PHYSICA ELSEVIER Physica A 224 (1996) 639 668

Green's function and position correlation function for a charged oscillator in a heat bath and a magnetic field

X.L. Li, R.F. O ' C o n n e l l *

Department of Physics and Astronomy, Louisiana State UniversiW, Baton Rouge, Louisiana 70803-4001, USA

Received 9 June 1995

Abstract

We formulate, in the framework of the generalized quantum Langevin equation approach, the retarded Green's functions and the symmetrized position correlation functions for the motion of a charged quantum-mechanical particle in a spatial harmonic potential, coupled linearly to a passive heat bath, and subject to a constant homogeneous magnetic field. General conclusions can then be reached by using only those properties of the generalized susceptibility tensor imposed by fundamental physical principles. Explicit calculations are made for the Ohmic heat bath. We next investigate the Brownian motion of a charged particle in an external magnetic field. We continue by proving general relations between the retarded Green's func- tions and displacement correlation functions in the limit of long times at both absolute zero and nonzero temperatures, and further evaluate the long-time asymptotic behaviors of the two functions, for both the Ohmic and a rather general class of heat baths discussed extensively in the literature.

1. Introduction

Dissipative systems in the presence of an external magnetic field is an important but difficult problem in solid state physics. Some of the early research topics include the

influence of collisions on the magnetic susceptibility of metals [1, 2], quantum theory

of transport for an electron gas in a magnetic field [3], magneto resistance on the

Fermi surface [4, 5], electronic conduction in a strong magnetic field [6, 7], nuclear

magnetic resonance (NMR) [8], relaxation and resonance of spins in zero or low external magnetic fields [9, 10], electron-hole pair production and recombination in semiconductors [11], diffusion of non degenerate charge carriers in a semiconductor [12], and magnetopolaron (i.e., the Fr6hlich polaron in the presence of an external magnetic field) [13]. The techniques employed in these studies are predominantly the

* Corresponding author.

0378-4371/96/$15.00 (C' 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 3 7 1 ( 9 5 ) 0 0 2 9 5 - 2

640 X..L. Li, R.F. O'Connell/ Physica A 224 (1996) 639-668

phase-space Fokker-Planck equation for the Wigner function, with the influence of the ambient medium being treated only phenomenologically 1-14].

The proper incorporation of dissipation into macroscopic systems, especially in the quantum domain, is by considering the coupled system of the particle involved and its environment, for which detailed microscopic modeling is necessary. Strong impetus to this field was initiated by the pioneering work of Caldeira and Leggett on dissipative quantum tunneling at zero temperature 1,15]. Since then, the Caldeira-Leggett (C-L) model has been applied to a variety of physical systems to investigate, among others, the asymptotic low temperature properties, which show anomalous behavior [-16].

Meanwhile, the subject of dissipation in a magnetic field has also received renewed interest over the last decade mainly due to the discovery of highly nonclassical transport of a degenerate Fermi gas in the presence of strong disorder in the quantized Hall effect (QHE) [17] and the temperature-dependent normal-state Hall effect in high-temperature superconductors 1-18]. To understand corrections to the classical form of magnetic properties in such systems, Hong and Wheatley have presented a magneto transport theory for a charged particle executing quantum diffusion in a two-dimensional translationally invariant system subject to an external magnetic field, using a somewhat complicated method of diagonalizing the underlying Hamil- tonian of the coupled system fi la Caldeira Leggett 1,19].

In this paper, we shall use the much simpler and more transparent approach of the generalized Langevin equation (GLE) based on the uncharged independent-oscillator (IO) model of the heat bath [20], which is equivalent to the translationally invariant version of the C-L model required for a free Brownian particle [21]. The problem of a charged quantum particle moving in a scalar potential V(r), coupled linearly to a passive heat bath, and in the presence of a static external magnetic field B, has recently been formulated based on the IO model 1,22]. The formulation fully incorpor- ates the effects of Landau orbit quantization and the associated Landau level struc- ture, thus rendering it unnecessary to make any semiclassical approximation. The linear coupling between particle and heat bath adopted in the IO model allows the magnetic field to be taken into account nonperturbatively. The ensuing GLE for an isotropic spatial (three-dimensional) harmonic potential as well as a uniform magnetic field has been solved exactly by means of the Fourier transformation, enabling us to obtain integral expressions for many physical quantities such as susceptibilities, position correlation functions, and free energies 1-23]. Here we shall expand that work and focus our attention on two important quantities frequently employed in the study of condensed matter: the retarded Green's functions and the symmetrized position correlation functions. They play prominent roles in the theoretical interpretation of experiments because of their direct relationship with measurable physical quantities and are the subject of much interest [24, 25].

The rest of this paper is organized as follows. In Section 2 we first introduce the general formalism and notation used in this paper. In particular, we establish several useful properties about the generalized susceptibility tensor ~p~(~o) obtained from the GLE for an isotropic harmonic oscillator. We then define the retarded Green's

X.L. Li, R.F. O'Connell / Physica A 224 (1996) 639-668 641

functions as the Fourier transform of the generalized susceptibility tensor and relate them to the nonequal-time commutators of position operators. In Section 3 we express, using the fluctuation-dissipation theorem (FD), the symmetrized position correlation functions in terms of the generalized susceptibility tensor and prove, based on the properties of ~f,~(co) just outlined in Section 2, two general theorems concerning the position autocorrelation functions (dispersions) of motion perpendicular to the external magnetic field that are true for any physical heat baths. In Section 4 we calculate explicitly the retarded Green's functions and the symmetrized position correlation functions for a harmonic oscillator in the Ohmic heat bath in both classical and quantum limits.

In Section 5 we extend the investigation to the Brownian motion of a charged particle in an external magnetic field. To extract finite results, we introduce the displacement correlation functions, which are related to the symmetrized position correlation functions but are more appropriate for studying the Brownian motion. We next give a formula for the self-diffusion constant and derive, in the limit of long times at both absolute zero (the quantum regime) and non-zero temperatures (the classical regime), two general relations between the retarded Green's functions and the displacement correlation functions, the classical version of which is a general- ization of the Einstein relation and can thus be cast into a form of the Green-Kubo formulae connecting transport coefficients with integrals of appropriate correlation functions. The formulae so developed are subsequently applied to analyze the long- time asymptotic expansion of the displacement correlation functions from that of the retarded Green's functions, for the Ohmic heat bath and a rather general class of frequency-dependent heat baths that correspond to many realistic microscopic models and have therefore been studied extensively, particularly in the context of dissipative quantum coherence [26]. Finally, in Section 6 we summarize our results and compare them with those without a magnetic field and present our conclusions.

2. The generalized susceptibility

The quantum Langevin equation for a particle of mass m in a potential V(r) and subject to a static external magnetic field B takes the form [22]

m~" + i dt'tt(t - t')i'(t') + VV(r) - e(~:XB)c =F( t ) , (2.1) - - c C

where the dot denotes differentiation with respect to t. The influence of the external magnetic field B is solely represented by the quantum version of the Lorentz force team, with both the Gaussian random operator-force F(t) and the memory function p(t)of the heat bath unchanged by the magnetic field.

