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Quantum sine Gordon thermodynamics
Volume 85A, number 1
PHYSICS LETFERS
7 September 1981
QUANTUM SINE GORDON THERMODYNAMICS Michael FOWLER Department of Physics, University of Virginia, Charlottesville, VA 22901, USA Received 28 April 1981
A Bethe ansatz description of the thermodynamics of the quantum sine Gordon system is shown to arise in the appropriate limit from Talcahashj’s analysis of the XYZ spin chain. Some simple thermodynamic properties of the quantum sine Gordon system follow.
It was first pointed out by Luther [1] that the one-dimensional quantum Sine Gordon (SG) system (or, equivalently, the Massive Thirring Model (MTM)) was a particular continuum limit of the XYZ (completely anisotropically coupled) spin one-half chain [2]. He found that the semiclassical quantum SG excitation spectrum of Dashen et al. [3] corresponded exactly with the appropriate limit of the XYZ chain spectrum, derived by Johnson et al. [4] using the methods of Baxter [2]. More recently, Bergknoff and Thacker [5] and Korepin [6] have given much more complete derivations of the quantum SG spectrum. The thermodynamic properties of the XYZ spin chain have been discussed by Takahashi and Suzuki [7], usmg a set of coupled integral equations derived from the Bethe ansatz by generalizing the work of Gaudin [81. Our purpose here is to point out that this gives a method (which is exact in principle) for analyzing the thermodynamics of the quantum SG system [9]. As an example of the method, we use Takahathi’s results to show that for the quantum SG system, the specific heat energy gap at low temperatures, expressed as a function of the coupling constant, has a discontinuity in slope at a particular value of the coupling, A simple physical explanation of this effect is given. It should be emphasized that the method is of wide potential application to the quantum SG system it appears to be a practical technique for evaluating thermodynamic quantities and, for instance, finite temperature excitation spectra. To set up the notation, we briefly review the Bethe —
ansatz analysis of the XYZ model [2,4], defined by the spin one-half chain hamiltonian N
H=
(J~S~S~÷i + ~Y~f~~’÷1
~
+J
2S~Sf+1).
(1)
1=1
The standard parametrization of the coupling constants [2] is j J J = 2 1 dn 2 1 2 X Y Z t~n so that for 1-÷0, J, cos 2~,J~,-÷ J~.The “magnons” on this spin chain are conveniently labelled by a “rapidity” variable ~z,related to the quasimomentum k by [7] .
-+
e
=
H(~(a.+i))/H(~(a. 1))
(3)
—
1
where Jacobi’s function H has modulus 1’, and the above expression has real period 2KJ. The reason for transforming to the a/s is that in these variables the magnon—magnon phase shift e
ji
H(~(a. a~+ 2i))/H(~(a.—a•— 2i)) —
/
‘
.‘
(4)
is a function only of the relative rapidity. This makes it possible to solve certain equations we shall discuss below by Fourier analysis. The XYZ ground state consists of a Dirac sea of N/2 of these interacting magnons, and to find the local density of (filled) states one must take into account the rapidity dependent phase shifting from all the other magnons present, leading in the usual Bethe ansatz fashion [4] to an integral equation for the density, soluble by Fourier transformation. The mag53
Volume 85A, number 1
PHYSICS LETTERS
non energy—rapidity dispersion curve is an elliptic function of the rapidity [7], real on the real axis and on Im(~a)= iK1, with real period 2K/~.It has extrema on these lines at Re ci = 0 and at Re(a~)= K. The ground state consists of a Dirac sea of magnons with rapidities filling one period 2K; either on the real axis or on Im(~a)= iK1, depending on ~ The equations of Yang and Yang [10] for the Heisenberg—lsing spin chain are recovered by taking the Dirac sea to be centered about Re a = 0 and letting K -+00 (1 0). However, since the magnon dispersion curve and the Dirac sea are also symmetneal about Re(~ci)= K, an equally natural / 0 limit is given by taking Kft as the origin in rapidity space before letting the period become infinite. As shown by Bergknoff and Thacker [5], this leads (with an appropriate overall renormalization) to the quantum SG system, with the magnons corresponding to the fermions in the MTM picture. The Dirac sea rapidities lie along lm(~a)= iK1. Excitations of the system are constructed by taking fermions a1 out of the sea and placing them on the real axis or in strings sets {~}with the same real part, successive imaginary parts differing by 2i symmetrically placed about the real axis or about Im(~)= iK1. Those strings centered about the real axis having Im(~)I
