Monte-Carlo rejection as a tool for measuring the energy landscape scaling of simple fluids

A simple modification of the Monte-Carlo algortihm is proposed to explore the topography and the scaling of the energy landscape. We apply this idea to a simple hard-core fluid. The results for different packing fractions show a power law scaling of the landscape boundary, with a characteristic scale that separates the values of the scaling exponents. Finally, it is shown how the topology determines the freezing point of the system due to the increasing importance and complexity of the boundary

Topological Map of the Hofstadter Butterfly and Van Hove Singularities

The Hofstadter butterfly is a quantum fractal with a highly complex nested set of gaps, where each gap represents a quantum Hall state whose quantized conductivity is characterized by topological invariants known as the Chern numbers. Here we obtain simple rules to determine the Chern numbers at all scales in the butterfly fractal and lay out a very detailed topological map of the butterfly. Our study reveals the existence of a set of critical points, each corresponding to a macroscopic annihilation of orderly patterns of both the positive and the negative Cherns that appears as a fine structure in the butterfly. Such topological collapses are identified with the Van Hove singularities that exists at every band center in the butterfly landscape. We thus associate a topological character to the Van Hove anomalies. Finally, we show that this fine structure is amplified under perturbation, inducing quantum phase transitions to higher Chern states in the system

A higher-dimensional quasicrystalline approach to the Hofstadter and Fibonacci butterflies topological phase diagram and band conductance: symbolic sequences, Sturmian coding and self-similar rules at all magnetic fluxes

The topological properties of the quantum Hall effect in a crystalline lattice, described by Chern numbers of the Hofstadter butterfly quantum phase diagram, are deduced by using a geometrical method to generate the structure of quasicrystals: the cut and projection method. Based on this, we provide a geometric unified approach to the Hofstadter topological phase diagram at all fluxes. Then we show that for any flux, the bands conductance follow a two letter symbolic sequence . As a result, bands conductance at different fluxes obey inflation/deflation rules as the ones observed to build quasicrystals. The bands conductance symbolic sequences are given by the Sturmian coding of the flux and can be found by considering a circle map, a billiard or trajectories on a torus. Simple and fast techniques are thus provided to obtain Chern numbers at any magnetic flux. This approach rationalize the previously observed topological equivalences between the Fibonacci and Harper potentials (also known as the almost Mathieu operator problem) or with other trigonometric potential, as well as the relationship with Farey sequences and trees

Chern and Majorana Modes of Quasiperiodic Systems

New types of self-similar states are found in quasiperiodic systems characterized by topological invariants-- the Chern numbers. We show that the topology introduces a competing length in the self-similar band edge states transforming peaks into doublets of size equal to the Chern number. This length intertwines with the quasiperiodicity and introduces an intrinsic scale, producing Chern-beats and nested regions where the fractal structure becomes smooth. Cherns also influence the zero-energy mode, that for quasiperiodic systems which exhibit exponential localization, is related to the ghost of the Majorana; the delocalized state at the onset to topological transition. The Chern and the Majorana, two distinct types of topological edge modes, exist in quasiperiodic superconducting wires.Comment: Revised versio

Spectral butterfly, mixed Dirac-Schr\"odinger fermion behavior and topological states in armchair uniaxial strained graphene

An exact mapping of the tight-binding Hamiltonian for a graphene's nanoribbon under any armchair uniaxial strain into an effective one-dimensional system is presented. As an application, for a periodic modulation we have found a gap opening at the Fermi level and a complex fractal spectrum, akin to the Hofstadter butterfly resulting from the Harper model. The latter can be explained by the commensurability or incommensurability nature of the resulting effective potential. When compared with the zig-zag uniaxial periodic strain, the spectrum shows much bigger gaps, although in general the states have a more extended nature. For a special critical value of the strain amplitude and wavelength, a gap is open. At this critical point, the electrons behave as relativistic Dirac femions in one direction, while in the other, a non-relativistic Schr\"odinger behavior is observed. Also, some topological states were observed which have the particularity of not being completly edge states since they present some amplitude in the bulk. However, these are edge states of the effective system due to a reduced dimensionality through decoupling. These states also present the fractal Chern beating observed recently in quasiperiodic systems.Comment: 10 pages, 8 figure

Comment on "Penrose Tilings as Jammed Solids"

