Matrix Quantum Research Articles

Combined tilings and separated set-systems

In 1998, Leclerc and Zelevinsky introduced the notion of weakly separated collections of subsets of the ordered $n$-element set $[n]$ (using this notion to give a combinatorial characterization for quasi-commuting minors of a quantum matrix). They conjectured the purity of certain natural domains $D\subseteq 2^{[n]}$ (in particular, of the hypercube $2^{[n]}$ itself, and the hyper-simplex $\{X\subseteq[n]\colon |X|=m\}$ for $m$ fixed), where $D$ is called pure if all maximal weakly separated collections in $D$ have the same cardinality. These conjectures have been answered affirmatively. In this paper, generalizing those earlier results, we reveal wider classes of pure domains in $2^{[n]}$. This is obtained as a consequence of our study of a novel geometric--combinatorial model for weakly separated set-systems, so-called \emph{combined (polygonal) tilings} on a zonogon, which yields a new insight in the area.

Matrix Infrared Spectra and Quantum Chemical Calculations of Ti, Zr, and Hf Dihydride Phosphinidene and Arsinidene Molecules.

Laser ablated Ti, Zr, and Hf atoms react with phosphine during condensation in excess argon or neon at 4 K to form metal hydride insertion phosphides (H2P-MH) and metal dihydride phosphinidenes (HP═MH2) with metal phosphorus double bonds, which are characterized by their intense metal-hydride stretching frequencies. Both products are formed spontaneously on annealing the solid matrix samples, which suggests that both products are relaxed from the initial higher energy M-PH3 intermediate complex, which is not observed. B3LYP (DFT) calculations show that these phosphinidenes are strongly agostic with acute H-P═M angles in the 60° range, even smaller than those for the analogous methylidenes (carbenes) (CH2═MH2) and in contrast to the almost linear H-N═Ti subunit in the imines (H-N═TiH2). Comparison of calculated agostic and terminal bond lengths and covalent bond radii for HP═TiH2 with computed bond lengths for Al2H6 finds that these strong agostic Ti-H bonds are 18% longer than single covalent bonds, and the bridged bonds in dialane are 10% longer than the terminal Al-H single bonds, which show that these agostic bonds can also be considered as bridged bonds. The analogous arsinidenes (HAs═MH2) have 4° smaller agostic angles and almost the same metal-hydride stretching frequencies and double bond orders. Calculations with fixed H-P-Ti and H-As-Ti angles (170.0°) and Cs symmetry find that electronic energies increased by 36 and 44 kJ/mol, respectively, which provide estimates for the agostic/bridged bonding energies.

Unitary Easy Quantum Groups: The Free Case and the Group Case

Easy quantum groups form a class of compact matrix quantum groups which are completely determined by set partitions. They have been defined in 2009 by Banica and Speicher in the orthogonal case (as quantum subgroups of Wang’s free orthogonal quantum group |$O_n^+$|⁠). We extend their definition to the unitary case (as quantum subgroups of Wang’s free unitary quantum group |$U_n^+$|⁠) using colored partitions. In the free case (i.e., restricting to noncrossing partitions), the corresponding categories of partitions have recently been classified by the authors by purely combinatorial means. We show how the quantum groups associated with them can be constructed from other known examples using generalizations of Banica’s free complexification. For doing so, we introduce new kinds of products between quantum groups.

Matrix model for non-Abelian quantum Hall states

We propose a matrix quantum mechanics for a class of non-Abelian quantum Hall states. The model describes electrons which carry an internal SU(p) spin. The ground states of the matrix model include spin-singlet generalisations of the Moore-Read and Read-Rezayi states and, in general, lie in a class previously introduced by Blok and Wen. The effective action for these states is a U(p) Chern-Simons theory. We show how the matrix model can be derived from quantisation of the vortices in this Chern-Simons theory and how the matrix model ground states can be reconstructed as correlation functions in the boundary WZW model.

A matrix model for WZW

We study a U(N) gauged matrix quantum mechanics which, in the large N limit, is closely related to the chiral WZW conformal field theory. This manifests itself in two ways. First, we construct the left-moving Kac-Moody algebra from matrix degrees of freedom. Secondly, we compute the partition function of the matrix model in terms of Schur and Kostka polynomials and show that, in the large $N$ limit, it coincides with the partition function of the WZW model. This same matrix model was recently shown to describe non-Abelian quantum Hall states and the relationship to the WZW model can be understood in this framework.

Enhanced Kerr nonlinearity in a quantized four-level graphene nanostructure

In this paper, a new model is proposed for manipulating the Kerr nonlinearity of right-hand circular probe light in a monolayer of graphene nanostructure. By using the density matrix equations and quantum optical approach, the third-order susceptibility of probe light is explored numerically. It is realized that the enhanced Kerr nonlinearity with zero linear absorption can be provided by selecting the appropriate quantities of controllable parameters, such as Rabi frequency and elliptical parameter of elliptical polarized coupling field. Our results may be useful applications in future all-optical system devices in nanostructures.

