- Email: [email protected]
When does the Rayleigh-Schrödinger perturbation series for the energy eigenvalue have nothing to do with the exact value?
Volume 104B, number 2
PHYSICS LETTERS
20 August 1981
WHEN DOES THE R A Y L E I G H - S C H R O D I N G E R PERTURBATION SERIES FOR THE ENERGY EIGENVALUE HAVE NOTHING TO DO WITH THE EXACT VALUE? S.N. BEHERA Institute of Physics, Bhubanesh war 751 O07, India and Avinash KHARE 1 Department of Theoretical Physics, The University, Manchester M13 9PL, UK Received 28 July 1980
On the basis of some exactly solvable models we suggest that the Rayleigh-Schr6dinger perturbation series for the energy eigenvalue may have nothing to do with the corresponding exact value in case the vacuum expectation value of the system changes discontinuously, i.e. the system undergoes first-order phase transition.
Recently a number of authors [1 ] have given counter-examples to Dyson's heuristic argument [2] that instability implies divergence of perturbation theory. In particular, it has been shown that for nonrelativistic quantum-mechanical systems having unstable vacua for suitable values of the coupling constant g, the Rayleigh-Schr6dinger (RS) perturbation series associated with the energy eigenvalue E ( g ) may (a) converge to a correct eigenvalue or diverge but be uniquely Borel summable to the correct eigenvalue, (b) converge to the wrong eigenvalue, or (c) converge or diverge but be not asymptotic to E(g), i.e. RS perturbation series has nothing to do with E(g). The possibilities (b) and (c) are quite disturbing particularly in the context of quantum field theory where perturbation theory is perhaps the only known way o f making systematic calculations. Thus it is o f the utmost importance to know what kind of theories are expected to exhibit features (b) and (c). The purpose of this note is to suggest a possible answer to this vitally important question. In particular with the help o f some exactly solvable models we pro-
1 Nuffield Foundation Fellow. On leave of absence from Institute of Physics, Bhubaneshwar 751007, India.
pose that the possibilities (b) and (c) may occur whenever the vacuum expectation value of a system changes discontinuously, i.e. it undergoes a first-order phase transition while for a second-order (or no) transition one has possibility (a). We start with the hamiltonian H=
fox l - l [~m(d~/dt) 2 + ½mc20(dqVdx)2
+ B~b2 - IA [~b4 + C~b6 ] ,
(1)
which is the continuum limit of the corresponding one-dimensional lattice problem [3]. Here I is the lattice spacing, c O is the velocity of sound and ~b(x, t) isa the displacement of the mass at the space-time point (x, t). In the mean-field approximation, the on-site potential given by V(~b)=B~b 2 - l A l q ~ 4 +C~b6 ,
(B,C>O,A
<0),
(2)
is a representative of a first-order phase transition; the transition taking place at a = 9/8 where a = 9BC/2 IA [2 To appreciate this let us notice that the potential (2) has three minima at [4]
0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
169
= 0,
~b+= +(IA [/3C) 1/2[1 + (1 - ~a) 1/2] 1/2 ,
V(4L+) = (2 IA 13/27C2)[a - 1 - (1 - 2a)3/2] ,
r/(~b) which we normalise by (3)
1(94
Thus whereas for a > 3/2 there is only a single well; at a = 3/2 we have two points o f inflation at 4~= +-([A 1/ 3C) 1/2 in addition to the minimum at ¢ = 0. For 9/8 < a < 3/2 we have two degenerate local minima and an absolute minimum at 4~ = 0. At a = 9/8 all the three minima o f the potential are degenerate while for 0 < a < 9/8 there are two degenerate absolute minima and a local minimum at ~ = 0. Finally for a ~< 0 one has two degenerate minima and q~ = 0 is now a maximum of the potential. Now as has been shown by Scalapino et al. [5 ] the classical partition function o f such systems can be exactly evaluated in the thermodynamic limit provided the ground-state energy e 0 of the Schr6dinger-like equation [ - ( 2 m ) - 1 d2/dq~ 2 + B~b2 _ ~41 q~4+Cq~6 ] ~n(~ b) = en((~),
(4) can be calculated. Thus the conclusions about the validity or otherwise o f the RS perturbation series will have a direct bearing on these lattice models which have applications in the theory of H e 3 - H e 4 admixture and metamagnets [6] as well as for structural phase transitions in ferroelectrics [4]. Let us first try to obtain e 0 for a > 9/8, i.e. above the first-order phase transition. It is not very difficult to show that if i.e.
