Continuous Translational Symmetry Research Articles

Magnetic translation groups in an n-dimensional torus and their representations

A charged particle in a uniform magnetic field in a two-dimensional torus has a discrete noncommutative translation symmetry instead of a continuous commutative translation symmetry. We study topology and symmetry of a particle in a magnetic field in a torus of arbitrary dimensions. The magnetic translation group (MTG) is defined as a group of translations that leave the gauge field invariant. We show that the MTG in an n-dimensional torus is isomorphic to a central extension of a cyclic group Zν1×⋯×Zν2l×Tm by U(1) with 2l+m=n. We construct and classify irreducible unitary representations of the MTG in a three-torus and apply the representation theory to three examples. We briefly describe a representation theory for a general n-torus. The MTG in an n-torus can be regarded as a generalization of the so-called noncommutative torus.

A study of a new class of discrete nonlinear Schr dinger equations

A new class of 1D discrete nonlinear Schr${\ddot{\rm{o}}}$dinger Hamiltonians with tunable nonlinerities is introduced, which includes the integrable Ablowitz-Ladik system as a limit. A new subset of equations, which are derived from these Hamiltonians using a generalized definition of Poisson brackets, and collectively refered to as the N-AL equation, is studied. The symmetry properties of the equation are discussed. These equations are shown to possess propagating localized solutions, having the continuous translational symmetry of the one-soliton solution of the Ablowitz-Ladik nonlinear Schr${\ddot{\rm{o}}}$dinger equation. The N-AL systems are shown to be suitable to study the combined effect of the dynamical imbalance of nonlinearity and dispersion and the Peierls-Nabarro potential, arising from the lattice discreteness, on the propagating solitary wave like profiles. A perturbative analysis shows that the N-AL systems can have discrete breather solutions, due to the presence of saddle center bifurcations in phase portraits. The unstaggered localized states are shown to have positive effective mass. On the other hand, large width but small amplitude staggered localized states have negative effective mass. The collison dynamics of two colliding solitary wave profiles are studied numerically. Notwithstanding colliding solitary wave profiles are seen to exhibit nontrivial nonsolitonic interactions, certain universal features are observed in the collison dynamics. Future scopes of this work and possible applications of the N-AL systems are discussed.

Surface reconstruction of an ordered fluid: an analogy with crystal surfaces.

The surface structure of a lamellar polystyrene-block-polybutadiene-block-polymethylmethacrylate (SBM) triblock copolymer forms a complex reconstruction, which breaks the two-dimensional continuous translational symmetry of an ideal (homogeneous) SBM surface. Despite the very different types of matter and order, our findings reveal a remarkable analogy with the well-known phenomenon of surface reconstruction of single crystals, in particular, with the (2x1) "buckling row" reconstruction of the Si(100) surface. Similarities and differences between both classes of materials are discussed on the basis of symmetry considerations.

ChemInform Abstract: Phase Fronts and Synchronization Patterns in Forced Oscillatory Systems

This is a review of recent studies of extended oscillatory systems that are subjected to periodic temporal forcing. The periodic forcing breaks the continuous time translation symmetry and leaves a discrete set ofstable uniform phase states. The multiplicity ofphase states allows for front structures that shift the oscillation phase by[n where n 1, 2,..., hereafter [nfronts. The main concern here is with front instabilities and their implications on pattern formation. Most theoretical studies have focused on the 2 1 resonance where the system oscillates at half the driving frequency. All front solutions in this case are -fronts. At high forcing strengths only stationary fronts exist. Upon decreasing the forcing strength the stationary fronts lose stability to pairs of counter-propagating fronts. The coexistence of counter-propagating fronts allows for traveling domains and spiral waves. In the 4:1 resonance stationary -fronts coexist with /2-fronts. At high forcing strengths the stationary -fronts are stable and standing two-phase waves, consisting of successive oscillatory domains whose phases differ by, prevail. Upon decreasing the forcing strength the stationary -fronts lose stability and decompose into pairs ofpropagating ]2-fronts. The instability designates a transition from standing two-phase waves to traveling four-phase waves. Analogous decomposition instabilities have been found numerically in higher 2n 1 resonances. The available theory is used to account for a few experimental observations made on the photosensitive Belousov-Zhabotinsky reaction subjected to periodic illumination. Observations not accounted for by the theory are pointed out.

Phase fronts and synchronization patterns in forced oscillatory systems

This is a review of recent studies of extended oscillatory systems that are subjected to periodic temporal forcing. The periodic forcing breaks the continuous time translation symmetry and leaves a discrete set of stable uniform phase states. The multiplicity of phase states allows for front structures that shift the oscillation phase by π/n where n = 1, 2, …, hereafter π/n‐fronts. The main concern here is with front instabilities and their implications on pattern formation. Most theoretical studies have focused on the 2 : 1 resonance where the system oscillates at half the driving frequency. All front solutions in this case are π‐fronts. At high forcing strengths only stationary fronts exist. Upon decreasing the forcing strength the stationary fronts lose stability to pairs of counter‐propagating fronts. The coexistence of counter‐propagating fronts allows for traveling domains and spiral waves. In the 4 : 1 resonance stationary π‐fronts coexist with π/2‐fronts. At high forcing strengths the stationary π‐fronts are stable and standing two‐phase waves, consisting of successive oscillatory domains whose phases differ by π, prevail. Upon decreasing the forcing strength the stationary π‐fronts lose stability and decompose into pairs of propagating π/2‐fronts. The instability designates a transition from standing two‐phase waves to traveling four‐phase waves. Analogous decomposition instabilities have been found numerically in higher 2n : 1 resonances. The available theory is used to account for a few experimental observations made on the photosensitive Belousov–Zhabotinsky reaction subjected to periodic illumination. Observations not accounted for by the theory are pointed out.

