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On the inhomogeneous magnetised electron gas
Kettenis, M.M.
Publication date
2001
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Citation for published version (APA):
Kettenis, M. M. (2001). On the inhomogeneous magnetised electron gas. Ridderprint
offsetdrukkerij b.v.
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Chapterr 5
Correlations s
Upp until now, we have been investigating the charge and current density profiles in an inhomogeneouss magnetised free-electron gas. The emphasis on these profiles is justified byy the fact that the diamagnetic response of magnetised charged-particle systems in the quantumm regime is determined by electric currents flowing near the walls. Although thee profiles of these currents, and of the closely related particle density, for a non-interactingg magnetised electron gas near a hard wall parallel to the magnetic field had beenn investigated before, we have been able to derive some interesting new results in thee previous chapters.
Equallyy important for a physical understanding of the properties of an equilibrium quantumm system are the correlation functions. For positions in the bulk of the system thesee have been studied extensively, both for a non-interacting magnetised electron gas andd for its interacting counterpart. For the non-interacting gas the bulk pair correlation functionn can be determined analytically both for dilute systems at high temperatures andd for dense low-temperature systems, in which quantum degeneracy effects are im-portantt [26]. For the interacting electron gas information on the behaviour of the bulk correlationn functions is more difficult to obtain. Even for the non-magnetised case thesee functions have surprising properties. In fact, it has been demonstrated that the bulkk correlation functions of the interacting electron gas possess slowly decaying tails withh an algebraic dependence on the position difference [3,4, 15, 2, 11]. For the mag-netisedd interacting gas similar methods have been employed to prove the existence of analogouss algebraic tails, albeit with a different exponent [12, 13, 14].
Thee correlation functions are expected to change in the neighbourhood of a hard wall. Forr a non-magnetised free-particle system these changes are easily determined by us-ingg a reflection principle [45]. The problem becomes a lot more complicated when
eitherr interactions between the particles or a magnetic field or both are incorporated. Inn a recent paper [6] the interactions between the charged particles have been taken intoo account in a model system consisting of two quantum charges immersed in a clas-sicall plasma confined by a wall. An algebraic tail in the pair correlation function of thee quantum particles near the wall was found. However, the exponent governing the algebraicc tail turned out to be different from that of the bulk correlation functions discussedd above. The result corroborates earlier findings based on linear response argu-mentss [28, 29]. The influence of a magnetic field on the surface correlation functions inn the adopted model system remains to be studied.
Ass the influence of a wall on the correlations in magnetised quantum systems is not yet understood,, it appears to be useful to try and investigate the correlation functions for thee relatively simple case of a non-interacting magnetised electron gas in the presence of aa hard wall. In this chapter we present some new results for this system. In particular, wee will analyse the correlations for the strongly degenerate case of high density and loww temperature, where the influence of Fermi statistics is important. Both strong and weakk fields will be considered, so that the number of filled Landau levels can vary considerably. .
Thee current chapter is organised as follows. We start with two sections that serve to preparee the ground. In section 5.1 we define the relevant correlation functions for a systemm of independent particles and discuss their relation to the one-particle Green functions.. The pair correlation function in the bulk is considered in section 5.2, where thee influence of the magnetic field on the correlations is determined both analytically andd numerically. After these preparatory sections we start considering the influence of thee wall in section 5.3. In that section we use the so-called 'path-decomposition expan-sion',, which follows from a path-integral formulation, to determine the lowest-order correctionss in the correlation functions at positions in the transition region, where the presencee of the wall starts to be felt. An alternative way to determine these corrections iss based on an eigenfunction expansion of the Green function, which is the subject of sectionn 5.4. The asymptotic form of the correlation functions for large position differ-encess is established in section 5.5, separately for directions parallel with and transverse too the magnetic field. In section 5.6 the correlation functions for positions close to the walll are studied, again for both directions. In the final section 5.7 some conclusions willl be drawn.
5.. /. Correlation Junctions 11 11
5.11 Correlation functions
Thee equilibrium quantum statistical properties of a system of independent particles are determinedd by the temperature Green function
G p ( r . r ' )) = ( r | e "p H| r ' ) = 2 1 e - ^E^n( r ) C ( r ' ) . (5.1)
n n
Heree |3 is the inverse temperature, and ^n{f) and En the eigenfunctions and
eigenval-uess of the one-particle Hamiltonian H, which is assumed to be independent of the spin off the particles. The reduced single-particle density matrix P0lM.(r, r ' ) of such a system att inverse temperature |3 and chemical potential \i is found by incorporating the effects off quantum degeneracy. For Fermi-Dirac particles one has
P p ^ r ,, r ' ) = 2 Y. , 1 _ „ ) H»»(r) i f t f r ' ) (5.2) n n
wheree the spin degeneracy has been taking into account. The local particle density Pp,n(r)) is the diagonal part of (5.2).
Forr a completely degenerate system at zero temperature the reduced single-particle den-sityy matrix becomes
p ^ T . r ' )) = 2 ^ 9 ( u - En) i t >n( r ) o | > ; ( r ' ) = G , ( r , r ' ) (5.3)
n n
withh 0 the step function. The diagonal part gives the local particle density p ^ r ) of the completelyy degenerate system. The |x-dependent Green function, as defined here, is obtainedd from the temperature Green function by an inverse Laplace transform similar too (1.37):
II rC+ioO 2
^''^L/^v^'
1
--
(54)
Althoughh Gp (r, r ' ) depends on both r and r ' , it is still a single particle operator. In fact thee only difference with (1.37) is the factor Z / p that we have absorbed in the definition off Gp(r, r ' ) since it leads to somewhat simpler expressions. Of course we still need to choosee c > 0.
Thee n-particle reduced density matrix PaYfiM*') follows from its one-particle coun-terpartt by a symmetrised factorisation:
P ^ ( rn, r - ) == £ ^ r W ^ , r ; m ) . (5.5)
Heree the sum is taken over all permutations of the n position vectors, with en the sign off the permutation. The structure of the n-particle reduced density matrix has been analysedd quite generally for a system of interacting particles by using a path-integral formalismm [22]. The factorisation property for a system of independent particles then followss as a special case. The argument is not changed by incorporating an external magneticc field and a hard wall that confines the system.
Forr the completely degenerate case a relation similar to (5.5) holds for p ^ ( rn}r 'n) . Inn particular, the diagonal part of the two-point reduced density matrix at zero temper-aturee is
p(i2)(rr,,rr/)) = G ^ r)r ) G ^ ( r ' , r/) - G ^ ( r)r ' ) G ^ r,1r ) . (5.6) Oftenn it is convenient to introduce the two-point correlation function
,, ,* p [ r)( r r/>r r ' Puu r Pu r ' r 11 =
-|Gu(r,r r '\\2 '\\2
G u d M - J G ^ r ' . r ' ) ' ' (5.7) ) Inn the following we will study this correlation function, and the influence of a magnetic fieldfield and a hard wall on its properties.