a = g9 [1 - (9C[2mB2)1/2] - 1 > 9/8 ,
(8)
is not square integrable but since r/(• = -+x/~A I/2C) = 1, dr/(q~ = -+x//A I/2C)/d~ = 0 hence
~bmax = +(IA I/3C) 1/2 [1 - (1 - ~a)l[2] 1/2 ,
0 < e 0 - ~(2B/m) 1/2 [1 - (9C/2mB2) 1/2 ] 1/2 <~Nlexp[-N2(IA 12Jl 6C3/2) x/-2-m] .
(9)
On the other hand, if we calculate e 0 perturbatively by expanding about an harmonic oscillator at 4~ = 0, we get
e 0 = ~(2B/m) 1/2 - ¼(C/m2B) 1/2 + .... i.e. the RS perturbation series converges to the wrong value since it fails to give the exponentially damped term. Ir is worth pointing out here that for 9/8 < a < 3/2, V(¢) as given by eq. (2) has two local minima and an absolute minimum at ~ = 0. For the particular case of C = g4, IA I = 2g 2, B = 1 + 3g 2, 2m = 1, the hamiltonian of eq. (4) is essentially H (3) of Herbst and Simon [1 ]. Can we conclude similarly for a < 9/8, i.e. below the phase transition? In this case it is easy to show [4, 7] that if B = [A 12/4C - 5(C/2m) 1/2
i.e.
a = 9 [1 + (25C/2mB2)1/21 - i < 9/8
(10)
then the energy of the first excited state is given by
e 1 = - ~(2B/m)l/2[1 + (25C/2mB2)1/2] 1/2 •
(11)
Since RS perturbation theory around 4~ = 0 would give (5)
then the hamiltonian of eq. (4) can be written as
H =A*A + {(2B/rn) 1/2 [1 - (9C/2mB2) 1/2] 1/2 , (6) where
A ((p) = (2m)- l/Z d/d~ + ( IA I/2C1/2)dP - C1/2~b 3. (7) This implies that
e 0 >1 ~(2B/m) 1/2 [1 - (9C/2mB2) 1/2] 1/2 , the equality being valid only and only if the solution r/(~b) o f A (~b)r/(~) = 0 is square integrable. The solution 170
n(~b) = e x p [ - ~1 [A 1(2m/c)l/2dp 2 + (2mC) 1/2
+ (IA 12/16C)(2m/C) 1/2 ] ,
and two maxima at
B = IA 12/4C + 3(C/2m) 1/2
20 August 1981
PHYSICS LETTERS
Volume 104B, number 2
e I = + ~(2B/m) 1/2 + ... hence it is not asymptotic to the exact value. One might argue that this breakdown of perturbation theory around q~ = 0 is because it is only a local minimum o f the potential (2) for 0 < a < 9/8. However, we now show that the breakdown o f perturbation theory persists even around 4~= qL_. On substituting 4~ = 4~+ + cr in the potential as given by eq. (2) we have
V(o) = V(q~±) + [B - 6 IA I~b2 + 15C~ 4] o 2 + ..., where V(~b±)is given by eq. (3). On using eq. (10), it is
Volume 104B, number 2
PHYSICS LETTERS
20 August 1981
not difficult to show that the perturbative value of e 1 obtained by expanding about o = 0 is
and Simon [ 1 ]. The point is that by a suitable substitution q~-+ ~b+ a this potential can be converted to
e 1 = ~ ( 2 B / m ) 1/2 + ... =/: _ 3 ( 2 B / m ) l / 2
V(qS) =g2q~4 + 2g,~b3 + (I +g")~b 2 + g ' " ,
+ ....
