The field intensities are changed in three different ways. First,E andB are changed from (1,1) to (0,0) keepingE =B and accordingly DSP is expected to change over to OP. The corresponding data will be represented by circles. Second, keepingB = 1, E is changed from 1 to 0, then DSP should change to SP. These data are represented by squares. Third, keepingE = 1,B is changed from 1 to 0, thus DP will be reached from DSP. Triangles represent the corresponding data. The percolation threshold pc, at which a spanning cluster appears for the first time in the system, has been calculated as function of the field intensitiesEandB. Following the same procedure of section 3, pcs are identified as the percolation probability p corresponding to the maximum slope of the plot of spanning probabilityPsp againstp. The values ofpcs are given in Tables 6.2 for (a) E =B, (b) E = 1 and (c) B = 1. In Figure 6.8, the values ofpcs are plotted against E and B for a square lattice of sizeL = 210. Each
6.4 Effect of field intensity
E =B pc
1.0 0.619
0.7 0.661
0.5 0.666
0.3 0.647
0.2 0.630
0.1 0.612
0.0 0.593
B(E = 1) pc
1.0 0.619
0.5 0.669
0.3 0.686
0.2 0.692
0.1 0.699
0.0 0.705
E(B = 1) pc
1.0 0.619
0.5 0.651
0.3 0.674
0.2 0.689
0.1 0.710
0.0 0.712
(a) (b) (c)
Table 6.2: Percolation threshold pc measured for (a) E=B, (b)E= 1 and (c) B = 1.
pc has been determined generating 105 spanning clusters. It can be seen that as E and B go to zero pc coincides with that of OP. Similarly, pc for DP and SP are also obtained for E = 1, B = 0 andE = 0, B = 1 respectively.
The percolation clusters are isotropic for OP and SP whereas they are anisotropic in the case of DP and DSP. For anisotropic clusters, there are two connectivity lengths ξ⊥ and ξk and for isotropic clusters there is only one connectivity length.
Sinceξ⊥andξkdiverge with two critical exponentsν⊥andνk respectively asp→pc, the ratio of two connectivity lengths ξk/ξ⊥ should diverge as |p −pc|−∆ν where
∆ν = νk − ν⊥. For isotropic clusters, νk is equal to ν⊥ since the clusters grow equally in all directions. Thus, ∆ν is zero for OP and SP clusters and ∆ν is finite for anisotropic clusters of DP and DSP. In DSP, ∆ν is approximately 0.21[1]whereas in DP, it is approximately 0.64[8,10,11]. The values of the connectivity exponents ν⊥
and νk are determined for different values of E and B. In Figure 6.9, ∆ν is plotted against E and B. As the field intensities are taken to E = 0 and B = 0 or E = 0 and B = 1 corresponding to OP and SP models the value of ∆ν continuously goes to zero. As the field intensities are taken to E = 1 and B = 0 corresponding to the DP model, the value of ∆ν approaches 0.64 as it is expected for DP. Not only the difference but also the absolute value of the connectivity exponent matches with the respective percolation models. Thus, there is a smooth crossover from DSP to OP, DP and SP at the appropriate values of E and B as they are changed continuously.
In order to determine other critical properties, fractal dimensiondf of the span- ning clusters and average cluster size exponent γ are also determined varying the field intensities and they are tabulated in Tables 6.3 and 6.4 respectively. In Figures 6.10(a) and (b),df andγare plotted againstEandBrespectively. The value ofdf in different percolation model are: df(OP) = 91/48≈ 1.896[6,7], df(DP) ≈1.765[8,11],
0.0
0.5
E 1.00.0
0.5
1.0
B
0.60 0.64 0.68 0.72
pc
Figure 6.8: Plot of percolation thresholdpcagainst field intensitiesEandB for a square lattice of size L = 210. For appropriate values of E and B, pcs correspond to that of respective percolation models.
