Critical fixed point

Critical fixed points that are undergoing bifurcation can be found by augmenting the set of fixed-point equations (5) with one of these Floquet multiplier conditions ... 

Infinite-Randomness Quantum Ising Critical Fixed Points. 

As shown in Chap. 5, the critical fixed point 0(0,0) lies in an invariant C" -smooth center manifold defined by the equation y = (x), where vanishes at the origin along with its first derivative. Moreover, the following reduction theorem holds ... 

The basic tool for studying stability of critical fixed points is the Lyapunov functions. 

Such a critical fixed point is called a complex degenerate) saddle. Its stable manifold is y = 0, and the unstable manifold is given by x = 0, as shown in Fig. 10.2.6(b). Here, in the critical case, the trajectories behave qualitatively identical to those nearby the rough unstable cycle shown in Fig. 10.2.7(b). 

We have seen in the previous sections that the qualitative behavior of a strongly resonant critical fixed point differs essentially from that of a non-resonant or a weakly resonant one. It is therefore natural to ask the question what happens at a strongly resonant point as the frequency varies In particular, in the case of the resonance a = 27t/3 the fixed point is a saddle with six separatrices in general, but when an arbitrarily small detuning is introduced the point becomes a weak focus (stable or unstable, depending on the sign of the first Lyapunov value). The question we seek to answer is how does the dynamics evolve before and after the critical moment ... 

We have seen in Sec. 10.4 that in the case of weak resonance cj = 2nM/N N > the stability of the critical fixed point is, in general, determined by the sign of the first non-zero Lyapunov value. The same situation applies to the critical case of an equilibrium state with a purely imaginary pair of characteristic exponents. However, there is an essential distinction, namely, for a resonant fixed point only a finite number which does not exceed N—3)/2 of the Lyapunov values is defined. The question of the structure of a small neighborhood of the fixed point in the case where all Lyapunov values vanish is difficult, so we do not study it here. Instead, we consider two examples. 

Figure A2.5.31. Calculated TIT, 0 2 phase diagram in the vicmity of the tricritical point for binary mixtures of ethane n = 2) witii a higher hydrocarbon of contmuous n. The system is in a sealed tube at fixed tricritical density and composition. The tricritical point is at the confluence of the four lines. Because of the fixing of the density and the composition, the system does not pass tiirough critical end points if the critical end-point lines were shown, the three-phase region would be larger. An experiment increasing the temperature in a closed tube would be represented by a vertical line on this diagram. Reproduced from [40], figure 8, by pennission of the American Institute of Physics.

At the critical value a = oi = 1, however, becomes unstable and the a-dependent fixed point becomes stable. This exchange of stability between two fixed points of a map is known as a transcritical bifurcation. By using the same linear-stability analysis as above, we see that remains stable if — 1 < a(l — Xjjj) < 1, or for all a such that 1 < a < 3. Something more interesting happens at a — 3. 

For small enough values of p so that pf p) < p for all 0 < p < 1, p = 0 will be the only fixed point. As p increases, there will eventually be some density p for which pf p ) > p in this case, we can expect there to be nonzero fixed point densities as well. Qualitatively, the mean-field-predicted behaviors will depend on the shape of the iterative map. If / has a concave downward profile, for example (i.e. if/" < 0 everywhere), then, as p decreases, Poo decreases continuously to zero at some critical value of p = Pc- Note also that the iterative map /jet for the deterministic rule associated with its minimally diluted probabilistic counterpart is given by /jet = //p-... 

Table 7.3 lists the four rules in this minimally-diluted rule-family, along with their corresponding iterative maps. Notice that since rules R, R2 and R3 do not have a linear term, / (p = 0) = 0 and mean-field-theory predicts a first-order phase transition. By first order we mean that the phase transition is discontinuous there is an abrupt, discontinuous change at a well defined critical probability Pc, at which the system suddenly goes from having poo = 0 as the only stable fixed point to having an asymptotic density Poo 7 0 as the only stable fixed point (see below). 

