Arbitrary Symmetry Research Articles

Parameterizing density operators with arbitrary symmetries to gain advantage in quantum state estimation

In this work, we show how to parameterize a density matrix that has an arbitrary symmetry, knowing the generators of the Lie algebra (if the symmetry group is a connected Lie group) or the generators of its underlying group (in case it is finite). This allows to pose MaxEnt and MaxLik estimation techniques as convex optimization problems with a substantial reduction in the number of parameters of the function involved. This implies that, apart from a computational advantage due to the fact that the optimization is performed in a reduced space, the amount of experimental data needed for a good estimation of the density matrix can be reduced as well. In addition, we run numerical experiments and apply these parameterizations to estimate quantum states with different symmetries.

Entanglement-Asymmetry Correspondence for Internal Quantum Reference Frames.

In the quantization of gauge theories and quantum gravity, it is crucial to treat reference frames such as rods or clocks not as idealized external classical relata, but as internal quantum subsystems. In the Page-Wootters formalism, for example, evolution of a quantum system S is described by a stationary joint state of S and a quantum clock, where time dependence of S arises from conditioning on the value of the clock. Here, we consider (possibly imperfect) internal quantum reference frames R for arbitrary compact symmetry groups, and show that there is an exact quantitative correspondence between the amount of entanglement in the invariant state on RS and the amount of asymmetry in the corresponding conditional state on S. Surprisingly, this duality holds exactly regardless of the choice of coherent state system used to condition on the reference frame. Averaging asymmetry over all conditional states, we obtain a simple representation-theoretic expression that admits the study of the quality of imperfect quantum reference frames, quantum speed limits for imperfect clocks, and typicality of asymmetry in a unified way. Our results shed light on the role of entanglement for establishing asymmetry in a fully symmetric quantum world.

Analytical and numerical modeling of optical second harmonic generation in anisotropic crystals using ♯SHAARP package

Electric-dipole optical second harmonic generation (SHG) is a second-order nonlinear process that is widely used as a sensitive probe to detect broken inversion symmetry and local polar order. Analytical modeling of the SHG polarimetry of a nonlinear optical material is essential to extract its point group symmetry and the absolute nonlinear susceptibilities. Current literature on SHG analysis involves numerous approximations and a wide range of (in)accuracies. We have developed an open-source package called the Second Harmonic Analysis of Anisotropic Rotational Polarimetry (♯SHAARP.si) which derives analytical and numerical solutions of reflection SHG polarimetry from a single interface (.si) for bulk homogeneous crystals with arbitrary symmetry group, arbitrary crystal orientation, complex and anisotropic linear dielectric tensor with frequency dispersion, a general SHG tensor and arbitrary light polarization. ♯SHAARP.si enables accurate modeling of polarimetry measurements in reflection geometry from highly absorbing crystals or wedge-shaped transparent crystals. The package is extendable to multiple interfaces.

Analysis of Acousto-Optic Figure of Merit in KGW and KYW Crystals.

Monoclinic potassium rare-earth crystals are known as efficient materials for solid-state lasers and acousto-optic modulators. A number of specific configurations for acousto-optic devices based on those crystals have recently been proposed, but the acousto-optic effect of those crystals has only been analyzed fragmentarily for some interaction directions. In this work, we numerically searched for the global maxima of an acousto-optic figure of merit for isotropic diffraction in KGd(WO4)2 and KY(WO4)2 crystals. It was demonstrated that the global maxima of the acousto-optic figure of merit in those crystals occur in the slow optical mode propagating along the crystal's twofold symmetry axis and in the acoustic wave propagating orthogonally, both for quasi-longitudinal and quasi-shear acoustic modes. The proposed calculation method can be readily used for the optimization of the acousto-optic interaction geometry in crystals with arbitrary symmetry.

Finite element method for the quasiclassical theory of superconductivity

The Eilenberger-Larkin-Ovchinnikov-Eliashberg quasiclassical theory of superconductivity is a powerful method enabling studies of a wide range of equilibrium and non-equilibrium phenomena in conventional and unconventional superconductors. We introduce here a finite element method, based on a discontinuous Galerkin approach, to self-consistently solve the underlying transport equations for general device geometries, arbitrary mean free path and symmetry of the superconducting order parameter. We present exemplary results on i) the influence of scalar impurity scattering on phase crystals in $d$-wave superconducting grains at low temperatures and ii) the current flow and focusing in $d$-wave superconducting weak links, modeling recent experimental realizations of grooved high-temperature superconducting Dayem bridges. The high adaptability of this finite element method for quasiclassical theory paves the way for future investigations of superconducting devices and new physical phenomena in unconventional superconductors.