642 X.L. LL R.F. O'Connell / Physica A 224 (1996) 639-668

For a spatial harmonic potential V(r) = ½ Kr 2 and a uniform magnetic field B, the resulting linear operator equation can be exactly solved by the Fourier transforma- tion method 1-23]:

~p(o)) = %~(co)P.(~o), (2.2)

~ ( ~ ) - [ D ( ~ ) - ' ] ; , ~

= [ 2 z 6 p ~ - ( ~ o ~ ) e B o B ~ - G ~ , B , 2 i m ~ ] / d e t D ( m ) , (2.3,

where

det D(co) = 2 [22 - (oa(e/c))ZB2], (2.4)

2(09) = - - m ~ 2 + K - - i o~ f i ( fo ) , (2.5)

and where 6o, is the Kronecker delta function and G-, the Levi-Civita symbol. Here we have used tensor notation and shall adopt the Einstein summation convention for repeated indices throughout this paper unless otherwise indicated. The Fourier transform is denoted by a tilde, e.g.,

Ft(m) = i" dt ei°~' #(t) , (2.6)

0

where, by convention, the memory function/~(t) vanishes for negative times. The c-number generalized susceptibility tensor %~(~o) uniquely determines the

dynamics of linear systems. It has the following two useful identities (see Appendix A):

~,~(o~)- * c~,,,,(og) = 2ioG~,(m)~*,.(m)mRe~(oa), (2.7)

~,~(co) - ~v(m) = 2i~,.~(~o)0~*~(co)mRefi(~o). (2.8)

As with the Fourier transform of the memory function fi(og) [20], ~p~(m) obeys several important properties required by general physical principles. First of all, ~p~(~o) satisfies the reality condition [23]

~}~(co) = ~p,( - ~o), (2.9)

which reflects the fact that r is a Hermitian operator. Thus the real and imaginary parts of %~(~o) are even and odd functions of m, respectively. Secondly, no element of the matrix ~p~(¢o) has poles in the upper half-plane (UHP) (see Appendix B). Further- more, for the three diagonal elements ~pp(~o) (p = 1, 2, 3),

Im ~pp(Co) > 0, for o~ > 0, (2.10)

thereby - iCO~pp(~o), p = 1,2, 3, are real positive functions (see Appendix C).

X..L. Li, R.F. O'Connell / Physica A 224 (1996) 639 668 643

The Fourier transform of ~((o) is related to the retarded Green's function G,,(t):

oc, 1; G,~(t) = ~ d~o e-i°":%(~o).

- o c

(2.11)

The causal Green's functions defined above are very useful for making calculations based on the equations of motion for the operators of interest [27]. They are to be distinguished from another type of Green's function commonly used in statistical physics called a time-ordered Green's function, suitable for the development of diagrammatic perturbation expansions [24].

Inverting the Fourier transform in (2.2) with the aid of (2.11) gives

r,(t) = i dr' Go~(t - t ' ) [L(t ' ) + F~(t ')].

- c o

(2.12)

Since ~p~(co) is analytic in the UHP, we see readily from (2.11) that

G,~(t) = 0, for t ~< 0. (2.13)

This causality property for the retarded Green's function ensures that a response of the system depends only upon the past perturbation.

The retarded Green's function is closely connected with the commutator of position operators. To this end, we need the formula for the commutator between the operator random forces [22]:

o o

[F,,( t) ,F.(t ' )] = 6,,~ 2 fde)Re[f i (e) + iO+)]hogsin[e)(t - t ' ) ] .

0

(2.14)

Thereupon, we derive the nonequal-time commutator of rp(t) and G(t') from (2.12):

1 [r,(t) (t')] , r o - ~ _

7 r

i &o ~,,,(o~)o~*,((o)Re fi((o)hcoe i,,,i,-,'l

- - c l o

c o

h I = ~ dooe-i"m-t ' )[%,((o) - 0~*,(e))] ,

- o o

(2.15)

where we have used the inverse Fourier transform of (2.11) and the second equality follows from (2.8).

Applying (2.9) and (2.11) in (2.15) results in

[r,,(t), G(t')] = (h/i)[Gt,~(t - t') -- G~p(t' -- t )] , (2.16)

644 X.L. Li, R.F. 0 'Connell / Physica A 224 (1996) 639-668

which may also be written, by (2.12), as

Gp~(t) = (i/h) O(t) [rp(t), r~(0)], (2.17)

where O(t) is the Heaviside unit step function. Eqs. (2.16) ad (2.17) are familiar in connection with the linear response theory and the fluctuation-dissipation theorem [28]. Note the commutators appearing here are all c-numbers, which is a consequence of the lineality of the system involved. In accordance, the Green's functions are temperature-independent.

3. The Position correlation function

The symmetrized position correlation functions may be obtained via the fluctu- ation-dissipation theorem [29, 30]:

_ 1 t t ~p~(t -- t') = ~ (rp(t)r~(t ) + r,(t )rp(t)>

1 i do)e-i°m-t ')(h/2i)c°th(ho)/2kT) 2~

-oo

x [ctp~(o) + i0 +) - ~*(o) + i0+)]

co

h f do) Im [c~ (o) + i0 ÷)] coth(ho)/2kT) cos [o)(t - t')] ~ - - t7

0

oo

h fdo)Re[ o(o) + io+)] coth(ho)/2kT)sin[o)(t - t ' ) ] , 7Z

0

where

(~pa (O) ) ~ 1 [ S p a ( O ) ) -I- 0 ~ a p ( O ) ) ] ~--- [ /~2 (~f,a - - (o)(e/c))2 BpB~]/det D(o)),

and

(3.1)

(3.2)

ct;~(o)) = ½ [ctp~(o)) -- c%,(~)3 = ( - ep,,B,2io)(e/c))/det D(o)) (3.3)

are the symmetric and antisymmetric parts of c%jo)), respectively, and k in front of temperature T denotes the Boltzmann constant, and where the last equality in (3.1) is obtained with use of the reality condition (2.9) on ~ j o ) ) and ~((~). We note here that c~Jo)) and ct~,o(o)) as defined in (3.2) and (3.3), respectively, possess the same properties as those for ~pjo)), namely (2.9), (2.10), and (2.11), which can easily be verified. For definiteness, we shall choose the direction of the magnetic field as the z direction in calculations throughout this paper. Then, from (2.3), the only non-zero elements of ~p~(o)) are cq 1,~22, ~33, ~12, and ~21, which, due to the cylindrical symmetry of the

X.L. LL R.F. O'Connell / Physica A 224 (1996) 639-668 645

system, are related to each other by

~11 (co) = ~22 (co) = 22/det O(co), (3.4)

~12(co) = - c~2a (co) = - ico(e/c)B2/det D(o)). (3.5)

There follows from (2.11), (3.1), (3.4), and (3.5) that the cross retarded Green's function G~z (t) and the position cross-correlation function I~12 (/~) are both identically zero if no magnetic field is present, as expected.

The position autocorrelation functions (also called dispersions) of the motions perpendicular and parallel to the magnetic field B are given by the equal-time values of ~9p~(t) in (3.1):

o o

(X 2) = (y2) = -nh f dcolmcql(co)coth(hco/2kT), 0

(3.6)

o o

<z2> = h fd ) h(h /2kT) - gO Im ~33(g0 cot co , 7~

0

(3.7)

and it is easy to verify that (Z 2 ) may simply be obtained by setting B to zero in (3.6) for (X 2) or (y2), which is a consequence of the fact that magnetic field does not affect motions parallel to it.