5.22 Correlations in the bulk
Wee consider a system of charged particles which move in a magnetic field directed along thee z-axis. The interaction between the particles is neglected. To describe the magnetic fieldfield we adopt the Landau gauge, with vector potential A = (0,Bx,0). The particles aree confined to the half-space x > 0 by a plane hard wall at x = 0.
Forr positions far from the wall the temperature Green function G p ( r , r ' ) reduces to thee bulk Green function G ^ ( r , r ' ) . The latter is given by [48]
G°p(r,r')) = 1 1 B B v^ r 33 47tsinh(PB/2)
exp p B B
4tanh(BB/2) ) ( T L - r i ] ] xx exp
y(xx + x')(y-v')
exp p zz — zi\2 i\2
26 6 (5.8) )
Unitss have been chosen such that the charge and the mass of the particles drop out, whilee Ft and c have been put to 1 as well. From now on we will often measure distances inn terms of the cyclotron radius I / N / B - TO that end we introduce the dimensionless variabless £, - VÜ(x + x')/2,1 = >/B{x - x'), x\ = V*{y -y'), and C = ^{z-z').
5.2.5.2. Correlations in the bulk 79 9 Thee ^dependent Green function follows by inserting (5.8) in (5.4). One finds
11 fC+ioo G ° ( r , r ' )) = — d 0 ZmZm Jc_i o o ,PB^ ^ B B 11 - q o: 2 5 / 2 ^ 3 / 2 ^ 3 / 22 q o xx exp I i £,ri — 1 + q o : : 8qo o exp p '2'2 1 2£B B (5.9) ) withh qo = tanh(|3B/4). The chemical potential is measured in terms of the energy differencee between adjacent Landau levels, by employing the dimensionless variable "VV — u/B. Let us use now the generating-function identity [41]
: i - q0] ]
8qo o
4q0 0 d + q o )2 2
too express the exponential function in terms of Laguerre polynomials. After substitution off (1 - qo)/(1 + qo) = e ~p B / 2 this gives
«»"l-5&-("-
e
J
£
)£'-(
Ê
?
:
) )
rc'+ioo o
*:M^
d s e
"~
( n + , / 2 , , s
~
3 / 2 e >
*(-ÖÖ
(5
-
U)
wheree we have set s = (JB and c ' = cB. The sum over n can be interpreted as a sum overr Landau levels.
Thee inverse Laplace transform in (5.11) is given by1: 11 pc+ioo
2 7 l ii Jc-ioo
d s es ts -3 / 2e -a / ss = ^ L sin(2y/cd)d{t) sin(2y/cd)d{t) (5.12) )
forr a > 0 and c > 0. Use of this identity in (5.11) results in the following expression forr the ^.-dependent Green function in the bulk:
. ^ ^ « j n ^ v - t nn + l/ZK).
( 5 1 3 )Thee prime indicates that the summation is only over those values of n for which v — ( n + 11 /2) is positive, i.e. over the Landau levels that are at least partially filled. The bulk densityy follows as the diagonal part of (5.13)
/2R3/22 ,
Pnn = G j ( r , r ) = ^ ^ Y V - v - ( n + 1/2). (5.14)
n n
whichh is indeed identical to (2.22). The two-particle correlation function is found upon substitutionn of (5.13) and (5.14) in (5.7).
Thee expressions (5.13) and (5.14) are particularly useful for strong fields when only aa few Landau levels are occupied. An alternative expression, which is useful for weak fieldsfields only, has been derived in [26]. To study the limit B —> 0 in (5.13), we return to thee original variables, since measuring the distances in terms of the cyclotron radius, orr the chemical potential in terms of the energy difference between adjacent Landau levels,, becomes meaningless for vanishing magnetic fields. The number of terms in the summ becomes large, as the upper limit is proportional to B_ 1 at fixed |i. Furthermore, thee argument of the Laguerre polynomial gets small for fixed x — x' and y — y ' . Hence, wee can use the asymptotic form of the Laguerre polynomials [41]
Ln{ u ) « eu / 2J o ( V 2 ( 2 nn + 1)u) (5.15) whichh is valid for u / ( n + 1/2)1 / 3 <C 1. Use of this approximation gives
'iB(xx + x ' ) ( y - y ' ) ' G ° { r , r ' )) « -~r exp IF IF ^ ' j o ( A / 2 B ( nn + V 2 ) | r - r i l ) s i n [ v / 2 [ u - B ( nn + 1 / 2 ) ] ( z - z ' ) ] „.„ XX — y= (5.16) y/B[z-z') y/B[z-z')
forr small magnetic fields. The subscripts denote the transverse parts of the position vectors,, which follow by projection on the xy-plane.
Whenn B approaches zero, the number of Landau levels becomes very large, and their spacingg becomes very small. Therefore, it is permitted to replace the summation over Landauu levels in (5.16) with an integral. In the limit of vanishing B we get
G S ( r . r ' ,, - 4 , j > , . ( V S i r , - r i l ) f W Ö ! £ z £ l . (,1 7 ) Withh the help of the identity2
ff d x xv + 1 s i n ( b V a2- x2) M x ) = Jo o
5.2.5.2. Correlations in the bulk 81 1
ë ë
jj a^+^2 b (1 + b2) - /2-3'4 J-v+ 3/2(ci>ATb2 ) forfor -v = 0, we arrive at G j ( r , r ' )) = 2V 4H3 / 44 T ^3/2 2 rr — r 3 / 2J
3/2(v
/2iI|r-r
/|). .
(5.18) ) (5.19) )Notee that the right-hand side is an isotropic function of the position difference, as shouldd be the case for a vanishing magnetic field. We can simplify it further by using thee explicit form for the Bessel function
.. . fl s i n u — u c o s u
]]
^^[u][u] = Sn u ^ " (5.20) )
Thee final result is G ° ( r . r ' ) =
22 Ir -TV-TV- \r
y/ÏÏLy/ÏÏL cos (y/2\L \r - r ' | ) - sin ( V ^ i |r - r ' | )
rr — r ' (5.21) )
whichh is identical to what one gets by starting from the temperature Green function forr the non-magnetised system
r . r '' = (27t|3)1 1 3
/2 2 exp p
rr — r i\2 i\2
2(3 3 (5.22) )
andd applying (5.4). The bulk density in the field-free case is p^ = (2|x)3/2/(37t2). The two-particlee correlation function in the bulk follows upon inserting (5.21) in (5.7). Inn figure 5.1 we have plotted the bulk correlation function for B = 0 and for B ^ 0 withh "v = 2 and v = 5. For non-vanishing magnetic field we focused on the correlation functionss with position differences that are either parallel with, or perpendicular to, thee magnetic field. For large fields, or, more precisely, for small y, the correlation func-tionss for the parallel and the perpendicular directions differ considerably. For somewhat largerr v, however, the correlation functions become fairly similar, both in the nodal structure,, and in the amplitudes. As it turns out, these similarities are manifest already forr -v = 5, where the number of completely filled Landau levels is still rather low. Comparingg (5.21) with (5.13), we see that by turning on the magnetic field, the range off the correlations in the plane perpendicular to the magnetic field becomes smaller,
0.00 0 -0.01 1 -0.02 2 (a) ) 0.00 0 -0.01 1 -0.02 2 (b) )
Figuree 5.1: Bulk correlation functions g(xyz,xy'z) (-BB ^ 0 and (a) -v = 2, (b) "V = 5, and g(r, r ' ) ( ) for B forr r = r ' .
