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On the critical behaviour of Bose systems at constant density
Volume 107A, n u m b e r 5
PHYSICS LETTERS
4 February 1985
ON THE CRITICAL BEHAVIOUR OF BOSE SYSTEMS AT CONSTANT DENSITY D.I. UZUNOV G. Nad/akov Institute o f Solid State Physics, Bulgarian Academy o f Sciences, 1184 Sofia, Bulgaria
and K. WALASEK Institute o f Physics, Technical University o f Wroc(aw, ICybrzeie Wyspiahskiego 27, 50-370 Wroci'aw, Poland Received 17 August 1984
The scaling behaviour of Bose systems at constant density is studied using the e-expansion method. The chemical potential and equation of state are obtained in terms o f t = (T - Tc)/T c to first order in e.
Usually, the critical behaviour of interacting Bose systems is studied by means of the grand canonical ensemble where the chemical potential/a ~< 0 is the quantity describing the distance Ij a - #cl from the critical point/~c < 0. For instance, the critical exponent connected with the susceptibility X is given by the relation X- 1 cx (/1c _ tz)~. In terms of this formalism, the renormalization-group (RG) analysis [ 1 - 3 ] reveals a universality o f the asymptotic scaling behaviour near finite-temperature critical points [ 4 - 7 ] (at which/a c corresponds to finite temperatures). The universality means that the exponents ~, ~, etc .... describing the critical behaviour of interacting Bose systems by [/a - gcl have the same values as those for the corresponding classical systems. But with this conclusion only the analysis is not completed. Besides, a critical behaviour expressed by I/z - / a c l cannot be experimentally checked in natural systems. A complete treatment requires the determination of the chemical potential as a function of the temperature T, using the conditions at which the system is studied, say, a constant pressure or a constant density. However, such conditions may essentially change the values o f the critical exponents. For example the critical behaviour of the ideal Bose gas at constant density and for any space dimension d > 2 coincides with that described by the n ~ ~ limit of the n-vector classical model (cf. refs. [8,9] ). On the one hand it is
not clear whether the interaction will affect the abovementioned coincidence. On the other hand, the RG resuits for the universality which are expressed in terms of the chemical potential are not sufficient for answering this question. In this letter we present results on the critical behaviour in the normal phase (#c - / a > 0) of interacting Bose systems at a constant density. We restrict ourselves to spatial dimensions 2 < d < 4. A brief discussion concerning a peculiarity of the case d = 4 will be given too. The calculations have been performed to first order in e = 4 - d. The grand canonical potential ~ = kBT In Tr {e S} is expressed by the euclidean action S = S O + SI, where SO = ~
q,ot
060 l - ek +/a)l~p~(q)[ 2
(1)
and °kBT
~
*
•
SI = 213Vql,q2,qa;a,i ~ ~oa(q 1) ~#(q 2 ) X ~oa(q3)~o#(q i + q 2 - q3)
(2)
are the free and interaction part of S, respectively. In eqs. (1) and (2), q = (k, ~t), c°t = 2~rlkBT(l = O, +-1...) denotes the Matsubara frequency, k is the wave vector, 207
Volume 107A, number 5
PHYSICS LETTERS
the energy spectrum is given by ek = ( h 2 / 2 m ) k 2 and v is interaction constant. In the functional formalism used here the fields s% (a = 1, ..., n/2) are c-numbers and the Tr symbol in f~ denotes a functional integration over ~pa. The density p = N / V is given by
P-
k B T ~ Ga(q) exp(ko/0+), V q,a
(3)
where Gc~(q ) = -([~%(q)l 2 ) is the correlation function. This function can be obtained using the direct method for calculations of the critical exponents [ 1,3]. To first order in e, Gc~(q) is expressed as Ga- l(q) = ieol _ ek + X- 1
(4)
The susceptibility X in eq. (4) is obtained in the form X-I =(Uc_U) ~,
(5)
where ~ = 1 + (n + 2)e/2(n + 8) is the universal critical exponent. As usual, #c is connected with the interaction parameter v. Eqs. (4), (5) yield O = ½nC (k B T ) d / 2 F d / 2 ( a ) , where C = ( m / 2 1 r h 2 ) d/2, a = (kBTX)
