HLRZ 65/93
INSTANTONS AND SURFACE TENSION AT A FIRSTORDER TRANSITION
Sourendu Gupta
HLRZ, c/o KFA Jülich, D5170 Jülich, Germany.
ABSTRACT
We study the dynamics of the first order phase transition in the two dimensional 15state Potts model, both at and off equilibrium. We find that phase changes take place through nucleation in both cases, and finite volume effects are described well through an instanton computation. Thus a dynamical measurement of the surface tension is possible. We find that the orderdisorder surface tension is compatible with perfect wetting. An accurate treatment of fluctuations about the instanton solution is seen to be of great importance.
First order phase transitions, i.e., phase coexistence points, have recently been subjected to intense analysis. Such systems are characterised by many different dimensionful quantities. It is useful to divide these into two classes of observables. The first pertains to properties of the pure phases. Such properties are obtained, as usual, by taking derivatives of the (extensive) free energy. These derivatives, or cumulants, are extracted through finitesize scaling. In recent years a full theory of such scaling has been developed [1,2] and tested [1–3]. The second class concerns coexistence. Foremost among such variables is the surface tension— the leading nonextensive part of the free energy. Most theories of the (canonical) dynamics at the phase transition involve the surface tension. The measurement of this quantity is usually approached through detailed investigations of the static system at the phase transition [4].
However, much of the recent attention enjoyed by firstorder phase transitions is due to the interesting dynamics of the transition. This is expected to be due to nucleation. We perform a careful analysis of finite size effects in the equilibrium dynamics and compare our observations with an instantonbased computation [5]. We also test nucleation theory through a nonequilibrium process— hysteresis. Certain scaling laws for this have been proposed recently [6]; we verify them for the first time. These two tests provide the justification for the use of dynamical techniques for the measurement of the surface tension.
Our numerical work is done with a simple model— the two dimensional 15 state Potts model. This model is defined through the partition function and Hamiltonian
where the spin sitting at the site of a (square) lattice can take one of 15 values and the angular brackets denote nearest neighbours. At , ordered and disordered phases coexist in the thermodynamic limit. The two phases can be distinguished either through a singlet magnetisation or the internal energy density
where the lattice size is . In the thermodynamic limit, the internal energies in the ordered and disorder phases are, respectively, and .
Recent analysis of correlation lengths in Potts models [7] have yielded an exact result for a spinspin correlation length. This has been identified with the correlation length in the disordered phase . A duality argument has been used [8] to relate the orderorder surface tension, , to this correlation length—
The perfect wetting conjecture would then imply the relation
between the orderdisorder surface tension, , and the rest of these quantities. Although proven only in the limit , it is strongly suspected that perfect wetting holds for two dimensional Potts models for all . From the formulæ of [7] one finds for ; implying . Thus a quantitative test of nucleation theory should yield this value for . Alternately, the argument can be turned around and the measurement of the single quantity, , can be used to check the perfect wetting conjecture.
One of the dynamical methods we use is a finitesize scaling of the exponential autocorrelation time, , of the energy density. Through a study of the 10state Potts model, numerical evidence was presented in [9] that this autocorrelation time is determined by the tunnelling rate between the coexisting ordered and disordered phases. As a result, the autocorrelation time can be identified with the outcome of an instanton computation [5] giving
In our case, of course, . The exponential factor is the saddlepoint result; and the dependence of the prefactor is obtained from a oneloop computation of the determinant of the fluctuations around the saddle point. The test of the instanton computation is in the dependence of the measured values of . Note that Eq. (5) describes dynamics in equilibrium.
We also study a particular example of offequilibrium dynamics, that of hysteresis. The coupling is cyclically varied about the critical coupling with a frequency and an amplitude . Hysteresis occurs as switches between the values and . The area of the hysteresis loop in the energy density, is studied as a function of . The system has a realtime excitation with a ‘mass’ given by Eq. (5). The measurement of is really a ‘lineshape’ measurement. When , the peak should be at and the shape should be approximately Lorentzian. This cannot be converted to an useful test because the functional form of the finite corrections is not known.
