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Project OPUS 21

Electronic Correlations, Superconductivity, and Quantum Fluctuations Combined: Theory and Quantitative Interpretation of Experiment

zdjęcie Profesora Józefa Spałka


Principal Investigator: Prof. dr hab. Józef Spałek

Project duration: 2022-2024

e-mail: jozef.spalek[at]




Physics deals with phenomena appearing at all energy scales and is concerned with their quantitative description. In the case of quantum many-particle (many-body) systems the two scales, high- and low-energy, emerge naturally in a single system. Here we concentrate on strongly correlated fermion (electron) systems, in which the interparticle (repulsive) interaction dominates over the representative (Fermi) energy of starting individual particles. In effect, this strong repulsive (Coulomb, van der Waals, etc.) interaction is the predominant high-energy factor, whereas the single-particle energy, and particularly the collective excitations, constitute the low-energy part of the system behavior. However, in the strongly correlated systems, the situation is not that simple, since the two energy scales are mutually coupled and combine into a coherent (effective) description. This situation takes place in high-temperature superconductors (high-Tc SC) and heavy-fermion (electron) compounds, to name just two examples, in which a new type of superconducting (non-Landau) Fermi liquid or Fermi-liquid with extremely heavy effective masses occur, respectively.

To approach theoretically those systems, we have developed in the last 5 years (2016-2020) the so-called Diagrammatic Expansion of Variational (e.g., Gutzwiller) Wave Function (DE-V(G)WF) method, for which we have succeeded in describing the principal equilibrium properties of high-temperature superconductors in a fully quantitative manner. This description is being summarized in an extensive review for Phys. Rep. (see: http://thwww.

The basic conceptual analysis of our whole approach is divided into two steps:

(i) the effective self-consistent single-particle states represented by effective Hamiltonian, which accounts well for
almost universal (independent of doping) Fermi velocity, Fermi wave-vector, effective mass, and effective pairing gap,

(ii) the development of the full correlated equilibrium state characterized by the correlated gap, kinetic-energy gain in
SC state (non-BCS feature), the observed kink in the highest-energy part of ARPES spectra, etc. It is crucial to note that our theory reduces in the lowest order to the celebrated Renormalized Mean Field Theory (RMFT).
We have shown earlier that RMFT must be reformulated to make it statistically consistent (the self-consistent and
variational solutions match each other, SGA method). In effect, we can pinpoint the features, where RMFT fails. DE-VWF approach goes systematically beyond SGA. 

Having a reliable formulation of the static properties, we are prepared to tackle next the recently observed and studied intensively ubiquitous quantum spin and charge dynamic fluctuations, i. e., paramagnons and plasmons. In the last year, we have been able to formulate a brand new and comprehensive approach unifying DE-GWF and field-theoretical (1/N) expansion and have obtained the first fully quantitative agreement with the observed excitation spectrum (cf. Phys. Rev. B 103, 105111 (2021) pp.1-27; arXiv: 2104.1281). This new development opens up a new opportunity to built in the fluctuations into consistent statistical thermodynamics of unconventional superconductors such as the cuprates and heavy-fermion systems (UGe2, URhGe, etc.). This is the subject and main goal of the proposed 2-year project. We are confident that the project will result in a unique description of the ground-state and thermal properties of those systems. This last opinion can be exemplified in the case of the effective gap, which exhibits a trend of the dopingdependence of observed pseudogap; the difference is in magnitude. The inclusion of the quantum fluctuations in the correlated state should improve the agreement decisively.


As a separate task, we plan to incorporate the fluctuations into the properties of the above heavy fermions in which weak ferromagnetism and spin-triplet superconductivity coexist. In this example, we should be able also to see how our theory of quantum fluctuations for correlated systems reduces to that of Moriya-Hertz-Millis when the moderateinteraction leads to weaker correlations. It should be emphasized that the above topics have been developing slowly, since they concern difficult, but profound aspects of the correlated quantum matter. We offer, in this respect, a comprehensive and consistent resolution of at least some of those longstanding fundamental problems.

picture: copper-oxygen plane diagram and schematic phase diagram of a high-temperature superconductor

Fig. 1. Left: Lattice of strongly-correlated electrons in two dimensions. Right: Generic hole-density vs. temperature phase diagram of a high-temperature superconductor. Supercoductivity appears in between metallic and Mott-insulating phases.