[Paper Review] Absence of renormalization group pathologies in some critical Dyson-Ising ferromagnets

Tom Kennedy|arXiv (Cornell University)|Jun 19, 2020

Theoretical and Computational Physics18 references4 citations

TL;DR

This paper establishes the rigorous existence of the first step of the renormalization group (RG) transformation—via decimation or majority rule—for a modified one-dimensional Dyson-Ising ferromagnet with long-range interactions. By introducing strong nearest-neighbor couplings ($\gamma$) relative to long-range couplings ($\epsilon$), the authors prove that the renormalized measure remains Gibbsian even near criticality, resolving a pathology identified by Van Enter and Le Ny in the standard model at low temperatures.

ABSTRACT

The Dyson-Ising ferromagnet is a one-dimensional Ising model with a power law interaction. When the power is between -1 and -2, the model has a phase transition. Van Enter and Le Ny proved that at sufficiently low temperatures the decimation renormalization group transformation is not defined in the sense that the renormalized measure is not a Gibbs measure. We consider a modified model in which the nearest neighbor couplings are much larger than the other couplings. For a family of Hamiltonians which includes critical cases, we prove that the first step of the renormalization group transformation can be rigorously defined for majority rule and decimation.

Motivation & Objective

• To address the failure of the decimation RG transformation in the standard critical Dyson-Ising model at low temperatures, where the renormalized measure is not Gibbsian.
• To show that introducing strong nearest-neighbor interactions ($\gamma$) relative to long-range couplings ($\epsilon$) restores the well-definedness of the first RG step.
• To extend the validity of RG methods to parameter regions including critical points, where non-rigorous treatments are most relevant.
• To prove that both decimation and majority rule transformations yield well-defined RG maps in the modified model for small $\epsilon$ and all $\gamma > 0$.

Proposed method

• Introduce a modified Hamiltonian with separate parameters $\gamma$ for nearest-neighbor and $\epsilon$ for long-range interactions, allowing independent tuning of coupling strengths.
• Use infrared bounds and Gaussian domination to control spin correlations and prove long-range order for $\gamma$ sufficiently large relative to $\epsilon$.
• Apply reflection positivity and integral representations for $J_{j,k} = |j-k|^{-\alpha}$ to establish the Gaussian domination bound, a key step in proving infrared bounds.
• Derive the infrared bound $g_m(p) \leq 1/(2E(p))$ using second-order expansion of the Gaussian domination inequality in $h_j = \delta e^{ipj}$.
• Use the dual lattice $\Lambda^*$ and sum over $p \in \Lambda^*$ to analyze the behavior of the susceptibility and detect long-range order via an atom at $p=0$.
• Prove that $\sup_m \frac{1}{2m} \sum_{p \in \Lambda^*} \frac{1}{R(p)} < \infty$ using a lower bound $R(p) \geq c p^{\alpha-1}$, ensuring integrability for $1 < \alpha < 2$.

Experimental results

Research questions

• RQ1 Can the first step of the RG transformation be rigorously defined in a critical Dyson-Ising model where previous methods fail?
• RQ2 Does introducing strong nearest-neighbor couplings ($\gamma$) prevent the renormalized measure from becoming non-Gibbsian at low temperatures?
• RQ3 Is the Gaussian domination bound valid for the modified Hamiltonian, enabling the derivation of infrared bounds?
• RQ4 Can the infrared bound detect long-range order in the modified model for all $\gamma > 0$ when $\epsilon$ is small?
• RQ5 Does the method extend to both decimation and majority rule transformations in the critical regime?

Key findings

• For any $\alpha \in (1,2)$, there exists $\epsilon_0 > 0$ such that for all $\epsilon < \epsilon_0$ and all $\gamma > 0$, the first RG step is well-defined via decimation or majority rule.
• The renormalized measure remains Gibbsian in the modified model, even near criticality, resolving a pathology in the standard model identified by Van Enter and Le Ny.
• The infrared bound $g_m(p) \leq 1/(2E(p))$ holds due to Gaussian domination, which is proven via reflection positivity and integral representations of $J_{j,k}$.
• The sum $\sup_m \frac{1}{2m} \sum_{p \in \Lambda^*} \frac{1}{R(p)} < \infty$ is finite because $R(p) \geq c p^{\alpha-1}$ and $\alpha - 1 < 1$, ensuring integrability near $p=0$.
• As $\gamma \to \infty$, the integral $\int \frac{dp}{E(p)} \to 0$, implying an atom at $p=0$ in the susceptibility, which signals long-range order.
• The proof relies on the reflection positivity of $N_{j,k}$ and $J_{j,k}$, and the existence of a positive measure $\mu$ such that $n^{-\alpha} = \int_0^1 \lambda^n \mu(d\lambda)$.