Universal Diamagnetism : a Diagnostic Test for Spinless Bosonic Excitations

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Indian J. Phys. 78(8), 713-718 (2(X)4)

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Universal diamagnetism : a diagnostic test for spinless bosonic excitations

Debnarayan Jana*

Department of Physics, Univcisity College of Science and Technology, 92 A I’ C Road, Kolkata-7(K) (X)9, India

Subhankar Ghc|sh

Depanmcni of Physics, Si. Xavier’s C oIle|c, Kolkata-7(K) 010, India

and ^ Debabrata Da®

Department ol Physics. Narendrapur R K M R College, Kolkata-7(K) 103, India

E mall ; d.janat^^cucc.crnel.in

Abstract : It is well known that a classical gas of point charged particles do not show any effect of magnetic field in thermal equilibrium. Magnetism is truly a quanlum phenomenon. Wc dcmonstiate explicitly that diamagnetism is a robust property of charged spinless bosc systems. It will be shown that diamagnetism is a universal phenomenon of spinicss hose systems at finite temperature regardless of their interactions. Starting from a single particle, the discussion will go on to N particles system and finally, end on to an interacting field theory in an arbitrary spatial dimension (I We will present an exact and non-perturbative result of the theory which is an interesting feature from the field theory aspect. In this theory, a.s a byproduct, we have also been able to develop a novel rcgulari/ation scheme, particularly useful for magnetic field problems. Finally, as an application wc will take up the cases of elementary bosonic type of excitations in condensed matter physics-one is the cooper pairs in superconductivity and other IS the composite bosons in Fractional Quantum Hall Fiffect.

Keywords : Diamagnetism, spmles.s bosons, cooper pairs, cornposiic bosons.

FACS Nos. : 72.80.Ng, 72.90.^-y

L Introduction behaviour of the complicated many bose systems in IV) understand a quantum many body system, elementary terms of these excitations. excitations [1~3] play an important role. It was Debye Elementary excitations can be classified as fermionic who first introduced the concept of elementary excitations and bosonic nature. The simplest example of fermionic in lattice vibration so called the phonon theory. The excitations are excitations in metals near the Fermi surface. elementary excitations have two essential characteristics In case of strongly correlated electron systems, it is as pointed out by Anderson [4]. In quantum many body possible to think in terms of elementary excitations very Iheory, one is interested on the relative position of lowest close to the Fermi surface. These quantum excitations excited states compared to the ground state. These lower termed as Landau Quasi particles interact very weakly slates are important because they can be easily excited at through an interaction fikjc*) In this theory, one can very low temperature. It happens so that the express quite a number of physically observable quantities thermodynamic and transport behaviour at low temperature such as specific heat, magnetic susceptibility and transport iire determined solely by these lowest excited stales. properties. The effective mass of the quasi particle is Secondly, these states in a large system behave modified by this interaction. independently and hence can be treated as noninteracting The second category of elementary excitations This fact simply helps one to understand the originates from electrons are the collective one which are Corresponding Author . Present address ; Center for Condensed Matter Sciences, National Taiwan University, Taipei, Taiwan © 2004 lACS 714 Dei narayan Jana, Subhankar GhOJt and Debabrata Das bosonic in nature. It is amazing to note that though the With change of momentum variables, it is easy to notice elementary particles in the system arc fermions in nature that but the collective excitations could be bosonic. The simplest example is the quantum of lattice vibrations- phonons. !=\ In this paper, we want to investigate the nature of these elementary excitations • by noting their behaviour xexpl -^1 =V(r,) under an external magnetic field. If the excitations show diamagnetic natuie then we can infer that the nature of the excitation is. -bo.sonic. We also give two simple xexp|-^5^w (r,-r;|) examples trom condensed matter systems at low imechanics. known as Bohr-van Lcewen theorem. Later, Peicr’s [6 ] gave a very siii.ple elegant argument baseJ on partition 3. Single particle case tunction to conclude that the partition function is The physics of a system in presence of an external unaffected by the .ipplication of an external magnetic magnetic field is interesting from the theoretical point ot field. A pictorial proof [61 in terms of closed orbits of view because of the invalidity of perturbation technique the inner electrons and the orbits at the boundary of the [8]. Before we go on to N particle systems, we bricriy systems also shows the cancellation of two types of summarise the single particle picture. In this single panicle currents resulting zero magnetisation. There also exists picture, statistics do not come into play. But the effect of another simple connection [7] between Brownian motion magnetic field on the systems is two fold~namely the and magnetism where magnetic field is used as a some orbital motion of the particle and the spin part. We kno^^ kind of counter in measuring the typical closed area in a that diamagnetic effect comes solely from the orbital pan .specific Brownian motion. In this description, a typical while paramagnetism from the spin part. A simple high temperature limit gives the Bohr-van-Leewen statistical mechanical calculation shows that the free energy thcr»*cm. Below, we present a simple proof of classical for the orbital part is given by interacting N particle system in terms of partition function.

