General Contact Interactions Research Articles

P -wave superconductivity and the axiplanar phase of triple-point fermions

We consider weak-coupling superconductivity in the inversion- and rotation-symmetric system of two pseudospin $s=1$ low-energy fermions of opposite chirality. General contact interactions can lead to Cooper instabilities toward $d$-wave or toward a novel $2\ifmmode\times\else\texttimes\fi{}3 p$-wave matrix order parameter. We compute the Ginzburg-Landau (GL) free energy for the latter. Remarkably, in this case the GL free energy can be minimized exactly, with the resulting ordered state being analogous to the ``axiplanar'' $p$ wave, which exhibits extra degeneracy and generally breaks time-reversal symmetry. Whereas a generic GL free energy for our order parameter has only $U(1)\ifmmode\times\else\texttimes\fi{}SO(2)\ifmmode\times\else\texttimes\fi{}SO(3)$ symmetry, at weak coupling we find it displaying the enlarged $U(1)\ifmmode\times\else\texttimes\fi{}SO(3)\ifmmode\times\else\texttimes\fi{}U(1)\ifmmode\times\else\texttimes\fi{}SO(3)$ symmetry broken down to $SO(2)\ifmmode\times\else\texttimes\fi{}SO(2)$ in the axiplanar ordered phase and leading therefore to six Goldstone bosons. We show how the lattice, once restored, fixes the allowed values of the magnetization in the superconducting state by locking the spatial directions implicit in the order parameter to its high-symmetry axes.

Effective field theories for interacting boundaries of 3D topological crystalline insulators through bosonisation

Here, we analyse two Dirac fermion species in two spatial dimensions in the presence of general quartic contact interactions. By employing functional bosonisation techniques, we demonstrate that depending on the couplings of the fermion interactions the system can be effectively described by a rich variety of topologically massive gauge theories. Among these effective theories, we obtain an extended Chern–Simons theory with higher order derivatives as well as two coupled Chern–Simons theories. Our formalism allows for a general description of interacting fermions emerging, for example, at the gapped boundary of three-dimensional topological crystalline insulators.

A smooth contact algorithm for the combined finite discrete element method

From its inception, the combined finite discrete element method has used a distributed potential contact force algorithm to resolve interaction between finite elements. The contact interaction algorithm relies on evaluation of the contact force potential field. The problem with existing algorithms is that the potential field introduces artificial numerical non-smoothness in the contact force. This work introduces a smooth potential field based on the finite element topology, and a generalized contact interaction law is constructed on top of the smooth potential field. A number of validation cases for the proposed algorithm, considering different shapes of discrete elements, are presented, and detailed aspects of the proposed contact interaction law are tested with numerical examples.

Contact interactions and Kronig–Penney models in Hermitian and symmetric quantum mechanics

The delta function potential is a simple model of zero-range contact interaction in non-relativistic quantum mechanics in one dimension. The Kronig–Penney model is a one-dimensional periodic array of delta functions and provides a simple illustration of energy bands in a crystal. Here we investigate contact interactions that generalize the delta function potential and corresponding generalizations of the Kronig–Penney model within conventional and symmetric quantum mechanics. In conventional Hermitian quantum mechanics we determine the most general contact interaction compatible with self-adjointness and in quantum mechanics we consider interactions that respect symmetry under the transformation where denotes parity and denotes time reversal. In both cases we find that the most general interaction has four independent real parameters and depending on the values of those parameters the contact interaction can support zero, one or two bound states. By contrast the conventional delta function can only support zero or one bound state. In the symmetric case moreover the two bound state energies can be both real or a complex conjugate pair. The transition from real to complex bound state energies corresponds to the spontaneous breaking of symmetry. The scattering states for the symmetric case are also found to exhibit spontaneous breaking of symmetry wherein the eigenvalues of the non-unitary S-matrix depart the unit circle in the complex plane. We also investigate the energy bands when the generalized contact interactions are repeated periodically in space in one dimension. In the Hermitian case we find that the two bound states result in two narrow bands generically separated by a gap. These bands intersect at a single point in the Brillouin zone as the interaction parameters are varied. Near the intersection the bands form a massless Dirac cone. In the symmetric case we find that as the parameters of the contact interaction are varied the two bound state bands undergo a symmetry breaking transition wherein the two band energies go from being real to being a complex conjugate pair. The symmetric Kronig–Penney model provides a simple soluble example of the transition which has the same form as in other models of symmetric crystals.

Traveling waves and conservation laws for highly nonlinear wave equations modeling Hertz chains

A highly nonlinear, fourth-order wave equation that models the continuum theory of long wavelength pulses in weakly compressed, homogeneous, discrete chains with a general power-law contact interaction is studied. For this wave equation, all solitary wave solutions and all nonlinear periodic wave solutions, along with all conservation laws, are derived. The solutions are explicitly parameterized in terms of the asymptotic value of the wave amplitude in the case of solitary waves and the peak of the wave amplitude in the case of nonlinear periodic waves. All cases in which the solution expressions can be stated in an explicit analytic form using elementary functions are worked out. In these cases, explicit expressions for the total energy and total momentum for all solutions are obtained as well. The derivation of the solutions uses the conservation laws combined with an energy analysis argument to reduce the wave equation directly to a separable first-order differential equation that determines the wave amplitude in terms of the traveling wave variable. This method can be applied more generally to other highly nonlinear wave equations.

