Two-dimensional Quantum Models Research Articles

Model and Energy Bounds for a Two-Dimensional System of Electrons Localized in Concentric Rings.

We study a two-dimensional system of interacting electrons confined in equidistant planar circular rings. The electrons are considered spinless and each of them is localized in one ring. While confined to such ring orbits, each electron interacts with the remaining ones by means of a standard Coulomb interaction potential. The classical version of this two-dimensional quantum model can be viewed as representing a system of electrons orbiting planar equidistant concentric rings where the kinetic energy may be discarded when one is searching for the lowest possible energy. Within this framework, the lowest possible energy of the system is the one that minimizes the total Coulomb interaction energy. This is the equilibrium energy that is numerically determined with high accuracy by using the simulated annealing method. This process allows us to obtain both the equilibrium energy and position configuration for different system sizes. The adopted semi-classical approach allows us to provide reliable approximations for the quantum ground state energy of the corresponding quantum system. The model considered in this work represents an interesting problem for studies of low-dimensional systems, with echoes that resonate with developments in nanoscience and nanomaterials.

On Hermite Functions, Integral Kernels, and Quantum Wires

In this note, we first evaluate and subsequently achieve a rather accurate approximation of a scalar product, the calculation of which is essential in order to determine the ground state energy in a two-dimensional quantum model. This scalar product involves an integral operator defined in terms of the eigenfunctions of the harmonic oscillator, expressed in terms of the well-known Hermite polynomials, so that some rather sophisticated mathematical tools are required.

Reinterpreting the vibrational structure in the electronic spectrum of the propargyl cation (H2C3H+) using an efficient and accurate quantum model.

The B̃1A1 ← X̃1A1 absorption spectra of propargyl cations H2C3H+ and D2C3D+ were simulated by an efficient two-dimensional (2D) quantum model, which includes the C-C stretch (v5) and the C≡C stretch (v3) vibrational modes. The choice of two modes was based on a scheme that can identify the active modes quantitively by examining the normal coordinate displacements (∆Q) directly based on the ab initio equilibrium geometries and frequencies of the X̃1A1 and B̃1A1 states of H2C3H+. The spectrum calculated by the 2D model was found to be very close to those calculated by all the higher three-dimensional (3D) quantum models (including v5, v3, and another one in 12 modes of H2C3H+), which validates the 2D model. The calculated B̃1A1 ← X̃1A1 absorption spectra of both H2C3H+ and D2C3D+ are in fairly good agreement with experimental results.

One- and two-dimensional quantum models: Quenches and the scaling of irreversible entropy.

Using the scaling relation of the ground state quantum fidelity, we propose the most generic scaling relations of the irreversible work (the residual energy) of a closed quantum system at absolute zero temperature when one of the parameters of its Hamiltonian is suddenly changed. We consider two extreme limits: the heat susceptibility limit and the thermodynamic limit. It is argued that the irreversible entropy generated for a thermal quench at low enough temperatures when the system is initially in a Gibbs state is likely to show a similar scaling behavior. To illustrate this proposition, we consider zero-temperature and thermal quenches in one-dimensional (1D) and 2D Dirac Hamiltonians where the exact estimation of the irreversible work and the irreversible entropy is possible. Exploiting these exact results, we then establish the following. (i) The irreversible work at zero temperature shows an appropriate scaling in the thermodynamic limit. (ii) The scaling of the irreversible work in the 1D Dirac model at zero temperature shows logarithmic corrections to the scaling, which is a signature of a marginal situation. (iii) Remarkably, the logarithmic corrections do indeed appear in the scaling of the entropy generated if the temperature is low enough while they disappear for high temperatures. For the 2D model, no such logarithmic correction is found to appear.