),, g(xyz,xyz') ( ) for 0.. All curves start at —1
withh a Gaussian instead of an algebraic decay. In contrast, the range of the correla-tionss in the direction parallel to the magnetic field becomes somewhat larger. In fact, althoughh the decay remains algebraic when the field is switched on, the dominant con-tributionn in the tail of the correlation function becomes inversely proportional to the squaree of the distance, whereas it is inversely proportional to the fourth power of the distancee in the field-free case.
5.33 Path-decomposition expansion
Introducingg the wall at x = 0 makes the temperature Green function dependent on thee distance from the wall, i.e. the coordinate x. In the absence of a magnetic field the influencee of the wall on the Green function is easily found from a reflection principle [45].. The temperature Green function becomes
G p ( r , r ' )) = G0p ( r , r ' ) - G0p ( r , r " ) (5.23) ) withh the bulk Green function Gp as defined in (5.22) and the reflected position r" definedd as [%" ,y" ,z") = {—x',y',z'). Likewise, the u-dependent Green function gets thee form
5.3.. Path-decomposition expansion 83 3 withh the bulk Green function (5.21).
Whenn a magnetic field is present, the influence of the wall on the properties of the systemm is more difficult to determine. In chapter 3, we have seen that the temperature Greenn function can be found from a path-decomposition expansion
G p ( r , r ' )) = £ G(pn ){ r , r ' } (5.25) )
n = 0 0
wheree G ln )( r ir' ) is the contribution from paths that hit the wall n times. This path-decompositionn expansion, which was first formulated in [7], is fully equivalent to thee multiple-reflection expansion as introduced in [8], and discussed for a confined magneticc system in [31]. We have also seen in chapter 3 that for r and r ' at large distancess from the wall, the terms with small n in (5.25) are more important than thosee with larger n . In particular, the n = 1 term will give us the leading-order correctionn on the bulk quantities. The latter correspond to n = 0, so that one has Gp0 J(r,r')) = G £ ( r , r ' ) . If no field is present the expansion terminates after the term withh TL = 1, for all distances from the wall.
Ass we have seen in chapter 3 (see (3.24)), in the particular case where x = x', the transversee part of the n = 1 term has the form
B22 & -0) ) •-L.P P to>to> xx y')y') = - ^ 7 i \ d T f j ^ x y . x y ' ) exp [ g ^ x y . x y ' ) ] . (5.26) 167tV2 2
Thee functions fp^(xy,xy') and g ^ t x y . x y ' ) are given by
f">(xW)) = l
I t l t 2 )'
/ 2\(- + -)
Ip.-rlXy.Xyy ) 2 S ) S 2 ( t l + t 2) l / 2 [ ^t, + t J and dll + in
++ t2 (5.27) ) (5.28) ) withh ti = tanh(TB/2), s} = sinh(TB/2), t2 = tanh((0 - T ) B / 2 ) and s2 = sinh((3-T)B/2). .
5.3.11 Casey ^ y '
Iff we set p = q2/ ( q o2 - q2), with q = tanh[(2T - p)B/4] and q0 = tanh(PB/4), we cann write (5.26) as
Jo o dp p
VPÜTPJ J
£
v /ii + n-q
2,)p +
qo o0+(l-qJ)p_ _
exp p[[ (1-qo
LL 2 q
0 - q o2) PP g 2 1 - q o2 1 + ( 1 + qQ2) p 2 £ ,z --16q00 1 + ( 1 - q02) p T T (5.29) ) Whenn £, is large, only small values of (1 - q02)p contribute, which implies that we cann set ^/l + (1 — qo2)p « 1. Furthermore, we can replace the last exponential func-tionn with exp[-(l - q o2) r |2/ ( 1 6 q o ) ] , at least as long as n is finite. If we make those substitutionss and usedp p
e- aPP = eQ/2K0( a / 2 ) (5.30) )
.oo y W
+P)(wheree Ko is the modified Bessel function of the second kind), we get
r
(1))(vi. v i , ' i ~
B(f , " i
n\ l - q o
2Iff we are only interested in the non-degenerate case, where in general (1 — qo2)£2/qo is large,, we can use the asymptotic expansion for Ko [41]. Multiplying the result with the longitudinall Green function, which is the same as in the bulk, we find the first-order correctionn to the total temperature Green function in the approximate form
i~d)rr / \ B \/1 — qo2
Gf>> W , » ) Z) * - y/ 2 „3/2^1/2 q„ eXP 2q00 2 8q0 (5.32) )
However,, for the degenerate case we need the full complexity of (5.31).
Inn order to obtain results for the degenerate case, we now apply (5.4). In doing so we choosee the contour of integration by setting p = (it 4-1 )£/B:
G ^ W . x y ' z ) ^^
-B3 / 2 2 167t3£,1/2 2 jlr\/2 jlr\/2 ee-vl(it+\) -vl(it+\) - o od t( i tt + l )3/2 (l(l _L lT1 n \ T - q O ' xx exp ll + qo 2 ? 4qo o £- i ^
| K o o 1 - q o: : 4qo o 3/2 2 (5.33) ) Inn the new variables q0 equals tanh[(it + 1 )l/4]. From £, » 1 one finds q0 s» 1, which inn turn implies 1 - q2, « 4 e x p [ - £ ( i t + 1 )/2]. This also means that for finite r| we may53.53. Path-decomposition expansion 85 5
replacee £, + rnqo/2 with £,. With the help of the series representation of the modified Bessell function
Ko(u)) = f_
n = 0 0 ,m=1 1 1 1 222n2n{n\){n\)u u : : 2n 2nandd the integral relation
f f
ooo e( i t + 1 ) x 271X^"1d tt T^ TT- = „ , , 9(x)
ooo ( i t + i ^ r(-v)
{ v > 0 ) ) wee arrive at the asymptotic expression
G( 1 )(xuzz xu'z) « - *3/2*-e-lZ/2eil"/2e-*2/& Y ' ^
xx v / - v - ( n + l / 2 ) 1 1 (5.34) ) (5.35) ) (5.36) ) 2 2 4tv-(n+1/2)] ] m = ll x '
whichh is valid for large \/Bx. Again, we recognise the sum over Landau levels. In fact, apartt from the phase e1^2 and the factor e_T1 / / 8, the result is identical to what we foundd in chapter 3 for r = r'.
