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 functions 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
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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 Hamiltonian 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 incorporates the effects of Landau orbit quantization and the associated Landau level structure, 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 generalization 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 longtime 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
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 ) ,
--
V(r) and
(2.1)
cC
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) = ½Kr 2 and a uniform magnetic field B, the
resulting linear operator equation can be exactly solved by the Fourier transformation method 1-23]:
~p(o)) = %~(co)P.(~o),
~(~)
(2.2)
- [D(~)-'];,~
=[2z6p~-(~o~)eBoB~-G~,B,2im~]/detD(m),
(2.3,
where
det D(co) = 2 [22 - (oa(e/c))ZB2],
(2.4)
2(09) =
(2.5)
-- m ~ 2 + K - - i o ~ f i ( f o ) ,
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),
*
~,~(co) - ~v(m) = 2i~,.~(~o)0~*~(co)mRefi(~o).
(2.7)
(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). Furthermore, for the three diagonal elements ~pp(~o) (p = 1, 2, 3),
Im ~pp(Co) > 0,
for o~ > 0,
thereby - iCO~pp(~o),p = 1,2, 3, are real positive functions (see Appendix C).
(2.10)
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):
1;
oc,
G,~(t) = ~
-
d~o e-i°":%(~o).
(2.11)
oc
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
(2.12)
r,(t) = i dr' Go~(t - t')[L(t') + F~(t')].
-co
Since ~p~(co) is analytic in the U H P , 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]:
oo
[F,,(t),F.(t')] = 6,,~ 2 fde)Re[fi(e) + iO+)]hogsin[e)(t - t')].
(2.14)
0
Thereupon, we derive the nonequal-time commutator of rp(t) and G(t') from (2.12):
[r,(t) , r o - (t')]
1
~
i &o ~,,,(o~)o~*,((o)Re fi((o)hcoe i,,,i,-,'l
_
7r
--
clo
co
= ~
h I
dooe-i"m-t')[%,((o)
- 0~*,(e))] ,
(2.15)
-oo
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 fluctuation-dissipation theorem [29, 30]:
~p~(t -- t') =_ ~1 (rp(t)r~(t t ) + r,(t 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~ t7(o) + i0 ÷)] coth(ho)/2kT) cos [o)(t - t')]
0
oo
h fdo)Re[ o(o) + io+)] coth(ho)/2kT)sin[o)(t
- t')],
7Z
0
(3.1)
where
(~pa(O)) ~
1 [Spa(O))
-I- 0 ~ a p ( O ) ) ]
~--- [/~2 (~f,a --
(o)(e/c))2 BpB~]/det D(o)),
(3.2)
and
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),
~12(co) = - c~2a(co) = -
ico(e/c)B2/det D(o)).
(3.4)
(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):
oo
(y2) = -nhf dcolmcql(co)coth(hco/2kT),
(X 2) =
(3.6)
0
oo
<z2> = h fd gOIm ~33(g0) cot h(hco/2kT) ,
-
7~
(3.7)
0
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 temperature 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,noting 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)
-oo
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)
= ~ - \~2o2 + 2 2.,
n=l /~2(Vn) nt- (Vnf'Oc)2J '
(3.10)
X.L. Li, R.F. 0 'Connell I Physica A 224 (1996) 639-668
646
where coc = e B / m c is the cyclotron frequency and
quency, and where
A
~.(v,)
-- 2(iv,)/m = v,2 +
+ v.
K / m is the oscillator fre-
COo
2 =
Cv.)
(3.11)
(3.12)
~(v,) - fi(iv,)/m.
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-independent. 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:
oo
1 f dco e- io;t
4rim
G11 (t) --
--
(
o0
,
1
)
x ~02 _ ~° 2 + iy~o + ~ 0 ~ + co 2 -- oJ 2 + iToo -- ~0~0
- 2mbl O ( t ) [ e x p ( - ~ 2 4 t )
x
\q-~--sin(f22t)
2
+
cos(~22t)
+
ac°s'O'")l
exp(
--
Q3t)
(4.1)
647
X.L. Li, R.F. O'Connell / Physica A 224 (1996) 639 668
1
4him i d¢o e - io,t
-oo
G12(t) --
X
1
1
~o2 -- Oo2 + i7~o + ¢o/o
~o z -
~o 2 +
)
i?o~ -
~o,,~
- 2mbl O(t)[exp(-f2,t)f\X/~-~ + acos(f22t)
- X/~-~
sin(f2: t)) - exp( - f23 t )
×
+
{4.2)
,
where
~c21.2 = N/(b q- a)/2 ___ ~o~/2,
I2s,4 = 7/2 --+ x ~ - -
a)/2
(4.3)
are four n o n - n e g a t i v e frequencies, a n d w h e r e
a = ffo~/2) 2 + ¢02 - (7/2) 2 ,
(4.4)
b = [a 2 + (7¢oc/2)2] 1/2 .
