## SciPost Submission Page

# The self-consistent quantum-electrostatic problem in strongly non-linear regime

### by P. Armagnat, A. Lacerda, B. Rossignol, C. Groth, X. Waintal

### Submission summary

As Contributors: | Xavier Waintal |

Arxiv Link: | https://arxiv.org/abs/1905.01271v1 |

Date submitted: | 2019-05-07 |

Submitted by: | Waintal, Xavier |

Submitted to: | SciPost Physics |

Domain(s): | Computational |

Subject area: | Condensed Matter Physics - Computational |

### Abstract

The self-consistent quantum-electrostatic (also known as Poisson-Schr\"odinger) problem is notoriously difficult in situations where the density of states varies rapidly with energy. At low temperatures, these fluctuations make the problem highly non-linear which renders iterative schemes deeply unstable. We present a stable algorithm that provides a solution to this problem with controlled accuracy. The technique is intrinsically convergent including in highly non-linear regimes. We illustrate our approach with (i) a calculation of the compressible and incompressible stripes in the integer quantum Hall regime and (ii) a calculation of the differential conductance of a quantum point contact geometry. Our technique provides a viable route for the predictive modeling of the transport properties of quantum nanoelectronics devices.

###### Current status:

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 1 on 2019-5-8 Invited Report

### Strengths

1. Clearly written

2. Well presented and documented

3. Detailed discussion of a difficult problem

4. Valuable information for researchers discussed in one place

5. New stable approach to a known problem

### Weaknesses

No obvious weakness

### Report

The authors propose an effective method to solve "the

the self-consistent quantum-electrostatic problem" at

zero temperature. Commonly, this has been avoided by

performing the calculations at a finite, but low

temperature. Indeed, the suggested methodology is

well explained, documented, and supported by examples.

All this together makes the manuscript very valuable

to many researchers that have to address this problem

in some form in their modeling of physical phenomena.

### Requested changes

For the authors to consider:

1.

The manuscript is well written, but I am note quite

sure if in the sentence:

"The technique is intrinsically convergent including

in highly non-linear regimes."

the word "including" is the best choice.

2.

The authors do investigate the "self-consistent

quantum-electrostatic problem", which they also

refer to as the "Poisson-Schrödinger problem".

In view of a difference in use of the terms within

the physics and the mathematical communities I would

like the authors briefly to relate this problem to

the Hartree Approximation (HA) which in the physics

and the quantum chemistry communities is also the

mentioned simultaneous solution of the Schrödinger

and the Poisson equations with the condition that

the wavefunctions are to be orthogonal.

In the mathematics community the problem is often

confronted without this last condition.

The different stand point can be referred back to

the need of physicists to relate the HA to higher

order ones, the Hartree-Fock Approximation (HFA), or

higher order Green functions schemes. This issue

is briefly mentioned in the Introduction of:

The European Physical Journal B 84, 699 (2011).