Exact Diagonalization Results Research Articles

Scaling of excitations in dimerized and frustrated spin-12chains

We study the finite-size behavior of the low-lying excitations of spin-1/2 Heisenberg chains with dimerization and next-to-nearest neighbors interaction, J_2. The numerical analysis, performed using density-matrix renormalization group, confirms previous exact diagonalization results, and shows that, for different values of the dimerization parameter \delta, the elementary triplet and singlet excitations present a clear scaling behavior in a wide range of \ell=L/\xi (where L is the length of the chain and \xi is the correlation length). At J_2=J_2c, where no logarithmic corrections are present, we compare the numerical results with finite-size predictions for the sine-Gordon model obtained using Luscher's theory. For small \delta we find a very good agreement for \ell > 4 or 7 depending on the excitation considered.

Symmetry-projected variational approach to the one-dimensional Hubbard model

We apply a variational method devised for the nuclear many-body problem to the one-dimensional Hubbard model with nearest neighbor hopping and periodic boundary conditions. The test wave function consist for each state out of a single Hartree--Fock determinant mixing all the sites (or momenta) as well as the spin projections of the electrons. Total spin and linear momentum are restored by projection methods before the variation. It is demonstrated that this approach reproduces the results of exact diagonalizations for half-filled $N=12$ and $N=14$ lattices not only for the energies and occupation numbers of the ground but also of the lowest excited states rather well. Furthermore, a system of ten electrons in an $N=12$ lattice is investigated and, finally, an $N=30$ lattice is studied. In addition to energies and occupation numbers we present the spectral functions computed with the help of the symmetry-projected wave functions as well.

NUMERICAL STABILITY AND THE SIGN PROBLEM IN THE DETERMINANT QUANTUM MONTE CARLO METHOD

A recent paper by Matuttis and Ito questions the numerical accuracy of a widely-used fermion Monte Carlo algorithm. They also claim that the increase in the d-wave pairfield susceptibility χd(T) of a doped 4×4 Hubbard model at low temperature, previously found using this algorithm, is an artifact due to numerical errors. Here, we provide tests which show that this algorithm is numerically accurate and show that the simulation of χd for a 2×2 lattice agrees with exact diagonalization results. We also provide more complete data for χd on a 4×4 lattice that is consistent with our previous results.

Nuclear Structure Calculations with Coupled Cluster Methods from Quantum Chemistry

We present several coupled-cluster calculations of ground and excited states of 4He and 16O employing methods from quantum chemistry. A comparison of coupled cluster results with the results of exact diagonalization of the hamiltonian in the same model space and other truncated shell-model calculations shows that the quantum chemistry inspired coupled cluster approximations provide an excellent description of ground and excited states of nuclei, with much less computational effort than traditional large-scale shell-model approaches. Unless truncations are made, for nuclei like 16O, full-fledged shell-model calculations with four or more major shells are not possible. However, these and even larger systems can be studied with the coupled cluster methods due to the polynomial rather than factorial scaling inherent in standard shell-model studies. This makes the coupled cluster approaches, developed in quantum chemistry, viable methods for describing weakly bound systems of interest for future nuclear facilities.

Ground state of two-dimensional quantum-dot helium in zero magnetic field: Perturbation, diagonalization, and variational theory

We study the ground-state properties of two-dimensional quantum-dot helium in zero external magnetic field (a system of two interacting electrons in a two-dimensional parabolic confinement potential) by using perturbation and variational theory. We introduce a family of ground-state trial wave functions with one, two, and three variational parameters. We compare the perturbation and variational energies with numerically exact diagonalization results and earlier unrestricted Hartree-Fock studies. We find that the three-parameter variational wave function is an excellent representation of the true ground state and argue on how to generalize such a wave function for larger quantum dots with arbitrary numbers of electrons.

