# Inhomogeneous systems with unusual critical behaviour

TL;DR: The phase transitions and critical properties of two types of inhomogeneous systems are reviewed in this article, where the authors combine mean field theory, scaling considerations, conformal transformations and perturbation theory.

Abstract: The phase transitions and critical properties of two types of inhomogeneous systems are reviewed. In one case, the local critical behaviour results from the particular shape of the system. Here scale-invariant forms like wedges or cones are considered as well as general parabolic shapes. In the other case the system contains defects, either narrow ones in the form of lines or stars, or extended ones where the couplings deviate from their bulk values according to power laws. In each case the perturbation may be irrelevant, marginal or relevant. In the marginal case one finds local exponents which depend on a parameter. In the relevant case unusual stretched exponential behaviour and/or local first-order transitions appear. The discussion combines mean field theory, scaling considerations, conformal transformations and perturbation theory. A number of examples are Ising models for which exact results can be obtained. Some walks and polymer problems are considered, too.

##### Citations

More filters

••

TL;DR: In this article, a review of recent developments in non-equilibrium statistical physics is presented, focusing on phase transitions from fluctuating phases into absorbing states, the universality class of directed percolation is investigated in detail.

Abstract: This review addresses recent developments in non-equilibrium statistical physics. Focusing on phase transitions from fluctuating phases into absorbing states, the universality class of directed percolation is investigated in detail. The survey gives a general introduction to various lattice models of directed percolation and studies their scaling properties, field-theoretic aspects, numerical techniques, as well as possible experimental realizations. In addition, several examples of absorbing-state transitions which do not belong to the directed percolation universality class will be discussed. As a closely related technique, we investigate the concept of damage spreading. It is shown that this technique is ambiguous to some extent, making it impossible to define chaotic and regular phases in stochastic non-equilibrium systems. Finally, we discuss various classes of depinning transitions in models for interface growth which are related to phase transitions into absorbing states.

1,475 citations

••

TL;DR: In this article, a review of recent developments in nonequilibrium statistical physics is presented, focusing on phase transitions from fluctuating phases into absorbing states, and several examples of absorbing-state transitions which do not belong to the directed percolation universality class are discussed.

Abstract: This review addresses recent developments in nonequilibrium statistical physics. Focusing on phase transitions from fluctuating phases into absorbing states, the universality class of directed percolation is investigated in detail. The survey gives a general introduction to various lattice models of directed percolation and studies their scaling properties, field-theoretic aspects, numerical techniques, as well as possible experimental realizations. In addition, several examples of absorbing-state transitions which do not belong to the directed percolation universality class will be discussed. As a closely related technique, we investigate the concept of damage spreading. It is shown that this technique is ambiguous to some extent, making it impossible to define chaotic and regular phases in stochastic nonequilibrium systems. Finally, we discuss various classes of depinning transitions in models for interface growth which are related to phase transitions into absorbing states.

1,081 citations

••

TL;DR: In this article, the authors present a review of universality classes in nonequilibrium systems defined on regular lattices and discuss the most important critical exponents and relations, as well as the field-theoretical formalism used in the text.

Abstract: This article reviews our present knowledge of universality classes in nonequilibrium systems defined on regular lattices. The first section presents the most important critical exponents and relations, as well as the field-theoretical formalism used in the text. The second section briefly addresses the question of scaling behavior at first-order phase transitions. In Sec. III the author looks at dynamical extensions of basic static classes, showing the effects of mixing dynamics and of percolation. The main body of the review begins in Sec. IV, where genuine, dynamical universality classes specific to nonequilibrium systems are introduced. Section V considers such nonequilibrium classes in coupled, multicomponent systems. Most of the known nonequilibrium transition classes are explored in low dimensions between active and absorbing states of reaction-diffusion-type systems. However, by mapping they can be related to the universal behavior of interface growth models, which are treated in Sec. VI. The review ends with a summary of the classes of absorbing-state and mean-field systems and discusses some possible directions for future research.

698 citations

••

TL;DR: The control of internal and motional quantum degrees of freedom of laser-cooled trapped ions has been subject to intense theoretical and experimental research for about three decades as discussed by the authors, which may be employed to investigate physical models conceived to describe systems that are not directly accessible for experimental investigations.

Abstract: The control of internal and motional quantum degrees of freedom of laser-cooled trapped ions has been subject to intense theoretical and experimental research for about three decades. In the realm of quantum information science, the ability to deterministically prepare and measure quantum states of trapped ions is unprecedented. This expertise may be employed to investigate physical models conceived to describe systems that are not directly accessible for experimental investigations. Here, we give an overview of current theoretical proposals and experiments for such quantum simulations with trapped ions. This includes various spin models (e.g. the quantum transverse Ising model or a neural network), the Bose?Hubbard Hamiltonian, the Frenkel?Kontorova model, and quantum fields and relativistic effects.