Bifurcation Research Two-Dimensional XYX Quantum Model Based on Fidelity

For two-dimensional XYX quantum model, iPEPS algorithm can select randomly initial state evolution, and get two degenerate symmetric broken ground state wave functions. In the quantum model, not only bifurcation behavior of ground state fidelity can be used, but bifurcation behavior of reduced density matrix fidelity can also be used to determine the phase transition point and its type caused by spontaneous broken symmetry of quantum of the system. Therefore, spontaneous symmetry breaking based on fidelity bifurcation can determine quantum phase transition one quantum system had gone through. This nature provides a method to further study quantum critical phenomena in quantum multibody system

Theoretical simulations of the vibrational predissociation spectra of H 5 + and D 5 + clusters

In the present study, the effect of the potential energy surface representation on the infrared spectra features of the H+ 5 and D+ 5 clusters is investigated. For the spectral simulations, we adopted a recently proposed (Sanz-Sanz et al. in Phys Rev A 84:060502-1–4, 2011) two-dimensional adiabatic quantum model to describe the proton-transfer motion between the two H2 or D2 units. The reported calculations make use of a reliable “on the fly” DFT-based potential surface and the corresponding new dipole moment surface. The results of the vibrational predissociation dynamics are compared with earlier and recent experimental data available from mass-selected photodissociation spectroscopy, as well as with previous theoretical calculations based on an analytical ab initio parameterized surfaces. The role of the potential topology on the spectral features is studied, and general trends are discussed.

Analytical solution of two-dimensional Scarf II model by means of SUSY methods

A new two-dimensional quantum model—a generalization of Scarf II—is completely solved analytically for integer values of the parameter. This model not being amenable to the conventional procedure of separation of variables is solved by a recently proposed method of supersymmetrical separation. The latter is based on two constituents of SUSY Quantum Mechanics: the intertwining relations with second order supercharges and the property of shape invariance. As a result, all energies of bound states were found, and the analytical expressions for corresponding wave functions were obtained.

New two-dimensional quantum models with shape invariance

Two-dimensional quantum models which obey the property of shape invariance are built in the framework of polynomial two-dimensional supersymmetric quantum mechanics. They are obtained using the expressions for known one-dimensional shape invariant potentials. The constructed Hamiltonians are integrable with symmetry operators of fourth order in momenta, and they are not amenable to the conventional separation of variables.

A new exactly solvable two-dimensional quantum model not amenable to separation of variables

The supersymmetric intertwining relations with second-order supercharges allow us to investigate a new two-dimensional model which is not amenable to standard separation of variables. The corresponding potential being the two-dimensional generalization of a well-known one-dimensional Pöschl–Teller model is proven to be exactly solvable for an arbitrary integer value of the parameter p: all its bound state energy eigenvalues are found analytically, and the algorithm for analytical calculation of all wavefunctions is given. The shape invariance of the model and its integrability are of essential importance to obtain these results.

Supersymmetrical Separation of Variables in Two-Dimensional Quantum Mechanics

Two different approaches are formulated to analyze two-dimensional quantum models which are not amenable to standard separation of variables. Both methods are essentially based on supersymmetrical second order intertwining relations and shape invariance - two main ingredients of the supersymmetrical quantum mechanics. The first method explores the opportunity to separate variables in the supercharge, and it allows to find a part of spectrum of the Schr\"odinger Hamiltonian. The second method works when the standard separation of variables procedure can be applied for one of the partner Hamiltonians. Then the spectrum and wave functions of the second partner can be found. Both methods are illustrated by the example of two-dimensional generalization of Morse potential for different values of parameters.

A class of partially solvable two-dimensional quantum models with periodic potentials

The supersymmetric approach is used to analyse a class of two-dimensional quantum systems with periodic potentials. In particular, the method of SUSY-separation of variables allowed us to find a part of the energy spectra and the corresponding wave functions (partial solvability) for several models. These models are not amenable to conventional separation of variables, and they can be considered as two-dimensional generalizations of Lamé, associated Lamé, and trigonometric Razavy potentials. All these models have the symmetry operators of fourth order in momenta, and one of them (the Lamé potential) obeys the property of self-isospectrality.