Amorphous solid state: A locally stable thermodynamic phase of randomly constrained systems

The question of the local stability of the (replica-symmetric) amorphous solid state is addressed for a class of systems undergoing a continuous liquid to amorphous-solid phase transition driven by the effect of random constraints. The Hessian matrix, associated with infinitesimal fluctuations around the stationary point corresponding to the amorphous solid state, is obtained. The eigenvalues of this Hessian matrix are all shown to be strictly positive near the transition, except for one--the zero mode associated with the spontaneously broken continuous translational symmetry of the system. Thus the local stability of the amorphous solid state is established.

Slow dynamics in condensed matter: spinodal decomposition and glass transition

A brief review is presented about our recent work on computer studies of a continuum model describing spinodal decomposition of binary fluid, which quantitatively accounts for corresponding experimental results. We then switch over to discussing problems of glass transitions. First we consider the ultraslow mode observed in fragile glass-formers, where the diffusive behavior of this mode is explained in terms of phase dynamics describing slow relaxation of spatial heterogeneous structure resulting from spontaneous break-down of continuous translational symmetry. Next we present a stochastic model for density fluctuation in supercooled liquids, which is mapped onto a certain kinetic lattice gas model which is convenient for Monte Carlo simulation studies.

Remarks on bifurcations with symmetry

Symmetries play a central role in the weak nonlinear theory of bifurcations. I present two examples of physical situations where symmetry is important. First, the alternating release of vortices in the Bénard-von Karman wake of a cylinder, then an implementation of the continuous translation symmetry in a modified version of the Lorenz model.

Perfect lattice actions for the Gross—Neveu model

We apply the method of Hasenfratz and Niedermayer to analytically construct perfect lattice actions for the Gross--Neveu model. In the large $N$ limit these actions display an exactly perfect scaling, i.e. cut-off artifacts are completely eliminated even at arbitrarily short correlation length. Also the energy spectrum coincides with the spectrum in the continuum and continuous translation and rotation symmetries are restored in physical observables. This is the first (analytic) construction of an exactly perfect lattice action at finite correlation length.

Restoration of the continuous phase transition in the vortex state due to the lattice translational symmetry: Large-N limit (abstract)

The critical behavior of type II superconductors (SC) near the upper critical magnetic field is studied. The order parameter fluctuations in type II SC near the upper critical magnetic field were shown to destroy the off-diagonal long-range order (ODLRO) in d<4 dimensions. In the report this result is proved to be valid in the 1/N expansion. Thus, there is a serious controversy between the experimentally observed phase coherence in SC and theoretical results forbidding a spontaneous breakdown of the U(1) symmetry. Lattice effects breaking the continuous translational magnetic symmetry and suppressing the dimensional reduction effect are shown to be responsible for the existence of the ODLRO. The lattice Hamiltonian of the uniformly frustrated O(2N) symmetric nonlinear sigma model is used to obtain the effective Ginzburg–Landau action which is analyzed in the large N limit. It is shown that the continuous phase transition does indeed occur with critical exponents being universal and identical to those of a superconductor with H=0.

Electromagnetic gauge invariance and orbital motion in field-induced spin-density-wave collective modes: Magnetorotons (abstract)

The metallic ground state of strongly anisotropic metals with an open Fermi surface is known to be destroyed by a magnetic field. A field-induced spin-density-wave (FISDW) phase appears. As the field varies a cascade of transitions between SDW subphases is observed. The physics of FISDW is characterized by two very different lengths. One, the modulation wavelength, is of the order of the interatomic distance. The other is the magnetic length x0, which is much larger and which is connected to electronic orbital motion in the presence of the magnetic field. The order parameters breaks the continuous translation and spin rotational symmetries. We present a theory of the collective excitation spectrum of the FISDW phases. We find that this spectrum exhibits, beside the Goldstone bosons related to the two continuous broken symmetries, a series of rotonlike modes within the single particle electronic gap at the Fermi level, both in the transverse (to the order parameter) spin-wave spectrum and the phase fluctuation spectrum. The magnetoroton minima are centered at quantized values of the wave vector in terms of x−10. The degeneracy between the modes is lifted by the interaction between fluctuations transverse and parallel to the magnetic field. The rotonlike structure is connected to the interplay between orbitally broken translation invariance and electromagnetic gauge invariance. The spectrum of collective modes proves that FISDW is a new type of electron-hole condensate. The theory applies to Bechgaard’s salts.

A one-dimensional fermion model exhibiting an anomalous type of phase transition

We construct a one-dimensional fermion model (rigorously reducible to a mean field theory). We show that this model exhibits a second order phase transition associated with a spontaneous breakdown of continuous space translational symmetry in favor of a periodic symmetry. However, Landau and Lifshitz conjectured that a phase transition in which there is a spontaneous breakdown of Euclidean symmetry in favor of a crystallographic symmetry must be a transition of the first order. Thus we obtain a counterexample to this conjecture in the case of one dimension.