- Email: [email protected]
On Bose-Einstein condensation in any dimension
8
September 1997 PHYSICS
ELSEYIER
Physics Letters A 234 (1997)
LETTERS
A
13-19
On Bose-Einstein condensation in any dimension H. Perez Rojas ’ International Centre for Theoretical Physics, Trieste, Italy Received
11 September
1996; revised manuscript received 14 May 1997; accepted Communicated by P.R. Holland
for publication
19 June 1997
Abstract A general property of an ideal Bose gas as temperature tends to zero and when conditions of degeneracy are satisfied is to have an arbitrarily large population in the ground state (or in its neighborhood) ; thus, condensation occurs in any dimension D but for D < 2 there is no critical temperature. Some astrophysical consequences, as well as the temperature-dependent mass case, are discussed. @ 1997 Published by Elsevier Science B.V.
1. What is Bose-Einstein
condensation?
At present there is a renewed interest in BoseEinstein condensation (BEC), particularly after its experimental realization [ 11. Actually, BEC is one of the most interesting problems of quantum statistics. It occurs in a free particle Bose gas at a critical temperature T,, and is a pure quantum phenomenon, in the sense that no interaction is assumed to exist among the particles. BEC is interesting for condensed matter (superfluidity, superconductivity) but it also has increasing interest in high energy physics (electroweak phase transition, superfluidity in neutron stars). The consequences of its occurrence, at dimensions different from D = 3, may be interesting in these two fields of physics. Bose-Einstein condensation is understood as the steady increase of particles in the state with zero energy [ 21, or as the macroscopically large number of particles accumulating in a single quantum ’ Permanent Address: Grupo de Fisica Te6rica. ICIMAF,Calle E No. 309, Vedado, La Habana inf.cu.
4, Cuba. E-mail: [email protected].
037%9601/97/$17.00 @ 1997 Published PII SO375-960 1(96)00509-4
state [ 31. For a free gas the Bose-Einstein distribution ( e(E-p)‘T - 1) -’ for ,X < 0 and y # 0 vanishes strictly at T = 0, which suggests that at T = 0 no excited states of a Bose gas can exist on the average, and condensation in the ground state seems to be a general property whenever the conditions of quantum degeneracy of the Bose-Einstein gas are satisjed. Quantum degeneracy is usually understood to be achieved when the De Broglie thermal wavelength A is greater that the mean interparticle separation N-‘j3. However, the remarkable discovery made by Einstein on the Bose distribution was that condensation may occur starting at some critical temperature T, different from zero. BEC leads to a second-order phase transition for dimensions D > 2, since by decreasing the temperature some critical value is found at which condensation starts. The general case of condensation in an arbitrary dimension was first studied by May [4] for D 3 2 and later by Ziff, Uhlenbeck and Kac [5] for D 2 1, and as a consequence of their results it is usually stated that in the thermodynamic limit BEC is not possible in D = 2 and that it neither occurs for D = 1. But in mathematical physics it has been
by Elsevier Science B.V. All rights reserved.
H. Perez Rojas/Physics Letters A 234 (1997) 13-19
14
shown that in presence of external field-like boundary conditions [6,7] critical conditions are found even forD= 1. Concerning what is to be understood as BEC there are two different ideas which are usually considered to be the same: ( 1) The existence of a critical temperature T,> 0 such that ,u(T,) = I&,EO being the single particle ground state energy. (This condition is usually taken as a necessary and sufficient condition for condensation; see e.g. Ref. [ 81) . Then for T 6 T,, some significant amount of particles starts to condense in the ground state. (2) The existence of a finite fraction of the total particle density in the ground state and in states in its neighborhood at some temperature T > 0.We shall call ( 1) the strong and (2) the weak criterion. From the point of view of finite-temperature quantum field theory, the strong criterion for BEC leads to the infrared k-* divergence of the Boson propagator,
,$
OG(k4 - ip,k,T)
1 N +2+@,
1 =
g’
which is cancelled by the density of states h-k* when calculating the particle density. High temperature radiative corrections usually have the effect of shifting the (longitudinal) boson mass in an amount SM* 'v T* (Debye screening), Spontaneous symmetry breaking (SSB) has a close connection with BEC and it plays an important role in statistical mechanics (see e.g. Ref. [ 91 and references quoted therein). An important relation between BEC and SSB was found by Fannes, Rule and Verbeure [lo], who by assuming two body interactions derived a generalization of the Bogoliubov condensate equation and showed that BEC occurs if and only if there is a spontaneous breaking of a (first kind) gauge symmetry. In SSB gauge field theories it is the vacuum which bears the property of a broken symmetry, and its analogy with a Bose-Einstein condensate leads to consider that temperature may provide a mechanism for symmetry restoration, as suggested by Kirzhnits and Linde [ 131. Some authors, as Haber and Weldon [ 111, and Kapusta [ 121 have investigated the relation between SSB and BEC in temperature quantum field models by including some chemical potential. But in quantum field theory the charge operator corresponding to a SSB symmetry does not annihilate the vac-
uum [ 141, it is an ill-defined operator and it is in general not conserved (see Ref. [ 151 and references quoted therein). This leads to conclude that in building the grand-canonical density matrix, the chemical potential corresponding to that charge must be actually taken as zero. We may understand the vanishing of such chemical potential as corresponding in some way to the condition for condensation of Goldstone bosons, the symmetry breaking parameter being the condensate. But this does not imply that such chemical potential is non-zero for temperatures above the critical temperature for symmetry restoration [ 161. In systems of low dimensionality, no SSB of a continuous symmetry occurs in one or two spatial dimensions D according to the Met-mitt-Wagner theorem [ 171; (see also Ref. [ 181 for a proof that there are no Goldstone bosons in one dimension). These results are, thus, in close correspondence with those obtained for BEC, e.g. in Ref. [ 51. In the present Letter we want to investigate again the occurrence of BEC in D = 1,2,3 and later in any dimension for the case of a free gas not in the thermodynamic limit, but having a finite volume and number of particles. Throughout all our considerations, we implicitly assume that our system is described by a Hamiltonian density written in terms of creation and annihilation field operators defined in a finite volume and satisfying periodic boundary conditions (boundary conditions determine the form in which the energy of excited states tend to the ground state value, which is critical for the appearance of BEC [ 191). We will adopt the procedure of investigating the microscopic behavior of the particle density in momentum space, which exhibits some interesting properties. We will discuss briefly at the end some specific features of the relativistic and the temperature-dependent mass cases.
