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Ramification of Liouville integrability: Statistical mechanics of the integrable sinh- and sine-Gordon fields
Physica 18D (1986) 368-370 North-Holland, Amsterdam
RAMIFICATION OF LIOUVILLE INTEGRABILITY: STATISTICAL MECHANICS OF T H E INTEGRABLE SINH- AND SINE-GORDON FIELDS R.K. BULLOUGH, D.J. P I L L I N G and M. STIRLAND Department of Mathematics, UMIST, PO Box 88, Manchester M60 1QD, UK
and J. T I M O N E N Department of Physics, University of Jyviiskylii, 40100 Jyviiskyli~, Finland
Extended abstract
The spectral transform discovered by Gardner, Greene, Kruskal and Miura (GGKM) is a canonical transformation. Functional integration on the classical action offers a direct map between the classical and quantum integrable systems; but in order to compare with the Bethe ansatz method and the quantum inverse method it is necessary to evaluate the functional integrals in terms of spectral data. We report on how this is done for the sinh-Gordon equation and draw comparisons with the similar calculations for the non-linear SchrSdinger and sine-Gordon equations.
Martin Kruskal is now quite rightly seeking a more satisfactory definition of "integrability". Nevertheless, this short note is written in admiration of the discovery he jointly made (GGKM) [1] that the Korteweg-de Vries equation was solvable by the inverse scattering method. This isolated fact was then brilliantly "explained" by Zakharov and Faddeev [2] who exhibited the KdV as a completely integrable Hamiltonian system with the required continuous infinity of constants of the motion in involution. The significance of both results is, of course, that they are in some sense universal. Recently this fact has exhibited itself again in the realisation that certain solvable 2-dimensional models in classical statistical mechanics owe their solvability to integrability conditions corresponding to the "Lax pair" [3] description of the KdV. Of these perhaps the 8-vertex model [4] is most significant. Its solution [4, 5] solves the spin-½ X Y Z quantum spin chain. In the continuum limit this chain is equivalent to the quantum massive Thirring and sine-Gordon
models [6]. This and especially Baxter's work [4], stimulated the quantum inverse method of the Leningrad School. In principle, from the classical integrable systems one can derive quantum integrable systems; for, sweeping aside the many problems associated with this view, a direct map from one to the other exists through the functional integral G(q~, qb; T) on the classical action S[q~],
a(q,, ¢o; T) = f~/7~,/,expi S[¢],
(1)
¢(x,0); -½L
L--*oo.
The one-sided Fourier transform of Tr G, the trace of G, yields the quantum eigenspectrum, for example. For comparison with the Bethe ansatz or the
0167-2789/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
369
R.K. Bullough et a l . / Ramification of Liouville integrability
quantum inverse method we need to evaluate G on a phase space described by spectral data a(k), b (k), etc., rather than H, 0. A concrete problem is this one: under the Wick rotation t ~ - i z , T ~ -iflh (fl-1 is the temperature), and in the classical limit h ~ 0, TrG is the classical partition function /,
z = J
n
oexp
-
flH[ep].
