296 research outputs found
Affine quantum super Schur-Weyl duality
The Schur-Weyl duality, which started as the study of the commuting actions of the symmetric group and on where , was extended by Drinfeld and Jimbo to the context of the finite Iwahori-Hecke algebra and quantum algebras , on using universal -matrices, which solve the Yang-Baxter equation. There were two extensions of this duality in the Hecke-quantum case: to the affine case, by Chari and Pressley, and to the super case, by Moon and by Mitsuhashi. We complete this chain of works by completing the cube, dealing with the general affine super case, relating the commuting actions of the affine Iwahori-Hecke algebra and of the affine quantum Lie superalgebra using the presentation by Yamane in terms of generators and relations, acting on the th tensor power of the superspace . Thus we construct a functor and show it is an equivalence of categories of and -modules when
Realizing Hopf Insulators in Dipolar Spin Systems
The Hopf insulator represents a topological state of matter that exists
outside the conventional ten-fold way classification of topological insulators.
Its topology is protected by a linking number invariant, which arises from the
unique topology of knots in three dimensions. We predict that three-dimensional
arrays of driven, dipolar-interacting spins are a natural platform to
experimentally realize the Hopf insulator. In particular, we demonstrate that
certain terms within the dipolar interaction elegantly generate the requisite
non-trivial topology, and that Floquet engineering can be used to optimize
dipolar Hopf insulators with large gaps. Moreover, we show that the Hopf
insulator's unconventional topology gives rise to a rich spectrum of edge mode
behaviors, which can be directly probed in experiments. Finally, we present a
detailed blueprint for realizing the Hopf insulator in lattice-trapped
ultracold dipolar molecules; focusing on the example of KRb, we
provide quantitative evidence for near-term experimental feasibility.Comment: 6 + 7 pages, 3 figure
Distinguished non-Archimedean representations
For a symmetric space (G,H), one is interested in understanding the vector
space of H-invariant linear forms on a representation \pi of G. In particular
an important question is whether or not the dimension of this space is bounded
by one. We cover the known results for the pair (G=R_{E/F}GL(n),H=GL(n)), and
then discuss the corresponding SL(n) case. In this paper, we show that
(G=R_{E/F}SL(n),H=SL(n)) is a Gelfand pair when n is odd. When is even, the
space of H-invariant forms on \pi can have dimension more than one even when
\pi is supercuspidal. The latter work is joint with Dipendra Prasad
Antiferromagnetism in the Exact Ground State of the Half Filled Hubbard Model on the Complete-Bipartite Graph
As a prototype model of antiferromagnetism, we propose a repulsive Hubbard
Hamiltonian defined on a graph \L={\cal A}\cup{\cal B} with and bonds connecting any element of with all the
elements of . Since all the hopping matrix elements associated with
each bond are equal, the model is invariant under an arbitrary permutation of
the -sites and/or of the -sites. This is the Hubbard model
defined on the so called -complete-bipartite graph,
() being the number of elements in (). In this
paper we analytically find the {\it exact} ground state for at
half filling for any ; the repulsion has a maximum at a critical
-dependent value of the on-site Hubbard . The wave function and the
energy of the unique, singlet ground state assume a particularly elegant form
for N \ra \inf. We also calculate the spin-spin correlation function and show
that the ground state exhibits an antiferromagnetic order for any non-zero
even in the thermodynamic limit. We are aware of no previous explicit analytic
example of an antiferromagnetic ground state in a Hubbard-like model of
itinerant electrons. The kinetic term induces non-trivial correlations among
the particles and an antiparallel spin configuration in the two sublattices
comes to be energetically favoured at zero Temperature. On the other hand, if
the thermodynamic limit is taken and then zero Temperature is approached, a
paramagnetic behavior results. The thermodynamic limit does not commute with
the zero-Temperature limit, and this fact can be made explicit by the analytic
solutions.Comment: 19 pages, 5 figures .ep
Global Jacquet-Langlands correspondence, multiplicity one and classification of automorphic representations
In this paper we show a local Jacquet-Langlands correspondence for all
unitary irreducible representations. We prove the global Jacquet-Langlands