Variational study for the ground state energy of the linear E– b1, b2 Jahn–Teller system

A new method for determining the ground state wave functions of the linear E– b 1, b 2 Jahn–Teller system is presented. Up to now all methods for determining the ground state wave functions lead to relatively large errors in determining the ground state energies of the linear E– e Jahn–Teller system in comparison with the exact diagonalization results. The presented method is based on a combination of unitary transformations and the variational principle, and the results of this method are presented on the linear E– e Jahn–Teller system. The ground state energies calculated by the ground state wave functions obtained by the presented method are in very good agreement with the exact diagonalization results and are substantially better than those from earlier studies. This method uses a linear combination of displacements of a vacuum state which can be considered as generalization of a two-mode squeezed vacuum state defined in the earlier studies. Further, this method is used to prove that the ground state energy of the linear E– b 1, b 2 Jahn–Teller system is twofold degenerate. This method provides not only the analytical ground state wave functions for the linear E– e Jahn–Teller system examined in the previous studies but also for the linear E– b 1, b 2 Jahn–Teller system which is more general than the linear E– e Jahn–Teller system.

Time-dependent Gutzwiller theory of magnetic excitations in the Hubbard model

We use a spin-rotational invariant Gutzwiller energy functional to compute random-phase-approximation-like (RPA) fluctuations on top of the Gutzwiller approximation (GA). The method can be viewed as an extension of the previously developed $\mathrm{GA}+\mathrm{RPA}$ approach for the charge sector [G. Seibold and J. Lorenzana, Phys. Rev. Lett. 86, 2605 (2001)] with respect to the inclusion of the magnetic excitations. Unlike the charge case, no assumptions about the time evolution of the double occupancy are needed in this case. Interestingly, in a spin-rotational invariant system, we find the correct degeneracy between triplet excitations, showing the consistency of both computations. Since no restrictions are imposed on the symmetry of the underlying saddle-point solution, our approach is suitable for the evaluation of the magnetic susceptibility and dynamical structure factor in strongly correlated inhomogeneous systems. We present a detailed study of the quality of our approach by comparing with exact diagonalization results and show its much higher accuracy compared to the conventional $\mathrm{Hartree}\ensuremath{-}\mathrm{Fock}+\mathrm{RPA}$ theory. In infinite dimensions, where the GA becomes exact for the Gutzwiller variational energy, we evaluate ferromagnetic and antiferromagnetic instabilities from the transverse magnetic susceptibility. The resulting phase diagram is in complete agreement with previous variational computations.

Impact of phonons on quantum phase transitions in nanorings of coupled quantum dots

CDW-to-CDW and CDW-to-SDW (CDW/SDW=charge-spin-density wave) quantum phase transitions have been recently reported for finite rings described by an extended Hubbard model [I. B\^aldea, H. K\oppel, and L. S. Cederbaum, Eur. Phys. J. B 20, 289 (2001)]. We present exact (Lanczos) diagonalization results which demonstrate that these transitions survive in the presence of a dynamic Su-Schrieffer-Heeger (SSH) electron-phonon coupling. The two transitions are affected in a different way discussed in detail. By treating the SSH electron-phonon coupling dynamically, two levels of different symmetries do cross, allowing one to define precisely a critical point. Because the ground-state symmetry changes at the critical points, we suggest to study the quantum phase transitions by optical methods. Molecular rings such as polyenes (annulenes) turn out to be too far from the critical points, where the most interesting phenomena occur. However, we present an analysis revealing that, if metallic quantum dots of the type already fabricated can be assembled in nanorings, model parameters can be tuned and quantum phase transitions can become observable.

Power-law dependence of the angular momentum transition fields in few-electron quantum dots

We show that the critical magnetic fields at which a few-electron quantum dot undergoes transitions between successive values of its angular momentum (M), for large M values follow a very simple power-law dependence on the effective inter-electron interaction strength. We obtain this power law analytically from a quasi-classical treatment and demonstrate its nearly-universal validity by comparison with the results of exact diagonalization.

Time-dependent density functional theory of excitation energies of closed-shell quantum dots

We study the low-energy excitations of two-dimensional elliptical quantum dots with closed-shell structures using time-dependent density functional theory (TDDFT). We find that TDDFT gives fairly accurate results for the low-energy excitations and describes the spin singlet–triplet transition due to the exchange energy. The deviations from exact-diagonalization results are less than 0.20 meV . We show the dependence of excitation energies on the strength of Coulomb interaction and the photoabsorption spectra.

