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arXiv:cond-mat/9801056 · Europhysics Letters (EPL)

Non-trivial fixed-point structure of the two-dimensional ± J 3-state Potts ferromagnet/spin glass

Erik Schwartz Sørensen, Michel J. P. Gingras, David A. Huse

Published Nov 15, 1998• 27 citations
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Abstract

The fixed-point structure of the 2D 3-state random-bond Potts model with a bimodal (± J ) distribution of couplings is fully determined using numerical renormalization group techniques. Along the paramagnet-to-ferromagnet critical line we find a total of four distinct fixed points: i) the pure critical fixed point, ii) the critical fixed point for the random-bond, but unfrustrated, ferromagnet, iii) a bicritical fixed point analogous to the bicritical Nishimori fixed point found in the random-bond frustrated Ising model, and iv) the zero-temperature spin-glass–to–ferromagnet critical fixed point. Estimates of the associated critical exponents are given for the various fixed points.

DOI
10.1209/epl/i1998-00502-1

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The fixed-point structure of the 2D 3-state random-bond Potts model with a bimodal (± J ) distribution of couplings is fully determined using numerical renormalization group techniques. Along the paramagnet-to-ferromagnet critical line we find a total of four distinct fixed points: i) the pure critical fixed point, ii) the critical fixed point for the random-bond, but unfrustrated, ferromagnet, iii) a bicritical fixed point analogous to the bicritical Nishimori fixed point found in the random-bond frustrated Ising model, and iv) the zero-temperature spin-glass–to–ferromagnet critical fixed point. Estimates of the associated critical exponents are given for the various fixed points.
The two-dimensional q-state Potts model is subjected to a Z(q) symmetric disorder that allows for the existence of a Nishimori line. At q=2, this model coincides with the +/- J random-bond Ising model. For q>2, apart from the usual pure- and zero-temperature fixed points, the ferro/paramagnetic phase boundary is controlled by two critical fixed points: a weak disorder point, whose universality class is that of the ferromagnetic bond-disordered Potts model, and a strong disorder point which generalizes the usual Nishimori point. We numerically study the case q=3, tracing out the phase diagram and precisely determining the critical exponents. The universality class of the Nishimori point is inconsistent with percolation on Potts clusters.
The fixed-point structure of three-dimensional bond-disordered Ising models is investigated using the numerical domain-wall renormalization-group method. It is found that, in the +/-J Ising model, there exists a non-trivial fixed point along the phase boundary between the paramagnetic and ferromagnetic phases. The fixed-point Hamiltonian of the +/-J model numerically coincides with that of the unfrustrated random Ising models, strongly suggesting that both belong to the same universality class. Another fixed point corresponding to the multicritical point is also found in the +/-J model. Critical properties associated with the fixed point are qualitatively consistent with theoretical predictions.
The random-bond Ising model on the square lattice has several disordered critical points, depending on the probability distribution of the bonds. There are a finite-temperature multicritical point, called the Nishimori point, and a zero-temperature fixed point, for both a binary distribution where the coupling constants take the values ± J and a Gaussian disorder distribution. Inclusion of dilution in the ± J distribution ( J = 0 for some bonds) gives rise to another zero-temperature fixed point which can be identified with percolation in the non-frustrated case ( J ≥ 0). We study these fixed points using numerical (transfer matrix) methods. We determine the location, critical exponents, and central charge of the different fixed points and study the spin–spin correlation functions. Our main findings are the following. (1) We confirm that the Nishimori point is universal with respect to the type of disorder, i.e. we obtain the same central charge and critical exponents for the ± J and Gaussian distributions of disorder. (2) The Nishimori point, the zero-temperature fixed point for the ± J and Gaussian distributions of disorder, and the percolation point in the diluted case all belong to mutually distinct universality classes. (3) The paramagnetic phase is re-entrant below the Nishimori point, i.e. the zero-temperature fixed points are not located exactly below the Nishimori point, either for the ± J distribution, or for the Gaussian distribution.
We study the criticality at the spin-glass-ferromagnetic-paramagnetic multicritical point in the d=3, \ifmmode\pm\else\textpm\fi{}J distribution, random-bond Ising model. Using high-temperature expansions to order ${\mathit{T}}^{\mathrm{\ensuremath{-}}34}$, we estimate that the multicritical point N lies on the Nishimori line at Tc{\mathit{T}}_{\mathit{c}}Tc​/J=1.690\ifmmode\pm\else\textpm\fi{}0.016. Along this line the critical exponents are found to be \ensuremath{\gamma}=1.80\ifmmode\pm\else\textpm\fi{}0.15 and \ensuremath{\nu}=0.85\ifmmode\pm\else\textpm\fi{}0.08. The latter is clearly consistent with the rigorous exponent inequality \ensuremath{\nu}\ensuremath{\ge}2/d. We also calculate the crossover exponent \ensuremath{\varphi} and show that the scaling axes at N are in agreement with the recent predictions of Le Doussal and Harris.
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.
The Ising model with a quenched random distribution of ferromagnetic and antiferromagnetic interactions has been investigated by tracking the probability distribution of interactions under rescaling. As the initial probability of a ferromagnetic interaction is increased, the phase diagram in three dimensions shows three ordered phases: antiferromagnetic, spin glass and ferromagnetic. We find that, within the spin-glass region, effective interactions between pairs of spins at successive length scales occur in a chaotic sequence, while the average magnitudes of the ferromagnetic and the antiferromagnetic interactions both flow to infinity under iteration. Our calculated spin-glass multicritical points are on the Nishimori line in cases of both bimodal (±J) and diluted (trimodal) distributions, and these points flow under rescaling to a common fixed distribution with a specific heat exponent α of approximately −3.7.