© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Journal of Physics C Solid State Physics

One-dimensional Ising system with random long-ranged interactions

Alba Theumann

Published Nov 14, 1978• 0 citations

Abstract

The author considers an Ising chain where the random interaction matrix has the symmetrised form J=PV+VP. The matrix elements of V and P are V ij =exp(- gamma mod i-j mod ) and P ij =P i delta j,i-1 +P i+1 delta j,i+1 , where i, j indicate sites on the chain. The quantities P i are independent random variables with the same probability distribution. Their mean and variance are fixed to be (P i )=J 0 gamma /4; (P i 2 )-(P i ) 2 =J 2 gamma /4, where gamma -1 is the range of the interaction. The author then obtains Sigma j (J ij )=2J 0 , Sigma j (J ij 2 )=J 2 in the limit gamma to 0, as in a model Hamiltonian that describes a spin-glass transition. The author evaluates exactly in this limit the density of eigenvalues of the random matrix PV. An integral equation for the distribution probability of new local random variables is derived by using the continued fraction method and the asymptotic solution in the limit gamma to 0 is obtained analytically. The largest eigenvalue of PV is found to be J 0 independent of J. However the largest eigenvalue of J=PV+VP is found to be mu M =J 0 +(J 0 2 +NJ 2 ) 1 2/, when gamma =0 and N is the total number of sites. An independent evaluation of the ground state energy shows that E 0 <-J square root NN, hence E 0 is not extensive. These results show that the scaling of the variance of the random interactions as J 2 N -1 gives highly unphysical results for the present model.

DOI: 10.1088/0022-3719/11/21/013

Related papers

One-dimensional Ising spin-glass model with long-range interactions

M A Moore

An Ising chain with Hamiltonian H=-J Sigma i<j epsilon ij S i S j / mod i-j mod (1+ sigma )/2 is considered where the epsilon ij are independent random variables. According to Kotliar et al. (1983) this model has a phrase transition with non-mean-field exponents when 1 / 3 < sigma <1 and mean-field exponents beyond the upper critical value of sigma = 1 / 3 . By means of an epsilon expansion about the lower critical value of sigma =1, the model is shown to be replica symmetric as epsilon to 0 and it is speculated that this result holds for 1 / 3 < sigma <1.

Journal of Physics A Mathematical and General•Mar 11, 1986•11 citations

Model of a quasi-one-dimensional spin glass

Matej Orešič, R. Pirc

We consider a system of linear chains of Ising spins with short-range nearest-neighbor ferromagnetic interactions of strength K along the chains, and with infinite-range random interchain interactions of a Sherrington-Kirkpatrick type. Also included in the model is a local Gaussian random field with variance \ensuremath{\Delta}. In the replica symmetric case, the system is mapped exactly on the one-dimensional Ising model in a random field and a recursion method for calculating the spin-correlation functions is developed. For \ensuremath{\Delta}=0, an analytic relation for freezing temperature ${\mathit{T}}_{\mathit{f}}$(K) is obtained. The replica-symmetric spin-glass phase is shown to be stable above the freezing temperature ${\mathit{T}}_{\mathit{f}}$(K,\ensuremath{\Delta}), which is determined numerically.

Physical review. B, Condensed matter•Feb 1, 1993•16 citations

One-dimensional spin-glass model with long-range random interactions

Gabriel Kotliar, Philip Warren Anderson, D. L. Stein

We consider an Ising chain with Hamiltonian $H=\ensuremath{\Sigma}{i>j}^{}\frac{({\ensuremath{\epsilon}}_{\mathrm{ij}}{S}_{i}{S}_{j})}{{(a|i\ensuremath{-}j|)}^{\ensuremath{\sigma}}}$, where the ${\ensuremath{\epsilon}}_{\mathrm{ij}}$ are independent random variables. We find a phase transition for $\frac{1}{2}<\ensuremath{\sigma}<1$. For $\frac{1}{2}<\ensuremath{\sigma}<\frac{2}{3}$ the critical exponents exhibit mean-field classical behavior. Near $\ensuremath{\sigma}=1$ we find a smoothly varying specific heat. We investigate the critical behavior near the upper and lower critical range by means of an $\ensuremath{\epsilon}$ expansion around $\ensuremath{\sigma}=1 \mathrm{and} \frac{2}{3}$.

