Internal Degrees Of Freedom Research Articles

Multifluid flows with weak and strong discontinuous interfaces using an elemental enriched space

SummaryIn a previous paper, the authors presented an elemental enriched space to be used in a finite‐element framework (EFEM) capable of reproducing kinks and jumps in an unknown function using a fixed mesh in which the jumps and kinks do not coincide with the interelement boundaries. In this previous publication, only scalar transport problems were solved (thermal problems). In the present work, these ideas are generalized to vectorial unknowns, in particular, the incompressible Navier‐Stokes equations for multifluid flows presenting internal moving interfaces. The advantage of the EFEM compared with global enrichment is the significant reduction in computing time when the internal interface is moving. In the EFEM, the matrix to be solved at each time step has not only the same amount of degrees of freedom (DOFs) but also the same connectivity between the DOFs. This frozen matrix graph enormously improves the efficiency of the solver. Another characteristic of the elemental enriched space presented here is that it allows a linear variation of the jump, thus improving the convergence rate, compared with other enriched spaces that have a constant variation of the jump. Furthermore, the implementation in any existing finite‐element code is extremely easy with the version presented here because the new shape functions are based on the usual finite‐element method shape functions for triangles or tetrahedrals, and once the internal DOFs are statically condensed, the resulting elements have exactly the same number of unknowns as the nonenriched finite elements.

The role of bulk viscosity on the decay of compressible, homogeneous, isotropic turbulence

While Stokes’ hypothesis of neglecting bulk viscous effects is exact for monatomic gases and unlikely to strongly affect the dynamics of fluids whose bulk-to-shear viscosity ratio is small and/or of weakly compressible turbulence, it is unclear to what extent this assumption holds for compressible, turbulent flows of gases whose bulk viscosity is orders of magnitude larger than their shear viscosities (e.g. $\text{CO}_{2}$). Our objective is to understand the mechanisms by which bulk viscosity and the associated phenomena affect moderately compressible turbulence, in particular energy transfer and dissipation. Using direct numerical simulations of the compressible Navier–Stokes equations, we study the decay of compressible, homogeneous, isotropic turbulence for ratios of bulk-to-shear viscosity ranging from 0 to 1000. Our simulations demonstrate that bulk viscosity increases the decay rate of turbulent kinetic energy; whereas enstrophy exhibits little sensitivity to bulk viscosity, dilatation is reduced by over two orders of magnitude within the first two eddy-turnover times. Via a Helmholtz decomposition of the flow, we determine that the primary action of bulk viscosity is to damp the dilatational velocity fluctuations and reduce dilatational–solenoidal exchanges, as well as pressure–dilatation coupling. In short, bulk viscosity renders compressible turbulence incompressible by reducing energy transfer between translational and internal degrees of freedom. Our results indicate that for gases whose bulk viscosity is of the order of their shear viscosity (e.g. hydrogen) the turbulence is not significantly affected by bulk viscous dissipation, in which case neglecting bulk viscosity is acceptable in practice. However, in problems involving compressible, turbulent flows of gases like $\text{CO}_{2}$ whose bulk viscosities are thousands of times greater than their shear viscosities, bulk viscosity cannot be ignored.

Real meromorphic differentials: a language for describing meron configurations in planar magnetic nanoelements

In this paper we use the language of real meromorphic differentials from the theory of Klein surfaces to describe the metastable states of multiply connected planar ferromagnetic nanoelements which minimize the exchange energy and have no side magnetic charges. Those solutions still have enough internal degrees of freedom which may serve as the Ritz parameters for minimization of further relevant energy terms or as the dynamical variables for the adiabatic approach. The nontrivial topology of the magnet itself brings us to several effects first described for the annulus and observed in the experiment. We explain the topological constraints on the numbers of vortexes and antivortexes in the magnet, as well as the algebraic constraints on their positions which stem from the Abel's theorem. The use of multivalued Prym differentials bring us to new meron configurations which were not considered in the seminal work of D.J.Gross.

Virtual Element approximation of 2D magnetostatic problems

We consider the use of nodal and edge Virtual Element spaces for the discretization of magnetostatic problems in two dimensions, following the variational formulation of Kikuchi. In addition, we present a novel Serendipity variant of the same spaces that allow to save many internal degrees of freedom. These Virtual Element Spaces of different type can be useful in applications where an exact sequence is particularly convenient, together with commuting-diagram interpolation operators, as is definitely the case in electromagnetic problems. We prove stability and optimal error estimates, and we check the performance with some academic numerical experiments.