Quantum Monte Carlo and variational approaches to the Holstein model

Based on the canonical Lang-Firsov transformation of the Hamiltonian we develop a very efficient quantum Monte Carlo algorithm for the Holstein model with one electron. Separation of the fermionic degrees of freedom by a reweighting of the probability distribution leads to a dramatic reduction in computational effort. A principal component representation of the phonon degrees of freedom allows to sample completely uncorrelated phonon configurations. The combination of these elements enables us to perform efficient simulations for a wide range of temperature, phonon frequency and electron-phonon coupling on clusters large enough to avoid finite-size effects. The algorithm is tested in one dimension and the data are compared with exact-diagonalization results and with existing work. Moreover, the ideas presented here can also be applied to the many-electron case. In the one-electron case considered here, the physics of the Holstein model can be described by a simple variational approach.

Stability of the quantized circulation of an attractive Bose-Einstein condensate in a rotating torus

We investigate rotational properties of a system of bosons with attractive interactions confined in a one-dimensional torus. Two kinds of ground states, uniform-density and bright-soliton states, are obtained analytically as functions of the strength of interaction and of the rotational frequency of the torus. The quantization of circulation appears in the uniform-density state, but disappears upon formation of the soliton. By comparison with the results of exact diagonalization of the many-body Hamiltonian, we show that the Bogoliubov theory is valid at absolute zero over a wide range of parameters. At finite temperature we employ the exact diagonalization method to examine how thermal fluctuations smear the plateaus of the quantized circulation. Finally, by rotating the system with an axisymmetry-breaking potential, we clarify the process in which the quantized circulation becomes thermodynamically stabilized.

Coupled cluster theory of strongly correlated spin- and electron-lattice systems: an illustration via a model exhibiting competition between magnetic order and dimerization

The coupled cluster method (CCM) of microscopic quantum many-body theory has become an ab initio method of first choice in quantum chemistry and many fields of nuclear, subnuclear and condensed matter physics, when results of high accuracy are required. In recent years it has begun to be applied with equal success to strongly correlated systems of electrons or quantum spins defined on a regular spatial lattice. One regularly finds that the CCM is able to describe accurately the various zero-temperature phases and the quantum phase transitions between them, even when frustration is present and other methods such as quantum Monte Carlo often fail. We illustrate the use and powerfulness of the method here by applying it to a square-lattice spin-half Heisenberg model where frustration is introduced by competing nearest neighbour bonds. The model exhibits the physically interesting phenomenon of competition between magnetic order and dimerization. Results obtained for the model with the CCM are compared with those found from spin-wave theory and from extrapolating the results of exact diagonalizations of small lattices. We show that the CCM is essentially unique among available methods in being able both to describe accurately all phases of this complex model and to provide accurate predictions of the various phase boundaries and the order of the corresponding transitions.

Two-dimensional quantum dots in high magnetic fields: Rotating-electron-molecule versus composite-fermion approach

Exact diagonalization results are reported for the lowest rotational band of N=6 electrons in strong magnetic fields in the range of high angular momenta 70 <= L <= 140 (covering the corresponding range of fractional filling factors 1/5 >= nu >= 1/9). A detailed comparison of energetic, spectral, and transport properties (specifically, magic angular momenta, radial electron densities, occupation number distributions, overlaps and total energies, and exponents of current-voltage power law) shows that the recently discovered rotating-electron-molecule wave functions [Phys. Rev. B 66, 115315 (2002)] provide a superior description compared to the composite-fermion/Jastrow-Laughlin ones.

Coexistence of composite bosons and composite fermions in nu = 1/2 + 1/2 quantum Hall bilayers.

In bilayer quantum Hall systems at filling fractions near nu=1/2+1/2, as the spacing d between the layers is continuously decreased, intralayer correlations must be replaced by interlayer correlations, and the composite fermion (CF) Fermi seas at large d must eventually be replaced by a composite boson (CB) condensate or "111 state" at small d. We propose a scenario where CBs and CFs coexist in two interpenetrating fluids in the transition. Trial wave functions describing these mixed CB-CF states compare very favorably with exact diagonalization results. A Chern-Simons transport theory is constructed that is compatible with experiment.

