Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Condensed matter

Nearest-neighbor frustrated random-bond model ind=2: Some exact results

R. Shankar, Ganpathy N. Murthy

Published Jul 1, 1987• 141 citations
Publisher

Abstract

We study a d=2 Ising model where the veritcal bonds are fixed and ferromagnetic and the horizontal bonds can vary randomly in sign and in magnitude (within some limits) but are same within each now. The model therefore generalizes that of McCoy and Wu since it allows for the interesting case of frustration. We use the transfer matrix to map our problem to a collection of random field d=1 problems about which a lot is known. We find generally three transitions: a Griffiths transition, its dual version, and one with infinite correlation length and index \ensuremath{\nu}=1. In all cases the free energy has infinitely differentiable singularities. In addition there are some zero-temperature transitions.

DOI
10.1103/physrevb.36.536

Related papers

Frustrated nearest-neighbord=2Ising model: Exact results

R. Shankar, Ganpathy N. Murthy

We study a d=2 Ising model with vertical coupling K>0 and random horizontal coupling ${K}_{H}$(${K}_{H}$:${K}_{1}$\ensuremath{\le}${K}_{H}$,\ensuremath{\le}${K}_{2}$), which is the same within a row. This generalizes the McCoy-Wu model since it allows for frustration. By mapping it into a random-field model in d=1, we show that the free energy F has infinitely differentiable singularities at ${K}^{\mathrm{*}}$=${K}_{2}$ (Griffith's singularity in temperature); ${K}^{\mathrm{*}}$=〈K〉, where 〈K〉 is the average horizontal bond (infinite correlation length and \ensuremath{\nu}=1), and ${K}^{\mathrm{*}}$=${K}_{1}$ (Griffith's singularity, if ${K}_{1}$0, its dual version if not). We show for the last case that F(t)\ensuremath{\approxeq}${t}^{1/t}$.

Physical review. B, Condensed matter•Mar 1, 1987•13 citations

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

Akshat Pandey, Aditya Mahadevan, A. Alan Middleton +1 more

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.

cond-mat.stat-mechCross-listedarXiv:2603.02308•Mar 4, 2026•0 citations

Thermodynamics of Ising models with layered randomness: Exact solutions on square and triangular lattices

Theo M. Nieuwenhuizen, Henri Orland

Two-dimensional Ising models are studied on square and triangular lattices with layered disordered bonds (McCoy-Wu-type models). It is shown that exact solutions exist for diluted exponential distributions of couplings. In the ferromagnetic case the free energy exhibits an essential singularity (slightly different from the result of McCoy and Wu) at the transition point. If frustration is present the transitions occurs at T=0, and the specific heat is linear.

Physical review. B, Condensed matter•Sep 1, 1989•7 citations

Two-dimensional random-bond Ising model, free fermions, and the network model

Florian Merz, J. T. Chalker

We develop a recently proposed mapping of the two-dimensional Ising model with random exchange (RBIM) via the transfer matrix, to a network model for a disordered system of noninteracting fermions. The RBIM transforms in this way to a localization problem belonging to one of a set of nonstandard symmetry classes, known as class D; the transition between paramagnet and ferromagnet is equivalent to a delocalization transition between an insulator and a quantum Hall conductor. We establish the mapping as an exact and efficient tool for numerical analysis: using it, the computational effort required to study a system of width M is proportional to ${M}^{3},$ and not exponential in M as with conventional algorithms. We show how the approach may be used to calculate for the RBIM the free energy, typical correlation lengths in quasi-one dimension for both the spin and the disorder operators, and the even powers of spin-spin correlation functions and their disorder averages. We examine in detail the square-lattice, nearest-neighbor $\ifmmode\pm\else\textpm\fi{}J$ RBIM, in which bonds are independently antiferromagnetic with probability p, and ferromagnetic with probability $1\ensuremath{-}p.$ Studying temperatures $T>~0.4J,$ we obtain precise coordinates in the $p\ensuremath{-}T$ plane for points on the phase boundary between ferromagnet and paramagnet, and for the multicritical (Nishimori) point. We demonstrate scaling flow towards the pure Ising fixed point at small p, and determine critical exponents at the multicritical point.

arXiv:cond-mat/0106023•Physical review. B, Condensed matter•Jan 15, 2002•143 citations

Griffiths singularities in two-dimensional random-bond Ising models: Relation with Lifshitz band tails

Theo M. Nieuwenhuizen

Ising models with correlated or uncorrelated random horizontal bonds are studied for square lattices in zero field. The partition sum is expressed in terms of a one-component field. Griffiths singularities in the free energy as function of temperature are derived. They are log-transforms of certain Lifshitz band tails. Griffiths singularities occur above as well as below the critical temperature.