7 September 1981
ularly complicated because it is necessary to include finite densities of the longer strings mentioned above phase shifting by these longer strings affects the distribution of other excitations in rapidity space. A full discussion of these technical details, and in particular how they affect the SG limit, will be given elsewhere. At sufficiently low temperatures, the coupled integral equations become more tractable, and Takahashi finds that the temperature dependence of the specific heat is dominated by an exponential term —
C~e_~T,
(6)
—~
—
where =
~ sn(2~l)Kk k’/2~
for (K1 =
—
~
[J~ sn(2~,l)K~,k’Rjsn(Kk(Kl
for (K1
—
(7)
sn’(~,k)/Kk —
~)/~ k’) ,
~)/~~ sn~(~, k’)/Kk~
and k is defined by
—
—
~,
~ (p + ~) ln(p + ~) ~ In ~ ~ ln ~ (5) Minimizing the free energy leads to a set of coupled integral equations for the densities of the various types of excitations. Takahashi’s equations are partic—
54
—
= K1,ft. (8) Thus the energy gap appearing in the specific heat cannot be analytically continued in the coupling through a certain value. This rather surprising sudden change in the behavior of the gap as a function of coupling is preserved iii the quantum SG limit. Taking the limit / 0, with an overall renormalization factor such that the dressed excitation spectrum remains finite (following Bergknoff and Thacker [51)the conditions (7) become:
Kk/Kk,
-~
=
m (soliton mass)
for ji
~
z~=2msin(~ir(ir/p—1)) forp~~ir,p=2~. (9) The soliton mass itself varies smoothly at ji = ~ir. In fact, this behavior can be understood in terms of a simple picture. At sufficiently low temperatures, the system is a dilute gas of solitons, antisolitons and the series of soliton—antisoliton bound states the lowest of which is equivalent to the phonon. Consider first the contribution to specific heat from soliton—antisoliton unbound pairs. The energy needed to create suJ~a pair is 2m. However, this creates two statistically independent particles so the specific heat ~emfu’T.
Volume 85A, number 1
PHYSICS LETFERS
(This is the same argument as that for a superconductor, where specific heat ‘--e ~11Teven though the excitation energy gap is 2~.)The leading contribution to the specific heat from the bound states is given by the phonon term ,~,e_mPh,~’T. Which term dominates in the overall specific heat thus depends on the relative masses of the phonon and the soliton. The crossover occurs at ~i = iiT. This same argument works for the general XYZ model, giving the crossover condition (7). For a particular limit of (7)— the Heisenberg—Isingferromagnetic spin chain withJ~: : = 5/3 : 1: 1 the crossover was recently discussed by Johnson and Bonner [11]. For that case, there is no free soliton, but an infinite sequence of soliton—antisoliton bound states corresponding to 1, 2, 3, ...n, ... overturned spins. (The “soliton” here is essentially a boundary between up spins and down spins. In the XYZ model it is presumably a similar boundary between segments of the two degenerate ground states.) The binding energy tends to zero as n -~00 so asymptotically the leading contribution to the free energy from the high n bound states is equivalent to that from an unbound soliton—antisoliton pair. This leads to the same crossover behavior as that discussed above. —
7 September 1981
References [1] A. Luther, Phys. Rev. B14 (1976) 2153. [2] R.J. Baxter, Ann. Phys. (NY) 70 (1972) 193; 212. [3] R. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. Dli (1975) 3424. [4] J.D. Johnson, S. Krinsky and B.M. McCoy, Phys. Rev. A8 (1973) 2526. [5] H. Bergknoff and H.B. Thacker, Phys. Rev. D19 (1979) 3666. [6] V.E. Korepin, Teor. Mat. Fiz. 41(1979)169. [71 M. Takahashi and M. Suzuki, Prog. Theor. Phys. 48 (1972) 2187; M. Takahashi, Prog. Theor. Phys. 50 (1973) 1519. [8] M. Gaudin, Phys, Rev. Lett. 26 (1971) 1301. [9] A different (but related) approach to this problem has been J.A. Krumhansl and J.R. Schrieffer, Phys.developed Rev. Bil in (1975) 3535; see J.F. Currie, J.A. Krumhansl, A.R. Bishop and i.E. Trullinger, Phys. Rev. B22 (1980) 477 for a review. [10] C.N. Yang and C.P. Yang, Phys. Rev. 150 (1966) 321, 327. [11] i.D. Johnson and J.C. Bonner, Phys. Rev. B22 (1980) 251.
