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172 KiB

Open AccessArticle

Symmetry, Symmetry Breaking and Topology

by Siddhartha Sen

Symmetry 2010, 2(3), 1401-1422; https://doi.org/10.3390/sym2031401 - 07 Jul 2010

Cited by 2 | Viewed by 6528

The ground state of a system with symmetry can be described by a group G. This symmetry group G can be discrete or continuous. Thus for a crystal G is a finite group while for the vacuum state of a grand unified [...] Read more.

The ground state of a system with symmetry can be described by a group G. This symmetry group G can be discrete or continuous. Thus for a crystal G is a finite group while for the vacuum state of a grand unified theory G is a continuous Lie group. The ground state symmetry described by G can change spontaneously from G to one of its subgroups H as the external parameters of the system are modified. Such a macroscopic change of the ground state symmetry of a system from G to H correspond to a “phase transition”. Such phase transitions have been extensively studied within a framework due to Landau. A vast range of systems can be described using Landau’s approach, however there are also systems where the framework does not work. Recently there has been growing interest in looking at such non-Landau type of phase transitions. For instance there are several “quantum phase transitions” that are not of the Landau type. In this short review we first describe a refined version of Landau’s approach in which topological ideas are used together with group theory. The combined use of group theory and topological arguments allows us to determine selection rule which forbid transitions from G to certain of its subgroups. We end by making a few brief remarks about non-Landau type of phase transition. Full article

(This article belongs to the Special Issue Symmetry Breaking Phenomena)

186 KiB

Open AccessArticle

Study of Dynamical Chiral Symmetry Breaking in (2 + 1) Dimensional Abelian Higgs Model

by Jian-Feng Li, Shi-Song Huang, Hong-Tao Feng, Wei-Min Sun and Hong-Shi Zong

Symmetry 2010, 2(2), 907-915; https://doi.org/10.3390/sym2020907 - 19 Apr 2010

Viewed by 7245

Abstract

In this paper, we study the dynamical mass generation in the Abelian Higgs model in 2 + 1 dimensions. Instead of adopting the approximations in [Jiang H et al., J. Phys. A 41 2008 255402.], we numerically solve the coupled Dyson–Schwinger Equations [...] Read more.

In this paper, we study the dynamical mass generation in the Abelian Higgs model in 2 + 1 dimensions. Instead of adopting the approximations in [Jiang H et al., J. Phys. A 41 2008 255402.], we numerically solve the coupled Dyson–Schwinger Equations (DSEs) for the fermion and gauge boson propagators using a specific truncation for the fermion-photon vertex ansatz and compare our results with the corresponding ones in the above mentioned paper. It is found that the results quoted in the above paper remain qualitatively unaffected by refining the truncation scheme of the DSEs, although there exist large quantitative differences between the results presented in the above paper and ours. In addition, our numerical results show that the critical number of fermion flavor N_c decreases steeply with the the gauge boson mass m_a (or the ratio of the Higgs mass m_h to the gauge boson mass m_a, r = m_h/m_a) increasing. It is thus easier to generate a finite fermion mass by the mechanism of DCSB for a small ratio r for a given m_a. Full article

(This article belongs to the Special Issue Symmetry Breaking Phenomena)

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937 KiB

Open AccessArticle

Hartee Fock Symmetry Breaking Effects in La₂CuO₄: Hints for connecting the Mott and Slater Pictures and Pseudogap Prediction

by Alejandro Cabo-Bizet and Alejandro Cabo Montes De Oca

Symmetry 2010, 2(1), 388-417; https://doi.org/10.3390/sym2010388 - 22 Mar 2010

Cited by 6 | Viewed by 6114

Abstract

This work expands the results and derivations presented in a recent letter. It is argued that symmetry breaking Hartree-Fock (HF) solutions of a simple model of the Cu-O planes in La₂CuO₄, are able to describe the insulator and antiferromagnetic [...] Read more.

