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Related Concept Videos

Cluster Sampling Method01:20

Cluster Sampling Method

Appropriate sampling methods ensure that samples are drawn without bias and accurately represent the population. Because measuring the entire population in a study is not practical, researchers use samples to represent the population of interest.
To choose a cluster sample, divide the population into clusters (groups) and then randomly select some of the clusters. All the members from these clusters are in the cluster sample. For example, if you randomly sample four departments from your...
Radius of Gyration of an Area01:12

Radius of Gyration of an Area

The second moment of area, also known as the moment of inertia of area, is a crucial factor in understanding an object's resistance against bending deformation, or stiffness. To accurately estimate the second moment of area along any axis, one needs to concentrate all areas associated with that object into a thin strip, which should be placed parallel to that particular axis.
Modified Boxplots00:57

Modified Boxplots

A standard box and whisker plot informs us about the spread of the data in a given sample. One can identify the minimum value, maximum value, first quartile value, second quartile or median value, and third quartile.
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Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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Linear Approximations01:23

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...
Extraction: Partition and Distribution Coefficients01:14

Extraction: Partition and Distribution Coefficients

The distribution law or Nernst's distribution law is the law that governs the distribution of a solute between two immiscible solvents. This law, also known as the partition law, states that if a solute is added to the mixture of two immiscible solvents at a constant temperature, the solute is distributed between the two solvents in such a way that the ratio of solute concentrations in the solvents remains constant at equilibrium.
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Related Experiment Video

Updated: Jun 18, 2026

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry
12:11

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Published on: April 8, 2020

Parquet approximation for the 4x4 Hubbard cluster.

S X Yang1, H Fotso, J Liu

  • 1Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 13, 2009
PubMed
Summary

We present a numerical solution for the parquet approximation, a conserving diagrammatic method. This approach shows improved accuracy for the Hubbard model compared to other approximations.

Related Experiment Videos

Last Updated: Jun 18, 2026

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry
12:11

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Published on: April 8, 2020

Area of Science:

  • Condensed matter physics
  • Quantum many-body theory
  • Computational physics

Background:

  • Diagrammatic techniques are crucial for solving quantum many-body problems.
  • The parquet approximation offers a conserving and self-consistent approach.
  • Accurate solutions for models like the Hubbard model are essential.

Purpose of the Study:

  • To numerically solve the parquet approximation for the Hubbard model.
  • To assess the accuracy and self-consistency of this approximation.
  • To compare its performance against established methods.

Main Methods:

  • Numerical solution of the parquet approximation.
  • Approximating the fully irreducible vertex with the bare interaction.
  • Application to a half-filled 4x4 Hubbard model cluster.

Main Results:

  • The parquet approximation provides a self-consistent solution at single-particle and two-particle levels.
  • Satisfactory agreement was found when comparing results with determinant quantum Monte Carlo (DQMC).
  • Significant improvements were observed over FLuctuation EXchange (FLEX) and self-consistent second-order approximation methods.

Conclusions:

  • The simplified parquet approximation is a viable and accurate method for the Hubbard model.
  • This approach offers a balance between complexity and predictive power.
  • It represents a valuable tool for studying strongly correlated electron systems.