(2)
where, for the integrable system with Hamiltonian H[~], H [ p ] is most conveniently that Hamiltonian expressed in action-angle variables: ~ # is a measure to be determined. We shall use the sinhGordon equation as example to show how this can be done. The classical sinh-G has _ -1 f+~L[l_
H[~]=r0
j
~ ~ro
-iL
2/-/2
ldp2
+~ x
+ m2(cosh¢ - 1)) dx,
(4)
where ( H , c k ) = 6 ( x - x ' ) , and Yo>0 is a coupling constant. Under the spectral transform for L ---, ~ (with necessary boundary conditions) H[~] is expressible as H[p] = f~_ooto(k)P(k)dk, where to(k) = ( m 2 + k 2 ) 1/2 > 0, 0 < P ( k ) < oo. The canonical Q ( k ) lie in 0 < Q ( k ) < 2~r, and ( P ( k ) , a ( k ' ) ) = 6 ( k - k'). Evidently, n [ p ] = Ento(k,)P~, (L/2~r)Pn = P(k~), k~ = 2~rnL -1 for L ~ o o (with ( P , , Q m } = 6 , m ) . Apparently, the linear Klein-Gordon equation has the same description. However, we have discovered how to impose periodic boundary conditions on the functional integral (3) without conceding the action-angle variables-for large enough L. We find through Floquet theory that acceptable k~, k~ are given by kn = kn - t - 1
E
m~n
A ( k n , kin)Pro,
(6)
-- ,rr/a
(3)
exp - #H[p],
F L -1 = - f l - l L - 1 In Z = (2~rB)-'f+~/~lnB~(k)dk;
We would want to express this in the form z =
where, for sinh-G, A(k, k') = - ¼3,omZ[kto( k ') k'to(k)] -1. Thus for L ~ o o , k,=--2~rnL - l ~ k and k , ~ k. For free K - G , 3'0 = 0 and ~:, = k,, all n, for L < oo. With the constraint (5) imposed on the functional integral (3) we find the free energy per unit length is
(5)
the cut-off Ikl-< ~ra -1 is needed in the classical limit for physical reasons (ultraviolet divergence, a ~ 0 if possible). The frequencies ~ in (6) are solutions of ~ ( k ) = to(k) +
1 [+~/~
dqa(k'q) dq InfiX(q). (7)
By transforming the functional integral (1) rather than its classical limit (2) we find under WKB quantisation that In/3~ ~ ln(1 - e -a'~) in (6) and (7). These expressions are then of the form given by Yang and Yang, in fermion description [7], for the free energy of a bose gas with repulsive 6-function interactions. Evidently, (6) and (7) are classical limits of a corresponding boson description for quantum sinh-G. By working from (1) for the integrable repulsive classical non-linear Schri3dinger equation with coupling constant c > 0, we regain the quantum form of (6) with (7) with, however, A = - 2 c ( k , k,,) -1. A chemical potential is included by imposing the further constraint EP, = N. To complete the quantum description for the NLS and find the Yangs' result we need the quantum form za = - 2 t a n - x[c ( k , kin) -1] and a careful choice of the branches of the tan -x function: then c = O is free bosons and c = oo is free fermions [7, 8]. This problem must be treated elsewhere. We merely note that under WKB quantisation (5) is the usual Bethe ansatz periodicity condition and revert to (6) with (7) which give the classical free energy for sinh-G: this proves to
370
R.K. Bullough et al. / Ramification of Liouville integrability
be
F=mfl l[¼(Mt~)-l-~(i~)-2q-3(ifl)-3 L -~(ifl)
-4 + . . . ]
+fl l a - l ( l n ( f l a - l ) + ½ m a } ,
(8)
M = 8mY0-1. T h e cut-off d e p e n d e n t term is found by using the dispersion relation for the infinite lattice [8], not ~ ( k ) = ( m 2 + k 2 ) 1/2, and the series (8) can be f o u n d b y integrating the integral equation (7). The result (8) is also found exactly by the transfer integral m e t h o d [8] applied to (2), not (3). The p r o b l e m reduces to finding the smallest eigenvalue e 0 of the S t u r m - L i o u v i l l e p r o b l e m [
1 2~ m
d~ ] d~b--5- + (coshd~ - 1) + Vo ~b,(O)
= cab,(+) ;
(9)
___ / ~ 2 m 2 y 0 2 ' Vo = f l - l Y o a - 1 m 2 ln(fl'rol/2rra) 1/2. The series (8) is asymptotic.
Our conclusion is that the functional integral (1) expressed in terms of appropriate spectral data is fully equivalent to the Bethe ansatz method and the quantum inverse method with the additional merit that at temperatures f l - l > 0 explicit asymptotic series can be found for the free energies. This is yet one more example of the universal features contained in the inverse scattering method (spectral transform method) discovered by G G K M in 1967 for KdV.
