Standard Basis Operators Research Articles

Hamiltonian simulation in the Pauli basis of multi-qubit clusters for condensed matter physics

We propose an efficient method for Hamiltonian simulation of multi-qubit quantum systems with special types of interaction. In our approach, the Hamiltonian of a \(n\)-qubit system should be represented as a linear combination of the standard Pauli basis operators, and then decomposed into a sum of partial Hamiltonians, which are, in general, not Pauli operators and satisfy some anticommutation relations. For three types of Hamiltonians, which are invariant with respect to permutations of qubits, the effectiveness of the main algorithm in the three-qubit cluster model is shown by calculating the operator exponentials for these Hamiltonians in an explicit analytical form. We also calculate the density operator, partition function, entropy, and free energy of the cluster weakly coupled to a thermal environment. In our model, the cluster is in the Gibbs state in the temperature interval \(0.1\!-\!2\:\!\text{K}\), which corresponds to the operating range of modern quantum processors. It follows from our analysis that the thermodynamic properties of such systems strongly depend on the type of internal interaction of qubits in the cluster.

Excitation spectrum and momentum distribution of the ionic Bose–Hubbard model

We investigate the excitation spectrum and momentum distribution of the ionic Bose–Hubbard model by the standard basis operator method. We derive Green’s functions in the random phase approximation in Mott insulator, superfluid, charge density wave, and supersolid phases. The excitation spectrum has gapped modes and gapless Goldstone modes in the superfluid and supersolid phases. We show that the momentum distribution has a peak at the zone corner in the supersolid phase and the charge density wave phase close to the phase boundary. In addition, we demonstrate that the momentum distribution can be explained by the excitation spectrum and spectral weights of hole excitation modes.

Phase transitions and spin excitations of spin-1 bosons in optical lattice

We investigate ground state properties of spin-1 bosonic system trapped in optical lattice with extended standard basis operator (SBO) method. For both ferromagnetic ($U_2<0$) and antiferromagnetic ($U_2>0$) systems, we analytically figure out the symmetry properties in Mott-insulator and superfluid phases, which would provide a deeper insight into the MI-SF phase transition process. Then by applying self-consistent approach to the method, we include the effect of quantum and thermal fluctuations and derive the MI-SF transition phase diagram, which is in quantitative agreement with recent Monte-Carlo simulation at zero temperature, and at finite temperature, we find the underestimation of finite-temperature-effect in the mean-field approximation method. If we further consider the spin excitations in the insulating states of spin-1 system in external field, distinct spin phases are expected. Therefore, in the Mott lobes with $n=1$ and $n=2$ atoms per site, we give analytical and numerical boundaries of the singlet, nematic, partially magnetic and ferromagnetic phases in the magnetic phase diagrams.

A standard basis operator equation of motion impurity solver for dynamical mean field theory

We present an efficient impurity solver for the dynamical mean-field theory (DMFT). It is based on the separation of bath degrees of freedom into the low energy and the high energy parts. The former is solved exactly using exact diagonalization and the latter is treated approximately using Green's function equation of motion decoupling approximation. The two parts are combined coherently under the standard basis operator formalism. The impurity solver is applied to the Anderson impurity model and, combined with DMFT, to the one-band Hubbard model. Qualitative agreement is found with other well established methods. Some promising features and possible improvements of the present solver are discussed.

Ground-state and finite-temperature properties of correlated ultracold bosons on optical lattices

We investigate the ground-state and finite-temperature properties of the Bose-Hubbard model using the standard basis operator method applied to the equation of motion for Green's functions. We show that by introducing the self-consistent approach, the theory goes significantly beyond the mean-field approximation, allowing more-precise description of quantum and thermal fluctuations. For optical lattice systems with ultracold bosons, we find relevant quantitative agreement with numerical and experimental data. In particular, the presented method reveals the importance of thermal fluctuations in time-of-flight experiments. For the most reliable fitting of the visibility analytical curve to the experimental data, the value of temperature is determined and proximity to the quantum regime is discussed. The problem of identification of the superfluid--Mott-insulator phase boundary from time-of-flight experiments is also considered.