This work expands the results and derivations presented in a recent letter. It is argued that symmetry breaking Hartree-Fock (HF) solutions of a simple model of the Cu-O planes in La₂CuO₄, are able to describe the insulator and antiferromagnetic characters of this material. Then, this classical primer of a Mott insulator is alternatively obtained here as an exact Slater insulator within the simplest of the first principles schemes. Moreover, pseudogap HF states are also predicted. The maximal energy gap of 100 meV over the Fermi surface of this wavefunction, reasonably well matches the ARPES upper pseudogap measurements for La₂CuO₄ in the zero doping limit. These surprising results followed after eliminating spin and crystal symmetry constraints usually imposed on the HF orbitals. The discussion helps to clarify the role of the antiferromagnetism and pseudogaps in the physics of the HTSC materials and indicates a promising way to start conciliating the Mott and Slater pictures for the description of the transition metal oxides. Full article

(This article belongs to the Special Issue Symmetry Breaking Phenomena)

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225 KiB

Open AccessArticle

Systems with Symmetry Breaking and Restoration

by Vyacheslav I. Yukalov

Symmetry 2010, 2(1), 40-68; https://doi.org/10.3390/sym2010040 - 11 Jan 2010

Cited by 14 | Viewed by 6562

Abstract

Statistical systems, in which spontaneous symmetry breaking can be accompanied by spontaneous local symmetry restoration, are considered. A general approach to describing such systems is formulated, based on the notion of weighted Hilbert spaces and configuration averaging. The approach is illustrated by the [...] Read more.

Statistical systems, in which spontaneous symmetry breaking can be accompanied by spontaneous local symmetry restoration, are considered. A general approach to describing such systems is formulated, based on the notion of weighted Hilbert spaces and configuration averaging. The approach is illustrated by the example of a ferroelectric with mesoscopic fluctuations of paraelectric phase. The influence of the local symmetry restoration on the system characteristics, such as sound velocity and Debye-Waller factor, is discussed. Full article

(This article belongs to the Special Issue Symmetry Breaking Phenomena)

210 KiB

Open AccessArticle

Phase Diffusion of a q-Deformed Oscillator

by Turan Birol and Özgür Esat Müstecaplıoğlu

Symmetry 2009, 1(2), 240-251; https://doi.org/10.3390/sym1020240 - 21 Dec 2009

Cited by 3 | Viewed by 6685

Abstract

We examine the problem of phase diffusion rate in a U(1) global phase symmetry broken system, from the perspective of q-deformed oscillators where the deformation parameter represents the anharmonicity. It is shown that broken phase symmetry states, described by deformed coherent states, suffer [...] Read more.

We examine the problem of phase diffusion rate in a U(1) global phase symmetry broken system, from the perspective of q-deformed oscillators where the deformation parameter represents the anharmonicity. It is shown that broken phase symmetry states, described by deformed coherent states, suffer phase diffusion at a rate determined by the deformation parameter. Analytical discussions are given for the case of weak deformations, while detailed numerical results are presented when strong anharmonicity is present in the system. Full article

(This article belongs to the Special Issue Symmetry Breaking Phenomena)

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Review

Jump to: Research

890 KiB

Open AccessReview

Phase Diagram and Critical Properties within an Effective Model of QCD: The Nambu–Jona-Lasinio Model Coupled to the Polyakov Loop

by Pedro Costa, Maria C. Ruivo, Célia A. De Sousa and Hubert Hansen

Symmetry 2010, 2(3), 1338-1374; https://doi.org/10.3390/sym2031338 - 06 Jul 2010

Cited by 83 | Viewed by 7636

Abstract

We investigate the phase diagram of the so-called Polyakov–Nambu–Jona-Lasinio model at finite temperature and non-zero chemical potential with three quark flavors. Chiral and deconfinement phase transitions are discussed and the relevant order-like parameters are analyzed. The results are compared with simple thermodynamic expectations [...] Read more.

We investigate the phase diagram of the so-called Polyakov–Nambu–Jona-Lasinio model at finite temperature and non-zero chemical potential with three quark flavors. Chiral and deconfinement phase transitions are discussed and the relevant order-like parameters are analyzed. The results are compared with simple thermodynamic expectations and lattice data. We present the phase diagram in the (T, μB) plane, paying special attention to the critical end point: as the strength of the flavor-mixing interaction becomes weaker, the critical end point moves to low temperatures and can even disappear. Full article

(This article belongs to the Special Issue Symmetry Breaking Phenomena)

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559 KiB

Open AccessReview

Spontaneous Symmetry Breaking and Nambu–Goldstone Bosons in Quantum Many-Body Systems

by Tomáš Brauner

Symmetry 2010, 2(2), 609-657; https://doi.org/10.3390/sym2020609 - 07 Apr 2010

Cited by 121 | Viewed by 14737

Abstract

Spontaneous symmetry breaking is a general principle that constitutes the underlying concept of a vast number of physical phenomena ranging from ferromagnetism and superconductivity in condensed matter physics to the Higgs mechanism in the standard model of elementary particles. I focus on manifestations [...] Read more.