(4.5)
Setting mc = 0, we arrive at the familiar result for a o n e - d i m e n s i o n a l
h a r m o n i c oscillator in the a b s e n c e of an external m a g n e t i c field:
r e ~ m - - ¼72 O(t)exp( - ½?t)sin(tx/¢o 2 - 1,/2),
damped
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 calculate the s y m m e t r i z e d position c o r r e l a t i o n functions ~t,,(t) in (3.1) by
the m e t h o d of c o n t o u r integration. T h e results are:
~11 (t) --
h I m (1[//J~coi '~ e -''1~ - cot
4m
[{~-L/C°t \2kTjl--I
h
1
-- - - F ( 1 ,
o31
1
(1 F 1
- O3G1
- 1
+
o3,;e-
(~l;e
-Vlr)
--
7
~o2
]}
,
e-,,,2~1}
~~)
1
--
+ - - F(1, -- o32; 1 - O32; e-"'*)
(.02
(2~)
F(1,O32; 1 + o32;e - ' ' ~ )
(4.7)
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
+ _-- F(1, - (51; 1 -
( 5 1 ; e -vl~) -
0)1
@ F(1,(52; 1 +
(52;e -v'r)
0)2
- - 1 -F ( (I ' -b (52;
0)1,2 = ½(7 -
1
(52; e-V'~)]} ,
(4.8)
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 + i Q 2 ,
(4.11)
0)2 = ~ 3 - - i 0 1 .
From (4.7), (4.8) and (4.11), we get the classical results
kT { [ (
coc /b+a
~,l(t)-2m0)2b
b-~2
T b-ab~a~
2X/
2
/ c°s(g21z)
+
+
+ ( 2 ~ b+a2
c°c
} ~ ) 2 sin(t22"r)l e-~*'°
(4.12)
~L. Li, R,F. O'Connell / Physica A 224 (1996) 639-668
012(t) - 2mo)2ob sign(t)
b - ~-
+¢0c b~~+ a
--
-}
~
2
649
2
~7 ~ ) sbl-na ( 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 - ' ° ~ ) ] } ,
(4.14)
012(0 = -- (h/ 4nm) R e { (1/ ~)[e"~" E l (oh t) - e-'°'t E l ( - 6o10
-- e"2tEl(o92 t) + e-°'2tEl( -- 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
X.L. Li, R.F. O'Connell / Physica A 224 (1996) 639-668
650
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
b - a (1y + ~ ( b - a ) ) ]
\~/----~--1 + x / ~ l n , ½ 7 _ ~ j j
_1~/2 b/~a~a~
(x2> =2-~-~mbL ~ / ~ t a n
+ ~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 ~ ( c o ) 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 contributions 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~+
B,,B. + ~,,~,B.,7(O ) ~cc {m~7(0)['Tz(0)+ ~ ° 2 ] } - ' '
~cc
(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~oo
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 corresponding 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.
X.L. Li, R.F. 0 'Connell / Physica A 224 (1996) 639-668
652
Substituting (3.1) in (5.9), we find, if kT >>ho),
oo
f
(vp(t)v~(O)) = ~ /
do) e-i~"e)[ap,(o)) - a*(o))]
--00
kT i
= 27t----i d°)e-i°"°oaP'(°))
--o0
kT
J d~ocos(ogt)toctp~(og)
for t ~> 0
(5.10)
--o0
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 representation of the Dirac delta function yields
00
f dt (vp(t)v~(O)) = - ikT i do) ¢o~p~(o))6(m) = k T lim ( - io))ap~(o)).
(5.11)
o) --* 0
0
--oo
Comparing (5.11) with (5.6) and (5.8), we obtain
o0
(5.12)
Dp~ = f dt (vp(t)v~(O)),
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
sin(ogt) = 2
f dy '
cos(~ot/y)
0
oo
-
y
7Z
0
sln(ogt/y),
(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
[yG,,~(t/y) + G,,o(t/y)],
dy
for T = 0,
(5.15)
0
where G~(t) and G~",~(t)are the symmetric and antisymmetric parts of G,~(t) corresponding, t h r o u g h (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 mobility as for the O h m i c heat bath, then,
upon splitting the integral in (5.15) into one from 0 to t and a remaining correction
term, one obtains to the leading order term
d,,(t) = (2h/n)G;~( + oo)ln(t),
for t --* oo, T = 0.