2. The critical temperature
in the D = 3 case
Let us recall the origin of the critical quantities
pc,
T,in the standard 3D BEC theory in which, as pointed out before, we assume periodic boundary value conditions. The chemical potential p = f( N, T) < 0 is a decreasing function of temperature at fixed density N, and for p = 0 one gets an equation defining Tc= fc(N) . For temperatures T < Tc, as P = 0, the expression for the density gives values N'(T)< N, and the
H. Perez Rojas/Physics
difference N - N’ = No is interpreted as the density of particles in the condensate. The mean interparticle separation is then 1= N-‘/3. In our considerations we will use integrals which, as it is usually done, must be interpreted as approximations of sums over discrete quantum states, which does not imply working at the thermodynamic limit. For usual macroscopic systems, as the separation between quantum states is hp = h/V113, the approximation of the sum by the integral is quite well justified. Now, above the critical temperature for condensation
Letters A 234 (1997) 13-19
15
In this sense f(x, p) has a similar behavior than the Maxwell-Boltzmann distribution of classical statistics. But as ii -+ 0, also xP + 0, and the maximum of the density, for strictly ,% = 0, is located at x = 0. The convergence to the limit x = 0 is not uniform. A finite fraction of the total density falls in the ground state. If we go back and substitute the original integral over momentum by a sum over shells of quantum states of momentum (energy states), we can write
Nc=$c
dAP . i=.ePTI2MT _ 1
(4)
m
N =
x2dx eX*+fi _ 1
J
~TA-~
(1)
0
(2)
= h-3&?3,*(ZL
where fi = -p/T( > 0), x = p/pr is the relative momentum, pr = m the characteristic thermal momentum, and A = h/(2mnT) ‘j2 the De Broglie thermal wavelength. The function g, (z ) is (see e.g. Ref. [31), co
_1’ J Z-leX x”--idx
g,(z)
= 1 r(n)
0
where z = efilT is the fugacity. At T = T,, we have g,( 1) = c(n) , and the density is then (3) or in other words, NA3 a microscopic look to this end we investigate defined as the particle space
= c( 3/2) N 2.612. Let us have BEC in the D = 3 case, and to in detail the following quantity density in relative momentum
x2 f3(x,bii)
= &+F
_ 1’
By calculating the first and second derivatives of this function, we find that for p # 0 it has a minimum at x = 0 and a maximum at x = xP, where xcL is the solution of _ e x2+p --
1 1 --x2’
For i = 0, by taking Ap N h/V’j3, where V is the volume of the vessel containing the gas, the contribution of the ground state density is No = 4/V’/3A2. We have thus a fraction of No/N, = 4Ac(3/2)/V”3
= 4A/l(3/2)213N’/3 (5)
particles in the ground state, which is the most populated, as described by the statistical distribution, at T = T,. A numerical estimation for one liter of He gas leads to No/N cv 10P6. In quantum states in a small neighborhood of the ground state, the momentum density has slightly lower values. Thus, at the critical temperature for BEC, there is a set of states close to the ground state, having relative large densities. Eq. (5) indicates an interesting relation: the larger the separation between quantum states, the larger the population of the ground state at the critical temperature. In the thermodynamic limit the quantum states form a continuum, and (5) has no meaning. However, most systems of physical interest, in laboratory as well as in astrophysical and cosmological contexts, have finite V and N. We conclude that at the critical temperature for condensation the density of particles in momentum space has its maximum at zero momentum, and by describing the densiq as a sum over quantum states, a macroscopic fraction is obtained for the density in the ground state and in neighbor states. Thus, at the critical temperature, the weak criterion is satisfied. For values of T < T,, the curve describing the density in momentum space flattens on the p (or x) axis, and its maximum decreases also. As conservation of particles is assumed, we get the ground state density by
H. Perez Rojas/Physics Letter.s A 234 (1997) 13-I9
16
adding to f(x) the quantity 2N[ 1 - (T/T,)3/2] 19(T, T) h3S(x) as an additional density. As T -+ 0, the density in the ground state increases at the expense of non-zero momentum states. This leads to the usual Bose-Einstein condensation. We must remark that in that case, for values of T smaller but close to T,, both the weak and the strong criteria are satisfied.
which indicates that they vanish as T/N + 0, the specific heat C,: = XJf8T N &5(3/2) + O(N/h*) decreasing as T’/* as T + 0. For high temperatures one can easily find that C,, is roughly constant; it indicates some correspondence with the behavior of C,, in the D = 3 case [3], but there is no discontinuity in its derivative with regard to T. By substituting the last expression for ji back in (7), one has
3. The D = 1 and D = 2 cases For a Bose-Einstein gas in the presence of a potential of form (xl/L)” it has been shown by Van den Berg and Lewis [ 6,7] that the critical conditions in dimension D are the same as for the free boson gas in dimension D + 2/cr. Thus, BEC occurs for D = 1,2,. . . in their case. In our model of free gas with periodic boundary conditions, let us see what happens for D = 1. In this case, the mean interparticle separation is 1 = N-l. The density in momentum space coincides with the Bose-Einstein distribution, P2 2MT-fiIT 1) -I. This function has ~I(P,T,P) = (e / only one extremum, a maximum, at p = 0. By using the previous change of variables, we have the expression for the density of particles as 0;)
N = 2A-tr-t/*
I
dx ex*+P - l = $,*w.
