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Semi-classical approximation to the octic double well oscillator
Volume lI9B, number 1,2,3
PHYSICS LETTERS
16 December 1982
SEMI-CLASSICAL APPROXIMATION TO THE OCTIC DOUBLE WELL OSCILLATOR
E. MOTTOLA and A. ROUET x
Institute for Advanced Study, Princeton,NJ, 08540, USA Received 30 July 1982
We show that it is possible to calculate the energy levels and splittings in the oetic double well potential by semi-classical methods - although dilute gas instanton techniques are not applicable. Numerical results are given and compared to other estimates.
It is now well known that the complex saddle point method is a powerful substitute to the dilute gas approximation. In fact, whenever the dilute gas approximation applies, as in the computation of the energy splitting due to tunnelling [1], the superposition of instantons and anti-instantons can be shown to mimic the exact complex classical solutions [2]. Moreover, the complex saddle point method allows one to compute tr exp ( - ~ H ) at Fmite temperature [3], which cannot be done using the dilute gas. In this letter, we use the complex saddle point method to compute the spectrum of the hamiltonian associated with the potential V = ½(~b2 , f 2 ) 4 .
(1)
This is a double well potential, but the bottom of each well is quartic. The absence of gaussian curvature makes any simple dilute gas approximation impossible [4]. Such a difficulty exists in gauge field theories like QCD or the nonlinear a-model which are classically conformal invariant. In order to compute the spectrum, we consider G(E) [3,5], the Legendre transform of the partition function
G(E)=tr~__E= ~n En -1E "
(2)
In the semi-classical approximation, it reads [3]
1 Albert Einstein Professor. Permanent address: CNRS-CPT, Luminy-Case 907 F-13288 MarseiUeCedex 2 - France. 162
G(E) = ~_l Tp(E)exp [-Wp(E)] pathsp
exp(i0p).
(3)
The sum runs over all the possible periodic paths p. Wp(E) is defined by
Wp(E) = Jd~b [2 g(q~) -
2E] 1/2,
(4)
where the contour of integration is the path p. The class of all possible paths includes multiple traversals. I f p denotes such a multiple traversal path, let's call PO the associated single traversal path. Then Tp(E) is defined as
rp(E) =
~Wpo/aE.
(s)
Op is equal to
-~r/2 times the number of turning points for the path. For the potential eq. (1) the integrand of eq. (4) has eight complex branch points: A+ = _+[f2 + (2E)1/4] 1/2, B± = +[f2 _ (2E)1/4] 1/2,
C± = +[[2 +i(2E)1/4]1/2, D± = +[[2 _i(2E)1/4]1/2, (6) where (2E) 1/4 is chosen to be real and positive. The four cuts can be chosen according to fig. 1. A general path can be analytically deformed to a combination of the basic paths represented in fig. 2, according to the number of times they surround the different cuts. In fig. 3, the same picture is sketched in a more readable way. In terms of this picture, a general path con0 031-9163[82/0000-0000/$02.75
© 1982
North-Holland
Volume 119B, number 1,2,3
PHYSICS LETTERS
P
D--
/
A-
16 December 1982
a2
p/
C÷
B+//A +-
/B-
Fig. 3. Schematic diagram of the same paths of fig. 2 with only their essential features shown: the points of intersection and multiplicative weights defined by eq. (7).
D+
C-
B÷
Fig. I. Complex z plane showing the locations of the eight complex branch points and cuts of (V - E ) 1/2 . tributing to eq. (3) is a path which starts and ends at some point o f the contour fig. 3, and runs over this contour following the arrows, traversing any loop an arbitrary number of times. The computation of G(E) involves some lengthy combinatorics for evaluating the contribution of these various paths. We shall present only the relevant result of the calculation here. An analogous but simpler application o f the method o f calculation can be found in ref. [3]. The result can be expressed in terms of the basic quantifies.