Thus we have seen that for first-order phase transition the RS perturbation series may have nothing to do with the exact eigenvalue both above and below the transition point. What can one say in the case of a second-order transition? Consider a model analogous to eq. (1) with V(40 given by V(~)=Bq~ 2 +Aq5 4 +C~ 6 ,
A,C>O
.
(12)
It is easy to convince oneself that in the mean-field approximation this model exhibits a second-order phase transition, the transition taking place at B = 0 [4]. Now if B = A 2/4C - 3 (C/2m) 1/2 > 0 ,
(13)
then the e 0 can be shown to be given by e 0 = ½ ( 2 B / m ) 1/2 [1 + ( 9 C / 2 m B 2 ) 1/2 ] 1/2 .
(14)
Further, following Herbst and Simon [ 1 ] it follows that the RS series for the ground-state energy is asymptotic to e 0 and correctly converges to it in the limit A -+ O, C --," O, A 2 / 4 C = B (finite). Thus above the second-order transition (B > 0) the RS perturbation series follow the possibility (a) rather than (b) or (c). For below the transition, one knows for the example of the q~4 potential that the RS series even though divergent, is uniquely Borel summable to the correct eigen-
which is known to have a discontinuous order parameter, i.e. is known to exhibit a first-order transition in the mean-field approximation [8]. The results obtained in this note have a direct bearing on some of the conclusions reached recently by Sobelman [9]. Assuming that the RS double series expansion for e 0 around ~ = 0 converges to the correct eigenvalue for the potential as given by eqs. (2) and (12) he has tried to obtain the expansion coefficients. As our analysis has shown, even though this assumption is valid for the potential o f e q . (12) it is not so for the one given by eq. (2) thereby invalidating most of his results f o r A < 0, a > 9/8 ( - 1 < 3' < 0 in his notation). The most important conclusion which emerges from our paper is that for systems undergoing a firstorder phase transition, the RS perturbation theory may not be valid. This is particularly disturbing in the context of nonabelian gauge theories since there is some indication that these theories undergo a phase transition even though the order of the transition is not clear as yet. In case it turns out that it is a firstorder phase transition, the whole perturbation theory programme in these theories will be highly suspect (unless one can prove otherwise). Thus the problem of the order of the phase transition is nonabelian gauge theories deserves immediate attention.
value. From these examples we see that whereas for a second-order (or no) transition one encounters the possibility (a) for the first-order case one may have the possibifity (b) or (c). In a way this is understandable. The point is that for a first-order transition (unlike the second order) the order parameter has a discontinuous behaviour and this discontinuous behaviour may be reflected in the expression for the free energy and hence in the energy eigenvalues of the system. We therefore believe that this explanation has a more general validity. We can substantiate this picture by explaining why the RS series for the energy eigenvalue of the potential V(~b) = g2~b4 + 2g~ 3 + q~2 _ 2g~b
(15)
converge to a wrong answer as first shown by Herbst
One of us (AK)is grateful to the members of the theoretical physics department of Manchester University for fruitful conversations. Further he wishes to thank Professor A. Donnachie for warm and cordial hospitality, and acknowledges a Nuffield Foundation Fellowship.
References
[1] I.W. Herbst and B. Simon, Phys. Lett. 78B (1978) 304; 80B (1979) 433(E); F. Calogero, Lett. Nuovo Cimento 25 (1979) 533; M.P. Fry, Phys. Lett. 86B (1979) 183. [2] F. Dyson, Phys. Rev. 85 (1952) 631. [3] J.A. Krumhausl and J.R. Schrieffer, Phys. Rev. B11 (1975) 3535. 171
Volume 104B, number 2
PHYSICS LETTERS
[4] A. Khare and S.N. Behera, Pramana, 14 (1980) 327; 15 (1980) 245; M.E. Lines and A.M. Glass, Principles and applications of ferroelectrics and related materials (Clarendon, Oxford, 1977).
172
[5] [6] [7] [8] [9]
20 August 1981
D.J. Scalapino et al., Phys. Rev. B6 (1972) 3409. J.M. Kincaid and E.G. Cohen, Phys. Rep. 22C (1975) 57. V. Singh et al., Phys. Rev. D18 (1978) 190. T. Sasada, J. Phys. A12 (1979) 2583. G.E. Sobelman, Phys. Rev. D19 (1979) 3754.