Correlated electronic states at domain walls of a Mott-charge-density-wave insulator 1T-TaS2

Domain walls in interacting electronic systems can have distinct localized states, which often govern physical properties and may lead to unprecedented functionalities and novel devices. However, electronic states within domain walls themselves have not been clearly identified and understood for strongly correlated electron systems. Here, we resolve the electronic states localized on domain walls in a Mott-charge-density-wave insulator 1T-TaS2 using scanning tunneling spectroscopy. We establish that the domain wall state decomposes into two nonconducting states located at the center of domain walls and edges of domains. Theoretical calculations reveal their atomistic origin as the local reconstruction of domain walls under the strong influence of electron correlation. Our results introduce a concept for the domain wall electronic property, the walls own internal degrees of freedom, which is potentially related to the controllability of domain wall electronic properties.

Emerging internal symmetries from effective spacetimes

Can global internal and spacetime symmetries be connected without supersymmetry? To answer this question, we investigate Minkowski spacetimes with d space-like extra dimensions and point out under which general conditions external symmetries induce internal symmetries in the effective 4-dimensional theories. We further discuss in this context how internal degrees of freedom and spacetime symmetries can mix without supersymmetry in agreement with the Coleman–Mandula theorem. We present some specific examples which rely on a direct product structure of spacetime such that orthogonal extra dimensions can have symmetries which mix with global internal symmetries. This mechanism opens up new opportunities to understand global symmetries in particle physics.

Round Robin Study: Molecular Simulation of Thermodynamic Properties from Models with Internal Degrees of Freedom.

Thermodynamic properties are often modeled by classical force fields which describe the interactions on the atomistic scale. Molecular simulations are used for retrieving thermodynamic data from such models, and many simulation techniques and computer codes are available for that purpose. In the present round robin study, the following fundamental question is addressed: Will different user groups working with different simulation codes obtain coinciding results within the statistical uncertainty of their data? A set of 24 simple simulation tasks is defined and solved by five user groups working with eight molecular simulation codes: DL_POLY, GROMACS, IMC, LAMMPS, ms2, NAMD, Tinker, and TOWHEE. Each task consists of the definition of (1) a pure fluid that is described by a force field and (2) the conditions under which that property is to be determined. The fluids are four simple alkanes: ethane, propane, n-butane, and iso-butane. All force fields consider internal degrees of freedom: OPLS, TraPPE, and a modified OPLS version with bond stretching vibrations. Density and potential energy are determined as a function of temperature and pressure on a grid which is specified such that all states are liquid. The user groups worked independently and reported their results to a central instance. The full set of results was disclosed to all user groups only at the end of the study. During the study, the central instance gave only qualitative feedback. The results reveal the challenges of carrying out molecular simulations. Several iterations were needed to eliminate gross errors. For most simulation tasks, the remaining deviations between the results of the different groups are acceptable from a practical standpoint, but they are often outside of the statistical errors of the individual simulation data. However, there are also cases where the deviations are unacceptable. This study highlights similarities between computer experiments and laboratory experiments, which are both subject not only to statistical error but also to systematic error.

Internal clock formulation of quantum mechanics

The basic tenet of the present work is the assumption of the lack of external and fixed time in the Universe. This assumption is best embodied by general relativity, which replaces the fixed space-time structure with the gravitational field, which is subject to dynamics. The lack of time does not imply the lack of evolution but rather brings to the forefront the role of internal clocks which are some largely arbitrary internal degrees of freedom with respect to which the evolution of timeless systems can be described. We take this idea seriously and try to understand what it implies for quantum mechanics when the fixed external time is replaced by an arbitrary internal clock. We put the issue in a solid, mathematically rigorous framework. We find that the dynamical interpretation of a quantum state of a timeless system depends on the employed internal clock. In particular, we find that the continuous spectra of well-known dynamical observables like the position of a free particle on the real line may turn discrete if measured in unusual clocks. We discuss the meaning of our result for attempts at quantization of global gravitational degrees of freedom.

