Nodal portraits of quantum billiards: Domains, lines, and statistics

This is a comprehensive review of the nodal domains and lines of quantum billiards, emphasizing a quantitative comparison of theoretical findings to experiments. The nodal statistics are shown to distinguish not only between regular and chaotic classical dynamics but also between different geometric shapes of the billiard system itself. How a random superposition of plane waves can model chaotic eigenfunctions is discussed and the connections of the complex morphology of the nodal lines thereof to percolation theory and Schramm-Loewner evolution are highlighted. Various approaches to counting the nodal domains—using trace formulas, graph theory, and difference equations—are also illustrated with examples. The nodal patterns addressed pertain to waves on vibrating plates and membranes, acoustic and electromagnetic modes, wave functions of a “particle in a box” as well as to percolating clusters, and domains in ferromagnets, thus underlining the diversity and far-reaching implications of the problem.

Figure 1

Experimental approaches for generating (a) classical and (b) modern Chladni figures. The resonant oscillations excited by the bow, in the original scheme, are eigenmodes of the plate. In modern experiments ([587], [588], [589]; [316]), the resonant modes are locally excited by driving the plate at varying frequencies with an electronically controlled mechanical oscillator. From [562].

Figure 2

Sound figures of a square plate. From [170].

Figure 3

A trajectory in a rectangular billiard. After the first reflection, it follows the dotted path. Instead, if we reflect the domain, the trajectory straightens into a domain with reflected orientation. On reaching $B$, the next reflected copy is identical to the one at the top left, so the trajectory resumes from $B^{\prime }$, and eventually, from 3 to 4 in the lower-left copy. For any given initial direction, there are a maximum of four directions generated by successive reflections. Four copies therefore make a fundamental domain, which can be stacked to tessellate the entire plane. Pairwise identification of the sides yields the two-dimensional manifold equivalent to a 2-torus.

Figure 4

The eigenstates of (a) the integrable circular billiard (nonergodic) and (b) the chaotic cardioid billiard (ergodic) reflect the structure of the corresponding classical dynamics. Shown is a density plot of $|\psi (\mathbf{r}){|}^2$ where black corresponds to high probability. Adapted from [43].

Figure 5

Upon three subsequent reflections, the rhombus billiard returns onto itself, albeit with a flipped orientation. This double valuedness is taken care of by constructing a fundamental domain with the six copies required for identification of the sides; the vertex is a “monkey saddle” ([207]). The manifold ${\mathcal{M}}_c$ is thus, topologically, a sphere with two handles or a double torus with genus $g=2$. Stacking the domains fills the plane, as seen in the lower figure. The bold line segments, although inaccessible, connect two branch points forming a branch cut. The branch cuts are arranged in a doubly periodic manner throughout the plane and a classical trajectory is a zigzag line reflecting from the cuts as the phase changes by $\pi$.

Figure 6

Superscars in a pseudointegrable barrier billiard experimentally obtained using the perturbing bead method (Sec. 7c). Observed is a clear wave function structure connected with the family of classical periodic orbits as well as a distinct localization of excitation strength ([481]). The bottom row indicates the corresponding classical orbits (dashed lines). From [125].

Figure 7

Chladni lines and vortices. (Top) Snapshots of the two-dimensional amplitude field $A(x,y,t)$ that represents nodal lines in the (a) $\widehat{x}$ and (b) $\widehat{y}$ directions along with (c) an in-phase and (d) a $\pi /2$-out-of-phase superposition, resulting in a realigned nodal line and a vortex, respectively. At the vortex, the amplitude $\sqrt{x^2+y^2}=0$ and the phase of $u$ is indeterminate. (Bottom) The amplitude fields $A(x,y,t)$ representing higher-charged vortices of strengths (a) $Q=1$ to (d) 4. During one oscillation period, the field rotates through an angle $360°/Q$. From [179].

Figure 8

Typical pattern of the nodal lines $\mathrm{Im}[\psi (x,y)]=0$ (black lines, $\eta \mathrm{NL}$) and $\mathrm{Re}[\psi (x,y)]=0$ (red lines, $\xi \mathrm{NL}$) for a random superposition of plane waves. Nodal lines in each set do not cross. Around the nodal points at which the two sets intersect, there is vortical flow in either clockwise (green dots) or anticlockwise (blue dots) directions. From [492].

Figure 9

A counterexample to the property of domain monotonicity for the Neumann boundary condition. Although ${\mathcal{D}}_1\subset {\mathcal{D}}_2$, the second eigenvalue ${\lambda }_2({\mathcal{D}}_1)={\pi }^2/c^2<{\lambda }_2({\mathcal{D}}_2)={\pi }^2/a^2$ (if $a>b$) when $c=\sqrt{(a-\alpha {)}^2+(b-\beta {)}^2}>a$. From [265]; figure by Naoki Saito.