PHYSICS LETTERS
20 August 1981
WHEN DOES THE R A Y L E I G H - S C H R O D I N G E R PERTURBATION SERIES FOR THE ENERGY EIGENVALUE HAVE NOTHING TO DO WITH THE EXACT VALUE? S.N. BEHERA Institute of Physics, Bhubanesh war 751 O07, India and Avinash KHARE 1 Department of Theoretical Physics, The University, Manchester M13 9PL, UK Received 28 July 1980
On the basis of some exactly solvable models we suggest that the Rayleigh-Schr6dinger perturbation series for the energy eigenvalue may have nothing to do with the corresponding exact value in case the vacuum expectation value of the system changes discontinuously, i.e. the system undergoes first-order phase transition.
Recently a number of authors [1 ] have given counter-examples to Dyson's heuristic argument [2] that instability implies divergence of perturbation theory. In particular, it has been shown that for nonrelativistic quantum-mechanical systems having unstable vacua for suitable values of the coupling constant g, the Rayleigh-Schr6dinger (RS) perturbation series associated with the energy eigenvalue E ( g ) may (a) converge to a correct eigenvalue or diverge but be uniquely Borel summable to the correct eigenvalue, (b) converge to the wrong eigenvalue, or (c) converge or diverge but be not asymptotic to E(g), i.e. RS perturbation series has nothing to do with E(g). The possibilities (b) and (c) are quite disturbing particularly in the context of quantum field theory where perturbation theory is perhaps the only known way o f making systematic calculations. Thus it is o f the utmost importance to know what kind of theories are expected to exhibit features (b) and (c). The purpose of this note is to suggest a possible answer to this vitally important question. In particular with the help o f some exactly solvable models we pro-
1 Nuffield Foundation Fellow. On leave of absence from Institute of Physics, Bhubaneshwar 751007, India.
pose that the possibilities (b) and (c) may occur whenever the vacuum expectation value of a system changes discontinuously, i.e. it undergoes a first-order phase transition while for a second-order (or no) transition one has possibility (a). We start with the hamiltonian H=
fox l - l [~m(d~/dt) 2 + ½mc20(dqVdx)2
+ B~b2 - IA [~b4 + C~b6 ] ,
(1)
which is the continuum limit of the corresponding one-dimensional lattice problem [3]. Here I is the lattice spacing, c O is the velocity of sound and ~b(x, t) isa the displacement of the mass at the space-time point (x, t). In the mean-field approximation, the on-site potential given by V(~b)=B~b 2 - l A l q ~ 4 +C~b6 ,
(B,C>O,A
<0),
(2)
is a representative of a first-order phase transition; the transition taking place at a = 9/8 where a = 9BC/2 IA [2 To appreciate this let us notice that the potential (2) has three minima at [4]
0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
169
= 0,
~b+= +(IA [/3C) 1/2[1 + (1 - ~a) 1/2] 1/2 ,
V(4L+) = (2 IA 13/27C2)[a - 1 - (1 - 2a)3/2] ,
r/(~b) which we normalise by (3)
1(94
Thus whereas for a > 3/2 there is only a single well; at a = 3/2 we have two points o f inflation at 4~= +-([A 1/ 3C) 1/2 in addition to the minimum at ¢ = 0. For 9/8 < a < 3/2 we have two degenerate local minima and an absolute minimum at 4~ = 0. At a = 9/8 all the three minima o f the potential are degenerate while for 0 < a < 9/8 there are two degenerate absolute minima and a local minimum at ~ = 0. Finally for a ~< 0 one has two degenerate minima and q~ = 0 is now a maximum of the potential. Now as has been shown by Scalapino et al. [5 ] the classical partition function o f such systems can be exactly evaluated in the thermodynamic limit provided the ground-state energy e 0 of the Schr6dinger-like equation [ - ( 2 m ) - 1 d2/dq~ 2 + B~b2 _ ~41 q~4+Cq~6 ] ~n(~ b) = en((~),
(4) can be calculated. Thus the conclusions about the validity or otherwise o f the RS perturbation series will have a direct bearing on these lattice models which have applications in the theory of H e 3 - H e 4 admixture and metamagnets [6] as well as for structural phase transitions in ferroelectrics [4]. Let us first try to obtain e 0 for a > 9/8, i.e. above the first-order phase transition. It is not very difficult to show that if i.e.
