Numerical Diagonalization Method Research Articles

The spin-S Heisenberg antiferromagnet on the orthogonal-dimer lattice

The spin-SHeisenberg antiferromagnet on the orthogonal-dimer lattice is studied. In most of the investigations of the antiferromagnet on this lattice beginning with the study by Shastry and Sutherland, theS=1/2case has intensively been treated. In the present study, the cases ofS = 1 and 3/2 are particularly treated by the numerical-diagonalization method based on the Lanczos algorithm applied to finite-size clusters. We successfully capture the edge of the exact-dimer phase and that of the Néel-ordered phase. A significant finding is the existence of an intermediate phase between both the edges irrespective of spin-S.

A semi-exact analytical study of the phase diagram of the two dimensional extended Holstein-Hubbard model

An extended Holstein-Hubbard model is studied in two dimensions analytically for both weak and strong correlations. A series of unitary transformations followed by an averaging with a fully-generalized phonon state is performed to obtain an effective electronic Hamiltonian. For weak correlation, the Hartree-Fock approximation is employed and for strong correlation, the electronic Hamiltonian is mapped on to an effective t−J model which is solved using the Gutzwiller approximation and the Zubarev technique. The variation of the effective Coulomb interaction strength is studied for different electron-electron and electron-phonon interaction coefficients and the SDW-CDW phase diagram is plotted using the Mott-Hubbard metallicity criteria. A metallic regime is shown to exist at the SDW-CDW cross-over region and the width of the metallic phase is found to be broader than the result obtained by numerical diagonalization method. The metallic phase is also found to be wider in two dimensions than in one dimension.

Josephson effects between the Kitaev ladder superconductors

The two-leg ladder system consisting of the Kitaev chains is known to exhibit a richer phase diagram than that of the single chain. We theoretically investigate the variety of the Josephson effects between the ladder systems. We consider the Josephson phase difference \thetaθ between these two ladder systems as well as the phase difference \phiϕ between the parallel chains in each ladder system. The total energy of the junction at T = 0T=0 is calculated by a numerical diagonalization method as functions of \thetaθ, \phiϕ, and also a transverse hopping t_{\perp}t⊥ in the ladders. We find that, by controlling t_{\perp}t⊥ and \phiϕ, the junction exhibits not only the fractional Josephson effect for the phase difference \thetaθ, but also the usual 0-junction and even \piπ-junction properties.

The quantum vortex states in extended Bose–Hubbard model: effects of lattice geometries, inter-particle interactions and spatial inhomogeneity

We study quantum vortex states of strongly interacting bosons in a two-dimensional rotating optical lattice. The system is modeled by Bose-Hubbard Hamiltonian with rotation. We consider lattices of different geometries, such as square, rectangular and triangular. Using numerical exact diagonalization method we show how the rotation introduces vortex states of different ground-state symmetries and the transition between these states at discrete rotation frequencies. We show how the geometry of the lattice plays crucial role in determining the maximum number of vortex states as well as the general characteristics of these states such as, the average angular momentum $<L_z>$, the current at the perimeter of the lattice, phase winding, the relation between the maximum phase difference, the maximum current and also the saturation of the current between the two neighboring lattice points. The effect of the two- and three-body interactions between the particles, both attractive and repulsive, also depends on the geometry of the lattice as the current flow or the lattice current depends on the interactions. We also consider the effect of the spatial inhomogeneity introduced by the presence of an additional confining harmonic trap potential. It is shown that the curvature of the trap potential and the position of the minimum of the trap potential with respect to the axis of rotation or the center of the lattice have a significant effect on the general characteristics these vortex states.

NUMERICAL STUDYING ENERGY PARAMETERS OF MULTIELECTRON ATOM IN A MAGNETIC FIELD: HELIUM

There are presented the results of calculating the energies of the helium atom energy in a homogeneous magnetic field on the basis of the new numerical quantum-mechanical approach. The approach is based on the numerical difference solution of the Schrödinger equation, the model potential method and the operator perturbation theory formalism. The obtained results on energy of the helium atom in dependence upon the magnetic field strength are compared with available theoretical results, obtained on the basis of alternative numerical Hartree-Fock and diagonalization methods.

