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Relativistic Landau quantization in nonuniform magnetic field and its applications to white dwarfs and quantum information
by Srishty Aggarwal (IISc), Banibrata Mukhopadhyay (IISc), Gianluca Gregori (Oxford)
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Submission summary
As Contributors:  Srishty Aggarwal · Banibrata Mukhopadhyay 
Preprint link:  scipost_202106_00030v1 
Date submitted:  20210618 08:12 
Submitted by:  Mukhopadhyay, Banibrata 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We investigate the twodimensional motion of relativistic cold electrons in the presence of ‘strictly’ spatially varying magnetic fields satisfying, however, no magnetic monopole condition. We find that the degeneracy of Landau levels, which arises in the case of the constant magnetic field, lifts out when the field is variable and the energy levels of spinup and spindown electrons align in an interesting way depending on the nature of change of field. Also the varying magnetic field splits Landau levels of electrons with zero angular momentum from positive angular momentum, unlike the constant field which only can split the levels between positive and negative angular momenta. Exploring Landau quantization in nonuniform magnetic fields is a unique venture on its own and has interdisciplinary implications in the fields ranging from condensed matter to astrophysics to quantum information. As examples, we show magnetized white dwarfs, with realistic nonuniform magnetic fields involved simultaneously with Lorentz force and Landau quantization affecting the underlying degenerate electron gas, exhibiting a significant violation of the Chandrasekhar masslimit; and an increase in quantum speed of electrons in the presence of a spatially growing magnetic field.
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Reports on this Submission
Anonymous Report 1 on 2021715 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202106_00030v1, delivered 20210715, doi: 10.21468/SciPost.Report.3240
Strengths
1. The manuscript contains new and Important findings.
2. Those can be applied cutting across different areas of physics.
Weaknesses
1. It is not evident from the manuscript how big is the effect of the Landau quantisation (LQ) of electrons in nonuniform magnetic fields on the structures of white dwarfs (WDs).
Report
Authors should quantify the impact of the LQ on the massradius relation in nonuniform magnetic fields compared with the previous results.
Author: Banibrata Mukhopadhyay on 20210816 [id 1673]
(in reply to Report 1 on 20210715)
AUTHORS:
First, we would like to thank the reviewer very much for carefully reading
our manuscript and assessing it with encouraging comments. Here we respond
their comments one by one mentioning changes made in the manuscript. All the
changes are kept in boldface and Figures 9 and 10 are replaced by new ones.
REVIEWER:
This work addresses the Landau Quantization (LQ) of electrons in nonuniform
magnetic fields and its applications to problems cutting across different
disciplines of physics, for example equation of state (EoS) and massradius
(MR) relationship of white dwarfs (WDs) as well as the quantum speed of
electrons which might be important for processing the speed of quantum information.
The nonuniform field has a simple power law variation as given by B = B_0 ρ^n ẑ
where n=0 represents the uniform magnetic field.
It is a well known fact that the zeroth Landau level is singly degenerate whereas
it is doubly degenerate for all other levels in a uniform magnetic field. Authors
in this manuscript demonstrate that the degeneracy of energy levels as found in
a uniform field, is lifted in nonuniform magnetic fields. They further note that
there is a definite alignment of energy levels of spinup and spindown electrons
as the magnitude of a nonuniform magnetic field changes at each point. These are
new findings. It is also observed that the quantum speed of electrons increases as
'n' grows.
The impact of the LQ of electrons in nonuniform magnetic fields is investigated
on the EoS and MR relationship of magnetized WDs. A specific example of the EoS for
n=0.3 is used to compute the MR curve in Figs. 9. If one carefully compares Fig. 9(a)
with Fig. 8(a), it is observed that the EoS of Fig. 8(a) for n=0.3 is distinctly
stiffer with respect to the case of the uniform field (n=0), but it is not the
case in Fig. 9(a). This needs an explanation.
AUTHORS:
In Figure 8(a), when we showed EoS for a uniform magnetic field, it is for n=0
throughout the star and field B=B_0. However, in Figure 9a, we showed two EoSs
for the same star: one is EoS of central region with constant magnetic field
where B is higher than B_0, according to the chosen profile in the previous
version, and another is EoS with varying magnetic field. See in Eq. (5.2),
how central field and B_0 are related in the central region of the star,
for the model considered in the previous version, with a varying magnetic
field other than central region. Hence this difference is between Figure 8(a)
and Figure 9(a). In the revised version, however, we have replaced these figures
with better ones for clarity with modified physics and changed the text
accordingly. Note the modifications in section 5.1 in the revised version,
part of which prompted due to reviewer's this question.
REVIEWER:
It is observed in Fig. 8(a) that less number of Landau levels is populated
when n=0.7 resulting in a softer EoS at lower densities and a stiffer EoS
at higher densities. It would be worth demonstrating the effect of the
nonuniform magnetic field for n=0.7 on the MR relation in a new plot.