Acknowledgements O n e of us ( R K B ) is grateful to the Institute for Theoretical Physics, University of California at Santa Barbara, for the opportunity to attend the meeting on 'Solitons and Coherent Structures'; and to t h e m also for their hospitality during the ' I n t e g r a b l e models' programme. This short note was written at that time.
m*
M e t h o d s of m a t c h e d asymptotic expansions allow c o n t i n u a t i o n in the coupling constant Y0. Since s i n h - G --, s i n - G by ~ ~ - iff, ~'0 ~ - 3'0, we can find for s i n - G that -~/0 replaces ~/0 ( - M replaces M ) in (8) but there is added -- j~ _ l m 8 ( m ~ / ~ ~0 )1/2
× e - ~ M [ 1 -- ~( Mfl) -1 + ... ]
(lO)
and terms in e-2tiM, etc. To find this result from the functional integral (3) we need to extend the constraint (5) to include the solitons. This has been done [8] but some details of the subsequent calculation are incomplete.
References [1] C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura (GGKM), Phys. Rev. Lett. 19 (1967) 1095. [2] V.E. Zakharov and L.D. Faddeev, Funct. Anal. Appl. 5 (1971) 280. [3] P.D. Lax, Comm. Pure. Appl. Math. 21 (1968) 467. [4] R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Adademic Press, New York, 1982). [5] JD. Johnson, S. Krinsky and B.M. McCoy, Phys. Rev. A8 (1973) 2526. [6] A. Luther, in: Solitons, Topics in Current Physics 17, R.K. Bullough and P.J. Caudrey, eds. (Springer, Heidelberg, 1980); and references. [7] C.N. Yang and C.P. Yang, J. Math. Phys. 10 (1969) 1115. [8] R.K. Bullough, Statistical Mechanics of the sine-Gordon Field, in: Nonlinear Phenomena in Physics, F. Claro, ed. (Springer, Heidelberg, 1985) pp. 70-102 and R.K. Bullough, D.J. Pilling and J. Timonen, ibid, pp. 103-128.
RAMIFICATION OF LIOUVILLE INTEGRABILITY: STATISTICAL MECHANICS OF T H E INTEGRABLE SINH- AND SINE-GORDON FIELDS R.K. BULLOUGH, D.J. P I L L I N G and M. STIRLAND Department of Mathematics, UMIST, PO Box 88, Manchester M60 1QD, UK
and J. T I M O N E N Department of Physics, University of Jyviiskylii, 40100 Jyviiskyli~, Finland
Extended abstract
The spectral transform discovered by Gardner, Greene, Kruskal and Miura (GGKM) is a canonical transformation. Functional integration on the classical action offers a direct map between the classical and quantum integrable systems; but in order to compare with the Bethe ansatz method and the quantum inverse method it is necessary to evaluate the functional integrals in terms of spectral data. We report on how this is done for the sinh-Gordon equation and draw comparisons with the similar calculations for the non-linear SchrSdinger and sine-Gordon equations.
Martin Kruskal is now quite rightly seeking a more satisfactory definition of "integrability". Nevertheless, this short note is written in admiration of the discovery he jointly made (GGKM) [1] that the Korteweg-de Vries equation was solvable by the inverse scattering method. This isolated fact was then brilliantly "explained" by Zakharov and Faddeev [2] who exhibited the KdV as a completely integrable Hamiltonian system with the required continuous infinity of constants of the motion in involution. The significance of both results is, of course, that they are in some sense universal. Recently this fact has exhibited itself again in the realisation that certain solvable 2-dimensional models in classical statistical mechanics owe their solvability to integrability conditions corresponding to the "Lax pair" [3] description of the KdV. Of these perhaps the 8-vertex model [4] is most significant. Its solution [4, 5] solves the spin-½ X Y Z quantum spin chain. In the continuum limit this chain is equivalent to the quantum massive Thirring and sine-Gordon
models [6]. This and especially Baxter's work [4], stimulated the quantum inverse method of the Leningrad School. In principle, from the classical integrable systems one can derive quantum integrable systems; for, sweeping aside the many problems associated with this view, a direct map from one to the other exists through the functional integral G(q~, qb; T) on the classical action S[q~],
a(q,, ¢o; T) = f~/7~,/,expi S[¢],
(1)
¢(x,0); -½L
L--*oo.