Strategies for Determining the Cascade Rate in MHD Turbulence: Isotropy, Anisotropy, and Spacecraft Sampling

Exact laws for evaluating cascade rates, tracing back to the Kolmogorov “4/5” law, have been extended to many systems of interest including magnetohydrodynamics (MHD), and compressible flows of the magnetofluid and ordinary fluid types. It is understood that implementations may be limited by the quantity of available data and by the lack of turbulence symmetry. Assessment of the accuracy and feasibility of such third-order (or Yaglom) relations is most effectively accomplished by examining the von Kármán–Howarth equation in increment form, a framework from which the third-order laws are derived as asymptotic approximations. Using this approach, we examine the context of third-order laws for incompressible MHD in some detail. The simplest versions rely on the assumption of isotropy and the presence of a well-defined inertial range, while related procedures generalize the same idea to arbitrary rotational symmetries. Conditions for obtaining correct and accurate values of the dissipation rate from these laws based on several sampling and fitting strategies are investigated using results from simulations. The questions we address are of particular relevance to sampling of solar wind turbulence by one or more spacecraft.

Analytical non-Hermitian description of photonic crystals with arbitrary lateral and transverse symmetry

In this work we propose a general theoretical approach to the modelling of complex dispersion characteristics of leaky optical modes operating in photonic crystal slab composed of two high-index contrast gratings, beyond the protection of the light cone. Opening access of wave-guided resonances to free space continuum provides large amount of extra degrees of freedom for mode coupling engineering. Not only can the two gratings communicate via near field coupling, but they are also allowed to couple via the propagating radiated field. Our analytical model, based on a non-Hermitian Hamiltonian, including both coupling schemes, allows for a unified description of the wide family of optical modes which may be generated within uni-dimensional photonic crystal. Through a variety of illustrative examples, we show that our theoretical approach provide a simplified categorization of these modes, but it is also a powerful enabler for the discovery of novel photonic species. Finally, as proof-of-concept, we demonstrate experimentally the formation of a Dirac point at the merging of three bound states in the continuum that is the most achieved photonic specie discussed in this work.

Finite electro-elasticity with physics-augmented neural networks

In the present work, a machine learning based constitutive model for electro-mechanically coupled material behavior at finite deformations is proposed. Using different sets of invariants as inputs, an internal energy density is formulated as a convex neural network. In this way, the model fulfills the polyconvexity condition which ensures material stability, as well as thermodynamic consistency, objectivity, material symmetry, and growth conditions. Depending on the considered invariants, this physics-augmented machine learning model can either be applied for compressible or nearly incompressible material behavior, as well as for arbitrary material symmetry classes. The applicability and versatility of the approach is demonstrated by calibrating it on transversely isotropic data generated with an analytical potential, as well as for the effective constitutive modeling of an analytically homogenized, transversely isotropic rank-one laminate composite and a numerically homogenized cubic metamaterial. These examinations show the excellent generalization properties that physics-augmented neural networks offer also for multi-physical material modeling such as nonlinear electro-elasticity.

A high-efficient strain-stress method for calculating higher-order elastic constants from first-principles

Though the method for calculating higher-order elastic constants (HOECs) have a long history, there still exist some unsolved issues (e.g. low efficiency), making the development of HOECs much slower than second-order elastic constants. In this paper, we present a general and efficient strain-stress method (SSM) for calculating HOECs from first-principles. In this method, the required number of strain modes is sharply cut down, which ensures a higher efficiency. The time spent on traditional methods is about 3-5 times that of SSM for calculating TOECs of diamond. By taking the HOECs into consideration, the convergence against maximum strain in SSM gets improved significantly, and the results of diamond, gold and magnesium obtained in SSM agree well with previous calculations by other methods. To accelerate the development of the calculation tools for HOECs, we present an algorithm, as well as an open source code, to deduce the strain modes and corresponding coefficients. Specially, we give an explicit expression of strain modes and corresponding coefficients for calculating TOECs in arbitrary symmetry and fifth-order elastic constants in CI (Laue group) symmetry. In addition, we make some extension, e.g. high-accurate numerical differentiation formula, of some existing methods.

Acoustic wave tunneling across a vacuum gap between two piezoelectric crystals with arbitrary symmetry and orientation

It is not widely appreciated that an acoustic wave can ``jump'' or ``tunnel'' across a vacuum gap between two piezoelectric solids, nor has the general case been formulated or studied in detail. Here, we remedy that situation, by presenting a general formalism and approach to study such an acoustic tunneling effect between two arbitrarily oriented anisotropic piezoelectric semi-infinite crystals. The approach allows one to solve for the reflection and transmission coefficients of all the partial-wave modes, and is amenable to practical numerical or even analytical implementation, as we demonstrate by a few chosen examples. The formalism can be used in the future for quantitative studies of the tunneling effect in connection not only with the manipulation of acoustic waves, but with many other areas of physics of vibrations such as heat transport, for example.

