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arXiv:2603.02308

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

Akshat Pandey, Aditya Mahadevan, A. Alan Middleton, Daniel S. Fisher

cond-mat.stat-mechcond-mat.dis-nncond-mat.str-elquant-phSubmitted Mar 2, 2026• Announced Mar 4, 2026• 0 citations
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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: $\log \varepsilon_{\it min}^{-1} \sim L^ψ$. The tunneling exponent $ψ$ is equal to the spin stiffness exponent $θ_c$ characterizing the zero-temperature fixed point.

Comments
7+10 pages, 4+4 figures

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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⁡εmin−1∼Lψ\log \varepsilon_{\it min}^{-1} \sim L^ψlogεmin−1​∼Lψ. The tunneling exponent ψψψ is equal to the spin stiffness exponent θcθ_cθc​ characterizing the zero-temperature fixed point.
Abstract Systems with quenched disorder possess complex energy landscapes that are challenging to explore under conventional Monte Carlo methods. In this work, we implement an efficient entropy sampling scheme for accurate computation of the entropy function in low-energy regions. The method is applied to the two-dimensional ± J random-bond Ising model, where frustration is controlled by the fraction p of ferromagnetic bonds. We investigate the low-temperature paramagnetic–ferromagnetic phase boundary below the multicritical point at T N = 0.9530(4), P N = 0.89078(8), as well as the zero-temperature ferromagnetic–spin-glass transition. Finite-size scaling analysis reveals that the phase boundary for T < T N exhibits reentrant behavior. By analyzing the evolution of the magnetization-resolved density of states g ( E , M ) and ground-state spin configurations against increasing frustration, we provide strong evidence that the zero-temperature transition is a mixed-order. Finite-size scaling conducted on the spin-glass side supports the validity of β = 0, where β is the magnetization exponent, with a correlation length exponent ν = 1.50(8). Our results provide new insights into the nature of the ferromagnetic-to-spin-glass phase transition in an extensively degenerate ground state.
Systems with quenched disorder possess complex energy landscapes that are challenging to explore under the conventional Monte Carlo method. In this work, we implement an efficient entropy sampling scheme for accurate computation of the entropy function in low-energy regions. The method is applied to the two-dimensional ±J\pm J±J random-bond Ising model, where frustration is controlled by the fraction ppp of ferromagnetic bonds. We investigate the low-temperature paramagnetic--ferromagnetic phase boundary below the multicritical point at TN=0.9530(4)T_N = 0.9530(4)TN​=0.9530(4), PN=0.89078(8)P_N = 0.89078(8)PN​=0.89078(8), as well as the zero-temperature ferromagnetic--spin-glass transition. Finite-size scaling analysis reveals that the phase boundary for T<TNT < T_NT<TN​ exhibits reentrant behavior. By analyzing the evolution of the magnetization-resolved density of states g(E,M)g(E, M)g(E,M) and ground-state spin configurations against increasing frustration, we provide strong evidence that the zero-temperature transition is quasi-first order. Finite-size scaling conducted on the spin-glass side supports the validity of β=0β= 0β=0, with a correlation length exponent ν=1.50(8)ν= 1.50(8)ν=1.50(8). Our results provide new insights into the nature of the ferromagnetic-to-spin-glass phase transition in an extensively degenerate ground state.
For the quantum Ising chain, the self-dual block renormalization procedure of Fernandez-Pacheco (1979 Phys. Rev. D 19 3173 ) is known to reproduce exactly the location of the zero-temperature critical point and the correlation length exponent ν = 1. Recently, Miyazaki and Nishimori (2013 Phys. Rev. E 87 032154 ) have proposed to study the disordered quantum Ising model in dimensions d > 1 by applying the Fernandez-Pacheco procedure successively in each direction. To avoid the inequivalence of directions of their approach, we propose here an alternative procedure where the d directions are treated on the same footing. For the pure model, this leads to the correlation length exponents ν ≃ 0.625 in d = 2 (to be compared with the 3D classical Ising model exponent ν ≃ 0.63) and ν ≃ 0.5018 (to be compared with the 4D classical Ising model mean-field exponent ν = 1/2). For the disordered model in dimension d = 2, either ferromagnetic or spin-glass, the numerical application of the renormalization rules to samples of linear size L = 4096 yields that the transition is governed by an Infinite Disorder Fixed Point, with the activated exponent ψ ≃ 0.65, the typical correlation exponent ν typ ≃ 0.44 and the finite-size correlation exponent ν FS ≃ 1.25. We discuss the similarities and differences with the Strong Disorder Renormalization results.
