Sf reeze is also a natural ob ject to consider from a geometrical point of view.
Freezing a Fluid Random Surface
Sf reeze is the equivalent of Stight for the intrinsic geometry of the surface and in what follows we will examine the effect of this term on both the intrinsic and extrinsic geometry of a dynamically triangulated random surface.
Freezing a Fluid Random Surface
As we are varying the coeﬃcient of a term in the intrinsic geometry, Sf reeze , we would expect most of the effects to be intrinsic rather than extrinsic.
Freezing a Fluid Random Surface
Sf reeze is independent of X and therefore trivially scale invariant.
Freezing a Fluid Random Surface
We have not tabulated the Euler characteristic obtained by dropping the modulus sign in Sf reeze but this also comes out correctly.
Freezing a Fluid Random Surface
The results for X 2 show that Sf reeze does not smooth out the extrinsic geometry as λ is increased indeed, if anything, it has the reverse effect – see Fig. 1.
Freezing a Fluid Random Surface
We have thus frozen the internal geometry by the addition of Sf reeze .
Freezing a Fluid Random Surface
There is still a gentle bump in the speciﬁc heat associated with Sf reeze which gives some indication that we may be seeing a genuine transition rather than a crossover, though it is diﬃcult to say for certain with the modest simulations we have carried out here.
Freezing a Fluid Random Surface
Sf reeze is most effective at freezing out the ﬂip moves on the surface, which reassures us that we are not seeing a ﬁnite size effect.
Freezing a Fluid Random Surface
The effect of the spherical topology is to reduce the average value of qi (an icosahedron, for example, has qi = 5 for all the points) so there must be defects of qi = 7 frozen in too, though there cannot be many as Sf reeze is close to its minimum bound of 4π .
Freezing a Fluid Random Surface
From these results it is clear that the addition of Sf reeze to a gaussian action allows us to interpolate between a dynamically triangulated surface with a ﬂuctuating internal geometry and a surface with a ﬁxed, regular (qi ≃ 6) geometry.
Freezing a Fluid Random Surface
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