Quantum walking in curved spacetime: (3+1) dimensions, and beyond

A discrete-time Quantum Walk (QW) is essentially an operator driving the evolution of a single particle on the lattice, through local unitaries. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation). Recently it was discovered that prior grouping and encoding allows for more general continuum limit equations (e.g. the Dirac equation in (1+ 1) curved spacetime). In this paper, we extend these results to arbitrary space dimension and internal degree of freedom. We recover an entire class of PDEs encompassing the massive Dirac equation in (3 + 1) curved spacetime. This means that the metric field can be represented by a field of local unitaries over a lattice.

Quantum friction imprints on the geometric phase of a moving atom in front of a dielectric plate

We compute the non-unitary geometric phase for the moving atom under the presence of the vacuum field and a dielectric mirror, analytically and numerically. We consider the atom (represented by a two-level system) moving in front of a dielectric plate, and study how decoherence of the particle’s internal degrees of freedom can be found in the corrections to the geometric phase accumulated by the atom. We consider the particle to follow a classical, macroscopically-fixed trajectory and by integrating over the vacuum field and the microscopic degrees of freedom of the plate we may calculate friction effects. We find a velocity dependance in the correction to the unitary geometric phase due to quantum frictional effects. We also show in which cases decoherence effects could, in principle, be controlled in order to perform a measurement of the geometric phase using standard interferometry procedures.

Semiclassics for matrix Hamiltonians: The Gutzwiller trace formula with applications to graphene-type systems

We present a tractable and physically transparent semiclassical theory of matrix-valued Hamiltonians, i.e., those that describe quantum systems with internal degrees of freedoms, based on a generalization of the Gutzwiller trace formula for a $n\ifmmode\times\else\texttimes\fi{}n$ dimensional Hamiltonian $H(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{p}},\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{q}})$. The classical dynamics is governed by $n$ Hamilton-Jacobi (HJ) equations that act in a phase space endowed with a classical Berry curvature encoding anholonomy in the parallel transport of the eigenvectors of $H(\mathbf{p},\mathbf{q})$; these vectors describe the internal structure of the semiclassical particles. At the $\mathcal{O}({\ensuremath{\hbar}}^{1})$ level and for nondegenerate HJ systems, this curvature results in an additional semiclassical phase composed of (i) a Berry phase and (ii) a dynamical phase resulting from the classical particles ``moving through the Berry curvature''. We show that the dynamical part of this semiclassical phase will, generally, be zero only for the case in which the Berry phase is topological (i.e., depends only on the winding number). We illustrate the method by calculating the Landau spectrum for monolayer graphene, the four-band model of AB bilayer graphene, and for a more complicated matrix Hamiltonian describing the silicene band structure. Finally, we apply our method to an inhomogeneous system consisting of a strain engineered one-dimensional moir\'e in bilayer graphene, finding localized states near the Dirac point that arise from electron trapping in a semiclassical moir\'e potential. The semiclassical density of states of these localized states we show to be in perfect agreement with an exact quantum mechanical calculation of the density of states.

{{SO(d,1)}}-Invariant Yang\u2013Baxter Operators and the dS/CFT Correspondence

We propose a model for the dS/CFT correspondence. The model is constructed in terms of a “Yang–Baxter operator” R for unitary representations of the de Sitter group {SO(d,1)}. This R-operator is shown to satisfy the Yang–Baxter equation, unitarity, as well as certain analyticity relations, including in particular a crossing symmetry. With the aid of this operator we construct: (a) a chiral (light-ray) conformal quantum field theory whose internal degrees of freedom transform under the given unitary representation of {SO(d,1)}. By analogy with the O(N) non-linear sigma model, this chiral CFT can be viewed as propagating in a de Sitter spacetime. (b) A (non-unitary) Euclidean conformal quantum field theory on {mathbb{R}^{d-1}}, where SO(d, 1) now acts by conformal transformations in (Euclidean) spacetime. These two theories can be viewed as dual to each other if we interpret {mathbb{R}^{d-1}} as conformal infinity of de Sitter spacetime. Our constructions use semi-local generator fields defined in terms of R and abstract methods from operator algebras.