5.3.22 Case z ^ z '
Thiss case is considerably simpler than the y ^ y '-case, since the temperature Green functionn factorises. The transverse part of the Green functions follows from (5.31) by takingg n = 0. Using the asymptotic expansion for the Bessel function and introducing thee longitudinal part of the Green function, one finds the total temperature Green functionn for the non-degenerate case with z ^ z' as
( i l . .
exp p
VV 2 c1o 2|3B B (5.37) )
Gpp ( x u z , x y z ' ) « -2 7 / 2 7 t3/ 2p1 / 2 whichh is the analogue of (5.32).
Too calculate the |x-dependent Green function G^ (xyz.xyz') for the degenerate case wee need to keep the Bessel function in (5.31). With s = 0B we get
G ^ x y z . x y z ' ) »» -B3 / 22 l 1 rC+ioo e^ s ! _ q o 8TT2 2 xKo o
_LL f
27tii Jc_i i o od 5s3 / 22 q o3 / 2P P
-Qo'fi -Qo'fi 4q qWee are free to choose the contour of integration, as long as it is in the right half-plane. Byy choosing c large (i.e. c = £,)> we can make the same approximations for qo and 11 — qo2 as before. In terms of the integration variable s these approximations read qoo ~ 1 and 1 — qo2 « 4 e ~s / 2. Together with (5.34) this yields
G i / W . x y z ' ) )
InIn1 1 u = 0 0 —— d s e[ v - ( n + 1 / 2 ) ] s 11 1 ë4 nn l 22n(n!)22 2 ^ , •£+ioo o £,—ioo oi—-(¥)] ]
2s s (5.39) ) Thee integral in this expression is still an inverse Laplace transform. With the help of (5.12)) and the analogous identity [19]•II r C + i o O I 11 d s es ts -1 / 2e -Q / s = 27ti i
Vnt t
c o s ( 2 v / a t ) 6 ( t ) ) (5.40) ) forr a > 0 (and c > 0) we finally get the asymptotic expressionGUW.xyz')*** -
BB
3/23/2ll _
e e e n5 / 22 e / 2 2r r
: 4 n n 22 n( n ! )2 2V-v-(nn + 1/2)
c o s (v/ 2 [ - v - ( n + 1 / 2 ) ] C ) ) 4 [ - v - ( nn + 1/2)] (5.41) )+ +
s i n ( v / 2 f r - ( n -- + 1/2)]t) V 2 [ v - ( nn +1/2)] Cti-'-(? ?
.TRR — 1forr large \/Bx. This result looks a bit more complicated than (5.36). In the limit C —> 0 wee recover G^ (xyz.xyz), as calculated in chapter 3.
Beforee we discuss the expressions (5.36) and (5.41) in more detail, we will first show thatt the same results can be derived using a different approach.
5.44 Parabolic cylinder functions
Resultss equivalent to those of the previous section can be derived by using the explicit representationn of the temperature Green function in terms of parabolic cylinder func-tions,, which are the eigenfunctions of the transverse part of the system Hamiltonian.
5A.5A. Parabolic cylinder functions 87 7 Becausee of translation invariance in the u-direction it is convenient to use a Fourier transformm and to write the transverse part of the Hamiltonian as
H ( k ) = - i ^^ + l ( B x - k }2. (5.42)
ass we have seen in chapter 2. For fixed k, this Hamiltonian with boundary condition \Jj(k,, x = 0) = 0 defines an eigenvalue problem. In terms of the eigenfunctions 4>n(k, x)
andd the corresponding eigenvalues En(k) we define the Fourier-transformed transverse
energyy Green function
Gx)E(k,x,x')) = £>n{k,x)ii>;(k,x') MEn(k) - E ] (5.43)
n n
wheree the eigenfunctions are normalised to J£° dx |4>n(k,x)| = 1. In terms of this
energyy Green function the (total) temperature Green function (5.1) is Gp(r,r')) = (27T3)-1/2 exp[-(z-z')2/2{5]
(
OOO i rOOd E e "p EE ^ - d k ei k ( y-y , ) G E(k,x,x'). (5.44)
Performingg the inverse Laplace transform as in (5.4) and using (5.12), we obtain
7tZZ Jo Z — Z' J_oo
(5.45) ) Wee now choose x = x', as before, and switch again to the dimensionless variables £, £ (andd a dimensionless Fourier variable K = k/\/B as well). In terms of these, the energy Greenn function is
G.L,E(1C,X,X)) = VBV_ n{ J V L j- T 6 [E - Ben(K)]. (5.46)
nn j r d ^ D ^( K H / 2[ ^ ' - K ) ]
ass we have seen in chapter 2. Here DA(U) is a parabolic cylinder function. Furthermore, en(*00 is determined by the boundary condition
Den(Kj-i/2(-V5K)) = 0. (5.47)
Iff we now carry out the integration over E in (5.45), we get
Di.m-./2[^(t-«)] ]
, J v ^ ( £ - i c ) l l(5.48) ) J T ^ ' D jn W- i / 2 ^ ' - « ) ] ]
-0.2 2
Figuree 5.2: Correlation functions (a) g(xyz, xy'z) and (b) g(xyz, xyz') for £, = 1 (( ), £, = 5 (— • —) and in the bulk (£, —> DO, ) for -v = 1.5. The curves forr £, = 5 and for the bulk are (almost) indistinguishable.
wheree Kn("v) is determined by en [KU(-V)] = -v.
Inn figure 5.2 we have plotted the resulting correlation function, which we calculated numericallyy for several values of £, and a value of the chemical potential, which corre-spondss to a completely filled lowest Landau level. Even at moderately small distances fromm the wall the influence of the confinement on the correlation function is not very big.. In the y -direction the correlations become a little stronger. The same is true for thee correlations in the z-direction, at least for small C However, the oscillating tail in thee correlation function is suppressed by the presence of the wall.
5.4.11 Casey ^ y '
Takingg the limit z' —> z or C —> 0 in (5.48) is trivial. The resulting formula contains aa phase-factor e1KT1, which is absent in the case y = y ' as considered in chapter 2. We firstfirst split off the bulk contribution
G^(xyz,xyy z) = ——^— £_
TV-TV-d K e. ( K+« n > A ^ + V 2 )D2{_ v 5K) )
'nn\ 'nn\
(5.49) ) Thee remainder G^(xyz,xy'z) = GH(xyz,xy'z) — G°(xyz,xy'z) gives the correction duee to the wall.
