Schrieffer-Wolff Transformation Research Articles

Decoherence of Charge States in Double Quantum Dots Due to Cotunneling

Solid state quantum bits are a promising candidate for the realization of a scalable quantum computer, however, they are usually strongly limited by decoherence. We consider a double quantum dot charge qubit, whose basis states are defined by the position of an additional electron in the system of two laterally coupled quantum dots. The coupling of these two states can be controlled externally by a quantum point contact between the two dots. We discuss the decoherence through coupling to the electronic leads due to cotunneling processes. We focus on a simple Gedanken experiment, where the system is initially brought into a superposition and then the inter-dot coupling is removed nonadiabatically. We treat the system by invoking the Schrieffer-Wolff transformation in order to obtain a transformed Hamiltonian describing the cotunneling, and then obtain the dynamics of the density matrix using the Bloch-Redfield theory. As a main result, we show that there is energy relaxation even in the absence of inter-dot coupling. This is in contrast to what would be expected from the Spin-Boson model and is due to the fact that a quantum dot is coupled to two distinct baths.

Local moments in an interacting environment

We discuss how local moment physics is modified by the presence of interactions in the conduction sea. Interactions in the conduction sea are shown to open up new symmetry channels for the exchange of spin with the localized moment. We illustrate this conclusion in the strong-coupling limit by carrying out a Schrieffer Wolff transformation for a local moment in an interacting electron sea, and show that these corrections become very severe in the approach to a Mott transition. As an example, we show how the Zhang Rice reduction of a two-band model is modified by these new effects.

Theory of the Anderson Impurity Model: The Schrieffer–Wolff Transformation Reexamined

We test the method of infinitesimal unitary transformations recently introduced by Wegner on the Anderson single impurity model. It is demonstrated that infinitesimal unitary transformations in contrast to the Schrieffer–Wolff transformation allow the construction of an effective Kondo Hamiltonian consistent with the established results in this well understood model. The main reason for this is the intrinsic energy scale separation of Wegner's approach with respect to arbitrary energy differences coupled by matrix elements. This allows the construction of an effective Hamiltonian without facing a vanishing energy denominator problem. Similar energy denominator problems are troublesome in many models. Infinitesimal unitary transformations have the potential to provide a general framework for the systematic derivation of effective Hamiltonians without such problems.

Enhanced local moment formation in a chiral Luttinger liquid.

We derive here a stability condition for a local moment in the presence of an interacting sea of conduction electrons. The conduction electrons are modeled as a Luttinger liquid in which chirality and spin are coupled. We show that an Anderson $U$ defect in such an interacting system can be transformed onto a nearly Fermi-liquid problem. We find that correlations among the conduction electrons stabilize the local moment phase. A Schrieffer-Wolff transformation is then performed which results in an anisotropic exchange interaction indicative of the Kondo effect in a Luttinger liquid. The ground-state properties of this model are then equivalent to those of the Kondo model in a Luttinger liquid.

Bi-/polaron formation and optical conductivity in the Holstein-Hubbard model

The polaron formation at finite densities is studied in the Holstein-Hubbard model transformed, by a modified Lang-Firsov transformation, to a polaron Hamiltonian. In a variational approach, an effective polaronic Hubbard model is derived, where squeezing effects are taken into account. The correlations are treated in the slave-boson saddle-point approximation. Including charge- and magnetically ordered states the phase diagram is calculated, and a self-trapping transition from light to heavy polarons is found. The optical conductivity of heavy polarons reveals correlation-induced structures. By a Schrieffer-Wolff transformation up to second order in the residual polaron-multiphonon interaction, an effective polaron model is derived, where for a polaronic attraction an on-site bipolaron model is obtained.

Interaction of a magnetic impurity with strongly correlated conduction electrons.