(5.16)
The contribution of G ~ ( + oo) to (5.16) is p r o p o r t i o n a l to t - 1
5.2. Ohmic heat bath
The results for a charged Brownian particle in an O h m i c heat bath and in the
presence of an external magnetic field m a y simply be derived by taking the limit
COo
2 ---,0 in the corresponding formulas for a charged oscillator in Section 4. F r o m (4.1)
and (4.2), the retarded 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)
C o m b i n i n g (4.7), (4.8), and (5.1), we find for the displacement correlation functions of
a Brownian particle
2kT:,z
dll(t) = m(:, 2 + (02)
2kT(? 2 -- a) 2)
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 ) -- c o s ( h ? / k T ) ] - 1
+ - -4 k T G
L
m
v, + :,
.=~ Vn[(V. +
4k T ~
:,)2 + {o2]
:,(v~ - 7 2 - o92)e -~"~
-777..2coc]
2 ,
+ - - m , = 1 v, [ (v~ -- ..--2
:, -+ o.~2~c ) -+4~,
(5.19)
654
X..L. Li, R.F. O'Connell / Physica A 224 (1996) 639-668
2k Tooct
d12(t) - m(y2 + 092)
4k T Te)c
he - ~
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
.
m ~=1 (v,2 - 7 2 + e)2) 2 + 472c°2
(5.20)
In the classical regime k T >>h7 and hooc, these simplify to the expressions
d11(t)
-
2kT
m(72 + co2) 7z
7 2 -- o9~
e -~
2
-t
72
72 + o)c
+ co~
2
× [(72 - 0)2) cos(co, z) - 27o9c sin(cocz)]),
(5.21)
2kT
(
209~7
e -'~
dl2(t) = m(72 + co2), og~t - sign(t) 72 + O)c2 + sign(t) 72 + (/)c2
× [(72 - co2) s i n ( ~ 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:
cos(coCt),
(5.23)
(vl(t)v2(O)) = ½d'12(t) = ( k r / m ) e -~'~sin(co~t).
(5.24)
( F 1 (t)v 1 ( 0 ) ) = 1 ~/'11(t) =- (kT/m)e-~'~
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
kT
7
Dl1 = m 72 + co~2'
kT
(5.25)
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 / k T , on the other hand, the series terms in (5.19) and
(5.20) are important. F r o m (4.14), (4.15), and (5.1), after taking co2 ~ 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)
1
he -.~
m(7 z + e92)
[m~ cos(o)~z) + 7 sin(~ocz)l + h Re I
rcm
~ + i~oc
x(e(~'+i°~'Et((7+ioo~)t)+e-(~+i'°°)'El(--(7+
io9~)t))],
(5.27)
h e - ~ ~ cos(ooCz) - ~o~sin(oocr)
dl 2(t) = sign(t) - -
2
7 2 -I-09 c
m
h Im(
1tin
1
7 -- i~o~ [e(~-i")~'Ex((7 - ion)t)
(5.28)
- e-(';-i'°c)'El( -- (7 - io9~)t)]),
where C = 0.577 ... is the Euler constant. The corresponding long-time behaviors of
d l l ( t ) and dl2(t ) are given by
d11(t)
--
2h
7
7rm 7 2 +
2
(O c
2h C7
ln(z~/72 + °)~2) + - -
+ ~o~ t a n - ~ (O~c/7)
~/2 + o92
+ O(t-2)
xm
(5.29)
4h
dl2(t ) =
_
7a~c
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) coth (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-time dependence f o r frequency-dependent memory function
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)
X..L. LL R.F. O'Connell/ Physica A 224 (1996) 639 668
656
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 frequencies 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
~(z) = m~(z) =--2z
z~
f
do9
Re/~(~o + i0 +)
c° 2 + zZ
0
-2m~(f2c~'-af2~F
~s kOS]
z
1,~;1+~;
zZ j
(5.35)
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-~)],
---vmU + ~/zZ),
if(z) =
Ir(s--~t~- )
0 < s < 2,
s = 2,
~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].
