Luttinger liquid, singular interaction and quantum criticality in cuprate materials

With particular reference to the role of the renormalization group approach and Ward identities, we start by recalling some old features of the one-dimensional Luttinger liquid as the prototype of non-Fermi-liquid behavior. Its dimensional crossover to the Landau normal Fermi liquid implies that a non-Fermi liquid, as, e.g., the normal phase of the cuprate high temperature superconductors, can be maintained in d>1, only in the presence of a sufficiently singular effective interaction among the charge carriers. This is the case when, nearby an instability, the interaction is mediated by fluctuations. We are then led to introduce the specific case of superconductivity in cuprates as an example of avoided quantum criticality. We will disentangle the fluctuations which act as mediators of singular electron-electron interaction, enlightening the possible order competing with superconductivity and a mechanism for the non-Fermi-liquid behavior of the metallic phase. This paper is not meant to be a comprehensive review. Many important contributions will not be considered. We will also avoid using extensive technicalities and making full calculations for which we refer to the original papers and to the many good available reviews. We will here only follow one line of reasoning which guided our research activity in this field.Comment: 23 pages, 10 figure

Phase separation frustrated by the long range Coulomb interaction II: Applications

The theory of first order density-driven phase transitions with frustration due to the long range Coulomb (LRC) interaction develop on paper I of this series is applied to the following physical systems: i) the low density electron gas ii) electronic phase separation in the low density three dimensional $t-J$ model iii) in the manganites near the charge ordered phase. We work in the approximation that the density within each phase is uniform and we assume that the system separates in spherical drops of one phase hosted by the other phase with the distance between drops and the drop radius much larger than the interparticle distance. For i) we study a well known apparent instability related to a negative compressibility at low densities. We show that this does not lead to macroscopic drop formation as one could expect naively and the system is stable from this point of view. For ii) we find that the LRC interaction significantly modifies the phase diagram favoring uniform phases and mixed states of antiferromagnetic (AF) regions surrounded by metallic regions over AF regions surrounded by empty space. For iii) we show that the dependence of local densities of the phases on the overall density found in paper I gives a non-monotonous behavior of the Curie temperature on doping in agreement with experiments.Comment: Second part of cond-mat/0010092 12 pages, 12 figure

Renormalization group and Ward identities in quantum liquid phases and in unconventional critical phenomena

By reviewing the application of the renormalization group to different theoretical problems, we emphasize the role played by the general symmetry properties in identifying the relevant running variables describing the behavior of a given physical system. In particular, we show how the constraints due to the Ward identities, which implement the conservation laws associated with the various symmetries, help to minimize the number of independent running variables. This use of the Ward identities is examined both in the case of a stable phase and of a critical phenomenon. In the first case we consider the problems of interacting fermions and bosons. In one dimension general and specific Ward identities are sufficient to show the non-Fermi-liquid character of the interacting fermion system, and also allow to describe the crossover to a Fermi liquid above one dimension. This crossover is examined both in the absence and presence of singular interaction. On the other hand, in the case of interacting bosons in the superfluid phase, the implementation of the Ward identities provides the asymptotically exact description of the acoustic low-energy excitation spectrum, and clarifies the subtle mechanism of how this is realized below and above three dimensions. As a critical phenomenon, we discuss the disorder-driven metal-insulator transition in a disordered interacting Fermi system. In this case, through the use of Ward identities, one is able to associate all the disorder effects to renormalizations of the Landau parameters. As a consequence, the occurrence of a metal-insulator transition is described as a critical breakdown of a Fermi liquid.Comment: 47 pages, 11 figure

Local behavior of fractional pp-minimizers

We extend the De Giorgi-Nash-Moser theory to nonlocal, possibly degenerate integro-differential operators.Comment: 26 pages. To appear in Ann. Inst. H. Poincare Anal. Non Lineaire. arXiv admin note: text overlap with arXiv:1405.784

Self-organized electronic superlattices in layered materials

We show that in layered systems with electronic phase separation tendency, the long-range Coulomb interaction can drive the spontaneous formation of unidirectional superlattices of electronic charge in a completely homogeneous crystalline background. In this self-organized electronic heterostructure, the ratio among the number of crystalline planes in the minority and majority electronic phases corresponds to Farey fractions with the superlattice period controlled by the background charge density and the frustrating Coulomb interaction strength. The phase diagram displays Arnold tongues obeying a modified Farey tree hierarchy and a devil's staircase typical of systems with frustration among different scales. We further discuss the competition of these electronic superlattices, recently observed in iron-based superconductors and mixed valence compounds, with in-plane electronically modulated phases.Comment: 5 pages, 4 figure