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A conventional approach to dynamic correlations in the Toda lattice
Volume 85A, number 4
A CONVENTIONAL
28 September 1981
PHYSICS LETTERS
APPROACH TO DYNAMIC CORRELATIONS
IN THE TODA LATTICE
S. DIEDERICH Institu t fiir Theoretische Physik, Justus-Liebig-Universit& Giessen, D-6300 Giessen, Fed. Rep. Germany Received 18 May 1981 The spectral density associated with correlations of relative displacements is calculated in the Toda chain. Strong nonlinear motion activated already at low temperatures gives rise to a shoulder on the high-frequency side of the phonon peak. This secondary structure is a reminiscence of the sol&on response obtained with a more refined untraditional approach.
Statistical mechanical properties of soliton-bearing systems are the subject of an increasing activity in the physics of nonlinear one-dimensional systems. In the special case of the exponential lattice dynamic correlation functions require particular attention to provide the basis for the construction of a phenomenological configuration space approach. This is because in this system it is very difficult to identify the known nonlinear excitations in the low-temperature properties of the static quantities alone. In a recent paper of Schneider and Stall [l] the dynamics of the linear exponential lattice is investigated, employing a molecular-dynamics technique. Although clear-cut evidence is obtained for the presence of thermally excited solitons in the low-temperature dynamics of the chain, the authors do not succeed in an identification of these solitons in the spectral densities associated with the force-force and displacement-displacement correlations. Phonons and solitons lend their weight to one and the same one-peak structure only. These results do not agree with those which we have obtained recently by an analytic procedure [2]. In the special case of the exponential lattice a closed system of equations for the displacement-displacement response has been derived which is valid at low temperatures. The associated spectral densities obtained by a numerical solution of these equations reveal a two-peak structure in certain regions of wave vector space. On the high-frequency side of the conventional phonon response we found a second peak which was traced back to soliton-like motion in the chain. The object of the present paper is to contribute to 0 03 l-9163/81/0000-0000/$02.75
0 1981 North-Holland
a clarification of this controversial point. We present results of a calculation of the spectral density which is associated with the displacement-displacement correlation function. The approximation procedure employed is a conventional one and corresponds to a fully renormalized three-phonon to second-order process to the phonon self-energy. It turns out that already in this low-order approximation - only three-phonon vertices are included - characteristic deviations from the conventional weakly damped phonon response occur. This seems to contradict the molecular-dynamics results. Before presenting the results we give a derivation of the dynamical equations which describe the applied approximation procedure employing the projection operator formalism [3,4]. We proceed from the equations of motion of the chain, which are written as [2]
WithIk = (2/??2)(1 - COSk). ~4; and /!!:k (-n
iv
Ai =7~ nFlexP(--~n)Rn Periodic boundary
conditions
(2)
. R 1 = RNI+ 1, R2 =
R1M+2, S1 = ,F$~+~etc. are imposed on the chain; M
denotes the number of independent
particles and @(R) 233
Volume 85A, number 4
PHYSICS LETTERS
is the interaction potential between nearest neighbours. Applying the projection operator formalism for the variables of inter& At, Ai, it can be shown [5] that the relaxation functions ti4 (t) = (A A4(0) Ai(t))/(kB T) (the average ( 1is defined with respect to the classical canonical ensemble) obey the following system of equations:
28 September 1981
pression for X in terms of fourth-order correlations which may be decoupled: %7,0=
LWq(~~2)2/(2W1 c v$,(W~_~W,@I P
where the summation (C,) has to be performed within the first Brillouin zone. The system of eqs. (9) (3) and (7) provides the basis for the numerical determination of the relaxation function G,(t). It can also be derived within the framework of our previous approach [2] if we restrict ourselves to three-phonon vertices only. The phonon self-energy E defined by eq. (9) is diagrammatically represented by
(d2,dt2+X11)~q(f)+SP(4,t--T)~y(7)d~=o. 0
(3)
We give the low-temperature approximations for G1 and the phonon self-energy E for the special case of the Toda potential G(R) = (a/b) exp [-b(R - D)] t a(R - 0) ,
(4)
where a, b are constants and D denotes the natural distance between adjacent particles. The susceptibility J/ (0) which represents an initial value for the solution o P eq. (3) is determined with the help of the exact second and fourth moments:
=abZi. v,co> =-zq, l?qo>
(596)
These relations are a consequence of the equations of motion (1) together with (4). The relaxation function S,(t) as calculated from eq. (3) has to satisfy eqs. (5) and (6) which results in II/,(O) =Z*xq = tab - X(q,
oYr,l-1 *
(7)
Up to this stage of our calculation the employed equations are still rigorous. The phonon self-energy E remains to be calculated. Restricting ourselves to contributions which are correct in the limit T + 0, the following approximate representation for this quantity is obtained by the application of the projection operator technique [ 5,2] : x(4, t) = (Z&T)-’ A&)] ) . (8) The “random forces” [zi(r> t -lAi(r)] /Zq appearing in this equation may be calcu2 ated approximately from the equations of motion (1) together with eqs. (2) and (4). At first we neglect higher than threephonon vertices of the interaction potential and restrict ourselves only to contributions up to linear order in the temperature. From eq. (8) we obtain an exX ([;if,(O)
234
+ x$4f,(O>]
[;l’,:(t> + g’
2:=<>.
