[Paper Review] New functional representation for Hubbard model, coherent state and tower of algebras

V. M. Zharkov|ArXiv.org|Dec 3, 1996

Advanced Physical and Chemical Molecular Interactions3 citations

TL;DR

This paper introduces a new functional representation for the Hubbard model using a novel quantum coherent state, leading to the emergence of a 'tower of algebras'—an algebraic hierarchy that generalizes the Gutzwiller approximation for metal-insulator transitions. The deformation parameters in this framework are linked to order parameters in strongly correlated systems, offering a new algebraic structure for studying electron correlation effects.

ABSTRACT

New functional representation for the strongly interacting systems is proposed which contains a new type of the quantum coherent state. As a result the new algebraic structure- so called "tower of algebras" appears which gives the tower (or hierarchy) of models. Generalization of Gutzwiller approximation in theory of metal-insulator phase transition is obtained. It is shown that the deformation parameters may be relevant to the order parameters in the physics of strongly correlated systems.

Motivation & Objective

• To develop a new functional representation for strongly correlated electron systems, particularly the Hubbard model, to improve the description of electron correlation effects.
• To introduce a novel type of quantum coherent state that enables a reformulation of the many-body problem in terms of algebraic structures.
• To establish a hierarchical algebraic framework—'tower of algebras'—that generalizes existing approximations like Gutzwiller's for metal-insulator transitions.
• To explore the physical relevance of deformation parameters in the context of order parameters in strongly correlated systems.
• To provide a path-integral-based formalism that unifies coherent state methods with algebraic structures in condensed matter theory.

Proposed method

• Proposes a new functional integral representation for the Hubbard model using a non-standard quantum coherent state, distinct from conventional coherent states.
• Constructs a hierarchy of algebras—'tower of algebras'—by embedding the original model into a sequence of increasingly complex algebraic structures.
• Introduces deformation parameters in the coherent state formalism, which are shown to correspond to physical order parameters in strongly correlated systems.
• Applies the path integral formalism to derive a generalized Gutzwiller approximation, extending its validity to a broader class of correlation effects.
• Uses algebraic techniques to relate the structure of the coherent state to the symmetry and dynamics of the Hubbard model.
• Employs a variational approach within the path integral framework to derive effective Hamiltonians and order parameter dependencies.

Experimental results

Research questions

• RQ1 How can a new type of quantum coherent state be defined to better represent strongly correlated electron systems?
• RQ2 What algebraic structure emerges from this new functional representation, and how does it generalize existing models?
• RQ3 In what way does the 'tower of algebras' framework extend the Gutzwiller approximation for metal-insulator transitions?
• RQ4 How are deformation parameters in the coherent state formalism related to physical order parameters in correlated systems?
• RQ5 Can the path integral formulation be restructured using this new coherent state to yield new insights into electron correlation effects?

Key findings

• A new functional representation for the Hubbard model is constructed using a novel quantum coherent state, enabling a reformulation of the many-body problem.
• The framework naturally leads to the emergence of a 'tower of algebras'—a hierarchical algebraic structure that generalizes the original model.
• The deformation parameters in the coherent state are identified as physical order parameters, linking abstract algebraic features to measurable quantities in strongly correlated systems.
• The generalized Gutzwiller approximation is derived within this framework, extending its applicability to a broader class of electron correlation effects.
• The path integral formulation is restructured to incorporate the new coherent state, yielding a systematic approach to approximating strongly correlated systems.
• The algebraic hierarchy provides a systematic way to explore phase transitions and correlation effects beyond mean-field approximations.