[Paper Review] Renormalization group analysis of multi-band many-electron systems at half-filling

Yohei Kashima|arXiv (Cornell University)|May 4, 2014

Physics of Superconductivity and Magnetism7 references3 citations

TL;DR

This paper develops a rigorous renormalization group (RG) framework for multi-band many-electron systems at half-filling and positive temperature, using finite-dimensional Grassmann integrals and multi-scale integration. It proves the existence of an analytic free energy density in the infinite-volume, zero-temperature limit for the half-filled Hubbard model on a square lattice with π flux per plaquette, extending Giuliani and Mastropietro's RG approach to a four-band formulation and establishing scale-dependent estimates to ensure convergence.

ABSTRACT

Renormalization group analysis for multi-band many-electron systems at half-filling at positive temperature is presented. The analysis includes the Matsubara ultra-violet integration and the infrared integration around the zero set of the dispersion relation. The multi-scale integration schemes are implemented in a finite-dimensional Grassmann algebra indexed by discrete position-time variables. In order that the multi-scale integrations are justified inductively, various scale-dependent estimates on Grassmann polynomials are established. We apply these theories in practice to prove that for the half-filled Hubbard model with nearest-neighbor hopping on a square lattice the infinite-volume, zero-temperature limit of the free energy density exists as an analytic function of the coupling constant in a neighborhood of the origin if the system contains the magnetic flux pi (mod 2 pi) per plaquette and 0 (mod 2 pi) through the large circles around the periodic lattice. Combined with Lieb's result on the flux phase problem ([Lieb, E. H., Phys. Rev. Lett. 73 (1994), 2158]), this theorem implies that the minimum free energy density of the flux phase problem converges to an analytic function of the coupling constant in the infinite-volume, zero-temperature limit. The proof of the theorem is based on a four-band formulation of the model Hamiltonian and an extension of Giuliani-Mastropietro's renormalization designed for the half-filled Hubbard model on the honeycomb lattice ([Giuliani, A. and V. Mastropietro, Commun. Math. Phys. 293 (2010), 301-346]).

Motivation & Objective

• To develop a systematic renormalization group framework for multi-band many-electron systems at half-filling and positive temperature.
• To establish scale-dependent estimates on Grassmann polynomials to justify multi-scale integration inductively.
• To apply the theory to the half-filled Hubbard model on a square lattice with π flux per plaquette.
• To prove the existence of an analytic free energy density in the infinite-volume, zero-temperature limit.
• To extend Giuliani-Mastropietro's RG method from the honeycomb lattice to the square lattice via a four-band formulation.

Proposed method

• Formulates the multi-band Hamiltonian in a finite-dimensional Grassmann algebra indexed by discrete space-time variables.
• Employs Matsubara ultra-violet and infrared integrations using cut-off functions to decompose the full covariance.
• Applies generalized multi-scale integration schemes with scale-dependent norms and semi-norms for Grassmann polynomials.
• Uses a four-band formulation to handle the Hubbard model with π flux per plaquette, enabling control of the Fermi surface structure.
• Establishes L1-estimates and Gevrey-class bounds for kernels of Grassmann polynomials to ensure convergence.
• Implements a time-continuum, infinite-volume limit via truncated Grassmann integral formulations and symmetry considerations.

Experimental results

Research questions

• RQ1 Can the infinite-volume, zero-temperature limit of the free energy density be shown to exist and be analytic for the half-filled Hubbard model on a square lattice?
• RQ2 How can multi-scale integration be justified inductively in a finite-dimensional Grassmann algebra for multi-band systems at half-filling?
• RQ3 What role does the π flux per plaquette play in enabling analyticity of the free energy density in the zero-temperature limit?
• RQ4 Can Giuliani-Mastropietro's RG method for the honeycomb lattice be extended to the square lattice via a four-band formulation?
• RQ5 What scale-dependent estimates are necessary to control the convergence of the multi-scale integration in the infinite-volume and zero-temperature limits?

Key findings

• The infinite-volume, zero-temperature limit of the free energy density exists as an analytic function of the coupling constant in a neighborhood of the origin for the half-filled Hubbard model with π flux per plaquette.
• The analyticity of the free energy density is established via a four-band formulation of the Hubbard model Hamiltonian and rigorous multi-scale integration in Grassmann space.
• The method ensures uniform convergence of the analytic function in the infinite-volume, zero-temperature limit through scale-dependent estimates on Grassmann polynomials.
• The proof relies on extending Giuliani and Mastropietro's RG approach from the honeycomb lattice to the square lattice, incorporating Matsubara UV and infrared integrations.
• The framework successfully controls temperature dependencies, enabling analyticity down to zero temperature, with the domain shrinking only exponentially in inverse temperature.
• The result, combined with Lieb's flux phase theorem, implies that the minimum free energy density of the flux phase problem converges to an analytic function in the infinite-volume, zero-temperature limit.