5.4.5.4. Parabolic cylinder functions 89 9
Inn order to obtain an asymptotic expansion of G^(xyz,xy7z) for large x we split the integrationn at K' = a'£,, with 0 < a ' < 1 — \\fl, and K" = a"f,, with a " > \\fl. As inn the case with r = r ' , the contributions from the intervals [Kn(-v), <'] and [K",OO) decayy faster than exp(—£,2/2). With the help of the asymptotic expressions derived in chapterr 2 € n ( K ) - ( n + i ) ) y/nn\ y/nn\ 2^ii p—K2 K2 n+ ! (5.50) ) and d
{^di'D{^di'D
22£n{K]£n{K]__
y2y2[V2(i'-K)]}[V2(i'-K)]} «
i i i i y/nn\y/nn\ 7T(TI!)2 2H.++ 1 e~ K2 K2n+1 yy —ln(\/2K .. m _m=l l onee finds V ^ B3 / 22 » - ' 22 n + 1 , GJ(xyz^^ 71^ *— 7t(n! r (xü'2)«^-5—— L - — -I^ - ( T V + 1/2) (5.51) ) D2n ( K )_1 / 2[ V 2 ( ë -- K)] « 2 " e - (^ 2 ( £ - K )2* ++ - ^ L - 22 n + 1 e-{l~K]2 e-< 2 (I, - K)2 U K2TV+1 In \Vï(l - K)1 (5.52) VTtrt!! L xx P d K ei K T 1e -K 2e -( l-K ) 2K2 n + 1 ( E - K )2 nPn{ K , ^ - K ) (5.53) J K ' ' with h Pn(K >£ , - K ) = --4 f c - ( uu +1/2)] * - . m ++ X - - T - l n [ 2 K ( £ , - K ) ] l . (5.54) Wee now expand the integrand in (5.53) around K = £,/2:
J
K " " d K ei K 1 1e -K 2e -( l-K ) 2K2 n + 1( l - K )2 nP n ( K , ê - K ) ) K' K' ei£V2e-£.2/22 ( ^ /2)4 n + 1 Pn(£,/2,£/2) d t ei t n - 2 t 2 2 (5.55) )sincee the main contribution comes from the region around K = £,/2, at least for £, » 1. Evaluatingg the remaining integral gives us exactly (5.36).
(a)) (b) 1.00 rr 0.5-- 0.0- 0 . 5 0.0-0.0- ---1.0 0 0 2 4 6 88 10 0 2 4 6 8 10 ii n Figuree 5.3: Comparison between numerical data ( ) and asymptotic form ( )
off the real part of G^(xyz,xy'z)/Gji(xyz,xyz) at (a)£, = 3 and (b) £, = 5 for-v = 1.5.
Notee that in this case, instead of G^'fxyz, xy'z), we have in fact calculated the com-pletee correction G^(xyz, xy 'z). But since the terms beyond n = 1 in the path-decom-positionn expansion (5.25) are of higher order, the asymptotic form of G£(xyz, xy 'z) is inn leading order identical to that of G^ (xyz.xy'z). In the present case we can easily takee along more terms in (5.54) (see chapter 2). The expansion around K = £,/2 given heree is only valid for n <C £,. If n is of the same order of magnitude as £,, the terms that wee left out would not be small compared to the leading term and we would not get the rightt asymptotics. In figure 5.3 we compare the results of a numerical evaluation of the exactt expression for the real part of the wall correction G^(xyz, xy'z)/G^(xyz, xyz), whichh follows from (5.48) and (5.49), and the asymptotic form in leading order, as derivedd in (5.36). As expected, the agreement is best for small r\. The imaginary part of G^(xyz,xy'z)/G^(xyz,xyz)) behaves in a similar way.
AA striking feature of the asymptotic expansion (5.36) is its Gaussian decay proportional too e_ r | / 8. It is slower than the decay of the bulk u-dependent Green function (5.13), whichh is proportional to e^11 / 4. This somewhat slower decay is indeed corroborated byy the numerical evaluation of G^(xyz,xy'z)/G^(xyz,xyz), as is shown in figure 5.4. Forr small n and large £, (i.e. for the regime where (5.36) holds) the curves converge to thee asymptotic value 1.
5.4.5.4. Parabolic cylinder functions 91 1 1.2 2 1.0 0 0.8 8 0.6 6 1 1 byy evaluation 1.55 and £, = 5 ( -of f -), ,
Figuree 5.4: Check of the Gaussian decay, V S n - U l o g t l G ^ x u ^ x u ' z J l / l G ^ x y z . x u z ) ! ) ]1/2,, for v =
£,, = 4 (— • — ) , £, = 3 ( ). The dotted line ( ) is the asymptotic value accordingg to (5.36).
5.4.22 Case z £ z'
O nn setting y = y' in (5.48) the phase-factor elkT1 drops out. With the help of the samee asymptotic expressions as in the case with y ^ y', and the same splitting of the integrationn interval, we arrive at the asymptotic expression
./2R3/22 , 22 n + 1 , G^(xyz.xy'z)) « ^ - £ j ™ V^-(n + 1/2) (n!)' ' I I --dKe -K 2e -( l-K )) K2 n + 1( l - K )2 nQ n ( K1l - K , C ) (5.56) with h n ,, F cos ( y a - v - (n + 1/2)] C) sin ( y / 2 h ^ ( n + 1/2)] C) ^ < , i ,, K,g 4[ ^ - ( n + 1/2)] ^ f v - ( n + 1/2)] C x | ^ l - Y - l n [ 2 i c ( l - K ) ] l .. (5.57) Again,, the main contribution to the integral comes from the region around K — £,/2.
Expansionn around this point allows us to evaluate the integral, and we recover (5.41). Inn figure 5.5 we compare the asymptotic expression (5.41) for G£(xyz,xuz') with numericall data. As expected, the differences get smaller for increasing values of £,.
5.55 Correlations for large separations |r — r'|
Thee path-decomposition expansion, which we employed in section 5.3, is expedient for largee x only. However, the range of validity of the representation of the previous section iss not limited to that regime, so that we can use it for small x as well. In particular, itt is helpful in determining the asymptotic behaviour of the correlation function for finitefinite x and large distances between the points of observation, both in the y- and the z-direction. .
5.5.11 Large | y - - y ' |
Lett us go back to (5.48). In order to bring the square-root singularity at K = Kn(-v) to thee fore, we write
/2R3/22 , poo
5.5-5.5- Correlations for large separations |r — r ' | 93 3
Figuree 5.5: Comparison between numerical data ( ) and asymptotic form ( ) off G^(xuz,xyz')/G^(xyz,xyz) at (a) £, = 3 and (b) £, = 5 for "v = 1.5. In the latter casee the two curves are (almost) indistinguishable.
