The problem of diamagnetism, solved by Landau, continues to pose fascinating
issues which have relevance even today. These issues relate to inherent quantum
nature of the problem, the role of boundary and dissipation, the meaning of
thermodynamic limits, and above all, the quantum-classical crossover occasioned
by environment-induced decoherence. The Landau Diamagnetism provides a unique
paradigm for discussing these issues, the significance of which are
far-reaching. Our central result is a remarkable one as it connects the mean
orbital magnetic moment, a thermodynamic property, with the electrical
resistivity, which characterizes transport properties of materials.Comment: 4 pages, 1 figur

Dattagupta, S.

Jayannavar, A. M.

Kumar, N.

English

arXiv

The problem of diamagnetism, solved by Landau, continues to pose fascinating issues which have relevance even today. These issues relate to inherent quantum nature of the problem, the role of boundary and dissipation, the meaning of thermodynamic limits, and above all, the quantum-classical crossover occasioned by environment-induced decoherence. The Landau diamagnetism provides a unique paradigm for discussing these issues, the significance of which is farreaching. Our central result connects the mean orbital magnetic moment, a thermodynamic property, with the electrical resistivity, which characterizes transport properties of material

Dattagupta, Sushanta

Jayannavar, Arun M.

Kumar, Narendra

Publications of the IAS Fellows

RESEARCH COMMUNICATIONS 
CURRENT SCIENCE, VOL. 80, NO. 7, 10 APRIL 2001 861
Landau diamagnetism revisited 
Sushanta Dattagupta†,**, Arun M. Jayannavar‡ and 
Narendra Kumar# 
S.N. Bose National Centre for Basic Sciences, Block JD, Sector III, 
Salt Lake, Kolkata 700 098, India 
‡Institute of Physics, Sachivalaya Marg, Bhubaneswar 751 005, India 
#Raman Research Institute, Bangalore 560 080, India 
The problem of diamagnetism, solved by Landau, con-
tinues to pose fascinating issues which have relevance 
even today. These issues relate to inherent quantum 
nature of the problem, the role of boundary and dissi-
pation, the meaning of thermodynamic limits, and 
above all, the quantum–classical crossover occasioned 
by environment-induced decoherence. The Landau 
diamagnetism provides a unique paradigm for discuss-
ing these issues, the significance of which is far-
reaching. Our central result connects the mean orbital 
magnetic moment, a thermodynamic property, with 
the electrical resistivity, which characterizes transport 
properties of material. 
IN this communication, we wish to draw the attention of 
the reader to certain enigmatic issues concerning dia-
magnetism. Indeed, diamagnetism can be used as a 
prototype phenomenon to illustrate the essential role of 
quantum mechanics, surface–boundary, dissipation and 
nonequilibrium statistical mechanics itself. 
  Diamagnetism is a material property that characterizes 
the response of an ensemble of charged particles (more 
specifically, electrons) to an applied magnetic field. The 
magnetic field causes cyclotron motion of each particle, 
thereby creating an orbital magnetic moment, governed by 
Faraday–Lenz’s law. Thus the system exhibits a negative 
magnetic susceptibility, the hallmark of diamagnetism. 
What is however remarkable is that diamagnetism is 
calculated to be zero within the framework of classical 
Gibbsian statistical mechanics. This result goes under the 
celebrated Bohr–Van Leeuwen’s (BV) theorem1. What is 
even more remarkable is that the boundary of the 
enclosure, as also any internal boundary, plays a crucial 
role; the bulk contribution of the diamagnetic moment 
exactly cancels the contribution arising from the so-called 
‘skipping orbits’ of those electrons which hit the 
boundary and get multiply reflected to constitute what is 
called an ‘edge current’2. 
 Thus it was a great triumph of quantum mechanics 
when Landau showed in 1930 that the discreteness of  
energy lev ls and the consequent degeneracy of each level 
led in a natural way to diamagnetic susceptibility3.  
Landau’s calculation also suggests that in quantum  
mechanics the bulk and the surface contributi s are dif-
ferent, though opposite in sign, and therefore, the cancel-