Figure 10

Consider the family of (linear superpositions of) eigenfunctions of the Laplacian on the square $[0,\pi {]}^2$ ${\mathrm{\Phi }}_{1,2r}^{\theta }(x,y)=\cos \theta \sin x\sin (2ry)+\sin \theta \sin (2rx)\sin y$, associated with the Dirichlet eigenvalue $1+4r^2$. For $\theta =\pi /4$ (left), the nodal pattern exhibits 12 domains whereas for $\theta \lesssim \pi /4$ (right), the double points all disappear and the nodal set (a single connected line) divides the square into only two domains ([82]). In fact, [542] established that there are actually infinitely many eigenfunctions having exactly two nodal domains. This example also illustrates why counting nodal domains is such a hard problem, requiring one to resort to powerful numerical techniques. From [178].

Figure 11

Nodal patterns of Neumann eigenfunctions in the square. The nodal sets are depicted for the four nontrivial Courant-sharp cases ${\lambda }_2$, ${\lambda }_4$, ${\lambda }_5$, and ${\lambda }_9$ corresponding to wave functions $\cos \theta \cos x+\sin \theta \cos y$ ($\theta =1$ here), $\cos x\cos y$, $\cos 2x+\cos 2y$, and $\cos 2x\cos 2y$, respectively. Regions where the wave function is positive (negative) are black (white).

Figure 12

Example of a random wave [Eq. (54)] in comparison to the 6000th eigenfunction of the chaotic cardioid billiard (of odd symmetry). Locally the appearance of the states is practically indistinguishable. [446] investigated the properties of the eigenfunctions constructed by such random superpositions and revealed the existence of structures resembling precursors of periodic orbit scar localization ([283]; [327]). From [43].

Figure 13

(Left) Nodal domains of the eigenfunction of a quarter of the stadium billiard with area $4\pi$ and energy $E=10092.029$. (Right) Nodal domains of a random wave function Eq. (56) with $k=100$. As before, black (white) regions represent nodal domains where the function is positive (negative). The two figures look very similar. From [131].

Figure 14

Nodal portrait of a Gaussian spherical harmonic of degree 40. From Alex Barnett.

Figure 15

Eigenstates of the limaçon billiard, formed by deforming the circular billiard ([482]), with its boundary defined in polar coordinates by $r(\varphi )=1+ϵ\cos \varphi$. This is an example of a system with a mixed phase space for $ϵ>0$. For $ϵ=0.3$ as pictured, the eigenstates either concentrate in the regular islands or extend over the chaotic region. (Left to right) $|{\psi }_n(\mathbf{r}){|}^2$, density plot, and quantum Poincaré-Husimi representation (embedding the wave functions into the classical phase space). From [44].

Figure 16

The percolation process on the triangular lattice at $p=1/2$. From [374].

Figure 17

The exploration process for the Ising model. At each step the walk turns left or right according to the value of the spin in front of it. The relative probabilities are determined by the expectation value of this spin given the fixed spins on either side of the walk up to this time. The walk never crosses itself and never gets trapped. From [156].

Figure 18

(a) True nodal crossing. (b), (c) Avoided nodal crossings. Typically, nodal lines do not intersect as this would imply that $\psi$, ${\psi }_x$, and ${\psi }_y$ are simultaneously zero at some point. With these three functions being independent Gaussians, the probability of this event is nearly zero. From [129].

Figure 19

Mean values of the nodal domains (dots) and their variances (squares) for random functions, normalized by and plotted against $N\equiv \overline{N}(E)$. The solid and dashed horizontal lines represent theoretical predictions of Eqs. (80) and (81), respectively. (Inset) A realization of a random percolationlike process. The centers of the positive (negative) regions, denoted by $+$ ($-$), constitute two dual lattices. Dashed (solid) lines indicate graphs for the positive (negative) dual lattice. Adapted from [129].

Figure 20

Bond percolation, with probability $p=1/2$, on a graph generated by the random plane wave. (a) The nodes of the graph (red) are local maxima and the edges are gradient streamlines passing through saddles. The dual graph (blue) is formed by local minima. (b) The percolation model is critical and clusters represent the connected nodal components. From Dmitry Belyaev.

Figure 21

Comparison between scale and conformal transformations. The latter are also angle preserving and basically correspond to a combination of a local rotation, local translation, and local dilatation. There is no a priori reason why a scale-invariant system must also be conformally invariant although it is often found to be so. From [434].

Figure 22

(a) An unbiased random walk on a domain $\mathcal{D}$ (a $20\times 20$ square lattice) that starts in the interior and terminates when it encounters the boundary. (b) The same, after loop erasure. (c) LERW on a $80\times 80$ grid (in red) together with its shadow. (d) Construction of the domain Markov property. From [6].

