Unitary Group Research Articles

Synthesis of Iron Nanoparticle Using Polyphenol Bioreductor from Red Pomegranate Extract

Nano zero valent iron (nZVI) has been utilized for environmental remediation and raw water treatment. The NaBH4 reductor, utilized in iron nanoparticle synthesis, yielded easily oxidized and agglomerated material. Using polyphenol as a bioreductor resulted in a more stable material. The purpose of this research is to synthesize and compare quality of nZVI utilizing NaBH4 (C-nZVI) to nZVI utilizing polyphenol from spent tea and the peel of pomegranate (G-nZVI) and determine the efficiency of pomegranate peel G-nZVI in reducing dissolved organic matter. The iron nanoparticle was synthesized by reducing FeCl3 salt in a water solvent. C-nZVI material was generated as a black powder, whereas G-nZVI materials were black flakes. The band gap energy values of these three materials indicate that they have reached the nanoscale. All three materials had entire nZVI unitary groups based on its FTIR spectrums. The X-ray diffractogram did not clearly show the core of the phase crystals. The G-nZVI from the peel of pomegranate had a greater distribution and mean material size than the spent tea G-nZVI. Both zeta potentials G-nZVI demonstrate that the materials were stable in the aqueous medium. After two hours of incubation, G-nZVI pomegranate peel at room temperature and dark conditions achieved an optimal dissolved organic matter breakdown rate of 98%.

Powers in finite unitary groups

Let U(n,Fq2) denote the subgroup of unitary matrices of the general linear group GL(n,Fq2) which fix a Hermitian form and let M≥2 be an integer. In earlier work, elements of the groups GL(n,Fq), Sp(2n,Fq), O±(2n,Fq) and O(2n+1,Fq) with an M-th root have been described. Here we will describe the M-th powers in unitary groups for the regular semisimple, semisimple and cyclic elements, under the assumption that M and q are coprime. We will further describe the generating functions in the corresponding cases.

Probing bad theories with the dualization algorithm. Part II.

We continue our analysis of bad theories initiated in [1], focusing on quiver theories with bad unitary and special unitary gauge groups in three dimensions. By extending the dualization algorithm we prove that the partition function of bad linear quivers can be written as a distribution, given by a sum of terms involving a product of delta functions times the partition function of a good quiver theory. We describe in detail the good quiver theories appearing in the partition function of the bad theory and discuss the brane interpretation of our result. We also discuss in detail the lift of these theories to 4d quivers with symplectic gauge groups, in which our results can be recovered by studying the Higgsing triggered by the expectation value for certain chiral operators. The paper is accompanied by a Mathematica file which implements the algorithm for an arbitrary unitary bad linear quiver.

Group Theory on Quasisymmetry and Protected Near Degeneracy.

In solid state systems, group representation theory is powerful in characterizing the behavior of quasiparticles, notably the energy degeneracy. While conventional group theory is effective in answering yes-or-no questions related to symmetry breaking, its application to determining the magnitude of energy splitting resulting from symmetry lowering is limited. Here, we propose a theory on quasisymmetry and near degeneracy, thereby expanding the applicability of group theory to address questions regarding large-or-small energy splitting. Defined within the degenerate subspace of an unperturbed Hamiltonian, quasisymmetries form an enlarged symmetry group eliminating the first-order splitting. This framework ensures that the magnitude of splitting arises as a second-order effect of symmetry-lowering perturbations, such as external fields and spin-orbit coupling. We systematically tabulate the quasisymmetry groups within 32 crystallographic point groups and find all the possible unitary quasisymmetry group structures regarding double degeneracy. Applying our theory to the realistic material AgLa, we predict a "quasi-Dirac semimetal" phase characterized by two tiny-gap band anticrossings.

Polynomial Equivalence of Complexity Geometries

This paper proves the polynomial equivalence of a broad class of definitions of quantum computational complexity. We study right-invariant metrics on the unitary group—often called `complexity geometries' following the definition of quantum complexity proposed by Nielsen—and delineate the equivalence class of metrics that have the same computational power as quantum circuits. Within this universality class, any unitary that can be reached in one metric can be approximated in any other metric in the class with a slowdown that is at-worst polynomial in the length and number of qubits and inverse-polynomial in the permitted error. We describe the equivalence classes for two different kinds of error we might tolerate: Killing-distance error, and operator-norm error. All metrics in both equivalence classes are shown to have exponential diameter; all metrics in the operator-norm equivalence class are also shown to give an alternative definition of the quantum complexity class BQP. My results extend those of Nielsen et al., who in 2006 proved that one particular metric is polynomially equivalent to quantum circuits. The Nielsen et al. metric is incredibly highly curved. I show that the greatly enlarged equivalence class established in this paper also includes metrics that have modest curvature. I argue that the modest curvature makes these metrics more amenable to the tools of differential geometry, and therefore makes them more promising starting points for Nielsen's program of using differential geometry to prove complexity lowerbounds. In a previous paper my collaborators and I—inspired by the UV/IR decoupling that happens in the phenomenon of renormalization—conjectured that high- dimensional metrics that look very different at short scales will often nevertheless give rise at long scales to the same emergent effective geometry. The results of this paper provide evidence for those conjectures, since many complexity metrics that have radically different penalty factors and therefore radically different short- distance properties are shown to belong to the same long-distance equivalence class.