The factor coth(ho~/2kT) in (3.6) is a monotonically increasing function of temper- ature T, so are (x 2) and ( y 2 ) as deduced from (3.6) and (2.10), i.e.,

(O/OT)(x 2) = (O/OT)(y 2) > O. (3.8)

The same holds for (z 2) as in the one-dimensional case [31]. The dispersions (x 2) or ( y 2 ) may also be expressed in a series form by means of

the theorem of residues from the theory of functions of a complex variable. First,not- ing that the integrand in (3.6) is an even function of o because of the reality condition (2.9) on ~11(c9), (3.6) can be rewritten as

ac~

(x2) =2-~i~i dc°~ll(c°)c°th(hc°/2kT)" (3.9)

- o o

We may now close the contour in the UHP, where only the factor coth(hoJ/2kT) in the integrand in (3.9) contributes simple poles at ~o = iv, (n = 1,2 . . . . ). Here v, = 2nkTn/h are the usual Matsubara frequencies [32]. The summation over the residues yields

(x 2) = ~ - \~2o 2 + 2 2., (3.10) n=l /~2(Vn) nt- (Vnf'Oc)2J '

646 X.L. Li, R.F. 0 'Connell I Physica A 224 (1996) 639-668

where coc = e B / m c is the cyclotron frequency and COo 2 = K / m is the oscillator fre- quency, and where

A 2 + v. Cv.) ~.(v,) -- 2( iv , ) /m = v, + ( 3 . 1 1 )

~(v,) - f i ( i v , ) /m . (3.12)

Since f i ( i z ) > 0 for z > 0 [20], it follows from (3.11) and (3.12) that ,~(v,)> 0 (n = 1,2, ... ). Therefore, ( x z ) decreases monotomically with increasing magnetic field strength:

( 8 / 8 B ) ( x 2) < O. (3.13)

We conclude this section by emphasizing that the results (3.8) and (3.13) hold for any strength of magnetic field and any type of heat bath restricted only by general physical principles. Eq. (3.13) is also closely related to the fact that the dissipative system of a charged quantum oscillator in an external magnetic field is generally diamagnetic (see Appendix D).

4. Charged oscillator in an Ohmic heat bath and a magnetic field

For a strict Ohmic heat bath, the memory function fi(co) = m7 is frequency-indepen- dent. Bearing in mind that figo) is a property of heat bath only, the frictional coefficient 7 thus defined is actually inversely proportional to the particle mass m. The retarded Green's functions defined in (2.11), with the aid of (2.3)-(2.5), may now be evaluated by the method of contour integration:

o o

1 f dco e- io;t G11 (t) - - 4rim

- - o 0

( , 1 ) + co 2 oJ 2 + iToo ~0~0 x ~0 2 _ ~° 2 + iy~o + ~ 0 ~ -- --

- 2mbl O ( t ) [ e x p ( - ~ 2 4 t )

x \ q - ~ - - s i n ( f 2 2 t ) + cos(~22t) + exp( - - Q3t) (4 .1 )

2 ac°s'O'")l

X.L. Li, R.F. O'Connell / Physica A 224 (1996) 639 668 647

1 G12(t) -- 4him i d¢o e - io, t

-oo

1 X

~o 2 -- Oo 2 + i7~o + ¢o/o

1 ) ~o z - ~o 2 + i ? o ~ - ~ o , , ~

- 2mbl O(t)[exp(-f2,t)f\X/~-~ + acos ( f22 t )

- X / ~ - ~ s in(f2: t ) ) - exp( - f23 t )

× + , {4.2)

whe re

~c21. 2 = N/(b q- a)/2 ___ ~o~/2, I2s,4 = 7/2 --+ x ~ - - a)/2

are four n o n - n e g a t i v e f requencies , a n d whe re

(4.3)

a = ffo~/2) 2 + ¢02 - (7/2) 2 , (4.4)

b = [a 2 + (7¢oc/2)2] 1/2 . (4.5)

Se t t ing mc = 0, we a r r ive a t the fami l ia r resul t for a o n e - d i m e n s i o n a l d a m p e d

h a r m o n i c osc i l l a to r in the a bse nc e of an ex te rna l m a g n e t i c field:

r e ~ m - - ¼72 O(t)exp( - ½? t ) s in ( tx /¢o 2 - 1,/2), if ¢00 > ½7

G33(t) = (4.6)

[ rex/¼72 _ o 2 0 ( t ) exp( -- ½7t)sin(tx/¼72 -- ~Oo2); if (Oo < ½~,,.

N e x t we ca lcu la te the s y m m e t r i z e d pos i t i on co r r e l a t i on func t ions ~t,,(t) in (3.1) by

the m e t h o d of c o n t o u r in tegra t ion . T h e resul ts are:

~11 (t) -- h (1[- //J~co i '~ ( 2 ~ ) 1 } 4m I m [{~- L/C°t \2kTjl--I e -''1~ - cot e-,,,2~

h (1 F 1 - 1 ~~) + o3,;e-

1 1 -- - - F ( 1 , - O3G1 - - ( ~ l ; e - V l r ) - - 7 F(1,O32; 1 + o32;e - ' ' ~ )

o31 ~o2

1 ]} + - - F(1, -- o32; 1 - O32; e - " ' * ) , (4.7)

(.0 2

648 X.L. Li, R.F. 0 "Connell / Physica A 224 (1996) 639-668

012(t'=Zsign(t)Re{~[c°t(h0)l~e-"'~-c°t(h2~T)e-°~2*l}\2kT/l

h sign(t)Re{~[_~lF(1,(51;l+(se;e_~.,~ ) 4~m

1 1 + _-- F(1, - (51; 1 - ( 5 1 ; e -vl~) - @ F(1,(52; 1 + ( 5 2 ; e - v ' r )

0)1 0)2

- - 1 F ( I ' - (52; 1 - ( b 2 (52; e-V'~)]} , (4.8)

0)1,2 = ½(7 - i0)~) +- i~, (4.9)

= x /~ + a)/2 + ix/(b - a)/2, (4.10)

where (51,2 = h0)1,2/2rckT are the corresponding temperature-reduced dimensionless frequencies; vl =2rckT/h;z is the absolute value of t : r= l t l ; sign(t) is the sign function: sign(t)= t/[tl; and F(a, b;c;z) is the Gauss hypergeometric function [33].

To simplify the above formulas, we now discuss the high-temperature limit kT ~> fi0)1.2, where the first two terms in (4.7) and (4.8) dominate. Comparing (4.3) with (4.9) and (4.10), we see that o)1,2 is connected with f21.2,3,4 by

0)1 = Q4 + iQ2 , 0)2 = ~ 3 - - i 0 1 . (4.11)

From (4.7), (4.8) and (4.11), we get the classical results

kT { [ ( coc /b+a T b-ab~a~ ~,l(t)-2m0)2b b - ~ - 2 2X/ 2 / c°s(g21z)

+

+

+ ( 2 ~ b+a2 c°c ~ ) sin(t22"r)l e- °'*~ } 2 (4.12)

~L. Li, R,F. O'Connell / Physica A 224 (1996) 639-668

012(t) - 2mo)2ob sign(t) b - ~- 2 2

649

012(0 = -- (h/ 4nm) Re { (1/ ~)[e"~" El (oh t) - e- ' ° ' t E l ( - 6o10

-- e"2tEl(o92 t) + e-° '2 tEl( -- coil)

+ in sign(t)(e -'°'~ + e - " ~ ) ] }, (4.15)

where E1 (z) _= ~ (e-t / t )dt (I arg z l < n) is the exponential integral function, which is single-valued with the cut line along the negative axis [33]. For t ~ 1/7 we recover the power law for the long-time tail characteristic of Ohmic dissipation [34].

~11(t) = -- h?/nmo)~t 2 + O(1/t4), (4.16)

O~2(t) = 4hTe)c/nme)6t 3 + O(1/ts). (4.17)

We note here that the leading order t -2 term in (4.16) for the symmetrized position autocorrelation functions in the plane perpendicular to B at zero temperature is unchanged by the magnetic field.