55
PHYSICS LETFERS
7 September 1981
QUANTUM SINE GORDON THERMODYNAMICS Michael FOWLER Department of Physics, University of Virginia, Charlottesville, VA 22901, USA Received 28 April 1981
A Bethe ansatz description of the thermodynamics of the quantum sine Gordon system is shown to arise in the appropriate limit from Talcahashj’s analysis of the XYZ spin chain. Some simple thermodynamic properties of the quantum sine Gordon system follow.
It was first pointed out by Luther [1] that the one-dimensional quantum Sine Gordon (SG) system (or, equivalently, the Massive Thirring Model (MTM)) was a particular continuum limit of the XYZ (completely anisotropically coupled) spin one-half chain [2]. He found that the semiclassical quantum SG excitation spectrum of Dashen et al. [3] corresponded exactly with the appropriate limit of the XYZ chain spectrum, derived by Johnson et al. [4] using the methods of Baxter [2]. More recently, Bergknoff and Thacker [5] and Korepin [6] have given much more complete derivations of the quantum SG spectrum. The thermodynamic properties of the XYZ spin chain have been discussed by Takahashi and Suzuki [7], usmg a set of coupled integral equations derived from the Bethe ansatz by generalizing the work of Gaudin [81. Our purpose here is to point out that this gives a method (which is exact in principle) for analyzing the thermodynamics of the quantum SG system [9]. As an example of the method, we use Takahathi’s results to show that for the quantum SG system, the specific heat energy gap at low temperatures, expressed as a function of the coupling constant, has a discontinuity in slope at a particular value of the coupling, A simple physical explanation of this effect is given. It should be emphasized that the method is of wide potential application to the quantum SG system it appears to be a practical technique for evaluating thermodynamic quantities and, for instance, finite temperature excitation spectra. To set up the notation, we briefly review the Bethe —
ansatz analysis of the XYZ model [2,4], defined by the spin one-half chain hamiltonian N
H=
(J~S~S~÷i + ~Y~f~~’÷1
~
+J
2S~Sf+1).
(1)
1=1
The standard parametrization of the coupling constants [2] is j J J = 2 1 dn 2 1 2 X Y Z t~n so that for 1-÷0, J, cos 2~,J~,-÷ J~.The “magnons” on this spin chain are conveniently labelled by a “rapidity” variable ~z,related to the quasimomentum k by [7] .
-+
e
=
H(~(a.+i))/H(~(a. 1))
(3)
—
1
where Jacobi’s function H has modulus 1’, and the above expression has real period 2KJ. The reason for transforming to the a/s is that in these variables the magnon—magnon phase shift e
ji
H(~(a. a~+ 2i))/H(~(a.—a•— 2i)) —
/
‘
.‘
(4)
is a function only of the relative rapidity. This makes it possible to solve certain equations we shall discuss below by Fourier analysis. The XYZ ground state consists of a Dirac sea of N/2 of these interacting magnons, and to find the local density of (filled) states one must take into account the rapidity dependent phase shifting from all the other magnons present, leading in the usual Bethe ansatz fashion [4] to an integral equation for the density, soluble by Fourier transformation. The mag53
Volume 85A, number 1
PHYSICS LETTERS
non energy—rapidity dispersion curve is an elliptic function of the rapidity [7], real on the real axis and on Im(~a)= iK1, with real period 2K/~.It has extrema on these lines at Re ci = 0 and at Re(a~)= K. The ground state consists of a Dirac sea of magnons with rapidities filling one period 2K; either on the real axis or on Im(~a)= iK1, depending on ~ The equations of Yang and Yang [10] for the Heisenberg—lsing spin chain are recovered by taking the Dirac sea to be centered about Re a = 0 and letting K -+00 (1 0). However, since the magnon dispersion curve and the Dirac sea are also symmetneal about Re(~ci)= K, an equally natural / 0 limit is given by taking Kft as the origin in rapidity space before letting the period become infinite. As shown by Bergknoff and Thacker [5], this leads (with an appropriate overall renormalization) to the quantum SG system, with the magnons corresponding to the fermions in the MTM picture. The Dirac sea rapidities lie along lm(~a)= iK1. Excitations of the system are constructed by taking fermions a1 out of the sea and placing them on the real axis or in strings sets {~}with the same real part, successive imaginary parts differing by 2i symmetrically placed about the real axis or about Im(~)= iK1. Those strings centered about the real axis having Im(~)I
