Spontaneously self-organized structures in electron plasma

Nuclear Physics B220 [FS8] (1983) 185-195 (~) North-Holland Publishing Company

S P O N T A N E O U S L Y S E L F - O R G A N I Z E D S T R U C T U R E S IN E L E C T R O N PLASMA M. NOGA ~ Research Institute for Theoretical Physics, University of Helsinki, Finland

Received 30 August 1982 Properties of phase transitions between disordered and ordered states in a system of interacting electrons are derived from first principles. It is shown that the formation of a static spin density wave state is due to a phase transition of the third kind.

T h e s p o n t a n e o u s formation of self-organized structures out of states without any o r d e r is one of the most interesting physical p h e n o m e n a . W e investigate this p h e n o m e n o n applying merely first principles to a system consisting of an electron gas, or m o r e precisely N interacting electrons confined to a box of volume V and having a b a c k g r o u n d of uniformly distributed positive charge to guarantee electrical neutrality. O v e r h a u s e r [1] has pointed out that such a system m a y b e c o m e unstable with respect to the formation of a static periodic magnetization. His conclusion is based mainly on the a r g u m e n t that this system e m b e d d e d in a given external static and periodic magnetic field has lower energy than the same system without the external field. This a r g u m e n t for the formation of a periodic structure would be absolutely convincing provided that one proves the following properties of the electron gas. That the interacting electron gas can indeed s p o n t a n e o u s l y g e n e r a t e a static m a c r o scopic and periodic electromagnetic field, and that the energy of the generated electromagnetic field does not exceed that portion of the energy by which the energy of the electron gas is decreased. F u r t h e r it is necessary to specify the conditions u n d e r which the periodic structure as a g r o u n d state is energetically preferable. These question will be analyzed below. To carry out this p r o g r a m it is very convenient to use the functional integration m e t h o d [2, 3] for the calculation of the partition function ~ of the considered system. T h e r e f o r e we start by considering the non-relativistic lagrangian density On leave of absence from the Department of Theoretical Physics, Comenius University, 842 15 Bratislava, Czechoslovakia. 185

M. Noga / Electron plasma

186

.Lf of electrons interacting with an electromagnetic field, ~=-h4'~

4"+(-ihV-e A

-

4"+~BB'4"+c,4'

c 3 ~

dx - ~1e 2t9 ~ (x, 7)~b(x, T) PI - [~--x'[ 4"+( x ',~ ' ) 4 ' ( x " r ) + r t ° t b + ~ b

+

1 8rr

(

" A te

1 /~2_B2)+ c

V

4~rc

f d3x ' ~

+ ' ~')~(x',r) 4' ( x , .

(1)

H e r e A ( x , r) is the vector potential of the electromagnetic field satisfying the radiation gauge condition

V. A =0.

(la)

B = V x A is the magnetic field, ~b+(x, ~-) and tb(x, r) are two c o m p o n e n t electron spinors, /zB is the B o h r m a g n e t o n , ~ are Pauli matrices and rt0 is the chemical potential of the electron gas. All fields are functions of the four vector x = (x, r) in the euclidean four-dimensional space which is related to Minkowski space by the relation t = - H , where r is a real parameter. T h e dot on the top of the functions denotes the derivatives with respect to the p a r a m e t e r r. Next we write the action S as the sum of three functionals S =

dr

d3x,.~=Se(~',~b)+Sf(A)+Sl(tll~,4',A)

)

c o r r e s p o n d i n g to the lagrangians of the free electron field ~=-h~,~$

~

1

~0+(-ihV)2~+ri ''4'~6,

free electromagnetic field

and to the interaction lagrangian

.~l =

ie h -2m-----~A • ~b+V~b

1 2---,

Az4'+4' +/x~B • 4'+~r4'

I

d3x'

-~_e ~ (X,'C)~(X, T) _ I--X-------_X,[4" ~(X', ~')4"(X','r) ie "4 " ~7 I d3x', tb+(x ', 7")~b(x', "r) + 47rc ~ in that order. H e r e /3 = 1/kBT, ka, is the B o l t z m a n n constant and T is the temperature.

