Searcharxiv⌕ Search

arXiv subjects

Makoto Yamashita

Publications and source records attributed to Makoto Yamashita.

At least 73 records · Page 4Linked to original sources
arXiv2015

Classification of non-Kac compact quantum groups of SU(n) type

We classify up to isomorphism all non-Kac compact quantum groups with the same fusion rules and dimension function as $SU(n)$. For this we first prove, using categorical Poisson boundary, the following general result. Let $G$ be a coamenable compact quantum group and $K$ be its maximal quantum subgroup of Kac type. Then any dimension-preserving unitary fiber functor $Rep\ G \to Hilb_f$ factors, uniquely up to isomorphism, through $Rep\ K$. Equivalently, we have a canonical bijection $H^2(\hat G; T) \cong H^2(\hat K; T)$. Next, we classify autoequivalences of the representation categories of twisted $q$-deformations of compact simple Lie groups.

math.QA↗
arXiv2015

Theoretical Analysis on Spectroscopy of Atomic Bose-Hubbard Systems

We provide a numerical method to calculate comprehensively the microwave and the laser spectra of ultracold bosonic atoms in optical lattices at finite temperatures. Our formulation is built up with the sum rules, up to the second order, derived from the general principle of spectroscopy. The sum rule approach allows us to discuss the physical origins of a spectral peak shift and also a peak broadening. We find that a spectral broadening of superfluid atoms can be determined from number fluctuations of atoms, while that of normal-state atoms is mainly attributed to quantum fluctuations resulting from hopping of atoms. To calculate spectra at finite temperatures, based on the sum rule approach, we provide a two-mode approximation assuming that spectra of the superfluid and normal state atoms can be calculated separately. Our method can properly deal with multi-peak structures of spectra resulting from thermal fluctuations and also coexisting of the superfluid and the normal states. By combining the two-mode approximation with a finite temperature Gutzwiller approximation, we calculate spectra at finite temperatures by considering realistic systems, and the calculated spectra show nice agreements with those in experiments.

cond-mat.quant-gas↗
arXiv2015

An efficient second-order cone programming approach for optimal selection in tree breeding

An important problem in tree breeding is optimal selection from candidate pedigree members to produce the highest performance in seed orchards, while conserving essential genetic diversity. The most beneficial members should contribute as much as possible, but such selection of orchard parents would reduce performance of the orchard progeny due to serious inbreeding. To avoid inbreeding, we should include a constraint on the numerator relationship matrix to keep a group coancestry under an appropriate threshold. Though an SDP (semidefinite programming) approach proposed by Pong-Wong and Woolliams gave an accurate optimal value, it required rather long computation time. In this paper, we propose an SOCP (second-order cone programming) approach to reduce this computation time. We demonstrate that the same solution is attained by the SOCP formulation, but requires much less time. Since a simple SOCP formulation is not much more efficient compared to the SDP approach, we exploit a sparsity structure of the numerator relationship matrix, and formulate the SOCP constraint using Henderson's algorithm. Numerical results show that the proposed SOCP approach reduced computation time in a case study from 39,200 seconds under the SDP approach to less than 2 seconds.

math.OC↗
arXiv2015

Magnetism in the three-dimensional layered Lieb lattice: Enhanced transition temperature via flat-band and Van Hove singularities

We describe the enhanced magnetic transition temperatures $T_c$ of two-component fermions in three-dimensional layered Lieb lattices, which are created in cold atom experiments. We determine the phase diagram at half-filling using the dynamical mean-field theory. The dominant mechanism of enhanced $T_c$ gradually changes from the (delta-functional) flat-band to the (logarithmic) Van Hove singularity as the interlayer hopping increases. We elucidate that the interaction induces an effective flat-band singularity from a dispersive flat (or narrow) band. We offer a general analytical framework for investigating the singularity effects, where a singularity is treated as one parameter in the density of states. This framework provides a unified description of the singularity-induced phase transitions, such as magnetism and superconductivity, where the weight of the singularity characterizes physical quantities. This treatment of the flat-band provides the transition temperature and magnetization as a universal form (i.e., including the Lambert function). We also elucidate a specific feature of the magnetic crossover in magnetization at finite temperatures.

cond-mat.quant-gas↗
arXiv2015

An iterative method using boundary distance for box-constrained nonlinear semidefinite programs

