[Paper Review] Small parameter for lattice models with strong interaction

A. N. Rubtsov|arXiv (Cornell University)|Jan 16, 2006

Theoretical and Computational Physics2 references3 citations

TL;DR

This paper introduces an exact renormalization of diagrammatic series for lattice models with localized strong interactions, mapping vertices to irreducible correlators of a self-consistent impurity model. The method yields a small parameter valid in both weak- and tight-binding limits, enabling accurate description of strong correlations; benchmark results for O(N) models show excellent agreement with numerics, especially at N=1,2,3, with α² corrections capturing critical behavior beyond Gaussian-bath approximations.

ABSTRACT

Diagram series expansion for lattice models with a localized nonlinearity can be renormalized so that diagram vertexes become irreducible vertex parts of certain impurity model. Thus renormalized series converges well in the very opposite cases of tight and weak binding and pretends to describe in a regular way strong-correlated systems with localized interaction. Benchmark results for the classical O(N) models on a cubic lattice are presented.

Motivation & Objective

• To develop a regular, non-perturbative framework for strongly correlated lattice systems with localized nonlinearity.
• To overcome the failure of standard perturbation theory in the strong-coupling regime where nonlinearity and binding are comparable.
• To unify weak- and tight-binding limits via a self-consistent impurity model with exact renormalization of diagram series.
• To provide a small-parameter expansion valid across the crossover between weak and strong binding, enabling systematic improvement.
• To benchmark the method on O(N) models and demonstrate its superiority over mean-field and Gaussian-bath approximations.

Proposed method

• The method performs an exact renormalization of the diagrammatic expansion so that vertices correspond to irreducible vertex parts of a single-site impurity model.
• It introduces an auxiliary ensemble with an x-diagonal tensor A and a shift vector φ⁰, decoupling sites and enabling solution of the impurity problem.
• Correlators and irreducible vertex parts (γ⁽ⁿ⁾) are defined in the auxiliary ensemble, serving as building blocks for the renormalized expansion.
• The small parameter α is introduced via a perturbative expansion in α², with α related to the strength of non-local correlations.
• The self-consistency condition links the effective bath (A, φ⁰) to the physical system through the impurity solution and the full lattice Green’s function.
• The method is applied to the O(N) model in 3D, with results obtained via iterative solution of the self-consistency loop and α² correction.

Experimental results

Research questions

• RQ1 Can a single small parameter be constructed that governs both weak- and tight-binding limits in strongly correlated lattice models?
• RQ2 Can the diagrammatic expansion be exactly renormalized so that vertices become irreducible correlators of a self-consistent impurity model?
• RQ3 Does the resulting theory describe critical behavior and symmetry-breaking phases accurately beyond mean-field and Gaussian-bath approximations?
• RQ4 How well does the α² correction capture the N-dependence of the soft-mode dispersion in O(N) models?
• RQ5 Can the method be applied to general models without requiring special symmetries like O(N)?

Key findings

• The α² approximation captures the N-dependence of the effective soft-mode dispersion Σ₀ for O(N) models, correctly describing the Ising (N=1), XY (N=2), and Heisenberg (N=3) cases.
• The method shows excellent agreement with numerical data across the critical region, with solid lines in Figure 2 closely matching symbols for N=1,2,3.
• The Gaussian-bath approximation fails to reproduce the N-dependence of Σ₀, as it predicts the same behavior for all N, while the α² result correctly captures the physical trend.
• The iteration procedure converges well except in a narrow region near the critical point, where the finite-order perturbation theory breaks down.
• The theory performs well deep inside the critical region, with the Ising model data following the critical scaling ⟨ϕ·ϕ⟩ ∝ (β−βc)^1.24 with high accuracy.
• The method is general and does not rely on special symmetries, unlike 1/N expansions, and becomes exact in the N→∞ limit, consistent with known results.