AUTHORS:
For n=0.7, the Lorentz force may turn out to be too strong for a realistic
central density of the star to destabilize it. Hence, we do not show
an MR relation for n=0.7. However, in the revised version, we have modified
the field profile for the MR relation to remove this confusion. Note,
generally, that the trend of deviation for EoSs in nonuniform magnetic fields
compared to that of the uniform field is same for different n. However, the
deviation is more prominent with decreasing n, hence you noticed it for n=0.7.
REVIEWER:
In the introduction, authors mention another alternative approach of the
nonuniform magnetic field in Ref.[27] extensively used in the investigations
of structures of neutron stars and WDs. Authors should compare their current
findings on the MR relation of WDs with their previous results following
the prescription of the magnetic field variation of Ref. [27] inside the WDs.
This would definitely indicate whether new findings related to WDs in this
manuscript are quantitatively significant or not.
AUTHORS:
The aim of this work is not to propose a field profile opposing a density
dependent profile proposed earlier in Ref. 27, which facilitates MR relations
influenced by magnetic effects (classical and quantum). The main aim is to show
the Landau quantization in nonuniform magnetic fields and how the quantization
effect changes w.r.to the uniform magnetic field case. For that we have
proposed a field profile, satisfying Maxwell's equations, and solved
the Dirac equation (as opposed to the case of uniform field, when the
Dirac equation has to be solved for a quadratic potential). Subsequently,
as one of the immediate applications, we have attempted to explore the
results in (highly) magnetized white dwarfs. Otherwise, the density dependent
field profile, as proposed in Ref. 27, is quite an effective one, at least
as long as the star is spherical and we quite like it. However, the solution
of Dirac equation in varying magnetic field can not be proceeded with a
density dependent magnetic field profile, which merely talks about field
magnitude at various density points. Nevertheless, in principle, as
indicated in the manuscript, by changing parameters, various field profiles
can be obtained from the density dependent profile as well and we may get
equivalent results of the present one as long as its application is to
white dwarfs. Unless the reviewer is very particular about it, we prefer to
undertake this venture in a future work, in order to avoid deviating from
the main flow of the present manuscript, which is Landau quantization in
varying magnetic fields.
Nevertheless, we have revised the manuscript throughout for a better clarity.
Anonymous on 20210719 [id 1584]
The authors investigate the twodimensional motion of relativistic cold electrons in the presence of spatially varying magnetic fields. In this way they explore Landau quantization in nonuniform magnetic fields. They apply this to magnetized white dwarfs, with realistic nonuniform magnetic fields involved simultaneously, with Lorentz force and Landau quantization affecting the underlying degenerate electron gas, exhibiting a significant violation of the Chandrasekhar masslimit.
The detailed analysis of the impact of nonuniform magnetic field is interesting. Its a
nice piece of work which I envisage could have repercussions into various fields, such as,
astrophysics, condensed matter systems and quantum information.
It would be nice if they could work out in some detail, the impact of the field profile on an application, for example, the magnetized white dwarfs.
Author: Banibrata Mukhopadhyay on 20210816 [id 1674]
(in reply to Anonymous Comment on 20210719 [id 1584])AUTHORS:
First, we would like to thank the reviewer very much for encouraging comments
and suggestions. Here we respond their comments one by one mentioning changes
made in the manuscript. All the changes are kept in boldface and Figures 9
and 10 are replaced by new ones.
REVIEWER:
The authors investigate the twodimensional motion of relativistic cold electrons
in the presence of spatially varying magnetic fields. In this way they explore
Landau quantization in nonuniform magnetic fields. They apply this to magnetized
white dwarfs, with realistic nonuniform magnetic fields involved simultaneously,
with Lorentz force and Landau quantization affecting the underlying degenerate electron
gas, exhibiting a significant violation of the Chandrasekhar masslimit.
The detailed analysis of the impact of nonuniform magnetic field is interesting.
Its a nice piece of work which I envisage could have repercussions into various fields,
such as, astrophysics, condensed matter systems and quantum information.
It would be nice if they could work out in some detail, the impact of the field
profile on an application, for example, the magnetized white dwarfs.
AUTHORS:
In Figures 8 and 9, we had shown that how the field profile for nonuniform
magnetic fields leads to modified EoS and subsequently which may lead to
superChandrasekhar white dwarfs. Interestingly, we had/have shown in Figure 10
that had we considered only Lorentz force along with Chandrasekhar's EoS
(but not Landau quantization based EoS), then the mass of white dwarfs
decreased significantly. This proves that Landau quantization has a significant
impact on the result. Nevertheless, in the revised version, we have included
a revised MR relations for clarity along with a modified field profile more
suitable for the present purpose of MR relations. See the replaced
Figures 9 and 10 and changes in the text in section 5.1. For further detailed
applications of Landau quantization obtained here and more detailed MR relations
of white dwarfs, quantum speed etc., we prefer to refrain us for future works.
This will help to keep the main flow of the present manuscript intact, which is
the modification of Landau quantization in varying magnetic fields.
Nevertheless, we have revised the manuscript throughout for a better clarity.