We propose an iterative method for nonlinear semidefinite programs with box constraints. The search direction in the proposed method utilizes the distance from the current point to the boundary of a feasible set. The computation of the search direction exploits the second derivative of the objective function only in a quadratic form, and this property saves the computation cost compared to an evaluation of the whole entries of the second derivative. We compute a step length in an interval determined by a radius and we update the radius using a quadratic approximation function. In this paper, we also discuss convergence properties of the proposed method based on structures of the search direction. Numerical tests show that the proposed method solves problems in which the size of a variable matrix is larger than 5,000 and that it is faster than a feasible direction method for objective functions with strong nonlinearity.

math.OC↗
arXiv2014

Density Modulations Associated with the Dynamical Instability in the Bose-Hubbard Model

A superfluid flow beyond a critical momentum in an optical lattice decays drastically by the interplay between nonlinearity due to the interparticle interactions in Bose-Einstein condensate and periodicity of the lattice; this instability is called dynamical instability. The complex density modulational profiles after the condensate becomes unstable observed experimentally is not completely understood, while the dynamical instability has been studied theoretically and experimentally. In this paper, we analyze the density modulation of condensates as a precursor of the dynamical instability in the two-dimensional Bose-Hubbard model. Our analysis has clarified the unexplored properties of the density modulations associated with the dynamical instability at low filling and in a wide range of interactions, while the previous works have analyzed the density modulation on the basis of Gross-Pitaevskii equation under the specific condition that one-dimensional optical lattice is very shallow and the filling is very large. The numerical simulations based on the dynamical Gutzwiller approximation elucidate that the principal mode of density modulation highly depends on interaction strength U and the momentum acceleration rate. We briefly discuss these features with the stability phase diagram calculated on the basis of the Bogoliubov theory.

cond-mat.quant-gas↗
arXiv2014

Categorical duality for Yetter-Drinfeld algebras

We study tensor structures on (Rep G)-module categories defined by actions of a compact quantum group G on unital C*-algebras. We show that having a tensor product which defines the module structure is equivalent to enriching the action of G to the structure of a braided-commutative Yetter-Drinfeld algebra. This shows that the category of braided-commutative Yetter-Drinfeld G-C*-algebras is equivalent to the category of generating unitary tensor functors from Rep G into C*-tensor categories. To illustrate this equivalence, we discuss coideals of quotient type in C(G), Hopf-Galois extensions and noncommutative Poisson boundaries.

math.OA↗
arXiv2014

Flat-band ferromagnetism in the multilayer Lieb optical lattice

We theoretically study magnetic properties of two-component cold fermions in half-filled multilayer Lieb optical lattices, i.e., two, three, and several layers, using the dynamical mean-field theory. We clarify that the magnetic properties of this system become quite different depending on whether the number of layers is odd or even. In odd-number-th layers in an odd-number-layer system, finite magnetization emerges even with an infinitesimal interaction. This is a striking feature of the flatband ferromagnetic state in multilayer systems as a consequence of the Lieb theorem. In contrast, in even-number layers, magnetization develops from zero on a finite interaction. These different magnetic behaviours are triggered by the flat bands in the local density of states and become identical in the limit of the infinite-layer (i.e., three-dimensional) system. We also address how interlayer hopping affects the magnetization process. Further, we point out that layer magnetization, which is a population imbalance between up and down atoms on a layer, can be employed to detect the emergence of the flat-band ferromagnetic state without addressing sublattice magnetization.

cond-mat.quant-gas↗
arXiv2014

Fast implementation for semidefinite programs with positive matrix completion

Solving semidefinite programs (SDP) in a short time is the key to managing various mathematical optimization problems. The matrix-completion primal-dual interior-point method (MC-PDIPM) extracts a sparse structure of input SDP by factorizing the variable matrices. In this paper, we propose a new factorization based on the inverse of the variable matrix to enhance the performance of MC-PDIPM. We also use multithreaded parallel computing to deal with the major bottlenecks in MC-PDIPM. Numerical results show that the new factorization and multithreaded computing reduce the computation time for SDPs that have structural sparsity.

math.OC↗
arXiv2014

High-fidelity cluster state generation for ultracold atoms in an optical lattice

We propose a method for generating high-fidelity multipartite spin-entanglement of ultracold atoms in an optical lattice in a short operation time with a scalable manner, which is suitable for measurement-based quantum computation. To perform the desired operations based on the perturbative spin-spin interactions, we propose to actively utilize the extra degrees of freedom (DOFs) usually neglected in the perturbative treatment but included in the Hubbard Hamiltonian of atoms, such as, (pseudo-)charge and orbital DOFs. Our method simultaneously achieves high fidelity, short operation time, and scalability by overcoming the following fundamental problem: enhancing the interaction strength for shortening operation time breaks the perturbative condition of the interaction and inevitably induces unwanted correlations among the spin and extra DOFs.