M. Noga/ Electronplasma

187

The partition function ~r of the system under consideration is given by the following functional integral [3] ~=

I I ~A

@0'NOexp

{1-h[S~(O ,O)+S,(A)+&(~+,O,A)] ) ,

(2)

over anticommuting g,* and ~ variables and over a space of functions A satisfying the gauge condition (la). Suppose one carries out the functional integration over the anti-commuting 0 ÷ and ~ variables in the functional integral (2). After this integration the functional integrand in (2) becomes dependent only on the vector potential A and can be written down in the following form ~=f~Aexp{~I

'

drfd3x~,(A)I=f~Aexp{-~S~,(A)],

(3)

where ~'¢~ is an effective lagrangian. The last relation tells us that the interacting electron gas can be described entirely by means of the effective lagrangian which is a function only of the electromagnetic potential A. All quantum effects due to Fermi statistics of fermions can be completely implemented in the effective lagrangian. If the fermion system under consideration is capable of giving rise to a self-organized macroscopic periodic electromagnetic field A then this field must be a solution to the classical equation of motion 8*,~ctt - 0. 8A

(4)

Therefore our next step is to calculate the effective lagrangian *L~'en. The aforementioned functional integration over anticommuting variables cannot be carried out exactly, but only by employing the perturbative expansion. For this purpose we introduce the partition function ~o of the non-interacting electron gas in the form ~.,,= f c~4~+~¢~ exp {~1 So(~+, ~0)}. The statistical mean value of any function F [ $ * ( x t ) , d,(x2)] with respect to the action S~ is given by

(F[~ +(xl), ~(x2)])~--7-~..f c~ ~ F [ & " (x~), tb(x2)]exp {~ Se(~b+, tb)} . We will frequently use the statistical mean value of the product ~,~, (xl)t~2(x2) which has the explicit form (g,~, (x 1)~,,Cx2)) = ,S..... D (x 1 - x 2), where 0"1 and 0"2 denote the spinorial components. The function

D(xl-x2)

is the

M. Noga / Electronplasma

188

statistical propagator having the following representation [3] +~ I dsk exp[iw,(rl-r2)+ik(xl-x2)]

'

15)

where we have used the following abbreviations: oJ. =(~/flh)(2n +1), e ( k ) = (h2/2m)k 2. The statistical propagator D (xl -x2) allows one to use a diagrammatical description of individual contributions to the functional S..(A) in terms of Feynman diagrams [3]. With this notation the partition function ~ given by the relation (2) takes the following form

,6, Next we use the cluster expansion theorem, namely /

exp

/~

Sx

})

=exp

/,~

1 } ~M!hM ($~)~ ,

(7)

where the subscript c denotes contributions only from the connected Feynman diagrams. From the relations (3), (6) and (7) we get the effective action Sc.(.4) in the form

&,(a)=s,(a)+h E

1

M lM!h M

(s~),.

(8)

For our purpose we have calculated contributions dependent of the vector potential A coming from the first four terms (S~)~ M = 1, 2, 3, 4 of the infinite series (8). For the next step in our consideration it is sufficient to give the explicit formulae only for (SI)¢ and (S~)¢

2mc2A2(x)n~+e2 I ~ fJh

13 fi

)

2

j

tl~l

D(x-x')D(x'-x)},

(9)

2 C

× [A(x2) " ~2D(xl -x2)] 4

-- [2~2C4 A 2(x I)A 2(x2) + 2/z 2B"(Xl) • "(x2)]O(x2 - X l)O(x l - x2) 2ie3h x A 2(xl)D(x2-xl)[A(x2) • V2D(xl -x2)] m c 4ie3h f d3x~ , + mc [A(Xl) "VlD(X2-XI)] j ix2_x,2lDtxl-x'2)Dtx2 -x2) 2e +mc A2(xl)