We consider the stochastic dynamics of Ising ferromagnets (either pure or random) near zero temperature. The master equation satisfying detailed balance can be mapped onto a quantum Hamiltonian which has an exact zero-energy ground state representing the thermal equilibrium. The largest relaxation time teqt_{eq}teq​ governing the convergence towards this Boltzmann equilibrium in finite-size systems is determined by the lowest non-vanishing eigenvalue E1=1/teqE_1=1/t_{eq}E1​=1/teq​ of the quantum Hamiltonian HHH. We introduce and study a real-space renormalization procedure for the quantum Hamiltonian associated to the single-spin-flip dynamics of Ising ferromagnets near zero temperature. We solve explicitly the renormalization flow for two cases. (i) For the one-dimensional random ferromagnetic chain with free boundary conditions, the largest relaxation time teqt_{eq}teq​ can be expressed in terms of the set of random couplings for various choices of the dynamical transition rates. The validity of these RG results in d=1d=1d=1 is checked by comparison with another approach. (ii) For the pure Ising model on a Cayley tree of branching ratio KKK, we compute the exponential growth of teq(N)t_{eq}(N)teq​(N) with the number NNN of generations.
We consider the renormalization of quenched bond disorder in the Ising model in the limit that it is sparse -- highly localized and vanishing in the thermodynamic limit. We begin in 1D with arbitrary disorder assigned to a finite number of bonds and study how the system renormalizes, finding non-trivial fixed point structure for any given bond with a separatrix at zero bond strength, equivalent to inserting a break in the chain. Either side of this critical line, renormalization group (RG) trajectories flow towards one of two attractors on which the disordered bonds settle onto ferromagnetic or anti-ferromagnetic (AF) couplings of equal and opposite magnitude. Bonds that settle on an AF attractor are equivalent to inserting a twist in the chain at the location of the bond, implying a multi-kink ground state solution. Qualitatively different behavior emerges at the RG step when bonds start to coalesce, with the chain `untwisting' whenever two AF bonds coalesce. Our findings generalize to higher dimensions for codimension one defects that are sparse from the perspective of the orthogonal complement lattice. In 2D, the Z2\mathbb Z_2Z2​ symmetry of the model has an IR manifestation where one can construct field strengths and Wilson loops (which characterize frustration) from fundamental plaquette variables. In the non-sparse limit where the disorder parameter is drawn from an arbitrary but homogeneously assigned probability distribution function, we recover previously found fixed distributions as special cases, but find only the trivial paramagnetic distribution to be an attractor. For non-homogeneously assigned disorder that is sufficiently dilute however, we find the Edwards-Anderson model with equal probability ±J\pm J±J bonds to also be an IR attractor.
For the Ising model with Gaussian random coupling of average J0{J}_{0}J0​ and unit variance, the zero-temperature spin-glass--ferromagnetic transition as a function of the control parameter J0{J}_{0}J0​ can be studied via the size-LLL-dependent renormalized coupling defined as the domain-wall energy ${J}^{R}(L)\ensuremath{\equiv}{E}_{\text{GS}}^{(\text{AF})}(L)\ensuremath{-}{E}_{\text{GS}}^{(\text{F})}(L)$ (i.e., the difference between the ground-state energies corresponding to antiferromagnetic and ferromagnetic boundary conditions in one direction). We study numerically the critical exponents of this zero-temperature transition within the Migdal-Kadanoff approximation as a function of the dimension d=2,3,4,5,6d=2,3,4,5,6d=2,3,4,5,6. We then compare with the mean-field spherical model. Our main conclusion is that in low dimensions, the critical stiffness exponent ${\ensuremath{\theta}}^{c}$ is clearly bigger than the spin-glass stiffness exponent ${\ensuremath{\theta}}^{\text{SG}}$, but that they turn out to coincide in high enough dimension and in the mean-field spherical model. We also discuss the finite-size scaling properties of the averaged value and of the width of the distribution of the renormalized couplings.