Physical review. B, Condensed matter•Jan 1, 1983•151 citations

A Numerical Investigation of a Certain One-Dimensional Ising Model

Jimmy Green

The model is a linear chain in which each spin interacts with its 2r nearest neighbors, the interaction energy being proportional to 1/r. Using a method similar to that of Montroll, the partition function of the model in the thermodynamic limit is shown to be related to the largest eigenvalue of a certain matrix. The largest eigenvalue of the matrix is determined numerically for 3 < r < 12. Also, a correct method is demonstrated for evaluating an improper limit of the model, in which the interaction range is set to the chain length before the limit of an infinite number of spins is taken. Previously published works have performed this calculation incorrectly. Although the numerical results show some evidence of convergence to the improper limit, the results are inconclusive and furthermore raise doubts about the practicality of the numerical method in this context.

•Jan 1, 1981•0 citations

Replica symmetry in a one-dimensional spin-glass model with long-range interactions

Anil Khurana

In the replica formalism for an Ising spin glass the interaction energy between two sites depends on the number of replicas of the two sites that have parallel or antiparallel spins; therefore, the energy may take n+1 different values even when permutation symmetry is assumed in the space of n replicas. This insight is used to propose a new replica-symmetric solution to the renormalization-group equations for a one-dimensional spin-glass model with long-range interactions. In a simple approximation this solution gives a value of (2\ensuremath{\epsilon}${)}^{\mathrm{\ensuremath{-}}1/2}$ for the correlation-length exponent [\ensuremath{\epsilon}=1-\ensuremath{\sigma} and (1+\ensuremath{\sigma})/2 is the range parameter for the random interactions]. The new solution is argued to be equivalent to the replica-symmetric ansatz for the Sherrington-Kirkpatrick model.

Physical review. B, Condensed matter•Aug 1, 1989•4 citations

Renormalisation group study of the random Ising model in a transverse field in one dimension

Katarina Uzelac, Rémi Jullien, Pierre M. Pfeuty

The random Ising model in a transverse field in one dimension with Hamiltonian H=- Sigma ( Gamma i S i z +J i S i x S i+1 x ) is studied at T=O from a real-space renormalisation group block method which preserves duality transformations. The ground state magnetisation and the ground state energy are determined for random distributions P(J)(P( Gamma ))=N J J NJ-1 /J 0 N(J) for 0<J<J 0 and P(J)=0 for J>J O . A new critical behaviour corresponding to a new fixed point takes place in the presence of disorder. The crossover exponent describing the departure from the pure system behaviour is calculated. The second derivative of the ground state energy delta 2 E/ delta Gamma 2 which diverges logarithmically for the pure system is rounded in the presence of disorder but a sharp transition field still exists where the magnetisation goes to zero with an exponent beta about twice as large as the pure system exponent. Comparison is made with the analytical results of McCoy and Wu (1969) for the classical equivalent random-striped Ising 2D model.

Journal of Physics A Mathematical and General•Dec 1, 1980•23 citations

The author considers an Ising chain where the random interaction matrix has the symmetrised form J=PV+VP. The matrix elements of V and P are V ij =exp(- gamma mod i-j mod ) and P ij =P i delta j,i-1 +P i+1 delta j,i+1 , where i, j indicate sites on the chain. The quantities P i are independent random variables with the same probability distribution. Their mean and variance are fixed to be (P i )=J 0 gamma /4; (P i 2 )-(P i ) 2 =J 2 gamma /4, where gamma -1 is the range of the interaction. The author then obtains Sigma j (J ij )=2J 0 , Sigma j (J ij 2 )=J 2 in the limit gamma to 0, as in a model Hamiltonian that describes a spin-glass transition. The author evaluates exactly in this limit the density of eigenvalues of the random matrix PV. An integral equation for the distribution probability of new local random variables is derived by using the continued fraction method and the asymptotic solution in the limit gamma to 0 is obtained analytically. The largest eigenvalue of PV is found to be J 0 independent of J. However the largest eigenvalue of J=PV+VP is found to be mu M =J 0 +(J 0 2 +NJ 2 ) 1 2/, when gamma =0 and N is the total number of sites. An independent evaluation of the ground state energy shows that E 0 <-J square root NN, hence E 0 is not extensive. These results show that the scaling of the variance of the random interactions as J 2 N -1 gives highly unphysical results for the present model.

