[Paper Review] New path integral representation for Hubbard model: II. Spinless case

V. M. Zharkov, Вячеслав Сергеевич Кирчанов|arXiv (Cornell University)|Jun 8, 2010

Algebraic structures and combinatorial models3 references3 citations

TL;DR

This paper introduces a novel path integral formulation for the spinless Hubbard model using a supercoherent state derived from the atomic limit of strongly correlated electrons. By constructing an exact nonlinear representation of a dynamic supergroup acting on a superspinor bundle, the authors derive an effective functional framework rooted in geometric quantization and nonlinear realizations of superconformal symmetry, paving the way for a field-theoretic description of strong electron correlations.

ABSTRACT

The Hubbard model is used to study an electronic system. In this paper we present the new path integral representation for Hubbard model. We have constructed the new supercoherent state for spinless electrons which appears from a set of eigenfunctions of atomic limit of strongly correlated systems. Exact calculation of nonlinear representation of a supergroup has been carried. This group defines the transformation of atomic base. The general formalism we elaborate for Hubbard model is the one widely used in the gauge field theory of the nonlinear representation of a superconformal group.

Motivation & Objective

• To develop a new path integral approach for the strongly correlated Hubbard model, focusing on the spinless case.
• To formulate a geometric quantization framework for strongly correlated electron systems using supercoherent states.
• To identify the dynamical supergroup underlying the atomic limit of the Hubbard model and its nonlinear representation.
• To establish a connection between the Hubbard operators and the nonlinear realization of a superconformal group in a superspinor bundle.
• To lay the foundation for deriving an effective action through exact computation of supergroup generators in the atomic limit.

Proposed method

• Construct a supercoherent state based on the atomic eigenstates of the Hubbard model in the large-U limit, using both bosonic and fermionic Grassmann variables.
• Define a local superbundle over a 4D spacetime manifold, with the supercoherent state parametrizing the local supercoordinates via dynamical fields.
• Implement a nonlinear representation of a superconformal group, where the even (bosonic) part corresponds to Lorentz or SO(4) generators, and the odd (fermionic) part extends it to a supergroup.
• Use Cartan's differential one-forms to derive the Lagrangian of the system from the supergroup structure.
• Perform exact calculations of the nonlinear representation of the supergroup generators, particularly for the spinless case.
• Derive closed-form analytical expressions for matrix elements of the supergroup generators via series expansions and functional differentiation, leading to exact functions in terms of hyperbolic and trigonometric functions.

Experimental results

Research questions

• RQ1 How can the Hubbard model in the atomic limit be reformulated using a supercoherent state that unifies bosonic and fermionic degrees of freedom?
• RQ2 What is the structure of the dynamical supergroup that emerges from the Hubbard operators in the strongly correlated regime?
• RQ3 How can the nonlinear representation of this supergroup be exactly computed for the spinless case?
• RQ4 What role does geometric quantization play in constructing the effective field theory of strongly correlated electrons?
• RQ5 How do the derived matrix elements of the supergroup generators relate to the physical observables of the Hubbard model?

Key findings

• The authors derive exact analytical expressions for the matrix elements of the supergroup generators, such as $ f_{1}EE = \frac{-2b\cosh(b) + 2b\cosh(h) + (b^{2}-h^{2})\sinh(b)}{2b(b^{2}-h^{2})^{2}} $, which describe the nonlinear representation of the supergroup.
• The function $ fEE = \frac{2\sinh(h)b^{3} + h(b^{2}-h^{2})\cosh(b)b + (h^{3}-3b^{2}h)\sinh(b)}{2b^{3}h(b^{2}-h^{2})^{2}} $ is obtained as a closed-form expression for the matrix element in the supercoherent state formalism.
• The second derivative $ f_{2}EE $ is given by $ \frac{b(b^{2}-h^{2})\cosh(b) - (b^{2}+h^{2})\sinh(b) + 2bh\sinh(h)}{2b(b^{2}-h^{2})^{2}} $, confirming the consistency of the nonlinear representation.
• The series expansions for $ f_{1}EE, f_{2}EE, f_{1}hh, f_{2}hh $, and higher derivatives are shown to converge to exact hyperbolic functions, validating the analytical approach.
• The derived functions $ f = \frac{\frac{\sinh(b)}{b^{3}} - \frac{\sinh(h)}{h^{3}}}{b^{2}-h^{2}} $, $ f_3 = \frac{\cosh(b) - \cosh(h)}{b^{2}-h^{2}} $, and $ f_4 = \frac{b\sinh(b) - h\sinh(h)}{b^{2}-h^{2}} $ are exact representations of the supergroup matrix elements.
• The results confirm that the nonlinear representation of the supergroup is consistent and closed-form, enabling the construction of an effective functional in future work.