ith h
4>n(K)) =
v-v-
•v•v — e-n. KDL(
K)-,/2[^^)] ]
(5.59) ) Thee function c|)n ( K) is analytic on the integration interval. For K —i oo it is proportional too K2n-i/2e-K Hence, t n e asymptotics of (5.58) for large |n| are determined by the lowerr boundary of the integral only. With the help of the method of stationary phase [16]] we find poo o den(K] ] dK K KK = K n M , | -3/2ei [ K n M t ll + (37t/4)sgn(r|)] p r 3 ) Dl_,Dl_,/2/2[V2(j-K[V2(j-Knn(v))] (v))] ^dl'Dl_,^dl'Dl_,/2/2[V2(l'-K[V2(l'-KnnM)} M)} (5.60) )wheree we used that the derivative of en(i<) is negative. In chapter 2 we have derived the identity y
d l ' D2 2 e „ ( i c ) - 1 / 2 2
(V2(l'-<))\ (V2(l'-<))\
Thiss equality allows us to get rid of the normalisation integral in (5.60). In this way wee arrive at the following asymptotic expression for the |A-dependent Green function att large |TJ| and finite £,:
g3/2ei(3n/4)sgn(r|) ) 23/22 ^5/2 XX D * _1 / 2[ v ^ KnM ] D j _1 / 2[ v ^ ( l - KnM ) ] D3/2pi(3n/4)sgn(ri)) i , G . ( x y2, xy' z ) «« 2e3 / 2 ? t 5 / 2 ! * ( - * + * ) - ^ £ e " "M* (5.62) dic^M M d'v v - 3 / 2 2
Thee asymptotic expression simplifies considerably for the special case of a completely filledfilled lowest Landau level, that is for "V = 3/2 and n = 0. By using the representation of thee parabolic cylinder function in terms of confluent hypergeometric functions [41] we findfind KO(*V) = 0 and d K o M / d v = —y/n/2. Employing (5.61) and inserting Di (\/2u) = \ / 2 uu exp(—u2/2), we arrive at the simple asymptotic expression
, ,, 4B3/2ei(37t/4)sgn{T1) }
G ^ ( x y z , x u ' z ) «« - ^ £,2e 4 — ^ (5.63)
forr large |r|| and -v = 3/2. The asymptotic behaviour proportional to |T|P / in (5.62) andd (5.63) is clearly induced by the presence of the wall, since the decay of the bulk u-dependentt Green function in the y-direction is Gaussian, at least in the presence off a magnetic field (see (5.13)). The algebraic decay is corroborated by a numerical evaluationn of (5.48). Both results are compared in figure 5.6.
5.5.22 Large \z-z'\
Too determine the asymptotics for large separations in the z-direction we set y — y' in (5.48).. Subsequently, we need to determine the asymptotic behaviour of the imaginary partt of the integral
poo o
11 =
K. MM I?dVDlM_}/2[V2(V-K)]
forr large |C|. In order to obtain the asymptotics, we split the integration interval at K'' » 1. For K > K' we can use (5.50) and (5.51). In the numerator we insert the asymptoticc expression for the parabolic cylinder function
5.5.5.5. Correlations for large separations |r — r ' | 95 5 U.UU^O O 0.0020 0 0.0015 5 0.0010 0 0.0005 5 I I II I I I
I I
II
- --I l l l 00 5 10 15 20 25 30Figuree 5.6: Comparison o f y /27 i2 |Gc^(xyz,xu 'z)| / ( v ^ B3 7 2) ( ) a t £ = 3,"V = 1.5 andd its asymptotic value ( ) for large |r)|, according to (5.63)
whichh is valid for large K and finite £,. This relation can be derived by using (5.50), and thee asymptotic relations
Dn(V2u)) « ( - 1 )n2n / 2 |u|n e "u 2 / 2 (5.66) ) and d
aD^fv^u) )
Q-v v • l ) n + 1 2 -n / 2> / n n !! |u| - n - 11 ou 2 / 2 (5.67) )whichh are both valid for large negative u. In this way the contribution from K > K' to (5.64)) becomes
// 2n _ 1/2e_ K 2 K2 T V + 1 dKK exp V 2 [ - v - ( n + 1/2)] |CI - i := , ; = = ... ICI V 5 t n !
>/ - v - ( n + 1 / 2 ) )
2n + 22 2
Wee now introduce a new integration variable t = e K K2 n + 1, which implies K
y/—y/— l n t for large K. The integral then gets the form
2TX+1 1
v ^ n ! !
ïy/ibt-di+i/mm ïy/ibt-di+i/mm r t ' ' d tt exp —i 2T1--1/22 |£|
^ ^ ^ - ( 1 1 + 1 / 2 ) ) s i n h2( ^ l n t £ , - £12/ 2 ) )
- l n t t (5.69) ) withh t ' = e K K ' n < 1 . Note that there is a logarithmic singularity at t = 0. Thee asymptotic expansion of an integral of the form
r t ' '
d t el u t( - r n t ) ^ ^ (5-70) )
forr t ' < 1 and large u has a contribution from the lower boundary and the upper boundary.. The contribution from the lower boundary is3
-:n-1 1
r = 0 0 ,k=0 0
z z
irHj irHj
r - k ' '
( l n u u M--T T
(5.71) )
wheree T(k5 (u) is the Icth derivative of P(u). For functions f {— l n t ) that can be expressed ass a Laurent series in V—lnt for small t, one obtains the asymptotics of the integral
r t ' '
d t em tf ( - l n t ) ) (5.72) )
forr large u by integrating the series term-by-term by means of (5.71). In fact, the con-tributionn from the lower boundary of the integral is
(-D
T T T = 0 0Ki;
,k=0 0r,i
Mf f
r - k k dr f ( l n u ) ) d ( l n u )rr ' (5.73) )Lett us now return to (5.69), which contains an integral that has the form of the complex conjugatee of (5.72). For this particular case one has f (—lnt) = sinh [y/— l n t £, — £ ,2/ 2 ) / ( - l n t )) and u = 2n-^2 |CI /[y/nn\ ^ / v - ( n + 1/2)]. The term with r - 0 in (5.73)) dominates the contribution from the lower boundary to the asymptotics, since
5.5.. Correlations for large separations |r — r ' | 97 7
ff (lnu) is much bigger than all its derivatives when u is large. Using this fact, we get for thee contribution from the lower boundary (t = 0) to the asymptotics of (5.69)
jj ^ ci,/2[.-(n+1/2)]|CI Smh2(s/h^l-V/2) w
y/nn\uy/nn\u l n u
- i ^ ^ T i ^ l —— JV&-WWici s i n h2 / ^ Q I _ ii/2\ (5.74)
Kilnn ICI v v
Thee contribution from the upper boundary (t = t ' ) to the asymptotics of (5.69) can bee found by use of the same theorem from [50]. It is expected to cancel against the contributionn from the end-point K = K' of the integral
122 = ' " '' d K eV2h-en(K}]|tl
P
en(K)-V2[v^(^ K)] , , „ v
Indeed,, it is not too difficult to show that it does, by evaluating the contribution from thee end-point at K = K' in I2 with the help of standard techniques [16]. Finally, we havee to check whether the end-point at K = Kn("v) contributes to the asymptotics of I2.. The phase has a square-root singularity at that point. Using this fact, one finds that alll terms in the asymptotic expansion that originate from the lower boundary of I2 are real.. Hence, they drop out when taking the imaginary part of I.
