28,162 research outputs found
Band Connectivity for Topological Quantum Chemistry: Band Structures As A Graph Theory Problem
The conventional theory of solids is well suited to describing band
structures locally near isolated points in momentum space, but struggles to
capture the full, global picture necessary for understanding topological
phenomena. In part of a recent paper [B. Bradlyn et al., Nature 547, 298
(2017)], we have introduced the way to overcome this difficulty by formulating
the problem of sewing together many disconnected local "k-dot-p" band
structures across the Brillouin zone in terms of graph theory. In the current
manuscript we give the details of our full theoretical construction. We show
that crystal symmetries strongly constrain the allowed connectivities of energy
bands, and we employ graph-theoretic techniques such as graph connectivity to
enumerate all the solutions to these constraints. The tools of graph theory
allow us to identify disconnected groups of bands in these solutions, and so
identify topologically distinct insulating phases.Comment: 19 pages. Companion paper to arXiv:1703.02050 and arXiv:1706.08529
v2: Accepted version, minor typos corrected and references added. Now
19+epsilon page
Graph Theory Data for Topological Quantum Chemistry
Topological phases of noninteracting particles are distinguished by global
properties of their band structure and eigenfunctions in momentum space. On the
other hand, group theory as conventionally applied to solid-state physics
focuses only on properties which are local (at high symmetry points, lines, and
planes) in the Brillouin zone. To bridge this gap, we have previously [B.
Bradlyn et al., Nature 547, 298--305 (2017)] mapped the problem of constructing
global band structures out of local data to a graph construction problem. In
this paper, we provide the explicit data and formulate the necessary algorithms
to produce all topologically distinct graphs. Furthermore, we show how to apply
these algorithms to certain "elementary" band structures highlighted in the
aforementioned reference, and so identified and tabulated all orbital types and
lattices that can give rise to topologically disconnected band structures.
Finally, we show how to use the newly developed BANDREP program on the Bilbao
Crystallographic Server to access the results of our computation.Comment: v1: 29 Pages, 13 Figures. Explains how to access the data presented
in arXiv:1703.02050 v2: Accepted version. References updated, figures
improve
Next nearest neighbour Ising models on random graphs
This paper develops results for the next nearest neighbour Ising model on
random graphs. Besides being an essential ingredient in classic models for
frustrated systems, second neighbour interactions interactions arise naturally
in several applications such as the colour diversity problem and graphical
games. We demonstrate ensembles of random graphs, including regular
connectivity graphs, that have a periodic variation of free energy, with either
the ratio of nearest to next nearest couplings, or the mean number of nearest
neighbours. When the coupling ratio is integer paramagnetic phases can be found
at zero temperature. This is shown to be related to the locked or unlocked
nature of the interactions. For anti-ferromagnetic couplings, spin glass phases
are demonstrated at low temperature. The interaction structure is formulated as
a factor graph, the solution on a tree is developed. The replica symmetric and
energetic one-step replica symmetry breaking solution is developed using the
cavity method. We calculate within these frameworks the phase diagram and
demonstrate the existence of dynamical transitions at zero temperature for
cases of anti-ferromagnetic coupling on regular and inhomogeneous random
graphs.Comment: 55 pages, 15 figures, version 2 with minor revisions, to be published
J. Stat. Mec
Regular quantum graphs
We introduce the concept of regular quantum graphs and construct connected
quantum graphs with discrete symmetries. The method is based on a decomposition
of the quantum propagator in terms of permutation matrices which control the
way incoming and outgoing channels at vertex scattering processes are
connected. Symmetry properties of the quantum graph as well as its spectral
statistics depend on the particular choice of permutation matrices, also called
connectivity matrices, and can now be easily controlled. The method may find
applications in the study of quantum random walks networks and may also prove
to be useful in analysing universality in spectral statistics.Comment: 12 pages, 3 figure
Minimal vertex covers on finite-connectivity random graphs - a hard-sphere lattice-gas picture
The minimal vertex-cover (or maximal independent-set) problem is studied on
random graphs of finite connectivity. Analytical results are obtained by a
mapping to a lattice gas of hard spheres of (chemical) radius one, and they are
found to be in excellent agreement with numerical simulations. We give a
detailed description of the replica-symmetric phase, including the size and the
entropy of the minimal vertex covers, and the structure of the unfrozen
component which is found to percolate at connectivity . The
replica-symmetric solution breaks down at . We give a simple
one-step replica symmetry broken solution, and discuss the problems in
interpretation and generalization of this solution.Comment: 32 pages, 9 eps figures, to app. in PRE (01 May 2001
Polynomial iterative algorithms for coloring and analyzing random graphs
We study the graph coloring problem over random graphs of finite average
connectivity . Given a number of available colors, we find that graphs
with low connectivity admit almost always a proper coloring whereas graphs with
high connectivity are uncolorable. Depending on , we find the precise value
of the critical average connectivity . Moreover, we show that below
there exist a clustering phase in which ground states
spontaneously divide into an exponential number of clusters. Furthermore, we
extended our considerations to the case of single instances showing consistent
results. This lead us to propose a new algorithm able to color in polynomial
time random graphs in the hard but colorable region, i.e when .Comment: 23 pages, 10 eps figure
Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs
We consider the problem of coloring Erdos-Renyi and regular random graphs of
finite connectivity using q colors. It has been studied so far using the cavity
approach within the so-called one-step replica symmetry breaking (1RSB) ansatz.
We derive a general criterion for the validity of this ansatz and, applying it
to the ground state, we provide evidence that the 1RSB solution gives exact
threshold values c_q for the q-COL/UNCOL phase transition. We also study the
asymptotic thresholds for q >> 1 finding c_q = 2qlog(q)-log(q)-1+o(1) in
perfect agreement with rigorous mathematical bounds, as well as the nature of
excited states, and give a global phase diagram of the problem.Comment: 23 pages, 10 figures. Replaced with accepted versio
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