(6)
0
We have thus that limgt/2( z ) diverges ment of the quantum actually means that for N constant, and approximately,
N N 2A-tr-‘/*
Nh N gt/2( z). The fact that as z -+ 1 indicates an enhancedegeneracy regime. But this fact p is a decreasing function of T for very small ji one can write,
s ‘*
dx -=EL’/2A. x2 + p
&Q (7)
0
where x0 = pa/MT, po being some characteristic momentum pe > PT. Thus, p does not vanish at T # 0, and for small T it is approximately given by ,G = r/N*A*. It is easy to obtain the pressure P (= force/lo) and energy density respectively as P = ~g3,2(2L
(I=$
(8)
(9) -=I
where y = &/*/2NA. Due to the properties Cauchy distribution, one can write
of the
Y
$=A-
ydx ___
Y J
x*+ y2'
-Y
We can also write 2NA f~(x,T,
N)T/N+o
N -S(x). 77
Thus, for small T/N all the population tends to be concentrated on the ground state, and although not an usual Bose-Einstein condensation, we claim that (as in the magnetic field case, [20] ) we have a diffuse condensation [ 211 (in the sense of not having a critical temperature) when T/N is low enough. There is no critical point; there is no discontinuity in the derivative of the specific heat in this case. The condensation is the outcome of a continuous process since the ground state density is always non-zero, and this quantity can be increased continuously to reach macroscopically signijcant values, by decreasing enough the ratio T/N, even for temperatures farfrom zero. The D = 1 case satisfies the weak criterion at any temperature T, but not the strong one. Closely connected with the D = 1 case is the problem of condensation of a gas of charged particles in the presence of a strong magnetic field [ 201, where all the previous considerations apply, by substituting y = MeBT/Zi*cN, where e is the electric charge and B the magnetic field. If eBfi/McT > 1, the system is confined to the Landau ground state n = 0, and the problem can be studied in close correspondence to D = 1 case. We will consider the model of a gas of kaons in a neutron star [ 221. By taking a density N N 1O44 crne3, T N lo* K and local magnetic
H. Perez Rojas/Physics
field B w 1014 G, the condition of quantum degeneracy is exceedingly satisfied. In that case y = 10m3’. By taking the dimensions of the star as lo7 cm, the discrete quantum states would be spaced by an amount Sp = 1O-34 g cm/s. One half of the total density would be distributed in 17 = 2y/Ap = lo4 quantum states. The ground state density with zero momentum p along the magnetic field, can then be estimated as AN = +‘N = 104o, leading to observable effects: superfluidity and strong diamagnetic response to the applied field. (The magnetization M = eBFi/2Mc N lOI6 G would exceed by two orders of magnitude the microscopic applied local field and would be the preponderating field.) However, the quantity AN/N = -1 = di3Nc/MeBT@/3 tends to zero in the ther77 modynamic limit, and some of the physical effects or condensation would be erased in that limit. Let us now turn our attention to the D = 2 case. Here I = N-l/*. The distribution is x .f*(x,T,fi)
=
exz+P
Letters A 234 (1997) 13-19
the weak and perature T # condensation in the ground can write for
strong criteria are satisfied at any tem0. Thus, there is no strict Bose-Einstein in the sense that the value of the density state is zero for any T/N # 0. As one x small,
limf2(x,T,~),,O=lim~
-&
=%LS
i.e. the density at x = 0 is non-zero only for ,& = 0. But as the maximum of the density increases continuously by decreasing T/N, being located at x,, + 0 (the convergence to x = 0 being non-uniform), we have a sort of quasi-condensate, in the spirit of the weak criterion, i.e. most of the density can be found concentrated in a small interval of values of momentum around the p = 0 state at arbitrary small temperatures. The pressure P (= force/l) and energy density in such case are given by zJ =
_
17
U=p,
$2(z).
(11)
1 ’
and has always an absolute minimum at x = 0 (the density vanishes in the ground state) and a maximum at a value of x > 0 being a solution of
and obviously vanish in the limit T -+ 0. For high T, these quantities behave as linear in T. The specific heat for T -+ 0 is C, N $/3A* + 0( NA* e-NA2/2) and is approximately linear in T.
1
e X2+/?=s-
4. The case
We can write the density as 00
N=2A-* J n
-(X2+p)XdX e = 2A-* 1 _ e-(X’+P)
D > 3
In the case D > 3, the density in momentum for p # 0 reads
In( 1 - e-“).
space
xD--I
(10) We have fi = - ln( 1 - e-NA2/2); thus as T/N --+ 0, P-+e -Nh14nMT. For ,LL= 0 the density N in (10) diverges, as in the D = 1 case, except for T = 0. At non-zero ,& the population in a closed small neighborhood of x = 0 is strictly zero, but the amplitude of this interval decreases as p -+ 0, that is, the maximum of the density in momentum space is reached at a value of momentum xmax # 0 which decreases as T/N --+ 0. (This behavior bears some analogy to the case of a two dimensional boson gas with an external potential studied in Ref. [ 71, where there is a macroscopic occupation of an infinite number of lowlying levels.) The maximum of the density is not in the ground state but in a neighborhood of it. None of
fo(x,T,fiL) =
&+ji
_ 1’
This function has an absolute minimum at x = 0 and an absolute maximum at x = x,,, # 0 given by the non-zero solution of the equation ex2+p = l/ [ 1 2x2 / (D - 1) 1. The density is zero at the ground state, and in a small neighborhood of it, and in this sense differs from the D = 3 and the D = 1 cases. The total density is given by A-D N = r( D/2)
m
J 0
fo dx = A-DgD,2( z).
(12)
In this case, the density p decreases as T --+ 0 and N converges for ,ii = 0. Thus, there is a non-zero
H. Perez Rojas/Physics
18
critical temperature T, such that ,ii(T,) = 0. Then for T < Tc ( 12) is unable to account for the total density and conservation of particles demands that the lacking density NO = N - N’ be exactly at the ground state. Thus, although the density of particles in an (open) neighborhood of the ground state is zero, exactly at the groundstate it is given by 2N[ 1- (T/Tc)D/2]ADS(~). At the critical temperature N,hD = l( D/2), which tends to unity with increasing D. We may conclude that the D-dimensional gas becomes less degenerate with increasing D. For any dimension, the relation between energy and pressure P = 2U/D holds. Also, below the critical temperature, C,, - To/*. We see that the D > 3 case satisfies the strong but not the weak criterion for T < T,. Our results for D Z 3 are in agreement with those obtained in Ref. [ 81. Expression ( 12) is valid for continuous D. It can be easily checked that for 1 > D > 0 the quantity fD ( X) diverges for x = 0, and N remains finite for p # 0. Thus the di$%se condensation takes place in the interval 1 > D > 0. The D = 2-like behavior occurs for 2 2 D > 1, whereas, as demonstrated by May [ 41, usual condensation occurs for D > 2. However, for 2 < D < 3, both criteria, weak and strong are satisfied, fD being divergent at x = 0, whereas N remains finite.
