Spectrum Of Hamiltonian Research Articles

Solvable Model of Quantum Random Optimization Problems

We study the quantum version of a simplified model of optimization problems, where quantum fluctuations are introduced by a transverse field acting on the qubits. We find a complex low-energy spectrum of the quantum Hamiltonian, characterized by an abrupt condensation transition and a continuum of level crossings as a function of the transverse field. We expect this complex structure to have deep consequences on the behavior of quantum algorithms attempting to find solutions to these problems.

Universality of the glassy transitions in the two-dimensional±JIsing model

We investigate the zero-temperature glassy transitions in the square-lattice ±J Ising model, with bond distribution P(J{xy})=pδ(J{xy}-J)+(1-p)δ(J{xy}+J) ; p=1 and p=1/2 correspond to the pure Ising model and to the Ising spin glass with symmetric bimodal distribution, respectively. We present finite-temperature Monte Carlo simulations at p=4/5 , which is close to the low-temperature paramagnetic-ferromagnetic transition line located at p≈0.89 , and at p=1/2 . Their comparison provides a strong evidence that the glassy critical behavior that occurs for 1-p{0}<p<p{0} , p{0}≈0.897 , is universal, i.e., independent of p . Moreover, we show that glassy and magnetic modes are not coupled at the multicritical zero-temperature point where the paramagnetic-ferromagnetic transition line and the T=0 glassy transition line meet. On the theoretical side we discuss the validity of finite-size scaling in glassy systems with a zero-temperature transition and a discrete Hamiltonian spectrum. Because of a freezing phenomenon which occurs in a finite volume at sufficiently low temperatures, the standard finite-size scaling limit in terms of TL(1/ν) does not exist; the renormalization-group invariant quantity ξ/L should be used instead as basic variable.

Resonance scattering and singularities of the scattering function

Recent studies of transport phenomena with complex potentials are explained by generic square root singularities of spectrum and eigenfunctions of non-Hermitian Hamiltonians. Using a two channel problem we demonstrate that such singularities produce a significant effect upon the pole behaviour of the scattering matrix, and more significantly upon the associated residues. This mechanism explains why by proper choice of the system parameters the resonance cross section is increased drastically in one channel and suppressed in the other channel.

Accumulation Point in the Spectrum of the Collision Operator for a One-Dimensional Polaron System

The spectrum of Hamiltonians has been well investigated in quantum mechanics, since it plays an essential role in analyzing dynamical properties of conservative systems. For example, we know there are Hamiltonians that have a discrete spectrum, or a continuous spectrum, or a band structure, and so on, each of which leads to a characteristic behavior of the system. However, our knowledge on the spectrum of the collision operators in kinetic equations in dissipative systems still remains in a poor stage in spite of the fact that their spectral properties play a fundamental role in non-equilibrium statistical physics. In this paper we give an example of quantum systems that has a rich structure of the spectrum, such as an accumulation point and band structures, in the collision operator for a momentum relaxation process. We consider a one-dimensional (1D) polaron system in which a quantum particle is weakly interacting with a thermal reservoir consisting of an acoustic phonon field. Similar systems have been studied in different contexts, see e.g. Refs. 1)–5). As a consequence of a constraint in 1D due to the resonance condition, the momenta of the particle related successively through the collision operator form a subset separated from other momenta. Momentum relaxation occurs among those momenta in such a subset independently of other momenta. In this case the collision operator is represented by a tridiagonal matrix for each such subset of momenta. Taking advantage of the tridiagonal nature of the matrix, we found a solution of the eigenvalue problem of the collision operator in terms of continued fractions. This paper is organized as follows. In §2 we introduce a model of a quantum particle weakly coupled with a phonon field, and present a kinetic equation for the

The effect of boundaries on the spectrum of a one-dimensional random mass Dirac Hamiltonian

The average density of states (DoS) of the one-dimensional Dirac Hamiltonian with a random mass on a finite interval [0, L] is derived. Our method relies on the eigenvalue distributions (extreme value statistics problem) which are obtained explicitly. The well-known Dyson singularity is recovered above the crossover energy . Below ϵc we find a log-normal suppression of the average DoS .

Wegner-type Bounds for a Multi-particle Continuous Anderson Model with an Alloy-type External Potential

We consider an N -particle quantum systems in R d , with interaction and in pres- ence of a random external alloy-type potential (a continuous N -particle Anderson model). We establish Wegner-type bounds (inequalities) for such models, giving upper bounds for the probability that random spectra of Hamiltonians in finite volumes intersect a given set.

Wave packet splitting and zitterbewegung in two-dimensional semiconductor structures with spin-orbit coupling

Dynamics of electron wave packets in an asymmetric quantum well in the presence of Rashba spinorbit coupling was analytically and numerically studied. Electron Green’s functions were introduced and the evolution of 1D and 2D wave packets was studied. The effect of packet splitting caused by the presence of two branches with different chiralities in the Rashba Hamiltonian spectrum and zitterbewegung, i.e., packet center’s jitter, was studied. Spatial components of the spin density were calculated. It was shown that the component of the spin density Sy in split parts of the wave packet has opposite signs, and two other spin density components oscillate in space between scattering packets.

