29,040 research outputs found
Orthogonal nets and Clifford algebras
A Clifford algebra model for M"obius geometry is presented. The notion of
Ribaucour pairs of orthogonal systems in arbitrary dimensions is introduced,
and the structure equations for adapted frames are derived. These equations are
discretized and the geometry of the occuring discrete nets and sphere
congruences is discussed in a conformal setting. This way, the notions of
``discrete Ribaucour congruences'' and ``discrete Ribaucour pairs of orthogonal
systems'' are obtained --- the latter as a generalization of discrete
orthogonal systems in Euclidean space. The relation of a Cauchy problem for
discrete orthogonal nets and a permutability theorem for the Ribaucour
transformation of smooth orthogonal systems is discussed.Comment: Plain TeX, 16 pages, 4 picture
Nonlinear Holomorphic Supersymmetry on Riemann Surfaces
We investigate the nonlinear holomorphic supersymmetry for quantum-mechanical
systems on Riemann surfaces subjected to an external magnetic field. The
realization is shown to be possible only for Riemann surfaces with constant
curvature metrics. The cases of the sphere and Lobachevski plane are elaborated
in detail. The partial algebraization of the spectrum of the corresponding
Hamiltonians is proved by the reduction to one-dimensional quasi-exactly
solvable sl(2,R) families. It is found that these families possess the
"duality" transformations, which form a discrete group of symmetries of the
corresponding 1D potentials and partially relate the spectra of different 2D
systems. The algebraic structure of the systems on the sphere and hyperbolic
plane is explored in the context of the Onsager algebra associated with the
nonlinear holomorphic supersymmetry. Inspired by this analysis, a general
algebraic method for obtaining the covariant form of integrals of motion of the
quantum systems in external fields is proposed.Comment: 24 pages, new section and refs added; to appear in Nucl. Phys.
Towards a complete classification of fermionic symmetry protected topological phases in 3D and a general group supercohomology theory
Classification and construction of symmetry protected topological (SPT)
phases in interacting boson and fermion systems have become a fascinating
theoretical direction in recent years. It has been shown that the (generalized)
group cohomology theory or cobordism theory can give rise to a complete
classification of SPT phases in interacting boson/spin systems. Nevertheless,
the construction and classification of SPT phases in interacting fermion
systems are much more complicated, especially in 3D. In this work, we revisit
this problem based on the equivalent class of fermionic symmetric local unitary
(FSLU) transformations. We construct very general fixed point SPT wavefunctions
for interacting fermion systems. We naturally reproduce the partial
classifications given by special group super-cohomology theory, and we show
that with an additional (the so-called
obstruction free subgroup of ) structure, a complete
classification of SPT phases for three-dimensional interacting fermion systems
with a total symmetry group can be obtained for
unitary symmetry group . We also discuss the procedure of deriving a
general group super-cohomology theory in arbitrary dimensions.Comment: 48 pages, 35 figures, published versio
Symmetries of microcanonical entropy surfaces
Symmetry properties of the microcanonical entropy surface as a function of
the energy and the order parameter are deduced from the invariance group of the
Hamiltonian of the physical system. The consequences of these symmetries for
the microcanonical order parameter in the high energy and in the low energy
phases are investigated. In particular the breaking of the symmetry of the
microcanonical entropy in the low energy regime is considered. The general
statements are corroborated by investigations of various examples of classical
spin systems.Comment: 15 pages, 5 figures include
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