(10)
The phonon lines used here are renormalized. Moreover, renormalized three-phononvertices and harmonic frequencies are included in our approach. This is a direct consequence of the presumed exponential interaction potential (4). By thermally averaging the identity bq+~“(R) = -@“‘(R) = -b2 [g’(R) - a] we obtain from (G’(R)) = 0 the equation: b(@“) = +“‘)
= ab2 .
(11)
(G’(R)) = 0 is a consequence of the vanishing external pressure subjected to the chain. Thus the prefactor in eq. (9) is proportional to (@“‘>2,and the representations (6) and (7) in reality include the frequency w9 of the renormalized harmonic approximation. The simple equation w4 = (G#?,Z$/2 = (abZq) ‘j2
(12)
for this frequency expresses the fact that we are actually including higher-order contributions in our ap-
proach arising from the four-phonon vertex for example. These four-phonon to first-order processes are already absorbed into the renor-malized frequency 04, which in the special case of the exponential chain at zero pressure is given by eq. (12). The spectral density Jq(w) = 7 dt eiW’J/,(t)/Jl,(t --m
= O),
obtained from the solution of eqs. (3) (7) and (9) for a chain of M = 240 particles is plotted for several 4-
PHYSICS LETTERS
Volume 85A, number 4
0.2
0.L
0.6
0.8
1.0
1.2
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28 September 1981
sion of the lattice. It is more pronounced than the corresponding one of ref. [2]. Thus in the present approach the lattice reacts softerat the chosen temperature then observed previously, which results from the restriction to cubic vertices ordy. At higher temperatures this restriction does not even lead to stable results. Especially at intermediate wave vectors a shoulder appears on the high-frequency side of the phonon response which reflects the strong anharmonicities already present in the chain at the chosen temperature value. It seems to correspond to similar pronounced effects of strong anharmonicities encountered in zerotemperature calculations for solid He [6], H2 and D2 [7], and in finite-temperature calculations for the alkali halides [ 81. This secondary structure is a reminiscence of the soliton-peak which we have found with the help of the more refined approach referred to above [2]. The spectral densities obtained by this earlier approach are included in fig. 1 for the purpose of comparison. Neglecting higher than three-phonon vertices of the interaction potential clearly produces a confinement in the calculation which oversees subtle details of the interaction which are necessary in order to build up a well-defined soliton response in the displace: ment-displacement correlation. Hence, only a shoulder remains in our present approach; this shoulder, however, does not even appear in Schneider and Stoll’s work. Thus the results of the present calculation support our earlier ones. I would like to thank H. Horner for critical comments concerning the two previous papers [2] which stimulated me to the present approach.
0.L
0.8
I2
1.6
2.0
References
w
Fig. 1. The spectral density I&(W) versus w. Solid lines: present work, dashed lines: our earlier approach [2]. Parameters are chosen identical to those of Schneider and StoB (a = b = = m = 1, kgT = 0.25). values in fig. 1. As expected, a phonon peak is observed for a given wave vector. The shift of this peak to lower frequency values with respect to the renormalized harmonic frequency o9 = (ab1,)‘i2 (marked by arrows 1 in fig. 1) results from the thermal expan-
[l] [2] [3] [4]
[5] [6] [7] [8]
T. Schneider and E. Stoll, Phys. Rev. Lett. 45 (1980) 997, S. Diederich, Phys. Rev. B24 (1981). H. Mori, Prog. Theor. Phys. 33 (1965) 423. W. Giitze and K.H. Michel, in: Dynamical properties of solids, Vol. 1, eds.G.K. Horton and A.A. Maradudin (North-Holland, Amsterdam, 1974). S.W. Lovesey, Condensed matter physics: dynamic correlations (Benjamin, London, 1980). H. Horner, J. Low Temp. Phys. 8 (1972) 511. V.V. Goldman, J. Low Temp. Phys. (1980) 149. E.R. Cowley and R.A. Cowley, Proc. R. Sot. London 287 (1965) 259.