Equivalence of Szegedy’s and coined quantum walks

Szegedy's quantum walk is a quantization of a classical random walk or Markov chain, where the walk occurs on the edges of the bipartite double cover of the original graph. To search, one can simply quantize a Markov chain with absorbing vertices. Recently, Santos proposed two alternative search algorithms that instead utilize the sign-flip oracle in Grover's algorithm rather than absorbing vertices. In this paper, we show that these two algorithms are exactly equivalent to two algorithms involving coined quantum walks, which are walks on the vertices of the original graph with an internal degree of freedom. The first scheme is equivalent to a coined quantum walk with one walk-step per query of Grover's oracle, and the second is equivalent to a coined quantum walk with two walk-steps per query of Grover's oracle. These equivalences lie outside the previously known equivalence of Szegedy's quantum walk with absorbing vertices and the coined quantum walk with the negative identity operator as the coin for marked vertices, whose precise relationships we also investigate.

Synthetic-lattice enabled all-optical devices based on orbital angular momentum of light

All-optical photonic devices are crucial for many important photonic technologies and applications, ranging from optical communication to quantum information processing. Conventional design of all-optical devices is based on photon propagation and interference in real space, which may rely on large numbers of optical elements, and the requirement of precise control makes this approach challenging. Here we propose an unconventional route for engineering all-optical devices using the photon’s internal degrees of freedom, which form photonic crystals in such synthetic dimensions for photon propagation and interference. We demonstrate this design concept by showing how important optical devices such as quantum memory and optical filters can be realized using synthetic orbital angular momentum (OAM) lattices in degenerate cavities. The design route utilizing synthetic photonic lattices may significantly reduce the requirement for numerous optical elements and their fine tuning in conventional design, paving the way for realistic all-optical photonic devices with novel functionalities.

On the Wegner Orbital Model

The Wegner orbital model is a class of random operators introduced by Wegner to model the motion of a quantum particle with many internal degrees of freedom (orbitals) in a disordered medium. We consider the case when the matrix potential is Gaussian, and prove three results: localisation at strong disorder, a Wegner-type estimate on the mean density of eigenvalues, and a Minami-type estimate on the probability of having multiple eigenvalues in a short interval. The last two results are proved in the more general setting of deformed block-Gaussian matrices, which includes a class of Gaussian band matrices as a special case. Emphasis is placed on the dependence of the bounds on the number of orbitals. As an additional application, we improve the upper bound on the localisation length for one-dimensional Gaussian band matrices.

MQCT. I. Inelastic Scattering of Two Asymmetric-Top Rotors with Application to H2O + H2O.

A mixed quantum/classical theory (MQCT) for the inelastic collision of two asymmetric-top rotor molecules is developed. In this method, the quantum state-to-state transitions between the rotational states of molecules (internal) are treated quantum mechanically using the time-dependent Schrodinger equation, whereas their relative translational motion (responsible for scattering) is treated classically, using the average trajectory approach. Two versions of the formula for transition matrix elements are presented: a straightforward approach that uses numerical multidimensional quadrature over all the internal degrees of freedom and a more standard analytic approach that uses the expansion of the PES over the basis set of spherical harmonics. Adaptation to the case of identical molecules scattering is presented and is applied to the rotational excitation of two water molecules, H2O + H2O, using the PES from recent literature. Calculations of collisional excitation from the ground state of the system into a number of low-lying excited rotational states are carried out in a broad range of energies. Analysis of computed opacity functions shows a rather unusual scattering regime, dominated by a strong anisotropic long-range interaction (dipole-dipole). The coupled-states (CS) approximation is tested and found to agree semiquantitatively (within a factor of 2) with the fully coupled version of the method. Differential cross sections for the elastic scattering indicate a very narrow forward scattering peak.

Fluctuation-induced forces in confined ideal and imperfect Bose gases.

