Symplectic Symmetry Research Articles

Strain-induced transitions to quantum chaos and effective time-reversal symmetry breaking in triangular graphene nanoflakes

We investigate the effect of strain-induced gauge fields on statistical distribution of energy levels of triangular graphene nanoflakes with zigzag edges. In the absence of strain fields but in the presence of weak potential disorder such systems were found in Ref. [1] to display the spectral statistics of the Gaussian unitary ensemble (GUE) due to the effective time-reversal (symplectic) symmetry breaking. Here show that, in the absence of disorder, strain fields may solely lead to spectral fluctuations of GUE providing a nanoflake is deformed such that all its geometric symmetries are broken. In a particular case when a single mirror symmetry is preserved the spectral statistics follow the Gaussian orthogonal ensemble (GOE) rather then GUE. The corresponding transitions to quantum chaos are rationalized by means of additive random-matrix models and the analogy between strain-induced gauge fields and real magnetic fields is discussed.

Symplectic realizations and symmetries of a Lotka-Volterra type system

In this paper a Lotka-Volterra type system is considered. For such a system, bi-Hamiltonian formulation, symplectic realizations and symmetries are presented.

Intermediate statistics for a system with symplectic symmetry: the Dirac rose graph

We study the spectral statistics of the Dirac operator on a rose-shaped graph—a graph with a single vertex and all bonds connected at both ends to the vertex. We formulate a secular equation that generically determines the eigenvalues of the Dirac rose graph, which is seen to generalize the secular equation for a star graph with Neumann boundary conditions. We derive approximations to the spectral pair correlation function at large and small values of spectral spacings, in the limit as the number of bonds approaches infinity, and compare these predictions with results of numerical calculations. Our results represent the first example of intermediate statistics from the symplectic symmetry class.

Global phase diagram of two-dimensional Dirac fermions in random potentials

Anderson localization is studied for two flavors of massless Dirac fermions in 2D space perturbed by static disorder that is invariant under a chiral symmetry (chS) and a time-reversal symmetry (TRS) operation which, when squared, is equal either to plus or minus the identity. The former TRS (symmetry class BDI) can for example be realized when the Dirac fermions emerge from spinless fermions hopping on a 2D lattice with a linear energy dispersion such as the honeycomb lattice (graphene) or the square lattice with $\pi$-flux per plaquette. The latter TRS is realized by the surface states of 3D $\mathbb{Z}_{2}$-topological band insulators in symmetry class CII. In the phase diagram parametrized by the disorder strengths, there is an infrared stable line of critical points for both symmetry classes BDI and CII. Here we discuss a "global phase diagram" in which disordered Dirac fermion systems in all three chiral symmetry classes, AIII, CII, and BDI, occur in 4 quadrants, sharing one corner which represents the clean Dirac fermion limit. This phase diagram also includes symmetry classes AII [e.g., appearing at the surface of a disordered 3D $\mathbb{Z}_2$-topological band insulator in the spin-orbit (symplectic) symmetry class] and D (e.g., the random bond Ising model in two dimensions) as boundaries separating regions of the phase diagram belonging to the three chS classes AIII, BDI, and CII. Moreover, we argue that physics of Anderson localization in the CII phase can be presented in terms of a non-linear-sigma model (NLsM) with a $\mathbb{Z}_{2}$-topological term. We thereby complete the derivation of topological or Wess-Zumino-Novikov-Witten terms in the NLsM description of disordered fermionic models in all 10 symmetry classes relevant to Anderson localization in two spatial dimensions.

Scaling at chiral quantum critical points in two dimensions

We study the localization properties of electrons moving on two-dimensional bi-partite lattices in the presence of disorder. The models investigated exhibit a chiral symmetry and belong to the chiral orthogonal (chO), chiral symplectic (chS) or chiral unitary (chU) symmetry class. The disorder is introduced via real random hopping terms for chO and chS, while complex random phases generate the disorder in the chiral unitary model. In the latter case an additional spatially constant, perpendicular magnetic field is also applied. Using a transfer-matrix-method, we numerically calculate the smallest Lyapunov exponents that are related to the localization length of the electronic eigenstates. From a finite-size scaling analysis, the logarithmic divergence of the localization length at the quantum critical point at E=0 is obtained. We always find for the critical exponent \kappa, which governs the energy dependence of the divergence, a value close to 2/3. This result differs from the exponent \kappa=1/2 found previously for a chiral unitary model in the absence of a constant magnetic field. We argue that a strong constant magnetic field changes the exponent \kappa within the chiral unitary symmetry class by effectively restoring particle-hole symmetry even though a magnetic field induced transition from the ballistic to the diffusive regime cannot be fully excluded.

