Numerical Diagonalization Research Articles

Quasiparticle representation of coherent nonlinear optical signals of multi-excitons

Multi-exciton Green's functions and scattering matrices in many-fermion systems are calculated using a quasiparticle approach based on a generalized Bethe–Salpeter equation. The simulation protocol only requires numerical diagonalization of the single-exciton manifold. Using coboson algebra all many-body effects are recast in terms of two tetradic exciton–exciton interactions: direct Coulomb scattering and Pauli exchange. The tedious equations-of-motion derivations and calculations of multi-exciton manifolds are avoided. The approach is applied to calculate the third- and fifth-order signals generated by sequences of femtosecond optical pulses. Several coherent fifth order optical signals that directly probe three-exciton states and their projections on double and single-exciton states are predicted.

Molecular Mechanisms, Thermodynamics, and Dissociation Kinetics of Knob-Hole Interactions in Fibrin

Polymerization of fibrin, the primary structural protein of blood clots and thrombi, occurs through binding of knobs 'A' and 'B' in the central nodule of fibrin monomer to complementary holes 'a' and 'b' in the γ- and β-nodules, respectively, of another monomer. We characterized the A:a and B:b knob-hole interactions under varying solution conditions using molecular dynamics simulations of the structural models of fibrin(ogen) fragment D complexed with synthetic peptides GPRP (knob 'A' mimetic) and GHRP (knob 'B' mimetic). The strength of A:a and B:b knob-hole complexes was roughly equal, decreasing with pulling force; however, the dissociation kinetics were sensitive to variations in acidity (pH 5-7) and temperature (T = 25-37 °C). There were similar structural changes in holes 'a' and 'b' during forced dissociation of the knob-hole complexes: elongation of loop I, stretching of the interior region, and translocation of the moveable flap. The disruption of the knob-hole interactions was not an "all-or-none" transition as it occurred through distinct two-step or single step pathways with or without intermediate states. The knob-hole bonds were stronger, tighter, and more brittle at pH 7 than at pH 5. The B:b knob-hole bonds were weaker, looser, and more compliant than the A:a knob-hole bonds at pH 7 but stronger, tighter, and less compliant at pH 5. Surprisingly, the knob-hole bonds were stronger, not weaker, at elevated temperature (T = 37 °C) compared with T = 25 °C due to the helix-to-coil transition in loop I that helps stabilize the bonds. These results provide detailed qualitative and quantitative characteristics underlying the most significant non-covalent interactions involved in fibrin polymerization.

Emergence of Majorana modes in cylindrical nanowires

We present calculations of Majorana edge modes in cylindrical nanowires of a semiconductor material with proximity-induced superconductivity. We consider a Rashba field along the transverse direction and an applied magnetic field in arbitrary orientation. Our analysis is based on exact numerical diagonalizations for the finite cylinder and on the complex band structure for the semi-infinite one. Orbital effects are responsible for a strong anisotropy of the critical field for which the effective gap vanishes. Robust Majorana modes are induced by the parallel-field component and we find regimes with more than one Majorana mode on the same edge. Experimentally, they would manifest as a specific sequence of zero-bias conductances as a function of the magnetic field. In the finite cylinder, a degradation of the Majorana modes due to interference of the two edges leads to oscillating non-zero energies for large enough fields.

Relation between Wigner energy and proton-neutron pairing

The linear term proportional to $|N-Z|$ in the nuclear symmetry energy (Wigner energy)is obtained in a model that uses isovector pairing on single particle levels from a deformed potential combined with a $\vec T^2$ interaction. The pairing correlations are calculated by numerical diagonalization of the pairing Hamiltonian acting on the six or seven levels nearest the $N=Z$ Fermi surface. The experimental binding energies of nuclei with $N\approx Z$ are well reproduced. The Wigner energy emerges as a consequence of restoring isospin symmetry. We have found the Wigner energy to be insensitive to the presence of moderate isoscalar pair correlations.

