifically, for parton-like states, we establish a link with tensor network structures of finite bond dimension that emerge via root level entanglement.

     
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  2. Abstract

    Advancing a microscopic framework that rigorously unveils the underlying topological hallmarks of fractional quantum Hall (FQH) fluids is a prerequisite for making progress in the classification of strongly-coupled topological matter. We present a second-quantization framework that reveals an exact fusion mechanism for particle fractionalization in FQH fluids, and uncovers the fundamental structure behind the condensation of non-local operators characterizing topological order in the lowest-Landau-level. We show the first exact analytic computation of the quasielectron Berry connections leading to its fractional charge and exchange statistics, and perform Monte Carlo simulations that numerically confirm the fusion mechanism for quasiparticles. We express the sequence of (bosonic and fermionic) Laughlin second-quantized states, highlighting the lack of local condensation, and present a rigorous constructive subspace bosonization dictionary for the bulk fluid. Finally, we establish universal long-distance behavior of edge excitations by formulating a conjecture based on the DNA, or root state, of the FQH fluid.

     
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  3. Sick individuals do not all respond to an infection in the same way. One individual may experience mild symptoms and recover easily, while another may suffer devastating illness or even death. A number of factors are often assumed to account for these differences, including the sex, age and genes of the individuals, and differences in the environments the individuals have been exposed to. However, random variations in how an individual’s immune system interacts with the infection could also play an important role in recovery. Duneau et al. have now studied how genetically identical fruit flies who were raised in the same environment respond to different bacterial infections. This enabled them to develop a mathematical model that describes how a bacterial infection develops in an individual. In an initial phase, the bacteria proliferate freely. If the immune defenses activate in time to control the infection, the number of bacteria in the fly decreases to a constant level and the infection enters a long-term, or chronic, phase. The sooner this happens the more likely it is that the fly will survive. If the immune control happens too late, the infection enters a terminal phase and the fly will die once the number of bacteria increases to a certain level. The model therefore reveals that the precise time at which the immune system takes control of the bacterial population – termed the “Time to Control” – determines the outcome of the infection. Duneau et al. confirmed this by injecting bacteria into identical flies. A small variation in the Time to Control was sometimes the difference between a fly living or dying. Understanding what controls this apparently random variation is key to understanding infection and potentially developing more efficient treatments for a wide range of diseases – not just those caused by bacteria, but also those caused by viruses and parasites, like HIV and malaria. 
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