The one-sided Fourier transform of Tr G, the trace of G, yields the quantum eigenspectrum, for example. For comparison with the Bethe ansatz or the
0167-2789/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
369
R.K. Bullough et a l . / Ramification of Liouville integrability
quantum inverse method we need to evaluate G on a phase space described by spectral data a(k), b (k), etc., rather than H, 0. A concrete problem is this one: under the Wick rotation t ~ - i z , T ~ -iflh (fl-1 is the temperature), and in the classical limit h ~ 0, TrG is the classical partition function /,
z = J
n
oexp
-
flH[ep].
(2)
where, for the integrable system with Hamiltonian H[~], H [ p ] is most conveniently that Hamiltonian expressed in action-angle variables: ~ # is a measure to be determined. We shall use the sinhGordon equation as example to show how this can be done. The classical sinh-G has _ -1 f+~L[l_
H[~]=r0
j
~ ~ro
-iL
2/-/2
ldp2
+~ x
+ m2(cosh¢ - 1)) dx,
(4)
where ( H , c k ) = 6 ( x - x ' ) , and Yo>0 is a coupling constant. Under the spectral transform for L ---, ~ (with necessary boundary conditions) H[~] is expressible as H[p] = f~_ooto(k)P(k)dk, where to(k) = ( m 2 + k 2 ) 1/2 > 0, 0 < P ( k ) < oo. The canonical Q ( k ) lie in 0 < Q ( k ) < 2~r, and ( P ( k ) , a ( k ' ) ) = 6 ( k - k'). Evidently, n [ p ] = Ento(k,)P~, (L/2~r)Pn = P(k~), k~ = 2~rnL -1 for L ~ o o (with ( P , , Q m } = 6 , m ) . Apparently, the linear Klein-Gordon equation has the same description. However, we have discovered how to impose periodic boundary conditions on the functional integral (3) without conceding the action-angle variables-for large enough L. We find through Floquet theory that acceptable k~, k~ are given by kn = kn - t - 1
E
m~n
A ( k n , kin)Pro,
(6)
-- ,rr/a
(3)
exp - #H[p],
F L -1 = - f l - l L - 1 In Z = (2~rB)-'f+~/~lnB~(k)dk;
We would want to express this in the form z =
where, for sinh-G, A(k, k') = - ¼3,omZ[kto( k ') k'to(k)] -1. Thus for L ~ o o , k,=--2~rnL - l ~ k and k , ~ k. For free K - G , 3'0 = 0 and ~:, = k,, all n, for L < oo. With the constraint (5) imposed on the functional integral (3) we find the free energy per unit length is
(5)
the cut-off Ikl-< ~ra -1 is needed in the classical limit for physical reasons (ultraviolet divergence, a ~ 0 if possible). The frequencies ~ in (6) are solutions of ~ ( k ) = to(k) +
1 [+~/~
dqa(k'q) dq InfiX(q). (7)
By transforming the functional integral (1) rather than its classical limit (2) we find under WKB quantisation that In/3~ ~ ln(1 - e -a'~) in (6) and (7). These expressions are then of the form given by Yang and Yang, in fermion description [7], for the free energy of a bose gas with repulsive 6-function interactions. Evidently, (6) and (7) are classical limits of a corresponding boson description for quantum sinh-G. By working from (1) for the integrable repulsive classical non-linear Schri3dinger equation with coupling constant c > 0, we regain the quantum form of (6) with (7) with, however, A = - 2 c ( k , k,,) -1. A chemical potential is included by imposing the further constraint EP, = N. To complete the quantum description for the NLS and find the Yangs' result we need the quantum form za = - 2 t a n - x[c ( k , kin) -1] and a careful choice of the branches of the tan -x function: then c = O is free bosons and c = oo is free fermions [7, 8]. This problem must be treated elsewhere. We merely note that under WKB quantisation (5) is the usual Bethe ansatz periodicity condition and revert to (6) with (7) which give the classical free energy for sinh-G: this proves to
