[Paper Review] Momenta fields and the derivative expansion

Luca Zambelli|arXiv (Cornell University)|Oct 30, 2015

Quantum chaos and dynamical systems3 citations

TL;DR

This paper introduces a novel approximation scheme for the functional renormalization group (FRG) by reformulating the Polchinski equation in a covariant Hamiltonian framework using an infinite tower of momenta fields with higher spin. The method enables a derivative expansion that is regulator-independent at first order and yields η = 0.03616(1) for the 3D Ising model critical exponent, offering a systematic, high-order alternative to standard derivative expansion with improved control over symmetries and regulator dependence.

ABSTRACT

The Polchinski exact renormalization group equation for a scalar field theory in arbitrary dimensions is translated, by means of a covariant Hamiltonian formalism, into a partial differential equation for an effective Hamiltonian density that depends on an infinite tower of momenta fields with higher spin. A natural approximation scheme is then expanding the Hamiltonian in momenta with increasing rank. The first order of this expansion, one next to the local potential approximation, is regulator-independent and already includes infinitely many derivative interactions. Further truncating this down to a quadratic dependence on the momenta leads to an alternative to the first order of the derivative expansion, which is used to compute $η=0.03616(1)$ for the critical exponent of the three dimensional Ising model.

Motivation & Objective

• To develop a systematic, regulator-independent approximation scheme for functional renormalization group (FRG) equations beyond the local potential approximation.
• To address the limitations of existing schemes—such as regulator dependence and symmetry breaking—by introducing a new formalism based on momenta fields.
• To enable higher-order derivative expansions in FRG by formulating the effective Hamiltonian in terms of an infinite tower of momenta fields with increasing spin.
• To provide a computationally tractable yet systematically improvable method for studying critical phenomena and nonperturbative quantum field theories.

Proposed method

• The Polchinski exact FRG equation is translated into a partial differential equation for an effective Hamiltonian density using a covariant Hamiltonian formalism.
• The Hamiltonian is expanded in an infinite tower of momenta fields, each corresponding to higher-spin operators, enabling a systematic derivative-like expansion.
• The first-order truncation of this expansion is shown to be regulator-independent and includes infinitely many derivative interactions.
• Further truncation to quadratic dependence on momenta fields yields an alternative to the standard first-order derivative expansion, simplifying computation while preserving key physical features.
• Large-field asymptotic expansions of the potential and wave function renormalization are used to extract critical exponents numerically.
• The method is applied to the 3D Ising model, with numerical integration guided by the location of zeros in the wave function renormalization function.

Experimental results

Research questions

• RQ1 Can a systematic, regulator-independent derivative expansion be constructed in the FRG framework beyond the local potential approximation?
• RQ2 How does the inclusion of momenta fields with higher spin improve the control over symmetry and regulator dependence in FRG approximations?
• RQ3 What is the quantitative performance of the proposed method in computing critical exponents for the 3D Ising model?
• RQ4 Can the method be systematically extended to higher orders while maintaining computational feasibility and physical consistency?
• RQ5 How does the new scheme compare to standard derivative expansion and other existing approximation schemes in terms of accuracy and convergence?

Key findings

• The first-order momenta field expansion is regulator-independent, providing a significant advantage over standard derivative expansion at the same order.
• The method yields a critical exponent η = 0.03616(1) for the 3D Ising model, consistent with high-precision numerical estimates.
• The inclusion of infinitely many derivative interactions at first order captures nontrivial momentum dependence beyond the local potential approximation.
• The quadratic truncation of momenta fields leads to a simplified yet physically meaningful alternative to the standard first-order derivative expansion.
• Large-field asymptotic expansions of the potential and wave function renormalization are successfully used to extract critical behavior and guide numerical integration.
• The method shows promise for systematic high-order computations in FRG, with improved control over symmetries and regulator artifacts.