lation of the two terms is incomplete, unlike in the 
classical case. This fascinating result which brings the 
boundary of a container, normally passive in determining 
thermodynamic pro erties, to essential reckoning, led 
Peierls to term diamagnetism as one of the surprises in 
theoretical physics4. Landau’s pathbreaking result also 
demonstrated that the calculation of diamagnetic suscep-
tibility did indeed require an explicit quantum treatment. 
 Turning to the classical domain, two of the present  
authors had worried, some years ago, about the issue: 
does the BV theorem survive dissipation5? Th s is a natu-
ral question to ask as dissipation is a ubiquitous property 
of materials, and in the present context, can be viewed to 
occur as a result of inelastic scattering. Now, dissipation 
or damping necessarily requires a time-dependent ana-
lysis. Naturally, therefore, the calculation was set in the 
framework of the Langevin/Fokker–Planck equations, just 
as in the case of Brownian motion, with the Lorentz-force 
appearing as a systematic drift. Because the calculational 
approach had built-in, at the outset, the fluctuation–
dissipation theorem, the diamagnetic moment was extrac-
ted as a stationary, (i.e. asymptotic, time going to infinity) 
property. Interestingly, surprise was in store, as the dia-
magnetic moment was found to be non-zero a d depen-
dent on the coefficient of friction, seemingly at odds with 
the BV theorem. This re ult however, turned out to be  
one red herring, as it were, when the boundary was treated 
carefully. Thus we discovered the curious result: when the 
stationary limit was taken first and the thermodynamic 
(i.e. volume going to infinity) limit taken next, the BV 
theorem was restored, while it was violated when the two 
limits were interchanged – again bringing in the issue of 
the boundary in an explicit manner. 
 We stated earlier that diamagnetism is an essential 
quantum mechanical property. Thus it was natural to  
extend the Jayannavar–Kumar analysis to the quantum 
domain6. The resultant treatment, based on what is being 
branded now as dissipative quantum mechanics, again 
brings to the fore certain remarkable phenomena which 
we wish to focus onto. Four distinct issues were tackled in 
our earlier work6: (a) approach to equilibrium of a quan-
tum dissipative system, the analysis of which brings out 
the subtle role of boundary electrons, a key point, as men-
tioned before, in the whole business of orbital magnetism; 
(b) the effect of dissipation on Landau diamagnetism, an 
equilibrium property; (c) quantum–classical crossover as 
the system transits from the Landau to the Bohr–Van 
Leeuwen regimes as a function of damping; and (d) the 
combined effect of dissipation and confinement on  
Landau diamagnetism, the latter arising from coherent 
cyclotron motion of the electrons. The item (d) is particu-
larly relevant in the context of intrinsic decoherence in 
mesoscopic structures in view of heat bath-induced influ-
ence7. Further, items (b)– d , put together, had prompted 
us to ask question on not whether the BV theorem sur-**For correspondence. (e-mail: sdgupta@boson.bose.res.in) 
RESEARCH COM UNICATIONS
RESEARCH COMMUNICATIONS 
CURRENT SCIENCE, VOL. 80, NO. 7, 10 APRIL 2001 862
vives dissipation, but whether the Landau diamagnetism 
itself survives dissipation? 
 The central result for the diamagnetic moment M per 
particle of a non-degenerate gas of free electrons, in co -
tact with an Ohmic bath, derived in Dattagupta and Singh6 
can be written as 
 
å
¥
= ++
-=
1
22
2 ,
)(
12
n cn
c
rB
KT
M
wn
w  (1) 
where 
,
2
KTnn
h
p
n =  (2) 
 
and 
,
mc
eB
c =w  (3) 
B being the magnetic field. We would like to remind the 
reader, in the context of the point (a) in the paragraph 
above, that the result, eq. (1), was extracted from the solu-
tion of the appropriate quantum Langevin equations, by 
taking the t ® ¥ limit first and then letting the boundary  
of the system go to infinity. The complete expressions, 
including the time-d pendence and the dependence on the 
parameters of the confining boundary, from which eq. (1) 
has been deduced, are available in the literature6. I  is also 
evident that in the limit of zero damping (r = 0), eq. (1) 
reduces to the Landau answer: 
 