Magnetization process of the S = 1/2 Heisenberg antiferromagnet on the floret pentagonal lattice

We study the S = 1/2 Heisenberg antiferromagnet on the floret pentagonal lattice by numerical diagonalization method. This system shows various behaviours that are different from that of the Cairo-pentagonal-lattice antiferromagnet. The ground-state energy without magnetic field and the magnetization process of this system are reported. Magnetization plateaux appear at one-ninth height of the saturation magnetization, at one-third height, and at seven-ninth height. The magnetization plateaux at one-third and seven-ninth heights come from interactions linking the sixfold-coordinated spin sites. A magnetization jump appears from the plateau at one-ninth height to the plateau at one-third height. Another magnetization jump is observed between the heights corresponding to the one-third and seven-ninth plateaux; however the jump is away from the two plateaux, namely, the jump is not accompanied with any magnetization plateaux. The jump is a peculiar phenomenon that has not been reported.

Ground-state properties of a quantum frustrated spin chain with side spins and arbitrary spin s

The properties of the spin-s frustrated antiferromagnetic Heisenberg chain with side spins are studied by means of the coupled cluster method and the numerical exact diagonalization method. The results of the two methods both manifest that the ground state quantum phase diagram of the model consists of the Néel phase and the canted phase characterized by two pitch angles, irrespective of the value of the spin quantum number s. Although the phenomenon of quantum melting of magnetic long-range order is absent in the model, quantum fluctuation exerts an important influence on the property of the model. In addition to the reduction of magnetization, the nature of the phase transition from the Néel state to the canted state transforms from a classical second-order transition to a quantum first-order transition at any finite value of s. It is observed that the ways one of the two pitch angles evolves with frustration in quantum and classical cases, are qualitatively different in the large frustration parameter region. The location of the critical point and the physical quantities, including the ground state energy per spin and the magnetic order parameter, move toward their classical values in the form of power series in s.

Itinerant Topological Magnons in SU(2) Symmetric Topological Hubbard Models with Nearly Flat Electronic BandsSupported by the National Natural Science Foundation of China (Grant No. 11774152), and National Key R&D Program of China (Grant No. 2016YFA0300401).

We show that a suitable combination of flat-band ferromagnetism, geometry and nontrivial electronic band topology can give rise to itinerant topological magnons. An SU(2) symmetric topological Hubbard model with nearly flat electronic bands, on a Kagome lattice, is considered as the prototype. This model exhibits ferromagnetic order when the lowest electronic band is half-filled. Using the numerical exact diagonalization method with a projection onto this nearly flat band, we can obtain the magnonic spectra. In the flat-band limit, the spectra exhibit distinct dispersions with Dirac points, similar to those of free electrons with isotropic hoppings, or a local spin magnet with pure ferromagnetic Heisenberg exchanges on the same geometry. Significantly, the non-flatness of the electronic band may induce a topological gap at the Dirac points, leading to a magnonic band with a nonzero Chern number. More intriguingly, this magnonic Chern number changes its sign when the topological index of the electronic band is reversed, suggesting that the nontrivial topology of the magnonic band is related to its underlying electronic band. Our work suggests interesting directions for the further exploration of, and searches for, itinerant topological magnons.

Fractionalized spin excitations in the ferromagnetic edge state of graphene: Signature of the ferromagnetic Luttinger liquid

The elementary excitations from the conventional magnetic ordered states, such as ferromagnets and antiferromagnets, are magnons. Here, we elaborate a case where the well-defined magnons are absent completely and the spin excitation spectra exhibit an entire continuum in the itinerant edge ferromagnetic state of graphene arising from the flatband edge electronic states. Based on the further studies of the entanglement entropy and finite-size analysis, we show that the continuum other than the Stoner part results from the spin-1/2 spinons deconfined from magnons. The spinon continuum in a magnetically ordered state is ascribed to a ferromagnetic Luttinger liquid in this edge ferromagnet. The investigation is carried out by using the numerical exact diagonalization method with a projection of the interacting Hamiltonian onto the flat band.

The energy spectrum of a new exponentially confining potential

In this work, we consider a general Morse-type confining potential that was introduced recently by Alhaidari (J Theor Math Phys 205, 2020. arXiv:2005.09080 ). This potential is completely confining and hence has an infinite discrete spectrum. We compute the energy spectrum associated with this confining potential using two different approaches so as to ensure the correctness of our results. We use both asymptotic iteration method and the numerical diagonalization method based on the tridiagonal representation approach of our unperturbed Hamiltonian to compute the energy spectrum. We found that both approaches resulted in bound state energies that are in agreement with each other to a high degree of accuracy.