Figure 23

Loewner evolution (topmost panel) and the schematic outline for the derivation of the Loewner differential equation (107). From [6].

Figure 24

The phases of ${\mathrm{SLE}}_{\kappa }$. For $\kappa \le 4$, the trace is a simple nonintersecting scale-invariant random curve from the origin to infinity that does not intersect the real line. When $4<\kappa <8$, the curve touches (but does not cross) itself and intersects part of the real line, i.e., $\gamma \cap \mathbb{R}⊊\mathbb{R}$. The trace is plane filling and self-osculating for $\kappa \ge 8$. From [219].

Figure 25

(Top) A nodal line of a random wave function. (Bottom) Image of the same nodal line under the map (116). In both figures, the dashed line indicates the absorbing boundary. From [126].

Figure 26

Numerically calculated limiting distributions of the nodal domain number for rectangular (blue) and circular (red) billiards in the spectral intervals $62500\le j\le 125000$ and $30000\le j\le 60000$, respectively. For the rectangle, the limiting distribution (smooth line) coincides with the analytical result of Eq. (130). The singularities seen in the numerical data mark the contributions of periodic orbits. From [119].

Figure 27

The nodal count distribution for the right-isosceles triangle, considering the energy interval $[9000^2,2\times 9000^2]$. The colors represent the proportions of wave functions with specific tiling behaviors; in particular, light green denotes no tiling. The structure of the distribution remains invariant on inclusion of the data from tiling eigenfunctions. From [29].

Figure 28

(a) Nodal domains for the cosine combination of the eigenfunctions of the equilateral triangle for $(m,n)=(10,3)$. (b) The tiling pattern of the domains (with four tiles) is seen for $(m,n)=(10,2)$. (c) The antisymmetric (sine) combination for $(m,n)=(10,3)$. The eigenfunctions are positive (negative) in the white (black) regions.

Figure 29

The probability distribution and the integrated distribution (inset) of the nodal domain number for the eigenfunctions of an equilateral triangle billiard corresponding to two spectral intervals, [10 000, 20 000] (blue) and [20 000, 40 000] (green) containing 3014 and 6028 eigenfunctions, respectively.

Figure 30

The distribution of the normalized numbers of boundary intersections for the rectangular and circular billiards, in the same spectral intervals as in Fig. 26. The smooth lines indicate the limiting distributions. From [119].

Figure 31

The distribution $P(\eta )$ of the normalized number of boundary intersections for the equilateral triangle, evaluated over 882 455 wave functions in the spectral interval $[2000^2,2\times 2000^2]$. The red curve shows the exact distribution as a function of $\eta$ whereas the blue curve depicts the smoothened histogram representation of the same. The dotted line marks the prediction of Eq. (140).

Figure 32

The number of boundary intersections ${\tilde{\nu }}_n$ for chaotic systems follows Eq. (143), where the numerical results are based on 1637 (1483) eigenfunctions of the stadium (Sinai) billiard. (Inset) The distribution $P(\eta \sqrt{\pi \mathcal{A}}/\mathcal{P})$. From [119].

Figure 33

(Top) The limiting distribution of $P(\rho )$ for a rectangular billiard, Eq. (147), compared to the calculated distribution for eigenfunctions with $E\times \mathcal{A}<{10}^5$. (Bottom) A comparison between the distribution function $P(\rho )$ calculated for the random wave ensemble and for the inner domains of a Sinai and a stadium billiard. From [210].

Figure 34

The normalized signed area variance for the equilateral triangle. (Inset) The same for the stadium and Sinai billiards but plotted as a function of $N$. The analytical expression and numerical simulation for random waves are depicted by smooth curves and triangles, respectively. Adapted from [119].

Figure 35

Mean nodal line density ${\rho }_L(Y)$ for Dirichlet (solid curve) and Neumann (dashed curve) boundary conditions.

Figure 36

Nodal line curvature distribution, $C(\kappa )$, over 201–600 eigenfunctions for pseudointegrable (two-step) and chaotic (Sinai, deformed Sinai, stadium) billiards shows comparable trends. From [520].

Figure 37

The number of avoided crossings normalized with the number of saddle points gives the cumulative distribution of the avoidance range $I(z)$. The case of the rectangular billiard is trivial with $I(z)=1$. The nodal lines of the equilateral triangle do display avoided crossings as it is a nonseparable billiard. The histogram thereof follows a trend matched closely by that for the chaotic Sinai billiard. A continuous curve drawn through the histogram for the Sinai billiard is well approximated by the result obtained for the random wave model. From [426].