We consider a magnetic impurity which interacts by hybridization with a system of strongly correlated conduction electrons. The latter are described by a Hubbard Hamiltonian. The charge degrees of freedom of the magnetic impurity are eliminated by means of a canonical transformation, which reduces to the Schrieffer-Wolff transformation in the case of uncorrelated conduction electrons. In the limit of strong correlations, the coefficient of the dominant exchange coupling of the magnetic impurity with the nearest lattice site is modified. When the system is hole doped, there is also an antiferromagnetic coupling to the nearest neighbors of that site involving, additionally, a hole. Furthermore, it is found that the magnetic impurity attracts a hole. In the case of electron doping, double occupancies are repelled by the impurity. In contrast to the hole-doped case, we find no magnetic coupling that additionally involves a doubly occupied site.

The Fano-Anderson model applied to electrons in MIS-inversion layers

Subband states in n-inversion layers on narrow gap semiconductors are resonances due to the Zener coupling to the continuum of the bulk valence band. Thus excitation between subband states resembles the autoionization process of atomic physics described long ago by Fano. Starting from Kane's model for narrow gap semiconductors to describe the subband states we map the problem onto the Fano-Anderson Hamiltonian by applying a Schrieffer-Wolff transformation. This problem is solved self-consistently using an analytical surface potential. We calculate the relative transmission rate Δ T(ω) T(ω) for inversion layers on narrow-gap Hg 1− x Cd x Te and discuss our results in comparison with experimental data.

Fourth-order interactions in the canonically transformed d-p model for Cu-O superconductors

The low-energy Hamiltonian for localized Cu spins and mobile O holes is rederived from the minimal version of the Emery model ascribed to a CuO2 layer in high-Tc superconductors. By means of a computational algorithm the Schrieffer-Wolff transformation is performed, at large but finite Hubbard repulsion U on Cu sites, up to fourth order in the d-p (Cu-O) hopping amplitude t. The resultant interactions in O(t4) involve spin-spin superexchange, a Kondo-like spin-hole exchange opposite in sign to that in O(t2), and a Hubbard-type attraction between symmetrized O holes within a CuO4 group. An effective tpp-Jdd-Jdp-Jpp model is established for the special case U=2 epsilon , where epsilon is the charge transfer energy.

The effect of charge fluctuation and transfer on the exchange interaction in rare-earth and actinide systems

Starting with the Anderson lattice model, an extended effective exchange interaction is derived by the Schrieffer-Wolff transformation. The effective Hamiltonian includes not only the usual s-f exchange term but also a second order interaction term due to the charge transfer effect. Within the Hartree-Fock approximation, the indirect exchange interaction between f-electrons includes both the terms depending on the number of f-electron and the traditional RKKY interaction. Numerical calculation of the effective exchange interaction is performed and discussed. Making use of the derived Hamiltonian, the magnetic susceptibilities of two typical heavy-fermion materials are studied.

The effect of a parasite hybridizer on the magnetic behavior of partially delocalized transition shell systems (abstract)

We have investigated how the behavior of a transition shell atomic species (species A) with orbital magnetism, driven by hybridization-mediated interactions via a sea of band electrons, is modified by the addition of a second parasite hybridizer (species B). Our approach involves a two-stage procedure. First, we calculate the modification of the band electron sea by hybridization with B by using a slave boson formalism. Second, the modifications in the A-A interionic interactions driving the orbital magnetic ordering are calculated by applying a Schrieffer–Wolff transformation on the renormalized Anderson lattice hamiltonian obtained from the first stage. The new A-A interactions have a different radial dependence (range factor) which depends in a nonlinear way on the band-B hybridization strength: and the consequences of this change on the magnetic ordering are studied using a mean-field approximation. This enables us to model the reduction in the magnetic ordering caused by competing parasite hybridization, and the dependence of this reduction on the relative hybridization strengths of the two species.

On a unified canonical transformation of the large-negative-(positive-) U Hubbard model

The Schrieffer-Wolff transformation of the large-negative-U Hubbard model is performed in detail. Similarities to and differences from the canonical transformation of the large-positive-U Hamiltonian into the t-J model are discussed.