Counting operator analysis of the discrete spectrum of some model Hamiltonians

The first step in the counting operator analysis of the spectrum of any model Hamiltonian H is the choice of a Hermitean operator M in such a way that the third commutator with H is proportional to the first commutator. Next one calculates operators R and R † which share some of the properties of creation and annihilation operators, and are such that M becomes a counting operator. The spectrum of H is then decomposed into multiplets, not determined by symmetries of H, but by those of a reference Hamiltonian H ref , which is defined by H ref = H − R − R † , and which commutes with M. Finally, we introduce the notion of stable eigenstates. It is shown that under rather weak conditions one stable eigenstate can be used to construct another one.

Analysis of the [formula omitted] Hubbard ring using counting operators

We prove three theorems about the use of a counting operator to study the spectrum of model Hamiltonians. We analytically calculate the eigenvalues of the Hubbard ring with four lattice positions and apply our theorems to describe the observed level crossings.

Comments on the Dynamics of the Pais-Uhlenbeck Oscillator

We discuss the quantum dynamics of the Pais-Uhlenbeck oscillator. The Lagrangian of this higher-derivative model depends on two frequencies. When the frequencies are different, the free PU oscillator has a pure point spectrum that is dense everywhere. When the frequencies are equal, the spectrum is continuous. It is not bounded from below, running from minus to plus infinity, but this is not disastrous as the Hamiltonian is still Hermitian and the evolution operator is still unitary. Generically, the inclusion of interaction terms breaks unitarity, but in some special cases unitarity is preserved. We discuss also the nonstandard realization of the PU oscillator suggested by Bender and Mannheim, where the spectrum of the free Hamiltonian is positive definite, but wave functions grow exponentially for large real values of canonical coordinates. The free nonstandard PU oscillator is unitary when the frequencies are different, but unitarity is broken in the equal frequencies limit.

Non-Hermitian Hamiltonians with space-time symmetry

The spectrum of non-Hermitian Hamiltonians is studied using the point-group symmetries of the Hermitian and non-Hermitian parts of the Hamiltonian. We show that, in principle, the symmetry properties of the Hamiltonian are responsible for the appearance of real eigenvalues in the spectrum of the non-Hermitian Hamiltonian. In practice, however, one must take into account the ``strength'' of the non-Hermitian part of the Hamiltonian. An effective energy-dependent Hermitian Hamiltonian is constructed that has the same real spectrum as the non-Hermitian Hamiltonian. Several properties, including the self-orthogonality of the states at branch points of the spectrum, are proven using this effective representation. An interesting possibility of complex eigenvalues returning to the real axis is exemplified along with several examples of non-Hermitian Hamiltonians with real eigenvalues.

The spectrum of a commutator Hamiltonian is hydrogenlike

For any finite group, an element (commutator Hamiltonian) is defined in its group algebra so that in any representation of that group the image of this element is diagonalizable and has the spectrum contained in the set {1/n2|n = 1,2,3,…}. The result is generalized onto an arbitrary compact group. In particular, it is pointed out that for the natural representation of the group SU(2, C) in the space of complex-valued functions with the square of absolute values integrable over the Haar measure the multiplicity of the eigenvalue 1/n2 of the commutator Hamiltonian is equal to n2.

Elliptical instability of a vortex tube and drift current induced by it

We revisit, with some extension to the nonlinear regime, the Moore–Saffman–Tsai–Widnall instability, which is the three-dimensional instability of a straight vortex tube elliptically strained by a pure shear flow. A circular cylindrical vortex tube supports neutrally stable three-dimensional waves called Kelvin waves. A pure shear breaks the circular symmetry, causing a parametric resonance between two Kelvin waves whose azimuthal wavenumbers are separated by two. The eigenvalues and functions are written out in full in terms of the Bessel and the modified Bessel functions, and so is the growth rate. That this instability has a link with the elliptical instability in the short-wave limit is established in this study. This result is studied in the light of Krein's theory for Hamiltonian spectra. We develop a Lagrangian (particle) formulation that makes feasible an efficient calculation of iso-vortical disturbances up to second order in amplitude and thereby facilitates calculation of the energy of Kelvin waves. It is found that a Kelvin wave induces a drift current along the vortex tube.