657
X.L. Li, R.F. O'Connell / Physica A 224 (1996) 639 668
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
1
dzeZ' z 2 + z~(z) + ie)cz
)
Z 2 4- Z~(Z) -- iO.)cZ '
(5.38)
Br
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]
(5.39)
1 fdze=,z_ ~ = t~_l/V(s)"
2rci
Br
The calculation, though tedious, is straightforward, yielding
r(s)
sin(Tea/2}._
.. .to)O
.... -1[
1 -- (~°¢~2sin2(Tts~(~gt)2s- 2 _ _
r(3s \TsJ
\2J
+ O(f2ct)S-
2, (~ty- 2)~,
2)
0<s<l,
szl,
6,1(t) =
1
m(o c
sin(oct) + O((eSt)l-~),
_me)~lsin ( -1 + (272/rc~)ln(O~t)/+ o(1)-~t
lme)csin t( )mk mc°c
,
+ O ((°St)x-') '
1 <s<2,
'
S=2,
s>2,
(5.40)
X.L. LL R.F. O'Connell / Physica A 224 (1996) 639-668
658
c°csin2(rrs/2)
- 2s-2
o~c
o(~Oc~
~f-~-2~- ~ (~ot)
m(y 2 + co~) +
G12(t) =
2
m(Dc
[1 + o((~t)- 1,(o~t)2,-~)], 0 < s < l ,
s=1,
\Oct/'
sin 2 (COct/2) + O((oSt)-~),
me)c2sin2 ( 1 + . _ _ - , c ° c-t / 2"]
(ln(Oct)'~,..,
(2yz/lrCo)lnfOct)] + 0 \ ~ ]
2 sin2 ( ~ m ~~Oct
m , ) + O((oSt)-'),
re(Pc
1 < s < 2,
(5.41)
s = 2,
s > 2,
where
(
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 temperatures may now be deduced from the first integral of (5.4):
t
dp~( t)=
2kTfd' t Go~(t'),
for t --* oo, T > 0.
(5.43)
Applied to (5.40) and (5.41), we then arrive at
2k T sin 0zs/2)
(esty [ 1 my~cbF(s + 1)
k-~-/(C°C]2sin2(2)(cSt)Es-2
F(s + 1)
1)]
× r(3s - 1~)+ o(g2ct)- 3 ,
2kT?lt
m(7~ + ~o~)
all(t)
+ O(1),
0< s< 1
s = 1,
(5.44)
=
4kT
2rnco~sin z (COct/2),
l<s<2,
4kTfl+2721n(~2ct))sin2(
(Oct~2
mo3~ \
no)
1 + (272/nrS)ln(flct)J'
S=2,
sin
m 60c
COct ,
s>2,
X.L. Li, R.F. O'Connell / Physica A 224 (1996) 639-668
d12(t) =
659
2k Te;~ sin 2 (ns/2)
mTg&F(2s)
(oSt):~- 1 [1 + 0((050-1,(05t)2~-2)-] ,
0<s<l,
2kT~oct
+ 0(1),
m(y 2 + o~2)
s = 1,
2k T [~o~t -
m~02
m~o--~k c
sin(co~t)] + 0((050 l - q ,
-
1 + no
t--sin
m2o~2[mr
/
~coct
1 < s < 2,
1 + (272/n~)ln(f2ct)
'
,
s>2.
(5.45)
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,
1)moot,
h/(s-
1 < s < 2,
s>~2,
(5.46)
O<s<½,
o(t-3),
hco~_
1
S--2,
nm72/2Cot '
~ 2 _
1) sm 2 (½ns) tan [ (1 -- s) n] (e)t)2~ 2,
4fi71o9c
,
s = 1,
- (h/m~o~) singoct),
h
m(.Oc
. /
kmr
1<s<2,
O, ct
/
½<s<l,
"~
S~2,
s>2.
(5.47)
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
G33(t)=
t (
&t
~ [ 1
1
+O(ln-
m
O<s<2,
S=2,
t)],
,~031-~t2-~
-2)
1 +~~_2--~+o((o~t)
) , 2<s<4,
t/m, + O((f2~t)-1),
s~>4,
(5.48)
2kTsin(zcs/2) (cbt)s + O(t~_ 1, t2s-2)
0 < S< 2
rny~cSF(s + 1)
'
'
7rkr (rbt) 2
~ { (o302 "~
- - - )
=
2m72cb ln(g2ct) + u i n 2 - ~ t ) ,
s 2,
d33(t) =
k T t2 + O(t4_~) '
(5.49)
s > 2,
mr
(
2
( sin, sj2,),2s,
-
(2 - s)sin(rc/(2 - s))
-
-
~'s
/(
-
too3'
O<s<l
'
2h
[ rcm71
[
[
~
In(t),
s =
sin2(g(2- s)/2)
h . . . . . -1
m~-~t°~r)
'
1,
1 < s < 2,
t rrZh ~ t
d33(t)lr=o = \ ~
lnE(t),
s = 2,
]~~T7,.----z~((St)
[ COSkn~Z- s)/zj 1(,~ - s~ m, co
[ 2hm73,.,
l ~ 2
m(t),
3-~,
2 <s<3,
s = 3,
do~,
s>3,
(5.50)
where
oo
(z ~ + z~(z))
0
m2z~
(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 (dispersions) 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 fluctuations 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 longitudinal 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 prototypes 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, - iz/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 - iz/2(z) > 0,
for Imz > 0.
