Off-diagonal terms

In this section we will present our first novel results of this thesis. We explore the effect of parametric correlations, for a general symmetry preserving parameter, on correlated periodic orbit pairs with self-encounters. The calculation, discussed in Chapter 2, of M¨uller et al. (2005) for the non-parametric form factor is performed using the uniformity of long periodic orbits via the equidistribution theorem. A na¨ıve extension to the parametric case, including the average of Qγ from equa-tion (3.1.14) directly, and treating it as independent, would simply multiply their result by e^−Bτ. However, when considering orbit pairs with self-encounters, in each encounter region we have orbit stretches that follow almost identical paths, and so cannot be considered as independent or uncorrelated. The change in action as the parameter is varied will be almost identical for all the encounter stretches, and this must be taken into account when performing the Gaussian average. This correction constitutes the novelty of what we present in this section, and means that we should consider the average overQγ for the links and encounter regions separately.

For orbit pairs with structures described by a certain vector v, there are V encounters, α, each with lα encounter stretches that last t^α_enc. The time that the orbit spends in the links is simply the orbit time minus the total time it spends in the encounters

T_links=Tγ−X

α

lαt^α_enc (3.1.19)

The contribution from the parametric velocities of the uncorrelated links ofγ is on average

D For each encounter region α, however, we have lα stretches that are close together and will be affected by the external parameter variations in the same way. As the links are correlated in this way, when we perform the Gaussian average over the encounter, the variance of the parametric velocities of the lα encounter stretches will be approximately l²_α times the variance of a single stretch. The contribution from crossing the encounter region lα times is then

D

e^{σ¯ixhlαQαenc}E

= e^{−βlα2tαenc} (3.1.21)

meaning that the contribution from the parametric velocity over the whole periodic orbit is now approximated by

De^σ¯ixhQγE

To calculate the semiclassical contribution, the important quantity is the aug-mented weight (see section 2.2.5). This includes the weight of encounters and for parametric correlations also the factor from the parametric velocities given above.

In total the weight is given by (cf equation (2.2.18)) zT(˜s,u)˜

whereαlabels theV different encounters, each being alα-encounter, andL=P

αlα. Only terms where the encounter times in the numerator and denominator cancel exactly contribute in the semiclassical limit, so we can expand the exponentials as a power series up to first order and the augmented weight becomes

zT(˜s,u)˜

The contribution of orbits with different types of encounters can then be calculated

following the methods in M¨uller et al. (2005) using the recipe given in section 2.2.5.

In essence, for each vector v, we find the terms in the augmented weight where all the encounter times cancel exactly and use the semiclassical result of the integral in equation (2.2.25) to find the contribution. The contribution to the form factor, for all orbits with L−V ≤ 4, is summarized in Table 3.1. We use a shorthand notation for the vectors where each term in brackets, (l)^vl, means that the vector has vl l-encounters. The central column is the contribution of each structure, so

v L V ^K_κN(v)^v(τ,x) N(v) N(v)

Table 3.1: Contribution of different types of orbit pairs to the form factor for para-metric correlations, along with the number of structures for systems with and with-out time reversal symmetry (TRS).

to find the contribution to the form factor of each vector, we now multiply the contribution of each type of orbit by the number of structures N(v) and κ. If we do that for all orbits withL−V ≤6, and add the diagonal contribution, we obtain the following result, up to 9th order, for the form factor for systems without time reversal symmetry (κ= 1)

K(τ, x) = e^−Bτ

It is noticeable that, when we sum over all vectors with the same value ofL−V, all

terms cancel apart from the highest order term of the orbits with only 2-encounters.

In fact we can show that they do cancel, to all orders, using a recurrence relation argument which is presented in the Appendix of Kuipers and Sieber (2007a) and repeated here in Appendix A. We also consider a different proof starting from the parametric correlation function in Appendix B. For orbits with V 2-encounters, L= 2V and the only term remaining (the highest order one) gives a contribution of

τ^2V+1(2B)^VN(v)

(2V)! e^−Bτ (3.1.26)

To calculate the number of orbits corresponding to vectors that only have V 2-encounters we can use equation (2.2.9). In this case, the only non-vanishing compo-nent of v is v2 =V, and the sum is over all vectors with component v⁰₂ =m where m= 0, . . . , V. The result is

N(v) = 1 2V + 1

V

X

m=0

(−1)^m(2m)! (2V −2m)!

2^V m! (V −m)! = (2V)!

2^V(V + 1)!

1 + (−1)^V

2 (3.1.27)

so that we can easily see that the contribution is

τ^2V+1 B^V

(V + 1)!e^−Bτ (3.1.28)

for evenV and zero for oddV. In fact because we have the contribution for allV we can get the form factor explicitly (including the diagonal term which corresponds to V = 0) as

K(τ, x) =

∞

X

m=0

τ^4m+1 B^2m

(2m+ 1)!e^−Bτ = sinh(Bτ²)

Bτ e^−Bτ (3.1.29)

For systems with time reversal symmetry, we have the same contribution from each structure (as given in Table 3.1), but a different number of structures corre-sponding to each vector. By multiplying the contribution by the number of struc-tures and by a factor of κ = 2, because we can also pair each orbit with its time reversal, we get the contribution of each vector v. By summing over all vectors v

withL−V ≤6 we can obtain the result for the form factor for systems with time reversal symmetry (κ= 2) up to 7th order, which is given by

K(τ, x) = e^−Bτ

The two-point correlation function integrals are given in Simons and Altshuler (1993a,b) in terms of the rescaled parameter x. Here we take the Fourier trans-form, so the RMT prediction for the GUE (no time reversal symmetry) case is given by the following integral The result is given in Sieber (2000), and for τ <1 it is calculated as follows. First the integral overω is performed which gives

K^GUE(τ, x) = 1 Because τ is positive and λ1 ≥ λ the second delta function does not contribute.

From the first delta function we get the relation 2τ =λ₁−λ. As we are considering the case whereτ <1, the domain of integration forλ1is reduced to 1≤λ1≤1 + 2τ.