;

i [x2d-!Z'-x2lD(xl-x2)D(x2-x'2)D(x'2-xl)+C } ,

(10)

M. Noga / Electron plasma

189

where C denotes those Coulomb exchange terms which are independent of A, and n~ is the mean density of the electron gas. Now for a moment we will consider only "classical" terms in the integrands (9) and (10) which are independent of the Planck constant h, i.e. we neglect the terms which are of the order of h. From the relation (9) and (10) we get the following classical contribution to the effective lagrangian 1 ( 1 2 _ B 2) [ e2/'/e q._ e4bl ~A 2(X} 5~,,,:-ff-~ \-~-~ A - \ 2 m c 2 2mc:} 1 ItJs 2h ~ dr1 I d3xl { e 2 ~ c 4 A 2 ( x ) A 2 ( x l ) D ( x - x l ) D ( x l - x ) + O ( h ) + C '

}, (11)

where O(h) includes terms of the order of the Planck constant, C' denotes Coulomb exchange contributions independent of A. The positive constant b~ is given by the integral bl=-~

~ d~'2 d3x2

8 7r3

/

~

3

dx2 , , Ix2-x21' D ( x l - x e ) D ( x e - x 2 ) D ( x 2 - x l )

r0/k,)r,,(k=)[i

wherc

fo(k) = {1 +exp fl[e(k) - no]}-',

(12)

is the well-known Fermi function. As we stressed above, the spontaneously generated electromagnetic field must satisfy the Lagrange equation (4). From the effective lagrangian (11) one derives the equation AA(x) =

4

~ , ~e 2nc + e 4,o l ) A ( x )

mc

4 4"n'e +~A(x)

ft3hf)

dr1

d3x1A2(xl)D(x-xl)D(xl-x),

(13)

for the macroscopic electromagnetic field. The last equation is a complicated non-linear integro-differential equation. However it has an exact non-trivial solution in the form of a static circularly polarized wave

A(x)=a(eeiq.~+e,

e iq.x),

A 2 ( x ) = 2 a 2,

(14)

where a is an arbitrary constant amplitude, q is the wave vector, e and e* are the polarization vectors satisfying the relations

e.e=e*.e*=e.q=e*.q=O,

e.e*=l.

190

M. Noga / Electron plasma

From eq. (12) we get the constraint on the wave vector in the form q2=_

4 2 -~ -~ b 2 ,

8rre a

47rz(e2ne+e4bl)_,..

mc

(15)

m c

where b2 is another positive constant given by the integral _lf~3h

b2=

hJo

dr' f d3x'D(x-x')O(x'-x)="f~ 3 f''(k)[1-/°(k)]"

The interacting electron gas may thus indeed generate a static periodic magnetic field provided that the right-hand side of eq. (15) is positive. This can be always achieved by choosing a sufficiently large amplitude a. Thus the first task of our program is fulfilled. Next we must find out the conditions under which the periodic structure of our system will correspond to the ground state, i.e. the state with the lowest energy. We evaluate the individual contributions in the sum (8) with the vector potential A ( x ) given by the relation (14). In the sum (8) we calculate the contributions to the fourth order of the electromagnetic effects. For practical calculations it is very convenient to transfer the interaction term 2

e

Z- m e 2

2

AZ(xRb+(x)~(x)_ "

2

e a tb+(x)O(x) m¢

(16)

'

from the interaction lagrangian , ~ I into the free electron lagrangian ~ . This transfer will only entail a shift of the chemical potential from the value ~7o to the value rio--' rt =rio

2 2 e a

mc

2,

(17)

in the Fermi function fo(k) given by relation (12). From now on we we will use the Fermi function

f ( k ) = {1 + e t~r"k)-'l} "1 ,

(18)

with the effective chemical potential (17). From the relations (6), (7) and (8) one derives the grand canonical potential .Q = _fl-1 In ~ of the system under consideration in the form l/l