We shall revisit the relativistic case. Here the conservation of particles must reflect some invariance property of the Lagrangian. We are keeping in mind the simplest case of a charged massive scalar field. In that case, the conserved quantity, derived from the Noether theorem is the charge in D spatial dimensions),
J DX, m
jo(n)d
served quantities depend usually on the difference of their average densities. After building the density matrix for the grand canonical ensemble, one can write the thermodynamic potential, and from it the conserved charge as an expression which contains the difference of average number of particles minus antiparticles, 27@*TD
(Q)= r(D/2)
(13)
0
where j, = I+V&@ - (G’,,+*)ti is the four-current, and as different from Ref. [4], we must include in the temperature charge average the contribution from antiparticles: this is a natural consequence of a relativistic finite-temperature treatment of the problem (see e.g. [ 23,11,12]. At high T one must consider the excitation of particle-antiparticle pairs, and the con-
JxD-l Dc)
cDFiD
dx(n,
-n,),
(14)
0
where np = (eE-fi-- 1)-l, n, = (eE+fi-- 1)-l are the particle and antiparticle densities, E = x2 + A?* , and x = PC/T, I@ = Mc*/T. In the present case ,G may be positive or negative, but I,&) < Iii. Condensation in D > 3 occurs for IjZ] = I@. For D < 3 the condensation is very well reproduced by the infrared (nonrelativistic) limit already seen, by taking the chemical potential as ,u’ = p - M. A very interesting case occurs when M = 0. In that case ( 14) demands p = 0 to avoid negative population densities of particles or antiparticles. All the charge must be concentrated in the condensate and the critical temperature for condensation, as suggested in Ref. [ll],isT=co.
6. The case M =
5. The relativistic case
Q =i
Letters A 234 (1997) 13-19
M(T)
In some systems the interactions at high temperature behave in such a way that can be described effectively as free particle systems with variable mass; i.e. the temperature-dependent interaction leads to the arising of a mass M = M(T). If M(T) + 0 we have BEC with increasing temperatures, as discussed in Refs. [ 24,161. We have in that case two regions for condensation: the low temperature and the extremely high one. It is interesting to consider also the case in which M(T) decreases enough to have conditions for condensation in some finite interval Tl < T 6 T2, where Tl # 0, T2 # 00 are the two critical points. For D = 3 we would have condensation in some hot interval of temperatures; i.e. superfluid or superconductive effects may appear in some intervals of temperature even far from T = 0. (Technically, it would be very interesting to search for materials having such properties.) All our previous considerations for condensation in D f 3, would also be valid in such a case.
H. Perez Rojas/Physics Letters A 234 (1997) 13-19
Acknowledgement
The author would like to thank Professor M. Virasoro, the ICTP High Energy Group, IAEA and UNESCO for hospitality at the International Centre for Theoretical Physics. He also thanks A. Cabo, J. Hirsch, K. Kirsten, J.R. Maldonado, L. Villegas for very interesting comments and suggestions and G. Senatore for a discussion.
References III M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman and E.A. Cornell, Science 269 (1995) 198. 121L.D. Landau and E.M. Lifshitz, Statistical Physics (Pergamon, Oxford, 1980). 131 R.K. Pathria, Statistical Mechanics (Pergamon, Oxford, 1972). [41 R.M. May, Phys. Rev. 135 (1964) A 1515. [51 M. Ziff, G.E. Uhlenbeck and M. Kac, Phys. Rep. 32 (1977) 169. [61 M. Van der Berg, Phys. Lett. A 78 (1980) 88. [71 M. Van der Berg and J.T. Lewis, Commun. Math. Phys.81 (1981) 475. [f31 K. K&ten and D.J. Toms Phys. Lett. B 368 (1996) 119. [91 Ph. Martin, Nuovo Cimento, B 68 ( 1982) 302.
19
[lo] M. Fannes, J.V. Pule and A. Verbeure, Helv. Phys. Acta 55 (1982) 391. [ 111 H.E. Habcr and H.A. Weldon, Phys. Rev. Len. 46 ( 1981) 1497; Phys. Rev. D 25 (1982) 502. [ 121 J. Kapusta, Phys. Rev. D 24 (1981) 426. [ 131 D.A. Kirzhnits and A.D. Linde, Phys. Lett. B 42 (1972) 47. [ 141 J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127 (1962) 965. [ 151 M. Chaichian, J. Gonzalez, C. Montonen and H. Perez Rojas. Phys. Lett. B 342 (1993) 118; M. Chaichian, J.L. Lucia, C. Montonen, H. Perez Rojas and M. Vargas Phys. Lett. B 342 (1995) 206. [ 161 M. Chaichian, R. Gonzalez Felipe and H. Perez Rojas. Phys. J.&t. B 342 (1995) 245. [I71 N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133. 1181 S. Coleman, Commun. Math. Phys. 31 (1973) 259. [I91 J.T. Lewis and J.V. Pule, Commun. Math. Phys. 36 (1974) I. (201 H. Perez Rojas, Phys. Len. B 379 (1996) 148. [211GA. Smolenski and V.A. Isupov, Sov. J. Techn. Phys. 24 (1954) 1375; R.L. Moreira and R.P.S.M. Lobo, J. Phys. Sot. Japan 61 (1992) 1992. [221J. Cleymans and D.W. Oertzen, Phys. I&t. B 249 (1990) 511. 1231E.S. Fradkin, Proc. from the PN. Lebedev Physical Institute No. 29 (Consultants Bureau, New York, 1967). [241 H. Perez Rojas and O.K. Kalashnikov, Nucl. Phys. B 293 (1987) 241; H. Perez Rojas, Phys. Len. A 137 (1989) 13.