Gauge symmetry and non-Abelian topological sectors in a geometrically constrained model on the honeycomb lattice

We study a constrained statistical-mechanical model in two dimensions that has three useful descriptions. They are (i) the Ising model on the honeycomb lattice, constrained to have three up spins and three down spins on every hexagon, (ii) the three-color and fully packed loop model on the links of the honeycomb lattice, with loops around a single hexagon forbidden, and (iii) three Ising models on interleaved triangular lattices, with domain walls of the different Ising models not allowed to cross. Unlike the three-color model, the configuration space on the sphere or plane is connected under local moves. On higher-genus surfaces there are infinitely many dynamical sectors, labeled by a noncontractible set of nonintersecting loops. We demonstrate that at infinite temperature the transfer matrix admits an unusual structure related to a gauge symmetry for the same model on an anisotropic lattice. This enables us to diagonalize the original transfer matrix for up to 36 sites, finding an entropy per plaquette S/k{B} approximately 0.3661 ... centered and substantial evidence that the model is not critical. We also find the striking property that the eigenvalues of the transfer matrix on an anisotropic lattice are given in terms of Fibonacci numbers. We comment on the possibility of a topological phase, with infinite topological degeneracy, in an associated two-dimensional quantum model.

Mixed quantum-classical molecular dynamics simulation of vibrational relaxation of ions in an electrostatic field

The vibrational relaxation of ions in low-density gases under the action of an electrostatic field is reproduced through a molecular dynamics simulation method. The vibration is treated though quantum mechanics and the remaining degrees of freedom are considered classical. The procedure is tested through comparison against analytic results for a two-dimensional quantum model and by studying energy exchange during binary ion-atom collisions. Finally, the method has been applied successfully to the calculation of the mobility and the vibrational relaxation rate of O2+ in Kr as a function of the mean collision energy using a model interaction potential that reproduces the potential minimum of a previously known ab initio potential surface. The calculation of the steady mean vibrational motion of the ions in (flow) drift tubes seems straightforward, though at the expense of large amounts of computer time.

DIMENSIONAL REDUCTION ON A SPHERE

The question of the dimensional reduction of two-dimensional (2d) quantum models on a sphere to one-dimensional (1d) models on a circle is addressed. A possible application is to look at a relation between the 2d anyon model and the 1d Calogero–Sutherland model, which would allow for a better understanding of the connection between 2d anyon exchange statistics and Haldane exclusion statistics. The latter is realized microscopically in the 2d LLL anyon model and in the 1d Calogero model. In a harmonic well of strength ω or on a circle of radius R — both parameters ω and R have to be viewed as long distance regulators — the Calogero spectrum is discrete. It is well known that by confining the anyon model in a 2d harmonic well and projecting it on a particular basis of the harmonic well eigenstates, one obtains the Calogero–Moser model. It is then natural to consider the anyon model on a sphere of radius R and look for a possible dimensional reduction to the Calogero–Sutherland model on a circle of radius R. First, the free one-body case is considered, where a mapping from the 2d sphere to the 1d chiral circle is established by projection on a special class of spherical harmonics. Second, the N-body interacting anyon model is considered: it happens that the standard anyon model on the sphere is not adequate for dimensional reduction. One is thus led to define a new spherical anyon-like model deduced from the Aharonov–Bohm problem on the sphere where each flux line pierces the sphere at one point and exits it at its antipode.

New two-dimensional integrable quantum models from SUSY intertwining

Supersymmetrical intertwining relations of second order in the derivatives are investigated for the case of supercharges with deformed hyperbolic metric $g_{ik}=diag(1,-a^2)$. Several classes of particular solutions of these relations are found. The corresponding Hamiltonians do not allow the conventional separation of variables, but they commute with symmetry operators of fourth order in momenta. For some of these models the specific SUSY procedure of separation of variables is applied.

Exactly solvable two-dimensional complex model with a real spectrum

Using supersymmetric intertwining relations of the second order in derivatives, we construct a two-dimensional quantum model with a complex potential for which all energy levels and the corresponding wave functions are obtained analytically. This model does not admit separation of variables and can be considered a complexified version of the generalized two-dimensional Morse model with an additional sinh −2 term. We prove that the energy spectrum of the model is purely real. To our knowledge, this is a rather rare example of a nontrivial exactly solvable model in two dimensions. We explicitly find the symmetry operator, describe the biorthogonal basis, and demonstrate the pseudo-Hermiticity of the Hamiltonian of the model. The obtained wave functions are simultaneously eigenfunctions of the symmetry operator.

