A new proof of the flat wall theorem

We give an elementary and self-contained proof, and a numerical improvement, of a weaker form of the excluded clique minor theorem of Robertson and Seymour, the following. Let t,r >= 1 be integers, and let R = 49152t(24) (40t(2) +r). An r-wall is obtained from a 2r x r-grid by deleting every odd vertical edge in every odd row and every even vertical edge in every even row, then deleting the two resulting vertices of degree one, and finally subdividing edges arbitrarily. The vertices of degree two that existed before the subdivision are called the pegs of the r-wall. Let G be a graph with no Kt minor, and let W be an R-wall in G. We prove that there exist a set A subset of V(G) of size at most 12288t(24) and an r-subwall W' of W such that V(W') n A = 0 and W' is a flat wall in G A in the following sense. There exists a separation (X, Y) of G A such that X boolean AND Y is a subset of the vertex set of the cycle C' that bounds the outer face of W', V(W') subset of Y, every peg of W' belongs to X and the graph G[Y] can almost be drawn in the unit disk with the vertices X n Y drawn on the boundary of the disk in the order determined by C'. Here almost means that the assertion holds after repeatedly removing parts of the graph separated from X n Y by a cutset Z of size at most three, and adding all edges with both ends in Z. Our proof gives rise to an algorithm that runs in polynomial time even when r and t are part of the input instance. The proof is self-contained in the sense that it uses only results whose proofs can be found in textbooks. (C) 2017 The Authors. Published by Elsevier Inc

Flavor Doubling and the Nature of Asymptopia

We consider the possibility that QCD with N flavors has a useful low-energy description with 2N flavors. Specifically, we investigate a free theory of 2N quarks. Although the free theory is U(N)_L X U(N)_R invariant, it admits a larger U(2N) invariance. However, when the axial anomaly is accounted for in the effective theory by a 't Hooft interaction, only SU(N)_L X SU(N)_R X U(1)_B \subset U(2N) survives. There is however a residual discrete symmetry that is not a symmetry of the QCD lagrangian. This S_2 subgroup of U(2N) has many interesting properties. For instance, when explicit chiral symmetry breaking effects are present, S_2 is broken unless \bar\theta=0 or pi. By expressing the free theory on the light-front, we show that flavor doubling implies several superconvergence relations in pion-hadron scattering. Implicit in the 2N-flavor effective theory is a Regge trajectory with vacuum quantum numbers and unit intercept whose behavior is constrained by S_2. In particular, S_2 implies that forward pion-hadron scattering becomes purely elastic at high-energies, in good agreement with experiment.Comment: 26 pages TeX, uses mtexsis.te

Novel Studies on the \eta' Effective Lagrangian

The effective Lagrangian for \eta' incorporating the effect of the QCD \theta-angle has been developed previously. We revisit this Lagrangian and carry out its canonical quantization with particular attention to the test function spaces of constraints and the topology of the \eta'-field. In this way, we discover a new chirally symmetric coupling of this field to chiral multiplets which involves in particular fermions. This coupling violates P and T symmetries. In a subsequent paper, we will evaluate its contribution to the electric dipole moment (EDM) of fermions. Our motivation is to test whether the use of mixed states restores P and T invariance, so that EDM vanishes. This calculation will be shown to have striking new physical consequences.Comment: 14 pages, 1 figure; V2: NEW TITLE; revised version to be published in JHEP; references adde

The Neutron Electric Dipole Moment in the Instanton Vacuum: Quenched Versus Unquenched Simulations

We investigate the role played by the fermionic determinant in the evaluation of the CP-violating neutron electric dipole moment (EDM) adopting the Instanton Liquid Model. Significant differences between quenched and unquenched calculations are found. In the case of unquenched simulations the neutron EDM decreases linearly with the quark mass and is expected to vanish in the chiral limit. On the contrary, within the quenched approximation, the neutron EDM increases as the quark mass decreases and is expected to diverge as (1/m)**Nf in the chiral limit. We argue that such a qualitatively different behavior is a parameter-free, semi-classical prediction and occurs because the neutron EDM is sensitive to the topological structure of the vacuum. The present analysis suggests that quenched and unquenched lattice QCD simulations of the neutron EDM as well as of other observables governed by topology might show up important differences in the quark mass dependence, for mq < Lambda(QCD).Comment: 8 pages, 3 figures, 2 table

Anomalous diffusion at the Anderson transitions

Diffusion of electrons in three dimensional disordered systems is investigated numerically for all the three universality classes, namely, orthogonal, unitary and symplectic ensembles. The second moment of the wave packet $$ at the Anderson transition is shown to behave as $\sim t^a (a\approx 2/3)$. From the temporal autocorrelation function $C(t)$, the fractal dimension $D_2$ is deduced, which is almost half the value of space dimension for all the universality classes.Comment: Revtex, 2 figures, to appear in J. Phys. Soc. Jpn.(1997) Fe

Electronic properties of disordered corner-sharing tetrahedral lattices

We have examined the behaviour of noninteracting electrons moving on a corner-sharing tetrahedral lattice into which we introduce a uniform (box) distribution, of width W, of random on-site energies. We have used both the relative localization length and the spectral rigidity to analyze the nature of the eigenstates, and have determined both the mobility edge trajectories as a function of W, and the critical disorder, Wc, beyond which all states are localized. We find (i) that the mobility edge trajectories (energies Ec vs. disorder W) are qualitatively different from those found for a simple cubic lattice, and (ii) that the spectral rigidity is scale invariant at Wc and thus provides a reliable method of estimating this quantity -- we find Wc/t=14.5. We discuss our results in the context of the metal-to-insulator transition undergone by LiAlyTi{2-y}O4 in a quantum site percolation model that also includes the above-mentioned Anderson disorder, and show that the effects produced by Anderson disorder are far less important than those produced by quantum site percolation, at least in the determination of the doping concentration at which the metal-to-insulator transition is predicted to occur

On vertex coloring without monochromatic triangles

We study a certain relaxation of the classic vertex coloring problem, namely, a coloring of vertices of undirected, simple graphs, such that there are no monochromatic triangles. We give the first classification of the problem in terms of classic and parametrized algorithms. Several computational complexity results are also presented, which improve on the previous results found in the literature. We propose the new structural parameter for undirected, simple graphs -- the triangle-free chromatic number $\chi_3$. We bound $\chi_3$ by other known structural parameters. We also present two classes of graphs with interesting coloring properties, that play pivotal role in proving useful observation about our problem. We give/ask several conjectures/questions throughout this paper to encourage new research in the area of graph coloring.Comment: Extended abstrac

Irreducible Triangulations are Small

A triangulation of a surface is \emph{irreducible} if there is no edge whose contraction produces another triangulation of the surface. We prove that every irreducible triangulation of a surface with Euler genus $g\geq1$ has at most $13g-4$ vertices. The best previous bound was $171g-72$.Comment: v2: Referees' comments incorporate