September 1997 PHYSICS
ELSEYIER
Physics Letters A 234 (1997)
LETTERS
A
13-19
On Bose-Einstein condensation in any dimension H. Perez Rojas ’ International Centre for Theoretical Physics, Trieste, Italy Received
11 September
1996; revised manuscript received 14 May 1997; accepted Communicated by P.R. Holland
for publication
19 June 1997
Abstract A general property of an ideal Bose gas as temperature tends to zero and when conditions of degeneracy are satisfied is to have an arbitrarily large population in the ground state (or in its neighborhood) ; thus, condensation occurs in any dimension D but for D < 2 there is no critical temperature. Some astrophysical consequences, as well as the temperature-dependent mass case, are discussed. @ 1997 Published by Elsevier Science B.V.
1. What is Bose-Einstein
condensation?
At present there is a renewed interest in BoseEinstein condensation (BEC), particularly after its experimental realization [ 11. Actually, BEC is one of the most interesting problems of quantum statistics. It occurs in a free particle Bose gas at a critical temperature T,, and is a pure quantum phenomenon, in the sense that no interaction is assumed to exist among the particles. BEC is interesting for condensed matter (superfluidity, superconductivity) but it also has increasing interest in high energy physics (electroweak phase transition, superfluidity in neutron stars). The consequences of its occurrence, at dimensions different from D = 3, may be interesting in these two fields of physics. Bose-Einstein condensation is understood as the steady increase of particles in the state with zero energy [ 21, or as the macroscopically large number of particles accumulating in a single quantum ’ Permanent Address: Grupo de Fisica Te6rica. ICIMAF,Calle E No. 309, Vedado, La Habana inf.cu.
4, Cuba. E-mail: [email protected].
037%9601/97/$17.00 @ 1997 Published PII SO375-960 1(96)00509-4
state [ 31. For a free gas the Bose-Einstein distribution ( e(E-p)‘T - 1) -’ for ,X < 0 and y # 0 vanishes strictly at T = 0, which suggests that at T = 0 no excited states of a Bose gas can exist on the average, and condensation in the ground state seems to be a general property whenever the conditions of quantum degeneracy of the Bose-Einstein gas are satisjed. Quantum degeneracy is usually understood to be achieved when the De Broglie thermal wavelength A is greater that the mean interparticle separation N-‘j3. However, the remarkable discovery made by Einstein on the Bose distribution was that condensation may occur starting at some critical temperature T, different from zero. BEC leads to a second-order phase transition for dimensions D > 2, since by decreasing the temperature some critical value is found at which condensation starts. The general case of condensation in an arbitrary dimension was first studied by May [4] for D 3 2 and later by Ziff, Uhlenbeck and Kac [5] for D 2 1, and as a consequence of their results it is usually stated that in the thermodynamic limit BEC is not possible in D = 2 and that it neither occurs for D = 1. But in mathematical physics it has been
by Elsevier Science B.V. All rights reserved.
H. Perez Rojas/Physics Letters A 234 (1997) 13-19
14
shown that in presence of external field-like boundary conditions [6,7] critical conditions are found even forD= 1. Concerning what is to be understood as BEC there are two different ideas which are usually considered to be the same: ( 1) The existence of a critical temperature T,> 0 such that ,u(T,) = I&,EO being the single particle ground state energy. (This condition is usually taken as a necessary and sufficient condition for condensation; see e.g. Ref. [ 81) . Then for T 6 T,, some significant amount of particles starts to condense in the ground state. (2) The existence of a finite fraction of the total particle density in the ground state and in states in its neighborhood at some temperature T > 0.We shall call ( 1) the strong and (2) the weak criterion. From the point of view of finite-temperature quantum field theory, the strong criterion for BEC leads to the infrared k-* divergence of the Boson propagator,
,$
OG(k4 - ip,k,T)
1 N +2+@,
1 =
g’
which is cancelled by the density of states h-k* when calculating the particle density. High temperature radiative corrections usually have the effect of shifting the (longitudinal) boson mass in an amount SM* 'v T* (Debye screening), Spontaneous symmetry breaking (SSB) has a close connection with BEC and it plays an important role in statistical mechanics (see e.g. Ref. [ 91 and references quoted therein). An important relation between BEC and SSB was found by Fannes, Rule and Verbeure [lo], who by assuming two body interactions derived a generalization of the Bogoliubov condensate equation and showed that BEC occurs if and only if there is a spontaneous breaking of a (first kind) gauge symmetry. In SSB gauge field theories it is the vacuum which bears the property of a broken symmetry, and its analogy with a Bose-Einstein condensate leads to consider that temperature may provide a mechanism for symmetry restoration, as suggested by Kirzhnits and Linde [ 131. Some authors, as Haber and Weldon [ 111, and Kapusta [ 121 have investigated the relation between SSB and BEC in temperature quantum field models by including some chemical potential. But in quantum field theory the charge operator corresponding to a SSB symmetry does not annihilate the vac-
uum [ 141, it is an ill-defined operator and it is in general not conserved (see Ref. [ 151 and references quoted therein). This leads to conclude that in building the grand-canonical density matrix, the chemical potential corresponding to that charge must be actually taken as zero. We may understand the vanishing of such chemical potential as corresponding in some way to the condition for condensation of Goldstone bosons, the symmetry breaking parameter being the condensate. But this does not imply that such chemical potential is non-zero for temperatures above the critical temperature for symmetry restoration [ 161. In systems of low dimensionality, no SSB of a continuous symmetry occurs in one or two spatial dimensions D according to the Met-mitt-Wagner theorem [ 171; (see also Ref. [ 181 for a proof that there are no Goldstone bosons in one dimension). These results are, thus, in close correspondence with those obtained for BEC, e.g. in Ref. [ 51. In the present Letter we want to investigate again the occurrence of BEC in D = 1,2,3 and later in any dimension for the case of a free gas not in the thermodynamic limit, but having a finite volume and number of particles. Throughout all our considerations, we implicitly assume that our system is described by a Hamiltonian density written in terms of creation and annihilation field operators defined in a finite volume and satisfying periodic boundary conditions (boundary conditions determine the form in which the energy of excited states tend to the ground state value, which is critical for the appearance of BEC [ 191). We will adopt the procedure of investigating the microscopic behavior of the particle density in momentum space, which exhibits some interesting properties. We will discuss briefly at the end some specific features of the relativistic and the temperature-dependent mass cases.
