One-particle Hamiltonian Research Articles

Gas of Particles Obeying the Monotone Statistics.

The present note is devoted to the detailed investigation of a concrete model satisfying the block-monotone statistics introduced in a previous paper (joint, with collaborators) of the author. The model under consideration indeed describes the free gas of massless particles in a one-dimensional environment. This investigation can have consequences in two fundamental respects. The first one concerns the applicability of the (block-)monotone statistics to concrete physical models, yet completely unknown. Since the formula for the degeneracy of the energy-levels of the one-particle Hamiltonian of a free particle is very involved, the second aspect might be related to the, highly nontrivial, investigation of the expected thermodynamics of the free gas of particles obeying the block-monotone statistics in arbitrary spatial dimensions. A final section contains a comparison between the various (block, strict, and weak) monotone schemes with the Boltzmann statistics, which describes the gas of classical particles. It is seen that the block-monotone statistics, which takes into account the degeneracy of the energy-levels, seems the unique one having realistic physical applications.

Bose–Einstein condensation for particles with repulsive short-range pair interactions in a Poisson random external potential in

AbstractWe study Bose gases in$d \ge 2$dimensions with short-range repulsive pair interactions at positive temperature, in the canonical ensemble and in the thermodynamic limit. We assume the presence of hard Poissonian obstacles and focus on the non-percolation regime. For sufficiently strong interparticle interactions, we show that almost surely there cannot be Bose–Einstein condensation into a sufficiently localized, normalized one-particle state. The results apply to the canonical eigenstates of the underlying one-particle Hamiltonian.

Analysis of unitarity in conformal quantum gravity

We perform a canonical quantization of Weyl’s conformal gravity by means of the covariant operator formalism and investigate the unitarity of the resulting quantum theory. After reducing the originally fourth-order theory to second-order in time derivatives via the introduction of an auxiliary tensor field, we identify the full Fock space of quantum states under a Becchi–Rouet–Stora–Tyutin (BRST) construction that includes Faddeev–Popov ghost fields corresponding to Weyl transformations. This second-order formulation allows the formal tools of operator-based quantum field theory to be applied to quadratic gravity for the first time. Using the Kugo–Ojima quartet mechanism, we identify the physical subspace of quantum states and find that the subspace containing the transverse spin-2 states comes equipped with an indefinite inner product metric and a one-particle Hamiltonian that possesses only a single eigenstate. We construct the Lehmann–Symanzik–Zimmermann (LSZ) reduction formula for the S-matrix in this spin-2 subspace and find that unitarity is violated in scattering events. The explicit way in which this violation occurs represents a new view on the ghost-problem in quadratic theories of quantum gravity.

3d mathcal{N} = 4 OPE coefficients from Fermi gas

The partition function of a 3d mathcal{N} = 4 gauge theory with rank N can be computed using supersymmetric localization in terms of a matrix model, which often can be formulated as an ideal Fermi gas with a non-trivial one-particle Hamiltonian. We show how OPE coefficients of protected operators correspond in this formalism to averages of n-body operators in the Fermi gas, which can be computed to all orders in 1/N using the WKB expansion. We use this formalism to compute OPE coefficients in the U(N)k× U(N)−k ABJM theory as well as the U(N ) theory with one adjoint and Nf fundamental hypermultiplets, both of which have weakly coupled M-theory duals in the large N and finite k or Nf regimes. For ABJM we reproduce known results, while for the Nf theory we compute the all orders in 1/N dependence at finite Nf for the coefficient cT of the stress tensor two-point function.

Role of nonlocality in exchange correlation for magnetic two-dimensional van der Waals materials

To obtain accurate independent-particle descriptions for ferromagnetic two-dimensional van der Waals materials, we apply the quasiparticle self-consistent $GW$ (QSGW) method to ${\mathrm{VI}}_{3}, {\mathrm{CrI}}_{3}, {\mathrm{CrGeTe}}_{3}$, and ${\mathrm{Fe}}_{3}{\mathrm{GeTe}}_{2}$. QSGW provides a description of the nonlocal exchange-correlation term in the one-particle Hamiltonian. The nonlocal term is important not only as the $U$ of density functional theory $(\mathrm{DFT})+U$ but also for differentiating occupied and unoccupied states in semiconductors. We show the limitations of $\mathrm{DFT}+U$ in mimicking QSGW.