We end this section by considering the dispersion of the position operator (x2), which could be obtained by putting t = 0 in (4.7):

(x 2) = me)2 + 2~mIm + 03~) - 0(1 + 032)] , (4.18)

where O(z) is the logarithmic derivative of the gamma function F(z) [33]. The equal-time value of the position cross-correlation function O~2(t), in comparison, is zero as seen from (3.1) and (3.6). In the high-temperature region k T >> hcol.2, by

(4.14)

7 b - a . +¢0c b ~ ~ + a ~ ~ ) s l n ( Q 2 Q

-- -} ~ Z }COS(f22Z) e (4.13)

In the low-temperature regime k T ~ h~01,2, on the other hand, the hypergeometric functions in (4.7) and (4.8) become important. At T = 0, the summations in the serial expansion of the F functions are replaced by continuous integrals and we find for 6,,~(t), from (3.1),

~911(t) = (1/4nm)Im{(1/~)[e '~2~El(~o2z) + e-")2~El( - (/)2"[')

-- e ..... Ex(o91r) -- e - " ~ E l ( -- ~olr) + in(e -'°'~ + e - ' ° ~ ) ] } ,

650 X.L. Li, R.F. O'Connell / Physica A 224 (1996) 639-668

expanding if(z) functions involved about 1, (4.18) reduces to

(x 2> = kT/m~o 2 + O(1/T) , (4.19)

in accord with the classical equipartition law, since the phenomenon of magnetism is quantum-mechanical in nature. While for low temperatures kT ~ hcol,2, we may insert the asymptotic expansion of ~b(z) in (4.18) and find

~2 lb Ara _1~/2 b/~a~ a~ b - a (1y + ~ ( b - a ) ) ] (x2> =2-~-~mbL ~ / ~ t a n \~ / - - - -~- -1 + x / ~ l n , ½ 7 _ ~ j j

+ ~7(kT)2/3mhco~ + O(kT) 4 , (4.20)

which has the T 2 power-law correction characteristic of Ohmic heat bath. We note in passing that this leading order correction term is independent of the magnetic field.

5. Quantum Brownian motion of a charged particle in a magnetic field

5.1. Relations between d~,~ (t) and Gp~ (t) at long times

The Brownian motion is a special case of damped harmonic oscillator considered previously. As we take the limit ~Oo = 0 in (3.1), the symmetrized position correlation functions ~tpa(t ) become infrared-divergent, reflecting the fact that the coordinates of a free particle are unbounded. To extract finite results, we introduce the displacement correlation functions according to

dp.(t) = 2 [~p.(O) - ~p~(t)], (5.1)

which is physically more meaningful here. Its diagonal elements, from (3.1), are the mean square displacements in each direction:

dpp(t) = <(rp(t) - rp(O))2>, for p = 1, 2, 3. (5.2)

Taking the time derivatives in (3.1) and (5.1), we then have

dp~(t) = ~ do)e-i'~thrncoth ~ [~oo-(o) + iO +) - c~*p(~ + i0+)]

¢X3 zr ~ [Im ~)~(co) sin(ogt) + Re ~(co) cos(c~t)] .

0 (5.3)

X.L. Li, R.F, O'Connell / Physica A 224 (1996) 639-668 651

In the long-time limit t ~ oo at finite temperature T, the small-frequency contribu-

tions dominate in (5.3). By expanding the factor coth(hoo/2kT) about o~ = 0 and employing the definition (2.11) for the retarded Green's function, we obtain this simple relation between dp~(t) and G~,~(t):

d,,~(t) = 2kTGt,~(t ) , for t ~ oo, T > 0, (5.4)

where we have used the fact that ep,(~o) is analytic in the U H P and so is ~*~((o) in the lower half-plane (LHP).

The significance of (5.4) may be appreciated by introducing linear dc mobility tensor (P~)t,, and diffusion-coefficient tensor Dp~ [16]. For a constant external force f switched on at t = 0, we get from the Fourier transform of(2.2), after adding f to its right-hand side and averaging out the random force F, (rp(t)) = (~oGp~(s)ds)f~, so that the drift velocity of the particle is directly related to Gt,,(t ) by

(~p(t)) = Gf,~(t)fo, for t > 0. (5.5)

The linear dc mobility tensor (p~)p~ is defined through the asymptotic relation lira, ~ ~ (~:,,(t)) = (U~)t,~ f~, yielding, from (5.5), (2.11), (2.3)-(2.5),

(p~)~,~ = lim Gp~(t) = lim ( - ioo)~,~((o) t ~ ~,o --* 0

= ~72(0)6P~ + ~cc B,,B. + ~,,~,B.,7(O ) ~cc {m~7(0)['Tz(0) + ~°2]} - ' '

(5.6)

where we have assumed in the last line that fi(0) # 0 and y(0) - ~(O)/m. On the other hand, the diffusion-coefficient tensor Dp~ is defined in the standard way by

D,,~ = 1 lim d~,~(t). (5.7) t ~ o o

With the aid of (5.6) and (5.7), (5.4) may be recast as

D,~ = kT(p~),~, for T =~ 0, (5.8)

which is a generalized version of the Einstein relation [35]. The diffusion constant could also be derived in another way. For this purpose, we

calculate the velocity correlation function in the classical regime from the correspond- ing position correlation function:

( v , , ( t ) v ~ ( t ' ) ) = ( ~ , , ( ~ , 4 , , , . ( t - t'), (5.9)

where 0, denotes partial derivative with respect to t and we have already exploited the fact that the commutator of two operators is of the order of h in reducing the symmetrized correlation function for quantum operators to the simple correlation function for the same variables in the classical reaime.

652 X.L. Li, R.F. 0 'Connell / Physica A 224 (1996) 639-668

Substituting (3.1) in (5.9), we find, if k T >> ho),

oo

f (vp(t)v~(O)) = ~ / do) e-i~"e)[ap,(o)) - a*(o))]

- - 0 0

kT i = 27t----i d°)e-i°"°oaP'(°)) - - o 0

k T J d~ocos(ogt)toctp~(og) for t ~> 0 (5.10)

- - o 0

where the two equalities are obtained by using the analyticity of c~po(og) and ct*p(og) in the UHP and LHP, respectively, when t is positive.

Integrating both sides of (5.10) from 0 to + oo and employing the integral repres- entation of the Dirac delta function yields

00

f dt (vp(t)v~(O)) = - ikT

0

i do) ¢o~p~(o))6(m) = k T lim ( - io))ap~(o)). o) --* 0

- - o o

Comparing (5.11) with (5.6) and (5.8), we obtain

(5.11)

o0

Dp~ = f dt (vp(t)v~(O)), (5.12)

which is just the Green-Kubo type formula connecting transport coefficients with integrals of appropriate correlation functions [36-38].

