We study the effects which immobile random impurities may have on magnetic properties and spin-spin correlation functions of a ferromagnetic system properties the Curie temperature. This is done within the context of the model introduced in the first paper of this series by computing the spin correlation functions on the boundary of a half-plane of random Ising spins, where the boundary row is allowed to intereact with a magnetic field $\mathfrak{H}$ . We find that as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$ the average boundary spontaneous magnetization vanishes as ${T}_{c}\ensuremath{-}T$. However, the average boundary magnetic susceptibility is shown not to exist for a finite range of temperature about ${T}_{c}$ . Furthermore, at ${T}_{c}$ the average boundary magnetization behaves as $\ensuremath{-}{N}^{\ensuremath{-}1}$ sign $(\mathfrak{H}){[\mathrm{ln} N{\ensuremath{\beta}}_{c}|\mathfrak{H}|]}^{\ensuremath{-}1}$, where ${N}^{\ensuremath{-}1}$ is a measure of the width of the distribution of random bonds. Whenever $T\ensuremath{-}{T}_{c}=O({N}^{\ensuremath{-}2})$, the average spin-spin correlation function for two spins on the boundary is shown to approach its limit at infinite separation as some inverse power of the separation instead of as an exponential. At ${T}_{c}$ this average correlation function behaves asympotically as ${N}^{\ensuremath{-}2} {(\mathrm{ln}m{N}^{\ensuremath{-}2})}^{\ensuremath{-}1}$ when $m$ the separation between boundary spins, is large. Finally, we make the probabilistic nature of the boundary spontaneous magnetization more precise by computing its probability distribution function.We study the effects which immobile random impurities may have on magnetic properties and spin-spin correlation functions of a ferromagnetic system properties the Curie temperature. This is done within the context of the model introduced in the first paper of this series by computing the spin correlation functions on the boundary of a half-plane of random Ising spins, where the boundary row is allowed to intereact with a magnetic field $\mathfrak{H}$. We find that as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$ the average boundary spontaneous magnetization vanishes as ${T}_{c}\ensuremath{-}T$. However, the average boundary magnetic susceptibility is shown not to exist for a finite range of temperature about ${T}_{c}$. Furthermore, at ${T}_{c}$ the average boundary magnetization behaves as $\ensuremath{-}{N}^{\ensuremath{-}1}$ sign $(\mathfrak{H}){[\mathrm{ln} N{\ensuremath{\beta}}_{c}|\mathfrak{H}|]}^{\ensuremath{-}1}$, where ${N}^{\ensuremath{-}1}$ is a measure of the width of the distribution of random bonds. Whenever $T\ensuremath{-}{T}_{c}=O({N}^{\ensuremath{-}2})$, the average spin-spin correlation function for two spins on the boundary is shown to approach its limit at infinite separation as some inverse power of the separation instead of as an exponential. At ${T}_{c}$ this average correlation function behaves asympotically as ${N}^{\ensuremath{-}2} {(\mathrm{ln}m{N}^{\ensuremath{-}2})}^{\ensuremath{-}1}$ when $m$ the separation between boundary spins, is large. Finally, we make the probabilistic nature of the boundary spontaneous magnetization more precise by computing its probability distribution function.