Performance of the quantum MaxEnt estimation in the presence of physical symmetries

When an informationally complete measurement is not available, the reconstruction of the density operator that describes the state of a quantum system can be obtained, in a reliable way, by adopting the maximum entropy principle (MaxEnt principle), as an additional criterion, to achieve the least biased estimation. In this paper, we study the performance of the MaxEnt method for quantum state estimation when there is prior information about symmetries of the unknown state. We explicitly describe how to work with this method in the most general case, and we present an algorithm that allows to improve the estimation of quantum states with arbitrary symmetries. Furthermore, we implement this algorithm to carry out numerical simulations estimating the density matrix of several three-qubit states of particular interest for quantum information tasks. We observed that, for most states, our approach allows to considerably reduce the number of independent measurements needed to obtain a sufficiently high fidelity in the reconstruction of the density matrix. Moreover, we analyze the performance of the method in realistic scenarios, showing that it is robust even when the effects of finite statistics and experimental noise are considered.

Constraint on parameters of a rotating black hole in Einstein-bumblebee theory by quasi-periodic oscillations

We have studied quasi-periodic oscillations frequencies in a rotating black hole with Lorentz symmetry breaking parameter in Einstein-bumblebee gravity by relativistic precession model. We find that in the rotating case with non-zero spin parameter both of the periastron and nodal precession frequencies increase with the Lorentz symmetry breaking parameter, but the azimuthal frequency decreases. In the non-rotating black hole case, the nodal precession frequency disappears for arbitrary Lorentz symmetry breaking parameter. With the observation data of GRO J1655-40, XTE J1550-564, and GRS 1915+105, we find that the constraint on the Lorentz symmetry breaking parameter is more precise with data of GRO J1655-40 in which the best-fit value of the Lorentz symmetry breaking parameter is negative. This could lead to that the rotating black hole in Einstein-bumblebee gravity owns the higher Hawking temperature and the stronger Hawking radiation, but the lower possibility of exacting energy by Penrose process. However, in the range of 1 sigma , we also find that general relativity remains to be consistent with the observation data of GRO J1655-40, XTE J1550-564 and GRS 1915+105.

An anisotropic equation of state for high-pressure, high-temperature applications

SUMMARY This paper presents a strategy for extending scalar (P–V–T) equations of state to self-consistently model anisotropic materials over a wide range of pressures and temperatures under nearly hydrostatic conditions. The method involves defining a conventional scalar equation of state (V(P, T) or P(V, T)) and a fourth-rank tensor state variable $\boldsymbol {\Psi }(V,T)$ whose derivatives can be used to determine the anisotropic properties of materials of arbitrary symmetry. This paper proposes two functional forms for $\boldsymbol {\Psi }(V,T)$ and provides expressions describing the relationship between $\boldsymbol {\Psi }$ and physical properties including the deformation gradient tensor, the lattice parameters, the isothermal elastic compliance tensor and thermal expansivity tensor. The isothermal and isentropic stiffness tensors, the Grüneisen tensor and anisotropic seismic velocities can be derived from these properties. To illustrate the use of the formulations, anisotropic models are parametrized using numerical simulations of cubic periclase and experimental data on orthorhombic San Carlos olivine.

A new algebraic approach to genome rearrangement models

We present a unified framework for modelling genomes and their rearrangements in a genome algebra, as elements that simultaneously incorporate all physical symmetries. Building on previous work utilising the group algebra of the symmetric group, we explicitly construct the genome algebra for the case of unsigned circular genomes with dihedral symmetry and show that the maximum likelihood estimate (MLE) of genome rearrangement distance can be validly and more efficiently performed in this setting. We then construct the genome algebra for a more general case, that is, for genomes that may be represented by elements of an arbitrary group and symmetry group, and show that the MLE computations can be performed entirely within this framework. There is no prescribed model in this framework; that is, it allows any choice of rearrangements that preserve the set of regions, along with arbitrary weights. Further, since the likelihood function is built from path probabilities—a generalisation of path counts—the framework may be utilised for any distance measure that is based on path probabilities.

On the Classification of Topological Orders

We axiomatize the extended operators in topological orders (possibly gravitationally anomalous, possibly with degenerate ground states) in terms of monoidal Karoubi-complete n-categories which are mildly dualizable and have trivial centre. Dualizability encodes the word “topological,” and we take it as the definition of “(separable) multifusion n-category”; triviality of the centre implements the physical principle of “remote detectability.” We show that such n-categorical algebras are Morita-invertible (in the appropriate higher Morita category), thereby identifying topological orders with anomalous fully-extended TQFTs. We identify centreless fusion n-categories (i.e. multifusion n-categories with indecomposable unit) with centreless braided fusion $$(n{-}1)$$ -categories. We then discuss the classification in low spacetime dimension, proving in particular that all $$(1{+}1)$$ - and $$(3{+}1)$$ -dimensional topological orders, with arbitrary symmetry enhancement, are suitably-generalized topological sigma models. These mathematical results confirm and extend a series of conjectures and results by L. Kong, X.G. Wen, and their collaborators.