Disorder-induced quantum phase transitions in three-dimensional topological insulators and superconductors

We discuss the effects of disorder in time-reversal invariant topological insulators and superconductors in three spatial dimensions. For three-dimensional topological insulator in symplectic (AII) symmetry class, the phase diagram in the presence of disorder and a mass term, which drives a transition between trivial and topological insulator phases, is computed numerically by the transfer matrix method. The numerics is supplemented by a field theory analysis (the large-$N_f$ expansion where $N_f$ is the number of valleys or Dirac cones), from which we obtain the correlation length exponent, and several anomalous dimensions at a non-trivial critical point separating a metallic phase and a Dirac semi-metal. A similar field theory approach is developed for disorder-driven transitions in symmetry class AIII, CI, and DIII. For these three symmetry classes, where topological superconductors are characterized by integer topological invariant, a complementary description is given in terms of the non-linear sigma model supplemented with a topological term which is a three-dimensional analogue of the Pruisken term in the integer quantum Hall effect.

Master symmetries, non-Hamiltonian symmetries and superintegrability of the generalized Smorodinsky–Winternitz system

The first part of the paper is devoted to the theory of master symmetries using the geometric formalism as an approach. It is shown that certain superintegrable systems are endowed with this property as a consequence of the existence of a family of master symmetries. In the second part, the properties of dynamical but non-Hamiltonian symmetries are studied. It is proved that the higher order superintegrability of the generalized Smorodinsky–Winternitz system is a consequence of the existence of symplectic symmetries not preserving the Hamiltonian function.

Disordered two-dimensional electron systems with chiral symmetry

We review the results of our recent numerical investigations on the electronic properties of disordered two dimensional systems with chiral unitary, chiral orthogonal, and chiral symplectic symmetry. Of particular interest is the behavior of the density of states and the logarithmic scaling of the smallest Lyapunov exponents in the vicinity of the chiral quantum critical point in the band center at E=0. The observed peaks or depressions in the density of states, the distribution of the critical conductances, and the possible non-universality of the critical exponents for certain chiral unitary models are discussed.

Finite dimensional Hamiltonian system related to a Lax pair with symplectic and cyclic symmetries

For the 1 + 1 dimensional Lax pair with a symplectic symmetry and cyclic symmetries, it is shown that there is a natural finite-dimensional Hamiltonian system related to it by presenting a unified Lax matrix. The Liouville integrability of the derived finite-dimensional Hamiltonian systems is proved in a unified way. Any solution of these Hamiltonian systems gives a solution of the original PDE. As an application, the two-dimensional hyperbolic Toda equation is considered and the finite-dimensional integrable Hamiltonian system related to it is obtained from the general results.

Beyond the Heisenberg time: semiclassical treatment of spectral correlations in chaotic systems with spin 1/2

The two-point correlation function of chaotic systems with spin 1/2 is evaluated using periodic orbits. The spectral form factor for all times thus becomes accessible. Equivalence with the predictions of random matrix theory for the Gaussian symplectic ensemble is demonstrated. A duality between the underlying generating functions of the orthogonal and symplectic symmetry classes is semiclassically established.

Quantum magnetism in ultracold alkali and alkaline-earth fermion systems with symplectic symmetry

We numerically study quantum magnetism of ultra-cold alkali and alkaline-earth fermion systems with large hyperfine spin $F=3/2$, which are characterized by a generic $Sp(N)$ symmetry with N=4. The methods of exact diagonalization (ED) and density-matrix-renormalization-group are employed for the large size one-dimensional (1D) systems, and ED is applied to a two-dimensional (2D) square lattice on small sizes. We focus on the magnetic exchange models in the Mott-insulating state at quarter-filling. Both 1D and 2D systems exhibit rich phase diagrams depending on the ratio between the spin exchanges $J_0$ and $J_2$ in the bond spin singlet and quintet channels, respectively. In 1D, the ground states exhibit a long-range-ordered dimerization with a finite spin gap at $J_0/J_2>1$, and a gapless spin liquid state at $J_0/J_2 \le 1$, respectively. In the former and latter cases, the correlation functions exhibit the two-site and four-site periodicities, respectively. In 2D, various spin correlation functions are calculated up to the size of $4\times 4$. The Neel-type spin correlation dominates at large values of $J_0/J_2$, while a $2\times 2$ plaquette correlation is prominent at small values of this ratio. Between them, a columnar spin-Peierls dimerization correlation peaks. We infer the competitions among the plaquette ordering, the dimer ordering, and the Neel ordering in the 2D system.

