Friedel Sum Rule Research Articles

Friedel sum rule at finite temperature in hot dense plasmas

We study the notion of Friedel sum rule at finite temperature in hot dense plasmas. Using the average-atom model, we establish expressions for the Friedel sum rule at zero and finite temperature using non-relativistic or relativistic approaches. Formulas are also given using the Born approximation for the phase shifts. Numerical examples are provided. The Friedel sum rule is a stringent test of the internal consistency of a quantum average-atom model. The question of normalization of free wavefunctions is also discussed.

Scenarios of local spectral property and multifunctional spin selecting for a triple quantum dot molecule: How do the bonding and antibonding orbitals contribute to the spin currents?

Molecular electronic device is considered as a promising candidate for next generation electronic component, where the basic challenge involves understanding the charge and spin transports through molecular objects. In this paper, with the help of the sophisticated numerical renormalization group technique, we study theoretically the spin selective transport in a parallel triple quantum dot molecular device pierced by a local magnetic field along z axis. Based on simplified parameters of real molecular system, we find such device acts as a multifunctional spin selector when the inter-molecule tunneling couplings are asymmetric, including two 100% polarized spin-up summits, and one 100% spin-down summit in the linear conductance. We show in detail the local density of states for the bonding and anti-boding orbitals, and attribute the spin selection to the orbital polarized Coulomb blockade effect. We demonstrate our numerical results are consistent with those estimated by the analytical techniques, including the Friedel sum rule and the energy level crossings of the isolated orbitals.

Spectral properties and quantum phase transitions in superconducting junctions with a ferromagnetic link

We study theoretically the spectral and transport properties of a superconducting wire with a magnetic defect. We start by modelling the system as a one dimensional magnetic Josephson junction and derive the equation determining the full subgap spectrum in terms of the normal-state transfer matrix for arbitrary length and exchange field of the magnetic region. We demonstrate that the quantum phase transition predicted for a short-range magnetic impurity, and associated with a change of the total spin of the system, also occurs in junctions of finite length. Specifically, we find that the total spin changes discontinuously by integer jumps when bounds states cross the Fermi level. The spin can be calculated by using a generalization of Friedel sum rule for the superconducting state, which we also derive. With these tools, we analyze the subgap spectrum of a junction with the length of the magnetic region smaller than the superconducting coherence length and demonstrate how phase transitions also manifest as change of the sign of the supercurrent.

T-matrix formulation of real-space dynamical mean-field theory and the Friedel sum rule for correlated lattice fermions

We formulate real-space dynamical mean-field theory within scattering theory. Thereby the Friedel sum rule is derived for interacting lattice fermions at zero temperature.Graphical abstract

Fermi Liquids and the Luttinger Integral

The Luttinger Theorem, which relates the electron density to the volume of the Fermi surface in an itinerant electron system, is taken to be one of the essential features of a Fermi liquid. The microscopic derivation of this result depends on the vanishing of a certain integral, the Luttinger integral IL, which is also the basis of the Friedel sum rule for impurity models, relating the impurity occupation number to the scattering phase shift of the conduction electrons. It is known that non-zero values of IL with IL = ±π/2, occur in impurity models classified as singular Fermi liquids. Here we show the same values, IL = ±π/2, occur in an impurity model in phases with regular low energy Fermi liquid behavior. Consequently the Luttinger integral can be taken to characterize these phases, and the quantum critical points separating them interpreted as topological.

Ground- and excited-state scattering potentials for the stopping of protons in an electron gas

The self-consistent electron–ion potential V(r) is calculated for H+ ions in an electron gas system as a function of the projectile energy to model the electronic stopping power for conduction-band electrons. The results show different self-consistent potentials at low projectile-energies, related to different degrees of excitation of the electron cloud surrounding the intruder ion. This behavior can explain the abrupt change of velocity dependent screening-length of the potential found by the use of the extended Friedel sum rule and the possible breakdown of the standard free electron gas model for the electronic stopping at low projectile energies. A dynamical interpolation of V(r) is proposed and used to calculate the stopping power for H+ interacting with the valence electrons of Al. The results are in good agreement with the TDDFT benchmark calculations as well as with experimental data.