Synthesis of band and model Hamiltonian theory for strongly hybridizing uranium systems

For a specified chemical environment (e.g., isostructural compound), uranium typically shows the most delocalized correlated f-electron behavior of the light rare earths and actinides. This provides perhaps the most interesting correlated f-electron behavior for phenomena such as heavy fermion behavior, but also provides the greatest difficulty for the theory. Here we are concerned with the synthesis of band and model(Anderson lattice) Hamiltonian theory to be able to evaluate the phenomenological behavior on a materially predictive ab initio basis. In previous studies for cerium and plutonium systems, use was made of U(the correlation energy) being rather large in dealing with the f-f banding term, derived from the Schrieffer–Wolff transformation of the Anderson lattice Hamiltonian. Since the bandwidth is suppressed by a factor of 1/U, it was justified to neglect the f-f banding term. This is not justified for uranium systems which have a much smaller U. Therefore, rather than treating f electrons as core(resonant) states during the self-consistent band calculation, we first treat 5f’s as valence electrons and determine the self-consistent potential. Then the density of states, and hence the f bandwidth (Γ1) which characterizes hybridization between all the band electrons, can be calculated. Then we let the 5f’s form bands among themselves, while suppressing hybridization with other bands during the determination of the self-consistent potential. We then use that potential as the basis to calculate the density of states and the f bandwidth (Γ2), which characterizes the hybridization between f’s. We then use the value of (Γ1−Γ2), the hybridization strength so obtained, to absolutely evaluate the magnetic ordering behavior and compare with experiment.

Effective strong-coupling Hamiltonians for bipolaron centers and magnetic impurities with on-site electron-phonon coupling.

We have studied a model of bipolaron (negative-U) centers and magnetic impurities which includes a repulsive on-site Coulomb interaction as well as a coupling of the electronic on-site energy to the displacement amplitude of a local phonon mode. By means of a generalized Schrieffer-Wolff transformation, we derive an effective strong-coupling (or weak-hybridization) Hamiltonian for the low-lying states of the system. In the case of a bipolaron center (where the phonon-mediated on-site attraction dominates over the Coulomb repulsion), the effective Hamiltonian describes conduction-electron scattering and electron-pair tunneling processes between the localized impurity orbital and the conduction band of the metallic host. In the limit of very strong electron-phonon coupling, the pair tunneling matrix elements become exponentially small, whereas conduction electron scattering is suppressed only algebraically. The impurity orbital occupation number is then almost ``frozen in'' and the bipolaron center becomes inefficient in enhancing the superconducting pairing correlations of the host. In the case of a magnetic impurity (with a predominant repulsive Coulomb interaction), the effective Hamiltonian reduces to the form of an ordinary Kondo spin-exchange model. The spin-exchange coupling constants are not affected by the presence of the electron-phonon interaction.

Schrieffer-Wolff transformation for the Anderson Hamiltonian in a superconductor.

A generalized Schrieffer-Wolff transformation is introduced to relate the Anderson impurity model in a superconductor to a Kondo-like Hamiltonian. The effective exchange interactions coupling a magnetic impurity to a Bardeen-Cooper-Schrieffer (BCS) spectrum of electronic excitations are expressed in terms of the superconducting coherence factors. New exchange mechanisms originate from the particle-hole coherence unique to superconductors. These interactions are relevant for large asymmetry in the magnetic limit; they vanish for nonmagnetic impurities and in the symmetric Anderson model, for which the ``normal'' Schrieffer-Wolff coupling constant is recovered. The magnetically induced equal-spin Cooper-pairing correlations are suggested to be relevant for superconducting valence fluctuators and within heavy-fermion superconductivity.