Fermi–Hubbard model on nonbipartite lattices: flux problem and emergent chirality

On several one-dimensional (1D) and 2D nonbipartite lattices, we study both free and Hubbard interacting lattice fermions when some magnetic fluxes are threaded or gauge fields coupled. First, we focus on finding out the optimal flux which minimizes the energy of fermions at specific fillings. For spin-1/2 fermions at half-filling on a ring lattice consisting of odd-numbered sites, the optimal flux turns out to be ±π/2. We prove this conclusion for Hubbard interacting fermions utilizing a generalized reflection positivity technique, which can lead to further applications on 2D nonbipartite lattices such as triangular and Kagome. At half-filling the optimal flux patterns on the triangular and Kagome lattice are ascertained to be ±[π/2, π/2], ±[π/2, π/2, 0], respectively (see the meaning of these notations in the main text). We also find that chirality emerges in these optimal flux states. Then, we verify these exact conclusions and further study some other fillings with the numerical exact diagonalization method. It is found that when it deviates from half-filling, Hubbard interactions can alter the optimal flux patterns on these lattices. Moreover, numerically observed emergent flux singularities driven by strong Hubbard interactions in the ground states— both in 1D and 2D—are discussed and interpreted as some kind of non-Fermi liquid feature.

Transition between dissipatively stabilized helical states

We analyze a $XXZ$ spin-1/2 chain which is driven dissipatively at its boundaries. The dissipative driving is modelled by Lindblad jump operators which only act on both boundary spins. In the limit of large dissipation, we find that the boundary spins are pinned to a certain value and at special values of the interaction anisotropy, the steady states are formed by a rank-2 mixture of helical states with opposite winding numbers. Contrarily to previous stabilization of topological states, these helical states are not protected by a gap in the spectrum of the Lindbladian. By changing the anisotropy, the transition between these steady states takes place via mixed states of higher rank. In particular, crossing the value of zero anisotropy a totally mixed state is found as the steady state. The transition between the different winding numbers via mixed states can be seen in the light of the transitions between different topological states in dissipatively driven systems. The results are obtained developing a perturbation theory in the inverse dissipative coupling strength and using the numerical exact diagonalization and matrix product state methods.

Haldane Gaps of Large-S Heisenberg Antiferromagnetic Chains and Asymptotic Behavior

The one-dimensional Heisenberg antiferromagnets of large-integer-$S$ spins are studied; their Haldane gaps are estimated by the numerical diagonalization method for $S=5$ and $6$. We successfully obtain a monotonically increasing sequence of finite-size energy difference data corresponding to the Haldane gaps from the huge-scale parallel calculations of diagonalization under the twisted boundary condition and create a monotonically decreasing sequence within the range of system sizes treated in this study from the monotonically increasing sequence. Consequently, the gaps for $S=5$ and $6$ are estimated to be $0.000050 \pm 0.000005$ and $0.0000030 \pm 0.0000005$, respectively. The asymptotic formula of the Haldane gap for $S\rightarrow\infty$ is examined from the new estimates to determine the coefficient in the formula more precisely.

Magnetic properties of GaAs parabolic quantum dot in the presence of donor impurity under the influence of external tilted electric and magnetic fields

The dependence of the magnetization and magnetic susceptibility of donor impurity in parabolic GaAs (Gallium Arsenide) quantum dot has been investigated, at finite temperature under the influence of external tilted electric and magnetic fields. Based on the effective mass approximation, the Hamiltonian of an electron confined in a parabolic quantum dot in the presence of donor impurity which is presented in electric and magnetic fields has been solved by employing the numerical diagonalization method of the Hamiltonian matrix. All the energy matrix elements have been obtained in analytical form. We have shown the variations of the statistical energy and binding energy of donor impurity with the electric and magnetic field strengths and tilt electric field angle. The computed results show that the electrical field can tune the magnetic properties of the QD GaAs medium by flipping the sign of its magnetic susceptibility from diamagnetic (χ < 0) to paramagnetic (χ > 0).

Magnetic moment and susceptibility of an impurity in a parabolic quantum dot

Abstract We study the problem of an electron, neutrally charged donor impurity ( D 0 ) and acceptor impurity in a two-dimensional (2D) semiconductor quantum dot (SQD) with parabolic potential in the presence of an external magnetic field within the effective-mass approximation by using the numerical matrix diagonalization method. The energy spectrum is calculated for an electron, donor impurity and acceptor impurity as a function of the magnetic field. We find the ground state (GS) magnetic moment ( μ gs ) and the GS magnetic susceptibility ( χ gs ) of an acceptor impurity system, It only shows zero temperature diamagnetic peaks, but an electron and the donor impurity does not. The position and the number of these zero temperature diamagnetic peaks depend on the strength of the confinement potential ( ω 0 ) . The binding energies of the few low-lying states are also obtained by directly calculating the energies of an electron and donor impurity and it shows that the donor impurity is stable in a parabolic quantum dot (PQD). The theory is applied to a GaAs PQD.