a = g9 [1 - (9C[2mB2)1/2] - 1 > 9/8 ,
(8)
is not square integrable but since r/(• = -+x/~A I/2C) = 1, dr/(q~ = -+x//A I/2C)/d~ = 0 hence
~bmax = +(IA I/3C) 1/2 [1 - (1 - ~a)l[2] 1/2 ,
0 < e 0 - ~(2B/m) 1/2 [1 - (9C/2mB2) 1/2 ] 1/2 <~Nlexp[-N2(IA 12Jl 6C3/2) x/-2-m] .
(9)
On the other hand, if we calculate e 0 perturbatively by expanding about an harmonic oscillator at 4~ = 0, we get
e 0 = ~(2B/m) 1/2 - ¼(C/m2B) 1/2 + .... i.e. the RS perturbation series converges to the wrong value since it fails to give the exponentially damped term. Ir is worth pointing out here that for 9/8 < a < 3/2, V(¢) as given by eq. (2) has two local minima and an absolute minimum at ~ = 0. For the particular case of C = g4, IA I = 2g 2, B = 1 + 3g 2, 2m = 1, the hamiltonian of eq. (4) is essentially H (3) of Herbst and Simon [1 ]. Can we conclude similarly for a < 9/8, i.e. below the phase transition? In this case it is easy to show [4, 7] that if B = [A 12/4C - 5(C/2m) 1/2
i.e.
a = 9 [1 + (25C/2mB2)1/21 - i < 9/8
(10)
then the energy of the first excited state is given by
e 1 = - ~(2B/m)l/2[1 + (25C/2mB2)1/2] 1/2 •
(11)
Since RS perturbation theory around 4~ = 0 would give (5)
then the hamiltonian of eq. (4) can be written as
H =A*A + {(2B/rn) 1/2 [1 - (9C/2mB2) 1/2] 1/2 , (6) where
A ((p) = (2m)- l/Z d/d~ + ( IA I/2C1/2)dP - C1/2~b 3. (7) This implies that
e 0 >1 ~(2B/m) 1/2 [1 - (9C/2mB2) 1/2] 1/2 , the equality being valid only and only if the solution r/(~b) o f A (~b)r/(~) = 0 is square integrable. The solution 170
n(~b) = e x p [ - ~1 [A 1(2m/c)l/2dp 2 + (2mC) 1/2
+ (IA 12/16C)(2m/C) 1/2 ] ,
and two maxima at
B = IA 12/4C + 3(C/2m) 1/2
20 August 1981
PHYSICS LETTERS
Volume 104B, number 2
e I = + ~(2B/m) 1/2 + ... hence it is not asymptotic to the exact value. One might argue that this breakdown of perturbation theory around q~ = 0 is because it is only a local minimum o f the potential (2) for 0 < a < 9/8. However, we now show that the breakdown o f perturbation theory persists even around 4~= qL_. On substituting 4~ = 4~+ + cr in the potential as given by eq. (2) we have
V(o) = V(q~±) + [B - 6 IA I~b2 + 15C~ 4] o 2 + ..., where V(~b±)is given by eq. (3). On using eq. (10), it is
Volume 104B, number 2
PHYSICS LETTERS
20 August 1981
not difficult to show that the perturbative value of e 1 obtained by expanding about o = 0 is
and Simon [ 1 ]. The point is that by a suitable substitution q~-+ ~b+ a this potential can be converted to
e 1 = ~ ( 2 B / m ) 1/2 + ... =/: _ 3 ( 2 B / m ) l / 2
V(qS) =g2q~4 + 2g,~b3 + (I +g")~b 2 + g ' " ,
+ ....
Thus we have seen that for first-order phase transition the RS perturbation series may have nothing to do with the exact eigenvalue both above and below the transition point. What can one say in the case of a second-order transition? Consider a model analogous to eq. (1) with V(40 given by V(~)=Bq~ 2 +Aq5 4 +C~ 6 ,
A,C>O
.
