Symplectic Symmetry Research Articles

Symplectic symmetry approach to clustering in atomic nuclei: The case of $^{24}$Mg

Abstract Symplectic symmetry approach to clustering (SSAC) in atomic nuclei,
recently proposed, is modified and further developed in more detail.
It is firstly applied to the light two-cluster $^{20}$Ne + $\alpha$
system of $^{24}$Mg, the latter exhibiting well developed low-energy
$K^{\pi} = 0^{+}_{1}$, $K^{\pi} = 2^{+}_{1}$ and $K^{\pi} =
0^{-}_{1}$ rotational bands in its spectrum. A simple algebraic
Hamiltonian, consisting of dynamical symmetry, residual and vertical
mixing parts is used to describe these three lowest rotational bands
of positive and negative parity in $^{24}$Mg. A good description of
the excitation energies is obtain by considering only the $SU(3)$
cluster states restricted to the stretched many-particle Hilbert
subspace, built on the leading Pauli allowed $SU(3)$ multiplet for
the positive- and negative-parity states, respectively. The coupling
to the higher cluster-model configurations allows to describe the
known low-lying experimentally observed $B(E2)$ transition
probabilities within and between the cluster states of the three
bands under consideration without the use of an effective charge.

Averages of products of characteristic polynomials and the law of real eigenvalues for the real Ginibre ensemble

We review an elementary derivation of the Borodin–Sinclair–Forrester–Nagao Pfaffian point process, which characterizes the law of real eigenvalues for the real Ginibre ensemble in the large matrix size limit, in terms of averages of products of characteristic polynomials. This derivation relies on a number of interesting structures associated with the real Ginibre ensemble such as the hidden symplectic symmetry of the statistics of real eigenvalues. It leads to a representation for the [Formula: see text]-point correlation function for any [Formula: see text] in terms of an integral over the symmetric space [Formula: see text] and this paper gives the proof of asymptotic exactness for this integral.

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.

Momentum-space signatures of the Anderson transition in a symplectic, two-dimensional, disordered ultracold gas

We study Anderson localization in two-dimensional, disordered, spin-orbit systems belonging to the symplectic symmetry class using momentum-space signatures such as the coherent backscattering antipeak and the coherent forward-scattering peak. Significantly, these momentum-space features are readily accessible in ultracold atom experiments through absorption imaging after time-of-flight expansion. Here, the critical exponent and mobility edge of the metal-insulator transition are successfully obtained through a finite-time analysis of the coherent backscattering width. An anomalous residual diffusion, unique to two dimensions, is identified at the transition point where the system changes from a metal to an insulator. A spin localization phenomenon is also observed in the deep localized regime. Published by the American Physical Society 2024

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The degeneracies of 1/4 BPS states with unit torsion in heterotic string theory compactified on a six torus are given in terms of the Fourier coefficients of the reciprocal of the Igusa cusp Siegel modular form Φ10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi _{10}$$\\end{document} of weight 10. We use the symplectic symmetries of the latter to construct a fine-grained Rademacher-type expansion which expresses these BPS degeneracies as a regularized sum over residues of the poles of 1/Φ10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/\\Phi _{10}$$\\end{document}. The construction uses two distinct SL(2,Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{SL}(2, {\\mathbb {Z}})$$\\end{document} subgroups of Sp(2,Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Sp}(2, {\\mathbb {Z}})$$\\end{document} which encode multiplier systems, Kloosterman sums and Eichler integrals appearing therein. Additionally, it shows how the polar data are explicitly built from the Fourier coefficients of 1/η24\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/\\eta ^{24}$$\\end{document} by means of a continued fraction structure.