(6) l and
1 7' xV-ldx Fv(a) = 0F(v) - J e x+a - 1
(7)
is the Bose integral [ 8 - 1 0 ] . Eq. (6) can be used for the derivation of the susceptibility × in terms of (T Tc) near the critical temperature T c. Neglecting possible corrections to the scaling we get the following result X - 1 = kB T ( A t )2 / ( d - 2 )
(8)
where A = (2rr) l d ~ ( d / 2 ) F ( d / 2 ) sin[rr(d - 2)/2] , ~'(y) is the zeta function and t = ( T - T c ) / T e. Eq. (8) shows that the susceptibility X has the same exponent 3' = 2/(d - 2) as for the ideal Bose gas [8,9]. To first order in e, 3' = 1 + e/2. Eqs. (5) and (8) give the t dependence of the chemical potential. Near four dimensions we have: la = lac - ( k B T ) l + e / Z ( A t ) 1+ 3e/(n+8) •
208
(9)
4 February 1985
Expression (9) for/l is essentially different from the corresponding one for the ideal Bose gas [9]. Despite this difference, it should be emphasized that in general within the first-order e-approximation the interacting Bose system has again the critical behaviour of the nvector model in the limit n ~ co. This is due to the effect of the constant density condition. The effect is so strong that it suppresses the interaction influence at this order of the theory. Moreover there are no reasons to expect that the interaction will not manifest itself in higher orders in the RG analysis. For instance, in higher than first order in e, owing to the interaction, the energy spectrum ek at the critical point will not be quadratic, and this should lead to a change of the above picture. The result given by eqs. (8), (9) is not valid for d = 4. In this case the expansion of F2(c0 for small a has a logarithmic term, which yields the following result: X 1 [ln(x
l/k
BT)[ = ~r 1 2k B T t .
(10)
This cannot be obtained in the limit e ~ 0 from eq. (8). If we replace approximately the logarithm in eq. (10) by unity we obtain the result presented in ref. [9]. Our final discussion concerns the equation of state. We follow here ref. [ 11 ]. The pressure p = -~2/V can be split into a regular Pr and a singular Ps part (p = Pr + Ps)" Using general RG scaling arguments [2] we have Ps (x (/1c - t2) vd, where u is the correlation length exponent. In our approximation u = 3'/2 and taking into account eq. (9) we obtain Ps cc t 2+O(e). The regular part Pr can be expanded around the point (/a c, Tc). To first order in t and (/l - #c) we have Pr = Pc + act + bc(/2 - # c ) ,
(11)
where Pc is the pressure at the critical point (T c,/~c) a c and b c are the derivatives of Pr with respect to T and/2 at (T c, ttc), respectively. Taking in mind the dependence of(/1 - / a e ) on t from eq. (9), we see that both t-dependent terms in eq. (1 1) are more important than the contribution from Ps- Thus, we obtain the equation of state in the form P - Pc
ac
Pc
Pc
t-
bc
_ ~ ( k B T c ) l + e / 2 ( A t ) l+3e/(n+8) . (12) Pc
For comparison we present the equation of state of the ideal Bose gas in the spatial dimension 2 < d < 4
Volume 107A, number 5
PHYSICS LETTERS
P - Pc P(-d/2)(At)d/(d - 2) p-----~ - ( 1 + d / 2 ) t + 3 ( d / 2 + 1) Eqs. (12) and (13) differ mainly in the corrections to the linear part in t.
References [1] K.G. Wilson andJ. Kogut, Phys. Rep~ 12 (1974) 75. [2] F.J. Wegner, in: Phase transitions and critical phenomena, Vol. 6, ed~ C. Domb and M.S. Green (Academic Press, New York, 1976) p. 8. [3] S. Ma, Modern theory of critical phenomena (Benjamin, New York, 1976).
4 February 1985
[4] K.K. Singh, Phys. Rev. B12 (1975) 2819. [5] A.L. Stella and F. Toigo, Nuovo Cimento B34 (1976) 207. [6] G. Busiello and L. De Cesare, Nuovo Cimento 59B (1980) 491. [7] D.I. Uzunov, Phys. Lett. 87A (1981) 11. [8] J.D. Gunton and M.J. Buckingham, Phys. Rev. 166 (1968) 152. [9] P. Lacour-Gayet and G. Toulouse, J. Phys. (Paris) 35 (1974) 425. [10] J.E. Robinson, Phys. Rev. 83 (1951) 678. [11] K. Walasek, Phys. Lett. 101A (1984) 343.
