0 200 400 600 800
1.560 1.565 1.570 1.575 1.580 1.585 1.590 cL
ε L=7
L=10 L=13
Figure 4.1: Specific heat cL = V hE2i − hEi2
/V against inverse temperature for the three-dimensional (3D) Lebwohl-Lasher model with p = 10 (see Eq.(4.4)) using system side lengths L = 7,10,13. The peaks grow in height and narrow in width with increasing system size. The peaks shift towards the thermodynamic limit value, shown by the vertical line at ∞ = 1.5864.
increases, the squared width ofχnarrows∝L−d, owing to the central limit theorem [93] obtaining theδ-function limit of Eq.(4.1) for an infinitely large system [94, 95].
It is not just the height and widths of the maxima that change with system size, but their positions are also shifted from their thermodynamic limit locations;
these shifts also have a 1/Lddependence [96]. Using finite-size scaling laws we would therefore expect to locate first-order phase transitions from finite systems with shifts vanishing with 1/Ld.
4.3 Simulations using Lebwohl-Lasher type models
In this chapter we use the (continuous spin) Lebwohl-Lasher model [64, 77] to analyse the first-order isotropic-nematic phase transition; the Lebwohl-Lasher model was described in detail in chapter 3. However, the vast majority of finite-size scaling results for first-order transitions were derived for the Potts model [97]. The Potts model is fundamentally different to the Lebwohl-Lasher model as it has discrete spins, i.e. the spins at the lattice sites are only allowed to assumeq possible values.
This has some interesting consequences when the corresponding scaling relations are applied to the Lebwohl-Lasher model. For example, in the Potts model, the specific heat peak shifts from its thermodynamic limit value in finite systems as
L,c=∞−APotts/Ld+O(1/L2d), (4.2)
with proportionality constant APotts, the thermodynamic limit transition inverse temperature∞, andL,c the inverse temperature where the specific heat maximum exists in a finite system of side lengthL [96, 98]. APotts is then related to the latent heat densityλL and the number of Potts states q as
APotts = lnq/λ∞. (4.3)
Because the Lebwohl-Lasher model has continuous spins, the significance of APotts
appears to be lost, although a numerical value for it could still be obtained by fitting to finite-size simulation data. This would enable us to assign an effective discrete number of states to the Lebwohl-Lasher model, even though the model is continuous.
The above subtlety appears to have gone unnoticed until now. Despite the scaling equation Eq.(4.2) being derived for the Potts model, it has been applied to the Lebwohl-Lasher model without question, and shown to work remarkably well for this model also [99, 80]. With this observation in mind, it could be hoped that other Potts model scaling relations also have significance for the Lebwohl-Lasher model.
One such case is a method of Borgs and Koteck´y for calculating the thermodynamic limit inverse temperature ∞ [98, 100, 101]. The appealing property of the latter method is that the finite-size effects decay exponentially, i.e. much faster than the power-law decay of Eq.(4.2), making it possible for∞ to be obtained using smaller systems and thus saving valuable computation time. In this chapter we investigate if this approach indeed works.
4.3.1 Model and simulation method
The original Lebwohl-Lasher model (described in chapter 3) is defined on a three-dimensional (3D) periodic lattice of formV =L×L×L, with volumeV and system side length L with periodic boundary conditions. A 3D unit vector d~i is placed on each lattice sitei and these interact with their nearest neighbors at sites j via
H=−X
hiji
|d~i·d~j|p, (4.4) with exponentp= 2 and coupling constant. Thehijidenotes the sum over nearest neighbors. The factor 1/kBT is incorporated into the coupling constant, with kB the Boltzmann constant and T the temperature, so that > 0 is playing the role of the inverse temperature. In this form the model undergoes a first-order phase transition, as has been previously discussed [77, 102, 80, 65, 99] at ≈1.34 [103].
As described in section 3.3, to observe the first-order isotropic-nematic transition of the “original” Lebwohl-Lasher model (i.e. withp= 2) one needs very large system sizes (L ≥ 70 in 3D [80]). For this reason, in this chapter we use the modification of section 3.4, increasing the value of the exponent p in Eq.(4.4) and thus making the phase transition more strongly first-order. For example, withp= 10 in three di-mensions, phase coexistence can be easily observed with systems as small asL= 10,
4.3 Simulations using Lebwohl-Lasher type models
0 20 40 60 80 100
0.5 1.0 1.5 2.0
ln PL,ε(E)
−ρ = −E / Ld
I N
∆ρ
∆F
L=10 L=15 L=20
Figure 4.2: Logarithm of the energy density probability distribution PL,(E) of Eq.(4.4) with p = 10 in 3D for several system sizes L. The distribu-tion is distinctly bimodal, characteristic of a first-order transidistribu-tion. The distributions are plotted as functions of−ρ=E/V. Therefore, the peak on the left labeled I refers to the isotropic phase, whereas the peak on the right labeled N refers to the nematic phase. The inverse tempera-ture has been chosen to give peaks of equal height, permitting an easy calculation for the free energy barrier ∆F between the two phases. The latent heat density ∆ρ and ∆F are marked for the L = 10 system by the horizontal and vertical arrows, respectively.
as shown in Fig. 4.2. Hence, to accurately study finite-size effects at the first-order isotropic-nematic transition we proceed with this modified version of the Lebwohl-Lasher model. We perform simulations not only on 3D systems, but also on 2D systems, which we model as a single layer of spins, i.e. with volumeV = 1×L×L, but still keeping the spin vectorsd~i three-dimensional.
Similar to previous simulations of the Lebwohl-Lasher model [99, 80], our simu-lations are based on the order parameter distribution. Using the energyE as order parameter, we use transmission matrix Wang-Landau sampling [69], as described in section 2.4.1 to measure PL,(E) as accurately as possible, where PL,(E) is the probability of observing energyE in a system of side length L at inverse tempera-ture . To “speed-up” our simulations, we often split the energy range of interest into many smaller energy intervals, with a single processor working on each interval.
Once all individual simulations have been completed we combine the results from all energy intervals together to reproduce the full distribution. Because we simulate systems as large as L = 25 in 3D and L = 100 in 2D, sub-division of the energy range turned out to be crucial. For 3D systems of side lengthL= 10 it is found that a single simulation lasting approximately one day on a 2.66 GHz processor suffices.
However, forL= 15 (already more than 3×the number of spins) it is more efficient to split the energy range into 5−10 intervals.