235
A CONVENTIONAL
28 September 1981
PHYSICS LETTERS
APPROACH TO DYNAMIC CORRELATIONS
IN THE TODA LATTICE
S. DIEDERICH Institu t fiir Theoretische Physik, Justus-Liebig-Universit& Giessen, D-6300 Giessen, Fed. Rep. Germany Received 18 May 1981 The spectral density associated with correlations of relative displacements is calculated in the Toda chain. Strong nonlinear motion activated already at low temperatures gives rise to a shoulder on the high-frequency side of the phonon peak. This secondary structure is a reminiscence of the sol&on response obtained with a more refined untraditional approach.
Statistical mechanical properties of soliton-bearing systems are the subject of an increasing activity in the physics of nonlinear one-dimensional systems. In the special case of the exponential lattice dynamic correlation functions require particular attention to provide the basis for the construction of a phenomenological configuration space approach. This is because in this system it is very difficult to identify the known nonlinear excitations in the low-temperature properties of the static quantities alone. In a recent paper of Schneider and Stall [l] the dynamics of the linear exponential lattice is investigated, employing a molecular-dynamics technique. Although clear-cut evidence is obtained for the presence of thermally excited solitons in the low-temperature dynamics of the chain, the authors do not succeed in an identification of these solitons in the spectral densities associated with the force-force and displacement-displacement correlations. Phonons and solitons lend their weight to one and the same one-peak structure only. These results do not agree with those which we have obtained recently by an analytic procedure [2]. In the special case of the exponential lattice a closed system of equations for the displacement-displacement response has been derived which is valid at low temperatures. The associated spectral densities obtained by a numerical solution of these equations reveal a two-peak structure in certain regions of wave vector space. On the high-frequency side of the conventional phonon response we found a second peak which was traced back to soliton-like motion in the chain. The object of the present paper is to contribute to 0 03 l-9163/81/0000-0000/$02.75
0 1981 North-Holland
a clarification of this controversial point. We present results of a calculation of the spectral density which is associated with the displacement-displacement correlation function. The approximation procedure employed is a conventional one and corresponds to a fully renormalized three-phonon to second-order process to the phonon self-energy. It turns out that already in this low-order approximation - only three-phonon vertices are included - characteristic deviations from the conventional weakly damped phonon response occur. This seems to contradict the molecular-dynamics results. Before presenting the results we give a derivation of the dynamical equations which describe the applied approximation procedure employing the projection operator formalism [3,4]. We proceed from the equations of motion of the chain, which are written as [2]
WithIk = (2/??2)(1 - COSk). ~4; and /!!:k (-n
iv
Ai =7~ nFlexP(--~n)Rn Periodic boundary
conditions
(2)
. R 1 = RNI+ 1, R2 =
R1M+2, S1 = ,F$~+~etc. are imposed on the chain; M
denotes the number of independent
particles and @(R) 233
Volume 85A, number 4
PHYSICS LETTERS
is the interaction potential between nearest neighbours. Applying the projection operator formalism for the variables of inter& At, Ai, it can be shown [5] that the relaxation functions ti4 (t) = (A A4(0) Ai(t))/(kB T) (the average ( 1is defined with respect to the classical canonical ensemble) obey the following system of equations:
28 September 1981
pression for X in terms of fourth-order correlations which may be decoupled: %7,0=
LWq(~~2)2/(2W1 c v$,(W~_~W,@I P
where the summation (C,) has to be performed within the first Brillouin zone. The system of eqs. (9) (3) and (7) provides the basis for the numerical determination of the relaxation function G,(t). It can also be derived within the framework of our previous approach [2] if we restrict ourselves to three-phonon vertices only. The phonon self-energy E defined by eq. (9) is diagrammatically represented by
(d2,dt2+X11)~q(f)+SP(4,t--T)~y(7)d~=o. 0
(3)
We give the low-temperature approximations for G1 and the phonon self-energy E for the special case of the Toda potential G(R) = (a/b) exp [-b(R - D)] t a(R - 0) ,
(4)
where a, b are constants and D denotes the natural distance between adjacent particles. The susceptibility J/ (0) which represents an initial value for the solution o P eq. (3) is determined with the help of the exact second and fourth moments:
=abZi. v,co> =-zq, l?qo>
(596)
These relations are a consequence of the equations of motion (1) together with (4). The relaxation function S,(t) as calculated from eq. (3) has to satisfy eqs. (5) and (6) which results in II/,(O) =Z*xq = tab - X(q,
oYr,l-1 *
(7)
Up to this stage of our calculation the employed equations are still rigorous. The phonon self-energy E remains to be calculated. Restricting ourselves to contributions which are correct in the limit T + 0, the following approximate representation for this quantity is obtained by the application of the projection operator technique [ 5,2] : x(4, t) = (Z&T)-’ A&)] ) . (8) The “random forces” [zi(r> t -lAi(r)] /Zq appearing in this equation may be calcu2 ated approximately from the equations of motion (1) together with eqs. (2) and (4). At first we neglect higher than threephonon vertices of the interaction potential and restrict ourselves only to contributions up to linear order in the temperature. From eq. (8) we obtain an exX ([;if,(O)
234
+ x$4f,(O>]
[;l’,:(t> + g’
2:=<>.
