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Rescaled coherent state theory of the λ-transition in liquid He4
Physica 116A (1982) 604-611 North-Holland
Publishing Co.
RESCALED COHERENT THE A-TRANSITION
STATE THEORY IN LIQUID He4
OF
J.C. LEE Department
of Chemistry and Physics, Northwestern State University of Louisiana, Natchitoches, LA 71457 USA
Received 11 June 1982
The quantum partition function for the interacting boson system i.s calculated using a complete set of resealed coherent states in which the helium atom occupation numbers are resealed with a large scale factor. There are terms in the statistical weights that do not vanish in the limit of infinite scale factor. These scale independent terms are shown to be in the familiar form of the Ginzburg-Landau model, and are explained as the result of condensation.
1. Introduction
The Ginzburg-Landau (GL) model of the X-transition in liquid He4 assumes that the critical behavior is determined by the fluctuation of the condensate with statistical weights given by \V$l’ plus a power series in 1+1’, where the order parameter field $ is the wave function of the condensate. As this implies that the quantum system has the same critical behavior as the classical system of spins with two real components, it was examined with particular caution by several authors. The model itself was derived by Langer’) (before the renormalization group theory was developed). The critical exponents were calculated by Singhz), Wiegel’) and this author4,5) without adopting the GL model. Wiegel used the method of functional integration to construct an operatorfree formalism for the quantum system. As far as the critical exponents are concerned, the equivalence is clearly established. The partition function turns out to be given by a fluctuating two-component classical field, but the field is a function of a dimensionless variable 8 in addition to the space variable. The variable 19may be eliminated by means of the renormalization group transformation, but it appears difficult to regard the resultant expression for the partition function as a sum of statistical weights of all possible fluctuations of the field. Instead, it is a ratio of two such sums: the numerator takes the form of the GL model, but it is divided by another GL-like form which contains only IV4l’. The thermodynamic effect implied by the presence of the 0378-4371/82/0000-0000/$02.75
0
1982 North-Holland
COHERENT
STATE THEORY
OF A-TRANSITION
IN He4
605
denominator was not discussed, but it appears to us that it has no effect on the equivalence in the critical region. Singh approached the problem differently. The renormalization group transformation was applied directly to a quantum hamiltonian and had the unique feature of resealing the mass of helium atoms. A fixed point was shown to exist with critical behavior identical to that of the two-component classical spin system up to the first order of E = d - 4. Previously’), we replaced the mass resealing with the occupation number resealing, and for T > T, it was shown5) that the identity extends to all orders of E (see footnote6)). In view of the abundant information learned from the GL model, it would be desirable to derive the GL model itself from a quantum mechanical expression of the partition function. Langer has shown, using the coherent state theory of Glauber’), that the partition function may indeed be represented as a sum of statistical weights of all fluctuations of an order parameter function. The order parameter function is identified as the wave function of the condensate (&p(r)). The free energy functional is in a generalized form of the GL model, in the sense that the derivative of the order parameter function enters in a more complicated manner than in the familiar form of the GL model. The free energy functional derived by Langer contains microscopic details that the GL model in general does not. The details are lost in the coarse graining of the latter. The nature of this coarse graining of spin system is well knowns): it is the block spin picture of Kadanoff. In this way one can speak of a continuous spin field with a GL model when, in fact, the spins in the corresponding Ising model can only be + 1 or -1. What then would be the corresponding picture of the coarse graining for the superlluid quantum system? Since the onset of the A-transition is believed to be due to a macroscopic occupation of a condensate, the corresponding coarse graining should “bring out” those helium atoms in the condensate and “smear out” those in the other states. This may be accomplished by resealing the helium atom occupation numbers’) with a large scale factor. It is this conceptual issue that we wish to pursue in this paper. Several identities of the resealed occupation number representation play important roles in the present analysis. For easy reference, we present them in section 2. The coherent state theory in the resealed occupation number representation is presented in section 3. Discussions follow in section 4.
606
J.C.LEE
2. Resealed occupation number representation If one wishes to rescale the occupation numbers with the scale factor A, i.e., n4 = Api, the creation and annihilation operators acting on the state vectors in the resealed occupation number space must be defined by4 a&l;>
=(n>“+;-h-‘>,
a;! n;>
= (n;+ A-‘)“*I,;+ A-‘.