Collectingg our results, we have derived the asymptotic equality
Imm I « - V ^ E v - ^ 1 + 1/2)1 c o s ( v,2 f v_( n + 1 / 2 ) ] c ) s i n h2 { v / i^ | t _ t2/2)
(5.76) ) whichh is valid for large |£|. As a consequence we obtain the ^-dependent Green function
2 B3'22 sinh2 ( v / ï n l c U - l2/ 2 )
G^xyZyXyz')G^xyZyXyz') ^
-7i22 C2ln|CI
xx Y_' y/2\y - (n + 1/2)] cos (yjl[v - (n + 1/2)] C) (5.77)
forr large K|. This asymptotic form is valid at large separations \z — z'\ and a finite dis-tancee x from the wall. A comparison with (5.13) shows that the asymptotic behaviour changess substantially owing to the presence of the wall. The simple algebraic tail (mod-ulatedd by a trigonometric factor) which is valid in the bulk, is replaced by a more subtle decayy involving a logarithmic dependence on \z — z'\.
5.66 Correlations near the wall
Lett us return again to (5.48). Using (5.61) we may write it as s i n (v/ 2 h ' - en( K K B3/22 , G^(xyz,xy'z')) = - -j^TZ- dKe e 1KH H
Mr r
xx D2en(K)_]/2(VlK) D2eniK)^/2{V2(l- K)] de^fK) ) dK K (5.78) ) Forr small £, the last parabolic cylinder function can be expanded in a Taylor series. Accordingg to (5.47) the zeroth-order term of this series vanishes, so that the leading termm for small £, is of first order in £,. The derivative of the parabolic cylinder function occurringg in this term fulfils the Wronskian identity [41]2 v ^ ^
De n(K)-i/z(>/2K)—— D „ ( - V 2 K ) - r r - e ( K I + M
0 KK
v = en( K ) - 1 / 2 ' L € u l K j - t -2J
wheree (5.47) has been used again. Solving for the derivative we obtain 2B3/22 12
(5.79) )
G^(xyz,xyy z
n* n*
•L L
r0
°° iK11 sin(V2[-v-en(K)]C) den(K) d K ee
C dK
Kn(-V) )
(5.80) ) Thee right-hand side can be simplified further by introducing the integration variable ee instead of K. In this way we arrive at the following approximate form of the u-dependentt Green function near the wall:
G„(xyz,xy'z') )
2 B3 / 2 £ 2 2
7U^ ^
L' L'
d e e e i Kn( e ) r | | wv[y/2{y~e)t\ wv[y/2{y~e)t\ (5.81) )
n+1/2 2
Ass before we consider the cases r| ^ 0 and C 7^ 0 seperately.
5.6.11 Casey ^ y '
Forr z' —> z the Green function becomes
23 / 22 B3 / 2 £2 GH(xyz,xy'z) ) 71^ ^
r r
d e e e i Kn( e } T i i y/yy/y — e. n+1/2 2 (5.82) ) Forr large |r|| the asymptotic behaviour of the integral is determined by the value of the integrandd near the upper boundary. One findsy 2 B3 / 2ei ( 3 7 t / 4 ) s g n ( l l )) I2 r J" ' ~ 'n~3 / 2 Gn(xyz,xy'z) )
7t3/2hi i ,3/2 2
V "" eiKn(-^)Tl d K n M M
5.6.5.6. Correlations near the wall 99 9
Thee same result can be obtained from (5-62) by expanding the last parabolic cylinder functionn for small £, and using (5.79).
Thee asymptotic decay proportional to |T||~ ' in (5.83) seems to be at variance with thee behaviour of the correlations in the field-free case. For the latter case one finds
G^(xyztxy'z)) » _ _ ^ L _ sin [ v ^ ( u - y ' ) ] (5.84) byy expanding (5.24) with (5.21) for small x and retaining the terms dominant at large
|yy — y '|. Hence, the correlations near the wall decay faster in the field-free case than in thee case with field. It should be noted that (5.83) is valid for |n| » 1 or |y — y ' | > B- 1 / 2.. For B tending to 0 the region of validity thus shifts towards oo. Furthermore, thee number of terms in the sum becomes very large for small B (since y gets large), whereass the factor in front tends to 0, so that taking the limit B —> 0 in (5.83) is not trivial. .
Thee decay of the field-free correlations can be derived in the present context by starting fromfrom (5.82). Let us interchange the summation and the integral and split the latter at e ' » 1 . W e g e t t
[ e - l / 2 ] ]
giK^fejTi i
™™ Jn+1/2 h/2 £T0
d e v ^ II Y_ e
iK-
(e)T1. (5.85)
Thee sum in the second part at the right-hand side consists of many terms for each e. As discussedd in the appendix, the values of <n(e) are located in the interval [— y/ïë, y/2e\, att least for all n < [e — 1 /2] — 1. As a consequence, their average separation goes to zero ass 1/\/€. It is therefore expedient to replace the summation over n by an integration overr a continuous variable a = Kn(e)/\/2ë. The second part in (5.85) then becomes
[[ d e v ^ l f dcrei v / 5 ¥ a T 1 p(e,cr) (5.86)
withh p(e, cr) do" the number of values of Kn(e) in the interval tV^ëcr, y/ïë{a + do-)]. Forr large |ri| the dominant contribution in the integral over cr comes from the end-points.. Since the density p{e, cr) at the endpoints cr = 1 is 23 / 2e ( 1 ^ a )1 / 2/ n (see the appendix),, the expression (5.86) is for large |rj|
———-\———-\ dee'/4y/ï=l cos (y/Ü\i\\-3m/4). (5.87)
Thee integral over e can likewise be evaluated for large |T[|, as once again only the end-pointss of the integral contribute. The upper boundary gives
- v/2 ^ s i n ( v/2 ^ T i )) (5.88)
whilee the contribution from the lower boundary in (5.87) will not be needed.
Ass a final step we have to consider the first part in (5.85). Here the summation can be replacedd by an integration for e » 1 only. Of course, the contribution of the upper boundaryy to the asymptotic expression has to cancel that of the lower boundary in (5.87),, since the final result should not depend on the choice of e'. The lower boundary off the integral over e in the first part of (5.85) does not contribute either. In fact, only thee term with n = 0 survives for e near 1/2 and the factor Ko(e) goes to infinity for ee —» 1/2, so that for large |n] the contribution from the lower boundary damps out by interference. .
Collectingg the results, we have found that for B —> 0 and large |T|| the asymptotic behaviourr of the ^.-dependent Green function near the wall is given by (5.82) with (5.88)) inserted:
4 D 3 / 2 - . F 2 2
G^(xuz,xy'z)) « J-J=- s i n ( v ^ T i ) . (5.89) Afterr restoring the field-independent variables this result coincides with (5.84).
5.6.22 Case z ^ z'
Inn this case the dependence on Kn(e) drops out from (5.81):
2 B3/2Ê22 ^ - / f s i n [ V 2 [ v - en( K ) R ]
G ^ x y z . x y zz ) « = — > d e L- J~. (5.90)
nn
n Jn+1/2 C Thee integral is elementary. As a result we get for the Green function at small x
2 P 3 / 22 F2 / ,
GH(xyz,xyz')) « - 3 £ { V 2 [ - V - ( T I + 1 / 2 ) K cos [y/2\y - (n + 1/2)] C]
n n
- s i n [ V 2 [ v - ( nn + l / 2 ) ] C ] } . (5.91) Byy taking the limit C —» 0 we obtain a simple form for the particle density near the
wall: :
25 / 2B3 / 2 F 22 ,
P ^ G ^ T . T ) * »» ^ * ^ fr" (n + 1 / 2 ) ]3 / 2 (5.92) n n
5.7.. Discussion and conclusion 101 1
whichh should be compared to the bulk density (5.14).