A Differential Calculus on Superspace $${{\mathbb R}_h(1|2)}$$ R h ( 1 | 2 ) and Related Topics

We present a noncommutative differential calculus on the h-superspace via a contraction of the q-superspace, and find a new quantum matrix superalgebra. We also give a new deformation of the super-Heisenberg algebra.

Grassmann matrix quantum mechanics

We explore quantum mechanical theories whose fundamental degrees of freedom are rectangular matrices with Grassmann valued matrix elements. We study particular models where the low energy sector can be described in terms of a bosonic Hermitian matrix quantum mechanics. We describe the classical curved phase space that emerges in the low energy sector. The phase space lives on a compact Kahler manifold parameterized by a complex matrix, of the type discovered some time ago by Berezin. The emergence of a semiclassical bosonic matrix quantum mechanics at low energies requires that the original Grassmann matrices be in the long rectangular limit. We discuss possible holographic interpretations of such matrix models which, by construction, are endowed with a finite dimensional Hilbert space.

High-density InAs/GaAs1−xSbx quantum-dot structures grown by molecular beam epitaxy for use in intermediate band solar cells

InAs quantum-dot structures were grown using a GaAs1−xSbx matrix on a GaAs(001) substrate. The use of GaAs1−xSbx for the buffer and cap layers effectively suppressed coalescence between dots and significantly increased the dot density. The highest density (∼3.5 × 1011/cm2) was obtained for a nominal 3.0 monolayer deposition of InAs with an Sb composition of x = 13–14% in the GaAs1−xSbx matrix. When the Sb composition was increased to 18%, the resulting large photoluminescent red shift (∼90 meV) indicated the release of compressive strain inside the quantum dots. For x > 13%, we observed a significant decrease in photoluminescence intensity and an increase in the carrier lifetime (≥4.0 ns). This is attributed to the type-II band alignment between the quantum dots and matrix material.

The quantum matrix and information from the hydrocarbon oil molecule

The quantum matrix of the hydrocarbon (HC) molecule is substantiated. On the basis of its properties and behavior, the genesis of oil is explained as a process of self-evolution of oil and preservation of molecules of different composition and generation time. Individual HC molecules are generated in nanoseconds, and the period of the genesis of oil is comparable with that of migration of the HC fluid from the mantle to the deposit. A model of subatomic abiogenic genesis of oil is presented. Hydrocarbon (HC) molecules of various structure and composition are formed due to interaction of the valency electron orbitals of C and H atoms, the elemental particles of which are quantum objects and carriers of information. On the basis of this, the term quantum matrix of the HC molecule, the properties and behavior of which explain the genesis of oil as a process of its self-evolution and preservation of the molecules of various composition and the period of generation of oil, is substantiated. It is proved that individual HC molecules are generated within nanoseconds and the period of origin of the entire assemblage of more than 500 molecules of oil of various types is comparable with the period of migration of the HC fluid from the mantle to the deposit.

Chaos in classical D0-brane mechanics

Abstract We study chaos in the classical limit of the matrix quantum mechanical system describing D0-brane dynamics. We determine a precise value of the largest Lyapunov exponent, and, with less precision, calculate the entire spectrum of Lyapunov exponents. We verify that these approach a smooth limit as N → ∞. We show that a classical analog of scrambling occurs with fast scrambling scaling, t ∗ ∼ log S. These results confirm the k-locality property of matrix mechanics discussed by Sekino and Susskind.

Free wreath product quantum groups: The monoidal category, approximation properties and free probability

In this paper, we find the fusion rules of the free wreath product quantum groups G≀⁎SN+ for all compact matrix quantum groups of Kac type G and N≥4. This is based on a combinatorial description of the intertwiner spaces between certain generating representations of G≀⁎SN+. The combinatorial properties of the intertwiner spaces in G≀⁎SN+ allow us to obtain several probabilistic applications. We prove also the monoidal equivalence between G≀⁎SN+ and a compact quantum group whose dual is a discrete quantum subgroup of the free product Gˆ⁎SUq(2)ˆ, for some 0<q≤1. We obtain as a corollary certain stability results for the operator algebras associated with the free wreath products of quantum groups such as the ACPAP property and exactness.

Concise quantum associative memories with nonlinear search algorithm

The model of quantum associative memories (QAM) we propose here consists in simplifying and generalizing that of Rigui Zhou \etal \cite{zhou2012} who uses the quantum matrix with binary decision diagram and nonlinear search algorithm in his model. It is worth noting that David Rosenbaum put forth the quantum matrix with binary decision diagram \cite{Rosenbaum2010} and Abrams and Llyod did the nonlinear algorithm. \cite{Abrams1998} Our model gives the possibility to retrieve one of the sought states in multi-values retrieving scheme when a measure on the first register is done. It is better than Grover's algorithm and its modified form which need $\mathcal{O}(\sqrt{\frac{2^n} {m}})$ steps when they are used as the retrieval algorithm. $n$ is the number of qubit of the first register and $m$ the number of values $x$ for which $f(x)=1$. As the nonlinearity makes the system highly susceptible to noise, an analysis of the influence of the single qubit noise channels on the Nonlinear Search Algorithm of our model of QAM, shows a fidelity of about $0.7$ whatever the number of qubits present in the first register.