Physical Review Letters•Oct 23, 1989•19 citations

Random Ising models: Convergence of successive approximations towards exact results

A. Benyoussef, N. Boccara

We studied a new and simple method which gives successive approximate expressions for the magnetization (or any order parameter) of a random Ising model as a function of temperature, site, or bond dilution, frustration, magnetic field,... To obtain the various approximations a trace is performed on finite clusters increasing in size. In particular, we investigated a honeycomb Ising model with random nearest-neighbor interactions and calculated sequences of critical temperatures, percolation thresholds,... which converge towards exact known results.

Journal of Applied Physics•Mar 15, 1984•0 citations
We study a d=2 Ising model where the veritcal bonds are fixed and ferromagnetic and the horizontal bonds can vary randomly in sign and in magnitude (within some limits) but are same within each now. The model therefore generalizes that of McCoy and Wu since it allows for the interesting case of frustration. We use the transfer matrix to map our problem to a collection of random field d=1 problems about which a lot is known. We find generally three transitions: a Griffiths transition, its dual version, and one with infinite correlation length and index \ensuremath{\nu}=1. In all cases the free energy has infinitely differentiable singularities. In addition there are some zero-temperature transitions.
We study a d=2 Ising model with vertical coupling K>0 and random horizontal coupling KH{K}_{H}KH​(KH{K}_{H}KH​:K1{K}_{1}K1​\ensuremath{\le}KH{K}_{H}KH​,\ensuremath{\le}K2{K}_{2}K2​), which is the same within a row. This generalizes the McCoy-Wu model since it allows for frustration. By mapping it into a random-field model in d=1, we show that the free energy F has infinitely differentiable singularities at K∗{K}^{\mathrm{*}}K∗=K2{K}_{2}K2​ (Griffith's singularity in temperature); K∗{K}^{\mathrm{*}}K∗=〈K〉, where 〈K〉 is the average horizontal bond (infinite correlation length and \ensuremath{\nu}=1), and K∗{K}^{\mathrm{*}}K∗=K1{K}_{1}K1​ (Griffith's singularity, if ${K}_{1}$0, its dual version if not). We show for the last case that F(t)\ensuremath{\approxeq}${t}^{1/t}$.
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.
We develop a recently proposed mapping of the two-dimensional Ising model with random exchange (RBIM) via the transfer matrix, to a network model for a disordered system of noninteracting fermions. The RBIM transforms in this way to a localization problem belonging to one of a set of nonstandard symmetry classes, known as class D; the transition between paramagnet and ferromagnet is equivalent to a delocalization transition between an insulator and a quantum Hall conductor. We establish the mapping as an exact and efficient tool for numerical analysis: using it, the computational effort required to study a system of width M is proportional to M3,{M}^{3},M3, and not exponential in M as with conventional algorithms. We show how the approach may be used to calculate for the RBIM the free energy, typical correlation lengths in quasi-one dimension for both the spin and the disorder operators, and the even powers of spin-spin correlation functions and their disorder averages. We examine in detail the square-lattice, nearest-neighbor $\ifmmode\pm\else\textpm\fi{}J$ RBIM, in which bonds are independently antiferromagnetic with probability p, and ferromagnetic with probability $1\ensuremath{-}p.$ Studying temperatures T> 0.4J,T>~0.4J,T> 0.4J, we obtain precise coordinates in the $p\ensuremath{-}T$ plane for points on the phase boundary between ferromagnet and paramagnet, and for the multicritical (Nishimori) point. We demonstrate scaling flow towards the pure Ising fixed point at small p, and determine critical exponents at the multicritical point.
We studied a new and simple method which gives successive approximate expressions for the magnetization (or any order parameter) of a random Ising model as a function of temperature, site, or bond dilution, frustration, magnetic field,... To obtain the various approximations a trace is performed on finite clusters increasing in size. In particular, we investigated a honeycomb Ising model with random nearest-neighbor interactions and calculated sequences of critical temperatures, percolation thresholds,... which converge towards exact known results.