2. The critical temperature
in the D = 3 case
Let us recall the origin of the critical quantities
pc,
T,in the standard 3D BEC theory in which, as pointed out before, we assume periodic boundary value conditions. The chemical potential p = f( N, T) < 0 is a decreasing function of temperature at fixed density N, and for p = 0 one gets an equation defining Tc= fc(N) . For temperatures T < Tc, as P = 0, the expression for the density gives values N'(T)< N, and the
H. Perez Rojas/Physics
difference N - N’ = No is interpreted as the density of particles in the condensate. The mean interparticle separation is then 1= N-‘/3. In our considerations we will use integrals which, as it is usually done, must be interpreted as approximations of sums over discrete quantum states, which does not imply working at the thermodynamic limit. For usual macroscopic systems, as the separation between quantum states is hp = h/V113, the approximation of the sum by the integral is quite well justified. Now, above the critical temperature for condensation
Letters A 234 (1997) 13-19
15
In this sense f(x, p) has a similar behavior than the Maxwell-Boltzmann distribution of classical statistics. But as ii -+ 0, also xP + 0, and the maximum of the density, for strictly ,% = 0, is located at x = 0. The convergence to the limit x = 0 is not uniform. A finite fraction of the total density falls in the ground state. If we go back and substitute the original integral over momentum by a sum over shells of quantum states of momentum (energy states), we can write
Nc=$c
dAP . i=.ePTI2MT _ 1
(4)
m
N =
x2dx eX*+fi _ 1
J
~TA-~
(1)
0
(2)
= h-3&?3,*(ZL
where fi = -p/T( > 0), x = p/pr is the relative momentum, pr = m the characteristic thermal momentum, and A = h/(2mnT) ‘j2 the De Broglie thermal wavelength. The function g, (z ) is (see e.g. Ref. [31), co
_1’ J Z-leX x”--idx
g,(z)
= 1 r(n)
0
where z = efilT is the fugacity. At T = T,, we have g,( 1) = c(n) , and the density is then (3) or in other words, NA3 a microscopic look to this end we investigate defined as the particle space
= c( 3/2) N 2.612. Let us have BEC in the D = 3 case, and to in detail the following quantity density in relative momentum
x2 f3(x,bii)
= &+F
_ 1’
By calculating the first and second derivatives of this function, we find that for p # 0 it has a minimum at x = 0 and a maximum at x = xP, where xcL is the solution of _ e x2+p --
1 1 --x2’
For i = 0, by taking Ap N h/V’j3, where V is the volume of the vessel containing the gas, the contribution of the ground state density is No = 4/V’/3A2. We have thus a fraction of No/N, = 4Ac(3/2)/V”3
= 4A/l(3/2)213N’/3 (5)
particles in the ground state, which is the most populated, as described by the statistical distribution, at T = T,. A numerical estimation for one liter of He gas leads to No/N cv 10P6. In quantum states in a small neighborhood of the ground state, the momentum density has slightly lower values. Thus, at the critical temperature for BEC, there is a set of states close to the ground state, having relative large densities. Eq. (5) indicates an interesting relation: the larger the separation between quantum states, the larger the population of the ground state at the critical temperature. In the thermodynamic limit the quantum states form a continuum, and (5) has no meaning. However, most systems of physical interest, in laboratory as well as in astrophysical and cosmological contexts, have finite V and N. We conclude that at the critical temperature for condensation the density of particles in momentum space has its maximum at zero momentum, and by describing the densiq as a sum over quantum states, a macroscopic fraction is obtained for the density in the ground state and in neighbor states. Thus, at the critical temperature, the weak criterion is satisfied. For values of T < T,, the curve describing the density in momentum space flattens on the p (or x) axis, and its maximum decreases also. As conservation of particles is assumed, we get the ground state density by
H. Perez Rojas/Physics Letter.s A 234 (1997) 13-I9
16
adding to f(x) the quantity 2N[ 1 - (T/T,)3/2] 19(T, T) h3S(x) as an additional density. As T -+ 0, the density in the ground state increases at the expense of non-zero momentum states. This leads to the usual Bose-Einstein condensation. We must remark that in that case, for values of T smaller but close to T,, both the weak and the strong criteria are satisfied.
which indicates that they vanish as T/N + 0, the specific heat C,: = XJf8T N &5(3/2) + O(N/h*) decreasing as T’/* as T + 0. For high temperatures one can easily find that C,, is roughly constant; it indicates some correspondence with the behavior of C,, in the D = 3 case [3], but there is no discontinuity in its derivative with regard to T. By substituting the last expression for ji back in (7), one has
3. The D = 1 and D = 2 cases For a Bose-Einstein gas in the presence of a potential of form (xl/L)” it has been shown by Van den Berg and Lewis [ 6,7] that the critical conditions in dimension D are the same as for the free boson gas in dimension D + 2/cr. Thus, BEC occurs for D = 1,2,. . . in their case. In our model of free gas with periodic boundary conditions, let us see what happens for D = 1. In this case, the mean interparticle separation is 1 = N-l. The density in momentum space coincides with the Bose-Einstein distribution, P2 2MT-fiIT 1) -I. This function has ~I(P,T,P) = (e / only one extremum, a maximum, at p = 0. By using the previous change of variables, we have the expression for the density of particles as 0;)
N = 2A-tr-t/*
I
dx ex*+P - l = $,*w.