(12)
It is easy to convince oneself that in the mean-field approximation this model exhibits a second-order phase transition, the transition taking place at B = 0 [4]. Now if B = A 2/4C - 3 (C/2m) 1/2 > 0 ,
(13)
then the e 0 can be shown to be given by e 0 = ½ ( 2 B / m ) 1/2 [1 + ( 9 C / 2 m B 2 ) 1/2 ] 1/2 .
(14)
Further, following Herbst and Simon [ 1 ] it follows that the RS series for the ground-state energy is asymptotic to e 0 and correctly converges to it in the limit A -+ O, C --," O, A 2 / 4 C = B (finite). Thus above the second-order transition (B > 0) the RS perturbation series follow the possibility (a) rather than (b) or (c). For below the transition, one knows for the example of the q~4 potential that the RS series even though divergent, is uniquely Borel summable to the correct eigen-
which is known to have a discontinuous order parameter, i.e. is known to exhibit a first-order transition in the mean-field approximation [8]. The results obtained in this note have a direct bearing on some of the conclusions reached recently by Sobelman [9]. Assuming that the RS double series expansion for e 0 around ~ = 0 converges to the correct eigenvalue for the potential as given by eqs. (2) and (12) he has tried to obtain the expansion coefficients. As our analysis has shown, even though this assumption is valid for the potential o f e q . (12) it is not so for the one given by eq. (2) thereby invalidating most of his results f o r A < 0, a > 9/8 ( - 1 < 3' < 0 in his notation). The most important conclusion which emerges from our paper is that for systems undergoing a firstorder phase transition, the RS perturbation theory may not be valid. This is particularly disturbing in the context of nonabelian gauge theories since there is some indication that these theories undergo a phase transition even though the order of the transition is not clear as yet. In case it turns out that it is a firstorder phase transition, the whole perturbation theory programme in these theories will be highly suspect (unless one can prove otherwise). Thus the problem of the order of the phase transition is nonabelian gauge theories deserves immediate attention.
value. From these examples we see that whereas for a second-order (or no) transition one encounters the possibility (a) for the first-order case one may have the possibifity (b) or (c). In a way this is understandable. The point is that for a first-order transition (unlike the second order) the order parameter has a discontinuous behaviour and this discontinuous behaviour may be reflected in the expression for the free energy and hence in the energy eigenvalues of the system. We therefore believe that this explanation has a more general validity. We can substantiate this picture by explaining why the RS series for the energy eigenvalue of the potential V(~b) = g2~b4 + 2g~ 3 + q~2 _ 2g~b
(15)
converge to a wrong answer as first shown by Herbst
One of us (AK)is grateful to the members of the theoretical physics department of Manchester University for fruitful conversations. Further he wishes to thank Professor A. Donnachie for warm and cordial hospitality, and acknowledges a Nuffield Foundation Fellowship.
References
[1] I.W. Herbst and B. Simon, Phys. Lett. 78B (1978) 304; 80B (1979) 433(E); F. Calogero, Lett. Nuovo Cimento 25 (1979) 533; M.P. Fry, Phys. Lett. 86B (1979) 183. [2] F. Dyson, Phys. Rev. 85 (1952) 631. [3] J.A. Krumhausl and J.R. Schrieffer, Phys. Rev. B11 (1975) 3535. 171
Volume 104B, number 2
PHYSICS LETTERS
[4] A. Khare and S.N. Behera, Pramana, 14 (1980) 327; 15 (1980) 245; M.E. Lines and A.M. Glass, Principles and applications of ferroelectrics and related materials (Clarendon, Oxford, 1977).
172
[5] [6] [7] [8] [9]
20 August 1981
D.J. Scalapino et al., Phys. Rev. B6 (1972) 3409. J.M. Kincaid and E.G. Cohen, Phys. Rep. 22C (1975) 57. V. Singh et al., Phys. Rev. D18 (1978) 190. T. Sasada, J. Phys. A12 (1979) 2583. G.E. Sobelman, Phys. Rev. D19 (1979) 3754.