B-
b 1 = exp /~2i f [ 2 ( E - V)] 1/2 /) e-i~r/2 , ReD-
(7)
5
5
-o*,-
R
2
B-
a3=exp(i f \ v
)
t 2 ( E - V ) ] 1/2 e-iTr/2,
/~_ //I~p/
G(E) can be expressed as a rational function of these quantities together with their derivatives with respect to E. Since we are interested in the poles of G(E), i t is sufficient to write down the denominator D(E) of G(E) - the numerator being non-singular. D ( E ) = [1 -
(a 3 + as)(a 3 + b s ) ] 2
- a22[1 - a3(a3 + a l ) ] 2 •
(8)
D ( E ) [a~fa3=0 = (1 - alb 1)2 .
o'T ,,r
L,_q
Fig. 2. The three independent paths from which the general closed periodic path contributing to eq. (3) can be constructed. The path segments which are dashed indicate they are on the second Riemann sheet of the function (V - E) 1/2 . Thus the points of intersection, P, Q, P', Q*, R, S are as shown.
(9)
The square shows that - as expected, the spectrum is doubly degenerate. Taking into account a 3 introduces the contributions of the complex turning points, in a way analogous to the calculation carried out by Balian et al. [6].
D(E)[a2= 0 = [ 1 - ( a 3 + a l ) ( a 3 + b l ) ] s
(7 ton'd)
C-
Let us emphasize the meaning of these contributions. If we neglect the asymptotically small contributions a 2 and a3, we get the WKB answer for the spectrum without taking into account tunnelling effects:
/ ReDa l = e x p t+2iAf- [ 2 ( E - IF)] 1/2) e-in/2 ,
P
o =oxp(-/i2 v_ jx,2) o-i ,2
2 .
(9')
The spectrum is still doubly degenerate. Taking into account a 2 introduces the tunnelling contribution which splits the spectrum. We are then in a position to compute the energy splitting due to the tunnelling effect. Before taking into account the tunnelling, we can compute the ground state energy E 0 as the lowest solution of 1 - (a 3 +
a l ) (a 3 + bl)IgffiEo
= 0.
(10)
Then, at lowest order, the energy splitting is given by 163
Volume 119B, number 1,2,3
PHYSICS LETTERS
Table 1 Our values for the ground state E 0 and energy splitting ~E =.E 1 - E o compared to those of ref. [7] .1 ,Eo and fiE, for various values of the parameter f2.
2 3 4
1.62 2.21 2.72
0.020 0.7 X 10 -6 2.2× 10 -14
1.87 2.64 3.26
0.028 < 10 --4 < 10 -4
I31 a 2 [ 1 - a3(a3 + a l ) ] I BE0 - (a/BE)) [1 - (a3+al)(a 3 + b l ) ] e=eo
(11)
If we neglect the exponentially Small correction a 3, we get simply
8Eo ~ ( a / a E )
i-i-- (albl) l E =E 0
(12)
In table 1 we present numerical results for the ground state and the energy splitting, for different values o f the parameter f 2. In this numerical computation, we have neglected the exponentially small corrections due to a 3. Our results are compared to the numerical calculations o f ref. [7]* t , which do not use the WKB approximation. We have thus shown that the semi-classical approximation (not dilute gas approximation) can be work,1 The authors of ref. [7] use a hamiltonian which is twice ours. In the comparison of results in table 1 we have taken this into account by dividing their results by two.
164
16 December 1982
ed out for the double well octic oscillator, despite the absence o f gaussian curvature at the b o t t o m o f each well. The numerical agreement is not unreasonable, if we take into account the fact that WKB is not expected to give accurate results for the ground state. Better numerical accuracy is obtained for this ground state in the strong-weak coupling approximation o f ref. [8]. We wish to t h ~ l k A. Patrascioiu for illuminating discussions and for informing us o f his current results. We feel indebted to the Institute for Advanced Study for its hospitality. One o f us (A.R.) acknowledges the support o f a grant from the Federal Republic o f Germany. This work was supported in part b y the Department o f Energy under Grant No. DE-AC0276ER02220.
References [1] E. Gildener and A. Patrascioiu, Phys. Rev. D16 (1977) 423. [2] J.L. Richard and A. Rouet, Nucl. Phys. B183 (1981) 251;B185 (1981) 47. [3] A. Lapedes and E. Mottola, Nucl. Phys. B203 (1982) 58. [4] A. Patrascioiu, Phys. Rev. D17 (1978) 2764. [5] R. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D11 (1975) 3424. [6] R. Balian, G. Parisi and A. Voros, Phys. Rev. Lett. 41 (1978) 1141. [7] R. Blankenbecler, T. DeGrand and R.L. Sugar, Phys. Rev. D21 (1980) 1055. [8] A.Patrascioiu, An improved semi-classical approximation.