Experimental Distributions of the Reflection Amplitude for Networks with Unitary and Symplectic Symmetries

Experimental Distributions of the Reflection Amplitude for Networks with Unitary and Symplectic Symmetries

Third quantization for bosons: symplectic diagonalization, non-Hermitian Hamiltonian, and symmetries

Open quantum systems that interact with a Markovian environment can be described by a Lindblad master equation. The generator of time-translation is given by a Liouvillian superoperator acting on the density matrix of the system. As the Fock space for a single bosonic mode is already infinite-dimensional, the diagonalization of the Liouvillian has to be done on the creation- and annihilation-superoperators, a process called ‘third quantization’. We propose a method to solve the Liouvillian for quadratic systems using a single symplectic transformation. We show that the non-Hermitian effective Hamiltonian of the system, next to incorporating the dynamics of the system, is a tool to analyze its symmetries. As an example, we use the effective Hamiltonian to formulate a -‘symmetry’ of an open system. We describe how the inclusion of source terms allows us to obtain the cumulant generating function for observables such as the photon current.

Symplectic splitting of Hamiltonian structures and reduced monodromy matrices

In Hamiltonian systems monodromy and reduced monodromy matrices of periodic orbits are symplectic. A symmetry is a symplectic or anti-symplectic involution which leaves the Hamiltonian invariant. A natural class to study are periodic orbits which are invariant under such involutions. Depending on whether the orbit is invariant under a symplectic or anti-symplectic symmetry, its monodromy matrix satisfies special properties. In the symplectic case, the monodromy matrix admits a symplectic decomposition, depending on the symplectic involution's fixed point set. In the anti-symplectic case, the monodromy matrix is of special symmetric form, which was constructed by Frauenfelder–Moreno. Such monodromy matrices are very beneficial in applied problems. The goal of this paper is to design a general construction behind the reduced monodromy's symplectic splitting, by using the notion of Hamiltonian manifolds, and to give a general framework for monodromy and reduced monodromy matrices of invariant periodic orbits in Hamiltonian systems with symmetries. In addition, we show an application related to contact forms, especially to the spatial circular restricted three-body problem for energy level sets of contact type.

Spherical Induced Ensembles with Symplectic Symmetry

We consider the complex eigenvalues of the induced spherical Ginibre ensemble with symplectic symmetry and establish the local universality of these point processes along the real axis. We derive scaling limits of all correlation functions at regular points both in the strong and weak non-unitary regimes as well as at the origin having spectral singularity. A key ingredient of our proof is a derivation of a differential equation satisfied by the correlation kernels of the associated Pfaffian point processes, thereby allowing us to perform asymptotic analysis.

Robustness of negative low-field magnetoconductance in ultrathin topological insulator films

In the ultrathin films of three-dimensional topological insulators (TIs), modification of the Berry phase caused by surface hybridization was predicted to allow for a crossover between weak antilocalization (WAL) and weak localization (WL). Here we report on a systematic study of the magnetotransport properties of ${({\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x})}_{2}{\mathrm{Te}}_{3}$ (BST) thin films with well-controlled thickness, doping level, and chemical potential. We found that the sign of magnetoconductance in perpendicular magnetic field remains negative in all circumstances, suggesting absence of global WL in the ultrathin BST films. This is attributed to long-range disorder, as well as conductance corrections by quantum interference and electron-electron interaction. Our work shows that the electron transport in ultrathin TIs is distinctively different from the WAL regime in the systems with symplectic symmetry, and the WL or weak insulator regime associated with the orthogonal symmetry class.

Ab initio symmetry-adapted emulator for studying emergent collectivity and clustering in nuclei

We discuss emulators from the ab initio symmetry-adapted no-core shell-model framework for studying the formation of alpha clustering and collective properties without effective charges. We present a new type of an emulator, one that utilizes the eigenvector continuation technique but is based on the use of symplectic symmetry considerations. This is achieved by using physically relevant degrees of freedom, namely, the symmetry-adapted basis, which exploits the almost perfect symplectic symmetry in nuclei. Specifically, we study excitation energies, point-proton root-mean-square radii, along with electric quadrupole moments and transitions for 6Li and 12C. We show that the set of parameterizations of the chiral potential used to train the emulators has no significant effect on predictions of dominant nuclear features, such as shape and the associated symplectic symmetry, along with cluster formation, but slightly varies details that affect collective quadrupole moments, asymptotic normalization coefficients, and alpha partial widths up to a factor of two. This makes these types of emulators important for further constraining the nuclear force for high-precision nuclear structure and reaction observables.