209
PHYSICS LETTERS
4 February 1985
ON THE CRITICAL BEHAVIOUR OF BOSE SYSTEMS AT CONSTANT DENSITY D.I. UZUNOV G. Nad/akov Institute o f Solid State Physics, Bulgarian Academy o f Sciences, 1184 Sofia, Bulgaria
and K. WALASEK Institute o f Physics, Technical University o f Wroc(aw, ICybrzeie Wyspiahskiego 27, 50-370 Wroci'aw, Poland Received 17 August 1984
The scaling behaviour of Bose systems at constant density is studied using the e-expansion method. The chemical potential and equation of state are obtained in terms o f t = (T - Tc)/T c to first order in e.
Usually, the critical behaviour of interacting Bose systems is studied by means of the grand canonical ensemble where the chemical potential/a ~< 0 is the quantity describing the distance Ij a - #cl from the critical point/~c < 0. For instance, the critical exponent connected with the susceptibility X is given by the relation X- 1 cx (/1c _ tz)~. In terms of this formalism, the renormalization-group (RG) analysis [ 1 - 3 ] reveals a universality o f the asymptotic scaling behaviour near finite-temperature critical points [ 4 - 7 ] (at which/a c corresponds to finite temperatures). The universality means that the exponents ~, ~, etc .... describing the critical behaviour of interacting Bose systems by [/a - gcl have the same values as those for the corresponding classical systems. But with this conclusion only the analysis is not completed. Besides, a critical behaviour expressed by I/z - / a c l cannot be experimentally checked in natural systems. A complete treatment requires the determination of the chemical potential as a function of the temperature T, using the conditions at which the system is studied, say, a constant pressure or a constant density. However, such conditions may essentially change the values o f the critical exponents. For example the critical behaviour of the ideal Bose gas at constant density and for any space dimension d > 2 coincides with that described by the n ~ ~ limit of the n-vector classical model (cf. refs. [8,9] ). On the one hand it is
not clear whether the interaction will affect the abovementioned coincidence. On the other hand, the RG resuits for the universality which are expressed in terms of the chemical potential are not sufficient for answering this question. In this letter we present results on the critical behaviour in the normal phase (#c - / a > 0) of interacting Bose systems at a constant density. We restrict ourselves to spatial dimensions 2 < d < 4. A brief discussion concerning a peculiarity of the case d = 4 will be given too. The calculations have been performed to first order in e = 4 - d. The grand canonical potential ~ = kBT In Tr {e S} is expressed by the euclidean action S = S O + SI, where SO = ~
q,ot
060 l - ek +/a)l~p~(q)[ 2
(1)
and °kBT
~
*
•
SI = 213Vql,q2,qa;a,i ~ ~oa(q 1) ~#(q 2 ) X ~oa(q3)~o#(q i + q 2 - q3)
(2)
are the free and interaction part of S, respectively. In eqs. (1) and (2), q = (k, ~t), c°t = 2~rlkBT(l = O, +-1...) denotes the Matsubara frequency, k is the wave vector, 207
Volume 107A, number 5
PHYSICS LETTERS
the energy spectrum is given by ek = ( h 2 / 2 m ) k 2 and v is interaction constant. In the functional formalism used here the fields s% (a = 1, ..., n/2) are c-numbers and the Tr symbol in f~ denotes a functional integration over ~pa. The density p = N / V is given by
P-
k B T ~ Ga(q) exp(ko/0+), V q,a
(3)
where Gc~(q ) = -([~%(q)l 2 ) is the correlation function. This function can be obtained using the direct method for calculations of the critical exponents [ 1,3]. To first order in e, Gc~(q) is expressed as Ga- l(q) = ieol _ ek + X- 1
(4)
The susceptibility X in eq. (4) is obtained in the form X-I =(Uc_U) ~,
(5)
where ~ = 1 + (n + 2)e/2(n + 8) is the universal critical exponent. As usual, #c is connected with the interaction parameter v. Eqs. (4), (5) yield O = ½nC (k B T ) d / 2 F d / 2 ( a ) , where C = ( m / 2 1 r h 2 ) d/2, a = (kBTX)
