Determinantal polynomial wave functions induced by random matrices

Random-matrix eigenvalues have a well-known interpretation as a gas of like-charge particles. We make use of this to introduce a model of vortex dynamics by defining a time-dependent wave function as the characteristic polynomial of a random matrix with a parameterized deformation, the zeros of which form a gas of interacting vortices in the phase. By the introduction of a quaternionic structure, these systems are generalized to include antivortices and nonvortical topological defects: phase maxima, phase minima, and phase saddles. The commutative group structure for complexes (which undergo topologically allowed reactions) generates a hierarchy. Several special cases, including defect-line bubbles and knots, are discussed from both an analytical and computational perspective. Finally, we return to the quaternion structures to provide an interpretation of two-vortex fundamental processes as states in a quaternionic space, where annihilation corresponds to scattering out of real space, and identify a time-energy uncertainty principle.

Figure 1

Generic scattering scenario for phase-defect collisions. Time $t$ runs from left to right. For $t$ large and negative, free, incident phase defects $I_1,I_2,\cdots$ converge with purely radial motion towards an interaction region $\mathcal{I}$. Within the space-time volume $\mathcal{I}$, for which the internal lines are not shown, various topological reactions of the phase defects may occur. After the interaction, when $t$ is large and positive, one has a series of scattered free phase defects $S_1,S_2,\cdots$ diverging with purely radial motion away from $\mathcal{I}$.

Figure 2

Chart outlining the correspondence between evolving matrices $\mathbf{M}\left(t\right)$ (or ensembles thereof) and associated evolving determinantal wave functions (or wave-function ensembles).

Figure 3

Phase of ${\mathrm{\Psi }}_{10,10}(x,y;0)$ for a random $10\times 10$ matrix as in Eq. (1). All zeros—which in this case are eigenvalues of $\mathbf{M}$—are vortices and are marked as blue dots. In this and all subsequent phase plots, phase $\mathrm{\Phi }$ is given modulo $2\pi$, with a linear grayscale between black ($\mathrm{\Phi }\mathrm{mod}2\pi =-\pi$) and white ($\mathrm{\Phi }\mathrm{mod}2\pi =\pi$).

Figure 4

Trajectories of the vortices of ${\mathrm{\Psi }}_{10,10}(x,y;t)$ from Fig. 3. Figure 3 is the bottom layer of this diagram ($t=0$), with additional phase maps corresponding to $t=0.5$ and $t=1$ also shown. Although there are some close collisions (highlighted by the dashed green circle), there are no annihilation events, consistent with the vanishing probability of eigenvalue degeneracy (instability of vortices with $m>1$). See Supplemental Material [49] for a video of this system.

Figure 5

Plot of ${\mathrm{\Phi }}_{10,0}(x,y;0.7485)=arg\left[{\mathrm{\Psi }}_{10,0}(x,y;0.7485)\right]$, where $\mathbf{M}\left(t\right)$ is from Eq. (10) and the matrix ${\mathbf{M}}_0$ is the same as that used in Figs. 3 and 4. The blue dots are the zeros of ${\mathrm{\Psi }}_{10,0}(x,y;0.7485)$.

Figure 6

Trajectories of the zeros of ${\mathrm{\Psi }}_{10,0}(x,y;t)$ from Fig. 5. See Supplemental Material [49] for a video of this system.

Figure 7

Plot of ${\mathrm{\Phi }}_{10,4}(x,y;0.499)=arg\left[{\mathrm{\Psi }}_{10,4}(x,y;0.499)\right]$ from Eq. (51) with $\xi =7$, where $\mathbf{M}\left(t\right)$ is from Eq. (10) and the matrix ${\mathbf{M}}_0$ is the same as that used in Figs. 3 and 4. At this time, there are four zeros (the blue dots), all of which are vortices. The kink in the line of phase discontinuity in the top left is indicative of a vortex-antivortex creation event happening in the near future. A local phase maximum can be seen nearby, which will become another vortex-antivortex pair.

Figure 8

Trajectories of the zeros of ${\mathrm{\Psi }}_{10,4}(x,y;t)$ from Fig. 7. Note that Fig. 7 is the middle layer of this diagram. (The figure has been rotated with respect to the orientation of Fig. 7 to make the hairpin structures clear.) Notice that there are four vortices and no antivortices at $t=0$, then two vortex-antivortex pairs are created at $t\approx 0.6$, which preserves the total winding (or topological charge) of $+4$. See Supplemental Material [49] for a video of this system.

Figure 9

All minimal interactions with $w,\chi \in \{-2,-1,0,1,2\}$ between the topological features of $arg\left(\mathrm{\Psi }\right)=\mathrm{\Phi }$. The time arrow runs from bottom to top. The colors match those of the nodal-line figures: vortices and antivortices are marked with arrows in blue (black), saddles are magenta (gray) lines, and maxima and minima (extrema) are yellow (light gray) lines.

