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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:
Date submitted: 2019-05-07
Submitted by: Waintal, Xavier
Submitted to: SciPost Physics
Domain(s): Computational
Subject area: Condensed Matter Physics - Computational


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:
Editor-in-charge assigned

Submission & Refereeing History

Submission 1905.01271v1 on 7 May 2019

Reports on this Submission

Anonymous Report 1 on 2019-5-8 Invited Report


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


No obvious weakness


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:

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.

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).

  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: excellent

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