236 research outputs found
Bott-Kitaev Periodic Table and the Diagonal Map
Building on the 10-way symmetry classification of disordered fermions, the
authors have recently given a homotopy-theoretic proof of Kitaev's "Periodic
Table" for topological insulators and superconductors. The present paper offers
an introduction to the physical setting and the mathematical model used. Basic
to the proof is the so-called Diagonal Map, a natural transformation akin to
the Bott map of algebraic topology, which increases by one unit both the
momentum-space dimension and the symmetry index of translation-invariant ground
states of gapped free-fermion systems. This mapping is illustrated here with a
few examples of interest.Comment: Based on a talk delivered by the senior author at the Nobel Symposium
on "New Forms of Matter: Topological Insulators and Superconductors"
(Stockholm, June 13-15, 2014
Pure scaling operators at the integer quantum Hall plateau transition
Stationary wave functions at the transition between plateaus of the integer
quantum Hall effect are known to exhibit multi-fractal statistics. Here we
explore this critical behavior for the case of scattering states of the
Chalker-Coddington model with point contacts. We argue that moments formed from
the wave amplitudes of critical scattering states decay as pure powers of the
distance between the points of contact and observation. These moments in the
continuum limit are proposed to be correlations functions of primary fields of
an underlying conformal field theory. We check this proposal numerically by
finite-size scaling. We also verify the CFT prediction for a 3-point function
involving two primary fields.Comment: Published version, 4 pages, 3 figure
Energy correlations for a random matrix model of disordered bosons
Linearizing the Heisenberg equations of motion around the ground state of an
interacting quantum many-body system, one gets a time-evolution generator in
the positive cone of a real symplectic Lie algebra. The presence of disorder in
the physical system determines a probability measure with support on this cone.
The present paper analyzes a discrete family of such measures of exponential
type, and does so in an attempt to capture, by a simple random matrix model,
some generic statistical features of the characteristic frequencies of
disordered bosonic quasi-particle systems. The level correlation functions of
the said measures are shown to be those of a determinantal process, and the
kernel of the process is expressed as a sum of bi-orthogonal polynomials. While
the correlations in the bulk scaling limit are in accord with sine-kernel or
GUE universality, at the low-frequency end of the spectrum an unusual type of
scaling behavior is found.Comment: 20 pages, 3 figures, references adde
A conjecture on Hubbard-Stratonovich transformations for the Pruisken-Sch\"afer parameterisations of real hyperbolic domains
Rigorous justification of the Hubbard-Stratonovich transformation for the
Pruisken-Sch\"afer type of parameterisations of real hyperbolic
O(m,n)-invariant domains remains a challenging problem. We show that a naive
choice of the volume element invalidates the transformation, and put forward a
conjecture about the correct form which ensures the desired structure. The
conjecture is supported by complete analytic solution of the problem for groups
O(1,1) and O(2,1), and by a method combining analytical calculations with a
simple numerical evaluation of a two-dimensional integral in the case of the
group O(2,2).Comment: Published versio
Lattice Dirac fermions in a non-Abelian random gauge potential: Many flavors, chiral symmetry restoration and localization
In the previous paper we studied Dirac fermions in a non-Abelian random
vector potential by using lattice supersymmetry. By the lattice regularization,
the system of disordered Dirac fermions is defined without any ambiguities. We
showed there that at strong-disorder limit correlation function of the fermion
local density of states decays algebraically at the band center. In this paper,
we shall reexamine the multi-flavor or multi-species case rather in detail and
argue that the correlator at the band center decays {\em exponentially} for the
case of a {\em large} number of flavors. This means that a
delocalization-localization phase transition occurs as the number of flavors is
increased. This discussion is supported by the recent numerical studies on
multi-flavor QCD at the strong-coupling limit, which shows that the phase
structure of QCD drastically changes depending on the number of flavors. The
above behaviour of the correlator of the random Dirac fermions is closely
related with how the chiral symmetry is realized in QCD.Comment: Version appears in Mod.Phys.Lett.A17(2002)135
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