(10)
The phonon lines used here are renormalized. Moreover, renormalized three-phononvertices and harmonic frequencies are included in our approach. This is a direct consequence of the presumed exponential interaction potential (4). By thermally averaging the identity bq+~“(R) = -@“‘(R) = -b2 [g’(R) - a] we obtain from (G’(R)) = 0 the equation: b(@“) = +“‘)
= ab2 .
(11)
(G’(R)) = 0 is a consequence of the vanishing external pressure subjected to the chain. Thus the prefactor in eq. (9) is proportional to (@“‘>2,and the representations (6) and (7) in reality include the frequency w9 of the renormalized harmonic approximation. The simple equation w4 = (G#?,Z$/2 = (abZq) ‘j2
(12)
for this frequency expresses the fact that we are actually including higher-order contributions in our ap-
proach arising from the four-phonon vertex for example. These four-phonon to first-order processes are already absorbed into the renor-malized frequency 04, which in the special case of the exponential chain at zero pressure is given by eq. (12). The spectral density Jq(w) = 7 dt eiW’J/,(t)/Jl,(t --m
= O),
obtained from the solution of eqs. (3) (7) and (9) for a chain of M = 240 particles is plotted for several 4-
PHYSICS LETTERS
Volume 85A, number 4
0.2
0.L
0.6
0.8
1.0
1.2
iI II
12 -
28 September 1981
sion of the lattice. It is more pronounced than the corresponding one of ref. [2]. Thus in the present approach the lattice reacts softerat the chosen temperature then observed previously, which results from the restriction to cubic vertices ordy. At higher temperatures this restriction does not even lead to stable results. Especially at intermediate wave vectors a shoulder appears on the high-frequency side of the phonon response which reflects the strong anharmonicities already present in the chain at the chosen temperature value. It seems to correspond to similar pronounced effects of strong anharmonicities encountered in zerotemperature calculations for solid He [6], H2 and D2 [7], and in finite-temperature calculations for the alkali halides [ 81. This secondary structure is a reminiscence of the soliton-peak which we have found with the help of the more refined approach referred to above [2]. The spectral densities obtained by this earlier approach are included in fig. 1 for the purpose of comparison. Neglecting higher than three-phonon vertices of the interaction potential clearly produces a confinement in the calculation which oversees subtle details of the interaction which are necessary in order to build up a well-defined soliton response in the displace: ment-displacement correlation. Hence, only a shoulder remains in our present approach; this shoulder, however, does not even appear in Schneider and Stoll’s work. Thus the results of the present calculation support our earlier ones. I would like to thank H. Horner for critical comments concerning the two previous papers [2] which stimulated me to the present approach.
0.L
0.8
I2
1.6
2.0
References
w
Fig. 1. The spectral density I&(W) versus w. Solid lines: present work, dashed lines: our earlier approach [2]. Parameters are chosen identical to those of Schneider and StoB (a = b = = m = 1, kgT = 0.25). values in fig. 1. As expected, a phonon peak is observed for a given wave vector. The shift of this peak to lower frequency values with respect to the renormalized harmonic frequency o9 = (ab1,)‘i2 (marked by arrows 1 in fig. 1) results from the thermal expan-
[l] [2] [3] [4]
[5] [6] [7] [8]
T. Schneider and E. Stoll, Phys. Rev. Lett. 45 (1980) 997, S. Diederich, Phys. Rev. B24 (1981). H. Mori, Prog. Theor. Phys. 33 (1965) 423. W. Giitze and K.H. Michel, in: Dynamical properties of solids, Vol. 1, eds.G.K. Horton and A.A. Maradudin (North-Holland, Amsterdam, 1974). S.W. Lovesey, Condensed matter physics: dynamic correlations (Benjamin, London, 1980). H. Horner, J. Low Temp. Phys. 8 (1972) 511. V.V. Goldman, J. Low Temp. Phys. (1980) 149. E.R. Cowley and R.A. Cowley, Proc. R. Sot. London 287 (1965) 259.
235