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QUANTUM THERMODYNAMICS OF
SINE-GORDON, ϕ4 AND DOUBLE SINE-GORDON CHAINS
R. Giachetti, V. Tognetti, R. Vaia
To cite this version:
R. Giachetti, V. Tognetti, R. Vaia. QUANTUM THERMODYNAMICS OF SINE-GORDON, ϕ4 AND DOUBLE SINE-GORDON CHAINS. Journal de Physique Colloques, 1989, 50 (C3), pp.C3-59-C3-64.
�10.1051/jphyscol:1989308�. �jpa-00229447�
QUANTUM THERMODYNAMICS OF SINE-GORDON, (p4 AND DOUBLE SINE-GORDON CHAINS
R. GIACHETTI, V. TOGNETTI* and R. V A I A * *
Dipartimento d i Matematica, ~ n i v e r ~ i t i d i Cagliari, Italy
" ~ i p a r t i m e n t o d i Fisica, Universita di Firenze, Italy
* * I E Q - C N R , Firenze, Italy
R e s u m e - La fonction de partition quantique des champs de Klein-Gordon non linkaires dans une dimension spatiale a kt6 kvaluke par une mkthode variationnelle amkliorke qui conduit B un potentiel efficace. De cette fason les propriktks thermodinamiques sont calculkes par des techniques classiques.
Des rksultats sont montrks pour les chaines discrktes de Sine-Gordon, p4 et Double Sine-Gordon. La limite continue peut gtre obtenue aiskment.
A b s t r a c t - The quantum partition function of non-linear one dimensional Klein-Gordon fields is evaluated by an improved variational method yielding an effective potential to be inserted in a configurational integral which takes into account quantum effects. Results are presented for Sine- Gordon, p4 and Double Sine-Gordon discrete chains. The continuum limit can be easily obtained.
1. I N T R O D U C T I O N
Great interest was recently devoted to the statistical mechanics of one- dimensional models which admit soliton or solitary-wave solutions 11-31. One important class of these systems is represented by the non-linear Klein-Gordon fields which can reproduce the behaviour of many nearly one-dimensional systems in condensed matter, in particular magnetic chains [4,5]. Their scalar field theories are described in the discrete version by the Lagrangian:
where Q, =
{a;)
are scalar variables on a periodic chain with N sites and spacing a,
and U(a;)
is alocal potential with unitary second derivative in its absolute minilrld.
The SG (Sine-Gordon), DSG (Double Sine-Gordon) and p4 theories are obtained for
1 tanh2 p
DSG: U(p)=- (1 - cos $0)
+ 7 (1 - cos 2p) cosh2 p
The DSG potential contains the parameter p E [0, w]
,
and for p -+ O(w) it reduces to a 2n - ( T - ) SG model.In the continuum limit (a -+ 0
,
R o a = co = 1,
h = 1 ), rescaling the field ( Q, = gQ? with g = A - ' / ~ ) and defining the mass ( m =nl
), the Lagrangian reads:Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989308
C3-60 JOURNAL DE PHYSIQUE
In this limit the SG is integrable while (04 is not. However kink-like excitations are always available with a rest energy in the classical case: E x = vAcoRl ,(SG: v = 8 ; DSG: v = 4[1+2p/ sinh2pl ; (04 : v = 213 ).
The DSG kink resembles a pair of T-SG kinks a t distance 2p [6,7].
The following dimensionless parameters are useful:
R
=
R o / R I,
measures the kink length in lattice units, and R + co in the continuum limit.Q h R 1 / E K = h / ( v A c o )
,
is the 'coupling constant' (the usual field theoretic definition is g 2 h = v Q ) . t r T I E K,
is the reduced temperature.The classical partition function of these fields can be always evaluated, a t least numerically, by transfer matrix [8]. However this method cannot, in principle, share between the contributions coming from linear and non-linear excitations, so that the interpretation of the results requires other tecniques 191.
The solitary-wave phenomenology permits this interpretation [9,10]. At the beginning it was limited to the dilute gas approximation (lowest temperatures), afterwards it has been extended a t higher temper- atures by accounting for interaction effects [ I l l . Recently, for integrable systems, action angle variables have been used after overcoming initial difficulties related with the boundary conditions [12].
The quantum partition function has been calculated exactly for the integrable systems like SG, using
~ e t h e - A h s a t z or quantum inverse scattering 113,141. Apparently the calculations are easier for strong coupling, whereas more and more difficulties arise when the classical limit is approached [13], i.e. for Q
5
0.25.The semiclassical approach can be used for any systems [15,16], but the calculations are limited to the self consistent gaussian approximation, neglecting the soliton-soliton interaction, which turns out to be essential, for instance, in the explanation of the behaviour of the specific heat with temperature, even in the classical case [II].
Coherent state approach [17] and Wigner expansion [18] have been proposed, but in both cases the quantum character of each degree of freedom is treated a t same level of h so that the convergency of the expansions is poor unless some uncontrolled rearrangements are done [19].
We think that the path integral approach can be an ideal tool t o calculate the quantum effects starting from the classical partition function. Using the variational method 1201, we can construct an effective potential t o be inserted in a configurational integral, reducing quantum statistical mechanics calculations to the classical ones. The details of the method have been the subject of a set of papers [21], and applications t o magnetic chains have been given 1221. We will give here some new results for the above Klein-Gordon fields, showing the validity of the approach a t all temperatures for low coupling and its applicability t o integrable and non integrable models.
2. THE EFFECTIVE POTENTIAL
The variational ,method [20,21] gives for the free-energy F:
where
defined by the self-consistent secular equation
where
a k / 2 is the difference between the total mean square fluctuation and the classical one of the k-component of the field, calculated in one loop approximation.