On the regime of localized excitations for disordered oscillator systems

We study quantum oscillator lattice systems with disorder, in arbitrary dimension, requiring only partial localization of the associated effective one-particle Hamiltonian. This leads to a many-body localized regime of excited states with arbitrarily large energy density. We prove zero-velocity Lieb-Robinson bounds for the dynamics of Weyl operators as well as for position and momentum operators restricted to this regime. Dynamical localization is also shown in the form of quasi-locality of the time evolution of local Weyl operators and through exponential clustering of the dynamic correlations of states with localized excitations.

Electronic States and Persistent Currents in Nanowire Quantum Ring

The new model of a quantum ring (QR) defined inside a nanowire (NW) is proposed. The one-particle Hamiltonian for electron in [111]-oriented NW QR is constructed taking into account both Rashba and Dresselhaus spin-orbit coupling (SOC). The energy levels as a function of magnetic field are found using the exact numerical diagonalization. The persistent currents (both charge and spin) are calculated. The specificity of SOC and arising anticrossings in energy spectrum lead to unusual features in persistent current behavior. The variation of magnetic field or carrier concentration by means of gate can lead to pure spin persistent current with the charge current being zero.

Electronic states and persistent currents in nanowire quantum ring

AbstractA new model of a quantum ring defined inside a nanowire is proposed. The one-particle Hamiltonian for electron in [111]-oriented nanowire quantum ring is constructed taking into account both Rashba and Dresselhaus spin-orbit coupling. The energy levels as a function of magnetic field are found using the exact numerical diagonalization. The persistent currents (both charge and spin) are calculated. The specificity of spin-orbit coupling and arising anticrossings in energy spectrum lead to unusual features in persistent current behavior. The variation of magnetic field or carrier concentration by means of gates can lead to pure spin persistent current with the charge current being zero.

Chiral anomaly in Dirac semimetals due to dislocations

The dislocation in Dirac semimetal carries an emergent magnetic flux parallel to the dislocation axis. We show that due to the emergent magnetic field the dislocation accommodates a single fermion massless mode of the corresponding low-energy one-particle Hamiltonian. The mode is propagating along the dislocation with its spin directed parallel to the dislocation axis. In agreement with the chiral anomaly observed in Dirac semimetals, an external electric field results in the spectral flow of the one-particle Hamiltonian, in pumping of the fermionic quasiparticles out from the vacuum, and in creating a nonzero axial (chiral) charge in the vicinity of the dislocation.

Regularized perturbative series for the ionization potential of atomic ions

We study N-electron atoms with nuclear charge Z. It is well known that, in the cationic (Z > N) large-Z region, the atom behaves as a weakly interacting system. The anionic (Z < N) regime, on the other hand, is characterized by an instability threshold at [Formula: see text], below which the atom spontaneously emits an electron. We construct a regularized perturbative series (RPS) for the ionization potential that is based on the behaviors for large Z and Z near Zc. The large-Z expansion coefficients are analytically computed from perturbation theory, whereas the slope of the energy curve at Z = N − 1 is computed from a kind of zero-range forces theory that uses as input the electron affinity and the covalent radius of the neutral atom with N − 1 electrons. Relativistic effects in the one-particle Hamiltonian are considered at the level of first-order perturbation theory. Our RPS formula is to be used to check the consistency of the ionization potential values for atomic ions contained in the NIST database.