We define the coercive coupling through the fact that hysteresis loops reach the value when the coupling is . Here the nucleation rate becomes larger than , and the probability of a flip into the stable phase exceeds . This argument was presented in [6] and was developed into the scaling law
The coercive coupling also obeys the same scaling law. Verification of this relation thus furnishes a test of nucleation theory. Furthermore, since the tunnelling rate is finite for any finite lattice, is zero at some nonzero frequency , and we have the relation
Thus the scaling of with is again given by Eq. (5), and constitutes yet another test of nucleation theory. Note that Eq. (6) refers to a slow nonequilibrium situation.
TABLE 1.
Run parameters for the twodimensional 15state Potts model. We show the lattice sizes , pseudocritical couplings , statistics used for the determination of autocorrelation times, , and the hysteresis parameters , and .

For Potts models at phase coexistence it was observed [9] that both local and SwendsonWang dynamics are dominated by tunnellings, and that the exponential autcorrelation times, with changing and , are related by a constant. In view of this, all our simulations were performed with the latter algorithm. Autocorrelation times were measured at the pseudocritical couplings, , defined by the maximum of the specific heat. For the values of and were obtained in [10]; this work verifies these measurements. Hysteresis was induced by cyclically changing from a maximum of down by an amount and back, in discrete steps of , running sweeps of a SwendsonWang update at each coupling. The runs were started by first thermalising a system at with cluster updates. Then for each (and fixed values of the other parameters) we ran through 200 hysteresis cycles. Since fairly large values of had to be used, this was by far the most CPUintensive part of these computations. The run parameters are shown in Table 1. Three different sets of parameters were used in order to check that the scaling law of Eq. (6) and the extrapolated values of were independent of .
For each hysteresis loop, we defined the area by the sum
where labels each of the values of in the cycle, the bar above denotes averaging over the sweeps performed at that . The value of was set to be equal to 0 in that half of the cycle with decreasing , and 1 in the other. The averages and errors were obtained by jackknife estimators over all the cycles.
A second measurement was of the cyclic response function
where the angular brackets denote averages over all measurements performed at a given coupling in the hysteresis cycle. This response function peaks twice during a cycle, at , and allows us to extract by searching for the maximum of . This procedure is the dynamical analogue of defining the transition coupling on a finite lattice, by the peak in usual response function .
The average and error were again estimated by a jackknife procedure. These measurements of have larger relative errors than . This is due to two reasons. The first is intrinsic. Since tunnellings occur at random times, there is a cycle to cycle variation in the coupling at which tunnellings occur. The other error is related to the statistics. The identification of the coercive coupling depends on the relative heights of the two peaks, and is subject to fairly strong errors. Due to these uncertainties, we decided not to use this quantity for our scaling tests.
The frequency should be identified with . However, for each , since as well as are fixed inside each set, these factors are not important when trying to check the scaling with . Furthermore, when comparing different values of , we are interested in time scales expressed directly in sweeps. Hence we shall use the convention . This is only a matter of convenience. When necessary, one should use the full definition of .
FIGURE 1
Hysteresis loop areas as functions of for the 15state Potts model on lattices. The data are for sets 1 (squares), 2 (circles) and 3 (triangles) of Table 1. The lines show the best Lorentzian fits.
For each and a set of at fixed and , we tried to fit the data on to a Lorentzian. We found that this description improves as decreases. This is illustrated in Fig. 1 for the lattice. At low frequencies, where the data deviates from the Lorentzian shape, is independent of . For large , when the data deviate from the Lorentzian shape, we fit a form
We found extremely good fits to this form and could certainly rule out any powerlaw behaviour. The data and fits are shown in Fig. 2. The fitted parameters and give estimates of — the frequency at which the loop area vanishes. Errors on were estimated from the covariance matrix between these parameters. We found good agreement between these values obtained indirectly and the direct measurements of . The scaling of with is consistent with Eq. (5), but does not provide a very stringent test. The direct measurements of are, of course, more accurate.