Exact boson mapping of the nuclear pairing Hamiltonian

An exact boson mapping of the deformed mean-field plus equal strength pairing Hamiltonian is considered. In the mapping, fermion pair operators are mapped exactly to the corresponding bosons. The mapping occurs at the level of the Richardson–Gaudin equations. The image of the mapping results is a Bose–Hubbard model with level-dependent hopping. Although the resultant Bose–Hubbard Hamiltonian is non-Hermitian, a part of spectrum of the Bose–Hubbard Hamiltonian with U/t = 1 determined by the corresponding Richardson–Gaudin (Bethe ansatz) equations corresponds exactly to the whole spectrum of the pairing Hamiltonian.

Directed Abelian algebras and their application to stochastic models

With each directed acyclic graph (this includes some D-dimensional lattices) one can associate some Abelian algebras that we call directed Abelian algebras (DAAs). On each site of the graph one attaches a generator of the algebra. These algebras depend on several parameters and are semisimple. Using any DAA, one can define a family of Hamiltonians which give the continuous time evolution of a stochastic process. The calculation of the spectra and ground-state wave functions (stationary state probability distributions) is an easy algebraic exercise. If one considers D-dimensional lattices and chooses Hamiltonians linear in the generators, in finite-size scaling the Hamiltonian spectrum is gapless with a critical dynamic exponent z=D. One possible application of the DAA is to sandpile models. In the paper we present this application, considering one- and two-dimensional lattices. In the one-dimensional case, when the DAA conserves the number of particles, the avalanches belong to the random walker universality class (critical exponent sigma_(tau)=32 ). We study the local density of particles inside large avalanches, showing a depletion of particles at the source of the avalanche and an enrichment at its end. In two dimensions we did extensive Monte-Carlo simulations and found sigma_(tau)=1.780+/-0.005 .

Lower spectral branches of a spin-boson model

We study the structure of the spectrum of a two-level quantum system weakly coupled to a boson field (spin-boson model). Our analysis allows to avoid the cutoff in the number of bosons, if their spectrum is bounded below by a positive constant. We show that, for small coupling constant, the lower part of the spectrum of the spin-boson Hamiltonian contains (one or two) isolated eigenvalues and (one or two) manifolds of atom +1-boson states indexed by the boson momentum q. The dispersion laws and generalized eigenfunctions of the latter are calculated.

Strongly coupled largeNspectrum of two matrices coupled via a Yang-Mills interaction

We consider the large N spectrum of the quantum mechanical Hamiltonian of two Hermitian matrices coupled via a Yang-Mills interaction. In a framework where one of the matrices is treated exactly and the other is treated as a creation operator impurity, the difference equation associated with the Yang-Mills interaction is derived and solved exactly for two impurities. In this case, the full string tension corrected spectrum depends on two momenta. For a specific value of one of these momenta, the spectrum has the same structure as that of giant magnon bound states. States with general number of impurities are also discussed.

Visualization of Branch Points inPT-Symmetric Waveguides

The visualization of an exceptional point in a PT-symmetric directional coupler (DC) is demonstrated. In such a system the exceptional point can be probed by varying only a single parameter. Using the Rayleigh-Schrödinger perturbation theory we prove that the spectrum of a PT-symmetric Hamiltonian is real as long as the radius of convergence has not been reached. We also show how one can use a PT-symmetric directional coupler to measure the radius of convergence for non-PT-symmetric structures. For such systems the physical meaning of the rather mathematical term radius of convergence is exemplified.

Intruder states in multireference perturbation theory: The ground state of manganese dimer

A detailed analysis of a severe intruder state problem in the multistate multireference perturbation theory (MS-MRPT) calculations on the ground state of manganese dimer is presented. An enormous number of detected intruder states (> 5000) do not permit finding even an approximate shape of the X(1)Sigma(g) (+) potential energy curve. The intruder states are explicitly demonstrated to originate from quasidegeneracies in the zeroth-order Hamiltonian spectrum. The electronic configurations responsible for appearance of the quasidegeneracies are identified as single and double excitations from the active orbitals to the external orbitals. It is shown that the quasidegeneracy problem can be completely eliminated using shift techniques despite of its severity. The resultant curves are smooth and continuous. Unfortunately, strong dependence of the spectroscopic parameters of the X(1)Sigma(g) (+) state on the shift parameter is observed. This finding rises serious controversies regarding validity of employing shift techniques for solving the intruder state problem in MS-MRPT. Various alternative approaches of removing intruder states (e.g., modification of the basis set or changing the active space) are tested. None of these conventional techniques is able to fully avoid the quasidegeneracies. We believe that the MS-MRPT calculations on the three lowest A(g) states of manganese dimer constitute a perfect benchmark case for studying the behavior of MRPT in extreme situations.

Gamow vectors: miscellaneous results

In this note, we discuss some mathematical properties of Gamow vectors. We start with the definition and properties of the approximate Gamow vectors that are Hilbert space vectors with similar properties to Gamow vectors and are defined by integration over the Hamiltonian spectrum and not over the whole real line, as is the case with ordinary Gamow vectors. We also discuss some properties of Gamow dyads, which are density operators constructed with Gamow vectors.