Self-bound many-body states of quasi-one-dimensional dipolar Fermi gases: Exploiting Bose-Fermi mappings for generalized contact interactions

Using a combination of results from exact mappings and from mean-field theory we explore the phase diagram of quasi-one-dimensional systems of identical fermions with attractive dipolar interactions. We demonstrate that at low density these systems provide a realization of a single-component one-dimensional Fermi gas with a generalized contact interaction. Using an exact duality between one-dimensional Fermi and Bose gases, we show that when the dipole moment is strong enough, bound many-body states exist, and we calculate the critical coupling strength for the emergence of these states. At higher densities, the Hartree-Fock approximation is accurate, and by combining the two approaches we determine the structure of the phase diagram. The many-body bound states should be accessible in future experiments with ultracold polar molecules.

Wigner functions, contact interactions, and matching

Quantum mechanics in phase space (or deformation quantization) appears to fail as an autonomous quantum method when infinite potential walls are present. The stationary physical Wigner functions do not satisfy the normal eigen equations, the ★-eigen equations, unless an ad hoc boundary potential is added [N.C. Dias, J.N. Prata, J. Math. Phys. 43 (2002) 4602 (quant-ph/0012140)]. Alternatively, they satisfy a different, higher-order, “★-eigen-★ equation”, locally, i.e. away from the walls [S. Kryukov, M.A. Walton, Ann. Phys. 317 (2005) 474 (quant-ph/0412007)]. Here we show that this substitute equation can be written in a very simple form, even in the presence of an additional, arbitrary, but regular potential. The more general applicability of the ★-eigen-★ equation is then demonstrated. First, using an idea from [D.B. Fairlie, C.A. Manogue, J. Phys. A 24 (1991) 3807], we extend it to a dynamical equation describing time evolution. We then show that also for general contact interactions, the ★-eigen-★ equation is satisfied locally. Specifically, we treat the most general possible (Robin) boundary conditions at an infinite wall, general one-dimensional point interactions, and a finite potential jump. Finally, we examine a smooth potential, that has simple but different expressions for x positive and negative. We find that the ★-eigen-★ equation is again satisfied locally. It seems, therefore, that the ★-eigen-★ equation is generally relevant to the matching of Wigner functions; it can be solved piece-wise and its solutions then matched.

Continuous Spectra of Generalized Kronig–Penney Model

The standard Kronig-Penney model with periodic $\delta$ potentials is extended to the cases with generalized contact interactions. The eigen equation which determines the dispersion relation for one-dimensional periodic array of the generalized contact interactions is deduced with the transfer matrix formalism. Numerical results are presented which reveal unexpected band spectra with broader band gap in higher energy region for generic model with generalized contact interaction.

Wave-packet dynamics for general contact interactions on a circular setup: Revivals, bouncing, and trapping

Here we study a one-dimensional finite lattice formed by generalized contact interactions in a circular setup, i.e., under periodic boundary conditions. Considering only four such potentials, we show the emergence of different behaviors as revivals, bouncing, and trapping for the time evolution of wave packets. This is done by properly choosing the parameters that characterize the contact interactions. We also discuss possible physical applications for this type of system, such as using it to split an initially localized state into spatially separated and dynamically independent parts.

Model independent constraints on contact interactions from LEP2

We quantitatively discuss the possibility of deriving model-independent constraints on the general four-fermion contact interaction couplings, from the currently available data on the two-fermion production processes e + e −→ μ + μ −, b b ̄ and c c ̄ with unpolarized initial beams. The method is essentially based on particular, simple, combinations of the measured total cross section and forward–backward asymmetry that allow partial separation of the helicity cross sections, and the combination of experimental data obtained at the different energies of TRISTAN, LEP1 and LEP2.

A REMARK ON ONE-DIMENSIONAL MANY-BODY PROBLEMS WITH POINT INTERACTIONS

The integrability of one-dimensional quantum mechanical many-body problems with general contact interactions is extensively studied. It is shown that besides the pure (repulsive or attractive) δ-function interaction there is another singular point interactions which gives rise to a new one-parameter family of integrable quantum mechanical many-body systems. The bound states and scattering matrices are calculated for both bosonic and fermionic statistics.

Equivalence of Local and Separable Realizations of Discontinuity-Inducing Contact Interaction

We prove that the separable and local approximations of the discontinuity-inducing zero-range interaction in one-dimensional quantum mechanics are equivalent. We further show that the interaction allows the perturbative treatment through the coupling renormalization. Keywords: one-dimensional system, generalized contact interaction, renormalization, perturbative expansion. PACS Nos: 3.65.-w, 11.10.Gh, 31.15.Md

Fermion-Boson Duality of One-Dimensional Quantum Particles with Generalized Contact Interactions

For a system of spinless one-dimensional fermions, the non-vanishing short-range limit of two-body interaction is shown to induce the wave-function discontinuity. We prove the equivalence of this fermionic system and the bosonic particle system with two-body $\delta$-function interaction with the reversed role of strong and weak couplings. KEYWORDS: one-dimensional system, $\epsilon$-interaction, solvable many-body problem, exact bosonization

Constraints on four-fermion contact interactions from precise electroweak measurements

We establish constraints on a general four-fermion contact interaction from precise measurements of electroweak parameters. We compute the one-loop contribution for the leptonic Z width, anomalous magnetic, weak-magnetic, electric and weak dipole moments of leptons in order to extract bounds on the energy scale of these effective interactions.

EFFECTS OF COMPOSITENESS IN TOP DECAY

We present predictions for single quark decay of toponium in the presence of general contact interactions attributed to third-family compositeness.