The situation at zero temperature, once again, has to be treated separately. From (3.1) and (5.1) one finds for the displacement correlation functions at T = 0

dp~(t) =--2hrt i dco [Ima),(og)[1 -cos(o~t)] + Re~(o~)sin(~t)] , (5.13) - 0

which, by virtue of the identities [39]

GO

f ' sin(ogt) = 2 dy cos(~ot/y)

0

oo

- y sln(ogt/y), 7Z

0

(5.14)

X.L. Li, R.F. O'Connell / Physica A 224 (1996) 639 668 653

can be related to G~,.(t) by

:t3

d,,( t) =--2h dy [yG,,~(t/y) + G,,o(t/y)],

0

for T = 0, (5.15)

where G~(t) and G~",~(t) are the symmetr ic and an t i symmetr ic parts of G,~(t) corres-

ponding, th rough (2.11), to ~ and c ~ defined in (3.2) and (3.3), respectively. If G~,~(t) and Gp~(t) are finite when t --* oo, i.e. finite mobil i ty as for the Ohmic heat bath, then, upon splitting the integral in (5.15) into one f rom 0 to t and a remaining correct ion term, one obta ins to the leading order term

d, ,( t) = (2h/n)G;~( + oo)ln(t), for t --* oo, T = 0 . (5.16)

The cont r ibut ion of G ~ ( + oo) to (5.16) is p ropor t iona l to t - 1

5.2. Ohmic heat bath

The results for a charged Brownian particle in an Ohmic heat ba th and in the presence of an external magnet ic field m a y simply be derived by taking the limit COo 2 ---, 0 in the cor responding formulas for a charged oscil lator in Section 4. F r o m (4.1) and (4.2), the re tarded Green ' s functions read:

1 Gll(t) -- m(o32 + ?2) {:, q'- e-;'t[Ogcsin(°~ t) -- :,cos(ooct)]}O(t), (5.17)

1 G12(t) - re(m2 + :,2) { (Dc -- e-"[o~cos(e)ct) + 7sin({o~t)]}O(t). (5.18)

Combin ing (4.7), (4.8), and (5.1), we find for the displacement corre la t ion functions of a Brownian particle

2kT:,z 2kT(? 2 -- a) 2) dll( t) = m(:, 2 + (02) m(:, 2 + 82)2

+ he-;"

m(y 2 + (o2) (sin(h?/k T )[:, cos(ocz) - c0c sin({ocz)]

-- sh(hoc/kT)[: , sin(o)cz) + coccos(m~z)])

x [ch(hmc/kT ) -- cos (h? /kT)] - 1

4 k T G v, + :, + - - L m .=~ Vn[(V. + :,)2 + {o2]

4k T ~ :,(v~ - 7 2 - o92)e -~"~

+ - - :, + o c ) +4~, coc] m , = 1 v, [ (v~ -- ..--2 - - .~2~- - - -777..2 2 , (5.19)

654 X..L. Li, R.F. O'Connell / Physica A 224 (1996) 639-668

2k Tooct 4k T Te) c he - ~ d12(t) - m(y2 + 092) sign(t) m(72 + 092) 2 + sign(t) m(72 + 002)

x {sin(hT/kT)[7 sin(cOcZ) + e)ccos(cocz)]

+ sh(ho)dkT) [7 cos(oo~z) - o)~ sin(co, z)] }

x [ch (hfodk T ) - cos (hT/k T ) ] - 1

- sign(t) --8kT ~ 7e)ce -vnT . (5.20) m ~=1 (v, 2 - 7 2 + e)2) 2 + 472c°2

In the classical regime k T >> h7 and hooc, these simplify to the expressions

2 k T 7 2 -- o9~ e - ~ - 2 -t 72 2 d11(t) m(72 + co2) 7z 72 + o)c + co~

× [(72 - 0) 2) cos(co, z) - 27o9c sin(cocz)]), (5.21)

2 k T ( 209~7 e -'~ = , 2 + sign(t) 72 2 dl2(t) m(72 + co2) og~t - sign(t) 72 + O)c + (/)c

× [(72 - co 2) s in (~z) + 27coccos(og~z)]), (5.22)

upon which, by inserting (5.1) into (5.9), we readily arrive at the velocity correlation functions at high temperatures:

( F 1 ( t )v 1 ( 0 ) ) = 1 ~/'1 1(t) =- (kT/m)e-~'~ cos(coCt), (5.23)

(vl(t)v2(O)) = ½d'12(t) = ( k r / m ) e -~'~ sin(co~t). (5.24)

The exponential decay for the velocity correlation functions is characteristic of the Langevin theory [40, 41], as long as the time t involved is not small [42]. Substituting (5.23) and (5.24) in (5.12) gives

k T 7 Dl1 = m 72 + co~2' (5.25)

k T oo~ D12 -- m 7 2 + co~2' (5.26)

which is, of course, in accord with the direct evaluation of (5.6) in (5.8). The magnetic field manifests itself as a multiplicative term oscillating with the cyclotron frequency for the velocity correlation functions of a charged Brownian particle in the plane perpendicular to it and the self-diffusion constants are reduced by a magnetic field dependent cofactor.

In the quantum regime t ,~ h /kT , on the other hand, the series terms in (5.19) and (5.20) are important . F rom (4.14), (4.15), and (5.1), after taking co 2 ~ 0 limit, we obtain

X.L. LL R.F. O'Connell / Physica A 224 (1996) 639-668 655

for a Brownian particle at T = 0

d l l ( t ) - ~m(7 2 + co2) yln(zx/Y 2 + ~o~ 2) + Cy + COctan-~(codT)

he -.~ [m~ cos(o)~z) + 7 sin(~ocz)l + h Re I

1

m(7 z + e92) rcm ~ + i~oc

x ( e ( ~ ' + i ° ~ ' E t ( ( 7 + i o o ~ ) t ) + e - ( ~ + i ' ° ° ) ' E l ( - - ( 7 + io9~)t))], (5.27)

h e - ~ ~ cos(ooCz) - ~o~ sin(oocr) dl 2 (t) = sign(t) - - 2

m 7 2 - I -09 c

h I m ( 1 1tin 7 -- i~o~ [e(~-i")~'Ex((7 - ion)t)

- e-( ';-i '°c)'El( -- (7 - io9~)t)]), (5.28)

where C = 0.577 ... is the Euler constant. The corresponding long-time behaviors of dl l ( t ) and dl2( t ) are given by

2h C7 + ~o~ t a n - ~ (O~c/7) d11(t) 2h 7 ln(z~/72 + °)~2) + - - ~/2 + o92 + O(t-2)

- - 2 7rm 7 2 + (O c x m

(5.29)

4h 7a~c d l 2 ( t ) = _ rrm (72 + ~o~)t + O(t-3)" (5.30)

It is clear that the oscillatory terms with cyclotron frequency are associated with the helical motions of a charged particle about the magnetic field. However, for times long enough, the time dependence of dl 1 (t) is not altered by the B field, with only a reduced overall coefficient [43].

For later comparisons, we conclude this subsection by writing down the results for a free charged particle in a magnetic field, deduced from (5.19) and (5.20) by taking the limit 1' --+ 0:

d l ~ (t) = (2h/m~oc) co th (h~oc/2k T ) sin2(~o¢ t /2) , (5.31)

dlz(t) = 2k Tt/m~oc - (h/mogc) coth(h~od2k T ) sin(~o~t). (5.32)

5.3. Long-t ime dependence f o r f requency-dependent memory funct ion

In this subsection we shall work with a class of the spectral distributions of the memory function popularized in the recent literature, namely [44]

Refi(~o) = m?s(o~/¢b) s- 10((2c -- ~ ) , (5.33)