The apparently complicated implicit dependence on the field configuration makes it necessary to resort t o approximation methods. It has been shown [21] that, a t lowest temperatures, the use of a self consistent steepest-descent method yields the known results of the semiclassical approximation [15,16].
In the 1ow-Q limit a very interesting expansion is obtained. Setting $2: = 4R$ sin2(ka/2)
+
R: :Hence, replacing (2.5) in the logarithmic term of Veff one gets
where Fk = PttRk/2
.
The logarithmic term in (2.7) translates the free energy of classical oscillators to the corresponding quantum one, i.e. it accounts in fully quantum way for the linear excitations with the frequencies Rk. D ( t ) represents the quantum renormalization parameter afld is consistently calculated by (2.4) using Rk instead of wk, so t h a t it is site- and configuration- independent and easily computable, since in this limit Uk; is a n ordinary real Fourier transformation [20]. D(0) = (27r)-l U Q [ln8R+
O ( R - 2 ) ] coincides with that previously found in the semiclassical renormalization theory [IS].The effective local potentials related to (1.2) are
1 tanh2 p - 2 0 1
+
3 tanh2DSG : ee p - Te-D/2(1 - C O S ~ )
+
- e (1 - cos2p)+
8 D 2 (2.9)
( - cosh p 4
1 3
p4 : Ueff ( p ) = -g(p2
-
1 -I- 3 0 ) ' - -D2 4The expansion (2.7) for the effective potential is surely valid if the condition Pti6w:/(4Rl)
<
1 is satisfied, which amounts t o requireC3-62 JOURNAL DE PHYSIQUE
where p,i, is an absolute minimum of U(p) ( U U ( ~ ) ( ~ , ~ , ) = 8 for SG, 114 for P ~ ) . The latter condition can be extimated in the most unfavourable conditions replacing D(t) with its maximum value D(0)
,
giving
Notice that the Wigner expansion is valid for PhRo/2 << 1
,
or t >> QR/2,
so that (2.7) constitutes a very significant improvement.The advantage of (2.7) is that the complicated implicit dependence on the field configuration is absent. Besides the logarithmic term, which accounts for the quantum character of the linear excitations and is p-independent, we are left with an effective potential which usually resembles V(p)
,
apart from the quantum renormalization of some parameters entering the single site potential. Therefore the configurational integral (2.1) can be calculated by any classical method, like temperature expansions [ll]or numerical transfer matrix [8]
,
and it accounts for the nonlinear contribution t o the free energy.3. RESULTS
In order to test the validity of eqs. (2.7) and (2.9) we have calculated, by numerical transfer matrix, some thermodynamic quantities, namely the non linear contribution to the specific heat per site Gc
,
the non linear contribution to the internal energy per site Gu,
and - for SG and DSG - the magnetization m = (cospi).
We have numerically verified that the classical non linear free energy per site multiplied by R (which gives a linear density) is insensitive t o R, a t Ieast on the scale of our figures, provided that R 2 4.For Sc in the SG case we have made a comparison with quantum Monte Carlo calculations [23], (figure 1). The SG magnetization is shown in figure 2 : note the lack of saturation a t t = 0 in the quantum case, which is due to the quantum vacuum fluctuations.
F i g u r e 1. SG chain: Sc. F i g u r e 2. SG chain: m = (cosp;) for R = 10.
Continuous line : classical. Continuous line : classical.
Dashed 1. and ( A ) : Q = 0.1, R = 2.9. Dashed 1.: quantum for Q = 0.1 and Q = 0.2.
Dash-dotted 1. and (m): Q = 0.1, R = 10.
Symbols are quantum MC data from [23].
In order to test the theory with the existing solutions for integrable system [13,14] we must perform the continuum limit. However for R -+ co D is a divergent quantity and one must replace
nk
with its zero-T renormalized counterpart in (2.5), and rederive (2.6-8). The effective potential turns out to look like (2.7),This has been shown in [21] for SG and p4. The comparison with the available Bethe-Ansatz data [13], as well as with the rescaled Monte CkrIo simulations [23], is made in fig.3 for the non linear specific heat density 6C
.
We notice that the coupling Q = a110 is rather high, since it produces a further renormalization of the quantum kink energy by a factor 1 - Q/r = 0.9,
which is not accounted for in the low coupling theory. I t follows that the agreement is fully satisfactory. Unfortunately exact data for lower coupling are not available, because of the difficulties mentioned in the introduction.Figure 3. SG field: non linear specific heat density 6C
.
E k is the zero-T renormalized soliton energy.
. Dashed line: Bethe-Ansatz results from (131.
Symbols: as in fig.1, with proper rescaling.
0 0.4
t
Figure 4. (04 chain: 6u for R = 5. Figure 5. (p4 chain: 6c for R = 5.
Figure 6. DSG chain: 6 c for R = 5. Figure 7. DSG chain: m = (cospi) for R = 5.
Continuous lines: classical. Continuous lines: classical.
Dashed lines: quantum for Q = 0.1. Dashed lines: quantum for Q = 0.1.
C3-64 JOURNAL DE PHYSIQUE
In the above figures we have also reported our results for the non integrable p4 and DSG chains, together with the corresponding classical results. Note that the Q = 0 results coincide with the classical ones, because from the classical point of view one can take EK = hR1/Q finite whenletting Q -t 0, h - + O .
The quantum effect on the excess specific heat of the p4 and the DSG chain does not differ significantly, as expected after the choice of the temperature scale and of the coupling parameter, both of which are weighted through the respective kink energy.
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