In a recent letter, Stenull and Lubensky claim that periodic approximants of Penrose tilings, which are generically isostatic, have a nonzero bulk modulus B when disordered, and, therefore, Penrose tilings are good models of jammed packings. The claim of a nonzero B, which is made on the basis of a normal mode analysis of periodic Penrose approximants for a single value of the disorder epsilon, is the central point of their letter: other properties of Penrose tilings, such as the vanishing of the shear modulus, and a flat density of vibrational states, are already shared by most geometrically disordered isostatic networks studied so far. In this comment, Conjugate Gradient is used to solve the elastic equations on approximants with up to 8x10^4 sites for several values of epsilon, to show beyond reasonable doubt that Stenull and Lubensky's claim is incorrect. The bulk modulus of generic Penrose tilings is zero asymptotically. According to our results, B grows as (epsilon^2 L^3) when (epsilon^2 L^3) << 10^2, then saturates, and finally decays as (epsilon^2 L^3)^{-2/3} ~ 1/L^2 for epsilon^2 L^3 >> 10^2. Stenull and Lubensky seem to have only analyzed one value of epsilon for which saturation is reached at the largest size studied. This led them to a wrong conclusion. We support our results by also considering generic Penrose approximants with fixed boundaries, whose bulk modulus constitutes a strict upper bound for that of periodic systems, finding that these have a vanishing B as well for large L. We conclude that the main point in Stenull and Lubensky letter is unjustified. Penrose tilings are no better models of jammed packings than any of the previously studied isostatic networks with geometric disorder.Comment: 1 page. 4 subfigures. Submitted to PR

Mapping of strained graphene into one-dimensional Hamiltonians: quasicrystals and modulated crystals

The electronic properties of graphene under any arbitrary uniaxial strain field are obtained by an exact mapping of the corresponding tight-binding Hamiltonian into an effective one-dimensional modulated chain. For a periodic modulation, the system displays a rich behavior, including quasicrystals and modulated crystals with a complex spectrum, gaps at the Fermi energy and interesting localization properties. These features are explained by the incommensurate or commensurate nature of the potential, which leads to a dense filling of the reciprocal space in the former case. Thus, the usual perturbation theory approach breaks down in some cases, as is proved by analyzing a special momentum that uncouples the model into dimers.Comment: 5 pages, 4 figure

Anisotropic AC conductivity of strained graphene

The density of states and the AC conductivity of graphene under uniform strain are calculated using a new Dirac Hamiltonian that takes into account the main three ingredients that change the electronic properties of strained graphene: the real displacement of the Fermi energy, the reciprocal lattice strain and the changes in the overlap of atomic orbitals. Our simple analytical expressions of the density of states and the AC conductivity generalizes previous expressions only available for uniaxial strain. The results suggest a way to measure the Gruneisen parameter that appears in any calculation of strained graphene, as well as the emergence of a sort of Hall effect due to shear strain.Comment: In previous version of our work, a term was missing in equation (17). In version 2, the expression for the AC conductivity of graphene under uniform strain (equation (17)) is corrected. (A corrigendum has been sent to J. Phys.: Condens. Matter

Escape time, relaxation and sticky states of a softened Henon-Heiles model: low-frequency vibrational modes effects

Here we study the relaxation of a chain consisting of 3 masses joined by non-linear springs and periodic conditions when the stiffness is weakened. This system, when expressed in their normal coordinates, yields a softened Henon-Heiles system. By reducing the stiffness of one low-frequency vibrational mode, a faster relaxation is enabeled. This is due to a reduction of the energy barrier heights along the softened normal mode as well as for a widening of the opening channels of the energy landscape in configurational space. The relaxation is for the most part exponential, and can be explained by a simple flux equation. Yet, for some initial conditions the relaxation follows as a power law and, and in many cases, there is a regime change from exponential to power law decay. We pin point the initial conditions for the power law decay, finding two regions of sticky states. For such states, quasiperiodic orbits are found since almost for all components of the initial momentum orientation, the system is trapped inside two pockets of configurational space. The softened Henon- Heiles model presented here is intended as the simplest model in order to understand the interplay of rigidity, non-linear interactions and relaxation for non-equilibrium systems like glass-forming melts or soft-matter.Comment: 11 + 5 (SM) pages. 11 + 4(SM) figure

Topological edge states on time-periodically strained armchair graphene nanoribbons

We report the emergence of electronic edge states in time-periodically driven strained armchair terminated graphene nanoribbons. This is done by considering a short-pulse spatial-periodic strain field. Then, the tight-binding Hamiltonian of the system is mapped into a one dimensional ladder. The time periodicity is considered within the Floquet formalism. Thus the quasienergy spectrum is found numerically by diagonalizing the evolution operator. For some particular cases, the quasienergy spectrum is found analytically. We find that the system is able to support gapless and gapped phases. Very different edge states emerge for both the gapless and the gapful phases. In the case of the gapped phase, edge states emerge at the gap centered at zero quasienergy, although the Chern number is zero due to the chiral symmetry of the system. For the gapless phase, besides edge states at zero quasienergy, cosine like edge states which merge and coexist with the bulk band are observed. To confirm the topological nature of these edge states, we analytically obtained the effective Hamiltonian and its spectrum for a particular case, finding that the edge states are topologically weak. Finally, we found analytically the evolution of band edges and its crossings as a function of the driven period. Topological modes arise at such crossings.Comment: 11 pages, 6 figures, 2 appendice