An Ising chain with Hamiltonian H=-J Sigma i<j epsilon ij S i S j / mod i-j mod (1+ sigma )/2 is considered where the epsilon ij are independent random variables. According to Kotliar et al. (1983) this model has a phrase transition with non-mean-field exponents when 1 / 3 < sigma <1 and mean-field exponents beyond the upper critical value of sigma = 1 / 3 . By means of an epsilon expansion about the lower critical value of sigma =1, the model is shown to be replica symmetric as epsilon to 0 and it is speculated that this result holds for 1 / 3 < sigma <1.

We consider a system of linear chains of Ising spins with short-range nearest-neighbor ferromagnetic interactions of strength K along the chains, and with infinite-range random interchain interactions of a Sherrington-Kirkpatrick type. Also included in the model is a local Gaussian random field with variance \ensuremath{\Delta}. In the replica symmetric case, the system is mapped exactly on the one-dimensional Ising model in a random field and a recursion method for calculating the spin-correlation functions is developed. For \ensuremath{\Delta}=0, an analytic relation for freezing temperature

{\mathit{T}}_{\mathit{f}}

(K) is obtained. The replica-symmetric spin-glass phase is shown to be stable above the freezing temperature

{\mathit{T}}_{\mathit{f}}

(K,\ensuremath{\Delta}), which is determined numerically.

We consider an Ising chain with Hamiltonian $H=\ensuremath{\Sigma}{i>j}^{}\frac{({\ensuremath{\epsilon}}_{\mathrm{ij}}{S}_{i}{S}_{j})}{{(a|i\ensuremath{-}j|)}^{\ensuremath{\sigma}}}$, where the ${\ensuremath{\epsilon}}_{\mathrm{ij}}$ are independent random variables. We find a phase transition for $\frac{1}{2}<\ensuremath{\sigma}<1$. For $\frac{1}{2}<\ensuremath{\sigma}<\frac{2}{3}$ the critical exponents exhibit mean-field classical behavior. Near $\ensuremath{\sigma}=1$ we find a smoothly varying specific heat. We investigate the critical behavior near the upper and lower critical range by means of an $\ensuremath{\epsilon}$ expansion around $\ensuremath{\sigma}=1 \mathrm{and} \frac{2}{3}$.

The model is a linear chain in which each spin interacts with its 2r nearest neighbors, the interaction energy being proportional to 1/r. Using a method similar to that of Montroll, the partition function of the model in the thermodynamic limit is shown to be related to the largest eigenvalue of a certain matrix. The largest eigenvalue of the matrix is determined numerically for 3 < r < 12. Also, a correct method is demonstrated for evaluating an improper limit of the model, in which the interaction range is set to the chain length before the limit of an infinite number of spins is taken. Previously published works have performed this calculation incorrectly. Although the numerical results show some evidence of convergence to the improper limit, the results are inconclusive and furthermore raise doubts about the practicality of the numerical method in this context.

In the replica formalism for an Ising spin glass the interaction energy between two sites depends on the number of replicas of the two sites that have parallel or antiparallel spins; therefore, the energy may take n+1 different values even when permutation symmetry is assumed in the space of n replicas. This insight is used to propose a new replica-symmetric solution to the renormalization-group equations for a one-dimensional spin-glass model with long-range interactions. In a simple approximation this solution gives a value of (2\ensuremath{\epsilon}${)}^{\mathrm{\ensuremath{-}}1/2}$ for the correlation-length exponent [\ensuremath{\epsilon}=1-\ensuremath{\sigma} and (1+\ensuremath{\sigma})/2 is the range parameter for the random interactions]. The new solution is argued to be equivalent to the replica-symmetric ansatz for the Sherrington-Kirkpatrick model.

The random Ising model in a transverse field in one dimension with Hamiltonian H=- Sigma ( Gamma i S i z +J i S i x S i+1 x ) is studied at T=O from a real-space renormalisation group block method which preserves duality transformations. The ground state magnetisation and the ground state energy are determined for random distributions P(J)(P( Gamma ))=N J J NJ-1 /J 0 N(J) for 0<J<J 0 and P(J)=0 for J>J O . A new critical behaviour corresponding to a new fixed point takes place in the presence of disorder. The crossover exponent describing the departure from the pure system behaviour is calculated. The second derivative of the ground state energy delta 2 E/ delta Gamma 2 which diverges logarithmically for the pure system is rounded in the presence of disorder but a sharp transition field still exists where the magnetisation goes to zero with an exponent beta about twice as large as the pure system exponent. Comparison is made with the analytical results of McCoy and Wu (1969) for the classical equivalent random-striped Ising 2D model.