1"2 = .Q~ - ~

1 1 -z 1 -~ 1 -~ Sf(A ) +-~ (S,)~ + 2 T ~ (SI )~ +3~_-_~(Si )~ + 4 T. ~ (S~ )c + "

where /2,

2vf

/3(27r)3

d3k In {1 + e a f ' - " ~ l } ,

}

,

(19)

(20a)

is the grand canonical potential of the ideal electron gas with the effective chemical

M. Noga / Electron plasma

191

potential (17), ~S,(A)=

(20b)

4 ~ q 2 a 2,

and S~ denotes the action due to the interaction lagrangian ~f'~ without the term (16) and evaluated for the electromagnetic field A(x) given by the relation (14). By straightforward, however algebraically cumbersome calculations, one gets for the individual terms in the sum (19) the following formulae 2e V d3kld3k2 ~-~ (S~)c = (~--~)5 ~-~S_~2)2f(k~)f(k2),

(21a)

1 (~)c= #2ma2V f d3k[2k2+q2_2 (k .q)2]

j

-~-2

2h2fl

"~e

1

<,.~3),; lz~eZrn~ :V f

3!h3fl

-

f(k)

q2 + 2k • q

+O(e 6)

(21b)

'

f(k2) J d3kl d3k2 (kl-k2) 2

~'Sh4

(k,. q)2][f(kt +q)-f_(k,)

[r

×/[2k{+q 2-2

q2

- [ 2 ( k 1 " k2)+q 2 - 2

J[

q2+2k,'q

(kl'q)(k2"q)] q2

flh 2 f(k,)[1 - f ( k , ) ] ] +2m

~+2k;:q

J

f(kl+q)-f(kl)

(q2+2kt " q)(q2+2k +2k • q)

+O(e6),

(21c) 4

1 (~)c = 4!h4-----fl

.}

4

3

i'T3h 6

~'

+2k-q

2 k 2-

q2

k2 +2q 2- ( k 'q2 q)2

x[q2flh 2 1 - f ( k ) 2 3 ] m q4-4(k • q)2 (q2+2 k . q)2 2(q2 2k . q)(q2+k . q) ,I-~h 2

1 - f(k)

+q [2--mmqr~-k_q

2 ] }

(q2+2 k .q)2

-FO(e6),

(21d)

where O(e 6) denotes terms which are either independent of the vector potential or includes electromagnetic processes which are at least of the 6th order. The formulae (20a) and (21) represent the contributions to the energy of the system from the electromagnetic processes depicted in fig. 1. In fig. 1. the solid lines represent the electron propagators, the dashed lines represent Coulomb interaction, the vertices with the wavy lines denote electrons interacting with the spontaneously organized static and periodic magnetic field. The Feynman diagrams (b), (d) and (e) represent corrections to the kinetic energy of the electrons while those (a) and (c) describe corrections to the Coulomb interaction. The Feynman diagrams (e) come from the term (20a).

M. Noga / Electronplasma

192

Ic)

(d}

{e)

Fig. 1. Lowest order diagrams contributing to the energy of the electron plasma. In the formulae (21) the ordinary integration cannot be carried out analytically, but only numerically. From numerical results it is very difficult to extract the behaviour of the system in the neighbourhood of the critical point which is always associated with a certain type of singularity of the thermodynamical potential. It is however very instructive to study the behaviour of the thermodynamical potential in the vicinity of the critical point analytically which requires appropriate approximations. Our approximation is to assume that at the conditions allowing for the spontaneous formation of the macroscopic periodic structure in the system, the interacting electron gas may be considered as a non-degenerate gas of fermions. In this approximation one may replace the Fermi function (18) by the Boltzmann distribution function f ( k ) = e ~("-'~k~l .