Fluctuation-induced ("Casimir") forces caused by thermal and quantum fluctuations are investigated for ideal and imperfect Bose gases confined to d-dimensional films of size ∞^{d-1}×D under periodic (P), antiperiodic (A), Dirichlet-Dirichlet (DD), Neumann-Neumann (NN), and Robin (R) boundary conditions (BCs). The full scaling functions Υ_{d}^{BC}(x_{λ}=D/λ_{th},x_{ξ}=D/ξ) of the residual reduced grand potential per area φ_{res,d}^{BC}(T,μ,D)=D^{-(d-1)}Υ_{d}^{BC}(x_{λ},x_{ξ}) are determined for the ideal gas case with these BCs, where λ_{th} and ξ are the thermal de Broglie wavelength and the bulk correlation length, respectively. The associated limiting scaling functions Θ_{d}^{BC}(x_{ξ})≡Υ_{d}^{BC}(∞,x_{ξ}) describing the critical behavior at the bulk condensation transition are shown to agree with those previously determined from a massive free O(2) theory for BC=P,A,DD,DN,NN. For d=3, they are expressed in closed analytical form in terms of polylogarithms. The analogous scaling functions Υ_{d}^{BC}(x_{λ},x_{ξ},c_{1}D,c_{2}D) and Θ_{d}^{R}(x_{ξ},c_{1}D,c_{2}D) under the RBCs (∂_{z}-c_{1})ϕ|_{z=0}=(∂_{z}+c_{2})ϕ|_{z=D}=0 with c_{1}≥0 and c_{2}≥0 are also determined. The corresponding scaling functions Υ_{∞,d}^{P}(x_{λ},x_{ξ}) and Θ_{∞,d}^{P}(x_{ξ}) for the imperfect Bose gas are shown to agree with those of the interacting Bose gas with n internal degrees of freedom in the limit n→∞. Hence, for d=3,Θ_{∞,d}^{P}(x_{ξ}) is known exactly in closed analytic form. To account for the breakdown of translation invariance in the direction perpendicular to the boundary planes implied by free BCs such as DDBCs, a modified imperfect Bose gas model is introduced that corresponds to the limit n→∞ of this interacting Bose gas. Numerically and analytically exact results for the scaling function Θ_{∞,3}^{DD}(x_{ξ}) therefore follow from those of the O(2n)ϕ^{4} model for n→∞.

Super Rogers–Szegö polynomials associated with BCN type of Polychronakos spin chains

As is well known, multivariate Rogers–Szegö polynomials are closely connected with the partition functions of the AN−1 type of Polychronakos spin chains having long-range interactions. Applying the ‘freezing trick’, here we derive the partition functions for a class of BCN type of Polychronakos spin chains containing supersymmetric analogues of polarized spin reversal operators and subsequently use those partition functions to obtain novel multivariate super Rogers–Szegö (SRS) polynomials depending on four types of variables. We construct the generating functions for such SRS polynomials and show that these polynomials can be written as some bilinear combinations of the AN−1 type of SRS polynomials. We also use the above mentioned generating functions to derive a set of recursion relations for the partition functions of the BCN type of Polychronakos spin chains involving different numbers of lattice sites and internal degrees of freedom.

Influence of topological constraints and topological excitations: Decomposition formulas for calculating homotopy groups of symmetry-broken phases

A symmetry broken phase of a system with internal degrees of freedom often features a complex order parameter, which generates a rich variety of topological excitations and imposes topological constraints on their interaction (topological influence); yet the very complexity of the order parameter makes it difficult to treat topological excitations and topological influence systematically. To overcome this problem, we develop a general method to calculate homotopy groups and derive decomposition formulas which express homotopy groups of the order parameter manifold $G/H$ in terms of those of the symmetry $G$ of a system and those of the remaining symmetry $H$ of the state. By applying these formulas to general monopoles and three-dimensional skyrmions, we show that their textures are obtained through substitution of the corresponding $\mathfrak{su}(2)$-subalgebra for the $\mathfrak{su}(2)$-spin. We also show that a discrete symmetry of $H$ is necessary for the presence of topological influence and find topological influence on a skyrmion characterized by a non-Abelian permutation group of three elements in the ground state of an SU(3)-Heisenberg model.

Quantum simulation of discrete-time Hamiltonians using directionally unbiased linear optical multiports

Recently, a generalization of the standard optical multiport was proposed [Phys. Rev. A 93, 043845 (2016)]. These directionally unbiased multiports allow photons to reverse direction and exit backwards from the input port, providing a realistic linear optical scattering vertex for quantum walks on arbitrary graph structures. Here, it is shown that arrays of these multiports allow the simulation of a range of discrete-time Hamiltonian systems. Examples are described, including a case where both spatial and internal degrees of freedom are simulated. Because input ports also double as output ports, there is substantial savings of resources compared to feed-forward networks carrying out the same functions. The simulation is implemented in a scalable manner using only linear optics, and can be generalized to higher dimensional systems in a straightforward fashion, thus offering a concrete experimentally achievable implementation of graphical models of discrete-time quantum systems.