Scattering and transport properties of the three classical Wigner-Dyson ensembles at the Anderson transition.

An extensive numerical analysis of the scattering and transport properties of the power-law banded random matrix model (PBRM) at criticality in the presence of orthogonal, unitary, and symplectic symmetries is presented. Our results show a good agreement with existing analytical expressions in the metallic regime and with heuristic relations widely used in studies of the PBRM model in the presence of orthogonal and unitary symmetries. Moreover, our results confirm that the multifractal behavior of disordered systems at criticality can be probed by measuring scattering and transport properties, which is of paramount importance from the experimental point of view. Thus, a full picture of the scattering and transport properties of the PBRM model at criticality corresponding to the three classical Wigner-Dyson ensembles is provided.

Distributions of the Wigner reaction matrix for microwave networks with symplectic symmetry in the presence of absorption.

We report on experimental studies of the distribution of the reflection coefficients, and the imaginary and real parts of Wigner's reaction (K) matrix employing open microwave networks with symplectic symmetry and varying size of absorption. The results are compared to analytical predictions derived for the single-channel scattering case within the framework of random-matrix theory (RMT). Furthermore, we performed Monte Carlo simulations based on the Heidelberg approach for the scattering (S) and K matrix of open quantum-chaotic systems and the two-point correlation function of the S-matrix elements. The analytical results and the Monte Carlo simulations depend on the size of absorption. To verify them, we performed experiments with microwave networks for various absorption strengths. We show that deviations from RMT predictions observed in the spectral properties of the corresponding closed quantum graph and attributed to the presence of nonuniversal short periodic orbits does not have any visible effects on the distributions of the reflection coefficients and the K and S matrices associated with the corresponding open quantum graph.

Exact solution of the topological symplectic Kondo problem

The Kondo effect is an archetypical phenomenon in the physics of strongly correlated electron systems. Recent attention has focused on the application of Kondo physics to quantum information science by exploiting overscreened Kondo impurities with residual anyon-like impurity entropy. While this physics was proposed in the fine-tuned multi-channel Kondo setup or in the Majorana-based topological Kondo effect, we here study the Kondo effect with symplectic symmetry Sp(2k) and present details about the implementation which importantly only involves conventional s-wave superconductivity coupled to an array of resonant levels and neither requires perfect channel symmetry nor Majorana fermions. We carefully discuss the role of perturbations and show that a global Zeeman drives the system to a 2-channel SU(k) fixed point. Exact results for the residual entropy, specific heat and magnetization are derived using the thermodynamic Bethe Ansatz for Sp(2k). This solution not only proves the existence of a quantum critical ground state with anyon-like Hilbert space dimension, but also a particularly weak non-Fermi liquid behavior at criticality. We interpret the weakness of non-analyticities as a manifestation of suppressed density of states at the impurity causing only a very weak connection of putative anyons and conduction electrons. Given this weak connection, the simplicity of the design and the stability of the effect, we conjecture that the symplectic Kondo effect may be particularly suitable for quantum information applications.

Wronskian structures of planar symplectic ensembles

We consider the eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble, which are known to form a Pfaffian point process in the plane. It was recently discovered that the limiting correlation kernel of the symplectic Ginibre ensemble in the vicinity of the real line can be expressed in a unified form of a Wronskian. We derive scaling limits for variations of the symplectic Ginibre ensemble and obtain such Wronskian structures for the associated universality classes. These include almost-Hermitian bulk/edge scaling limits of the elliptic symplectic Ginibre ensemble and edge scaling limits of the symplectic Ginibre ensemble with boundary confinement. Our proofs follow from the generalised Christoffel–Darboux formula for the former and from the Laplace method for the latter. Based on such a unified integrable structure of Wronskian form, we also provide an intimate relation between the function in the argument of the Wronskian in the symplectic symmetry class and the kernel in the complex symmetry class which form determinantal point processes in the plane.