Piezoelectricity in asymmetrically strained bilayer graphene

We study the electronic properties of commensurate faulted bilayer graphene by diagonalizing the one-particle Hamiltonian of the bilayer system in a complete basis of Bloch states of the individual graphene layers. Our novel approach is very general and can be easily extended to any commensurate graphene-based heterostructure. Here, we consider three cases: (i) twisted bilayer graphene, (ii) bilayer graphene where triaxial stress is applied to one layer and (iii) bilayer graphene where uniaxial stress is applied to one layer. We show that the resulting superstructures can be divided into distinct classes, depending on the twist angle or the magnitude of the induced strain. The different classes are distinguished from each other by the interlayer coupling mechanism, resulting in fundamentally different low-energy physics. For the cases of triaxial and uniaxial stress, the individual graphene layers tend to decouple and we find significant charge transfer between the layers. In addition, this piezoelectric effect can be tuned by applying a perpendicular electric field. Finally, we show how our approach can be generalized to multilayer systems.

The Quantum Group and Harper Equation on a Honeycomb Lattice

The tight-binding model of quantum particle on a honeycomb lattice is investigated in the presence of homogeneous magnetic field. The one-particle Hamiltonian is expressed in terms of the generators of the quantum group Uq(sl2). The corresponding Harper equation is rewritten as a system of two coupled functional equations in the complex plane. The system is shown to exhibit certain symmetry that allows one to resolve the entanglement, and the basic single equation determining the eigenvalues and eigenstates is obtained. Equations specifying the roots of eigenstates in the complex plane are found.

The spectral problem of the ABJ Fermi gas

The partition function on the three-sphere of ABJ theory can be rewritten into a partition function of a non-interacting Fermi gas, with an accompanying one-particle Hamiltonian. We study the spectral problem defined by this Hamiltonian. We determine the exact WKB quantization condition, which involves quantities from refined topological string theory, and test it successfully against numerical calculations of the spectrum.

Density of one-particle states for 2-D electron gas in magnetic field

The density of states of a particle in a 2D area is independent both of the energy and form of the area only at the region of large values of energy. If energy is small, the density of states in the rectangular potential well essentially depends on the form of the area. If the bottom of the potential well has a potential relief, it can define the small eigenvalues as the discrete levels. In this case, dimensions and form of the area would not have any importance. If the conservation of zero value of the angular momentum is taken into account, the effective one-particle Hamiltonian for the 2D electron gas in the magnetic field in the circle is the Hamiltonian with the parabolic potential and the reflecting bounds. It is supposed that in the square, the Hamiltonian has the same view. The 2D density of states in the square can be computed as the convolution of 1D densities. The density of one-particle states for 2D electron gas in the magnetic field is obtained. It consists of three regions. There is a discrete spectrum at the smallest energy. In the intervening region the density of states is the sum of the piecewise continuous function and the density of the discrete spectrum. At great energies, the density of states is a continuous function. The Fermi energy dependence on the magnetic field is obtained when the field is small and the Fermi energy is located in the region of continuous spectrum. The Fermi energy has the oscillating correction and in the average it increases proportionally to the square of the magnetic induction. Total energy of electron gas in magnetic field also oscillates and increases when the magnetic field increases monotonously.

Aharony–Bergman–Jafferis–Maldacena Wilson Loops in the Fermi Gas Approach

The matrix model of the Aharony-Bergman-Jafferis-Maldacena theory can be formulated in terms of an ideal Fermi gas with a non-trivial one-particle Hamiltonian. We show that, in this formalism, vacuum expectation values (vevs) of Wilson loops correspond to averages of operators in the statistical-mechanical problem. This makes it possible to calculate these vevs at all orders in 1/N, up to exponentially small corrections, and for arbitrary Chern-Simons coupling, by using the Wentzel- Kramer-Brillouin expansion.We present explicit results for the vevs of 1/6 and the 1/2 Bogomolnyi- Prasad-Sommerfield Wilson loops, at any winding number, in terms of Airy functions. Our expressions are shown to reproduce the low genus results obtained previously in the ’t Hooft expansion.

