# Magnetic and superconducting systems in low spatial dimensions

**Advisor:** Václav Janiš (IP CAS)

**Funding:** Fully funded

**Website:** https://www.fzu.cz/~janis/

**Contact:** janis@fzu.cz

Magnetism and superconductivity are two collective phenomena that are caused by electron correlations in solids. Their microscopic description demands the application of quantum many-body techniques. The many-body theory is essentially based on the perturbation expansion in the interaction strength. One must sum infinite-many diagrams to deal with collective phenomena. It is achieved by a self-consistency determining the relevant physical parameters controlling the long-range fluctuations. Presently, the most advanced self-consistent theory of interacting electrons is the Dynamical Mean-Field Approximation (DMFT) [1].It is restricted, however, only to local dynamical quantum fluctuations. One has to include also spatial fluctuations if one wants to describe collective phenomena in low spatial dimensions.

There is an alternative approach we have developed recently that includes a new type of mean-field-like self-consistency beyond the DMFT in the many-body perturbation theory [2]. This construction controls and keeps consistent both thermodynamic and spectral properties in regions of quantum critical points. It moreover allows us to include spatial fluctuations in an affordable way [3]. It offers new opportunities to study the critical behavior in systems with correlated electrons, whereby the repulsive interaction leads to the magnetic criticality and attractive to superconductivity. It is one of the few approaches enabling distinguish the quantum critical behavior in different spatial dimensions.

The objective
of this project is to study the low-temperature behavior of correlated electrons
in solids with both repulsive and attractive interaction in low-spatial
dimensions *d*=1,2,3 by means of the many-body Green functions and the
self-consistent approximate approach from Refs. [2] and [3]. The problems to be
addressed are

- Breakdown of the Fermi-liquid behavior in dimensions
*d*=1,2, - Differences in the quantum critical behavior at zero and non-zero temperatures,
- Crossover behavior from weak to strong coupling regimes,
- Spectral properties in regions of quantum criticality,
- Extension of the developed approximations on multi-orbital models.

The work on this project demands a good knowledge of quantum statistical physics and complex analysis, experience with many-body Green functions, and good programming skills.

References:

[1] Georges, A.,
et al., Dynamical mean-field theory of strongly correlated fermion systems and
the limit of infinite dimensions. Reviews of Modern Physics, 1996. 68(1): p.
13--125.

[2] Janiš, V., et al., Strongly correlated electrons: Analytic
mean-field theories with two-particle self-consistency. Physical Review B, 2019.
100: p. 195114.

[3] Janiš, V., A. Klíč, and J. Yan, Antiferromagnetic
fluctuations in the one-dimensional Hubbard model. AIP Advances, 2020. 10(12):
p. 125127.