(B.1)
Let us now suppose that
2(z) = 4-mcocz
(B.2)
X.L. LL R.F. 0 "Connell / Physica A 224 (1996) 639-668
664
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.
AppendixC
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]:
W =
dt£(t)(vo(t))
--00
i
= ~
dco fp(o))(~.( - co)),
(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.2)
C~3
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
co
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))
--oo
(c.5)
X.L. Li, R.F. 0 'Connell / Physica A 224 (1996) 639-668
665
Assembling (C.4), (C.5), (2.7), (2.2), and (C.3), one finally obtains
1
~* oJ)
W = ~1 i do~o)~ii[~,.,,(~o)-~*,.(oJ)]f,(~o)f,.(
2g
~
-co
1 f dco Re fi(~o) ~ ] (~.(co))l 2
g
(C.6)
a
0
which is positive as demanded by the second law of thermodynamics.
Eq. (C.4) may also be written as
1
id
{I
3
( )R
[f.(
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~ + 0of,(~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
7[
0
ln[det~(o + i0 +
,
(D.1)
X.L. Li, R.F. 0 "Connell / Physica A 224 (1996) 639-668
666
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]
(D.4)
M = - OFo/dB.
Substituting (D.1)-(D.3) in (D.4) and integrating by parts once gives
o(3
M = B-rcc2
dcofo 2cOth(hc0/2kT)Im
2
- (e/cB) 2 o 2
0
= B
~
d~°°92c°th(h~°/2kT) Im
22 _ (efcB)2092
,
(D.5)
-oo
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 c o t h ( h o g / 2 k T ) for small h and exploiting the analyticity of the
integrand in the U H P (cf. Appendix B), we get
M = B
2__kT
rri
7 (
d~o~o 22 _ (e~B)2 ¢°j
--or2
)
= 0,
(D.6)
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 U H P and
by using the partial fractional expansion of coth (z) [39]
coth(z) =
~
1
(D.7)
z + inn"
The resulting serial expression for M is
v,
M = - 2kTB
.= ~ U(v.)
+ (v.oc) 2
< 0
(D.8)
'
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].
:
_
b
1
2~(b-
+
°> +
.),.(i, +
a)
7 - ~(b
.>)] 0
a)
,°9)
For a charged Brownian particle, this reduces in the limit cog -~ 0 to
M = - (he/r~mc) t a n - 1(coo/7) •
(D. 10)
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[191
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
R.B. Dingle, Proc. R. Soc. London A 211 (1952) 517.
R.B. Thomas, JR., Phys. Rev. 155 (1967) 577; 185 (1969) 823.
P.N. Argyres, Phys. Rev. 117 (1960) 315.
R.G. Chambers, Magnetoresistance in The Fermi Surface, eds W.A. Harrison and M.B. Webb (Wiley,
New York, 1960) p. 100.
E. Fawcett, Advan. Phys. 13 (1964) 139.
R. Kubo, H. Hasegawa, N. Hashitsume, J. Phys. Soc. Jpn. 14 (1959) 56.
R. Kubo, S.J. Miyake, N. Hashitsume, Solid State Phys. 17 (1965) 269.
R. Kubo and K. Tomita, J. Phys. Soc. Jpn. 9 (1954) 888.
R. Kubo and T. Toyabe, in: Magnetic resonance and Relaxation, ed. R. Blinc (North Holland,
Amsterdam, 1967) p. 810.
R.S. Hayano, Y.J. Uemura, J. Imazato, N. Nishida, T. Yamazaki, and R. Kubo, Phys. Rev. B 20 (1979)
850.
A.M. Jayannavar and N. Kumar, J. Phys. A 14 (1981) 1399.
A.K. Das, Z. Phys. B 40 (1981) 353; Physica A 110 (1982) 489.