Q-Deformation of symplectic dynamical symmetries in algebraic models of nuclear structure

With a view toward further nuclear structure applications of approaches based on quantum-deformed (or q-deformed) algebras, introduced to the authors by Yu.F. Smirnov, we construct a q analog of a boson realization of the symplectic noncompact sp(4, R) algebra together with a q analog of a fermion realization of the symplectic compact sp(4) algebra. The first study, on the q-deformed Sp(4,R) symmetry, is applied to the development of a q analog of the two-dimensional Interacting Boson Model with q-deformed SU(3) the underpinning dynamical symmetry group. An explicit realization in terms of q-tensor operators with respect to the standard suq(2) algebra is given. The group-subgroup structure of this framework yields the physical interpretation of the generators of the groups under consideration. The second symplectic algebra, the q-deformed sp(4), is applied to studying isovector pairing correlations in atomic nuclei. A specific q deformation of the sp(4) algebra is realized in terms of q deformed fermion creation and annihilation operators of the shell model. The generators of the algebra close on four distinct realizations of the uq(2) subalgebra. These reductions, which correspond to different types of pairing interactions, yield a complete classification of the basis states. An analysis of the role of the q deformation is based on a comparison of the results for energies of the lowest isovector-paired 0+ states in the deformed and nondeformed cases.

Orthogonality of Hermite polynomials in superspace and Mehler type formulae

In this paper, Hermite polynomials related to quantum systems with orthogonal O(m)-symmetry, finite reflection group symmetry 𝒢<O(m), symplectic symmetry Sp(2n) and superspace symmetry O(m)×Sp(2n) are considered. After an overview of the results for O(m) and 𝒢, the orthogonality of the Hermite polynomials related to Sp(2n) is obtained with respect to the Berezin integral. As a consequence, an extension of the Mehler formula for the classical Hermite polynomials to Grassmann algebras is proved. Next, Hermite polynomials in a full superspace with O(m)×Sp(2n)-symmetry are considered. It is shown that they are not orthogonal with respect to the canonically defined inner product. However, a new inner product is introduced, which behaves correctly with respect to the structure of harmonic polynomials on superspace. This inner product allows to restore the orthogonality of the Hermite polynomials and also restores the hermiticity of a class of Schrödinger operators in superspace. Subsequently, a Mehler formula for the full superspace is obtained, thus yielding an eigenfunction decomposition of the super Fourier transform. Finally, the new results for the Sp(2n)- and O(m)×Sp(2n)-symmetry are compared with the results in the different types of symmetry.

Symplectic Dynamical Symmetries in Algebraic Models of Nuclear Structure

Based on a generalized reduction scheme for boson representations of symplectic algebras of the type Sp(4k,R), we consider the symplectic extension of a boson realization of compact unitary algebras for the k = 1, k = 3 and k = 6 cases, which have relevance in nuclear structure theory. First we review an application of the k = 1 case for the creation of a Sp(4, R) classification scheme, which is used for obtaining a generalized phenomenological description of important nuclear characteristics in terms of the classification quantum numbers for large sets of nuclei. Then for the k = 3 and k = 6 cases we outline some of the new possibilities that appear in the symplectic extensions of the Interacting Vector Boson Model (IVBM) and the Interacting Boson Model (IBM-2), respectively. The examples presented are used to describe the collective modes of the nuclear spectra in individual nuclei as well as in sequences of nuclei.

Symmetric symplectic homotopy K 3 surfaces

A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is shown that an effective action by various maximal symplectic K3 groups forces the corresponding homotopy K3 surface to be minimally exotic with respect to our measure. (However, the standard K3 is the only known example of such minimally exotic homotopy K3 surfaces.) The possible structure of a finite group of symplectic symmetries of a minimally exotic homotopy K3 surface is determined and future research directions are indicated.