Wave functions and phase shifts of amplified modes within a vibrating metallic photonic crystal

We have so far investigated the optical properties within a one-dimensional photonic crystal whose stacked metallic plates are artificially driven by using actuators. A simple model was proposed and numerically analyzed, and the following novel phenomena were found out: The lattice vibration generates the light of frequency which added the integral multiple of the vibration frequency to that of the incident wave and also amplifies the incident wave resonantly. On a resonance, the amplification factor increases very rapidly with the number of layers. Resonance frequencies vary with the phases of lattice vibration. The amplification phenomenon was analytically discussed for low frequency of the lattice vibration and is confirmed by numerical works. The conditions of the resonance, however, have not been discovered yet. The lattice vibration is different from the usual phonon, since the photonic crystal is vibrated artificially, so that the dispersion relation of the lattice vibration does not exit. This fact is one of the causes which complicate the problem. In the present study, we compute the wave functions and phase shifts between the transmission and reflection coefficients around the amplifying resonance, and approach and discuss the resonance conditions with the photonic band structures and the Friedel sum rule in one dimension.

Kondo physics of the Anderson impurity model by distributional exact diagonalization

The Distributional Exact Diagonalization (DED) scheme is applied to the description of Kondo physics in the Anderson impurity model. DED maps Anderson's problem of an interacting impurity level coupled to an infinite bath onto an ensemble of finite Anderson models, each of which can be solved by exact diagonalization. An approximation to the self-energy of the original infinite model is then obtained from the ensemble averaged self-energy. Using Friedel's sum rule, we show that the particle number constraint, a central ingredient of the DED scheme, ultimately imposes Fermi liquid behavior on the ensemble averaged self-energy, and thus is essential for the description of Kondo physics within DED. Using the Numerical Renormalization Group (NRG) method as a benchmark, we show that DED yields excellent spectra, both inside and outside the Kondo regime for a moderate number of bath sites. Only for very strong correlations ($U/\Gamma\gg10$) does the number of bath sites needed to achieve good quantitative agreement become too large to be computationally feasible.

Zero-field splitting of the Kondo resonance and quantum criticality in triple quantum dots

We consider a triple-quantum-dot (TQD) system composed by an interacting quantum dot connected to two effectively non-interacting dots, which in turn are both connected in parallel to metallic leads. As we show, this system can be mapped onto a single-impurity Anderson model with a non-trivial density of states. The TQD's transport properties are investigated under a continuous tuning of the non-interacting dots' energy-levels, employing the Numerical Renormalization Group technique. Interference between single and many-particle resonances splits the Kondo peak, fulfilling a generalized Friedel sum rule. In addition, a particular configuration in which one of the non-interacting dots is held out of resonance with the leads allows to access a pseudogap regime where a Kosterlitz-Thouless type quantum-phase-transition (QPT) occur, separating the Kondo and non-Kondo behavior. Within this same configuration, the TQD exhibits traces of the Fano-Kondo effect, which is in turn, strongly affected by the QPT. Signatures of all these phenomena are neatly displayed by the calculated linear conductance.

Negative partial density of states in mesoscopic systems

Since the experimental observation of quantum mechanical scattering phase shift in mesoscopic systems, several aspects of it have not yet been understood. The experimental observations have also accentuated many theoretical problems related to Friedel sum rule and negativity of partial density of states. We address these problems using the concepts of Argand diagram and Burgers circuit. We can prove the possibility of negative partial density of states in mesoscopic systems. Such a conclusive and general evidence cannot be given in one, two or three dimensions. We can show a general connection between phase drops and exactness of semi classical Friedel sum rule. We also show Argand diagram for a scattering matrix element can be of few classes based on their topology and all observations can be classified accordingly.

Partial self-consistency and analyticity in many-body perturbation theory: Particle number conservation and a generalized sum rule

We consider a general class of approximations which guarantees the conservation of particle number in many-body perturbation theory. To do this we extend the concept of $\Phi$-derivability for the self-energy $\Sigma$ to a larger class of diagrammatic terms in which only some of the Green's function lines contain the fully dressed Green's function $G$. We call the corresponding approximations for $\Sigma$ partially $\Phi$-derivable. A special subclass of such approximations, which are gauge-invariant, is obtained by dressing loops in the diagrammatic expansion of $\Phi$ consistently with $G$. These approximations are number conserving but do not have to fulfill other conservation laws, such as the conservation of energy and momentum. From our formalism we can easily deduce if commonly used approximations will fulfill the continuity equation, which implies particle number conservation. We further show how the concept of partial $\Phi$-derivability plays an important role in the derivation of a generalized sum rule for the particle number, which reduces to the Luttinger-Ward theorem in the case of a homogeneous electron gas, and the Friedel sum rule in the case of the Anderson model. To do this we need to ensure that the Green's function has certain complex analytic properties, which can be guaranteed if the spectral function is positive semi-definite.The latter property can be ensured for a subset of partially $\Phi$-derivable approximations for the self-energy, namely those that can be constructed from squares of so-called half-diagrams. In case the analytic requirements are not fulfilled we highlight a number of subtle issues related to branch cuts, pole structure and multi-valuedness. We also show that various schemes of computing the particle number are consistent for particle number conserving approximations.