Random matrix theory and moments of moments of L-functions

In this paper, we give an analytic proof of the asymptotic behavior of the moments of moments of the characteristic polynomials of random symplectic and orthogonal matrices. We therefore obtain alternate, integral expressions for the leading order coefficients previously found by Assiotis, Bailey and Keating. We also discuss the conjectures of Bailey and Keating for the corresponding moments of moments of [Formula: see text]-functions with symplectic and orthogonal symmetry. Specifically, we show that these conjectures follow from the shifted moments conjecture of Conrey, Farmer, Keating, Rubinstein and Snaith.

On the Dominance of Deformation in Nuclear Physics, its Link to Symplectic Symmetry, and its Roots in an Effective Field Theory

A brief historical review of the important role symmetries have played in our gaining a deeper understanding nuclear structure is presented.We then focus our attention on the special role that symplectic symmetry plays in exposing a “Dominance of Deformation” that is found through enhanced B(E2) rates across the Chart of the Nuclides, and in addition, we will show how the No-Core Symplectic Shell Model (NC-SpSM) seems to emerge from a Symplectic Effective Field Theory(Sp-EFT), where the latter prepares away forward for gaining a truly ab initio understanding of the structure of atomic nuclei. As space permits, various spectra, B(E2) transition rates, and nuclear radii of selected light to medium-mass nuclei are shown.

Symplectic effective field theory for nuclear structure studies

A symplectic effective field theory that unveils the observed emergence of symplectic symmetry in atomic nuclei is advanced. Specifically, starting from a simple extension of the harmonic-oscillator Lagrangian, an effective field theory applied against symplectic basis states is shown to yield a Hamiltonian system with one fitted parameter. The scale of the system can be determined self-consistently as the ratio of the average volume of a nucleus assumed to be spherical to its volume as determined by the average number of oscillator quanta, which is stretched by the fact that the plane-wave solution satisfies the equations of motion at every order without the need for perturbative corrections. As an application of the theory, results for $^{20}\mathrm{Ne}$, $^{22}\mathrm{Ne}$, and $^{22}\mathrm{Mg}$ are presented that yield energy spectra, $B(E2)$ values, and matter radii in good agreement with experimentally measured results.

Supercell symmetry modified spectral statistics of Kramers–Weyl fermions* *Contribution to the special issue of J. Phys. A in honour of the life and work of Fritz Haake.

We calculate the spectral statistics of the Kramers–Weyl Hamiltonian H = v∑ α σ α sin p α + tσ 0∑ α cos p α in a chaotic quantum dot. The Hamiltonian has symplectic time-reversal symmetry (H is invariant when spin σ α and momentum p α both change sign), and yet for small t the level spacing distributionP(s) ∝s β follows the β = 1 orthogonal ensemble instead of the β = 4 symplectic ensemble. We identify a supercell symmetry of H that explains this finding. The supercell symmetry is broken by the spin-independent hopping energy ∝t cos p, which induces a transition from β = 1 to β = 4 statistics that shows up in the conductance as a transition from weak localization to weak antilocalization.

Ab initio View of Emergent Symplectic Symmetry and Its Crutial Role in Nuclear Dynamics

Published from 1956, the Bulgarian Journal of Physics accepts articles in the fields of: General and Mathematical Physics; Particle Physics and Field Theory; Atomic and Molecular Physics; Nuclear Physics; Fundamental Areas of Phenomenology; Theory of Relativity and Astrophysics; Condensed Matter, Statistical Physics; Cross-Disciplinary Physics; and Related Areas of Science and Technology.