5,442 research outputs found
Quantum integrability of sigma models on AII and CII symmetric spaces
Exact massive S-matrices for two dimensional sigma models on symmetric spaces
SU(2N)/Sp(N) and Sp(2P)/Sp(P)*Sp(P) are conjectured. They are checked by
comparison of perturbative and non perturbative TBA calculations of free energy
in a strong external field. We find the mass spectrum of the models and
calculate their exact mass gap.Comment: 11 p., minor correction
Automorphisms and a Cartography of the Solution Space for Vacuum Bianchi Cosmologies: The Type III Case
The theory of symmetries of systems of coupled, ordinary differential
equations (ODE's) is used to develop a concise algorithm for cartographing the
space of solutions to vacuum Bianchi Einstein's Field Equations (EFE). The
symmetries used are the well known automorphisms of the Lie algebra for the
corresponding isometry group of each Bianchi Type, as well as the scaling and
the time eparameterization symmetry. Application of the method to Type III
results in: a) the recovery of all known solutions without prior assumption of
any extra symmetry, b) the enclosure of the entire unknown part of the solution
space into a single, second order ODE in terms of one dependent variable and c)
a partial solution to this ODE. It is also worth-mentioning the fact that the
solution space is seen to be naturally partitioned into three distinct,
disconnected pieces: one consisting of the known Siklos (pp-wave) solution,
another occupied by the Type III member of the known Ellis-MacCallum family and
the third described by the aforementioned ODE in which an one parameter
subfamily of the known Kinnersley geometries resides. Lastly, preliminary
results reported show that the unknown part of the solution space for other
Bianchi Types is described by a strikingly similar ODE, pointing to a natural
operational unification as far as the problem of solving the cosmological EFE's
is concerned.Comment: 19 pages, LatTex source file, no figures, accepted in JM
Symplectic Symmetry of the Neutrino Mass For Many Neutrino Flavors
The algebraic structure of the neutrino mass Hamiltonian is presented for two
neutrino flavors considering both Dirac and Majorana mass terms. It is shown
that the algebra is Sp(8) and also discussed how the algebraic structure
generalizes for the case of more than two neutrino flavors.Comment: 6 pages, LaTeX, presented at Scandinavian Neutrino Workshop, Uppsala,
February 8-10, 2001, to appear in the proceedings, Physica Script
On the classification of q-algebras
The problem is the classification of the ideals of ``free differential
algebras", or the associated quotient algebras, the q-algebras; being finitely
generated, unital C-algebras with homogeneous relations and a q-differential
structure. This family of algebras includes the quantum groups, or at least
those that are based on simple (super) Lie or Kac-Moody algebras. Their
classification would encompass the so far incompleted classification of
quantized (super) Kac-Moody algebras and of the (super) Kac-Moody algebras
themselves. These can be defined as singular limits of q-algebras, and it is
evident that to deal with the q-algebras in their full generality is more
rational than the examination of each singular limit separately. This is not
just because quantization unifies algebras and superalgebras, but also because
the points "q = 1" and "q = -1" are the most singular points in parameter
space. In this paper one of two major hurdles in this classification program
has been overcome. Fix a set of integers n_1,...,n_k, and consider the space
B_Q of homogeneous polynomials of degree n_1 in the generator e_1, and so on.
Assume that there are no constants among the polynomials of lower degree, in
any one of the generators; in this case all constants in the space B_Q have
been classified. The task that remains, the more formidable one, is to remove
the stipulation that there are no constants of lower degree.Comment: 15 pages, plain TeX, to be published in Lett. Math. Phy
Unfolding the color code
The topological color code and the toric code are two leading candidates for
realizing fault-tolerant quantum computation. Here we show that the color code
on a -dimensional closed manifold is equivalent to multiple decoupled copies
of the -dimensional toric code up to local unitary transformations and
adding or removing ancilla qubits. Our result not only generalizes the proven
equivalence for , but also provides an explicit recipe of how to decouple
independent components of the color code, highlighting the importance of
colorability in the construction of the code. Moreover, for the -dimensional
color code with boundaries of distinct colors, we find that the
code is equivalent to multiple copies of the -dimensional toric code which
are attached along a -dimensional boundary. In particular, for , we
show that the (triangular) color code with boundaries is equivalent to the
(folded) toric code with boundaries. We also find that the -dimensional
toric code admits logical non-Pauli gates from the -th level of the Clifford
hierarchy, and thus saturates the bound by Bravyi and K\"{o}nig. In particular,
we show that the -qubit control- logical gate can be fault-tolerantly
implemented on the stack of copies of the toric code by a local unitary
transformation.Comment: 46 pages, 15 figure
Fredkin Spin Chain
We introduce a new model of interacting spin 1/2. It describes interaction of
three nearest neighbors. The Hamiltonian can be expressed in terms of Fredkin
gates. The Fredkin gate (also known as the CSWAP gate) is a computational
circuit suitable for reversible computing. Our construction generalizes the
work of Ramis Movassagh and Peter Shor.
Our model can be solved by means of Catalan combinatorics in the form of
random walks on the upper half of a square lattice [Dyck walks]. Each Dyck path
can be mapped to a wave function of the spins. The ground state is an equally
weighted superposition of Dyck walks [instead of Motzkin walks]. We can also
express it as a matrix product state. We further construct the model of
interacting spins 3/2 and greater half-integer spins. The models with higher
spins require coloring of Dyck walks. We construct SU(k) symmetric model [here
k is the number of colors]. The leading term of the entanglement entropy is
then proportional to the square root of the length of the lattice [like in
Shor-Movassagh model]. The gap closes as a high power of the length of the
lattice.Comment: 20 pages, 7 figure
An affine generalization of evacuation
We establish the existence of an involution on tabloids that is analogous to
Schutzenberger's evacuation map on standard Young tableaux. We find that the
number of its fixed points is given by evaluating a certain Green's polynomial
at , and satisfies a "domino-like" recurrence relation.Comment: 32 pages, 7 figure
Diagrams of affine permutations, balanced labellings, and symmetric functions
We generalize the work of Fomin, Greene, Reiner, and Shimozono on balanced
labellings in two directions: (1) we define the diagrams of affine permutations
and the balanced labellings on them; (2) we define the set-valued version of
the balanced labellings. We show that the column-strict balanced labellings on
the diagram of an affine permutation yield the affine Stanley symmetric
function defined by Lam, and that the column-strict set-valued balanced
labellings yield the affine stable Grothendieck polynomial of Lam. Moreover,
once we impose suitable flag conditions, the flagged column-strict set-valued
balanced labellings on the diagram of a finite permutation give a monomial
expansion of the Grothendieck polynomial of Lascoux and Sch\"{u}tzenberger. We
also give a necessary and sufficient condition for a diagram to be an affine
permutation diagram.Comment: 24 pages, 9 figures, part of the paper is submitted to FPSAC 2013 as
an extended abstrac
Linear code-based vector quantization for independent random variables
In this paper we analyze the rate-distortion function R(D) achievable using
linear codes over GF(q), where q is a prime number.Comment: 16 pages, 3 figure
BRST analysis of N=2 superconformal minimal unitary models in Coulomb gas formalism
We perform a BRST analysis of the N=2 superconformal minimal unitary models.
A bosonic as well as fermionic BRST operators are used to construct irreducible
representations of the N=2 superconformal algebra on the Fock space as BRST
cohomology classes of the BRST operators. Also a character formula is rederived
by using the BRST analysis.Comment: 34 pages, 2 eps figure
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