Effect of Random Impurities on Fluctuation-Driven First Order Transitions

We analyse the effect of quenched uncorrelated randomness coupling to the local energy density of a model consisting of N coupled two-dimensional Ising models. For N>2 the pure model exhibits a fluctuation-driven first-order transition, characterized by runaway renormalization-group behaviour. We show that the addition of weak randomness acts to stabilize these flows, in such a way that the trajectories ultimately flow back towards the pure decoupled Ising fixed point, with the usual critical exponents , , apart from logarithmic corrections. We also show by examples that, in higher dimensions, such transitions may either become continuous or remain first-order in the presence of randomness.We analyse the effect of quenched uncorrelated randomness coupling to the local energy density of a model consisting of N coupled two-dimensional Ising models. For N>2 the pure model exhibits a fluctuation-driven first-order transition, characterized by runaway renormalization-group behaviour. We show that the addition of weak randomness acts to stabilize these flows, in such a way that the trajectories ultimately flow back towards the pure decoupled Ising fixed point, with the usual critical exponents , , apart from logarithmic corrections. We also show by examples that, in higher dimensions, such transitions may either become continuous or remain first-order in the presence of randomness.

arXiv:cond-mat/9511112Journal of Physics A Mathematical and GeneralMay 7, 199652 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

Ising universality in the two-dimensional Blume-Capel model with quenched random crystal field

Erol Vatansever, Zeynep Demir Vatansever, Panagiotis E. Theodorakis +1 more

Using high-precision Monte Carlo simulations based on a parallel version of the Wang-Landau algorithm and finite-size scaling techniques, we study the effect of quenched disorder in the crystal-field coupling of the Blume-Capel model on a square lattice. We mainly focus on the part of the phase diagram where the pure model undergoes a continuous transition, known to fall into the universality class of a pure Ising ferromagnet. A dedicated scaling analysis reveals concrete evidence in favor of the strong universality hypothesis with the presence of additional logarithmic corrections in the scaling of the specific heat. Our results are in agreement with an early real-space renormalization-group study of the model as well as a very recent numerical work where quenched randomness was introduced in the energy exchange coupling. Finally, by properly fine tuning the control parameters of the randomness distribution we also qualitatively investigate the part of the phase diagram where the pure model undergoes a first-order phase transition. For this region, preliminary evidence indicate a smoothing of the transition to second-order with the presence of strong scaling corrections.Using high-precision Monte Carlo simulations based on a parallel version of the Wang-Landau algorithm and finite-size scaling techniques, we study the effect of quenched disorder in the crystal-field coupling of the Blume-Capel model on a square lattice. We mainly focus on the part of the phase diagram where the pure model undergoes a continuous transition, known to fall into the universality class of a pure Ising ferromagnet. A dedicated scaling analysis reveals concrete evidence in favor of the strong universality hypothesis with the presence of additional logarithmic corrections in the scaling of the specific heat. Our results are in agreement with an early real-space renormalization-group study of the model as well as a very recent numerical work where quenched randomness was introduced in the energy exchange coupling. Finally, by properly fine tuning the control parameters of the randomness distribution we also qualitatively investigate the part of the phase diagram where the pure model undergoes a first-order phase transition. For this region, preliminary evidence indicate a smoothing of the transition to second-order with the presence of strong scaling corrections.

arXiv:2012.02437Physical review. EDec 22, 202017 citations

Effect of randomness in many coupled Potts models.

Pierre Pujol

Using 2-loop renormalisation group calculations, we study a system of $N$ two-dimensional Potts models with random bonds coupled together by their local energy density. This model can be seen as a generalization of the random Ashkin-Teller model. We found that, depending on the sign of the coupling term, the universality class of the system in the presence of randomness is different. Under particular consideration, this model presents an example of a first order phase transition rounded by randomness.Using 2-loop renormalisation group calculations, we study a system of $N$ two-dimensional Potts models with random bonds coupled together by their local energy density. This model can be seen as a generalization of the random Ashkin-Teller model. We found that, depending on the sign of the coupling term, the universality class of the system in the presence of randomness is different. Under particular consideration, this model presents an example of a first order phase transition rounded by randomness.

Jan 1, 19964 citations

Rounding of a first-order quantum phase transition to a strong-coupling critical point

Fawaz Y. Hrahsheh, José A. Hoyos, Thomas Vojta

We investigate the effects of quenched disorder on first-order quantum phase transitions on the example of the $N$ -color quantum Ashkin-Teller model. By means of a strong-disorder renormalization group, we demonstrate that quenched disorder rounds the first-order quantum phase transition to a continuous one for both weak and strong coupling between the colors. In the strong-coupling case, we find a distinct type of infinite-randomness critical point characterized by additional internal degrees of freedom. We investigate its critical properties in detail and find stronger thermodynamic singularities than in the random transverse field Ising chain. We also discuss the implications for higher spatial dimensions as well as unusual aspects of our renormalization-group scheme.We investigate the effects of quenched disorder on first-order quantum phase transitions on the example of the $N$-color quantum Ashkin-Teller model. By means of a strong-disorder renormalization group, we demonstrate that quenched disorder rounds the first-order quantum phase transition to a continuous one for both weak and strong coupling between the colors. In the strong-coupling case, we find a distinct type of infinite-randomness critical point characterized by additional internal degrees of freedom. We investigate its critical properties in detail and find stronger thermodynamic singularities than in the random transverse field Ising chain. We also discuss the implications for higher spatial dimensions as well as unusual aspects of our renormalization-group scheme.

arXiv:1208.0471Physical Review BDec 10, 201221 citations

Frozen Disorder in a Driven System

B. Schmittmann, Kevin E. Bassler

We investigate the effects of quenched disorder on the universal properties of a randomly driven Ising lattice gas. The Hamiltonian fixed point of the pure system becomes unstable in the presence of a quenched local bias, giving rise to a new fixed point which controls a novel universality class. We determine the associated scaling forms of correlation and response functions, quoting critical exponents to two-loop order in an expansion around the upper critical dimension ${d}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5$ .We investigate the effects of quenched disorder on the universal properties of a randomly driven Ising lattice gas. The Hamiltonian fixed point of the pure system becomes unstable in the presence of a quenched local bias, giving rise to a new fixed point which controls a novel universality class. We determine the associated scaling forms of correlation and response functions, quoting critical exponents to two-loop order in an expansion around the upper critical dimension ${d}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5$.

arXiv:cond-mat/9609166Physical Review LettersOct 21, 199610 citations

\\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.

Using 2-loop renormalisation group calculations, we study a system of

N

two-dimensional Potts models with random bonds coupled together by their local energy density. This model can be seen as a generalization of the random Ashkin-Teller model. We found that, depending on the sign of the coupling term, the universality class of the system in the presence of randomness is different. Under particular consideration, this model presents an example of a first order phase transition rounded by randomness.

We investigate the effects of quenched disorder on first-order quantum phase transitions on the example of the

N

-color quantum Ashkin-Teller model. By means of a strong-disorder renormalization group, we demonstrate that quenched disorder rounds the first-order quantum phase transition to a continuous one for both weak and strong coupling between the colors. In the strong-coupling case, we find a distinct type of infinite-randomness critical point characterized by additional internal degrees of freedom. We investigate its critical properties in detail and find stronger thermodynamic singularities than in the random transverse field Ising chain. We also discuss the implications for higher spatial dimensions as well as unusual aspects of our renormalization-group scheme.

{d}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5