209 citations

••

TL;DR: In this paper, the authors generalized the scattering theory of the integrable statistical models to the case of systems with extended lines of defect, by adding the reflection and transmission amplitudes for the interactions with the line of inhomogeneity to the scattering amplitudes in the bulk.

168 citations

##### References

More filters

•

01 Jan 1972

TL;DR: The field of phase transitions and critical phenomena continues to be active in research, producing a steady stream of interesting and fruitful results as discussed by the authors, and the major aim of this serial is to provide review articles that can serve as standard references for research workers in the field.

Abstract: The field of phase transitions and critical phenomena continues to be active in research, producing a steady stream of interesting and fruitful results. It has moved into a central place in condensed matter studies. Statistical physics, and more specifically, the theory of transitions between states of matter, more or less defines what we know about 'everyday' matter and its transformations. The major aim of this serial is to provide review articles that can serve as standard references for research workers in the field, and for graduate students and others wishing to obtain reliable information on important recent developments.

12,039 citations

••

Yale University

^{1}TL;DR: In this article, the eigenwert problem involved in the corresponding computation for a long strip crystal of finite width, joined straight to itself around a cylinder, is solved by direct product decomposition; in the special case $n=\ensuremath{\infty}$ an integral replaces a sum.

Abstract: The partition function of a two-dimensional "ferromagnetic" with scalar "spins" (Ising model) is computed rigorously for the case of vanishing field. The eigenwert problem involved in the corresponding computation for a long strip crystal of finite width ($n$ atoms), joined straight to itself around a cylinder, is solved by direct product decomposition; in the special case $n=\ensuremath{\infty}$ an integral replaces a sum. The choice of different interaction energies ($\ifmmode\pm\else\textpm\fi{}J,\ifmmode\pm\else\textpm\fi{}{J}^{\ensuremath{'}}$) in the (0 1) and (1 0) directions does not complicate the problem. The two-way infinite crystal has an order-disorder transition at a temperature $T={T}_{c}$ given by the condition $sinh(\frac{2J}{k{T}_{c}}) sinh(\frac{2{J}^{\ensuremath{'}}}{k{T}_{c}})=1.$ The energy is a continuous function of $T$; but the specific heat becomes infinite as $\ensuremath{-}log |T\ensuremath{-}{T}_{c}|$. For strips of finite width, the maximum of the specific heat increases linearly with $log n$. The order-converting dual transformation invented by Kramers and Wannier effects a simple automorphism of the basis of the quaternion algebra which is natural to the problem in hand. In addition to the thermodynamic properties of the massive crystal, the free energy of a (0 1) boundary between areas of opposite order is computed; on this basis the mean ordered length of a strip crystal is ${(\mathrm{exp} (\frac{2J}{\mathrm{kT}}) tanh(\frac{2{J}^{\ensuremath{'}}}{\mathrm{kT}}))}^{n}.$

5,081 citations

••

IBM

^{1}TL;DR: In this article, two genuinely quantum models for an antiferromagnetic linear chain with nearest neighbor interactions are constructed and solved exactly, in the sense that the ground state, all the elementary excitations and the free energy are found.

3,382 citations

••

TL;DR: In this paper, a quantenmechanik der Gasentartung is discussed, e.g., ein ideales oder nichtideales Gase, der Paulischen Aquivalenzverbot unterworfenes Gas zu beschreiben with begriffen, die keinen Bezug nehmen auf den abstrakten Koordinatenraum der Atomgesamtheit des Gases, sondern nur den gewohnlichen dreidimensionalen Raum benut

Abstract: Die Arbeit enthalt eine Fortsetzung der kurzlich von einem der Verfasser vorgelegten Note „Zur Quantenmechanik der Gasentartung“, deren Ergebnisse hier wesentlich erweitert werden. Es handelt sich darum, ein ideales oder nichtideales, dem Paulischen Aquivalenzverbot unterworfenes Gas zu beschreiben mit Begriffen, die keinen Bezug nehmen auf den abstrakten Koordinatenraum der Atomgesamtheit des Gases, sondern nur den gewohnlichen dreidimensionalen Raum benutzen. Das wird ermoglicht durch die Darstellung des Gases vermittelst eines gequantelten dreidimensionalen Wellenfeldes, wobei die besonderen nichtkommutativen Multiplikationseigenschaften der Wellenamplitude gleichzeitig fur die Existenz korpus-kularer Gasatome und fur die Gultigkeit des Paulischen Aquivalenzverbots verantwortlich sind. Die Einzelheiten der Theorie besitzen enge Analogien zu der entsprechenden Theorie fur Einsteinsche ideale oder nichtideale Gase, wie sie von Dirac, Klein und Jordan ausgefuhrt wurde.

1,971 citations