(1)
The net effect of these operators is that A particles appear as one particle and one particle appears as a fraction of particle A-‘, i.e., they are partially blind. Consequently, the boson commutation rules become [a,,
a,3 = ia:, a’,,1= 0, [a,, a;] = A-S,,,.
The resealed stat evectors
are normalized
(2)
by
(n; 1 n;) = A-‘,
(3)
Finally, in the thermal interaction a,(r) = a, e-+‘*,
picture, the operators
take the form
a:(7) = aj)eQ’*,
(4)
where Z, = q2/2 - p, and p is the chemical potential.
3. Resealed coherent states We start with a hamiltonian H =x
+:a,
1 +-2 2v
given by
u,a:_,ai+,apak
= I&+&I,.
(9
and wish to evaluate the grand partition function? Z = Tr exp( - @!I). The trace is to be evaluated using the complete set of resealed coherent defined by a4 I 4
= aq I Q,
(6) states (7)
where ag is any complex number and a, is the resealed annihilation operator. The eigenstates ((Y,)may be constructed from the resealed occupation number states. Using (l), (3) and (7), one can obtain
(8)
COHERENT
The coherent
STATE THEORY
OF A-TRANSITION
states ICXJform a complete
~/d20,(a,)(a,l
601
IN He4
set as they satisfy (9)
= 1.
They are orthogonal in the limit of infinite h, but that has no bearing here. First let us calculate 2, for H = HO, using (10)
20 = v ($ j d’a,)(Iol}l exp( - PHJl{oD, where ~{cx})= III, la,). From (l), (3) and (8), it follows that 20 = v ($jd2aq)(a,l = y
(if
exp(-
p#,a,)(~J
(11)
d201,) expla,12h(e-8’q’A - 1).
The free energy functional
is therefore
given by
Fo{a} = - kT 2 Iaq12h(e-@‘q’r - 1). Setting h = 1 gives the microscopic
(12)
result of Langer,
Fo{a} = - kT 2 Iaq12(e-P& - 1)
(13)
and A + CQgives the coarse grained result, (14) which is the familar Gaussian part of the GL model. Next we include the interaction term in I-I. The density written as emBH= e+“S(p),
operator
is then
(15)
where S(p) is given by”‘) S(p)=
l_jdr~~~(?,)+jdi,jd7,H,oH,(r31 0
0
...
(16)
0
and takes care of the non-commutativity of operators. As eq. (2) suggests, all operators commute in the limit of infinite A, but the limit has to be taken at the end. In the perturbation theory for “fixed number” states (as opposed to the “unsharp number” coherent states”)) these vanishing commutators multiply with the occupation numbers of the states very near the condensate. As the
608
J.C. LEE
limit of infinite A is taken, these states are pushed to the condensate where the occupation numbers diverge. Whether or not the same situation will occur in the coherent state formalism is to be seen, but there is clearly a danger in speaking of vanishing commutators at an early stage of perturbation. Now returning to eq. (16) we carry out the r-integrations. The resultant S(p) shows the following A-dependence: @Hr)“[l+
0(X-‘) + @A-*) + * * -1,
(17)
where HI is now in the Schroedinger picture and the unspecified coefficients of various powers of A-’ are functions of lq only. Clearly, these terms cannot compete with the first term in the large A limit. Thus, they are all dropped and the partition function becomes
(18) A single term in the sum, say (HI)“, contains operators of different modes. Those in the same mode, say (a,)‘(a)“, will pair up with exp( - pa&) in exp( - /3Ho) and get “sandwiched” between ((~~1and ]a,). Thus the matrix element in (18) separates into factors each of which contains only one mode and is of the form (a,] exp( - p~~ba~as)(a~“(a4)‘I~,) + O(A-‘) + O(A-2) + . - .,
(1%
where the creation operators have been moved to the left using the commutation rule of (2) resmting in the additional terms of various orders of A-‘. The unspecified coefficients are the same as the matrix element in the first term but with s and t reduced properly. These may all be neglected in the limit of large A, leaving only the first term, which gives A-‘((Y*,)’exp( - P$/A)(a,)’
exp[)aq(2A(e-Pi~A - l)],
where we have used eqs. (l), (3) and (8). When this is substituted the A-limit is taken, it yields (o*q)YaqY exp( -
(20) into (18) and
(21)
b,l*P~,).