quant-ph↗
arXiv2013

CCAP for universal discrete quantum groups

We show that the discrete duals of the free orthogonal quantum groups have the Haagerup property and the completely contractive approximation property. Analogous results hold for the free unitary quantum groups and the quantum automorphism groups of finite-dimensional C*-algebras. The proof relies on the monoidal equivalence between free orthogonal quantum groups and SUq(2) quantum groups, on the construction of a sufficient supply of bounded central functionals for SUq(2) quantum groups, and on the free product techniques of Ricard and Xu. Our results generalize previous work in the Kac setting due to Brannan on the Haagerup property, and due to the second author on the CCAP.

math.OA↗
arXiv2013

Twisted Cyclic Cohomology and Modular Fredholm Modules

Connes and Cuntz showed in [Comm. Math. Phys. 114 (1988), 515-526] that suitable cyclic cocycles can be represented as Chern characters of finitely summable semifinite Fredholm modules. We show an analogous result in twisted cyclic cohomology using Chern characters of modular Fredholm modules. We present examples of modular Fredholm modules arising from Podleś spheres and from $SU_q(2)$.

math.KT↗
arXiv2013

Twisting the q-deformations of compact semisimple Lie groups

Given a compact semisimple Lie group $G$ of rank $r$, and a parameter $q>0$, we can define new associativity morphisms in Rep(Gq) using a 3-cocycle $Φ$ on the dual of the center of G, thus getting a new tensor category Rep(Gq)$^Φ$. For a class of cocycles $Φ$ we construct compact quantum groups $G^τ_q$ with representation categories Rep(Gq)$^Φ$. The construction depends on the choice of an r-tuple $τ$ of elements in the center of G. In the simplest case of G=SU(2) and $τ=-1$, our construction produces Woronowicz's quantum group SU_{-q}(2) out of SUq(2). More generally, for G=SU(n), we get quantum group realizations of the Kazhdan-Wenzl categories.

math.OA↗
arXiv2013

Equivariant comparison of quantum homogeneous spaces

We prove the deformation invariance of the quantum homogeneous spaces of the q-deformation of simply connected simple compact Lie groups over the Poisson-Lie quantum subgroups, in the equivariant KK-theory with respect to the translation action by maximal tori. This extends a result of Neshveyev-Tuset to the equivariant setting. As applications, we prove the ring isomorphism of the K-group of Gq with respect to the coproduct of C(Gq), and an analogue of the Borsuk-Ulam theorem for quantum spheres.

math.OA↗
arXiv2013

Operator algebra of foliations with projectively invariant transverse measure

We study the structure of operator algebras associated with the foliations which have projectively invariant measures. When a certain ergodicity condition on the measure preserving holonomies holds, the lack of holonomy invariant transverse measure can be established in terms of a cyclic cohomology class associated with the transverse fundamental cocycle and the modular automorphism group.

math.OA↗
arXiv2013

Tannaka-Krein duality for compact quantum homogeneous spaces. I. General theory

An ergodic action of a compact quantum group G on an operator algebra A can be interpreted as a quantum homogeneous space for G. Such an action gives rise to the category of finite equivariant Hilbert modules over A, which has a module structure over the tensor category Rep(G) of finite dimensional representations of G. We show that there is a one-to-one correspondence between the quantum G-homogeneous spaces up to equivariant Morita equivalence, and indecomposable module C*-categories over Rep(G) up to natural equivalence. This gives a global approach to the duality theory for ergodic actions as developed by C. Pinzari and J. Roberts.

math.OA↗
arXiv2013

Tannaka-Krein duality for compact quantum homogeneous spaces. II. Classification of quantum homogeneous spaces for quantum SU(2)

We apply the Tannaka-Krein duality theory for quantum homogeneous spaces, developed in the first part of this series of papers, to the case of the quantum SU(2) groups. We obtain a classification of their quantum homogeneous spaces in terms of weighted oriented graphs. The equivariant maps between these quantum homogeneous spaces can be characterized by certain quadratic equations associated with the braiding on the representations of SUq(2). We show that, for |q| close to 1, all quantum homogeneous spaces are realized by coideals.

math.OA↗