The N particle Hamiltonian is given by s i n h f ^ " (4) ejMr,) /=’orb..(5./3) = F„bi.(0) + /3- //v = Pi ■ S v ( „ 2nij \ c 1=1

( 1 ) while for that of spin part is simply f

d^riCXpi-pHf/). (2) If one looks carefully to the Hamiltonian of single Universal diamagnetism : a diagnostic test for spinless bosonic excitations 715 particle in an external uniform magnetic field, one finds It follows automatically then that that there are two competing terms given by jek (( F I y/p + v(^)| y/|: + vy(.v)| < eBtiL, e'B^ / 2 2^ H ------+------rlv + )’ ). O') 2m 2mc Hmc^ Jdjcf|(F _ ieA/c)^\- + V(.v)lt»|2 + (11) A quick look at the above equation reveals that the Now, if one chooses V' to be the ground stale wave second term is responsible for paramagnetism while the function for then the right hand side equal to E{A\ third term is for diamagnetism. Note carefully the change of sign before the two terms. In most atomic situations whereas by the variational principle for the ground state the term is unimportant because of the fact that only energiais, the left hand side is greater than £(0). Hence, ai fields of K f Tesia or higher (this field is much larger we ge| than the fields produced in the laboratory), the two terms £’((!> < E(A), (12) are comparable. The ground state of many systems has ll ma>^. be noted that the above proof fails for fermions. I no angular momentum and for them the first order The replacement of V'by |y| used above is not allowed Zeeman effect vanishes exactly. But the first order by Feimi statistics. Even spinlcss fermions do not obey correction to orbital part is always non-zero this inequality. For a counter example, we choose a and positive and almost same order of magnitude for all spherically symmetric potential V and concentrate on a alomic systems. However, for non-z.cro angular momentum particular eigenstate n = 1, Then, energies of / ^ 0 states of the ground .stale of many particle system, it is not a decrease in lowest order perturbation tlieory Ibr a suitable priori clear which term dominates over the other to choice of A. This shows that the above inequality fails decide the response of the systems to an external uniform for fermions. The above proof does not depend on the magnetic field. However, for any quantum .systems, in­ nature of applied external magnetic field and is also spile of its very low magnitude, the existence of its independent of the nature of interaction between the diamagnetism is definite resulting a universal phenomena. panicles. Later on, this proof was again confirmed via 4. N (finite) particle system the connection between Brownian motion and magnetism [7]. Diamagnetism was also shown to he an integral Simon [9] in 1876 first demonstrated through Kato’s property of hard core bosonsflO], It was shown in all the inequality that N ,spinlc.ss bosons in an external magnetic above proofs of the spinless bosons systems that the free field show diamagnetism. For the sake of completeness, energy of N particles in an external magnetic field is we reproduce briefly Simon’s arguments below for the ground state of the quantum system. The starting greater than the free energy without an external magnetic ilumiltonian is field. Thus, the free energy of such a system always increases with the magnetic field resulting the negative y /V susceptibility. However, it is not clear whether the tree + ) 1=1 energy is a monotonically increasing (unction of the magnetic field H or not.

+ £ w ( r , - r , | ) . (8) All the above systems deal with N finite paiheie i

Let ri...... be the wave function. Since | 'F\' in a continuum field theory the degrees of freedom being - (y'* we note that using 3N dimensional gradients infinite, one has to be very cardbl about the compai fson of the two energies. The energies arc divergent m natuic (|y'lF(|y'|) = (in presence or absence of magnetic field] and one lias = iRelV'nf' - ieA/c)'F]\ to develop a suitable regularization scheme in such a s |y'||(F - (

scheme! 12-14] particularly suitable for dealing this 6. Diagnostic test of two systems magnetic field problem. Wc here discuss the two systems where elementary excitations are charged and bosonic in nature. One is the S, Many particle systems-bosonic field tlieroy cooper pair in superconductivity and another is the There are several reasons for studying the scalar fields in composite bosons in Fractional Quantum Hall cffcci an external magnetic field at any finite temperature. First [16,17]. of all the results of N particle system do not always Cooper parts in superconductivity : translate in same form to a corresponding field theory We have used the concept of universal diamagnetism to [I4j. Secondly, in field theory one has to develop a compute the diamagnetic susceptibility of anisotropic [15| proper regularisation scheme to have a finite value of the as well as isotropic [12] superconductor above The observable (in this case the energy). Thirdly, to study the low temperature superconductors arc in general isotropic elementary excitations in the system in the long in nature while the high temperature superconductors are wavelength limit, it is better to consider the corresponding highly anisotropic. The relevant Landau-Ginzburgh free field theoretic version rather than the particle picture. energy functional is given by More ambitiously, to look into the various symmetries being broken, it is easier to understand from the field ■ j d “x ^|(-;7iV-iM >|^ theoretic version rather than the N particle picture. The relativistic action of complex charged scalar field (16) in the imaginary time formalism can be written as