Thus we have a*, for each a:, and (Yefor each aq ! The sum in (18) then may be evaluated easily with the final result (22)
Z = v (i 1 d’a,) exp( - PWQN
and = Id3r[$(V$]*-
~~~~z]+fld3r~d3rf~~(r)~2~~(r’)~2~(r-
r’),
(23)
COHERENT
STATE THEORY
OF A-TRANSITION
IN He4
609
where JI is the wave function of the condensate. This is the familiar form of the GL model. Here, we have terms only up to the fourth order. Higher order terms may be obtained by including higher order interaction terms in the hamiltonian.
4. Discussion In deriving the result of (23), we were cautious about the non-commutativity of operators. This was because, although the commutators vanish in the limit of infinite A, they might be multiplied by some quantity which diverges in the same limit. But that did not happen, consequently, all operators, ai and a, were effectively replaced by the c-number field a?) and aq This is a unique feature of the coherent state theory in the limit of infinite scale factor. The fact that the coherent state theory is a convenient method for the classical limit of quantum theory is well known’2). When it is formulated in the resealed occupation number representation, it shows the classical nature in a most drastic way. Nowhere in the calculation of 2, did we find anything characteristic of the quantum theory. We have even summed up the entire series ! The A-limit, on the other hand, was introduced on the basis of the conjecture that there is a macroscopic occupation of a single state. The statistical weight in this limit has an interacting feature. There are terms that vanish in that limit. The presence of these terms in the microscopic result account for the particles not in the condensate. However, the number of occupations of each of these states is not macroscopic and therefore becomes negligible. This is the desired coarse graining. There are also terms that do not vanish in the same limit and thus survive the process of the coarse graining. Clearly these terms account for the macroscopic number of particles in the condensate. If there is no condensation, everything should be washed out in the limit of infinite A. The fact that there is something that survives in that limit is a strong suggestion of the condensation. Whether or not the fluctuation of the corresponding wave function takes place around a finite value is determined by (q,,(r)), where the bracket represents the trace of the field operator with the density operator given by (15). One can go through-the same arguments that we have presented for 2 and find out that it is given, in the limit of infinite A, by the average value of Jl(r) with the statistical weight given by exp( - pF{$}), where e(r) is again the resealed eigenvalue of the field operator. Clearly a finite limit exists only when the statistical weight has a symmetry-breaking term which favors one particular phase in the complex plane of the random variable 4(r). This is just
610
J.C. LEE
the c-number equivalence of the idea of Bogoliubov’3) breaking the gauge symmetry of the hamiltonian. Since the c-number version has been amply worked out for magnetic systems, it is fair to claim that there exists a finite limit below the transition temperature. Bose condensation does not require anymore theoretical foundation than does the spontaneous magnetization of a two-component magnetic system. Just as eq. (14) is a coarse grained version of eq. (13) for an ideal boson gas, the free energy given by (23) is a coarse grained version of a much more complicated microscopic result for interacting systems. The latter starts with the idsal gas part given by eq. (13) followed by interaction terms involving all powers in \+I’. The two-particle interaction in the hamiltonian introduces interaction terms of all orders in the statistical weights. Thus, there is a difference between the two results”). The coarse grained result is in the familiar form of the GL model, but the microscopic result is not, except when the infinite h limit is taken. The important question then is: Do those terms eliminated by the A-limit not contribute to the critical behavior? According to the renormalization group transformation (RGT) that we introduced previously4*5,6) for the superfluid helium, the same A-limit has to be taken to determine the critical behavior. The RGT consists of the elimination of the short-wavelength modes followed by the resealing of length and occupation numbers. Therefore those terms (eliminated by the h-limit) are eliminated by the RGT-even if they are retained in the free-energy-and do not contribute to the critical behavior. We made a speculation in a previous paper’) that the critical behavior of the h-transition may be different in some unknown minute detail from that of a two-component classical spin system. In view of the present derivation, there is no such possibility. Since liquid 4He is a more suitable substance to work with experimentally15) than magnetic systems, this should be regarded as good news.