Tuning the transmission phase by the dot size

The phase shift of the electron's wave function after a tunneling event (i.e. the transmission phase) was at first measured for its fundamental or applicative relevance for quantum circuitry, but later the phase study self-motivated due to a number of unexpected results. One such result was the reduced increment of the phase on some resonances - with only fractions of π - in the few-electrons “mesoscopic” regime. In this paper we address such a regime for a rectangular quantum dot and compute the total phase increase on the first four resonances by means of accurate configuration–interaction method and a generalized Friedel sum rule as proposed by Rontani (2006) [17]. Our findings confirm that the electronic correlations reduce the on-resonance phase growth which is also found to decrease quasi–linearly with the dot size, the decrease being more pronounced as the number of electrons on the dot is raised. Sudden jumps (of small amplitude) of the phase are found to accompany ground states spin transitions.

Low-temperature behavior of transmission phase shift across a Kondo correlated quantum dot

We study the transmission phase shift across a Kondo correlated quantum dot in a GaAs heterostructure at temperatures below the Kondo temperature ($T < T_{\rm K}$), where the phase shift is expected to show a plateau at $\pi/2$ for an ideal Kondo singlet ground state. Our device is tuned such that the ratio $\Gamma/U$ of level width $\Gamma$ to charging energy $U$ is quite large ($\lesssim 0.5$ rather than $\ll 1$). This situation is commonly used in GaAs quantum dots to ensure Kondo temperatures large enough ($\simeq 100$ mK here) to be experimentally accessible; however it also implies that charge fluctuations are more pronounced than typically assumed in theoretical studies focusing on the regime $\Gamma/U \ll 1$ needed to ensure a well-defined local moment. Our measured phase evolves monotonically by $\pi$ across the two Coulomb peaks, but without being locked at $\pi/2$ in the Kondo valley for $T \ll T_{\rm K}$, due to a significant influence of large $\Gamma/U$. Only when $\Gamma/U$ is reduced sufficiently does the phase start to be locked around $\pi/2$ and develops into a plateau at $\pi/2$. Our observations are consistent with numerical renormalization group calculations, and can be understood as a direct consequence of the Friedel sum rule that relates the transmission phase shift to the local occupancy of the dot, and thermal average of a transmission coefficient through a resonance level near the Fermi energy.

Conditions for observing emergent SU(4) symmetry in a double quantum dot

We analyze conditions for the observation of a low energy SU(4) fixed point in capacitively coupled quantum dots. One problem, due to dots with different couplings to their baths, has been considered by Tosi, Roura-Bas and Aligia (2015). They showed how symmetry can be effectively restored via the adjustment of individual gates voltages, but they make the assumption of infinite on-dot and inter-dot interaction strengths. A related problem is the difference in the magnitudes between the on-dot and interdot strengths for capacitively coupled quantum dots. Here we examine both factors, based on a two site Anderson model, using the numerical renormalization group to calculate the local spectral densities on the dots and the renormalized parameters that specify the low energy fixed point. Our results support the conclusions of Tosi et al. that low energy SU(4) symmetry can be restored, but asymptotically achieved only if the inter-dot interaction $U_{12}$ is greater than or of the order of the band width of the coupled conduction bath $D$, which might be difficult to achieve experimentally. By comparing the SU(4) Kondo results for a total dot occupation $n_{\rm tot}=1$ and $n_{\rm tot}=2$ we conclude that the temperature dependence of the conductance is largely determined by the constraints of the Friedel sum rule rather than the SU(4) symmetry and suggest that an initial increase of the conductance with temperature is a distinguishing characteristic feature of an $n_{\rm tot}=1$ universal SU(4) fixed point.

Reprint of : Regular and singular Fermi liquid in triple quantum dots: Coherent transport studies

A system of three coupled quantum dots in a triangular geometry (TQD) with electron–electron interaction and symmetrically coupled to two leads is analyzed with respect to the electron transport by means of the numerical renormalization group. Varying gate potentials this system exhibits extremely rich range of regimes with different many-electron states with various local spin orderings. It is demonstrated how the Luttinger phase changes in a controlled manner which then via the Friedel sum rule formula exactly reproduces the conductance through the TQD system. The analysis of the uncoupled TQD molecule from the leads gives a reliable qualitative understanding of various relevant regimes and an insight into the phase diagram with the regular Fermi liquid and singular-Fermi liquid phases.

