Statics and dynamics of incommensurate lattices

Theodorou, G.; Rice, T. M.

doi:10.1103/physrevb.18.2840

Cited by 110 publications

(49 citation statements)

References 11 publications

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“…, consistent with some previous results [7,13,23]. The unperturbed normal mode frequencies for the linear chain are given by…”

Section: ⅲ Applicationssupporting

confidence: 79%

“…The lower branch represents the collective motions of the lattice dislocations, as "acoustical" phonons, and the upper branch corresponds to renormalized "optic" phonons [13,23]. We cannot readily distinguish the two phonons in our calculation; perhaps this is a limitation of the continuum limit approximation.…”

Section: ⅲ Applicationsmentioning

confidence: 91%

“…For example, Sutherland [21,22] found that the phonon spectrum separates simply into two branches in the continuumlimit approximation. The lower branch represents the collective modes of lattice dislocations, while the upper one corresponds to renormalized phonons [13,23]. Following standard procedures in the theory of lattice dynamics, the phonon spectrum may include multiple branches [7,23].…”

Section: Introductionmentioning

confidence: 99%

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Lattice vibrations in the Frenkel-Kontorova model. I. Phonon dispersion, number density, and energy

2015

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We studied the lattice vibrations of two inter-penetrating atomic sublattices via the Frenkel-Kontorova (FK) model of a linear chain of harmonically interacting atoms subjected to an on-site potential, using the technique of thermodynamic Green's functions based on quantum field-theoretical methods. General expressions were deduced for the phonon frequency-wave-vector dispersion relations, number density, and energy of the FK model system. As the application of the theory, we investigated in detail cases of linear chains with various periods of the on-site potential of the FK model. Some unusual but interesting features for different amplitudes of the on-site potential of the FK model are discussed. In the commensurate structure, the phonon spectrum always starts at a finite frequency, and the gaps of the spectrum are true ones with a zero density of modes. In the incommensurate structure, the phonon spectrum starts from zero frequency but at a nonzero wave vector; there are some modes inside these gap regions, but their density is very low. In our approximation, the energy of a higher-order commensurate state of the onedimensional system at finite temperature may become indefinitely close to the energy of an incommensurate state. This finding implies that the higher-order incommensuratecommensurate transitions are continuous ones, and that the phase transition may exhibit a "devil's staircase" behavior at a finite temperature.

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“…, consistent with some previous results [7,13,23]. The unperturbed normal mode frequencies for the linear chain are given by…”

Section: ⅲ Applicationssupporting

confidence: 79%

Section: ⅲ Applicationsmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

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Lattice vibrations in the Frenkel-Kontorova model. I. Phonon dispersion, number density, and energy

2015

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“…[24], let us immediately turn to multi-kink solutions as candidates for baryonic matter. Luckily, the sine-Gordon kink crystal has already been studied thoroughly in the literature, first in solid state physics [44,45] and more recently as a toy model for the Skyrme crystal [46], in terms of Jacobi elliptic functions and elliptic integrals [47]. We take over the results from Ref.…”

Section: ) Non-vanishing Bare Quark Massesmentioning

confidence: 99%

Emergence of the Skyrme crystal in Gross-Neveu and ’t Hooft models at finite density

2000

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We study two-dimensional, large N field theoretic models (Gross-Neveu model, 't Hooft model) at finite baryon density near the chiral limit. The same mechanism which leads to massless baryons in these models induces a breakdown of translational invariance at any finite density. In the chiral limit baryonic matter is characterized by a spatially varying chiral angle with a wave number depending only on the density. For small bare quark masses a sine-Gordon kink chain is obtained which may be regarded as simplest realization of the Skyrme crystal for nuclear matter. Characteristic differences between confining and non-confining models are pointed out. 1) IntroductionThe description of baryonic matter on the basis of QCD remains a theoretical challenge, especially since lattice gauge calculations have so far been of little help for this problem. Nuclear physics, relativistic heavy-ion physics and astrophysics are some of the fields which would greatly benefit from any progress on this issue. Recently, a new surge of interest has been triggered by the suggestion that at high density, the novel phenomenon of color superconductivity might set in [1, 2]. This development has highlighted how little is known reliably about dense, strongly interacting matter.Here, we address a much simpler finite density problem where a full analytic solution can be found: We consider two-dimensional model field theories with interacting fermions at or near the chiral limit. Specifically, we have in mind the two-dimensional version of the Nambu-Jona-Lasinio model [3], i.e., the chiral Gross-Neveu model [4], and QCD 2 with fundamental quarks, the 't Hooft model [5]. In both cases, one considers a large number N of fermion species and investigates the limit N → ∞, keeping Ng 2 fixed [6]. These two models are quite similar as far as their chiral properties are concerned but differ with respect to confinement of quarks which is only exhibited by the 't Hooft model. So far, the phase diagram of the Gross-Neveu model has been studied extensively as function of temperature and chemical potential [7, 8, 9, 10, 11], and the results seem to be uncontroversial. The 't Hooft model on the other hand has been investigated only sporadically at finite temperature [20,21], most recently in Ref. [22], but hardly anything is known yet about its properties at finite density [21].Our point of departure is the following observation: Both of these models possess light baryons whose mass vanishes in the chiral limit [23,24,25] (by light, we mean that M B /N is small on the relevant physical scale). This is of course no accident, but a generic feature of models with broken chiral symmetry in 1+1 dimension -the baryons are topologically non-trivial excitations of the Goldstone boson ("pion") field. By contrast, the Gross-Neveu model with discrete chiral symmetry can only accommodate baryons whose mass scales with the physical fermion mass and hence stays finite in the chiral limit. Nevertheless, it has been argued that both variants of the Gross-Neveu mode...

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“…When we transfer our system from a commensurate one to an incommensurate one, the number of resonance frequencies increases from a finite number to infinity. However, we emphasize that such a commensurateincommensurate transition depends on both the interaction strength between the two sublattices, and their lattice constants [24]. Changing the strength of the interaction can significantly alter phonon dispersion and the density of states.…”

Section: B Resonance Scatteringmentioning

confidence: 93%

Lattice vibrations in the Frenkel-Kontorova Model. II. Thermal conductivity

2015

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We applied the formulae for the phonon spectral-density function that we presented in the previous paper of this series to analyze the thermal conductivity of the lattice in the framework of the Frenkel-Kontorova (FK) model. We found that two extra mechanisms of phonon scattering (different from the point impurities, three-phonon processes, and boundary scattering typical of all crystals), viz., resonance, and anharmonic scattering, that mainly influences the thermal conductivity of the lattice. The frequencies of resonance scattering are discrete, and their number increases from a finite number to infinity with their transition from the commensurate-to the incommensurate-state.Changing the amplitude and period of the FK model changes the frequencies and the frequency number of resonance scattering, and the intensity of anharmonic scattering.We analyze these changes in detail. Our theory can explain all existing numerical results on this problem, and also suggest strategies to reduce the thermal conductivity of the lattice of layered materials.

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Statics and dynamics of incommensurate lattices

Cited by 110 publications

References 11 publications

Lattice vibrations in the Frenkel-Kontorova model. I. Phonon dispersion, number density, and energy

Lattice vibrations in the Frenkel-Kontorova model. I. Phonon dispersion, number density, and energy

Emergence of the Skyrme crystal in Gross-Neveu and ’t Hooft models at finite density

Lattice vibrations in the Frenkel-Kontorova Model. II. Thermal conductivity

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