RECONSTRUCTION OF FRACTIONAL QUANTUM HALL EDGES

Edge reconstruction has been a well-known effect for integer quantum Hall liquids in the presence of both electron interactions and a confining potential generated by charged background. At more generic fractional fillings, we point out that confined two-dimensional interacting electrons can exhibit the similar reconstruction effect. Our exact diagonalization results show that, in a fractional quantum Hall system with a sharp cleaved edge potential, the electron density oscillation near the edge increases with the distance between the electron gas and the background charge layer. As a results, the outermost hump can detach from the bulk beyond certain point. We suggest that the edge reconstruction effect is relevant to the recent edge tunneling experiments,1 as well as the microwave absorption experiment on two-dimensional electrons in an antidot array.2 Calculating the finite-temperature density profiles, we estimate the temperature above which the edge reconstruction disappears to further discuss the relevance to the microwave absorption experiment.

Quantum Monte Carlo and exact diagonalization study of a dynamic Hubbard model

A one-dimensional model of electrons locally coupled to spin-1/2 degrees of freedom is studied by numerical techniques. The model is one in the class of $dynamic$ $Hubbard$ $models$ that describe the relaxation of an atomic orbital upon double electron occupancy due to electron-electron interactions. We study the parameter regime where pairing occurs in this model by exact diagonalization of small clusters. World line quantum Monte Carlo simulations support the results of exact diagonalization for larger systems and show that kinetic energy is lowered when pairing occurs. The qualitative physics of this model and others in its class, obtained through approximate analytic calculations, is that superconductivity occurs through hole undressing even in parameter regimes where the effective on-site interaction is strongly repulsive. Our numerical results confirm the expected qualitative behavior, and show that pairing will occur in a substantially larger parameter regime than predicted by the approximate low energy effective Hamiltonian.

Static overscreening and nonlinear response in the Hubbard model

We investigate the static charge response for the Hubbard model. Using the Slave-Boson method in the saddle-point approximation we calculate the charge susceptibility. We find that RPA works quite well close to half-filling, breaking, of course, down close to the Mott transition. Away from half filling RPA is much less reliable: Already for very small values of the Hubbard interaction U, the linear response becomes much more efficient than RPA, eventually leading to overscreening already beyond quite moderate values of U. To understand this behavior we give a simple argument, which implies that the response to an external perturbation at large U should actually be strongly non-linear. This prediction is confirmed by the results of exact diagonalization.

Interacting electrons in quantum dots

The ground states of N-electron parabolic quantum dots in the presence of a perpendicular magnetic field are investigated. Rigorous lower bounds to the ground-state energies are obtained. It is shown that our lower bounds agree well with the results of exact diagonalization. Analytic results for the lower bounds to the ground-state energies of the quantum dots in a strong magnetic field (known as electron molecule) agree very well with numerically calculated lower bounds.

Accurate results from perturbation theory for strongly frustratedS=12Heisenberg spin clusters

We investigate the use of perturbation theory in finite-sized frustrated spin systems by calculating the effect of quantum fluctuations on coherent states derived from the classical ground state. We first calculate the ground- and first-excited-state wave functions as a function of applied field for a 12-site system and compare with the results of exact diagonalization. We then apply the technique to a 20-site system with the same threefold site coordination as the 12-site system. Frustration results in asymptotically convergent series for both systems which are summed with Pad\'e approximants. We find that at zero magnetic field the different connectivity of the two systems leads to a triplet first excited state in the 12-site system and a singlet first excited state in the 20-site system, while the ground state is a singlet for both. We also show how the analytic structure of the Pad\'e approximants at $|\ensuremath{\lambda}|\ensuremath{\simeq}1$ evolves in the complex $\ensuremath{\lambda}$ plane at the values of the applied field where the ground state switches between spin sectors and how this is connected with the nontrivial dependence of the $〈{S}^{z}〉$ number on the strength of quantum fluctuations. We discuss the origin of this difference in the energy spectra and in the analytic structures. We also characterize the ground and first excited states according to the values of the various spin correlation functions.