FIGURE 2
Hysteresis loop areas as functions of for the 15state Potts model for (filled circles), 12 (filled circles), 16 (filled squares) and 20 (open squares). The lines show the best fits.
The direct measurements of were obtained by constructing the autocorrelation function and fitting its longdistance form to an exponential. As a cross check, we measured local masses and looked for plateaus as a signal that a single mass fit over a given range was reasonable. The measurement procedure remains the same as in [9]. In all cases the fits were performed over a range which turned out to be . The errors on were, of course, reflections of the errors on the autocorrelation function. These were obtained as the dispersion between jackknife blocks. We varied the number of jackknife blocks between 5 and 25. We took the lack of sensitivity of the means and errors to the number of blocks as an indication that our error estimates are reliable.
In order to test Eq. (5) and measure , we fitted the data on to the form
Note that is given by . On the left hand side of Eq. (11), the division by takes care of the effects of fluctuations around the instanton solution. It turns out that this term in the fit is quite crucial. An attempt to perform the fit without this factor was completely unsuccessful; values obtained increased by almost an order of magnitude. In principle, one could perform a threeparameter fit, leaving the power of in the preexponential factor to be determined by the data. Unfortunately this requires more lattice sizes than we had in this study. Our fits gave
This result is compatible with perfect wetting.
FIGURE 3
We show the scaling of (open circles) and directly measured values of the autocorrelation time at , , (filled circles) against the lattice size . The lines show the best fits of the form shown in Eq. (13). The values of have been multiplied by 2 for visibility.
Although the difference between the perfect wetting result for and our measurement is not statistically significant at the level, we believe it deserves comment. We find it difficult to regard seriously the possibility that perfect wetting begins to break down when drops to a number close to 15. More likely is that Eq. (5) has to be supplemented with a higher loop computation. It has been argued [5] that a loopwise expansion of the preexponential factor yields a power series in . It is a reasonable guess that the twoloop term is marginally important for the lattice sizes we have worked with. Then our observations would imply that the coefficient of the term in is positive. A computation of this term would certainly be useful.
Finally we comment on previous numerical tests of Eqs. (5) and (6). A highstatistics study of the 10state Potts model [9] had established that the autocorrelation time in equilibrium was determined by the tunnelling phenomenon. However, this study had not been able to observe even the dominant exponential behaviour in Eq. (5). It was conjectured there that values used there were too small. The recent work of [7,11] shows that this is indeed correct. In that study the largest values of used were about , whereas this study uses between and .
Earlier studies of the scaling of with had parametrised the variation by power laws. This is presumably correct for some systems, but the arguments of [6] must hold whenever the dominant dynamical mechanism is nucleation and tunnelling. Magnetic hysteresis in the Ising model, or the onecomponent theory should therefore be described by Eq. (6). The contradictory results of [12] were obtained with values of much smaller than the ones we use. The implication is that Eq. (6) is not applicable to these.
We summarise the main results obtained in this study. The scaling law of Eq. (6) for the frequency dependence of hysteresis loop areas is found to hold extremely well over three decades in frequency, and for a variety of lattice sizes. This is strong evidence that in the particular nonequilibrium situation at a firstorder phase transition exemplified by hysteresis, the dynamics is of nucleation. Furthermore, the expression in Eq. (5) is found to describe the finitesize scaling of the autocorrelation times, showing that an instanton based description of the equilibrium dynamics is valid. A proper treatment of fluctuations around the instanton is observed to be crucial for the description of the data. The dynamics then allows the extraction of the surface tension. For the 15state Potts model we find that perfect wetting holds. A statistically insignificant discrepancy can be attributed to the neglect of twoloop terms in the treatment of the fluctuation determinant. It should be emphasised that this makes the present computation one of the most accurate measurements of a surface tension to date.
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