Criticalities of the transverse- and longitudinal-field fidelity susceptibilities for thed=2quantum Ising model

The inner product between the ground-state eigenvectors with proximate interaction parameters, namely, the fidelity, plays a significant role in the quantum dynamics. In this paper, the critical behaviors of the transverse- and longitudinal-field fidelity susceptibilities for the d=2 quantum (transverse-field) Ising model are investigated by means of the numerical-diagonalization method; the former susceptibility has been investigated rather extensively. The critical exponents for these fidelity susceptibilities are estimated as α(F)((t))=0.752(24) and α(F)((h))=1.81(13), respectively. These indices are independent, and suffice for obtaining conventional critical indices such as ν=0.624(12) and γ=1.19(13).

Effect of Spin–Orbit Interaction on (4d)3- and (5d)3-Based Transition-Metal Oxides

The effects of a spin-orbit interaction on transition-metal ions of (4d)^3- and (5d)^3 -based oxides in which three electrons occupy t_{2g} orbitals are studied. The amplitude of the magnetic moment of d electrons on the 5d and 4d orbitals is estimated by numerical diagonalization. It is found that the magnetic moment is reduced by the spin-orbit interaction. It is suggested that (4d)^3- and (5d)^3 -based oxides are located in the middle of the L-S and J-J coupling schemes.

Gutzwiller variational method for intersite Coulomb interactions: The spinless fermion model in one dimension

We study the Gutzwiller method for the spinless fermion model in one dimension, which is one of the simplest models that incorporates the intersite Coulomb interaction. The Gutzwiller solution of this model has been studied in the literature but with differing results. We obtain the Gutzwiller solution of the problem by a careful enumeration of the many-particle configurations and explain the origin of the discrepancy in the literature to be due to the neglect of the correlation between the neighbouring bond occupancies. The correct implementation of the Gutzwiller approach yields results different from the slave-boson mean-field theory, unlike for the Hubbard model with on-site interaction, where both methods are known to be equivalent. The slave-boson and the Gutzwiller results are compared to the exact solution, available for the half-filled case, and to the numerical exact diagonalization results for the case of general filling.

Synthesis, molecular and crystalline architectures, and properties of a mononuclear complex [Co II (benzidine) 2 (NCS) 2 (OH 2 ) 2

One mononuclear cobalt(II) compound of the type [Co(bnzd)2(NCS)2(OH2)2] (1) (bnzd = benzidine) has been isolated through a single-pot reaction of the molecular building components in MeOH–H2O solvent mixture at room temperature and characterized by physico-chemical and spectroscopic methods. The crystal structure of complex 1 has an octahedral geometry with trans, trans, trans orientations in the order (Na,Na), (Nt,Nt), (Ow,Ow) [Na = N(amine), Nt = N(thiocyanate) and Ow = O(water)]. In crystalline state, individual units of 1 are associated by weak cooperative O–H...N, O–H...S and N–H...S hydrogen bonds resulting in a 3D network structure. The compound is redox active and shows luminescence in MeOH solution. Thermal decomposition pattern of 1 reveals the presence of two coordinated water molecules. Variable-temperature magnetic susceptibility measurement shows significant orbital contribution and numerical matrix diagonalization method gives the best fit parameters: α = A*k = 1.43; λ = −130 cm − 1; Δ = − 864 cm − 1; TIP = 0.000975; R = 1.19 × 10 − 4. Mononuclear compound [Co(benzidine)2(NCS)2(OH2)2] (1) was isolated and characterized. Cobalt(II) center adopts an octahedral geometry and individual units of 1 are associated by weak O–H…N/S and N–H…S hydrogen bonds resulting in a 3D network structure. Complex 1 is electroactive and shows luminescence property. Variable-temperature magnetic property shows significant orbital contribution.