Figure 10

The blue (black) lines are the eigenvalues of a $4\times 4$ matrix [i.e., the zeros of ${\mathrm{\Psi }}_{4,4}(x,y;t)$], which are seen to be all vortices. The magenta (gray) lines are the zeros of $\mathbf{\nabla }\mathrm{\Phi }$, all of which are saddles.

Figure 11

(a) Same matrix as Fig. 10, but with ${\mathrm{\Psi }}_{4,0}(x,y;t)$, giving vortex-antivortex pairs. Blue (black) lines are the vortices-antivortices (zeros of $\mathrm{\Psi }$), magenta (gray) lines are the saddles of $\mathrm{\Phi }$, and yellow (light gray) lines are the maxima and minima of $\mathrm{\Phi }$. See Supplemental Material [49] for a video of this system. (b) Representation of the same topological reaction using a planar graph.

Figure 12

Same matrix as Fig. 10, but with $\xi =3$. Blue (black) lines are the vortices and antivortices (zeros of $\mathrm{\Psi }$), magenta (gray) lines are the saddles of $\mathrm{\Phi }$, and yellow (light gray) lines are the maxima and minima of $\mathrm{\Phi }$.

Figure 13

Defect-line knot, plotting the defect lines of the wave function given in [14, Eq. (17)]. The closed blue (black) nodal line, which forms a trefoil knot, traces the zeros of the wave function (which are the same points as plotted in Fig. 5 of [14]), the magenta (gray) points are the saddles of the phase, and the yellow (light gray) points are maxima and minima of the phase.

Figure 14

Topological-vacuum fluctuation (vacuum diagram), namely, a defect-line complex containing no external lines. This is obtained as a plot of the zeros of $\mathrm{\Psi }(x,y)$ [blue (black)] and $\mathbf{\nabla }\mathrm{\Phi }(x,y)$ [magenta (gray)] from Eq. (59), with $T=1$, $-1.1\le t\le 1.1$.

Figure 15

Vector representation of the topological charge ($m$) and topological (Poincaré) index ($n$) of each zero in $\mathrm{\Psi }$ (the $v$ and $v^{*}$) and those in $\nabla \mathrm{\Phi }$ (the $e$ and $s$). The vortices and antivortices ($v$ and $v^{*}$) are in blue (black), the saddles ($s$) are in magenta (gray), and the extrema ($e$) are yellow (light gray).

Figure 16

All possible $[w,\chi ,P]$ defect complexes from the group $D$ with $P\le 3$, arranged according to ascending $P$: (a) $P=1$ quartet; (b) $P=2$ decuplet; (c) $P=3$ 20-plet. Defect complexes with one species of defect are in black, those with two species of defect are in magenta (gray), and those with three species of defect are in yellow (light gray). (Note that we have suppressed the “+” signs of the complexes in the figure to save space.)

Figure 17

Using Eq. (50). Blue (black): ${\mathrm{\Psi }}_{2,0}(x,y)=0$. Yellow (light gray): $\mathbf{\nabla }{\mathrm{\Phi }}_{2,0}(x,y)=0$, where the stationary point is a maximum or a minimum. The green (gray) line is a representation of a virtual particle, and the color matches that of the quaternionic zeros in Fig. 18 as this virtual particle is (in some sense) a shadow of those zeros.

Figure 18

Solving for quaternionic zeros $\lambda =x+iy+jz+kw$ of Eq. (64) with the matrix that was used to generate Fig. 17. The blue (black) lines in this one-loop diagram correspond to solutions where $z=w=0$, which are the blue (black) solutions plotted in Fig. 17. The green (gray) lines correspond to the solutions with $z\ne 0$ and $w\ne 0$, which correspond to the yellow (light gray) points in Fig. 17. The $z,w$ components come in $\pm$ pairs so the quaternionic solutions give the points $\left(\right|x+iy|,|z+iw|,t)$ and $\left(\right|x+iy|,-|z+iw|,t)$.

Figure 19

${\overline{t}}_{max}\left(\sigma \right)$ is the mean of the lifetimes $t_{max}$ of the quaternionic-zero transients sketched in Figs. 17 and 18. For each $\sigma$, an ensemble of 5000 instances was generated, of $2\times 2$ matrices with entries having real and imaginary parts independently and identically distributed as a normal distribution of mean zero and standard deviation $\sigma$. The line of best fit is given by ${\overline{t}}_{max}=0.027266+1.63642\sigma$.

Figure 20

A schematic of the phase surface given by Eq. (C1), showing the saddle point in magenta (gray) and a local maximum in yellow (light gray) before collision.

Figure 21

Plots of the real and imaginary parts of Eq. (D1). When $\epsilon ={\epsilon }_{>}$, there are two intersections between the curves, which correspond to a vortex-antivortex pair, marked by small circles. When $\epsilon ={\epsilon }_{<}$, there is no intersection and we obtain a maximum-minimum pair in the phase.