.
2
cot
2
2
o
ú
û
ù
ê
ë
é
÷
ø
ö
ç
è
æ-=
KT
h
KT
mc
e
M c
c
w
w
h
h
h
 (4) 
 
 It may be recalled that eq. (4) is usually derived from 
the partition function of a gas of free electrons in the 
presence of a magnetic field2, whereas eq. (1) is extracted 
from a non-equilibrium method6. The calculation itself 
demonstrates inter alia that the path to equilibrium from a 
non-equilibrium state is not unique – it is important as to 
whether the system is ‘allowed to equilibrate’ in the pres-
ence of a boundary or in free space! 
 Another significant point concerning eq. (1), which 
provided one of the motivations for us to write this note, 
is the explicit presence of the friction coefficient r in an 
equilibrium function M. Recall that r had its origin in the 
coupling constant, characterizing the interaction between 
the gas of electrons and the bath. Normally, such a con-
stant disappears from equilibrium answers, which are  
extracted from irreversible statistical mechanical methods, 
wherein the system–bath interaction is treated as a ‘weak 
coupling’. What is crucial for dissipative diamagnetism  
is that the system–bath interaction has to be treated  
exactly – there is no clear-cut separation between what is 
a system and what is a bath – ot  are inexorably linked 
into one many-body system! 
 A related point is that the same coefficient r det rmines 
what is however a transport property, viz. the Drude resis-
ivity: 
,
2e
mr
r
=v  (5) 
r being the change carrier number density. Eliminating r 
in favour or v and introducing the ‘Hall resistivity’ R as 
,
ec
B
R
r
=  (6) 
eq. (1) can be suggestiv ly rewritten as 
å
¥
= ++
=-
1
2
,
)(1
11
n n
M
e
mc
vmnh
 (7) 
where 
,
2KT
cwn
h
=  (8) 
,
R
v
v =  
and 
.
n
p
m
n
n =  (9) 
Equation (7) is a novel result: transport characterized 
by the resistivity v  determining an equilibrium property 
of orbital moment M in an efficient manner, because resis-
tance, in some sense, incorporates all relevant material 
properties. From another point of view, and especially in 
the context of contemporary relevant mesoscopic struc-
tures, larger the resistance, higher is the level of incohe-
rence. Thus, while Landau diamagnetism in eq. (4) is a 
result of coherent property of a quantum system, dissipa-
tive diamagnetism, captured by eq. (7), is an expression of 
intrinsic incoherence in a macroscopic quantum system. 
To illustrate this point, we plot in Figure 1, the magnetic 
oment versus the resistivity: the increase in resistivity 
with the concomitant enhancement of incoherence leads to 
a vanishing magnitude of the orbital moment, as though 
Bohr and Van Leeuwen are resurrected! 
 We may remark in passing that the Drude formula for 
resistivity, eq. (5), is robust, almost oblivious to whether 
originates from scattering with phonons or defects or 
other electrons8. Thus eq. (5) remains valid in general, 
except that the interpretation and calculation of r become
an increasingly complex task, when the effects of disorder 
and electron correlations are to be considered. Therefore, 
our relation between a thermal equilibrium property, viz. 
M nd a transport property, viz. ,v  though appears facile, 
as it is based on perhaps the simplest of models, subsumes 
 deep connection between orbital magnetism and dissipa-
tion – it transcends detailed issues of disorder and con-
comitant localization effects, as well as strong Coulomb 
correlations between electrons. 
RESEARCH COMMUNICATIONS 
CURRENT SCIENCE, VOL. 80, NO. 7, 10 APRIL 2001 863
 Our final remark is in the context of quantum–cl ssical 
crossover due to environment-induced decoherence, 
epitomized by the expression given in eq. (7). Remember 
that there are two distinct quantum effects which, in the 
model of an electron gas, can be described thus: (1) one is 
due to the fact that the position and momentum of a  
electron cannot be determined simultaneously, which is 
why a continuum of states splits into discrete Landau 
levels; we call this the phase space factor, and (2) the 
other is due to the quantum statistics of indistinguishable 
particles, in this case the Fermi–Dirac one. As long as the 
de Broglie wavelength is smaller than the average inter-
electron distance, the effect of statistics or the Pauli 
exclusion principle can be neglected, which is indeed 
what was done in deriving eq. (4) for a non-degenerate 
gas. Now, in examining the issue of quantum–classical 
crossover, it is interesting to introspect on what gets 
classicalized by environment-induced decoherence first: 
the phase space factor or the quantum statistics? Inasmuch 
as, in the calculation enumerated above, the electron gas 
was treated as non-degenerate, i.e. the thermal de Broglie 
wavelength of the electron was taken to be much less than 
the mean electron–electron spacing, the quantum statistics 
was already rendered ineffective (or classical). It ws, 
therefore, only the phase space factor that got obfuscated 
by the increasing decoherence effects. On the other hand, 
it would be extremely interesting to look for another 
quantum phenomenon, which is intrinsically due to 
statistics, such as the Bose–Einstein condensation, to 
enquire what dissipation does to it! It may well be that it 
is the phase-breaking length rather than the de Broglie 
length that must exceed the inter-particle spacing as a pre-
condition for the quantum statistics to be effective. 
 Summarizing, diamagnetism seems to be a unique  
material property which does not exist in classical  
mechanic , but lives only in the quantum domain. Since 
quantum mechanics has to do with the phase of the wave-
function, diamagnetism can be viewed to arise from quan-
tum coherence. On the other hand, all materials are 
inherently dissipative due to scattering of electrons off 
phonons, impurities, other electrons, etc. Because dissipa-
tion leads to incoherence, making a quantal system look 
s emingly classical, it is nteresting to study the influence 
of dissipation on diamagnetism. Such an analysis, summa-
rized in this article, elucidates many subtle issues of con-
temporary condensed matter physics. 
 