370
R.K. Bullough et al. / Ramification of Liouville integrability
be
F=mfl l[¼(Mt~)-l-~(i~)-2q-3(ifl)-3 L -~(ifl)
-4 + . . . ]
+fl l a - l ( l n ( f l a - l ) + ½ m a } ,
(8)
M = 8mY0-1. T h e cut-off d e p e n d e n t term is found by using the dispersion relation for the infinite lattice [8], not ~ ( k ) = ( m 2 + k 2 ) 1/2, and the series (8) can be f o u n d b y integrating the integral equation (7). The result (8) is also found exactly by the transfer integral m e t h o d [8] applied to (2), not (3). The p r o b l e m reduces to finding the smallest eigenvalue e 0 of the S t u r m - L i o u v i l l e p r o b l e m [
1 2~ m
d~ ] d~b--5- + (coshd~ - 1) + Vo ~b,(O)
= cab,(+) ;
(9)
___ / ~ 2 m 2 y 0 2 ' Vo = f l - l Y o a - 1 m 2 ln(fl'rol/2rra) 1/2. The series (8) is asymptotic.
Our conclusion is that the functional integral (1) expressed in terms of appropriate spectral data is fully equivalent to the Bethe ansatz method and the quantum inverse method with the additional merit that at temperatures f l - l > 0 explicit asymptotic series can be found for the free energies. This is yet one more example of the universal features contained in the inverse scattering method (spectral transform method) discovered by G G K M in 1967 for KdV.
Acknowledgements O n e of us ( R K B ) is grateful to the Institute for Theoretical Physics, University of California at Santa Barbara, for the opportunity to attend the meeting on 'Solitons and Coherent Structures'; and to t h e m also for their hospitality during the ' I n t e g r a b l e models' programme. This short note was written at that time.
m*
M e t h o d s of m a t c h e d asymptotic expansions allow c o n t i n u a t i o n in the coupling constant Y0. Since s i n h - G --, s i n - G by ~ ~ - iff, ~'0 ~ - 3'0, we can find for s i n - G that -~/0 replaces ~/0 ( - M replaces M ) in (8) but there is added -- j~ _ l m 8 ( m ~ / ~ ~0 )1/2
× e - ~ M [ 1 -- ~( Mfl) -1 + ... ]
(lO)
and terms in e-2tiM, etc. To find this result from the functional integral (3) we need to extend the constraint (5) to include the solitons. This has been done [8] but some details of the subsequent calculation are incomplete.
References [1] C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura (GGKM), Phys. Rev. Lett. 19 (1967) 1095. [2] V.E. Zakharov and L.D. Faddeev, Funct. Anal. Appl. 5 (1971) 280. [3] P.D. Lax, Comm. Pure. Appl. Math. 21 (1968) 467. [4] R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Adademic Press, New York, 1982). [5] JD. Johnson, S. Krinsky and B.M. McCoy, Phys. Rev. A8 (1973) 2526. [6] A. Luther, in: Solitons, Topics in Current Physics 17, R.K. Bullough and P.J. Caudrey, eds. (Springer, Heidelberg, 1980); and references. [7] C.N. Yang and C.P. Yang, J. Math. Phys. 10 (1969) 1115. [8] R.K. Bullough, Statistical Mechanics of the sine-Gordon Field, in: Nonlinear Phenomena in Physics, F. Claro, ed. (Springer, Heidelberg, 1985) pp. 70-102 and R.K. Bullough, D.J. Pilling and J. Timonen, ibid, pp. 103-128.