Direct overlapping effect of f orbitals for valence fluctuating materials in Kondo regime

In Rare-Earth (RE) systems, direct overlapping of 4f orbitals plays a role if the distance between RE ions is sufficiently small. Especially, such a situation happens under pressure for RE metals and RE compounds. Furthermore, in Actinide systems, 5f orbitals are more extended than the previously considered 4f orbitals. Therefore, in this case, direct overlapping of 5f orbitals should be taken into account even without applying pressure. In the present work, using Standard Basis Operators, we study direct overlapping effect of f orbitals for valence fluctuating materials in Kondo regime. An important result within our present model is that direct f orbital overlapping reduces f electron effective mass.

A model of spin-polarised charge transport in magnetic oxides

We propose a new model of the spin-polarised charge transport in magnetic oxides. A formula for the electric current is derived within the framework of a Green function formalism [1,2]. The Green functions are constructed from two types of operators. The first are the standard-basis operators [3,4], defined in a basis of p(s) - d hybridised eigen-states of central ionic clusters [5]. These operators are used on equal footing with the fermion creation and annihilation operators, which are defined for the band states (k⃗) of the metallic electrodes. The proposed model is extended by an interaction with an external magnetic field in a way, which enables calculations of magnetoresistance. Special emphasis is put on a coupling of the centre and the attached electrodes. The method is tested on nano-structures of the transition metal oxides such as some of the magnetic garnets and/or manganites. The process is mediated both by the hybridisation between the extended states of the electrodes and localised states of the centre, as well as by the hybridisation at the centre between the atomic p(s) states of the anions and the nd states of the transition metal.

Model for magnetic–nonmagnetic binary alloys

AbstractAn extension of a mean‐field approximation (MFA) developed within standard basis operators (SBO) is used to study magnetism in magnetic–nonmagnetic binary alloys. The Curie temperature is calculated from the free energy within the framework of the present approach. The calculated results are in fair agreement with the theoretical results of other research groups for the same problem but utilizing other methods. Finally, the case of NiPt alloys is briefly examined as an example test for the comparison with experiment. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Itinerant-localized dual character of a strongly correlated superfluid Bose gas in an optical lattice

We investigate a strongly-correlated Bose gas in an optical lattice. Extending the standard-basis operator method developed by Haley and Erdos to a boson Hubbard model, we calculate excitation spectra in the superfluid phase, as well as in the Mott insulating phase, at T=0. In the Mott phase, the excitation spectrum has a finite energy gap, reflecting the localized character of atoms. In the superfluid phase, the excitation spectrum is shown to have an itinerant-localized dual structure, where the gapless Bogoliubov mode (which describes the itinerant character of superfluid atoms) and a band with a finite energy gap coexist. We also show that the rf-tunneling current measurement would give a useful information about the duality of a strongly-correlated superfluid Bose gas near the superfluid-insulator transition.

Elementary Excitations in Antiquadrupolar, Antiferrimagnetic and Antiferromagnetic Biquadratic Spin Systems (S = 3/2)

The magnon and quadron spectrum is calculated in a two-sublattice system of localized spins S = 3/2 with the bilinear Heisenberg and isotropic biquadratic interactions. The calculations are performed in the (1/z)0 approximation in the high-density scheme (z is the number of spins interacting with any given spin) using diagrammatic methods for the Racah and standard basis operators. In the antiquadrupolar phase two positive and two negative branches of magnons and quadrons are obtained. In the antiferrimagnetic phase there are two positive and one negative magnon branches as well as one positive and one negative quadron branch. In the antiferromagnetic phase there is one positive and one negative branch of magnons and quadrons. The negative spectra, more pronounced here than in the (considered elsewhere) ferrimagnetic and ferromagnetic phases, give rise to the zero-point corrections at T = 0 in higher-order approximations.