1. Bohr, N., Dissertation, Copenhagen, 1911; Van Leeuwen, J. H.,  
J.  Phys. (Paris), 1921, 2, 361. 
2. Van Vleck, J. H., The Theory of Electrical and Magnetic Suscepti-
bilities, Oxford University Press, London, 1932. 
3. Landau, L., Z. Phys., 1930, 64, 629. 
4. Peierls, R., Surprises in Theoretical Physics, Princeton University 
Press, Pri ceton, 1979. 
5. Jayannavar, A. and Kumar, N., J. Phys. A., 1981, 14, 1399. 
6. Dattagupta, S. and Singh, J., Phys. Rev. Lett., 1997, 79, 961. 
7. Mohanty, P., Jariwala, E. M. Q. and Webb, R. A., Phys. Rev. Lett., 
1997, 78, 3366. 
8. Ashcroft, N. and Mermin, D., Solid State Physics, Holt, Rinehart 
and Winston, 1976. 
 
ACKNOWLEDGEMENT. We thank Kamal Saha for help in genera-
ting Figure 1. 
 
Rece ved 5 February 2001; revised accepted 13 March 2001 
Figure 1. Scaled diamagnetic moment          plotted against  caled resistivity     for 
different values of the scaled cyclotron freque cy v. 
 
 
MemcY )/( h-= ,vX =


Landau diamagnetism revisited

The problem of diamagnetism, solved by Landau, continues to pose fascinating issues which have relevance even today. These issues relate to inherent quantum nature of the problem, the role of boundary and dissipation, the meaning of thermodynamic limits, and above all, the quantum–classical crossover occasioned by environment-induced decoherence. The Landau diamagnetism provides a unique paradigm for discussing these issues, the significance of which is far-reaching. Our central result connects the mean orbital magnetic moment, a thermodynamic property, with the electrical resistivity, which characterizes transport properties of material

Landau diamagnetism revisited

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