Asymptotic Analysis of Nonlinear Resonanсe Interaction between Capillary Waves with Arbitrary Symmetry on a Moving Charged Jet at Multimode Initial Deformation

Asymptotic Analysis of Nonlinear Resonanсe Interaction between Capillary Waves with Arbitrary Symmetry on a Moving Charged Jet at Multimode Initial Deformation

J-states and quantum channels between indefinite metric spaces

In the present work, we introduce and study the concepts of state and quantum channel on spaces equipped with an indefinite metric. Exclusively, we will limit our analysis to the matricial framework. As it will be confirmed below, from our research it is noticed that, when passing to the spaces with indefinite metric, the use of the adjoint of a matrix with respect to the indefinite metric is required in the construction of states and quantum channels; which prevents us to consider the space of matrices of certain order \(M_{n}({\mathbb {C}})\) as a \(C^{*}\)-algebra. In our case, this adjoint is defined through a J-metric, where the matrix J is a fundamental symmetry of \(M_{n}({\mathbb {C}})\). In our paper, for quantum operators, we include the general setting in the which, these operators map \(J_{1}\)-states into \(J_{2}\)-states, where \(J_{2}\ne \pm J_{1}\) are two arbitrary fundamental symmetries. In the middle of this program, we carry out a study of the completely positive maps between two different positive matrices spaces by considering two different indefinite metrics on \({\mathbb {C}}^{n}\).

Spontaneous symmetry breaking, spectral statistics, and the ramp

Ensembles of quantum chaotic systems are expected to exhibit energy eigenvalues with random-matrix-like level repulsion between pairs of energies separated by less than the inverse Thouless time. Recent research has shown that exact and approximate global symmetries of a system have clear signatures in these spectral statistics, enhancing the spectral form factor or correspondingly weakening level repulsion. This paper extends those results to the case of spontaneous symmetry breaking and shows that, surprisingly, spontaneously breaking a symmetry further enhances the spectral form factor. For both random-matrix-theory--inspired toy models and models where the symmetry breaking has a description in terms of fluctuating hydrodynamics, we obtain formulas for this enhancement for arbitrary symmetry breaking patterns, including broken Abelian symmetries ${Z}_{n}$ and $U(1)$ and partially or fully broken non-Abelian symmetries.

Weak-anisotropy approximation of P-wave reflection coefficient in anisotropic media of arbitrary symmetry and tilt

Applicability of the approximate expression for the P-wave reflection coefficient at a weak-contrast reflector separating two weakly anisotropic half-spaces of arbitrary symmetry is extended to media of arbitrary symmetry and tilt. The reflection coefficient consists of the approximate P-wave reflection coefficient at a weak-contrast interface separating two reference isotropic half-spaces and a correction term due to anisotropy. Along an arbitrarily chosen profile, the “isotropic” term depends on the density and P- and S-wave contrasts, and the correction term depends linearly on the contrasts of four profile weak-anisotropy (WA) parameters specifying anisotropy along the profile. In addition, both terms depend on the polar angle of incidence. In each half-space, the four profile WA parameters can be expressed as a linear combination of 12 of 21 global WA parameters specifying anisotropy of the half-space. Coefficients of this linear combination are functions of the azimuth of the profile. WA parameters are a generalization of Thomsen parameters to arbitrary anisotropy and represent an alternative to 21 independent elements of the stiffness tensor. WA parameters can be used for the approximation of other related concepts such as the reflection moveout or the geometric spreading. Presented tests illustrate high accuracy and flexibility of the proposed formula for the P-wave reflection coefficient. They also show that very accurate results can be obtained for contrasts and anisotropy, which cannot be considered weak.

Inverse design of two-dimensional structure by self-assembly of patchy particles.

We propose an optimization method for the inverse structural design of self-assembly of anisotropic patchy particles. The anisotropic interaction can be expressed by the spherical harmonics of the surface pattern on a patchy particle, and thus, arbitrary symmetries of the patch can be treated. The pairwise interaction potential includes several to-be-optimized parameters, which are the coefficients of each term in the spherical harmonics. We use the optimization method based on the relative entropy approach and generate structures by Brownian dynamics simulations. Our method successfully estimates the parameters in the potential for the target structures, such as square lattice, kagome lattice, and dodecagonal quasicrystal.