(6)
0
We have thus that limgt/2( z ) diverges ment of the quantum actually means that for N constant, and approximately,
N N 2A-tr-‘/*
Nh N gt/2( z). The fact that as z -+ 1 indicates an enhancedegeneracy regime. But this fact p is a decreasing function of T for very small ji one can write,
s ‘*
dx -=EL’/2A. x2 + p
&Q (7)
0
where x0 = pa/MT, po being some characteristic momentum pe > PT. Thus, p does not vanish at T # 0, and for small T it is approximately given by ,G = r/N*A*. It is easy to obtain the pressure P (= force/lo) and energy density respectively as P = ~g3,2(2L
(I=$
(8)
(9) -=I
where y = &/*/2NA. Due to the properties Cauchy distribution, one can write
of the
Y
$=A-
ydx ___
Y J
x*+ y2'
-Y
We can also write 2NA f~(x,T,
N)T/N+o
N -S(x). 77
Thus, for small T/N all the population tends to be concentrated on the ground state, and although not an usual Bose-Einstein condensation, we claim that (as in the magnetic field case, [20] ) we have a diffuse condensation [ 211 (in the sense of not having a critical temperature) when T/N is low enough. There is no critical point; there is no discontinuity in the derivative of the specific heat in this case. The condensation is the outcome of a continuous process since the ground state density is always non-zero, and this quantity can be increased continuously to reach macroscopically signijcant values, by decreasing enough the ratio T/N, even for temperatures farfrom zero. The D = 1 case satisfies the weak criterion at any temperature T, but not the strong one. Closely connected with the D = 1 case is the problem of condensation of a gas of charged particles in the presence of a strong magnetic field [ 201, where all the previous considerations apply, by substituting y = MeBT/Zi*cN, where e is the electric charge and B the magnetic field. If eBfi/McT > 1, the system is confined to the Landau ground state n = 0, and the problem can be studied in close correspondence to D = 1 case. We will consider the model of a gas of kaons in a neutron star [ 221. By taking a density N N 1O44 crne3, T N lo* K and local magnetic
H. Perez Rojas/Physics
field B w 1014 G, the condition of quantum degeneracy is exceedingly satisfied. In that case y = 10m3’. By taking the dimensions of the star as lo7 cm, the discrete quantum states would be spaced by an amount Sp = 1O-34 g cm/s. One half of the total density would be distributed in 17 = 2y/Ap = lo4 quantum states. The ground state density with zero momentum p along the magnetic field, can then be estimated as AN = +‘N = 104o, leading to observable effects: superfluidity and strong diamagnetic response to the applied field. (The magnetization M = eBFi/2Mc N lOI6 G would exceed by two orders of magnitude the microscopic applied local field and would be the preponderating field.) However, the quantity AN/N = -1 = di3Nc/MeBT@/3 tends to zero in the ther77 modynamic limit, and some of the physical effects or condensation would be erased in that limit. Let us now turn our attention to the D = 2 case. Here I = N-l/*. The distribution is x .f*(x,T,fi)
=
exz+P
Letters A 234 (1997) 13-19
the weak and perature T # condensation in the ground can write for
strong criteria are satisfied at any tem0. Thus, there is no strict Bose-Einstein in the sense that the value of the density state is zero for any T/N # 0. As one x small,
limf2(x,T,~),,O=lim~
-&
=%LS
i.e. the density at x = 0 is non-zero only for ,& = 0. But as the maximum of the density increases continuously by decreasing T/N, being located at x,, + 0 (the convergence to x = 0 being non-uniform), we have a sort of quasi-condensate, in the spirit of the weak criterion, i.e. most of the density can be found concentrated in a small interval of values of momentum around the p = 0 state at arbitrary small temperatures. The pressure P (= force/l) and energy density in such case are given by zJ =
_
17
U=p,
$2(z).
(11)
1 ’
and has always an absolute minimum at x = 0 (the density vanishes in the ground state) and a maximum at a value of x > 0 being a solution of
and obviously vanish in the limit T -+ 0. For high T, these quantities behave as linear in T. The specific heat for T -+ 0 is C, N $/3A* + 0( NA* e-NA2/2) and is approximately linear in T.
1
e X2+/?=s-
4. The case
We can write the density as 00
N=2A-* J n
-(X2+p)XdX e = 2A-* 1 _ e-(X’+P)
D > 3
In the case D > 3, the density in momentum for p # 0 reads
In( 1 - e-“).
space
xD--I
(10) We have fi = - ln( 1 - e-NA2/2); thus as T/N --+ 0, P-+e -Nh14nMT. For ,LL= 0 the density N in (10) diverges, as in the D = 1 case, except for T = 0. At non-zero ,& the population in a closed small neighborhood of x = 0 is strictly zero, but the amplitude of this interval decreases as p -+ 0, that is, the maximum of the density in momentum space is reached at a value of momentum xmax # 0 which decreases as T/N --+ 0. (This behavior bears some analogy to the case of a two dimensional boson gas with an external potential studied in Ref. [ 71, where there is a macroscopic occupation of an infinite number of lowlying levels.) The maximum of the density is not in the ground state but in a neighborhood of it. None of
fo(x,T,fiL) =
&+ji
_ 1’
This function has an absolute minimum at x = 0 and an absolute maximum at x = x,,, # 0 given by the non-zero solution of the equation ex2+p = l/ [ 1 2x2 / (D - 1) 1. The density is zero at the ground state, and in a small neighborhood of it, and in this sense differs from the D = 3 and the D = 1 cases. The total density is given by A-D N = r( D/2)
m
J 0
fo dx = A-DgD,2( z).
(12)
In this case, the density p decreases as T --+ 0 and N converges for ,ii = 0. Thus, there is a non-zero
H. Perez Rojas/Physics
18
critical temperature T, such that ,ii(T,) = 0. Then for T < Tc ( 12) is unable to account for the total density and conservation of particles demands that the lacking density NO = N - N’ be exactly at the ground state. Thus, although the density of particles in an (open) neighborhood of the ground state is zero, exactly at the groundstate it is given by 2N[ 1- (T/Tc)D/2]ADS(~). At the critical temperature N,hD = l( D/2), which tends to unity with increasing D. We may conclude that the D-dimensional gas becomes less degenerate with increasing D. For any dimension, the relation between energy and pressure P = 2U/D holds. Also, below the critical temperature, C,, - To/*. We see that the D > 3 case satisfies the strong but not the weak criterion for T < T,. Our results for D Z 3 are in agreement with those obtained in Ref. [ 81. Expression ( 12) is valid for continuous D. It can be easily checked that for 1 > D > 0 the quantity fD ( X) diverges for x = 0, and N remains finite for p # 0. Thus the di$%se condensation takes place in the interval 1 > D > 0. The D = 2-like behavior occurs for 2 2 D > 1, whereas, as demonstrated by May [ 41, usual condensation occurs for D > 2. However, for 2 < D < 3, both criteria, weak and strong are satisfied, fD being divergent at x = 0, whereas N remains finite.
