Dyson Hierarchical Long-Ranged Quantum Spin-Glass via real-space renormalization

TL;DR

This paper investigates the quantum phase transition in a Dyson hierarchical quantum spin-glass model with long-range interactions, using real-space renormalization to analyze critical behavior and exponents.

Contribution

It introduces a real-space renormalization approach to study the quantum spin-glass phase transition in a Dyson hierarchical model with long-range couplings, providing numerical estimates of critical exponents.

Findings

The typical renormalized coupling grows as a power-law with length scale in the spin-glass phase.

The correlation length diverges at the critical point following a power-law.

At criticality, the renormalized coupling and field decay as a power-law with a finite dynamical exponent.

Abstract

We consider the Dyson hierarchical version of the quantum Spin-Glass with random Gaussian couplings characterized by the power-law decaying variance $\overline{J^2(r)} \propto r^{-2\sigma}$ and a uniform transverse field $h$. The ground state is studied via real-space renormalization to characterize the spinglass-paramagnetic zero temperature quantum phase transition as a function of the control parameter $h$. In the spinglass phase $h<h_c$, the typical renormalized coupling grows with the length scale $L$ as the power-law $J_L^{typ}(h) \propto \Upsilon(h) L^{\theta}$ with the classical droplet exponent $\theta=1-\sigma$, where the stiffness modulus vanishes at criticality $\Upsilon(h) \propto (h_c-h)^{\mu} $, whereas the typical renormalized transverse field decays exponentially $ h^{typ}_L(h) \propto e^{- \frac{L}{\xi}}$ where the correlation length diverges at the transition $\xi \propto (h_c-h)^{-\nu}$. At the critical point $h=h_c$, the typical renormalized coupling $J_L^{typ}(h_c) $ and the typical renormalized transverse field $ h^{typ}_L(h_c)$ display the same power-law behavior $L^{-z}$ with a finite dynamical exponent $z$. The RG rules are applied numerically to chains containing $L=2^{12}=4096 $ spins in order to measure these critical exponents for various values of $\sigma$ in the region $1/2<\sigma<1$.