Self-consistent-field Ensembles of Disordered Hamiltonians: Efficient Solver and Application to Superconducting Films

Principal Investigator:
Ferdinand Evers

Institute of Theoretical Physics, University of Regensburg

Local Project ID:

HPC Platform used:
SuperMUC of LRZ

Date published:


Our general interest is in self-consistent-field (scf) theories of disordered fermions. They generate physically relevant sub-ensembles for models of dirty metals (“scf-ensembles”) within the standard Altland-Zirnbauer classification. We are motivated to investigate such ensembles (i) by the possibility to discover new phase transitions due to (long-range) interactions; ii) by other (analytical) scf-theories that rely on partial self-consistency approximations awaiting a numerical validation; (iii) by the overall importance of scf-theories for the understanding of complex interaction-mediated phenomena in terms of effective single-particle pictures.

In particular, we investigate disordered thin film superconductors within the  Boguliubov-deGennes (BdG) theory of the attractive-U Hubbard model in the presence of on-site disorder; the sc-fields are the particle density n(r) and the gap function ∆(r). For this case, we reach system sizes unprecedented in earlier work. They allow us to study phenomena emerging at scales substantially larger than the lattice constant, such as the interplay of multifractality and interactions, or the formation of superconducting islands.

Results and Methods

We observe that the superconducting coherence length exhibits a non-monotonic behavior with increasing disorder strength already at moderate interaction strength as seen in Fig. 1.

In Fig. 2 and 3 both the local density of states and the pairing amplitude respectively are seen to exhibit very broad (log-normal) distribution functions, consistent with the strong mesoscopic fluctuations predicted by analytical theories.

With respect to methodology our results are important because we establish that partial self- consistency (”energy-only”) schemes as typically employed in analytical approaches tend to miss qualitative physics such as island formation.

The fully self-consistent, energy-only treatment with Hartree potential and energy-only schemes are compared in Fig. 4. For a more comprehensive presentation of our results please refer to [2].

To compute the self-consistent fields, we devised an algorithm based on the kernel polynomial method (KPM) [3].  With a KPM based code, we were able to reduce the asymptotic runtime from a cubic to a quadratic dependence on the size of the lattice with respect to conventional full-diagonalization approaches. Through this we were able to achieve system sizes unprecedented in earlier works.

We achieved a close to perfect parallelization (a) through the computation of multiple impurity configurations and (b) by computation of the self-consistent fields on different points in space with negligible communication between processes. Our code is written in a combination of Python and C, combining the performance of a highly optimized C-kernel with the adaptability and convenience of a high-level language for the non performance critical sections.

It is computationally expensive to find a single solution of the self-consistent fields (~100-1000 core-h) for one disorder configuration. We need a large number (~100000) of these configurations to achieve a reliable disorder average and to investigate different regimes of disorder strength, interaction strength, filling fraction and system size.

In total we used 33,350,298 core-h producing approximately 1 TB of self-consistent mean-field Hamiltonians across the parameter and impurity configuration space. We generated ~20000 hdf5 files, where multiple parameter configurations can be saved in the same file for the same impurity configuration. For a typical job we used ~20000 cores.

On-going Research / Outlook

Without the tremendous resources and support of SuperMUC, we couldn’t have met the high computational demands that our project entailed. Since the start of the project we have continually improved and optimized our code, most notably with a matrix-free implementation of the Chebyshev expansion [2]. Some very interesting related questions remain open. In a follow-up project we will include spin-orbit coupling to our system. In particular we are interested in the effect of self-consistency on the pile-up of close to zero modes in a disordered 2D topological superconductor. In closing we would like to mention that this project and the future program proposed was recognized as scientifically important enough in order to warrant further funding within a binational initiative of Germany and Russia (RFBR-DFG EV30/14-1).

References and Links

[1] https://www.lrz.de/projekte/hlrb-projects/0000000000F43A36.html

[2] M. Stosiek, B. Lang and F. Evers. 2019. Self-consistent-field ensembles of disordered Hamiltonians: Efficient solver and application to superconducting films. arXiv:1903.10395 (Mar. 2019). DOI: https://arxiv.org/abs/1903.10395 (accepted for publication in Physical Review B)

[3] A. Weiße, G. Wellein, A. Alvermann, and H. Fehske. 2006.  Rev Mod Phys 78, 275 (Mar. 2006). DOI: https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.78.275


Matthias Stosiek, Ferdinand Evers (PI)

Research Partner

Bruno Lang, Institute of Applied Informatics, University of Wuppertal

Scientific Contact

Prof. Dr. Ferdinand Evers
Department of Physics
Institute of Theoretical Physics
University of Regensburg
Universitätsstraße 31, D-93053 Regensburg (Germany)
e-mail: ferdinand.evers [@] ur.de

LRZ Project ID: pr53lu

July 2020

Tags: Universität Regensburg LRZ QCD