PHYSICS LETTERS
16 December 1982
SEMI-CLASSICAL APPROXIMATION TO THE OCTIC DOUBLE WELL OSCILLATOR
E. MOTTOLA and A. ROUET x
Institute for Advanced Study, Princeton,NJ, 08540, USA Received 30 July 1982
We show that it is possible to calculate the energy levels and splittings in the oetic double well potential by semi-classical methods - although dilute gas instanton techniques are not applicable. Numerical results are given and compared to other estimates.
It is now well known that the complex saddle point method is a powerful substitute to the dilute gas approximation. In fact, whenever the dilute gas approximation applies, as in the computation of the energy splitting due to tunnelling [1], the superposition of instantons and anti-instantons can be shown to mimic the exact complex classical solutions [2]. Moreover, the complex saddle point method allows one to compute tr exp ( - ~ H ) at Fmite temperature [3], which cannot be done using the dilute gas. In this letter, we use the complex saddle point method to compute the spectrum of the hamiltonian associated with the potential V = ½(~b2 , f 2 ) 4 .
(1)
This is a double well potential, but the bottom of each well is quartic. The absence of gaussian curvature makes any simple dilute gas approximation impossible [4]. Such a difficulty exists in gauge field theories like QCD or the nonlinear a-model which are classically conformal invariant. In order to compute the spectrum, we consider G(E) [3,5], the Legendre transform of the partition function
G(E)=tr~__E= ~n En -1E "
(2)
In the semi-classical approximation, it reads [3]
1 Albert Einstein Professor. Permanent address: CNRS-CPT, Luminy-Case 907 F-13288 MarseiUeCedex 2 - France. 162
G(E) = ~_l Tp(E)exp [-Wp(E)] pathsp
exp(i0p).
(3)
The sum runs over all the possible periodic paths p. Wp(E) is defined by
Wp(E) = Jd~b [2 g(q~) -
2E] 1/2,
(4)
where the contour of integration is the path p. The class of all possible paths includes multiple traversals. I f p denotes such a multiple traversal path, let's call PO the associated single traversal path. Then Tp(E) is defined as
rp(E) =
~Wpo/aE.
(s)
Op is equal to
-~r/2 times the number of turning points for the path. For the potential eq. (1) the integrand of eq. (4) has eight complex branch points: A+ = _+[f2 + (2E)1/4] 1/2, B± = +[f2 _ (2E)1/4] 1/2,
C± = +[[2 +i(2E)1/4]1/2, D± = +[[2 _i(2E)1/4]1/2, (6) where (2E) 1/4 is chosen to be real and positive. The four cuts can be chosen according to fig. 1. A general path can be analytically deformed to a combination of the basic paths represented in fig. 2, according to the number of times they surround the different cuts. In fig. 3, the same picture is sketched in a more readable way. In terms of this picture, a general path con0 031-9163[82/0000-0000/$02.75
© 1982
North-Holland
Volume 119B, number 1,2,3
PHYSICS LETTERS
P
D--
/
A-
16 December 1982
a2
p/
C÷
B+//A +-
/B-
Fig. 3. Schematic diagram of the same paths of fig. 2 with only their essential features shown: the points of intersection and multiplicative weights defined by eq. (7).
D+
C-
B÷
Fig. I. Complex z plane showing the locations of the eight complex branch points and cuts of (V - E ) 1/2 . tributing to eq. (3) is a path which starts and ends at some point o f the contour fig. 3, and runs over this contour following the arrows, traversing any loop an arbitrary number of times. The computation of G(E) involves some lengthy combinatorics for evaluating the contribution of these various paths. We shall present only the relevant result of the calculation here. An analogous but simpler application o f the method o f calculation can be found in ref. [3]. The result can be expressed in terms of the basic quantifies.