(6) l and
1 7' xV-ldx Fv(a) = 0F(v) - J e x+a - 1
(7)
is the Bose integral [ 8 - 1 0 ] . Eq. (6) can be used for the derivation of the susceptibility × in terms of (T Tc) near the critical temperature T c. Neglecting possible corrections to the scaling we get the following result X - 1 = kB T ( A t )2 / ( d - 2 )
(8)
where A = (2rr) l d ~ ( d / 2 ) F ( d / 2 ) sin[rr(d - 2)/2] , ~'(y) is the zeta function and t = ( T - T c ) / T e. Eq. (8) shows that the susceptibility X has the same exponent 3' = 2/(d - 2) as for the ideal Bose gas [8,9]. To first order in e, 3' = 1 + e/2. Eqs. (5) and (8) give the t dependence of the chemical potential. Near four dimensions we have: la = lac - ( k B T ) l + e / Z ( A t ) 1+ 3e/(n+8) •
208
(9)
4 February 1985
Expression (9) for/l is essentially different from the corresponding one for the ideal Bose gas [9]. Despite this difference, it should be emphasized that in general within the first-order e-approximation the interacting Bose system has again the critical behaviour of the nvector model in the limit n ~ co. This is due to the effect of the constant density condition. The effect is so strong that it suppresses the interaction influence at this order of the theory. Moreover there are no reasons to expect that the interaction will not manifest itself in higher orders in the RG analysis. For instance, in higher than first order in e, owing to the interaction, the energy spectrum ek at the critical point will not be quadratic, and this should lead to a change of the above picture. The result given by eqs. (8), (9) is not valid for d = 4. In this case the expansion of F2(c0 for small a has a logarithmic term, which yields the following result: X 1 [ln(x
l/k
BT)[ = ~r 1 2k B T t .
(10)
This cannot be obtained in the limit e ~ 0 from eq. (8). If we replace approximately the logarithm in eq. (10) by unity we obtain the result presented in ref. [9]. Our final discussion concerns the equation of state. We follow here ref. [ 11 ]. The pressure p = -~2/V can be split into a regular Pr and a singular Ps part (p = Pr + Ps)" Using general RG scaling arguments [2] we have Ps (x (/1c - t2) vd, where u is the correlation length exponent. In our approximation u = 3'/2 and taking into account eq. (9) we obtain Ps cc t 2+O(e). The regular part Pr can be expanded around the point (/a c, Tc). To first order in t and (/l - #c) we have Pr = Pc + act + bc(/2 - # c ) ,
(11)
where Pc is the pressure at the critical point (T c,/~c) a c and b c are the derivatives of Pr with respect to T and/2 at (T c, ttc), respectively. Taking in mind the dependence of(/1 - / a e ) on t from eq. (9), we see that both t-dependent terms in eq. (1 1) are more important than the contribution from Ps- Thus, we obtain the equation of state in the form P - Pc
ac
Pc
Pc
t-
bc
_ ~ ( k B T c ) l + e / 2 ( A t ) l+3e/(n+8) . (12) Pc
For comparison we present the equation of state of the ideal Bose gas in the spatial dimension 2 < d < 4
Volume 107A, number 5
PHYSICS LETTERS
P - Pc P(-d/2)(At)d/(d - 2) p-----~ - ( 1 + d / 2 ) t + 3 ( d / 2 + 1) Eqs. (12) and (13) differ mainly in the corrections to the linear part in t.
References [1] K.G. Wilson andJ. Kogut, Phys. Rep~ 12 (1974) 75. [2] F.J. Wegner, in: Phase transitions and critical phenomena, Vol. 6, ed~ C. Domb and M.S. Green (Academic Press, New York, 1976) p. 8. [3] S. Ma, Modern theory of critical phenomena (Benjamin, New York, 1976).
4 February 1985
[4] K.K. Singh, Phys. Rev. B12 (1975) 2819. [5] A.L. Stella and F. Toigo, Nuovo Cimento B34 (1976) 207. [6] G. Busiello and L. De Cesare, Nuovo Cimento 59B (1980) 491. [7] D.I. Uzunov, Phys. Lett. 87A (1981) 11. [8] J.D. Gunton and M.J. Buckingham, Phys. Rev. 166 (1968) 152. [9] P. Lacour-Gayet and G. Toulouse, J. Phys. (Paris) 35 (1974) 425. [10] J.E. Robinson, Phys. Rev. 83 (1951) 678. [11] K. Walasek, Phys. Lett. 101A (1984) 343.
209