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Related Concept Videos

Probability Distributions01:32

Probability Distributions

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 The probability of a random variable x  is the likelihood of its occurrence. A probability distribution represents the probabilities of a random variable using a formula, graph, or table. There are two types of probability distribution– discrete probability distribution and continuous probability distribution.
A discrete probability distribution is a probability distribution of discrete random variables. It can be categorized into binomial probability distribution and Poisson...
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A Poisson probability distribution is a discrete probability distribution. It gives the probability of a number of events occurring in a fixed interval of time or space if these events happen at a known average rate and independently of the time since the last event. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on average, there are five words spelled incorrectly in 100 pages. The interval is 100 pages.
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A binomial distribution is a probability distribution for a procedure with a fixed number of trials, where each trial can have only two outcomes.
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Lattice Centering and Coordination Number02:33

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The structure of a crystalline solid, whether a metal or not, is best described by considering its simplest repeating unit, which is referred to as its unit cell. The unit cell consists of lattice points that represent the locations of atoms or ions. The entire structure then consists of this unit cell repeating in three dimensions. The three different types of unit cells present in the cubic lattice are illustrated in Figure 1.
Types of Unit Cells
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Probability Laws01:49

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Trends in Lattice Energy: Ion Size and Charge02:54

Trends in Lattice Energy: Ion Size and Charge

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An ionic compound is stable because of the electrostatic attraction between its positive and negative ions. The lattice energy of a compound is a measure of the strength of this attraction. The lattice energy (ΔHlattice) of an ionic compound is defined as the energy required to separate one mole of the solid into its component gaseous ions. For the ionic solid sodium chloride, the lattice energy is the enthalpy change of the process:
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Simple probability distributions on a Fock-space lattice.

Staszek Welsh1, David E Logan

  • 1Department of Chemistry, Physical and Theoretical Chemistry, Oxford University, South Parks Road, Oxford, OX1 3QZ, United Kingdom.

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|August 29, 2018
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Summary

This study analyzes interacting spinless fermions with disorder, mapping the many-body localization model to a Fock-space (FS) lattice. Exact results for FS distributions are obtained, aiding in identifying mobility edges.

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Area of Science:

  • Condensed Matter Physics
  • Quantum Mechanics
  • Statistical Mechanics

Background:

  • Many-body localization (MBL) is crucial for understanding quantum systems' thermalization properties.
  • Disordered interacting fermion systems are standard models for MBL studies.
  • Understanding the transition from localized to delocalized states is key.

Purpose of the Study:

  • To analyze a standard model for many-body localization: interacting spinless fermions with quenched disorder.
  • To investigate the model's behavior on d-dimensional hypercubic lattices at non-zero filling fractions.
  • To explore the mapping to a Fock-space (FS) lattice and its implications.

Main Methods:

  • Recasting the fermion model into an equivalent tight-binding model on a Fock-space (FS) lattice.
  • Obtaining exact results in the thermodynamic limit for distributions of local FS coordination numbers, FS site-energies, and density of many-body states.
  • Utilizing exact diagonalization for numerical analysis on modest system sizes.

Main Results:

  • Distributions of local FS coordination numbers, FS site-energies, and density of many-body states were accurately determined.
  • These distributions are well-captured by exact diagonalization on numerically tractable system sizes.
  • The importance of choosing the correct variance for eigenvalue distributions for mobility edge identification was highlighted.

Conclusions:

  • The Fock-space lattice approach provides an effective framework for studying disordered interacting fermion systems.
  • Exact results and numerical methods confirm the model's behavior and distributions.
  • Careful analysis of eigenvalue distributions is essential for robust identification of mobility edges in MBL studies.