where af =ao{T-T,y' and af =ao(T-Tj- and S = I J J + (0 0 ’ ) + (*4>) being a constant independent of temperature. The (13) expression of the susceptibility obtained in this anisotropic case [15] as Here // = 0, 1, 2 and ieA^. The imaginary time 1 T has the range from 0 to P = —^ . The vector potential ^amso i^d ~~ 1 ){ 4 ~ d )i2 kgT Xd is A and the respective magnetic field 5 = V x A, The [Md)(T - T , B ( d ) ( T - T , )-‘'^ I'’"''*'* bosonic condition is imposed via the boundary condition (17) Without the interaction, the partition Here, A{d) = *]exp(-5) (14) temperature with exponents v', and V2 respectively. The isotropic case follows from the above expression when can be evaluated exactly. With proper regularization V/, = V2 = V. Note the symmetric nature of the expression scheme, we have shown explicitly [12] that in two of the susceptibility with interchange of Vi to It dimensions the free energy of this charged scalar Helds be interesting to study the fluctuation conductivity of in presence of a uniform magnetic field is greater than superconductors in this model. that without the magnetic field. This establishes the Composite bosons in FQHE : diamagnetic character of the charged scalar fields. In an The composite bosons [18] arc the another interesting arbitrary dimension d, for an interacting case, we have charged elementary excitations in 2d electron system in a followed a non-perturbative approach [12,15] to write the strong magnetic field at low temperature. Electrons rrioving partition function as in two dimensions in such a strong magnetic field are mathematically equivalent to a collection of composite Z{A) bosons in a much weaker magnetic field. In two = ((exp(-5,„)))< (15) Z(0) dimensions a fermion can be treated as a boson to which where the action S has been divided into parts So which an odd number of flux quantum ^^0 ~ j is attached. does not contain A and containing A only. From the above eq. (15), it is easy to notice that F(B) > F(0) This magnetic field is introduced via the statistical gauge Universal diamagnetism : a diagnostic test for spinless bosonic excitations 717 field. This is statistical because of the fact that the Meissner effect [201 (PME) is found in the field-cooling statistical nature of the particles is changed on introduction mode below a field of order 1 Oe. PME has also been of this gauge field. The binding of the statistical flux to observed in Nb, YBCO and LCO. Contrary to ordinary ihc boson is achieved by the Chem-Simons interaction. Meissner effect, they seem to turn on below T, without Ihus, apart from the external magnetic field in this the need of an external magnetic field. This PME is not svstem. we have another magnetic field originated by the caused by paramagnetic or ferromagnetic impurities. This Statistical vector potential a. This statistical magnetic is because of the fact that no large PME magnetisation is field is proportional to the density of electrons and is seen to start abruptly at and no Cunc-Weiss type generated internally. Mathematically, the statistical susceptibility is seen above As has been pointed out magnetic field b can be written as earlicf that the free energy could in principle dccrea.ses within a very narrow window of magnetic field retaining jf = {7 X a = (*8) the higher value in compared to at the origin, one might where the density pi.r) = '^.5~(.r^-r) and k = 2n + 1 . sort |n 0 ** field theory pcrlurbativcly to compute the „ 0, 1, 2...... Thcsc Chem-Simon bosons dp obey the partition function. This may shed some light on the usual diamagnetic inequality [15]. The advantage of lypiciil elementary excitations in these materials. mapping is clear if one considers the magic Landau level To summarise, we have shown here that if the filing fractions where the fractional quantum Hall effects elementary excitations of a system arc bosonic in nature are observed. At all these filling factors, the Chem- then Iheir response towards an external magnetic field is Simons gauge field cancels on an average the external diamagnetic. Wc have given here two examples irom applied magnetic field fi. Thus, the composite bosons in strongly cofrclaled electron systems to justify the above this situation experience a net zero magnetic field. As statement. one changes the magnetic field from the value corresponding to magic filling factors (where composite Acknowledgments bosons condensate), the condensed state try to repel the This work is supported by the University Grants magnetic field so that the condensations remain. 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