Acknowledgements I wish to thank Dr. Donald Kobe for a very helpful discussion, and Dr. Robert Roger for having helped make the final manuscript more readable. Mitzi Lee and Sue Lee translated the French article in ref. 14 into English for me.
COHERENT
STATE THEORY OF h-TRANSITION
IN He4
611
References 1) J.S. Langer, Phys. Rev. 167 (1968) 183. 2) K.K. Singh, Phys. Rev. 12 (1975) 2819; 13 (1976) 3192; 17 (1978) 324. 3) F.W. Wiegel, Physica 91A (1978) 139; F.W. Wiegel in: Path Integrals, G.J. Papadopoulos and J.T. Devreese, eds. (Plenum, New York, 1978). 4) J.C. Lee, Phys. Rev. 20 (1979) 1277. 5) J.C. Lee, Physica 104A (1980) 189. 6) The analysis promised for T < ‘I, in ref. 5 is straightforward as long as one introduces a symmetry-breaking term in the hamiltonian. However, we do not report it here because it offers little or no additional insight into the problem. My original intention was to find a way to avoid the introduction of the symmetry-breaking term, but I have not succeeded in it yet. 7) R.J. Glauber, Phys. Rev. 131 (1963) 2766. 8) See, for example, S.-K. Ma, Modern Theory of Critical Phenomena (Benjamin, New York 1976). 9) Here we will only present the track of the scale factor A. The algebraic details go parallel to those in refs. 7, 1, and 12. 10) See, for example, A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York 1971). 11) A good review on this aspect is: J. deBoer, Physica 69 (1973) 193. 12) See, for example, P. Carruthers and M.M. Nieto, Am. J. Phys. 33 (1965) 537. 13) N.N. Bogoliubov, Physica 26 (1960) Sl; P.C. Hohenberg and P.C. Martin, Ann. Phys. (N.Y.) 34 (1965) 291. 14) The difference in the ideal boson gas was discussed by S.P. Ohanessian and A. Quattropani, Helv. Phys. Act. 46 (1973) 473. 15) G. Ahlers, Rev. Mod. Phys. 52 (1980) 489.
Publishing Co.
RESCALED COHERENT THE A-TRANSITION
STATE THEORY IN LIQUID He4
OF
J.C. LEE Department
of Chemistry and Physics, Northwestern State University of Louisiana, Natchitoches, LA 71457 USA
Received 11 June 1982
The quantum partition function for the interacting boson system i.s calculated using a complete set of resealed coherent states in which the helium atom occupation numbers are resealed with a large scale factor. There are terms in the statistical weights that do not vanish in the limit of infinite scale factor. These scale independent terms are shown to be in the familiar form of the Ginzburg-Landau model, and are explained as the result of condensation.
1. Introduction
The Ginzburg-Landau (GL) model of the X-transition in liquid He4 assumes that the critical behavior is determined by the fluctuation of the condensate with statistical weights given by \V$l’ plus a power series in 1+1’, where the order parameter field $ is the wave function of the condensate. As this implies that the quantum system has the same critical behavior as the classical system of spins with two real components, it was examined with particular caution by several authors. The model itself was derived by Langer’) (before the renormalization group theory was developed). The critical exponents were calculated by Singhz), Wiegel’) and this author4,5) without adopting the GL model. Wiegel used the method of functional integration to construct an operatorfree formalism for the quantum system. As far as the critical exponents are concerned, the equivalence is clearly established. The partition function turns out to be given by a fluctuating two-component classical field, but the field is a function of a dimensionless variable 8 in addition to the space variable. The variable 19may be eliminated by means of the renormalization group transformation, but it appears difficult to regard the resultant expression for the partition function as a sum of statistical weights of all possible fluctuations of the field. Instead, it is a ratio of two such sums: the numerator takes the form of the GL model, but it is divided by another GL-like form which contains only IV4l’. The thermodynamic effect implied by the presence of the 0378-4371/82/0000-0000/$02.75
0
1982 North-Holland
COHERENT
STATE THEORY
OF A-TRANSITION
IN He4
605
