Low-temperature transition of 2d random-bond Ising model and quantum infinite randomness

TL;DR

This paper investigates the low-temperature phase transition in the 2D random-bond Ising model, revealing a connection to quantum infinite randomness through a renormalization group approach and spectral analysis.

Contribution

It introduces a novel RG transformation linking the classical Ising transition to quantum spectral properties, demonstrating the flow toward infinite randomness at the critical point.

Findings

Critical point characterized by infinite randomness in quantum spectrum

Log gap scales as a power of system size, log ε_min^{-1} ∼ L^ψ

Tunneling exponent matches the spin stiffness exponent θ_c

Abstract

At low temperatures, the classical two-dimensional random bond Ising model undergoes a frustration-driven ferromagnet-to-paramagnet transition controlled by a zero-temperature fixed point separating ferromagnet and spin glass phases. We show that this critical point can be understood through a renormalization group transformation that constructs the ground state of the Ising model through a sequence of Hamiltonians that, starting with an unfrustrated model, iteratively adds in frustration until the target Hamiltonian is reached. Via a mapping of the thermodynamics of the 2d Ising model to the spectral properties of a related Hermitian matrix -- the Hamiltonian of a noninteracting quantum problem -- this RG procedure corresponds to an iterative diagonalization of the quantum Hamiltonian. The flow toward zero temperature in the Ising picture manifests as a flow toward infinite randomness in the spectrum of the quantum Hamiltonian, with the log gap of the Hamiltonian scaling as a power of the system size: $\log \varepsilon_{\it min}^{-1} \sim L^\psi$. The tunneling exponent $\psi$ is equal to the spin stiffness exponent $\theta_c$ characterizing the zero-temperature fixed point.