Moments of Wishart-Laguerre and Jacobi ensembles of random matrices: application to the quantum transport problem in chaotic cavities

We collect explicit and user-friendly expressions for one-point densities of the real eigenvalues $\{\lambda_i\}$ of $N\times N$ Wishart-Laguerre and Jacobi random matrices with orthogonal, unitary and symplectic symmetry. Using these formulae, we compute integer moments $\tau_n=<\sum_{i=1}^N\lambda_i^n>$ for all symmetry classes without any large $N$ approximation. In particular, our results provide exact expressions for moments of transmission eigenvalues in chaotic cavities with time-reversal or spin-flip symmetry and supporting a finite and arbitrary number of electronic channels in the two incoming leads.

Symplectic symmetry feature of thermoacoustic network

Symplectic mathematics is introduced into the thermoacoustic network model. The transferring matrix of thermoacoustic system is analyzed, and the transferring matrix of working gas in isothermal fluid pipe of thermoacoustic system is a symplectic matrix. The transferring matrix of working gas in regenerator of thermoacoustic system is not a symplectic matrix, but it can be converted into a symplectic matrix by variable transformation. With variable transformation, the whole transferring matrix of thermoacoustic system can be represented by a symplectic matrix. The form of symplectic matrix is conducible to analyzing and calculating the thermoacoutic network model.

Optics and nonlinear effects in repetitive systems

The weakly nonlinear dynamics in repetitive systems like synchrotrons, storage rings, linacs, and many new compact accelerators in the class of FFAGs is determined by criteria different from those in single pass optical systems. In the latter case, a clean description is obtained by the aberrations, i.e. nonlinear terms in the Taylor representation of the transfer map, and possible connections between them because of the symplectic symmetry of Hamiltonian systems or geometric symmetries of the system under consideration. For repetitive systems other criteria are relevant because conventional aberrations can be either enhanced or weakened by the repeated passage through the same system.We begin with methods that provide highly accurate representations of fields, a pre-requisite for a detailed analysis of nonlinear motion, and discuss surface-based methods that assure accurate field representation far from midplane or axis. We then discuss methods to enforce the symplectic symmetry of the motion by minimal deformations in a suitable metric of Hamiltonian spaces, which is important in the attempt of long-term tracking of the dynamics. We then develop an analytical mechanism most closely related to the conventional aberration correction in conventional systems, the method of normal forms. As a result, quantitative measures of those linearities that matter are obtained in the form of resonance strengths and tune shifts. Residual effects can be described quantitatively using normal form defect methods, which allow rigorous predictions of long-term stability.For practical calculations, the above methods can be expressed seamlessly in terms of differential algebraic structures on Taylor representations, either in a truncated representation in the conventional DA framework, or under consideration of remainder error terms in a fully rigorous setting.

Classification of K3-surfaces with involution and maximal symplectic symmetry

K3-surfaces with antisymplectic involution and compatible symplectic actions of finite groups are considered. In this situation actions of large finite groups of symplectic transformations are shown to arise via double covers of Del Pezzo surfaces, and a complete classification of K3-surfaces with maximal symplectic symmetry is obtained.

Conformal invariance, multifractality, and finite-size scaling at Anderson localization transitions in two dimensions

We generalize universal relations between the multifractal exponent \alpha_0 for the scaling of the typical wave function magnitude at a (Anderson) localization-delocalization transition in two dimensions and the corresponding critical finite size scaling (FSS) amplitude \Lambda_c of the typical localization length in quasi-one-dimensional (Q1D) geometry: (i) When open boundary conditions are imposed in the transverse direction of Q1D samples (strip geometry), we show that the corresponding critical FSS amplitude \Lambda_c^o is universally related to the boundary multifractal exponent \alpha_0^s for the typical wave function amplitude along a straight boundary (surface). (ii) We further propose a generalization of these universal relations to those symmetry classes whose density of states vanishes at the transition. (iii) We verify our generalized relations [Eqs. (6) and (7)] numerically for the following four types of two-dimensional Anderson transitions: (a) the metal-to-(ordinary insulator) transition in the spin-orbit (symplectic) symmetry class, (b) the metal-to-(Z_2 topological insulator) transition which is also in the spin-orbit (symplectic) class, (c) the integer quantum Hall plateau transition, and (d) the spin quantum Hall plateau transition.