Superconductivity in the generalized periodic Anderson model with strong local attraction

We study a generalized periodic Anderson model with on-site hybridization between wide- and narrow-band electrons and strong and local coupling with the lattice deformation. Provided that the interaction with the lattice is strong enough, the narrow-band electrons will be turned into small polarons which interact attractively with each other over short distances, leading to the formation of local pairs of narrow-band electrons. This leads to a pinning of the Fermi level which is due to the fact that narrow-band electrons exist only in pair states. By means of a generalized Schrieffer-Wolff transformation we eliminate hybridization and obtain an effective Hamiltonian which describes a contact interaction between local pairs and wide-band electrons as well as the direct hopping of local pairs and interparticle Coulomb interactions. In such a system the two types of mechanisms which can lead to superconductivity have been studied. The first one is due to direct local pair hopping and involves exclusively the narrow-band subsystem giving rise to a superconductivity analogous to superfluidity in /sup 4/He. The second one is due to a contact interaction between local pairs and pairs of wide-band electrons. This leads to a superconducting state involving both subsystems where the local pairs of themore » narrow-band subsystem induce Cooper pairing amongst the electrons of the wide-band subsystem. Consequently, the single-particle spectrum of the wide-band electrons opens up a gap around the position of the narrow band of electrons in pair states.« less

Theoretical study on the anomalous electrical resistivity in the Kondo lattice intermetallic compound

A canonical transformation which is an extension of the Schrieffer-Wolff transformation is used to relate the Anderson lattice Hamiltonian with f-f electron coupling to that of the Kondo lattice. An important feature of our resultant Kondo lattice Hamiltonian is that only the spin-flip scattering between the conduction electron and the f electron exists under the assumption of neglecting the f-electron density fluctuation on the conduction electron. Based on the Kondo lattice Hamiltonian a simplified model Hamiltonian has been suggested to calculate the anomalous temperature dependence of the electrical resistivity in the Kondo lattice intermetallic compounds (e.g. CeAl 3, CeCu 6). The Green's functions for the conduction electron have been calculated by the equation of motion approach. Following the decoupling procedure introduced by Nagaoka and Hamann's approximation for the singular integrations we have obtained a nonlinear differential equation for the t-matrix, t r( z), which has been solved by numerical calculations. The result is consistent with the experimental data of CeCu 6.

A possible model for the anomalous electrical resistivity in the periodic dense Kondo system

A model Hamiltonian resulting from an extended Schrieffer-Wolff transformation on the Anderson lattice Hamiltonian with f-electron hopping has been used to calculate the anomalous temperature dependence of the electrical resistivity in the Kondo lattice compounds (e.g., CeAl3 and CeCu6). The Green's functions for the conduction electron have been calculated by the equation of motion approach. The numerical calculated results of the electrical resistivity of CeCu6 is consistent with the experimental data.

Self-consistent band calculations for an Anderson Hamiltonian treatment of anisotropic f-electron systems

The unusual magnetic ordering, magnetic excitation behavior, and anomalous crystal-field splitting in CeBi and CeSb can be explained using a model Hamiltonian obtained by applying the Schrieffer–Wolff transformation to the Anderson Hamiltonian. It is highly desirable to obtain a first principles prediction of the parameters of the theory in order to extend this understanding quantitatively to other cerium and light actinide materials. To this end, we have calculated the electronic structure of CeBi and CeSb with the linear muffin-tin orbital (LMTO) method, using a self-consistent fully warped potential, and have obtained the coupling between non-f band states and f states localized in the sense of the Anderson Hamiltonian. This is applied to a calculation of the anomalous crystal-field splitting in CeBi and CeSb, and application to calculating the anisotropic two-ion interaction is discussed. We find that the dominant contribution to the anomalous crystal-field effect (about two thirds) comes from hybridization with d bands.

A new pairing possibility in heavy fermion superconductors

Abstract Hybridization coupled bands of lozalized and itinerant states can produce pairing by double hybridization. The mechanism is illustrated for a periodic Anderson model. The effective Hamiltonian generated in a second order Schrieffer-Wolff transformation contains an exchange interaction which reduces the intra-atomic Coulomb repulsion and favours pairing.

Theoretical study of the Kondo lattice: A reexamination

In the case of a lattice, the Schrieffer-Wolff transformation yields extra terms in addition to the classical $s\ensuremath{-}f$ exchange Hamiltonian. The study of the Kondo-lattice Hamiltonian including these new terms by the renormalization-group method shows the possibility of a magnetic-nonmagnetic transition versus exchange constant, in good agreement with the experimental situation of cerium compounds.