Easy quantum groups and quantum subgroups of a semi-direct product quantum group

We consider homogeneous compact matrix quantum groups whose fundamental corepresentation matrix has entries which are partial symmetries with central support.We show that such quantum groups have a simple presentation as semi-direct product quantum groups of a group dual quantum group by an action of a permutation group. This general result allows us to completely classify easy quantum groups with the above property by certain reflection groups.We give four applications of our result. First, there are uncountably many easy quantum groups. Second, there are non-easy homogeneous hyperoctahedral quantum groups. Third, we study operator algebraic properties of the hyperoctahedral series. Finally, we prove a generalised de Finetti theorem for those easy quantum groups in the scope of this article.

Low-energy excited states of divanadium: a matrix isolation and MRCI study.

The ground and excited electronic states of the vanadium dimer (V2) have been studied using Ne matrix isolation experiments and quantum chemical calculations (multireference configuration interaction based on complete active space self-consistent orbitals). In the near infrared absorption spectrum, two vibrational progressions of a new electronic term with a large number of members have been observed with the origin at 1.08 eV and a fundamental vibrational quantum of 475 cm(-1). With the aid of calculations, it has been assigned to a (3)Πu electronic term. The calculations yield potential energy curves for a large number of singlet, triplet, and quintet electronic terms.

Extended supersymmetry and hidden symmetries in one-dimensional matrix quantum mechanics

We study properties of nonlinear supersymmetry algebras realized in the one-dimensional quantum mechanics of matrix systems. Supercharges of these algebras are differential operators of a finite order in derivatives. In special cases, there exist independent supercharges realizing an (extended) supersymmetry of the same super-Hamiltonian. The extended supersymmetry generates hidden symmetries of the super-Hamiltonian. Such symmetries have been found in models with (2×2)-matrix potentials.

Lie Subalgebras of the Matrix Quantum Pseudodifferential Operators

We give a complete description of the anti-involutions that preserve the principal gradation of the algebra of matrix quantum pseudodifferential operators and we describe the Lie subalgebras of their minus fixed points.

On the ground state wave function of matrix theory

We propose an explicit construction of the leading terms in the asymptotic expansion of the ground state wave function of BFSS SU(N ) matrix quantum mechanics. Our proposal is consistent with the expected factorization property in various limits of the Coulomb branch, and involves a different scaling behavior from previous suggestions. We comment on some possible physical implications.

Poisson and Hochschild cohomology and the semiclassical limit

Let \mathsf A be a quantum algebra possessing a semiclassical limit A . We show that under certain hypotheses \mathsf A^e can be thought of as a deformation of the Poisson enveloping algebra of A , and we give a criterion for the Hochschild cohomology of \mathsf A to be a deformation of the Poisson cohomology of A in the case that \mathsf A is Koszul. We verify that condition for the algebra of 2\times 2 quantum matrices and calculate its Hochschild cohomology and the Poisson cohomology of its semiclassical limit.

Upconversion luminescence of Eu 3+ ion in aluminoborate/NaYF 4 vitreous composite matrix induced by 800 nm laser

Abstract The upconversion luminescence (UCL) of Eu 3+ ion in calcium aluminoborate glass matrix with the composition of CaO–Al 2 O 3 –B 2 O 3 (CaAlB) under 800 nm laser excitation has been demonstrated and the structural, thermal and optical properties of Eu 3+ -doped glasses were investigated by X-ray diffractometry (XRD), Fourier transform infrared (FT-IR), thermogravimetry-differential scanning calorimetry (TG-DSC) and photoluminescence (PL) analyses. The results revealed that these glasses possess good thermal, chemical and mechanical stabilities. For improving the UCL of Eu 3+ , β-NaYF 4 crystal was successfully melted into CaAlB matrix to form a homogeneous vitreous material (CaAlB/NaYF 4 ). The result indicated that the UC emission of Eu 3+ in CaAlB/NaYF 4 composite matrix observably increases due to reduced phonon energy of CaAlB matrix and high UC quantum efficiency of NaYF 4 matrix. Similarly, additive Bi 3+ ion can improve the UC emission of Eu 3+ ion via reducing the phonon energy of the host. The UC emission of Eu 3+ in CaAlB/NaYF 4 composite matrix can be induced by two-photon simultaneous absorption. The results imply that these investigated glasses may be potential candidates for UCL and fiber materials.