656 X..L. LL R.F. O'Connell/ Physica A 224 (1996) 639 668

where f2c is a cutoff frequency that is very large compared with all relevant frequency scales of the dissipative system, but much less than other characteristic cutoff frequen- cies such as the Drude, Debye, or Fermi frequency etc., depending on the physical model involved. 05 denotes an appropriate reference frequency so that ~ has the usual dimension of frequency for all s. To avoid the pathological divergence of #(0), s is restricted to be positive. The Fourier transform of the memory function /~(o9) is connected with its spectral distribution by [20]

cx3

2ico fdco' Refi(co' + i0 +) rt j ~ - ~ ~ - ~ / ~ ' (5.34)

o

For convenience, we base the following calculations on the Laplace integral representation rather than the Fourier integral representation that has been employed so far. The two are related through an analytic continuation, e.g.,/7(m) =/~(z = -i~o), where, by convention, the Fourier transform is denoted by a tilde whereas the Laplace transform by a hat. From (5.33) and (5.34), the corresponding Laplace transform of the memory function is given by

oo

f Re/~(~o + i0 +) ~(z) = m~(z) =--2z do9 c° 2 zZ z~ +

0

- 2 m ~ ( f 2 c ~ ' - a f 2 ~ F 1 , ~ ; 1 + ~ ; zZ j (5.35) ~s kOS] z

where ~(z) is the associated friction coefficient introduced in (3.12), with its asymptotic expansion for small frequencies z ~ Oc being [16]

sin(~s/2) [1 + O(z/~2~,(z/O~)2-~)], 0 < s < 2,

- - - v m U + ~ / z Z ) , s = 2,

if(z) = I r ( s - - ~ t ~ - ) ~L1 + 0 ,

2 < s < 4 ,

~c(:---~|G-.] ~[1 + O , s >~ 4. (5.36)

The case of/~(z) = (2/lr)myatan-l(f2~/z) for s = 1, from (5.35), corresponds to the Ohmic heat bath in the limit f2~/z ~ ~ , while the cases of 0 < s < 1 and s > 1 have been referred to as sub-Ohmic and super-Ohmic, respectively [26].

X.L. Li, R.F. O'Connell / Physica A 224 (1996) 639 668 657

For general frequency-dependent memory functions like the ones in (5.35), only the long-time behaviors of the system can be solved analytically in terms of known functions, with the dominant contributions coming from the small-frequency regions in the integrals involved. Assembling (2.11), (3.4), (3.5), and (2.5) with e)o set to zero, we then find for the retarded Green's functions in the plane perpendicular to the magnetic field, in terms of the Laplace integral,

G11(t) ~-- O(t)-~--~m ~i~i dz \zZ + z¢(z) + io)cz -¢ z2 + z,2(z) _ iog,.z , (5.37) Br

Gxz(t) = O(t)~m ~xi dzeZ' z 2 + z~(z) + ie)cz Br

1 ) Z 2 4- Z~(Z) -- iO.)cZ ' (5.38)

where the symbol Br stands for the Bromwich path, which goes upward parallel to the imaginary axis and with positive real part. The integrals in (5.37) and (5.38) for long times can be evaluated by expanding the fractions in the brackets about z = 0 and using Hankel's formula [33]

1 fdze=,z_ ~ = t~_l/V(s)" 2rci

Br

The calculation, though tedious, is straightforward, yielding

6,1(t) =

sin(Tea/2}._ . . . . . . . - 1 [ (~°¢~2s in2(T t s~(~g t )2s - to)O 1 -- 2 _ _ \ T s J \ 2 J

+ O(f2ct)S- 2, (~ty- 2)~,

1 sin(oct) + O((eSt)l-~),

m(o c

_ ( o(1) me)~l sin -1 + (272/rc~)ln(O~t)/+ -~t '

lme)c sin ( m c°c t ) k m , + O ((°St) x - ' ) '

(5.39)

r(s )

r ( 3 s - 2)

0 < s < l ,

s z l ,

1 < s < 2 ,

S = 2 ,

s > 2 ,

(5.40)

658 X.L. LL R.F. O'Connell / Physica A 224 (1996) 639-668

G12(t) =

where

c°csin2(rrs/2) - 2s-2 ~f-~-2~- ~ (~ot) [1 + o((~t)- 1, (o~t)2,- ~)],

o~c o(~Oc~ m(y 2 + co~) + \Oct/'

0 < s < l ,

s = 1 ,

2 sin 2 (COct/2) + O((oSt)-~), 1 < s < 2,

m(Dc

me)c2 sin2 ( 1 + . _ _ - - - , c ° c t / 2 "] (ln(Oct)'~,.., (2yz/lrCo)lnfOct)] + 0 \ ~ ] s = 2,

2 sin2 ( ~ m ~ m , ) ~Oct + O((oSt)-'), s > 2, re(Pc

(5.41)

( 2 ) mr = m 1 + z(s - 2) ?~051-~0~-2 (5.42)

is the renormalized mass for s > 2 [44]. The long-time dependence of the displacement correlation functions at finite tem-

peratures may now be deduced from the first integral of (5.4):

t

( 2kTfd' ' dp~ t)= t Go~(t ), for t --* oo, T > 0. (5.43)

Applied to (5.40) and (5.41), we then arrive at

a l l ( t ) =

2k T sin 0zs/2) my~cbF(s + 1)

(esty [ 1 - k-~-/(C°C]2sin2(2)(cSt)Es-2

F(s + 1) 1)] 0 < s < 1 × r(3s - 1~) + o(g2ct)- 3 ,

2kT?l t + O(1), s = 1, m(7~ + ~o~)

4kT 2rnco~ sin z (COc t/2),

4kT f l+2721n(~2c t ) ) s in2( (Oct~2 mo3~ \ no) 1 + (272/nrS)ln(flct)J'

s in COct , m 60 c

l < s < 2 ,

S = 2 ,

s > 2 ,

(5.44)

X.L. Li, R.F. O'Connell / Physica A 224 (1996) 639-668 659

d12(t) =

2k Te;~ sin 2 (ns /2) mTg&F(2s ) (oSt):~- 1 [1 + 0((050-1,(05t)2~-2)-] ,

2kT~oct + 0(1) , s = 1,

m(y 2 + o~ 2)

2k T [~o~t - m~02 sin(co~t)] + 0((050 l - q , 1 < s < 2,

m~o--~ k c - 1 + n o / 1 + (272/n~) ln( f2c t ) '

m 2 o ~ 2 [ m r t - - s i n ~ c o c t , s > 2 . (5.45)

0 < s < l ,

At zero temperature, we insert (5.40) and (5.41) into (5.15) and obtain, to leading order term in the long-time expansion,

(2 - s) sin[n/(2 - s)]

(s + 1)(2s - 1) hco 2 {~bsin(ns/2) '] 3/~2-~)

(2 -- s) ~ s i ~ - 2 - - s)] rueS---5 \ ~ ] + 0(c°4)'

O < s < l ,

dl 1(0 = 2h71 In(t) nm(y 2 + ~o2), s = 1,

h / ( s - 1)moot, 1 < s < 2,

o(t-3), h c o ~ _ nm72/2Cot '

~ 2 _ 1) sm 2 (½ ns) tan [ (1 -- s) n] (e)t)2~ 2,

4fi71o9c ,

- (h/m~o~) singoct),

h . / O, ct "~

m(.Oc kmr /

s>~2,

(5.46)

O<s<½, 1

S - - 2 ,

½ < s < l ,

s = 1, (5.47)

1 < s < 2 ,

S ~ 2 ,

s > 2 .