Elementary Excitations in Antiquadrupolar, Antiferrimagnetic and Antiferromagnetic Biquadratic Spin Systems (S = 3/2)

The magnon and quadron spectrum is calculated in a two-sublattice system of localized spins S = 3/2 with the bilinear Heisenberg and isotropic biquadratic interactions. The calculations are performed in the (1/z) 0 approximation in the high-density scheme (z is the number of spins interacting with any given spin) using diagrammatic methods for the Racah and standard basis operators. In the antiquadrupolar phase two positive and two negative branches of magnons and quadrons are obtained. In the antiferrimagnetic phase there are two positive and one negative magnon branches as well as one positive and one negative quadron branch. In the antiferromagnetic phase there is one positive and one negative branch of magnons and quadrons. The negative spectra, more pronounced here than in the (considered elsewhere) ferrimagnetic and ferromagnetic phases, give rise to the zero-point corrections at T = 0 in higher-order approximations.

Analytical block-diagonalization of matrices using unit spherical tensors

A new method for block-diagonalizing large Hamiltonian matrices, in closed form, is described. The method is based on (i) a general unitary transformation due to Slichter, and (ii) Fano's unit spherical operatorsU Q K (Ii,Ii). The method is illustrated with a simple three spin 1/2 dipolar coupled spin system, characterized by off-block-diagonal unit spherical tensorsU 0 2 (3/2,1/2,α) andU 0 2 (3/2,1 /2′,α). In addition, it is pointed out that any Hamiltonian matrix can be re-labelled in terms of fictitious spin labels, enabling a wide variety of unit spherical tensors to be used in block-diagonalization. For example, a single spin 5/2 matrix can be re-labelled using three spin labels 1/2, 1/2′, and 1/2″, respectively. Thus the tensor algebra required to block-diagonalize a 6 x 6 matrix is determined solely by the properties of the Pauli spin matrices. Finally, it is shown that re-labelling within the unit spherical tensor framework provides a unifying framework for standard basis operators, fictitious spin 1/2 and 1 operators, and others. The fictitious spin 1 / 2 unit spherical operators discussed in this paper differ from those of Vega and Pines.

Thermodynamical and Green function many-body Wick theorems

Abstract The thermodynamical and Green function many-body reduction theorems of Wick type are proved for the arbitrary mixtures of the fermion, boson and spin systems. “Many-body” means that the operators used are the products of the arbitrary number of one-body standard basis operators [of the fermion or (and) spin types] with different site (wave vector) indices, but having the same “time” (in the interaction representation). The method of proving is based on: 1) the first-order differential equation of Schwinger type for: 1a) T -product of operators; 1b) its average value; 2) KMS boundary conditions for this average. It is shown that the fermion, boson and spin systems can be unified in the many-body formulation (bosonification of the fermion systems). It is impossible in the one-body approach. Both of the many-body versions of the Wick theorem have the recurrent feature: nth order moment diagrams for the free energy or Green functions can be expressed by the (n −1)th order ones. This property corresponds to the automatic realization of: (i) summations over Bose-Einstein or (and) Fermi-Dirac frequencies; (ii) elimination of Bose-Einstein or (and) Fermi-Dirac distributions. The procedures (i) and (ii), being the results of using the Green function one-body reduction theorem, have constituted the significant difficulty up to now in the treatment of quantum systems.

Magnetic Excitons in a Spin‐Cluster System

AbstractMagnetic properties are calculated in the effective‐field approximation for a four‐level (singlet‐triplet) spin‐cluster system. Furthermore the spectrum of magnetic excitons propagating in this interacting system having in zero order discrete energy levels is studied. For the treatment of the magnetic spin‐cluster crystal the standard‐basis operator (SBO) formalism is applied. The model Hamiltonian used by Heber, Nara, and Freudenhammer is transformed in terms of the SBO's; and the eigen‐values are determined. The thermodynamic properties are obtained from the SBO Green's function equations of motion developing in the random‐phase approximation (RPA). A formula for the magnetization and susceptibility is derived.