Exciton states of II–VI tetrapod-shaped nanocrystals

Excitons of tetrapod-shaped nanocrystals made of CdTe, CdS, CdSe, ZnTe, and ZnSe were investigated systematically by numerical diagonalization of configuration interaction Hamiltonian based on single-particle states obtained by finite-element method. Both one-particle and exciton wave functions have high spatial symmetries due to the tetrahedral symmetry of the nanocrystals, which leads to a distinct selection rule for optical absorption and emission. The absorption spectra thus calculated were compared with available experimental data and good agreement was found.

Symmetric excitation and de-excitation of a cavity QED system

We calculate the time evolution of a cavity-QED system subject to a time dependent sinusoidal drive. The drive is modulated by an envelope function with the shape of a pulse. The system consists of electrons embedded in a semiconductor nanostructure which is coupled to a single mode quantized electromagnetic field. The electron-electron as well as photon-electron interaction is treated exactly using “exact numerical diagonalization” and the time evolution is calculated by numerically solving the equation of motion for the system’s density matrix. We find that the drive causes symmetric excitation and de-excitation where the system climbs up the Jaynes-Cummings ladder and descends back down symmetrically into its original state. This effect is known at low electron-photon coupling strengths but our main finding is how robust the effect is even at ultra-strong coupling strength where the JC-model does not give qualitatively correct results. We investigate the robustness of this symmetric behavior with respect to the drive de-tuning and pulse duration.

Critical behavior of the fidelity susceptibility for the [formula omitted] transverse-field Ising model

The overlap (inner product) between the ground-state eigenvectors with proximate interaction parameters, the so-called fidelity, plays a significant role in the quantum-information theory. In this paper, the critical behavior of the fidelity susceptibility is investigated for a two-dimensional transverse-field (quantum) Ising model by means of the numerical diagonalization method. In order to treat a variety of system sizes N=12,14,…,32, we adopt the screw-boundary condition. Finite-size artifacts (scaling corrections) of the fidelity susceptibility appear to be suppressed, as compared to those of the Binder parameter. As a result, we estimate the fidelity susceptibility critical exponent as αF=0.715(20).

Relaxational Modes and Aging in the Glauber Dynamics of the Sherrington-Kirkpatrick Model

The relaxational modes and aging of the Glauber dynamics of the mean-field spin-glass model, SK model, are studied by a numerical diagonalization technique and Monte Carlo simulations. We found that the aging process of the model is understood as hierarchical growth of quasi-equilibrium domain in the phase-space.

Simulation of the Jahn–Teller–Dicke magnetic structural phase transition with trapped ions

We study theoretically the collective E⊗e Jahn–Teller–Dicke distortion in a system of trapped ions. In a previous work (Porras et al 2012 Phys. Rev. Lett. 108 235701) we have focused on cooperative Jahn–Teller models consisting of an ensemble of effective spins coupled to a set of many vibrational modes. Here, we show that under suitable conditions the spin ensemble may interact only with a single vibrational mode in each radial direction with U(1) symmetric couplings. Our model is exactly solvable in the thermodynamical limit and it is amenable to be solved by exact numerical diagonalization for a moderate number of ions. We show that trapped ions are ideally suited to study the spontaneous breaking of a continuous symmetry as well as magnetic structural phase transitions in a mesoscopic spin–boson system.

Self-consistent tight-binding model of B and N doping in graphene

Boron and nitrogen substitutional impurities in graphene are analyzed using a self-consistent tight-binding approach. An analytical result for the impurity Green's function is derived taking broken electron-hole symmetry into account and validated by comparison to numerical diagonalization. The impurity potential depends sensitively on the impurity occupancy, leading to a self-consistency requirement. We solve this problem using the impurity Green's function and determine the self-consistent local density of states at the impurity site and, thereby, identify acceptor and donor energy resonances.

Universal spatial correlations in random spinor fields

We identify universal spatial fluctuations in systems with nontrivial spin dynamics. To this end we calculate by exact numerical diagonalization a variety of experimentally relevant correlations between spinor amplitudes, spin polarizations, and spin currents, both in the bulk and near the boundary of a confined two-dimensional clean electron gas in the presence of spin-orbit interaction. We support our claim of universality with the excellent agreement between the numerical results and system-independent spatial correlations of a random field defined on both the spatial and spin degrees of freedom. A rigorous identity relating our universal predictions with response functions provides a direct physical interpretation of our results in the framework of linear response theory.