660 X..L. Li, R.F. 0 'Connell / Physica A 224 (1996) 639-668

Since the magnetic field does not affect particle motion parallel to it, the results for G33(t) and d33(t) are the same as in the one-dimensional case [16, 44]. We list them here for completeness and for later comparison with the results for motions in the plane normal to the magnetic field.

sin(~s)(o3t)~-~ [1 + O((Oct)- ' , (O~ty-2)] , myA(s]

zc & t 1 ~ [ 1 + O ( l n - t ) ] ,

G33(t)= t ( m ,~031-~t2-~ - 2 ) 1 + ~ ~ _ 2 - - ~ + o ( ( o ~ t ) ) ,

t/m, + O((f2~t)-1),

2kTsin(zcs/2) (cbt)s + O(t~_ 1, t2s-2) 0 < S < 2 rny~cSF(s + 1) ' '

7rkr (rbt) 2 ~ { (o30 2 "~ - - - - ) =

d33(t) = 2m72cb ln(g2ct) + u i n2 -~ t ) , s 2,

k T t2 + O(t4_~) ' s > 2, mr

O < s < 2 ,

S = 2 ,

2 < s < 4 ,

s~>4,

(5.48)

(5.49)

( 2 ( sin, sj2,),2s, - - - / ( - O < s < l

(2 - s)sin(rc/(2 - s)) ~'s too3' '

I n ( t ) , s = 1 , 2h

[ rcm71 [ s in2(g(2- s)/2) h . . . . . -1

[ ~ m~-~ t°~r) ' 1 < s < 2,

t rrZh ~ t d33(t)lr=o = \ ~ lnE(t), s = 2,

] ~ ~ T 7 , . - - - - z ~ ( ( S t ) 3-~, 2 < s < 3 , [ COSkn~Z - s)/zj 1(,~ - s~ m, co

[ 2 h m 7 3 , . , l ~ 2 m(t), s = 3,

do~, s > 3 ,

where oo

(z ~ + z~(z)) m2z ~ 0

(5.50)

(5.51)

32L. Li, R.F. O'Connell / Physica A 224 (1996) 639 668 661

is a constant depending on high-frequency, as well as low-frequency, properties of the memory function.

A comparison of these results for the motions of a charged quantum particle perpendicular and parallel to an external magnetic field enables us to see the influence of the magnetic field on the Brownian motion. For the retarded Green's functions at long times in the sub-Ohmic case (0 < s < 1), with the magnetic field B set along the z axis,

G~l(t) is the same as G33(t) to the leading-order term in t. In the Ohmic case (s = 1), as shown in Section 4, G11(t) is qualitatively the same as G33(t), with only a smaller mobility coefficient reduced by the magnetic field. In the super-Ohmic case (s > 1), however, GI x(t) is completely different from G33(t) ever increasing with t. The particle responds to a constant driving force with a bounded simple harmonic oscillation in the plane normal to B. In that plane, the dampling now effectively vanishes for long times, except for the special case of s = 2 arising from the corresponding non-analytic logarithmic term in (5.36) for ~(z), and for s > 2, the free particle's mass m is replaced by the renormalized mass mr and the quantity mco,./mr = eB/mrc in Eqs. (5.40)-(5.45) and (5.47) is merely the cyclotron frequency for a particle of the renormalized mass mr.

As for the long-time dependence of the displacement correlation functions in the

sub-Ohmic case (0 < s < 1), did(t) has the same subdiffusive behavior as d33(t) at non-zero temperatures. On the other hand, the long-time constant limit of dl~(t) is reduced by the magnetic field from that of d33(t) at T = 0. In the Ohmic case (s = 1), the magnetic field simply decreases the diffusion coefficient in the expression for dl 1 (t) 21, 43). For s > 1, in contrast to the unbounded growth at long times of d33(t) except for s > 3 at T = 0, d~(t) approaches a constant at zero temperature, while displays bounded oscillations, except for s = 2 again, at non-zero temperatures.

6. Summary and Discussion

We have considered the problem of calculating the retarded Green's functions and the symmetrized position correlation functions for a charged quantum oscillator linearly coupled to a heat bath and in the presence of a constant homogeneous magnetic field. The retarded Green's functions are shown, as in the linear-response theory, to be related to the commutators (i.e., antisymmetrized correlation functions) of the position operators at different times, which are c-number quantities here owing to the linear coupling between particle and bath in the IO model. In correspondence, the retarded Green's functions are temperature-independent and are connected with the symmetrized position correlation functions by the fluctuation-dissipation theorem (FD). For linear systems as are discussed here, all higher-order correlation functions can simply be factorized into pair correlation functions due to the Gaussian properties of the underlying stochastic processes [-20].

We have started off by examining some general properties of the generalized susceptibility tensor of the dynamical system involved, which in turn has enabled us to reach two general conclusions about the position autocorrelation functions (disper- sions) of the magnetic system in an arbitrary heat bath. In addition to the transversal

662 X.L. LL R.F. O'Connell / Physica A 224 (1996) 639-668

dispersions of a charged quantum particle, the free energy of such a system has also been shown to be decreasing monotonically with increasing magnetic field strength, hence indicating the diamagnetism of the system even in the presence of a physical heat bath. The generality of these theorems stems from the fact that, because of the charge neutrality of the heat bath independent oscillators implied in the IO model, the magnetic field enters into the GLE only through the Lorentz force term so that the external field and the dissipation do not affect each other. It may be of interest to note in passing a similar theorem on the magnetoconductivity of metals which states under rather general assumptions that if an external magnetic field has no bearing on scattering mechanisms, then the electric conductivity of metals is a monotonically nonincreasing function of the magnitude of the magnetic field [45].

We have also investigated the quantum diffusion of a charged Brownian particle in a uniform magnetic field for a variety of heat baths. As in the nonmagnetic case, well-separated time scales, essential for the interpretation in terms of a standard Brownian motion, only emerge in the high-temperature (classical) regime. In the opposite low-temperature limit, the interplay between quantum and thermal fluctu- ations dominates, leading to long-time tails of the form t-z in the time correlation functions [46]. For the Ohmic heat bath, both the friction and the Lorentz force terms depend linearly on the instantaneous velocity of the charged particle. Accordingly, the functional dependencies on time of both the retarded Green's functions and the displacement correlation functions are qualitatively the same as those for a free particle; they are unchanged by the magnetic field, with only the overall coefficients reduced by a magnetic-field-dependent factor for motions normal to it. Hence, a static magnetic field can not confine a charged particle coupled to an Ohmic heat bath, not even at absolute zero temperature. It only slows down the transverse diffusion [43]. For the sub-Ohmic case where damping dominates at low frequencies (equivalently at long times), an initially localized state remains localized at zero temperature even without an external confining potential because of a finite variance a(t) here [44]:

a(t) ==- ( (x -- ( x ) t ) 2 ) t = a(0) -- d(t)/2 + h2G2(t)/4a(O) .

Thereby, the transverse localization length al /2( t ~ oo) is shorter than the longitudi- nal one. For the super-Ohmic case, the magnetic field dominates at long times. As a result, the traverse localization lengths are bounded except for the case of s = 2 at T ¢ 0, whereas the longitudinal one is infinite. Therefore, an initially localized state will eventually spread out along the direction of the magnetic field.

We conclude the discussion by noting that the method and resutls presented here may be useful in studying magnetic properties such as the diamagnetic susceptibility, magnetoconductivity, and Hall coefficient for a two-dimensional (2D) system of charged particles in the dissipative (or incoherent) regime: h z - ~ ~> k T o , where z is the inelastic scattering life time and To is the bare degeneracy temperature. Two proto- types of quasi-two-dimensional system are the degenerate Fermi gas in the presence of strong disorder associated with the quantized Hall effect and the normal state in

XL. Li, R.F. O'Connell / Physica A 224 (1996) 639-668 663

cuperate superconductors. It has been argued that quantum statistics (Bose or Fermi) presents only quantitative corrections in the dissipative regime [47], and it is well known that two-body interactions do not alter the amplitude and period of the de Haas-van Alphen oscillations as well as the total magnetic moment of a system of finteracting ferminons [48]. Therefore, the GLE approach developed for the problem of a single charged Brownian particle might be applicable to such systems as well.