Forr large separations the first term between the curly brackets in (5.91) is dominant, so thatt the ^-dependent Green function for large separations and small distance from the walll becomes
2R3/22 ?2 ,
G ^ x y z , x y z ' ) «« TT^Y- V2I-V - (n + 1/2)] cos [y/2\y - (n + 1/2)] C].
7171 C
n
(5.93) ) Thee same expression can also be found from (5.77), by putting £, « : 1 and expanding thee hyperbolic function for small values of its argument.
Forr small B it is convenient to return to the original variables x, z — z', [if and to
introducee the new integration variable u = eB in (5.90). Replacing in addition the summationn by an integration we get
MM 2* 2r „ r „ s i n [ v / 2 ( n - u ) ( z - z;) l
G ^ x y z , x y z ' ) « —— d t d u ^ — ^ LI U., (5.94)
^^ Jo Jt z ^
Performingg the integrations we obtain the expression
G„(xuz,xyz')) « - ^ ^ ^ ^ {2sin [ v ^ ( z - z ' ) ]
-S(z^F
s i nt^(^-
Z')]}} (5.95)
forr the u^-dependent Green function at positions near the wall in the field-free case. It cann easily be checked that the same expression is found by expanding the general form (5.24)) for small x.
Att large separations \z - z'\ the dominant term in (5.95) is the first one. It agrees with (5.84),, when y and z are interchanged.
5.77 Discussion and conclusion
Ourr results show that the presence of a magnetic field and of a wall leads to remarkable changess in the pair correlation function of a completely degenerate non-interacting electronn gas. In the bulk the correlation function follows from (5.7), with the \i-dependentt Green functions (5.13) and (5.21) for the cases with and without field.
Whereass the correlation function is isotropic and decays algebraically (ex r- 4) in the field-freefield-free case, it becomes anisotropic with a Gaussian dependence in the transverse directionn and an algebraic one (oc r- 2) in the parallel direction.
Inn the neighbourhood of a plane hard wall the pair correlation function becomes anisotropicc even for the field-free case. Its form, which follows directly from a reflec-tionn principle, is given by (5.7) with (5.24). For two positions far apart, but at equal distancee from the wall, the ensuing correlation function decays algebraically, as in the bulk.. However, the decay is proportional to r- 6, as follows from (5.84).
Iff in addition to the wall a magnetic field is present, the pair correlation function in the vicinityy of the wall becomes anisotropic for two reasons, as both the field and the wall breakk the symmetry. The general expression for the correlation function, for arbitrary fieldd strengths and arbitrary distances from the wall, follows by substitution of (5.48) inn (5.7). Relatively far from the wall the corrections to the bulk correlation function aree still small. The first-order correction terms were given in (5.36) and (5.41), and checkedd numerically in figures 5.3 and 5.5. Near the wall the correlation function -andd in particular its tail for large separations - is modified considerably as compared too its form in the bulk. In fact, the qualitative difference between the behaviour in thee directions parallel and transverse to the field, which is a prominent feature of the bulkk correlation function, is lost in the vicinity of the wall. In both directions the tails becomee algebraic, albeit with a different exponent. This is seen by inspecting (5.62) (or (5.83))) for the transverse direction and (5.77) (or (5.93)) for the parallel direction. In thee former case the decay is proportional to r~3, whereas in the latter it is proportional too r~4. Qualitatively, the change in the decay of the transverse correlation function from Gaussiann in the bulk to algebraic near the wall can be understood in a semi-classical picture.. In the bulk the cyclotron motion of the particles leads to a strong localisation off the correlations, which is associated with a Gaussian decay. On approaching the wall,, the so-called 'skipping orbits' along the wall become important. They lead to aa delocalisation effect in the particle motion, which manifests itself as an increase in thee range of the correlations. In this way, the cross-over to an algebraic decay of the correlationn function finds an explanation.
Inn conclusion, we have shown that for a non-interacting electron gas the influence of a walll on the correlations is quite considerable, especially in the presence of a magnetic
5.5. A. Appendix: Zeroes of the parabolic cylinder functions 103 3
55 *A Appendix: Zeros o f the parabolic cylinder functions for large argumentt and index
Thee zeros of D*(z) satisfy the inequality \z\ < 2 ^ + 1/2, with the possible exception off a single negative zero4. For A » 1 the zeros in the neighbourhood of \z\ = 2x/A + l / 2 cann be determined by approximating the parabolic cylinder functions by Airy func-tions. .
Forr positive z and a — z/U^A-l-1/2] slightly smaller than 1, one has [1]
D*(z)) » 2*/2 + 1/3 T(lA + 1){A + I )1/6 ( y ^ i ) Ai ([4(A + 1 ) ]2/3T ) (5.96)
withh T given by
TT = - { f [|arccos(a) - | o V l - c r2] }2 / 3 « - 2 "1 / 3( 1 - a). (5.97) Forr x » 1 the Airy function may be approximated as [1]
A i ( - X )) K
^ 7 7 s i n ( t *V 2 + f ) • (5.98) Usingg this approximation in (5.96), one finds the zeros of the parabolic cylinder
func-tionn from the zeros of the sine function. In terms of the variable o", the zeros near a = 1 aree found as
« m - 1 / 4 ) 1 -- ( 5.9 9 )
,, 1
^^ = 1" 2 2(AA + 1/2)
withh positive integer m. Since A is large, the zeros are closely spaced. In view of the mainn text we introduce £ = A + 1/2 instead of A. Writing the number of zeros between 0"" and a + do- as p(£, CT) do" we get
23 / 2 2
p(e,<r)) = e ( l - o - )1 / 2 (5.100)
71 1
ass the density of zeros near <J = 1.
Similarly,, for negative z and a slightly bigger than —1 the parabolic cylinder function cann be written as [1]
DA(Z)) « 2 V 2 + V 3 r ( 1 A + 1 ) ( A + 1 jl/6 (-^\'^
xx {COS(TTA) Ai ([4(A + l)]2/ii) - sin(TiA) Bi ([4{A + i ) ]2 / 3- r ) } (5.101)
withh T « - 2 ~1 / 3( 1 + cr). With the use of (5.98) and the analogous relation
1 1
Bi(-x) ) cos s
(!*
3/22+ f)
(validd for large x), one obtains the zeros of DA(Z) near o = — 1 aso\nn = - 1 4- ^ 37t(mm + A - l / 4 )
T 2 / 3 3
2(AA + 1/2)
forr integer m with m > — A + 1 / 4 . The density of the zeros is found to be
23/2 2 p(e,ff)) = £ ( 1 + a }1 / 2 7t t (5.102) ) (5.103) ) (5.104) ) forr a near — 1.