(22)

At the phase transition point the wave vector q and the amplitude a associated with the periodic structure (14) are very small entities. This allows us to expand the functions (21) as a series in powers of q2 and a 2. It should be stressed that the integrals (21) give functions of the variable q2 which are analytic at the point q = 0. For the sake of a short notataion we introduce the abbreviations c~-=

~Sh 2 , 2m

2

u-~c~q ,

v=

J8e 2a 2 2 • mc

(23)

Retaining terms to the fourth order of the electromagnetic effects and to the third power of small quantities q2 and a 2 in the expansion of the functions (20a) and (21) we carry out the integration with the distribution function (22) to get the following result W e t3n'' .C/e- 4fl(,trot)3/2 (1 - v +½v2) , (24)

1 ~ ~(

e2V ,)_16~.2

2h2B 1

3! h3/~

2e2°",,(1-2v),

4~ (o~)3/2 v{1 +~u - ~ u 2 (g~)¢_

(25a) 1

v

-.~uL') + O(e6) ,

e2-----V-V e2m~,,2v(l+~Ju-u2)+O(e 6)

16"tr 2o~2

(25b) (25c)

M. Noga / Electron plasma

193

1 (~)= V e t3,''' 4! h 4 / 3 4/3(Tree)3/2 ~v 2(1 - ~ u ) + O ( e 6 ) .

(25d)

By making use of the relations (19), (20b), (24) and (25) we obtain for the grand canonical potential 1"2 of the system at the critical point the following expression 12 = 1 2 o + ~ q 2 a

2

V et3'~"v 1 l 2 V e 2 2 e2tJn"2t'(~ ~u - u 2 ) • 4O(rrot)3/2 ()u - lsu - ~ ! u v ) - 16¢r2ct

(26)

Here 12o is the grand canonical potential of the electronic gas in the absence of the spontaneously generated periodic electromagnetic field. Note that all terms in (23) and (24) which are independent of q2 have cancelled among themselves in the sum (26). The form of the grand canonical potential (26) is indeed such that its correction term due to the interaction of electrons with the generated magnetic field vanishes if any of the parameters q2 or a 2 is zero. The grand canonical potential (26) at the critical point is expressed as a sum of /20 plus small increments. From the given grand canonical potential (26) we can express any thermodynamical potential by choosing the proper thermodynamical variables. The small increments to all thermodynamical potentials, expressed in terms of the corresponding thermodynamical variables are equal. Hence expressing the correction to 1"2 in (26) in terms of the temperature T, volume V and the electron density nc by using the relation e t~''' = 4(rra

)3/2n e ,

we obtain directly the correction to the free energy F in the form F = F o + V ( y t J v +61V/g2 +62UV2) ,

(27)

when we have used the following abbreviation 2m2c 2

n¢

Y =4rr•2e2h 2

3fl

2~.2

81 =

62 =

20rr e2h2n~j8 9

m

(28a)

2~

ne ~re n ne/J + , 15B rn llne

4t3

(28b)

(28c)

The free energy (27) at a given temperature T, volume V and density nc is a function of two non-negative parameters u and v. The ground state of the system corresponds to the absolute minimum of the free energy with respect to any free parameter. Hence we get the relations y v + 261uv +621) 2 = 0 ,

(29) y u + 61u2 + 262uv = 0 .

M. Noga / Electron plasma

194

There are two solutions to the last equations, namely u=t~=0,

(30)

if 3' > 0 and 3' u = ---, 38x

v-

3' 382 '

(31)

if 3' < 0. For non-degenerate electron gas with n~ ~ 10 ~8 cm-3the linear terms in n~ entering the relations (28a) and (28b) can be neglected with respect to the quadratic terms in he. By introducing the electron plasma frequency o~p and the critical temperature Tc by the relations

2 4"tre2nc oJp = m

(kBT¢)~_ 5 (hoop)4 18 mc ~ '

(32)

we express the coefficient 3' at the critical point by the formula

3m2c2 2 3" - ~ e ~ k a T ~ ( T - To).