Acknowledgement

The work was supported in part by the U.S. Army Research Office under Grant No. DAAHO4-94-G-0333.

Appendix A

Let us denote the inverse matrix of c~p~(co) by Dt,~(o)) [23]:

Df,~(e)) = 26~,~ + i (e /c )co~p~B~, (A.1)

which, by definition, is related to ape(e)) through

Dp,(~J)~,~(co) = 6p~, (A.2)

~F,,(co)D,~(co) = 6po, (A.3)

where the Kronecker delta function 60~ is unity for p = a, and zero otherwise. From (A.1) and (2.5), we have

D*~(co) - D~,,(co) = 2i6,,~o0 Refi(co). (A.4)

Multiplying (A.4) by ~p~,(co)~*v(co) and using (A.2), we obtain (2.7) and, multiplying (A.4) by ~vo(e))c~*o(co ) and with the aid of (A.3), (2.8).

Appendix B

Since 1/2---~o) is simply the generalized susceptibility for a one-dimensional oscillator, - i z /2 ( z ) = - ize~°)(z) is a positive real function for K > 0 [49], and thus its real part everywhere in the U H P is positive [20]:

Re - i z /2 (z ) > 0, for Imz > 0. (B.1)

Let us now suppose that

2(z) = 4-mcocz (B.2)

664 X.L. LL R.F. 0 "Connell / Physica A 224 (1996) 639-668

for some z in the UHP. Then we would get

- iz/2(z) = T-i/me)c, (B.3)

which contradicts (B.1). Therefore (B.2) has no roots in the UHP. It follows that c~/,~(e)), from (2.3), has no poles in the UHP.

Appendix C

To prove (2.10), we start by calculating the work done by an external c-number force f (aside from the magnetic field) in a complete cycle on an otherwise isolated system [20]:

i W = d t £ ( t ) ( v o ( t ) ) = ~ dco fp(o))(~.( - co)),

- - 0 0 - - ~

(c.i)

where the second equality is obtained by using the Parseval theorem [50], vp(t) is the velocity operator of the particle, and f ( t ) is assumed to be arbitrary except for the requirement that it vanish at the distant past and future, and where tilde denotes the Fourier transform as usual, e.g.,

oo

~(co) = f dtei°~tv(t).

- - C~3

(C.2)

From (C.2), one can readily see that

~Tp(o)) = -- icoFp(co). (C.3)

Putting (C.3) and (2.2), with )7 added to F, in (C.1), and averaging out the random force ff gives

c o

W = f ~ i f dcococe.(co)jT(co)jT.(co ) (C.4)

where we have used the reality condition on ~7(e)): ~7( - 09) = 7*(09). Forming complex conjugate of (C.4) and interchanging the dummy indices It and v, one then finds

W- 2~i i doom~"~(e))L(e))f*(e)) - - o o

(c.5)

X.L. Li, R.F. 0 'Connell / Physica A 224 (1996) 639-668 6 6 5

Assembling (C.4), (C.5), (2.7), (2.2), and (C.3), one finally obtains

1 i 1 ~* oJ W = ~ do~o)~ii[~,.,,(~o)-~*,.(oJ)]f,(~o)f,.( )

2g ~ - c o

1 f dco Re fi(~o) ~ ] (~.(co))l 2 g a

0

(C.6)

which is positive as demanded by the second law of thermodynamics. Eq. (C.4) may also be written as

1 i d {I ( )R [f.( 3 0

- Re ~,v(co) Im [ f,(co) f~(co)] } . (C.7)

where we have used the fact that, due to the reality conditions on ~,,(~o) and f,(o~), Re ~,,(o~) and Im ~,,(m) as well as Re jT~(~o) and Im f,(~) are even and odd functions of m, respectively. Since .~,(co) are arbitrary other than the boundary conditions lim,o~ + 0o f,(~o)= o,L(~o ) (/t = 1,2, 3) may well be chosen all real (and thus even functions of o~). Then the integrand in (C.7), according to (C.6), must be positive for all (D:

Im ~,,,,(~) fi,(~o) j~.(~ ) Im ~ ~ ~ = ~,,,,(~o)f~(co)/v(o~) > 0, for ~o > 0, (C.8)

where ~,.(oJ), given by (3.2), is the symmetric part of ~uv(o~). Hence, Im ~v(~o) must be a positive definite matrix for all co > 0, and (2.10) readily follows as a corollary.

Appendix D

The free energy of a charged quantum oscillator linearly coupled to a neutral heat bath and in a magnetic field, defined as the free energy of the composite system of the oscillator interacting with the heat bath minus that of the bath alone, assumes the form [23]

Fo(T,B) = - d~of(co, T ) I m ln[de t~(o + i0 + , 7[

0

(D.1)

666 X.L. Li, R.F. 0 "Connell / Physica A 224 (1996) 639-668

wheref(o, T) is the free energy (including the zero-point energy) of a free oscillator of frequency o:

f ( tn, T ) = k T In [2 sinh ( h~o /2k T ) ] , (D.2)

and where

1 1 det ~ (o) - det D (~) - 2 [22 - (~o (e/c) 2 ) B 2] (D. 3)

is the determinant of the matrix ~,(~o) given in (2.3). Since the heat bath is neutral, the magnetic movement M of the charged oscillator is

related to the free energy F o ( T , B) through the equation [51]

M = - OFo /dB . (D.4)

Substituting (D.1)-(D.3) in (D.4) and integrating by parts once gives

o(3

M = B - - dcofo 2cOth(hc0/2kT)Im 2 rcc 2 - (e/cB) 2 o 2

0

= B ~ d ~ ° ° 9 2 c ° t h ( h ~ ° / 2 k T ) I m 22 _ (efcB)2092 , (D.5)

- o o

where we have used in the last line the reality condition on the quantity in the brackets. Before we move on, it would be of interest to check the classical limit of (D.5). Expanding co th (hog /2kT) for small h and exploiting the analyticity of the integrand in the UHP (cf. Appendix B), we get

7 ( ) 2__kT d~o~o 22 = , M = B rri _ (e~B)2 ¢° j 0 (D.6) --or2

which is expected on account of the quantum nature of magnetism (the Bohr-van Leeuwen theorem) [52].

The integration in (D.5) may be performed by closing the contour in the UHP and by using the partial fractional expansion of coth (z) [39]

coth(z) = ~ 1 z + inn" (D.7)

The resulting serial expression for M is

M = - 2 k T B v, < 0 (D.8) . = ~ U ( v . ) + ( v . o c ) 2 '

where v, = 2~rkTn/h are again the Martsubra frequencies. Hence, the magnetic moment due to the orbital motion of a charged oscillator is still diamagnetic,

)£L. Li, R.F. O'Connell / Physica A 224 (1996) 639-668 667

unaltered by the presence of an arbitrary heat bath. The same holds for a charged

Brownian particle as one takes the limit cog ~ 0 in (D.8). For an Ohmic heat bath at zero temperature, the magnetic moment of a charged

oscillator can be calculated explicitly [23].

: _ 1 b + °> +

.),.(i, + .>)] 0 ,°9) 2 ~ ( b - a) 7 - ~ ( b a)

For a charged Brownian particle, this reduces in the limit cog -~ 0 to

M = - (he/r~mc) t an - 1 (coo/7) • (D. 10)

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