The temperature dependence of inelastic neutron scattering in rotational tunnelling systems. II. Calculation of the time-dependent correlation functions

For pt.I see ibid., vol.15, p.3841 (1982). The problem of explaining the temperature dependence observed in the inelastic neutron scattering spectrum of rotational tunnelling systems was considered in an earlier paper and based on a model in which a threefold molecular group in a hindering potential is coupled to a spectrum of phonon modes. An expression was derived for the neutron scattering function in terms of the correlation functions of the standard basis operators of the uncoupled system. Here the authors extends the perturbational approach for the calculation of these correlation functions that was developed earlier to include the effects of line broadening. The temperature dependence of the shifting and broadening of the peaks in the neutron spectrum is then calculated for different assumed forms for the density of the coupled-phonon states. Not enough is known about the coupling to the phonon spectrum for a complete quantitative comparison with experiment for any one system. However, the author shows that the theory can be used to explain the range of the experimentally observed behaviour, and the form of the temperature dependence of the shifting and broadening of the peaks.

The temperature dependence of inelastic neutron scattering in rotational tunnelling systems. I. Formulation and perturbation theory

In recent years inelastic neutron scattering experiments have revealed marked variations in the temperature dependence of the rotational tunnelling spectra of molecular groups in various local environments. A new theoretical approach is developed here to explain these results. A microscopic model of a rigid molecular group, whose rotational degree of freedom is coupled to lattice modes, is formulated in terms of standard basis operators for states of the uncoupled system. It is shown that a complete description of the neutron spectrum can be found in terms of the time-dependent correlation functions of these operators. As a preliminary to a fuller treatment of this problem a perturbational approach is developed, which allows the conditions for a treatment-dependent shift to lower tunnelling frequencies, which is commonly observed experimentally, to be thoroughly examined.

A theory of the temperature dependence of the inelastic neutron scattering spectrum for rotational tunnelling systems

A model of a three-fold rotational tunnelling system, such as methyl group, is reformulated as a finite level system interacting with phonons. The time-dependent correlation functions of the standard basis operators for these levels are calculated for a description of the inelastic neutron scattering spectrum. The theory relates the temperature dependence of the line shifts and their broadening to the form of the hindering potential and to the features of the coupled phonon spectrum.

A model for electrons existing in hybridized narrow and wide bands

We propose a model for electrons existing in hybridized narrow and wide bands which occur in the transition metals, the mixed valence systems of rare earth metals and their compounds, and the doped semiconductors, etc. The model is derived from the standard basis operator (SBO) form of the extended Anderson model, originally proposed for the dilute magnetic alloys. The equation of motion method has been used to obtain the SBO's Green's functions and, thereby, the spectrum of the quasi-particle excitations. For translationally invariant systems, we found the possibility of the nonmetallic behavior and the metal-nonmetal transition for a nonintegral number of electrons per lattice site.

Fourth-order effective Hamiltonian for the Anderson lattice

A canonical transformation of the Schrieffer-Wolff type is applied to the periodic Anderson model. The case of infinitely correlated $f$-like orbitals is dealt with, and these are described through standard basis operators, including spin degeneracy only. First- and third-order terms in the mixing parameter are eliminated. The resulting Hamiltonian contains terms up to fourth order. These describe several interaction processes including intersite spin and charge correlations, conduction-electron scattering, and excitonic interactions. A detailed analysis of the modified Ruderman-Kittel-Kasuya-Yosida interaction obtained is given. Possible extensions of some of the results to higher orders are discussed. The method provides a way to evaluate the spin and charge susceptibilities of the conduction-electron gas in this model. The possible relevance of these functions to a theory of the phonon spectrum and of magnetic ordering of intermediate-valence systems is mentioned.

A general dispersion relation for spin-phonon excitations in paramagnets

The advantage of using standard basis operators in evaluating the semi-invariants occurring in the diagrammatic theory of spin-phonon modes is discussed. The dispersion relation of the coupled-mode excitations is derived for paramagnetic ions of arbitrary spin having a general static spin hamiltonian.