denominator was not discussed, but it appears to us that it has no effect on the equivalence in the critical region. Singh approached the problem differently. The renormalization group transformation was applied directly to a quantum hamiltonian and had the unique feature of resealing the mass of helium atoms. A fixed point was shown to exist with critical behavior identical to that of the two-component classical spin system up to the first order of E = d - 4. Previously’), we replaced the mass resealing with the occupation number resealing, and for T > T, it was shown5) that the identity extends to all orders of E (see footnote6)). In view of the abundant information learned from the GL model, it would be desirable to derive the GL model itself from a quantum mechanical expression of the partition function. Langer has shown, using the coherent state theory of Glauber’), that the partition function may indeed be represented as a sum of statistical weights of all fluctuations of an order parameter function. The order parameter function is identified as the wave function of the condensate (&p(r)). The free energy functional is in a generalized form of the GL model, in the sense that the derivative of the order parameter function enters in a more complicated manner than in the familiar form of the GL model. The free energy functional derived by Langer contains microscopic details that the GL model in general does not. The details are lost in the coarse graining of the latter. The nature of this coarse graining of spin system is well knowns): it is the block spin picture of Kadanoff. In this way one can speak of a continuous spin field with a GL model when, in fact, the spins in the corresponding Ising model can only be + 1 or -1. What then would be the corresponding picture of the coarse graining for the superlluid quantum system? Since the onset of the A-transition is believed to be due to a macroscopic occupation of a condensate, the corresponding coarse graining should “bring out” those helium atoms in the condensate and “smear out” those in the other states. This may be accomplished by resealing the helium atom occupation numbers’) with a large scale factor. It is this conceptual issue that we wish to pursue in this paper. Several identities of the resealed occupation number representation play important roles in the present analysis. For easy reference, we present them in section 2. The coherent state theory in the resealed occupation number representation is presented in section 3. Discussions follow in section 4.
606
J.C.LEE
2. Resealed occupation number representation If one wishes to rescale the occupation numbers with the scale factor A, i.e., n4 = Api, the creation and annihilation operators acting on the state vectors in the resealed occupation number space must be defined by4 a&l;>
=(n>“+;-h-‘>,
a;! n;>
= (n;+ A-‘)“*I,;+ A-‘.
(1)
The net effect of these operators is that A particles appear as one particle and one particle appears as a fraction of particle A-‘, i.e., they are partially blind. Consequently, the boson commutation rules become [a,,
a,3 = ia:, a’,,1= 0, [a,, a;] = A-S,,,.
The resealed stat evectors
are normalized
(2)
by
(n; 1 n;) = A-‘,
(3)
Finally, in the thermal interaction a,(r) = a, e-+‘*,
picture, the operators
take the form
a:(7) = aj)eQ’*,
(4)
where Z, = q2/2 - p, and p is the chemical potential.
3. Resealed coherent states We start with a hamiltonian H =x
+:a,
1 +-2 2v
given by
u,a:_,ai+,apak
= I&+&I,.
(9
and wish to evaluate the grand partition function? Z = Tr exp( - @!I). The trace is to be evaluated using the complete set of resealed coherent defined by a4 I 4
= aq I Q,
(6) states (7)
where ag is any complex number and a, is the resealed annihilation operator. The eigenstates ((Y,)may be constructed from the resealed occupation number states. Using (l), (3) and (7), one can obtain
(8)
COHERENT
The coherent
STATE THEORY
OF A-TRANSITION
states ICXJform a complete
~/d20,(a,)(a,l
601
IN He4
set as they satisfy (9)
= 1.
They are orthogonal in the limit of infinite h, but that has no bearing here. First let us calculate 2, for H = HO, using (10)
20 = v ($ j d’a,)(Iol}l exp( - PHJl{oD, where ~{cx})= III, la,). From (l), (3) and (8), it follows that 20 = v ($jd2aq)(a,l = y
(if
exp(-
p#,a,)(~J
(11)
d201,) expla,12h(e-8’q’A - 1).
The free energy functional
is therefore
given by
Fo{a} = - kT 2 Iaq12h(e-@‘q’r - 1). Setting h = 1 gives the microscopic
(12)
result of Langer,
Fo{a} = - kT 2 Iaq12(e-P& - 1)
(13)
and A + CQgives the coarse grained result, (14) which is the familar Gaussian part of the GL model. Next we include the interaction term in I-I. The density written as emBH= e+“S(p),
operator
is then
(15)
where S(p) is given by”‘) S(p)=
l_jdr~~~(?,)+jdi,jd7,H,oH,(r31 0
0
...
