Dynamics of Uniform Quantum Gases, II: Magnetic Susceptibility

where M, H, ǫ 0 , c, ω p , and χC ⊥ (q) denote, respectively, magnetization, magnetic field, permittivity of vacuum, speed of light, plasma frequency, and static transverse chargecurrent susceptibility. For the mono-plasma, one has ω 2 p =(Ze) 2 n/(ǫ 0 m) with mass m and average number density n=N /V of particles.

Choosing a cartesian coordinate system {e x , e y , e z } such that q=qe z , the static susceptibility of the transverse charge-current density is given by χC

where ĴC q, x denotes the operator of transverse charge-current density. In the last line, χC ⊥ (q) has been approximated in terms of the transverse charge-current susceptibility χC ⊥ (q) (0) of a fictitious similar system in which interactions between particles have been switched off [3]. This generalized random phase approximation (GRPA) correctly accounts for the transverse electromagnetic shielding effects in systems of particles which carry a charge and/or a spin. The conventional RPA expression for the transverse particle-current susceptibility of a system of neutral and spinless particles, Eq.(45) of Ref. [1], is easily recovered from Eq.(3) (taking into account Eq.( 4) below) for s=0 and Z → 0.

There are three contributions to the operator of charge-current density [4] resulting in the transverse component ĴC q, x = (Ze) Ĵq, xiqγ Ŝq,y -

with Ĵq denoting the number-current density, Ŝq = kσσ ′ ŝσσ ′ a † kσ a k+qσ ′ the spin density, γ=gµ B /h the gyromagnetic ratio with g as the g-factor and µ B =eh/(2m e ) the Bohr magneton, Nq the particle-number density and its fluctuation δ Nq = Nq -Nq , and A q = d 3 r e -iq•r A(r) the vector potential associated with the moving charges. However, in a uniform system only the first two terms on the right-hand side will have main contributions to the transverse charge-current susceptibility in Eq.( 2), because the contribution due to the vector potential A k produced by the moving charges will be negligibly small in a non-relativistic gas. Moreover, cross correlations between Ĵq, x and Ŝq, y will also vanish due to rotational symmetry. Therefore, we expect only two contributions to the static charge-current susceptibility: an ‘orbital contribution’ induced by the macroscopic current density due to moving charges and a ‘spin contribution’ induced by the magnetization-current density due to magnetic moments associated with particle’s spin.

Consequently, we express the transverse charge-current susceptibility of the noninteracting system, which is needed in Eq.( 3), as

Here χ⊥ (q) and χS yy (q) denote the ideal-gas susceptibilities of the particle-current density and the spin density, respectively. The latter is easily shown to be proportional to the number-density susceptibility χ(q) of an ideal quantum gas. The expressions for ideal-gas static susceptibilities χ⊥ (q) and χ(q) have been obtained in Paper I. For notational simplicity, superscripts (0) on these quantities are being omitted now onwards.

The expression for the magnetic susceptibility within GRPA is finally obtained -using Eqs. ( 1), ( 3) and ( 5) -in the form

where

It is to be noted that Ξ(T ) may be interpreted as a static susceptibility, too, which describes the linear response of the magnetization M to the magnetic induction field B. From Eqs.( 1) and ( 6) in conjunction with the general relation B=µ 0 (1 + χ m )H, where µ 0 =1/(ǫ 0 c 2 ) is the free-space permeability, one finds

In order to evaluate Ξ(T ), we read for the small-q region from Eqs. (34) and (36) of Paper I:

and

Here β=1/(k B T ) and λ= exp[βµ η (n, T )] denotes the fugacity with chemical potential µ η (n, T ), where η= -1, 0, +1 for the gas obeying FD, MB, BE statistics, respectively. C 0 = lim N →∞ N 0 /N, the fraction of particles occupying the zero-momentum singleparticle state, is given by C 0 =δ η, 1 Θ (T BEC -T ) 1 -(T /T BEC ) 3/2 with Θ(x) denoting the unit-step function and T BEC the BEC critical temperature. Also, ζ ν (x)= ∞ i=1 x i /i ν denotes the polylogarithm.

Inserting Eqs.( 9) and (10) into Eq.( 7), one gets

with the “Curie constant”

and the parameter

determining whether the gas will show paramagnetic (ξ > 0) or diamagnetic (ξ < 0) behaviour.

Thus for neutral particles of nonzero spin (Z=0, s ≥ 1/2 =⇒ ξ > 0, paramagnetic) one finds for the “Curie constant” the well known expression

which does not depend on particle’s mass m. The Curie law

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