Renormalization-group analysis of the one-dimensional extended Hubbard model with a single impurity

We analyze the one-dimensional extended Hubbard model with a single static impurity by using a computational technique based on the functional renormalization group. This extends previous work for spinless fermions to spin-1/2 fermions. The underlying approximations are devised for weak interactions and arbitrary impurity strengths, and have been checked by comparing with density-matrix renormalization-group data. We present results for the density of states, the density profile and the linear conductance. Two-particle backscattering leads to striking effects, which are not captured if the bulk system is approximated by its low-energy fixed point, the Luttinger model. In particular, the expected decrease of spectral weight near the impurity and of the conductance at low energy scales is often preceded by a pronounced increase, and the asymptotic power laws are modified by logarithmic corrections.Comment: 36 pages, 13 figures, revised version as publishe

A gentle introduction to the functional renormalization group: the Kondo effect in quantum dots

The functional renormalization group provides an efficient description of the interplay and competition of correlations on different energy scales in interacting Fermi systems. An exact hierarchy of flow equations yields the gradual evolution from a microscopic model Hamiltonian to the effective action as a function of a continuously decreasing energy cutoff. Practical implementations rely on suitable truncations of the hierarchy, which capture nonuniversal properties at higher energy scales in addition to the universal low-energy asymptotics. As a specific example we study transport properties through a single-level quantum dot coupled to Fermi liquid leads. In particular, we focus on the temperature T=0 gate voltage dependence of the linear conductance. A comparison with exact results shows that the functional renormalization group approach captures the broad resonance plateau as well as the emergence of the Kondo scale. It can be easily extended to more complex setups of quantum dots.Comment: contribution to Les Houches proceedings 2006, Springer styl

Functional renormalization group for Luttinger liquids with impurities

We improve the recently developed functional renormalization group (fRG) for impurities and boundaries in Luttinger liquids by including renormalization of the two-particle interaction, in addition to renormalization of the impurity potential. Explicit flow-equations are derived for spinless lattice fermions with nearest neighbor interaction at zero temperature, and a fast algorithm for solving these equations for very large systems is presented. We compute spectral properties of single-particle excitations, and the oscillations in the density profile induced by impurities or boundaries for chains with up to 1000000 lattice sites. The expected asymptotic power-laws at low energy or long distance are fully captured by the fRG. Results on the relevant energy scales and crossover phenomena at intermediate scales are also obtained. A comparison with numerical density matrix renormalization results for systems with up to 1000 sites shows that the fRG with the inclusion of vertex renormalization is remarkably accurate even for intermediate interaction strengths.Comment: 35 pages, 16 figures, revised version as publishe

Functional renormalization group approach to zero-dimensional interacting systems

We apply the functional renormalization group method to the calculation of dynamical properties of zero-dimensional interacting quantum systems. As case studies we discuss the anharmonic oscillator and the single impurity Anderson model. We truncate the hierarchy of flow equations such that the results are at least correct up to second order perturbation theory in the coupling. For the anharmonic oscillator energies and spectra obtained within two different functional renormalization group schemes are compared to numerically exact results, perturbation theory, and the mean field approximation. Even at large coupling the results obtained using the functional renormalization group agree quite well with the numerical exact solution. The better of the two schemes is used to calculate spectra of the single impurity Anderson model, which then are compared to the results of perturbation theory and the numerical renormalization group. For small to intermediate couplings the functional renormalization group gives results which are close to the ones obtained using the very accurate numerical renormalization group method. In particulare the low-energy scale (Kondo temperature) extracted from the functional renormalization group results shows the expected behavior.Comment: 22 pages, 8 figures include

Correlation effects on electronic transport through dots and wires

We investigate how two-particle interactions affect the electronic transport through meso- and nanoscopic systems of two different types: quantum dots with local Coulomb correlations and quasi one-dimensional quantum wires of interacting electrons. A recently developed functional renormalization group scheme is used that allows to investigate systems of complex geometry. Considering simple setups we show that the method includes the essential aspects of Luttinger liquid physics (one-dimensional wires) as well as of the physics of local correlations, with the Kondo effect being an important example. For more complex systems of coupled dots and Y-junctions of interacting wires we find surprising new correlation effects.Comment: to appear in "Advances in Solid State Physics" Volume 46, Ed. R. Haug (Springer, 2006

Nonequilibrium functional RG with frequency dependent vertex function: A study of the single impurity Anderson model

We investigate nonequilibrium properties of the single impurity Anderson model by means of the functional renormalization group (fRG) within Keldysh formalism. We present how the level broadening Gamma/2 can be used as flow parameter for the fRG. This choice preserves important aspects of the Fermi liquid behaviour that the model exhibits in case of particle-hole symmetry. An approximation scheme for the Keldysh fRG is developed which accounts for the frequency dependence of the two-particle vertex in a way similar but not equivalent to a recently published approximation to the equilibrium Matsubara fRG. Our method turns out to be a flexible tool for the study of weak to intermediate on-site interactions U <= 3 Gamma. In equilibrium we find excellent agreement with NRG results for the linear conductance at finite gate voltage, magnetic field, and temperature. In nonequilibrium, our results for the current agree well with TD-DMRG. For the nonlinear conductance as function of the bias voltage, we propose reliable results at finite magnetic field and finite temperature. Furthermore, we demonstrate the exponentially small scale of the Kondo temperature to appear in the second order derivative of the self-energy. We show that the approximation is, however, not able to reproduce the scaling of the effective mass at large interactions.Comment: [v2] - minor changes throughout the text; added new Fig. 3; corrected pert.-theory data in Figs. 10, 11; published versio