The beauty of impurities: Two revivals of Friedel's virtual bound-state concept

Abstract Jacques Friedel pioneered the theoretical study of impurities and magnetic impurities in metals. He discovered Friedel oscillations, introduced the concept of virtual bound-state, and demonstrated that the charge on the impurity is related to the scattering phase-shift at the Fermi level (Friedel sum-rule). After a brief review of some of these concepts, I describe how they proved useful in two new contexts. The first one concerns the Coulomb blockade in quantum dots, and its suppression by the Kondo effect. The second one is the dynamical mean-field theory of strong electronic correlations.

Regular and singular Fermi liquid in triple quantum dots: Coherent transport studies

A system of three coupled quantum dots in a triangular geometry (TQD) with electron–electron interaction and symmetrically coupled to two leads is analyzed with respect to the electron transport by means of the numerical renormalization group. Varying gate potentials this system exhibits extremely rich range of regimes with different many-electron states with various local spin orderings. It is demonstrated how the Luttinger phase changes in a controlled manner which then via the Friedel sum rule formula exactly reproduces the conductance through the TQD system. The analysis of the uncoupled TQD molecule from the leads gives a reliable qualitative understanding of various relevant regimes and an insight into the phase diagram with the regular Fermi liquid and singular-Fermi liquid phases.

Coulomb screening in graphene with topological defects

We analyze the screening of an external Coulomb charge in gapless graphene cone, which is taken as a prototype of a topological defect. In the subcritical regime, the induced charge is calculated using both the Green’s function and the Friedel sum rule. The dependence of the polarization charge on the Coulomb strength obtained from the Green’s function clearly shows the effect of the conical defect and indicates that the critical charge itself depends on the sample topology. Similar analysis using the Friedel sum rule indicates that the two results agree for low values of the Coulomb charge but differ for the higher strengths, especially in the presence of the conical defect. For a given subcritical charge, the transport cross-section has a higher value in the presence of the conical defect. In the supercritical regime we show that the coefficient of the power law tail of polarization charge density can be expressed as a summation of functions which vary log periodically with the distance from the Coulomb impurity. The period of variation depends on the conical defect. In the presence of the conical defect, the Fano resonances begin to appear in the transport cross-section for a lower value of the Coulomb charge. For both sub and supercritical regime we derive the dependence of LDOS on the conical defect. The effects of generalized boundary condition on the physical observables are also discussed.

Dynamic screening of an ion in a degenerate electron gas within the second-order Born approximation

The dynamic Friedel sum rule (FSR) is derived within the second-order Born (B2) approximation for an ion that moves in a fully degenerate electron gas and for an arbitrary spherically-symmetric electron–ion interaction potential. This results in an implicit equation for the dynamic B2 screening parameter which depends on the ion atomic number Z1 unlike the first-order Born (B1) dynamic screening parameter reported earlier by some authors. Furthermore, for typical metallic densities our analytical results for the Yukawa and hydrogenic potentials are compared, for both positive and negative ions, to the exact screening parameters calculated self-consistently by imposing the exact dynamic FSR requirement to the scattering phase shifts. The B1 and B2 screening parameters agree excellently with the exact values at large velocities, while at moderate and low velocities the B1 approximation deviates from the exact solution whereas the B2 approximation still remains close to it. In addition, a Padé approximant to the Born series yields a further improvement of the perturbative approach, showing an excellent agreement on the whole velocity range in the case of antiprotons.

Common non-Fermi liquid phases in quantum impurity physics

We study correlated quantum impurity models that undergo a local quantum phase transition (QPT) from a strong coupling, Fermi liquid phase to a non-Fermi liquid phase with a globally doubly degenerate ground state. Our aim is to establish what can be shown exactly about such ``local moment'' (LM) phases, of which the permanent (zero-field) local magnetization is a hallmark, and an order parameter for the QPT. A description of the zero-field LM phase is shown to require two distinct self-energies, which reflect the broken symmetry nature of the phase and together determine the single self-energy of standard field theory. Distinct Friedel sum rules for each phase are obtained, via a Luttinger theorem embodied in the vanishing of appropriate Luttinger integrals. By contrast, the standard Luttinger integral is nonzero in the LM phase but found to have universal magnitude. A range of spin susceptibilites are also considered, including that corresponding to the local order parameter, whose exact form is shown to be RPA-like, and to diverge as the QPT is approached. Particular attention is given to the pseudogap Anderson model, including the basic physical picture of the transition, the low-energy behavior of single-particle dynamics, the quantum critical point itself, and the rather subtle effect of an applied local field. A two-level impurity model that undergoes a QPT (``singlet-triplet'') to an underscreened LM phase is also considered, for which we derive on general grounds some key results for the zero-bias conductance in both phases.