Figure 7.8 shows a plot of the iterative map /2(p) for rule R2 as a function of p for four different values of p p = 1 (top curve), p > Pc, P = Pc and p < Pc, where Pc 0.5347. Notice that all four curves have zero first and second derivatives at the origin. This ensures the existence of some critical value Pc such that for all p < Pc, p t + 1) < p t) and thus that limt->oo p t) = 0. In fact, for all 0 < p < Pc the origin is the only stable fixed point. At p = Pc, another stable fixed point ps 0.373 appears via a tangent bifurcation. For values of p greater than Pc, /2 undergoes a... 

It is evident that the projections of the fixed points A and B, in the EJ map, always lie on relative equilibria of type I, but that the position of the overlapping projections of C and D depends on the sign of b — a. If > b the double point is isolated between the two type I equilibria, and quantum monodromy is expected, for a sufficiently dense quantum lattice. If, on the other hand, the critical point lies on the type II relative equilibrium line and... 

Temperature measurement(s), 24 433-467, 75 469, 77 783-784 of critical current density, 23 847-848 fixed-point thermometer calibration,... 

If we choose a generic element we have exp R = T. Therefore in such a case, the critical point is the same as the fixed point of the torus action. In the following, we assume is generic, and hence Crit(/) =... 

For critical process control parameters specified by fixed points, the value ought to be challenged within an acceptable tolerance, typically 1 unit. 

We are now in the position to sketch the RG flow globally. A schematic picture of the flow in the critical manifold is shown in Fig. 10.2, where the manifold is approximated as a plane parameterized by wfy, wfy J. Assuming that is the irrelevant field of smallest fixed point dimension we note that... 

Fig. 10.2. Schematic flow diagram on the critical surface close to a fixed point. Full lines show the flow of nonlinear scaling fields, where > A0 > 0. That relation is signalled by the broken lines which illustrate that the flow from a generic point on the critical surface approaches the fixed point from direction. The tangents of...

Up to now we tacitly assumed that the single fixed point t> dominates the critical manifold. This is not the full story. In the applications of interest there are at least two fixed points located on the critical manifold. Besides t)J there is a fixed point vj at which an additional coordinate... 

This structure of the critical manifold is valid for d < 4. For d —> 4 the nontrivial fixed point merges into the Gaussian fixed point. For d > 4 only the Gaussian fixed point is physical, governing the system for all ft > 0. In RG language this is the reason for the triviality of the results for d > 4, pointed out in Chap. 6. 

In the construction of the RGf dimension d = 4 plays a special role as upper critical dimension of the thebry. This for instance shows up in the estimate of the nonuniversal corrections to the theorem of renormalizability, or in the feature that the nontrivial fixed point u merges with the Gaussian fixed point for d — 4. It naturally leads to the e-expansion. However, the RG mapping constructed in minimal subtraction only trivially depends on e. Also results of renormalized perturbation theory do not necessarily ask for further expansion in e. Equation (12.25) gives an example. We should thus consider the practical implications of the -expansion in some more detail. 

The critical behavior is, however, the same there is a Kosterlitz-Thouless (KT) transition at the phase boundary Ku between a disorder dominated, pinned and a free, unpinned phase which terminates in the fixed point K = 6/p2. One can derive an implicit equation for Ku by combining (23a) and (23b) to a differential equation... 

As 9 continues to increase (1 +< 9), the characteristic multipliers 2C an(l 2d Pa-ss through = — 1, and so these 2-period solutions become unstable. At this stage, we look at the fourth iterate and we find, as might now be expected, that a 4-cycle periodic solution appears (Figure 3.3 C, fixed point of period 4 for 9 = 3.5). The period doubles repeatedly and goes to infinity as one approaches a critical point 9C at which instability sets in for all periodic solutions, e.g., for the model (3.1), 9C 3.5699456. Above 9C all fixed points are unstable and the system is chaotic. 

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