We shall revisit the relativistic case. Here the conservation of particles must reflect some invariance property of the Lagrangian. We are keeping in mind the simplest case of a charged massive scalar field. In that case, the conserved quantity, derived from the Noether theorem is the charge in D spatial dimensions),
J DX, m
jo(n)d
served quantities depend usually on the difference of their average densities. After building the density matrix for the grand canonical ensemble, one can write the thermodynamic potential, and from it the conserved charge as an expression which contains the difference of average number of particles minus antiparticles, 27@*TD
(Q)= r(D/2)
(13)
0
where j, = I+V&@ - (G’,,+*)ti is the four-current, and as different from Ref. [4], we must include in the temperature charge average the contribution from antiparticles: this is a natural consequence of a relativistic finite-temperature treatment of the problem (see e.g. [ 23,11,12]. At high T one must consider the excitation of particle-antiparticle pairs, and the con-
JxD-l Dc)
cDFiD
dx(n,
-n,),
(14)
0
where np = (eE-fi-- 1)-l, n, = (eE+fi-- 1)-l are the particle and antiparticle densities, E = x2 + A?* , and x = PC/T, I@ = Mc*/T. In the present case ,G may be positive or negative, but I,&) < Iii. Condensation in D > 3 occurs for IjZ] = I@. For D < 3 the condensation is very well reproduced by the infrared (nonrelativistic) limit already seen, by taking the chemical potential as ,u’ = p - M. A very interesting case occurs when M = 0. In that case ( 14) demands p = 0 to avoid negative population densities of particles or antiparticles. All the charge must be concentrated in the condensate and the critical temperature for condensation, as suggested in Ref. [ll],isT=co.
6. The case M =
5. The relativistic case
Q =i
Letters A 234 (1997) 13-19
M(T)
In some systems the interactions at high temperature behave in such a way that can be described effectively as free particle systems with variable mass; i.e. the temperature-dependent interaction leads to the arising of a mass M = M(T). If M(T) + 0 we have BEC with increasing temperatures, as discussed in Refs. [ 24,161. We have in that case two regions for condensation: the low temperature and the extremely high one. It is interesting to consider also the case in which M(T) decreases enough to have conditions for condensation in some finite interval Tl < T 6 T2, where Tl # 0, T2 # 00 are the two critical points. For D = 3 we would have condensation in some hot interval of temperatures; i.e. superfluid or superconductive effects may appear in some intervals of temperature even far from T = 0. (Technically, it would be very interesting to search for materials having such properties.) All our previous considerations for condensation in D f 3, would also be valid in such a case.
H. Perez Rojas/Physics Letters A 234 (1997) 13-19
Acknowledgement
The author would like to thank Professor M. Virasoro, the ICTP High Energy Group, IAEA and UNESCO for hospitality at the International Centre for Theoretical Physics. He also thanks A. Cabo, J. Hirsch, K. Kirsten, J.R. Maldonado, L. Villegas for very interesting comments and suggestions and G. Senatore for a discussion.
References III M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman and E.A. Cornell, Science 269 (1995) 198. 121L.D. Landau and E.M. Lifshitz, Statistical Physics (Pergamon, Oxford, 1980). 131 R.K. Pathria, Statistical Mechanics (Pergamon, Oxford, 1972). [41 R.M. May, Phys. Rev. 135 (1964) A 1515. [51 M. Ziff, G.E. Uhlenbeck and M. Kac, Phys. Rep. 32 (1977) 169. [61 M. Van der Berg, Phys. Lett. A 78 (1980) 88. [71 M. Van der Berg and J.T. Lewis, Commun. Math. Phys.81 (1981) 475. [f31 K. K&ten and D.J. Toms Phys. Lett. B 368 (1996) 119. [91 Ph. Martin, Nuovo Cimento, B 68 ( 1982) 302.
19
[lo] M. Fannes, J.V. Pule and A. Verbeure, Helv. Phys. Acta 55 (1982) 391. [ 111 H.E. Habcr and H.A. Weldon, Phys. Rev. Len. 46 ( 1981) 1497; Phys. Rev. D 25 (1982) 502. [ 121 J. Kapusta, Phys. Rev. D 24 (1981) 426. [ 131 D.A. Kirzhnits and A.D. Linde, Phys. Lett. B 42 (1972) 47. [ 141 J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127 (1962) 965. [ 151 M. Chaichian, J. Gonzalez, C. Montonen and H. Perez Rojas. Phys. Lett. B 342 (1993) 118; M. Chaichian, J.L. Lucia, C. Montonen, H. Perez Rojas and M. Vargas Phys. Lett. B 342 (1995) 206. [ 161 M. Chaichian, R. Gonzalez Felipe and H. Perez Rojas. Phys. J.&t. B 342 (1995) 245. [I71 N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133. 1181 S. Coleman, Commun. Math. Phys. 31 (1973) 259. [I91 J.T. Lewis and J.V. Pule, Commun. Math. Phys. 36 (1974) I. (201 H. Perez Rojas, Phys. Len. B 379 (1996) 148. [211GA. Smolenski and V.A. Isupov, Sov. J. Techn. Phys. 24 (1954) 1375; R.L. Moreira and R.P.S.M. Lobo, J. Phys. Sot. Japan 61 (1992) 1992. [221J. Cleymans and D.W. Oertzen, Phys. I&t. B 249 (1990) 511. 1231E.S. Fradkin, Proc. from the PN. Lebedev Physical Institute No. 29 (Consultants Bureau, New York, 1967). [241 H. Perez Rojas and O.K. Kalashnikov, Nucl. Phys. B 293 (1987) 241; H. Perez Rojas, Phys. Len. A 137 (1989) 13.