The Loschmidt-echo dynamics in a quantum chaos model

We have considered the spin-1/2 XXZ chain model which is exactly solvable with the Bethe anzats method. It is shown adding a single defect away from edges is sufficient to see a chaos signature in the model. In this work, we have studied the time behavior of the Loschmidt-echo (LE) by considering domain-wall and Neel states as two product initial states and also a Bell state as an entangled initial state. We have used the numerical full diagonalization method to have full spectrum of the model. It is shown that very short times behavior of the LE is independent from defect and initial state, and the decay is quadratic. Moreover, results show that in the chaotic regime and for intermediate times, the exponential decay behavior of the LE shows some changes by removing defect from the chain in the domain-wall and Bell states, while no changes is seen in the Neel state. To get some intuition of these changes, we have addressed the relation between the dynamics of the LE and the local density of states features.

Ferromagnetism and spin excitations in topological Hubbard models with a flat band

We study the spin-1 excitation spectra of the flatband ferromagnetic phases in interacting topological insulators. As a paradigm, we consider a quarter filled square lattice Hubbard model whose free part is the $\pi$ flux state with topologically nontrivial and nearly-flat electron bands, which can realize either the Chern or $Z_2$ Hubbard model. By using the numerical exact diagonalization method with a projection onto the nearly-flat band, we obtain the ferromagnetic spin-1 excitation spectra for both the Chern and $Z_2$ Hubbard models, consisting of spin waves and Stoner continuum. The spectra exhibit quite distinct dispersions for both cases, in particular the spin wave is gapless for the Chern Hubbard model, while gapped for the $Z_2$ Hubbard model. Remarkably, in both cases, the nonflatness of the free electron bands introduces dips in the lower boundary of the Stoner continuum. It significantly renormalizes the energies of the spin waves around these dips downward and leads to roton-like spin excitations. We elaborate that it is the softening of the roton-like modes that destabilizes the ferromagnetic phase, and determine the parameter region where the ferromagnetic phase is stable.

Third Boundary of the Shastry–Sutherland Model by Numerical Diagonalization

The Shastry-Sutherland model --- the $S=1/2$ Heisenberg antiferromagnet on the square lattice accompanied by orthogonal dimerized interactions --- is studied by the numerical-diagonalization method. Large-scale calculations provide results for larger clusters that have not been reported yet. The present study successfully captures the phase boundary between the dimer and plaquette-singlet phases and clarifies that the spin gap increases once when the interaction forming the square lattice is increased from the boundary. Our calculations strongly suggest that in addition to the edge of the dimer phase given by $J_{2}/J_{1}\sim 0.675$ and the edge of the N$\acute{\rm e}$el-ordered phase given by $J_{2}/J_{1}\sim 0.76$, there exists a third boundary ratio $J_{2}/J_{1}\sim 0.70$ that divides the intermediate region into two parts, where $J_{1}$ and $J_{2}$ denote dimer and square-lattice interactions, respectively.

Integrable-chaos crossover in the spin-[formula omitted] XXZ chain with cluster interaction

Recent progress in the field of quantum magnets has shown that a wide variety of novel spin model can be generated with exploiting the optical lattices. Among them the spin cluster interaction gets some attention. In this work, we have considered a spin-1∕2 XXZ Heisenberg chain with added cluster interaction. In the absence of the cluster interaction, the XXZ chain is solvable by Bethe ansatz. Introducing a small defect in the middle of chain drives the XXZ model to the chaotic phase not for strong anisotropic parameter. Using the numerical full diagonalization method, we have shown that in presence of a cluster interaction, how quantum chaos may develop even for strong Ising interaction region. In addition, the effect of the cluster length and strength on the crossover from the integrable to the chaotic regime is also addressed.

Precise Estimation of the S = 2 Haldane Gap by Numerical Diagonalization

The Haldane gap of the S=2 Heisenberg antiferromagnet in a one-dimensional linear chain is examined by a numerical-diagonalization method. A precise estimate for the magnitude of the gap is successfully obtained by a multistep convergence-acceleration procedure applied to finite-size diagonalization data under the twisted boundary condition.