0.0
0.5
1.00.0
0.5
1.0 0.0
0.2 0.4 0.6
ν||-ν⊥
E
B ν||-ν⊥
Figure 6.9: Plot of ∆ν = νk−ν⊥ against field intensities E and B. ∆ν continuously goes to zero as the field intensities goes to E =B = 0 for OP andB = 1,E = 0 for SP respectively. ∆ν is also approaching 0.64 corresponding to DP for E= 1 and B = 0.
df(SP) ≈ 1.957[9] and df(DSP) ≈ 1.733[1]. The values of df for OP, DP and SP are marked by crosses in Figure 6.10. It can be seen that the value of df
changes continuously and approaches to the expected values ofdf of the respective percolation models as the intensities of E and B are changed appropriately. In
6.4 Effect of field intensity
E =B df
1.0 1.732
0.7 1.743
0.5 1.751
0.3 1.773
0.2 1.786
0.1 1.823
0.0 1.896
B(E= 1) df
1.0 1.732
0.5 1.734
0.3 1.735
0.2 1.736
0.1 1.745
0.0 1.765
E(B = 1) df
1.0 1.732
0.5 1.755
0.3 1.770
0.2 1.790
0.1 1.818
0.0 1.957
(a) (b) (c)
Table 6.3: Fractal dimensiondf measured for (a) E =B, (b)E = 1 and (c) B = 1.
E =B γ
1.0 1.87
0.7 1.89
0.5 1.90
0.3 1.95
0.2 1.97
0.1 2.08
0.0 2.39
B(E= 1) γ
1.0 1.87
0.5 1.95
0.3 2.02
0.2 2.07
0.1 2.14
0.0 2.2772
E(B = 1) γ
1.0 1.85
0.5 1.86
0.3 1.87
0.2 1.90
0.1 1.96
0.0 2.19
(a) (b) (c)
Table 6.4: Average cluster size exponents γ measured for (a) E =B, (b) E= 1 and (c) B = 1.
Figure 6.10(b), the values of γ of different percolation models are also indicated by crosses, γ(OP) = 43/18 ≈ 2.389[6,7], γ(DP) ≈ 2.2772[8], γ(SP) ≈ 2.19[9] and γ(DSP) = 1.85[1]. Similar to the fractal dimension, the value of the critical expo- nent γ is also changing continuously to that of the other percolation models as the intensities ofE andB are changed appropriately. However, for most of theE andB values, the values of df and other critical exponents are found close to that of DSP.
It could also be noticed that in the presence of small field intensities of E or B, the universality class of OP is changed. Similarly for DP and SP, a small intensity of the other field is able to change the universality class of the models to that of DSP.
This is a new and important observation.
A phase diagram for percolation models under external bias fields now can be plotted in the space of directional (E) and rotational (B) bias fields. Different percolation models are identified on the E−B space by black circles in Figure 6.11.
Each point in this space corresponds to a second order phase transition point. The thick lines from left lower to right upper diagonal and parallel to the E and B axes
0.0
0.5
1.00.0 0.5 1.6 1.0
1.7 1.8 1.9 2.0
df
E
B df
0.0
0.5
1.00.0 0.5 1.8 1.0
2.0 2.2 2.4
γ
E
B γ
(a) (b)
Figure 6.10: Plot of (a) fractal dimension df of the spanning clusters and (b) average cluster size exponent γ against field intensities E and B. As E and B change continu- ously, the values ofdf andγ approach to that of the respective percolation models at the appropriate values ofE andB.
SP DSP
DSP
SP
DP
OP DP
B E
(0,1)
(1,0) (0,0)
(1,1) DSP
Figure 6.11: Phase diagram of the percola- tion models under crossed external bias fields.
Different percolation models are represented by black circles. DSP changes continuously to OP, DP or SP for the field intensities ofE andBare changed to the appropriate values.
of the phase diagram represent lines of second order phase transition points. The two ends of the diagonal line correspond to OP and DSP. The line parallel to E connects SP and DSP whereas the line parallel to B connects DP and DSP. The dotted lines define the regions of DP and SP as indicated. DSP model then can be considered as the most generalized model of percolation under external bias fields whose different limiting situations correspond to other percolation models like OP, DP or SP.