(33)

Now we substitute the solutions (30) and (31) into the expression (27) to obtain •the free energy of the system above and below the critical temperature in the forms F = Fo, 3

T > T~,

(34a)

4t3

80 m c K a F = F o - V 9 9 ¢-r e 2 1n. 4 wp2 (T~- T) s '

T
(34b)

Thus the electron gas exhibits indeed a phase transition from the disordered state into a state with a static and periodic magnetization. From (34b) one sees that this phase transition is of the third kind and not of the second kind as was conjectured in [1]. The temeperature dependence of the wave vector q and of the amplitude M of the spontaneous magnetization below the critical temperature are 2[10mk~]1/2

q = ~L~J

M=']2qa-3

( T , . - T) w2 ~ ( T ¢ - T) ~ ,

8 mc2kB [ 1 0 k n T c ] l / 2 hop [ l l e 2 h 2 J (T~-T)~(T~-T)~'

K = ½,

(35a)

/3=1.

(35b)

The critical exponent associated with the wave vector q is K = ~ and that associated with the amplitude of spontaneous magnetization is /3 = 1. Both q and M are monotonically decreasing functions of temperature. Qualitative arguments used in [1] have led Overhauser to an incorrect conclusion that the wave vector q is monotonically increasing function of temperature. Once the static period magnetic field is spontaneously generated it will influence the energy spectrum of electrons. The energy spectrum of electrons will have

M. Noga / Electron plasma

195

essentially the same band structure as that described in [1] and [4]. In this case the band structure will exhibit a variation with temperature. In this c o m m u n i c a t i o n we have c o n c e n t r a t e d only on the s p o n t a n e o u s formation of the periodic structure in a system of interacting electrons. We should h o w e v e r note that the integro-differential equation (13) allows for a n o t h e r non-trivial solution, namely A = (0, B x , 0),

(36)

giving rise to a ferromagnetic g r o u n d state of the electron gas with the constant magnetic field B. T h e m o r e detailed analysis of the s p o n t a n e o u s l y generated magnetic structures (14) and (36) and of phase transitions between disordered, ferromagnetic and periodic g r o u n d states of the system will be given in a f o r t h c o m i n g c o m m u n i c a t i o n [5]. It is clear that a similar p r o b l e m can be formulated for c o n d e n s e d systems of c o l o u r e d quarks interacting via non-abelian gauge fields in q u a n t u m c h r o m o d y namics (QCD). Interest to study such a system is e n h a n c e d by the fact that there are indications [6] that the Q C D g r o u n d state corresponds to a constant colour magnetic field. The a u t h o r wishes to express his sincere indebtedness and gratitude to Professor C. C r o n s t r 6 m for m a n y enlightening discussions on self-organizations of o r d e r e d states and on possibilities to describe fermion systems entirely in terms of gauge fields. He is very grateful to Professor Stig Stenholm, the director of the Research Institute for Theoretical Physics at the University of Helsinki for hospitality and stimulating discussions.

References Ill A.W. Overhauser, Phys. Rcv. 128 q1962) 1437 [2] F.A. Berezin, The method of second quantization (Acad. Press, New York, 1966); I..D. Faddecv and A.A. Slavnov, Gauge fields: introduction to quantum theory (Benjamin Cummings, Reading, 1980) [3] V.N. Popov, Functional integrals in quantum field theory and statistical physics /in Russian) (Atomizdat, Moscow, 19761 [4] C. Cronstr6m and M. Noga, Phys. Lett. 60A (1977i 137 [5] M. Noga and 1.. Vodn~i, in preparation [6] N.K. Nielsen and P. Olescn, Nucl. Phys. B144 (1978) 376: ll.B. Nielscn and M. Ninomiya, Nucl. Phys. B156 (1979) 1