correspondence in characteristic zero. As consequences we obtain the
multiplicity one and strong multiplicity one theorems for inner forms of GL(n)
as well as a classification of the residual spectrum and automorphic
representations in analogy with results proved by Moeglin-Waldspurger and
Jacquet-Shalika for GL(n).Comment: 49 pages; Appendix by N. Grba
Floquet engineering ultracold polar molecules to simulate topological insulators
We present a quantitative, near-term experimental blueprint for the quantum simulation of topological insulators using lattice-trapped ultracold polar molecules. In particular, we focus on the so-called Hopf insulator, which represents a three-dimensional topological state of matter existing outside the conventional tenfold way and crystalline-symmetry-based classifications of topological insulators. Its topology is protected by a linking number invariant, which necessitates long-range spin-orbit-coupled hoppings for its realization. While these ingredients have so far precluded its realization in solid-state systems and other quantum simulation architectures, in an accompanying Letter [T. Schuster et al., Phys. Rev. Lett. 127, 015301 (2021)], we predict that Hopf insulators can arise naturally from the dipolar interaction. Here, we investigate a specific polar molecule architecture, where the effective âspinâ is formed from sublattice degrees of freedom. We introduce two techniques that allow one to optimize dipolar Hopf insulators with large band gaps, and which should also be readily applicable to the simulation of other exotic band structures. First, we describe the use of Floquet engineering to control the range and functional form of dipolar hoppings and, second, we demonstrate that molecular AC polarizabilities (under circularly polarized light) can be used to precisely tune the resonance condition between different rotational states. To verify that this latter technique is amenable to current-generation experiments, we calculate, from first principles, the AC polarizability for
Ï+ light for 40K 87Rb. Finally, we show that experiments are capable of detecting the unconventional topology of the Hopf insulator by varying the termination of the lattice at its edges, which gives rise to three distinct classes of edge mode spectra
One dimensional SU(3) bosons with function interaction
In this paper we solve one dimensional SU(3) bosons with repulsive
-function interaction by means of Bethe ansatz method. The features of
ground state and low-lying excited states are studied by both numerical and
analytic methods. We show that the ground state is a SU(3) color ferromagnetic
state. The configurations of quantum numbers for the ground state are given
explicitly. For finite system the spectra of low-lying excitations and the
dispersion relations of four possible elementary particles (holon, antiholon,
-coloron and -coloron) are obtained by solving Bethe-ansatz
equation numerically. The thermodynamic equilibrium of the system at finite
temperature is studied by using the strategy of thermodynamic Bethe ansatz, a
revised Gaudin-Takahashi equation which is useful for numerical method are
given . The thermodynamic quantities, such as specific heat, are obtain for
some special cases. We find that the magnetic property of the model in high
temperature regime is dominated by Curie's law: and the
system has Fermi-liquid like specific heat in the strong coupling limit at low
temperature.Comment: RevTex 28 pages, 10 figure
Approximation of integral operators using product-convolution expansions
International audienceWe consider a class of linear integral operators with impulse responses varying regularly in time or space. These operators appear in a large number of applications ranging from signal/image processing to biology. Evaluating their action on functions is a computationally intensive problem necessary for many practical problems. We analyze a technique called product-convolution expansion: the operator is locally approximated by a convolution, allowing to design fast numerical algorithms based on the fast Fourier transform. We design various types of expansions, provide their explicit rates of approximation and their complexity depending on the time varying impulse response smoothness. This analysis suggests novel wavelet based implementations of the method with numerous assets such as optimal approximation rates, low complexity and storage requirements as well as adaptivity to the kernels regularity. The proposed methods are an alternative to more standard procedures such as panel clustering, cross approximations, wavelet expansions or hierarchical matrices
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