Figure 38

Density $p(s)$ of the nearest-neighbor spacings $s$ between the points where a reference curve intersects a network of nodal lines created by a random wave (RW), the normal derivative of a random wave (NRW), or the Wigner surmise of RMT. For small distances, $p(s)$ grows linearly. On closer inspection, one can observe persistent oscillations relative to the mean decaying curve for the RW and the NRW. From [30].

Figure 39

Normalized correlation: nodal intersections of RW vs RMT level spacings: GOE ($\beta =1$), Gaussian unitary ensemble ($\beta =2$), and Gaussian symplectic ensemble ($\beta =4$). From [30].

Figure 40

A section of the nodal set of a random wave function. One of its nodal lines lies within the prescribed thin circular tube, i.e., this configuration contributes to the density $\rho$. From [224].

Figure 41

Two domains that are isospectral but not isometric in the sense of geometric congruence. The (top) 19th and (bottom) 20th eigenfunctions are shown. Isospectraity can be proved by the transplantation technique ([81]). Note that the isospectral billiards have the same area and perimeter. From [423].

Figure 42

The absolute value of the length spectra for ellipsoids and flat tori. The ellipsoids correspond to surfaces of revolution for $f(x)=R\sqrt{1-x^2}$ with (a) $R=2$ and (b) $1/2$. For the tori chosen, (c) ${\tau }^2=2$ and (d) $\sqrt{2}$. Points represent the numerical data of the length spectra. The continuous line is obtained by the Fourier transform of the trace formulas, Eqs. (210) and (220). From [250].

Figure 43

The power spectrum of $c_{\mathrm{osc}}(K)$ for the (top) right-angled isosceles and (bottom) equilateral triangles. The lengths of some periodic orbits are identified on the $l$ axis. From [29].

Figure 44

(a) The pattern of the nodal set of ${\psi }_{9,4}$ for the right-isosceles triangle. The shading highlights the cells that the nodal lines pass over; here ${\psi }_{mn}^{(1)}$ and ${\psi }_{mn}^{(2)}$ have the same sign. (b) The graph $G_{9,4}$ produced by the counting algorithm. From [29].

Figure 45

Schematic illustration of a realization of the quantum analog of a classical billiard through a flat electromagnetic cavity, taking a quarter of the Bunimovich stadium as an example. From [480].

Figure 46

Measured eigenmode spectrum between 17 and 18 GHz for a cavity in the shape of a quarter stadium billiard. Each maximum in the transmitted microwave power corresponds to an eigenfrequency of the resonator. The upper part is taken at room temperature (normally conducting), the lower part at 2 K (superconducting). (Inset) The shape of the resonator and the positions of the antennae. From [480].

Figure 47

Number of nodal domains ${\nu }_n$ as a function of a global phase ${\varphi }_g$ for a wave function at $n_{\text{Weyl}}\approx 223$. Arrows denote the phases ${\varphi }_g$ for which the nodal domains are shown, corresponding to ${\varphi }_g/\pi =0$, 0.225, 0.325, 0.475, 0.5, and 0.5525. Areas where appearances and disappearances of nodal domains can be observed are highlighted. The insets show the corresponding plots of ${\psi }_{\mathrm{I}}$ vs ${\psi }_{\mathrm{R}}$. From [356].

Figure 48

(Left to right) The experimentally measured number of nodal domains, its scaled variance, and the normalized area distribution for two different billiards (inset). The predictions of the Bogomolny-Schmit percolation model (or the linear fits) as solid and dashed lines for comparison. (Top) The billiard is a quantum dotlike structure of a rectangular shape with rounded corners and two attached leads. Data points for the real (imaginary) part are represented by crosses and solid histogram (diamonds, red dotted histogram). (Bottom) The corresponding data for the chaotic half-circular microwave ray-splitting rough billiard as solid circles. Adapted from [356], and [304].

Figure 49

(a) Map of the current in a ferrite billiard at 6.41 GHz. The lengths of the arrows correspond to the magnitude of the Poynting vector. (b) Distributions of $|j|$ (inset $j_x$) and (c) $\mathrm{\Omega }$, with $\sigma$ and $\lambda$ shorthand for $\sqrt{⟨j_x^2⟩}$ and $⟨{\mathrm{\Omega }}^2⟩$, respectively. Dashed lines indicate the theoretical predictions of Eqs. (264)–(267). Adapted from [62].

Figure 50

(Top) Probability current density $\mathbf{j}$ in an open microwave billiard. The vortex (full disk) and saddle (crosses) structure is clearly observed in the enlarged area, showing clockwise and anticlockwise rotation, respectively. (Bottom) Vortex pair correlation function $g$ and charge correlation function $g_Q$ for vortices taken from low [(a) 5–9 GHz] and high [(b) 15–18.6 GHz] frequency regimes. The slight mismatch in the oscillation length is due to boundary effects in the low-frequency limit ([45]; [296]), where the wavelength is no longer much smaller than the system size. From [355].