B-
b 1 = exp /~2i f [ 2 ( E - V)] 1/2 /) e-i~r/2 , ReD-
(7)
5
5
-o*,-
R
2
B-
a3=exp(i f \ v
)
t 2 ( E - V ) ] 1/2 e-iTr/2,
/~_ //I~p/
G(E) can be expressed as a rational function of these quantities together with their derivatives with respect to E. Since we are interested in the poles of G(E), i t is sufficient to write down the denominator D(E) of G(E) - the numerator being non-singular. D ( E ) = [1 -
(a 3 + as)(a 3 + b s ) ] 2
- a22[1 - a3(a3 + a l ) ] 2 •
(8)
D ( E ) [a~fa3=0 = (1 - alb 1)2 .
o'T ,,r
L,_q
Fig. 2. The three independent paths from which the general closed periodic path contributing to eq. (3) can be constructed. The path segments which are dashed indicate they are on the second Riemann sheet of the function (V - E) 1/2 . Thus the points of intersection, P, Q, P', Q*, R, S are as shown.
(9)
The square shows that - as expected, the spectrum is doubly degenerate. Taking into account a 3 introduces the contributions of the complex turning points, in a way analogous to the calculation carried out by Balian et al. [6].
D(E)[a2= 0 = [ 1 - ( a 3 + a l ) ( a 3 + b l ) ] s
(7 ton'd)
C-
Let us emphasize the meaning of these contributions. If we neglect the asymptotically small contributions a 2 and a3, we get the WKB answer for the spectrum without taking into account tunnelling effects:
/ ReDa l = e x p t+2iAf- [ 2 ( E - IF)] 1/2) e-in/2 ,
P
o =oxp(-/i2 v_ jx,2) o-i ,2
2 .
(9')
The spectrum is still doubly degenerate. Taking into account a 2 introduces the tunnelling contribution which splits the spectrum. We are then in a position to compute the energy splitting due to the tunnelling effect. Before taking into account the tunnelling, we can compute the ground state energy E 0 as the lowest solution of 1 - (a 3 +
a l ) (a 3 + bl)IgffiEo
= 0.
(10)
Then, at lowest order, the energy splitting is given by 163
Volume 119B, number 1,2,3
PHYSICS LETTERS
Table 1 Our values for the ground state E 0 and energy splitting ~E =.E 1 - E o compared to those of ref. [7] .1 ,Eo and fiE, for various values of the parameter f2.
2 3 4
1.62 2.21 2.72
0.020 0.7 X 10 -6 2.2× 10 -14
1.87 2.64 3.26
0.028 < 10 --4 < 10 -4
I31 a 2 [ 1 - a3(a3 + a l ) ] I BE0 - (a/BE)) [1 - (a3+al)(a 3 + b l ) ] e=eo
(11)
If we neglect the exponentially Small correction a 3, we get simply
8Eo ~ ( a / a E )
i-i-- (albl) l E =E 0
(12)
In table 1 we present numerical results for the ground state and the energy splitting, for different values o f the parameter f 2. In this numerical computation, we have neglected the exponentially small corrections due to a 3. Our results are compared to the numerical calculations o f ref. [7]* t , which do not use the WKB approximation. We have thus shown that the semi-classical approximation (not dilute gas approximation) can be work,1 The authors of ref. [7] use a hamiltonian which is twice ours. In the comparison of results in table 1 we have taken this into account by dividing their results by two.
164
16 December 1982
ed out for the double well octic oscillator, despite the absence o f gaussian curvature at the b o t t o m o f each well. The numerical agreement is not unreasonable, if we take into account the fact that WKB is not expected to give accurate results for the ground state. Better numerical accuracy is obtained for this ground state in the strong-weak coupling approximation o f ref. [8]. We wish to t h ~ l k A. Patrascioiu for illuminating discussions and for informing us o f his current results. We feel indebted to the Institute for Advanced Study for its hospitality. One o f us (A.R.) acknowledges the support o f a grant from the Federal Republic o f Germany. This work was supported in part b y the Department o f Energy under Grant No. DE-AC0276ER02220.
References [1] E. Gildener and A. Patrascioiu, Phys. Rev. D16 (1977) 423. [2] J.L. Richard and A. Rouet, Nucl. Phys. B183 (1981) 251;B185 (1981) 47. [3] A. Lapedes and E. Mottola, Nucl. Phys. B203 (1982) 58. [4] A. Patrascioiu, Phys. Rev. D17 (1978) 2764. [5] R. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D11 (1975) 3424. [6] R. Balian, G. Parisi and A. Voros, Phys. Rev. Lett. 41 (1978) 1141. [7] R. Blankenbecler, T. DeGrand and R.L. Sugar, Phys. Rev. D21 (1980) 1055. [8] A.Patrascioiu, An improved semi-classical approximation.