Quantum Harmonic Oscillator Systems with Disorder

We study many-body properties of quantum harmonic oscillator lattices with disorder. A sufficient condition for dynamical localization, expressed as a zero-velocity Lieb-Robinson bound, is formulated in terms of the decay of the eigenfunction correlators for an effective one-particle Hamiltonian. We show how state-of-the-art techniques for proving Anderson localization can be used to prove that these properties hold in a number of standard models. We also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of one-particle eigenfunction correlators. In particular, we show that static correlations decay exponentially fast if the corresponding effective one-particle Hamiltonian exhibits localization at low energies, regardless of whether there is a gap in the spectrum above the ground state or not. Our results apply to finite as well as to infinite oscillator systems. The eigenfunction correlators that appear are more general than those previously studied in the literature. In particular, we must allow for functions of the Hamiltonian that have a singularity at the bottom of the spectrum. We prove exponential bounds for such correlators for some of the standard models.

Local unitary transformation method for large-scale two-component relativistic calculations. II. Extension to two-electron Coulomb interaction

The local unitary transformation (LUT) scheme at the spin-free infinite-order Douglas-Kroll-Hess (IODKH) level [J. Seino and H. Nakai, J. Chem. Phys. 136, 244102 (2012)], which is based on the locality of relativistic effects, has been extended to a four-component Dirac-Coulomb Hamiltonian. In the previous study, the LUT scheme was applied only to a one-particle IODKH Hamiltonian with non-relativistic two-electron Coulomb interaction, termed IODKH/C. The current study extends the LUT scheme to a two-particle IODKH Hamiltonian as well as one-particle one, termed IODKH/IODKH, which has been a real bottleneck in numerical calculation. The LUT scheme with the IODKH/IODKH Hamiltonian was numerically assessed in the diatomic molecules HX and X(2) and hydrogen halide molecules, (HX)(n) (X = F, Cl, Br, and I). The total Hartree-Fock energies calculated by the LUT method agree well with conventional IODKH/IODKH results. The computational cost of the LUT method is reduced drastically compared with that of the conventional method. In addition, the LUT method achieves linear-scaling with respect to the system size and a small prefactor.

The quantum group, Harper equation and structure of Bloch eigenstates on a honeycomb lattice

The tight-binding model of quantum particles on a honeycomb lattice is investigated in the presence of a homogeneous magnetic field. Provided the magnetic flux per unit hexagon is a rational of the elementary flux, the one-particle Hamiltonian is expressed in terms of the generators of the quantum group Uq(sl2). Employing the functional representation of the quantum group Uq(sl2), the Harper equation is rewritten as a system of two coupled functional equations in the complex plane. For the special values of quasi-momentum, the entangled system admits solutions in terms of polynomials. The system is shown to exhibit a certain symmetry allowing us to resolve the entanglement, and a basic single equation determining the eigenvalues and eigenstates (polynomials) is obtained. Equations specifying the locations of the roots of polynomials in the complex plane are found. Employing numerical analysis, the roots of polynomials corresponding to different eigenstates are solved and diagrams exhibiting the ordered structure of one-particle eigenstates are depicted.

Two interacting particles in deformed nanolayer: discrete spectrum and particle storage

The problem of particle storage in a nanolayered structure is considered. Local perturbations of the nanolayers can lead to the appearance of eigenvalues of the corresponding one-particle Hamiltonian. To study the particle storage it is necessary to deal with a multi-particle problem. The Hartree method and the finite element method are used. The discrete spectrum of the Hamiltonian of two interacting particles is considered. Two different types of the perturbation are treated: deformation of the layer boundary and a small window in a wall between two layers. The relation between the system parameters (interaction intensity–waveguide deformation) ensuring the existence of a non-empty discrete spectrum is studied. Comparison of particle storage efficiencies in these two cases is made.

Dynamical Localization in Disordered Quantum Spin Systems

We say that a quantum spin system is dynamically localized if the time-evolution of local observables satisfies a zero-velocity Lieb-Robinson bound. In terms of this definition we have the following main results: First, for general systems with short range interactions, dynamical localization implies exponential decay of ground state correlations, up to an explicit correction. Second, the dynamical localization of random xy spin chains can be reduced to dynamical localization of an effective one-particle Hamiltonian. In particular, the isotropic xy chain in random exterior magnetic field is dynamically localized.