Entanglement and Factorized Ground States in Two-Dimensional Quantum Antiferromagnets

Making use of exact results and quantum Monte Carlo data for the entanglement of formation, we show that the ground state of anisotropic two-dimensional S=1/2 antiferromagnets in a uniform field takes the classical-like form of a product state for a particular value and orientation of the field, at which the purely quantum correlations due to entanglement disappear. Analytical expressions for the energy and the form of such states are given, and a novel type of exactly solvable two-dimensional quantum models is therefore singled out. Moreover, we show that the field-induced quantum phase transition present in the models is unambiguously characterized by a cusp minimum in the pairwise-to-global entanglement ratio R, marking the quantum-critical enhancement of multipartite entanglement.

New two-dimensional quantum models partially solvable by the supersymmetrical approach

New solutions for second-order intertwining relations in two-dimensional SUSY QM are found via the repeated use of the first order supersymmetrical transformations with intermediate constant unitary rotation. Potentials obtained by this method - two-dimensional generalized P\"oschl-Teller potentials - appear to be shape-invariant. The recently proposed method of $SUSY-$separation of variables is implemented to obtain a part of their spectra, including the ground state. Explicit expressions for energy eigenvalues and corresponding normalizable eigenfunctions are given in analytic form. Intertwining relations of higher orders are discussed.

Vibrationally mediated photolysis dynamics of H2O in the vOH=3 manifold: Far off resonance photodissociation cross sections and OH product state distributions

Vibrationally mediated photodissociation dynamics of water on the first excited electronic state surface (Ã) has been studied with slit jet-cooled H2O prepared in the complete polyad of vOH=3 overtone stretch levels (|03+〉, |12+〉, |12−〉, and |03−〉). (Notationally, |n1n2±〉 refers to symmetric/antisymmetric combinations of local mode OH stretch excitation, roughly corresponding to n1 and n2 quanta in the spectator and photolyzed OH bond, respectively.) At 248 nm photolysis wavelength the Condon point for bond cleavage occurs in the classically forbidden region, primarily sampling highly asymmetric H+OH exit valley geometries of the potential energy surface. Rotational, vibrational, spin orbit, and lambda doublet distributions resulting from this “far off resonance” photodissociation process are probed via laser induced fluorescence, exploiting the high efficiency laser excitation and light collection properties of the slit jet expansion geometry. Only vibrationally unexcited OH products are observed for both |12±〉 and |03±〉 initial excitation of H2O, despite different levels of vibration in the spectator OH bond. This is in contrast with “near-resonance” vibrationally mediated photolysis studies by Crim and co-workers in the |04−〉 and |13−〉 manifold, but entirely consistent with theoretical predictions from a simple two-dimensional quantum model. Photolysis out of the rotational ground H2O state (i.e., JKaKc=000) yields OH product state distributions that demonstrate remarkable insensitivity to the initial choice of H2O vibrational stretch state, in good agreement with rotational Franck–Condon models. However, this simple trend is not followed uniformly for rotationally excited H2O precursors, which indicates that these Franck–Condon models are insufficient and suggests that exit channel interactions do play a significant role in photodissociation dynamics of H2O at the fully state-to-state level.

Single-electron tunnelling transistor in SiGe/Si double-barrier structures

A single-electron transistor (SET) on an SiGe/Si double-barrier structure has been studied with side contact. In Si/SiGe vertical structure transistors we can more easily control the depletion region by gate voltage than in any other three-terminal device. Drain and source regions are one-dimensional and the well region (gate region) of the device is zero-dimensional with reverse gate bias. The resonant tunnelling (one dimension to zero dimension, zero dimension to one dimension) and current-voltage curves are investigated in a two-dimensional quantum model. The effect of delta doping results in shifting and reduction of the peak value of transmission probabilities. We found current enhancement in delta-doped heterostructures at low temperatures. We proved the possibility of an Si single-electron transistor in a two-dimensional simulation of the resonant tunnelling transistor.