7 September 1981
ularly complicated because it is necessary to include finite densities of the longer strings mentioned above phase shifting by these longer strings affects the distribution of other excitations in rapidity space. A full discussion of these technical details, and in particular how they affect the SG limit, will be given elsewhere. At sufficiently low temperatures, the coupled integral equations become more tractable, and Takahashi finds that the temperature dependence of the specific heat is dominated by an exponential term —
C~e_~T,
(6)
—~
—
where =
~ sn(2~l)Kk k’/2~
for (K1 =
—
~
[J~ sn(2~,l)K~,k’Rjsn(Kk(Kl
for (K1
—
(7)
sn’(~,k)/Kk —
~)/~ k’) ,
~)/~~ sn~(~, k’)/Kk~
and k is defined by
—
—
~,
~ (p + ~) ln(p + ~) ~ In ~ ~ ln ~ (5) Minimizing the free energy leads to a set of coupled integral equations for the densities of the various types of excitations. Takahashi’s equations are partic—
54
—
= K1,ft. (8) Thus the energy gap appearing in the specific heat cannot be analytically continued in the coupling through a certain value. This rather surprising sudden change in the behavior of the gap as a function of coupling is preserved iii the quantum SG limit. Taking the limit / 0, with an overall renormalization factor such that the dressed excitation spectrum remains finite (following Bergknoff and Thacker [51)the conditions (7) become:
Kk/Kk,
-~
=
m (soliton mass)
for ji
~
z~=2msin(~ir(ir/p—1)) forp~~ir,p=2~. (9) The soliton mass itself varies smoothly at ji = ~ir. In fact, this behavior can be understood in terms of a simple picture. At sufficiently low temperatures, the system is a dilute gas of solitons, antisolitons and the series of soliton—antisoliton bound states the lowest of which is equivalent to the phonon. Consider first the contribution to specific heat from soliton—antisoliton unbound pairs. The energy needed to create suJ~a pair is 2m. However, this creates two statistically independent particles so the specific heat ~emfu’T.
Volume 85A, number 1
PHYSICS LETFERS
(This is the same argument as that for a superconductor, where specific heat ‘--e ~11Teven though the excitation energy gap is 2~.)The leading contribution to the specific heat from the bound states is given by the phonon term ,~,e_mPh,~’T. Which term dominates in the overall specific heat thus depends on the relative masses of the phonon and the soliton. The crossover occurs at ~i = iiT. This same argument works for the general XYZ model, giving the crossover condition (7). For a particular limit of (7)— the Heisenberg—Isingferromagnetic spin chain withJ~: : = 5/3 : 1: 1 the crossover was recently discussed by Johnson and Bonner [11]. For that case, there is no free soliton, but an infinite sequence of soliton—antisoliton bound states corresponding to 1, 2, 3, ...n, ... overturned spins. (The “soliton” here is essentially a boundary between up spins and down spins. In the XYZ model it is presumably a similar boundary between segments of the two degenerate ground states.) The binding energy tends to zero as n -~00 so asymptotically the leading contribution to the free energy from the high n bound states is equivalent to that from an unbound soliton—antisoliton pair. This leads to the same crossover behavior as that discussed above. —
7 September 1981
References [1] A. Luther, Phys. Rev. B14 (1976) 2153. [2] R.J. Baxter, Ann. Phys. (NY) 70 (1972) 193; 212. [3] R. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. Dli (1975) 3424. [4] J.D. Johnson, S. Krinsky and B.M. McCoy, Phys. Rev. A8 (1973) 2526. [5] H. Bergknoff and H.B. Thacker, Phys. Rev. D19 (1979) 3666. [6] V.E. Korepin, Teor. Mat. Fiz. 41(1979)169. [71 M. Takahashi and M. Suzuki, Prog. Theor. Phys. 48 (1972) 2187; M. Takahashi, Prog. Theor. Phys. 50 (1973) 1519. [8] M. Gaudin, Phys, Rev. Lett. 26 (1971) 1301. [9] A different (but related) approach to this problem has been J.A. Krumhansl and J.R. Schrieffer, Phys.developed Rev. Bil in (1975) 3535; see J.F. Currie, J.A. Krumhansl, A.R. Bishop and i.E. Trullinger, Phys. Rev. B22 (1980) 477 for a review. [10] C.N. Yang and C.P. Yang, Phys. Rev. 150 (1966) 321, 327. [11] i.D. Johnson and J.C. Bonner, Phys. Rev. B22 (1980) 251.
55