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Dyson hierarchical long-ranged quantum spin-glass via real-space renormalization

Abstract : We consider the Dyson hierarchical version of the quantum Spin-Glass with random Gaussian couplings characterized by the power-law decaying variance J 2 (r) ∝ r −2σ 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 < hc, the typical renormalized coupling grows with the length scale L as the power-law J typ L (h) ∝ Υ(h)L θ with the classical droplet exponent θ = 1 − σ, where the stiffness modulus vanishes at criticality Υ(h) ∝ (hc − h) µ , whereas the typical renormalized transverse field decays exponentially h typ L (h) ∝ e − L ξ in terms of the diverging correlation length ξ ∝ (hc − h) −ν. At the critical point h = hc, the typical renormalized coupling J typ L (hc) and the typical renormalized transverse field h typ L (hc) 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 σ in the region 1/2 < σ < 1.
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Cécile Monthus. Dyson hierarchical long-ranged quantum spin-glass via real-space renormalization. Journal of Statistical Mechanics, 2015, 2015 (10), pp.10024. ⟨10.1088/1742-5468/2015/10/P10024⟩. ⟨cea-01321792⟩

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