(16)
0
and takes care of the non-commutativity of operators. As eq. (2) suggests, all operators commute in the limit of infinite A, but the limit has to be taken at the end. In the perturbation theory for “fixed number” states (as opposed to the “unsharp number” coherent states”)) these vanishing commutators multiply with the occupation numbers of the states very near the condensate. As the
608
J.C. LEE
limit of infinite A is taken, these states are pushed to the condensate where the occupation numbers diverge. Whether or not the same situation will occur in the coherent state formalism is to be seen, but there is clearly a danger in speaking of vanishing commutators at an early stage of perturbation. Now returning to eq. (16) we carry out the r-integrations. The resultant S(p) shows the following A-dependence: @Hr)“[l+
0(X-‘) + @A-*) + * * -1,
(17)
where HI is now in the Schroedinger picture and the unspecified coefficients of various powers of A-’ are functions of lq only. Clearly, these terms cannot compete with the first term in the large A limit. Thus, they are all dropped and the partition function becomes
(18) A single term in the sum, say (HI)“, contains operators of different modes. Those in the same mode, say (a,)‘(a)“, will pair up with exp( - pa&) in exp( - /3Ho) and get “sandwiched” between ((~~1and ]a,). Thus the matrix element in (18) separates into factors each of which contains only one mode and is of the form (a,] exp( - p~~ba~as)(a~“(a4)‘I~,) + O(A-‘) + O(A-2) + . - .,
(1%
where the creation operators have been moved to the left using the commutation rule of (2) resmting in the additional terms of various orders of A-‘. The unspecified coefficients are the same as the matrix element in the first term but with s and t reduced properly. These may all be neglected in the limit of large A, leaving only the first term, which gives A-‘((Y*,)’exp( - P$/A)(a,)’
exp[)aq(2A(e-Pi~A - l)],
where we have used eqs. (l), (3) and (8). When this is substituted the A-limit is taken, it yields (o*q)YaqY exp( -
(20) into (18) and
(21)
b,l*P~,).
Thus we have a*, for each a:, and (Yefor each aq ! The sum in (18) then may be evaluated easily with the final result (22)
Z = v (i 1 d’a,) exp( - PWQN
and = Id3r[$(V$]*-
~~~~z]+fld3r~d3rf~~(r)~2~~(r’)~2~(r-
r’),
(23)
COHERENT
STATE THEORY
OF A-TRANSITION
IN He4
609
where JI is the wave function of the condensate. This is the familiar form of the GL model. Here, we have terms only up to the fourth order. Higher order terms may be obtained by including higher order interaction terms in the hamiltonian.
4. Discussion In deriving the result of (23), we were cautious about the non-commutativity of operators. This was because, although the commutators vanish in the limit of infinite A, they might be multiplied by some quantity which diverges in the same limit. But that did not happen, consequently, all operators, ai and a, were effectively replaced by the c-number field a?) and aq This is a unique feature of the coherent state theory in the limit of infinite scale factor. The fact that the coherent state theory is a convenient method for the classical limit of quantum theory is well known’2). When it is formulated in the resealed occupation number representation, it shows the classical nature in a most drastic way. Nowhere in the calculation of 2, did we find anything characteristic of the quantum theory. We have even summed up the entire series ! The A-limit, on the other hand, was introduced on the basis of the conjecture that there is a macroscopic occupation of a single state. The statistical weight in this limit has an interacting feature. There are terms that vanish in that limit. The presence of these terms in the microscopic result account for the particles not in the condensate. However, the number of occupations of each of these states is not macroscopic and therefore becomes negligible. This is the desired coarse graining. There are also terms that do not vanish in the same limit and thus survive the process of the coarse graining. Clearly these terms account for the macroscopic number of particles in the condensate. If there is no condensation, everything should be washed out in the limit of infinite A. The fact that there is something that survives in that limit is a strong suggestion of the condensation. Whether or not the fluctuation of the corresponding wave function takes place around a finite value is determined by (q,,(r)), where the bracket represents the trace of the field operator with the density operator given by (15). One can go through-the same arguments that we have presented for 2 and find out that it is given, in the limit of infinite A, by the average value of Jl(r) with the statistical weight given by exp( - pF{$}), where e(r) is again the resealed eigenvalue of the field operator. Clearly a finite limit exists only when the statistical weight has a symmetry-breaking term which favors one particular phase in the complex plane of the random variable 4(r). This is just