Sarnak's conjecture in quantum computing, cyclotomic unitary group coranks, and Shimura curves

AbstractSarnak's conjecture in quantum computing concerns when the groups and over cyclotomic rings with , , are generated by the Clifford‐cyclotomic gate set. We previously settled this using Euler–Poincaré characteristics. A generalization of Sarnak's conjecture is to ask when these groups are generated by torsion elements. An obstruction to this is provided by the corank: a group has only if is not generated by torsion elements. In this paper, we study the corank of these cyclotomic unitary groups in the families and , , by letting them act on Bruhat–Tits trees. The quotients by this action are finite graphs whose first Betti number is the corank of the group. Our main result is that for the families and , the corank grows doubly exponentially in as ; it is 0 precisely when , and indeed, the cyclotomic unitary groups are generated by torsion elements (in fact by Clifford‐cyclotomic gates) for these . We give explicit lower bounds for the corank in two different ways. The first is to bound the isotropy subgroups in the action on the tree by explicit cyclotomy. The second is to relate our graphs to Shimura curves over via interchanging local invariants and applying a result of Selberg and Zograf. We show that the cyclotomy arguments give the stronger bounds. In a final section, we execute a program of Sarnak to show that our results for the and families are sufficient to give a second proof of Sarnak's conjecture.

Archimedean zeta integrals for unitary groups

Archimedean zeta integrals for unitary groups

Generalized unistochastic matrices

We study a class of bistochastic matrices generalizing unistochastic matrices. Given a complex bipartite unitary operator, we construct a bistochastic matrix having as entries the normalized squared Frobenius norm of the blocks. We show that the closure of the set of generalized unistochastic matrices is the whole Birkhoff polytope. We characterize the points on the edges of the Birkhoff polytope that belong to a given level of our family of sets, proving that the different (non-convex) levels have a rich inclusion structure. We also study the corresponding generalization of orthostochastic matrices. Finally, we introduce and study the natural probability measures induced on our sets by the Haar measure of the unitary group. These probability measures interpolate between the natural measure on the set of unistochastic matrices and the Dirac measure supported on the van der Waerden matrix.

Character rigidity for lattices and commensurators

abstract: We prove an operator algebraic superrigidity statement for homomorphisms of irreducible lattices, and also their commensurators, from certain higher-rank groups into unitary groups of finite factors. This extends the authors' previous work regarding non-free measure-preserving actions, and also answers a question of Connes for such groups.

The Endoscopic Classification of Representations of Non-Quasi-Split Odd Special Orthogonal Groups

Abstract In an earlier book of Arthur, the endoscopic classification of representations of quasi-split orthogonal and symplectic groups was established. Later Mok gave that of quasi-split unitary groups. After that, Kaletha, Minguez, Shin, and White gave that of non-quasi-split unitary groups for generic parameters. In this paper we prove the endoscopic classification of representations of non-quasi-split odd special orthogonal groups for generic parameters, following Kaletha, Minguez, Shin, and White.

Spectrum of mesons in quenched Sp(2N) gauge theories

We report the findings of our extensive study of the spectra of flavored mesons in lattice gauge theories with symplectic gauge group and fermion matter content treated in the quenched approximation. For the Sp(4), Sp(6), and Sp(8) gauge groups, the (Dirac) fermions transform in either the fundamental, or the 2-index, antisymmetric or symmetric, representations. This study sets the stage for future precision calculations with dynamical fermions in the low-mass region of lattice parameter space. Our results have potential phenomenological applications ranging from composite Higgs models, to top (partial) compositeness, to dark matter models with composite, strong-coupling dynamical origin. Having adopted the Wilson flow as a scale-setting procedure, we apply Wilson chiral perturbation theory to extract the continuum and massless limits for the observables of interest. The resulting measurements are used to perform a simplified extrapolation to the large-N limit, hence drawing a preliminary connection with gauge theories with unitary groups. We conclude with a brief discussion of the Weinberg sum rules. Published by the American Physical Society 2024

Algebraic homotopy classes

We prove several positive results regarding representation of homotopy classes of spheres and algebraic groups by regular mappings. Most importantly we show that every mapping from a sphere to an orthogonal or a unitary group is homotopic to a regular one. Furthermore we prove that algebraic homotopy classes of spheres form a subgroup of the homotopy group, and that a similar result holds also for cohomotopy groups of arbitrary varieties.

