3,994 research outputs found
Linear-time algorithms for scattering number and Hamilton-connectivity of interval graphs.
We prove that for all inline image an interval graph is inline image-Hamilton-connected if and only if its scattering number is at most k. This complements a previously known fact that an interval graph has a nonnegative scattering number if and only if it contains a Hamilton cycle, as well as a characterization of interval graphs with positive scattering numbers in terms of the minimum size of a path cover. We also give an inline image time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the previously best-known inline image time bound for solving this problem. As a consequence of our two results, the maximum k for which an interval graph is k-Hamilton-connected can be computed in inline image time
Hopping on the Bethe lattice: Exact results for densities of states and dynamical mean-field theory
We derive an operator identity which relates tight-binding Hamiltonians with
arbitrary hopping on the Bethe lattice to the Hamiltonian with nearest-neighbor
hopping. This provides an exact expression for the density of states (DOS) of a
non-interacting quantum-mechanical particle for any hopping. We present
analytic results for the DOS corresponding to hopping between nearest and
next-nearest neighbors, and also for exponentially decreasing hopping
amplitudes. Conversely it is possible to construct a hopping Hamiltonian on the
Bethe lattice for any given DOS. These methods are based only on the so-called
distance regularity of the infinite Bethe lattice, and not on the absence of
loops. Results are also obtained for the triangular Husimi cactus, a recursive
lattice with loops. Furthermore we derive the exact self-consistency equations
arising in the context of dynamical mean-field theory, which serve as a
starting point for studies of Hubbard-type models with frustration.Comment: 14 pages, 9 figures; introduction expanded, references added;
published versio
Area law in the exact solution of many-body localized systems
Many-body localization was proven under realistic assumptions by constructing
a quasi-local unitary rotation that diagonalizes the Hamiltonian (Imbrie,
2016). A natural generalization is to consider all unitaries that have a
similar structure. We bound entanglement for states generated by such
unitaries, thus providing an independent proof of area law in eigenstates of
many-body localized systems. An error of approximating the unitary by a
finite-depth local circuit is obtained. We connect the defined family of
unitaries to other results about many-body localization (Kim et al, 2014), in
particular Lieb-Robinson bound. Finally we argue that any Hamiltonian can be
diagonalized by such a unitary, given it has a slow enough logarithmic
lightcone in its Lieb-Robinson bound.Comment: 21 page, 3 figure
Index theorems for quantum graphs
In geometric analysis, an index theorem relates the difference of the numbers
of solutions of two differential equations to the topological structure of the
manifold or bundle concerned, sometimes using the heat kernels of two
higher-order differential operators as an intermediary. In this paper, the case
of quantum graphs is addressed. A quantum graph is a graph considered as a
(singular) one-dimensional variety and equipped with a second-order
differential Hamiltonian H (a "Laplacian") with suitable conditions at
vertices. For the case of scale-invariant vertex conditions (i.e., conditions
that do not mix the values of functions and of their derivatives), the constant
term of the heat-kernel expansion is shown to be proportional to the trace of
the internal scattering matrix of the graph. This observation is placed into
the index-theory context by factoring the Laplacian into two first-order
operators, H =A*A, and relating the constant term to the index of A. An
independent consideration provides an index formula for any differential
operator on a finite quantum graph in terms of the vertex conditions. It is
found also that the algebraic multiplicity of 0 as a root of the secular
determinant of H is the sum of the nullities of A and A*.Comment: 19 pages, Institute of Physics LaTe
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