Solving for three-dimensional central potentials using numerical matrix methods

We present a numerical matrix method to find quantum stationary states and energies for three-dimensional central forces. The method can be used both for familiar, exactly solvable potentials and for those that are not solvable by analytical methods. As examples, we include the Coulomb potential, the finite spherical well, and the Yukawa potential. This method requires much less mathematical expertise than traditional analytical methods, although it does require some familiarity with numerical matrix diagonalization software.

Long-Range Order of the Three-Sublattice Structure in the S=1 Heisenberg Antiferromagnet on a Spatially Anisotropic Triangular Lattice

We study the S=1 Heisenberg antiferromagnet on a spatially anisotropic triangular lattice by the numerical diagonalization method. We examine the stability of the long-range order of a three-sublattice structure observed in the isotropic system between the isotropic case and the case of isolated one-dimensional chains. It is found that the long-range-ordered ground state with this structure exists in the range of 0.7 \simle J_2/J_1 \le 1, where J_1 is the interaction amplitude along the chains and J_2 is the amplitude of other interactions.

Even-odd effect in short antiferromagnetic Heisenberg chains

Motivated by recent experiments on chemically synthesized magnetic molecular chains, we investigate the lowest-lying energy band of short spin-$s$ antiferromagnetic Heisenberg chains focusing on effects of open boundaries. By numerical diagonalization we find that the Land\'e pattern in the energy levels, i.e., $E(S)\ensuremath{\propto}S(S+1)$ for total spin $S$, known from, e.g., ring-shaped nanomagnets, can be recovered in odd-membered chains, while strong deviations are found for the lowest excitations in chains with an even number of sites. This particular even-odd effect in the short Heisenberg chains cannot be explained by simple effective Hamiltonians and symmetry arguments. We go beyond these approaches, taking into account quantum fluctuations by means of a path-integral description and the valence bond basis, but the resulting quantum edge-spin picture which is known to work well for long chains does not agree with the numerical results for short chains and cannot explain the even-odd effect. Instead, by analyzing also the classical chain model, we show that spatial fluctuations dominate the physical behavior in short chains, with length $N\ensuremath{\lesssim}{e}^{\ensuremath{\pi}s}$, for any spin $s$. Such short chains are found to display a unique behavior, which is not related to the thermodynamic limit and cannot be described well by theories developed for this regime.

Phase transitions in a spinless, extended Falicov–Kimball model on the triangular lattice

A numerical diagonalization technique with canonical Monte-Carlo simulation algorithm is used to study the phase transitions from low temperature (ordered) phase to high temperature (disordered) phase of spinless Falicov–Kimball model on a triangular lattice with correlated hopping (t′). It is observed that the low temperature ordered phases (i.e. regular, bounded and segregated) persist up to a finite critical temperature (Tc). In addition, we observe that the critical temperature decreases with increasing the correlated hopping in regular and bounded phases whereas it increases in the segregated phase. Single and multi peak patterns seen in the temperature dependence of specific heat (Cv) and charge susceptibility (χ) for different values of parameters like on-site Coulomb correlation strength (U), correlated hopping (t′) and filling of localized electrons (nf) are also discussed.

Confinement of an exciton in a quantum dot: effect of modified Kratzer potential

In the present work, we have considered an exciton confined in a quantum dot with modified Kratzer potential. We have studied the electronic and optical properties of the system by using the numerical diagonalization of the Hamiltonian matrix. For this purpose, we have calculated the binding energies of the ground and first excited states as functions of the quantum dot size. We have also computed the linear, nonlinear and total absorption coefficients between ground and first excited states. It is found that the quantum dot radius has an important role on the binding energy and absorption coefficient.