Related Experiment Video
Updated: Apr 29, 2026
06:42
Generation and Coherent Control of Pulsed Quantum Frequency Combs
Published on: June 8, 2018
9.0K
Complete spectrum of the infinite-U Hubbard ring using group theory
Alessandro Soncini1, Willem Van den Heuvel1
1School of Chemistry, The University of Melbourne, VIC 3010, Australia.
The Journal of Chemical Physics
|May 17, 2014
Summary
This study provides the first analytical solution for the degeneracy of the N-Hubbard ring model with N-1 electrons. It uses combinatorial and group theory methods to solve strongly correlated mixed-valence problems.
Area of Science:
- Quantum Chemistry
- Condensed Matter Physics
- Mathematical Physics
Background:
- The N-Hubbard model describes strongly correlated electrons in materials.
- Mixed-valence systems exhibit complex electronic behaviors.
- Analytical solutions for degeneracy in such systems are challenging.
Purpose of the Study:
- To provide a full analytical solution for the degeneracy of the N-Hubbard ring model with N-1 electrons and infinite on-site repulsion.
- To present the analytical determination of degeneracy for the first time.
- To develop a general strategy applicable to arbitrary electron counts.
Main Methods:
- Mapping the Hubbard model to Hückel-annulene problems.
- Solving combinatorial enumeration problems (necklace problem) for N-1 beads and two colors.
- Applying the subduction of coset representation technique from group theory.
Main Results:
- A complete analytical solution for the Hubbard model, including degeneracy count, is achieved.
- The number and size of effective Hückel annulenes are determined.
- A general group theoretical strategy is established for solving the one-hole infinite-U Hubbard problem.
Conclusions:
- The study offers a novel and elegant group theoretical strategy for solving the infinite-U Hubbard model.
- The method provides a general approach for analyzing systems with arbitrary electron counts.
- This work advances the understanding of strongly correlated electron systems and their degeneracies.
Related Concept Videos
01:30Fundamental Theorem of Algebra
501
The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as: with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the...
501
01:29Indeterminate Products
139
Indeterminate forms also arise in the evaluation of limits involving products, particularly when one factor approaches zero while the other tends to positive or negative infinity. This situation, commonly described as a zero-times-infinity form, does not have an immediately interpretable outcome. Depending on how the factors behave relative to one another, the limit of such a product may be zero, infinite, or a finite nonzero value.Product Limits and Algebraic RewritingTo analyze limits of this...
139
01:08Hückel's Rule Diagram of π MOs: Frost Circle
4.2K
The Frost circle or the inscribed polygon method is a graphical method for determining the relative energies of π molecular orbitals (MOs) for planar, fully conjugated, and monocyclic compounds. This method was first described by A. A. Frost and Boris Musulin in 1953.
A Frost circle is constructed by drawing a polygon whose number of edges is equal to the number of carbons of the given cyclic system, with one of the vertices pointing down. Then, a circle is drawn enclosing the polygon so...
A Frost circle is constructed by drawing a polygon whose number of edges is equal to the number of carbons of the given cyclic system, with one of the vertices pointing down. Then, a circle is drawn enclosing the polygon so...
4.2K
01:09Rolle’s Theorem
222
Rolle’s Theorem states that if a real-valued function is continuous on a closed interval, differentiable on the open interval, and takes equal values at both endpoints, then there is at least one point within the open interval where the derivative of the function is zero.Rolle’s Theorem describes an important property of differentiable functions, this theorem applies to a real-valued function defined on a closed interval, provided three specific conditions are met. First, the...
222
01:16Second Uniqueness Theorem
1.4K
Consider a region consisting of several individual conductors with a definite charge density in the region between these conductors. The second uniqueness theorem states that if the total charge on each conductor and the charge density in the in-between region are known, then the electric field can be uniquely determined.
In contrast, consider that the electric field is non-unique and apply Gauss's law in divergence form in the region between the conductors and the integral form to the surface...
In contrast, consider that the electric field is non-unique and apply Gauss's law in divergence form in the region between the conductors and the integral form to the surface...
1.4K
02:45The Quantum-Mechanical Model of an Atom
47.1K
Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing...
47.1K