Spontaneous symmetry breaking is a general principle that constitutes the underlying concept of a vast number of physical phenomena ranging from ferromagnetism and superconductivity in condensed matter physics to the Higgs mechanism in the standard model of elementary particles. I focus on manifestations of spontaneously broken symmetries in systems that are not Lorentz invariant, which include both nonrelativistic systems as well as relativistic systems at nonzero density, providing a self-contained review of the properties of spontaneously broken symmetries specific to such theories. Topics covered include: (i) Introduction to the mathematics of spontaneous symmetry breaking and the Goldstone theorem. (ii) Minimization of Higgs-type potentials for higher-dimensional representations. (iii) Counting rules for Nambu–Goldstone bosons and their dispersion relations. (iv) Construction of effective Lagrangians. Specific examples in both relativistic and nonrelativistic physics are worked out in detail. Full article

(This article belongs to the Special Issue Symmetry Breaking Phenomena)

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394 KiB

Open AccessReview

Broken Time Translation Symmetry as a Model for Quantum State Reduction

by Jasper Van Wezel

Symmetry 2010, 2(2), 582-608; https://doi.org/10.3390/sym2020582 - 01 Apr 2010

Cited by 13 | Viewed by 9315

Abstract

The symmetries that govern the laws of nature can be spontaneously broken, enabling the occurrence of ordered states. Crystals arise from the breaking of translation symmetry, magnets from broken spin rotation symmetry and massive particles break a phase rotation symmetry. Time translation symmetry [...] Read more.

The symmetries that govern the laws of nature can be spontaneously broken, enabling the occurrence of ordered states. Crystals arise from the breaking of translation symmetry, magnets from broken spin rotation symmetry and massive particles break a phase rotation symmetry. Time translation symmetry can be spontaneously broken in exactly the same way. The order associated with this form of spontaneous symmetry breaking is characterised by the emergence of quantum state reduction: systems which spontaneously break time translation symmetry act as ideal measurement machines. In this review the breaking of time translation symmetry is first compared to that of other symmetries such as spatial translations and rotations. It is then discussed how broken time translation symmetry gives rise to the process of quantum state reduction and how it generates a pointer basis, Born’s rule, etc. After a comparison between this model and alternative approaches to the problem of quantum state reduction, the experimental implications and possible tests of broken time translation symmetry in realistic experimental settings are discussed. Full article

(This article belongs to the Special Issue Symmetry Breaking Phenomena)

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2513 KiB

Open AccessReview

Chiral Symmetry Breaking Phenomenon Caused by a Phase Transition

by Rui Tamura, Sekai Iwama and Hiroki Takahashi

Symmetry 2010, 2(1), 112-135; https://doi.org/10.3390/sym2010112 - 17 Feb 2010

Cited by 23 | Viewed by 6998

Abstract

We report the mechanism and scope of “preferential enrichment”, which is an unusual symmetry-breaking enantiomeric resolution phenomenon that is initiated by the solvent-assisted solid-to-solid transformation of a metastable polymorphic form into a thermodynamically stable one during crystallization from the supersaturated solution of certain [...] Read more.

We report the mechanism and scope of “preferential enrichment”, which is an unusual symmetry-breaking enantiomeric resolution phenomenon that is initiated by the solvent-assisted solid-to-solid transformation of a metastable polymorphic form into a thermodynamically stable one during crystallization from the supersaturated solution of certain kinds of racemic mixed crystals (i.e., solid solutions or pseudoracemates) composed of two enantiomers. The mechanism can well be interpreted in terms of a symmetrybreaking complexity phenomenon involving multistage processes that affect each other. Full article

(This article belongs to the Special Issue Symmetry Breaking Phenomena)

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