Strongly coupled quantum criticality with a Fermi surface in two dimensions: fractionalization of spin and charge collective modes

We describe two dimensional models with a metallic Fermi surface which display quantum phase transitions controlled by strongly interacting critical field theories below their upper critical dimension. The primary examples involve transitions with a topological order parameter associated with dislocations in collinear spin density wave ("stripe") correlations: the gapping of the order parameter fluctuations leads to a fractionalization of spin and charge collective modes, and this transition has been proposed as a candidate for the cuprates near optimal doping. The coupling between the order parameter and long-wavelength volume and shape deformations of the Fermi surface is analyzed by the renormalization group, and a runaway flow to a non-perturbative regime is found in most cases. A phenomenological scaling analysis of simple observable properties of possible second order quantum critical points is presented, with results quite similar to those near quantum spin glass transitions and to phenomenological forms proposed by Schroeder et al. (cond-mat/0011002).Comment: 16 pages, 4 figures; (v2) additional clarifying remark

The HER2 phenotype of circulating tumor cells in HER2-positive early breast cancer: A translational research project of a prospective randomized phase III trial

Background: HER2 is one of the predominant therapeutic targets in breast cancer. The metastatic selection process may lead to discrepancies between the HER2 status of the primary tumor and circulating tumor cells (CTCs). This study analyzed the HER2 status of CTCs in patients with HER2-positive primary breast cancer at the time of diagnosis. Aim of the study was to assess potential discordance of HER2 status between primary tumor and CTCs, as this may have important implications for the use of HER2-targeted therapy. Methods: The number and HER2 status of CTCs out of 30ml peripheral blood were assessed in 642 patients using the CellSearch System (Janssen Diagnostics, USA). The cutoff for CTC positivity was the presence of at least 1 CTC, and the cutoff for HER2 positivity of CTCs was the presence of at least 1 CTC with a strong HER2 staining. Results 258 (40.2%) of the 642 patients were positive for CTCs (median 2;range 1-1,689). 149 (57.8%) of these 258 patients had at least 1 CTC with strong HER2 staining. The presence of HER2-positive CTCs was not associated with tumor size (p = 0.335), histopathological grading (p = 0.976), hormone receptor status (ER: p = 0.626, PR: p = 0.263) or axillary lymph node involvement (p = 0.430). Overall, 83 (32.2%) of the CTC-positive patients exclusively had CTCs with strong HER2 staining, whereas 31 (12.0%) had only CTCs with negative HER2 staining. Within-sample variation in the HER2 status of CTCs was found in 86 (57.8%) of the 149 patients with more than 1 CTC. Conclusion: This study demonstrated that discordance between the HER2 expression of CTCs and that of the primary tumor frequently occurs in early breast cancer. Future follow-up evaluation will assess whether this discrepancy may contribute to trastuzumab resistance

International study on inter-reader variability for circulating tumor cells in breast cancer

Introduction: Circulating tumor cells (CTCs) have been studied in breast cancer with the CellSearch® system. Given the low CTC counts in non-metastatic breast cancer, it is important to evaluate the inter-reader agreement.Methods: CellSearch® images (N = 272) of either CTCs or white blood cells or artifacts from 109 non-metastatic (M0) and 22 metastatic (M1) breast cancer patients from reported studies were sent to 22 readers from 15 academic laboratories and 8 readers from two Veridex laboratories. Each image was scored as No CTC vs CTC HER2- vs CTC HER2+. The 8 Veridex readers were summarized to a Veridex Consensus (VC) to compare each academic reader using % agreement and kappa (κ) statistics. Agreement was compared according to disease stage and CTC counts using the Wilcoxon signed rank test.Results: For CTC definition (No CTC vs CTC), the median agreement between academic readers and VC was 92% (range 69 to 97%) with a median κ of 0.83 (range 0.37 to 0.93). Lower agreement was observed in images from M0 (median 91%, range 70 to 96%) compared to M1 (median 98%, range 64 to 100%) patients (P < 0.001) and from M0 and <3CTCs (median 87%, range 66 to 95%) compared to M0 and ≥3CTCs samples (median 95%, range 77 to 99%), (P < 0.001). For CTC HER2 expression (HER2- vs HER2+), the median agreement was 87% (range 51 to 95%) with a median κ of 0.74 (range 0.25 to 0.90).Conclusions: The inter-reader agreement for CTC definition was high. Reduced agreement was observed in M0 patients with low CTC counts. Continuous training and independent image review are required