610
J.C. LEE
the c-number equivalence of the idea of Bogoliubov’3) breaking the gauge symmetry of the hamiltonian. Since the c-number version has been amply worked out for magnetic systems, it is fair to claim that there exists a finite limit below the transition temperature. Bose condensation does not require anymore theoretical foundation than does the spontaneous magnetization of a two-component magnetic system. Just as eq. (14) is a coarse grained version of eq. (13) for an ideal boson gas, the free energy given by (23) is a coarse grained version of a much more complicated microscopic result for interacting systems. The latter starts with the idsal gas part given by eq. (13) followed by interaction terms involving all powers in \+I’. The two-particle interaction in the hamiltonian introduces interaction terms of all orders in the statistical weights. Thus, there is a difference between the two results”). The coarse grained result is in the familiar form of the GL model, but the microscopic result is not, except when the infinite h limit is taken. The important question then is: Do those terms eliminated by the A-limit not contribute to the critical behavior? According to the renormalization group transformation (RGT) that we introduced previously4*5,6) for the superfluid helium, the same A-limit has to be taken to determine the critical behavior. The RGT consists of the elimination of the short-wavelength modes followed by the resealing of length and occupation numbers. Therefore those terms (eliminated by the h-limit) are eliminated by the RGT-even if they are retained in the free-energy-and do not contribute to the critical behavior. We made a speculation in a previous paper’) that the critical behavior of the h-transition may be different in some unknown minute detail from that of a two-component classical spin system. In view of the present derivation, there is no such possibility. Since liquid 4He is a more suitable substance to work with experimentally15) than magnetic systems, this should be regarded as good news.
Acknowledgements I wish to thank Dr. Donald Kobe for a very helpful discussion, and Dr. Robert Roger for having helped make the final manuscript more readable. Mitzi Lee and Sue Lee translated the French article in ref. 14 into English for me.
COHERENT
STATE THEORY OF h-TRANSITION
IN He4
611
References 1) J.S. Langer, Phys. Rev. 167 (1968) 183. 2) K.K. Singh, Phys. Rev. 12 (1975) 2819; 13 (1976) 3192; 17 (1978) 324. 3) F.W. Wiegel, Physica 91A (1978) 139; F.W. Wiegel in: Path Integrals, G.J. Papadopoulos and J.T. Devreese, eds. (Plenum, New York, 1978). 4) J.C. Lee, Phys. Rev. 20 (1979) 1277. 5) J.C. Lee, Physica 104A (1980) 189. 6) The analysis promised for T < ‘I, in ref. 5 is straightforward as long as one introduces a symmetry-breaking term in the hamiltonian. However, we do not report it here because it offers little or no additional insight into the problem. My original intention was to find a way to avoid the introduction of the symmetry-breaking term, but I have not succeeded in it yet. 7) R.J. Glauber, Phys. Rev. 131 (1963) 2766. 8) See, for example, S.-K. Ma, Modern Theory of Critical Phenomena (Benjamin, New York 1976). 9) Here we will only present the track of the scale factor A. The algebraic details go parallel to those in refs. 7, 1, and 12. 10) See, for example, A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York 1971). 11) A good review on this aspect is: J. deBoer, Physica 69 (1973) 193. 12) See, for example, P. Carruthers and M.M. Nieto, Am. J. Phys. 33 (1965) 537. 13) N.N. Bogoliubov, Physica 26 (1960) Sl; P.C. Hohenberg and P.C. Martin, Ann. Phys. (N.Y.) 34 (1965) 291. 14) The difference in the ideal boson gas was discussed by S.P. Ohanessian and A. Quattropani, Helv. Phys. Act. 46 (1973) 473. 15) G. Ahlers, Rev. Mod. Phys. 52 (1980) 489.