TWISTED GAN–GROSS–PRASAD CONJECTURE FOR CERTAIN TEMPERED L-PACKETS

Abstract In this paper, we investigate the twisted GGP conjecture for certain tempered representations using the theta correspondence and establish some special cases, namely when the L-parameter of the unitary group is the sum of conjugate-dual characters of the appropriate sign.

A decomposition theorem for unitary group representations on Kaplansky–Hilbert modules and the Furstenberg–Zimmer structure theorem

A decomposition theorem for unitary group representations on Kaplansky–Hilbert modules and the Furstenberg–Zimmer structure theorem

Word Measures on Unitary Groups: Improved Bounds for Small Representations

Abstract Let $F$ be a free group of rank $r$ and fix some $w\in F$. For any compact group $G$ we can define a measure $\mu _{w,G}$ on $G$ by (Haar-)uniformly sampling $g_{1},...,g_{r}\in G$ and evaluating $w(g_{1},...,g_{r})$. In [23], Magee and Puder study the behavior of the moments of $\mu _{w,U(n)}$ as a function of $n$, establishing a connection between their asymptotic behavior and certain algebraic invariants of $w$, such as its commutator length. We employ geometric insights to refine their analysis, and show that the asymptotic behavior of the moments is also governed by the primitivity rank of $w$. Additionally, we also apply our methods to prove a special case of a conjecture of Hanany and Puder [13, Conjecture 1.13] regarding the asymptotic behavior of expected values of irreducible characters of $U(n)$ under $\mu _{w,U(n)}$.

Symbols of mixed unitary quantum channels generated by finite unitary groups

We study the symbols of linear operators generated by a finite unitary group. Under some assumptions, it is shown that the operator can be restored from its symbol. We also introduce symbols of mixed unitary quantum channels generated by finite unitary groups. An example illustrating the technique is given.

Spin textures of coherent photons with SU(4) symmetry

We show coherent photons with spin and orbital angular momentum have symmetry of special unitary group of degree four, SU(4), both theoretically and experimentally. We show the degree of polarisation could be a measure to characterise the quantum coherency of a spin state, which was significantly affected by the partial trace out of orbital degree of freedom. By changing the amplitudes of the classically entangled state, we observed a crossover from quantum Heisenberg spin to classical Ising spin. We also demonstrated projections of coherent states, which recover the coherency of the spin state, while the SU(4) state is projected to be a simple product state of SU(2)×SU(2). Based on the developed platform, we have explored spin texture of coherent photons, and found a unique spin profile reminiscent of the Bengal cat upon wrapping to a three-dimensional sphere. We have controlled the phase between singlet and triplet states and found the spin texture rotated upon increasing the phase.

Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices

We investigate the joint moments of derivatives of characteristic polynomials over the unitary symplectic group Sp(2N) and the orthogonal ensembles SO(2N) and O−(2N) . We prove asymptotic formulae for the joint moments of the n 1th and n 2th derivatives of the characteristic polynomials for all three matrix ensembles. Our results give two explicit formulae for each of the leading order coefficients, one in terms of determinants of hypergeometric functions and the other as combinatorial sums over partitions. We use our results to put forward conjectures on the joint moments of derivatives of L-functions with symplectic and orthogonal symmetry.

Limits of Bessel functions for root systems as the rank tends to infinity

We study the asymptotic behaviour of Bessel functions associated to root systems of type An−1 and type Bn with positive multiplicities as the rank n tends to infinity. In both cases, we characterize the possible limit functions and the Vershik–Kerov type sequences of spectral parameters for which such limits exist. In the type A case, this gives a new and very natural approach to recent results by Assiotis and Najnudel in the context of β-ensembles in random matrix theory. These results generalize known facts about the approximation of the positive-definite Olshanski spherical functions of the space of infinite-dimensional Hermitian matrices over F=R,ℂ,H (with the action of the associated infinite unitary group) by spherical functions of finite-dimensional spaces of Hermitian matrices. In the type B case, our results include asymptotic results for the spherical functions associated with the Cartan motion groups of non-compact Grassmannians as the rank goes to infinity, and a classification of the Olshanski spherical functions of the associated inductive limits.

Emergent behaviours of a non-abelian quantum synchronisation model over the unitary group

Abstract We introduce a new non-abelian quantum synchronisation model over the unitary group, represented as a gradient flow, where state matrices asymptotically converge to a common one up to phase translation. We provide a sufficient framework leading to quantum synchronisation based on Riccati-type differential inequalities. In addition, uniform time-delayed interaction is considered for modelling realistic communication, and we demonstrate that quantum synchronisation is persistent when a small time delay is allowed. Finally, numerical simulation is performed to visualise qualitative behaviours and support theoretical results.