Periodic-orbit theory of universality in quantum chaos

We argue semiclassically, on the basis of Gutzwiller’s periodic-orbit theory, that full classical chaos is paralleled by quantum energy spectra with universal spectral statistics, in agreement with random-matrix theory. For dynamics from all three Wigner-Dyson symmetry classes, we calculate the small-time spectral form factor $K\left(\tau \right)$ as power series in the time $\tau$. Each term ${\tau }^n$ of that series is provided by specific families of pairs of periodic orbits. The contributing pairs are classified in terms of close self-encounters in phase space. The frequency of occurrence of self-encounters is calculated by invoking ergodicity. Combinatorial rules for building pairs involve nontrivial properties of permutations. We show our series to be equivalent to perturbative implementations of the nonlinear $\sigma$ models for the Wigner-Dyson ensembles of random matrices and for disordered systems; our families of orbit pairs have a one-to-one relationship with Feynman diagrams known from the $\sigma$ model.

Figure 1

(Color online) Sketch of a Sieber-Richter pair in configuration space. The partner orbits, depicted by solid and dashed lines, differ noticeably only inside an encounter of two orbit stretches (marked by antiparallel arrows, indicating the direction of motion). The sketch greatly exaggerates the difference between the two partner orbits in the loops outside the encounter and depicts the loops disproportionally short; similar remarks apply to all subsequent sketches of orbit pairs.

Figure 2

(Color online) (a) Solid line: Periodic orbit $\gamma$ with one 4-encounter and one 2-encounter highlighted by bold arrows. Dashed line: Partner ${\gamma }^{\prime }$ differing from $\gamma$ by connections in the encounters. (b) Other reconnections yield a pseudo-orbit decomposing into three periodic orbits (dashed, dotted, and dash-dotted).

Figure 3

(Color online) Periodic orbit $\gamma$ with one 2- and one 3-encounter highlighted, and a partner ${\gamma }^{\prime }$ obtained by reconnection; the encounters depicted exist only in $\mathcal{T}$-invariant systems.

Figure 4

Connections between left and right ports in partner orbit ${\gamma }^{\prime }$. In (b), the encounter splits into three pieces, respectively, containing the two upper, middle, and lower stretches.

Figure 5

(Color online) Piercings ${\mathbf{x}}_1,{\mathbf{y}}_2$ of $\gamma$ (full line) and piercings ${\mathbf{y}}_1^{\prime },{\mathbf{y}}_2^{\prime }$ of ${\gamma }^{\prime }$ (dashed line) for a Sieber-Richter pair, depicted (a) in configuration space, with arrows indicating the momentum of the above phase-space points, and (b) in the Poincaré section $\mathcal{P}$ parametrized by stable and unstable coordinates. The symplectic area of the rectangle is the action difference $\Delta S$.

Figure 6

(Color online) Piercing points of $\gamma ,{\gamma }^{\prime }$ differing in a 3-encounter; inside ${\gamma }^{\prime }$, left ports 1,2,3 are connected to right ports 2,3,1, respectively.

Figure 7

Steps from connections in $\gamma$ [depicted in (a)] to those in ${\gamma }^{\prime }$ [shown in (d)], in each step interchanging right ports of two encounter stretches.

Figure 8

(Color online) Two 2-encounters (marked by boxes) overlap in one stretch and thus merge to a 3-encounter (solid bold arrows)

Figure 9

(Color online) Reconnections inside a parallel 2-encounter yield a pseudo-orbit decomposing into two separate periodic orbits.

Figure 10

(Color online) Two families of pairs of orbits differing in parallel encounters; both exist for systems with and without $\mathcal{T}$ invariance; each contributes to ${\tau }^3$, but the contributions mutually cancel. For labels see text.

Figure 11

(Color online) Entrance-to-exit connections for a Sieber-Richter pair, in an orbit $\gamma$, its time-reversed $\overline{\gamma }$, and the partners ${\gamma }^{\prime },{\overline{\gamma }}^{\prime }$.

Figure 12

(Color online) Pairs of orbits existing only for systems with $\mathcal{T}$-invariance and giving all of ${\tau }^3$. For labels see text. Two further families do not require $\mathcal{T}$-invariance, see Fig. 10.

Figure 13

(Color online) Motion of piercing points through Poincaré section $\mathcal{P}$ of a 2-encounter. As $\mathcal{P}$ is shifted, unstable components grow and stable ones shrink, traveling on a hyperbola $\Delta S=su$; arrows denote the direction of motion. At the end of the encounter, piercing points traverse the line $u=c$. Negative $s,u$ are not shown.

Figure 14

Encounters with almost self-retracing reflections: (a) orbit scheme in configuration space; (b) example for cardioid billiard.

Figure 15

(Color online) 3-encounter (in the box) with a “fringe” where only two antiparallel stretches remain close. Stretches are separated by a loop in (a) but not in (b); an almost self-retracing reflection arises in the latter case. A partner orbit which reconnects all three ports (dashed line) is obtained only in case (a).

Figure 16

(Color online) Two parallel encounter stretches (thick lines) (a) separated by a comparatively long loop (thin line), (b) separated by a short loop, and (c) overlapping in a region depicted by two very close thick lines. The long orbit respectively follows (a) slightly more than one, (b) slightly less than two, and (c) more than two periods of the shorter orbit $\tilde{\gamma }$ (dotted line). Also depicted are Poincaré sections approximately in the center of the encounters, each with two piercing points.

Figure 17

(Color online) (a) An orbit $\gamma$ approaches a shorter orbit $\tilde{\gamma }$ (dotted line) in two regions (marked by thick full and dash-dotted lines). It pierces three times through each of the Poincaré sections $\mathcal{P}$ and ${\mathcal{P}}^{\prime }$; the two sets of piercings belong to two different encounters. (b) Reconnections in either encounter lead to the same partner ${\gamma }^{\prime }$, with one revolution of $\tilde{\gamma }$ transposed between the full and dash-dotted regions.