Physical ReviewDec 10, 1969118 citations

Critical Properties of Ising Models Containing Dilute Impurities. II

K. Sawada, Takeo Osawa

Progress of Theoretical PhysicsOct 1, 19733 citations

Relaxation times in a finite Ising system with random impurities

Klaus Oerding

Journal of Statistical PhysicsFeb 1, 199513 citations

Effect of disorder on first-order phase transitions

Arash Bellafard

Disorder is an inevitable part of any condensed matter system and therefore its study has always been of great importance. The effect of quenched randomness on a system that exhibits a continuous phase transition in the absence of any impurity has been studied in the past and the results are relatively well understood. However, the effect of quenched randomness on \\emph{first-order} phase transitions is still not well understood. In this dissertation, we study the effect of quenched bond-randomness on the classical and quantum first-order phase transitions.In Chapter 2, we study the effect of the disordered three-color Ashkin-Teller model, whose pure version undergoes a first-order phase transition. We show that the rounding of the first-order transition of the pure model due to the impurities is manifested as a critical point. We conclusively rule out that the model belongs to the universality class of the two-dimensional Ising model. Furthermore, we find that the exponents $\\beta$ and $\\nu$ vary with disorder and the four-spin coupling strength.In Chapter 3, we extend our study of the disordered three-color Ashkin-Teller model. Utilizing extensive cluster Monte Carlo simulations on large lattice sizes of up to $128 \\times 128$ spins, each of which is represented by three colors taking values $\\pm 1$ , we show that the rounding of the first-order phase transition is an emergent criticality.We find that the critical exponents, $\\nu$ and $\\beta$ , change as the strength of disorder or the four-spin coupling varies, and we show that the correlation length critical exponent violates the lower bound $2/D \\le \\nu$ , where $D$ is the dimension of the system.In Chapter 4, we study the \\emph{quantum} three-color Ashkin--Teller model and show that the quantum critical point in $(1+1)$ dimension is an unusual one, with activated scaling at the critical point and Griffiths-McCoy phase away from it. We find that the behavior is similar to the transverse random field Ising model, even though the pure system has a first-order transition in this case.Disorder is an inevitable part of any condensed matter system and therefore its study has always been of great importance. The effect of quenched randomness on a system that exhibits a continuous phase transition in the absence of any impurity has been studied in the past and the results are relatively well understood. However, the effect of quenched randomness on \\emph{first-order} phase transitions is still not well understood. In this dissertation, we study the effect of quenched bond-randomness on the classical and quantum first-order phase transitions.In Chapter 2, we study the effect of the disordered three-color Ashkin-Teller model, whose pure version undergoes a first-order phase transition. We show that the rounding of the first-order transition of the pure model due to the impurities is manifested as a critical point. We conclusively rule out that the model belongs to the universality class of the two-dimensional Ising model. Furthermore, we find that the exponents $\\beta$ and $\\nu$ vary with disorder and the four-spin coupling strength.In Chapter 3, we extend our study of the disordered three-color Ashkin-Teller model. Utilizing extensive cluster Monte Carlo simulations on large lattice sizes of up to $128 \\times 128$ spins, each of which is represented by three colors taking values $\\pm 1$, we show that the rounding of the first-order phase transition is an emergent criticality.We find that the critical exponents, $\\nu$ and $\\beta$, change as the strength of disorder or the four-spin coupling varies, and we show that the correlation length critical exponent violates the lower bound $2/D \\le \\nu$, where $D$ is the dimension of the system.In Chapter 4, we study the \\emph{quantum} three-color Ashkin--Teller model and show that the quantum critical point in $(1+1)$ dimension is an unusual one, with activated scaling at the critical point and Griffiths-McCoy phase away from it. We find that the behavior is similar to the transverse random field Ising model, even though the pure system has a first-order transition in this case.

Jan 1, 20150 citations

On the boundary Ising model with disorder operators

G.M.T. Watts

arXiv:hep-th/0002218Nuclear Physics BMar 1, 20019 citations

\mathfrak{H}

. We find that as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$ the average boundary spontaneous magnetization vanishes as ${T}_{c}\ensuremath{-}T$. However, the average boundary magnetic susceptibility is shown not to exist for a finite range of temperature about

{T}_{c}

. Furthermore, at

{T}_{c}

the average boundary magnetization behaves as $\ensuremath{-}{N}^{\ensuremath{-}1}$ sign $(\mathfrak{H}){[\mathrm{ln} N{\ensuremath{\beta}}_{c}|\mathfrak{H}|]}^{\ensuremath{-}1}$, where ${N}^{\ensuremath{-}1}$ is a measure of the width of the distribution of random bonds. Whenever $T\ensuremath{-}{T}_{c}=O({N}^{\ensuremath{-}2})$, the average spin-spin correlation function for two spins on the boundary is shown to approach its limit at infinite separation as some inverse power of the separation instead of as an exponential. At

{T}_{c}

this average correlation function behaves asympotically as ${N}^{\ensuremath{-}2} {(\mathrm{ln}m{N}^{\ensuremath{-}2})}^{\ensuremath{-}1}$ when

m

the separation between boundary spins, is large. Finally, we make the probabilistic nature of the boundary spontaneous magnetization more precise by computing its probability distribution function.

\\beta

and

\\nu

vary with disorder and the four-spin coupling strength.In Chapter 3, we extend our study of the disordered three-color Ashkin-Teller model. Utilizing extensive cluster Monte Carlo simulations on large lattice sizes of up to

128 \\times 128

spins, each of which is represented by three colors taking values

\\pm 1

, we show that the rounding of the first-order phase transition is an emergent criticality.We find that the critical exponents,

\\nu

and

\\beta

, change as the strength of disorder or the four-spin coupling varies, and we show that the correlation length critical exponent violates the lower bound

2/D \\le \\nu

, where

D

is the dimension of the system.In Chapter 4, we study the \\emph{quantum} three-color Ashkin--Teller model and show that the quantum critical point in

(1+1)

dimension is an unusual one, with activated scaling at the critical point and Griffiths-McCoy phase away from it. We find that the behavior is similar to the transverse random field Ising model, even though the pure system has a first-order transition in this case.