F.M. Peeters and J.T. Devreese, Phys. Rev B 25 (1982) 7281, 7302.
V.V. Dodonov and O.V. Manko, Physica A 130 (1985) 353.
A.O. Caldeira and A.J. Leggett, Phys. Rev. Lett. 46 (1981) 211, Ann. Phys. (NY) 149 (1983) 374; 153
(1984) 445 (E).
U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 1993).
R.E. Prange and S.M. Girvin, eds., The Quantum Hall Effect (Springer, Berlin, 1987).
N.P. Ong, in: Physical properties of High Temperature Superconductors, ed. D.M. Ginsberg (World
Scientific, Singapore, 1990), Vol. 2.
T.M. Hong and J.M. Wheatley, Phys. Rev. B 42 (1990) 6492; B 43 (1991) 5702.
G.W. Ford, J.T. Lewis and R.F. O'Connell, Phys. Rev. A 37 (1988) 4419.
V. Hakim and V. Ambegaokar, Phys. Rev. A 32 (1985) 423.
X.L. Li, G.W. Ford and R.F. O'Connell, Phys. Rev. A 41 (1990) 5287.
X.L Li, G.W. Ford and R.F. O'Connell, Phys. Rev. A 42 (1990) 4519.
G. Rickayzen, Green's Functions and Condensed Matter (Academic Press, New York, 1980).
S.W. Lovesey, Condensed Matter Physics: Dynamic Correlations, 2nd Ed. (Benjamin/Cummings,
California, 1986).
A.J. Leggett, S. Chakravarty, A.D. Dorsey, M.P.A. Fisher, A. Garg and W. Zwerger, Rev. Mod. Phys.
59 (1987) 1.
D.N. Zubarev, Sov. Phys. Usp. 3 (19601 320.
668
[283
[29]
[303
[313
[32]
[333
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[423
[43]
[44]
[45]
[463
[47]
[48]
[493
[50]
[51]
[52]
X.L. Li, R.F. O'Connell / Physica A 224 (1996) 639-668
R. Kubo, J. Phys. Soc. Jpn. 12 (1957) 570.
H.B. Callen and T.A. Welton, Phys. Rev. 83 (1951) 34.
G.W. Ford, J.T. Lewis and R.F. O'Connell, Ann. Phys. (NY) 185 (1988) 270.
X.L. Li, G.W. Ford and R.F. O'Connell, Physica A 193 (1993) 575.
G.D. Mahan, Many-Particle Physics (Plenum Press, New York and London, 1990).
M. AbramowitzandI. Stegun, Handbook of Mathematical Functions (Dover Publ. NewYork, 1971).
M. Sassetti and U. Weiss, Phys. Rev. Lett. 65 (1990) 2262.
A. Einstein, Investigations on the Theory of the Brownian Movement (Dover Publ. New York, 1956).
M.S. Green, J. Chem. Phys. 20 (1952) 1281.
R. Kubo, Rep. Prog. Phys. (London) 29 (1966) 255.
R. Kubo, M. Toda and N. Hashitsume, Statistical Physics II, Solid-State Sciences, Vol. 31 (Springer,
Berlin, 1985).
I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, London,
1965) pp. 36,424.
M.C. Wang and G.E. Uhlenbech, Rev. Mod. Phys. 17 (1945) 323 [reprinted in Noise and Stochastic
Processes, ed. N. Wax (Dover, Publ., New York, 1954)].
S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1.
J.-P. Boon and S. Yip, Molecular Hydrodynamics (McGraw-Hill, New York, 1980).
Y. Marathe, Phys. Rev. A 39 (1989) 5927.
H. Grabert, P. Schramm and G.-L. Ingold, Phys. Rep. 168 (1988) 115.
G.H. Wannier, Phys. Rev. B 5, 3836 (1972).
C. Aslangul, N. Pottier, and D. Saint-James, J. Star. Phys. 40, 167 (1985).
J.M. Wheatley, Phys. Rev. B 41, (1990) 7301.
J.M. Luttinger, Phys. Rev. 121, (1961) 1251.
G.W. Ford, J.T. Lewis and R.F. O'Connell, Phys. Rev. Lett. 55 (1985) 2273.
E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford Univ. Press, Oxford 1931).
L.D. Landau and E.M. Lifshitz, Statistical Physics, 3rd Ed., Part 1, (Pergamon, Oxford, 1980).
J.H. van Vleck, Theory of Electric and Magnetic Susceptibilities (Oxford Univ. Press, Oxford, 1932).