Abstract
Tangles of string typically become knotted, from macroscopic twine down to longchain macromolecules such as DNA. Here, we demonstrate that knotting also occurs in quantum wavefunctions, where the tangled filaments are vortices (nodal lines/phase singularities). The probability that a vortex loop is knotted is found to increase with its length, and a wide gamut of knots from standard tabulations occur. The results follow from computer simulations of random superpositions of degenerate eigenstates of three simple quantum systems: a cube with periodic boundaries, the isotropic threedimensional harmonic oscillator and the 3sphere. In the latter two cases, vortex knots occur frequently, even in random eigenfunctions at relatively low energy, and are constrained by the spatial symmetries of the modes. The results suggest that knotted vortex structures are generic in complex threedimensional wave systems, establishing a topological commonality between wave chaos, polymers and turbulent Bose–Einstein condensates.
Introduction
Complexity in physical systems is often revealed in the subtle structures within spatial disorder. In particular, the complex modes of typical threedimensional (3D) domains are not usually regular, and at high energies behave according to the principles of quantum ergodicity^{1,2}. The differences between chaotic and regular wave dynamics can be seen clearly in the Chladni patterns of a vibrating twodimensional plate^{2}, in which the zeros (nodes) of the vibration accumulate small sand particles to appear as random curved lines when the plate is irregular. The spatial distribution of these nodal lines is statistically isotropic at high energies^{2}, and this irregularity is typical for chaotic systems, whose ergodic dynamics are determined only by the energy and there are no other constants of motion.
Understanding the spatial structure of these wavefunctions can be challenging. Following the hydrodynamic interpretation of singleparticle quantum mechanics^{3}, the zeros of 3D complexvalued scalar fields are in general lines; vortex filaments around which the phase, local velocity and probability current (in a quantum wavefunction) circulate^{4,5,6}. This is analogous to the vortices of a classical fluid, although the phase change around the vortex line is quantized in units of 2π, and is singular at the vortex core where the amplitude is zero. A similar vortex topology occurs in condensates of many quantum particles^{7}. The pattern of their vortex lines provides a structural skeleton to wavefunctions^{4,6,8}. In modes above the lowest energies, the vortex pattern is far from regular and they are densely intertwined. A natural 3D measure of the most extreme spatial irregularity is when the vortex filaments are knotted, and here we study the occurrence of knotted nodal vortex lines in three model systems of wave chaos, as a natural extension of the Chladni problem to three dimensions.
Despite numerous investigations of the physics of diverse random filamentary tangles, including quantum turbulence^{9}, loop soups^{10}, cosmic strings in the early universe^{11} and optical vortices in 3D laser speckle patterns^{12}, the presence of knotted structures in generic random fields has not been previously emphasized or systematically studied. Vortices and defects which are knotted have been successfully embedded in a controlled way in various 3D fields, such as vortex knots in water^{13}, knotted defects in liquid crystals^{14,15} and knotted optical vortices in laser beams^{16}, and theoretically as vortex lines in complex scalar fields, including superfluid flows^{17}, and superpositions of energy eigenstates of the quantum hydrogen atom^{18} and other wave fields^{19}, but rigorous mathematical techniques to resolve the statistical topology of random fields are limited^{20}. With modern highperformance computers, the structure can be explored using largescale simulations.
In the following, we investigate the knottedness of the nodal vortex structures in typical chaotic eigenfunctions of comparable size (that is, energy and total nodal line length) for three model systems. The chaotic eigenfunctions are represented as superpositions of degenerate energy eigenfunctions weighted by complex, Gaussian random amplitudes; such superpositions are established as good models of wave chaos in the semiclassical limit of high energy^{1,2,21}, since the wave pattern is determined only by their energy (unlike a plane wave which has a welldefined momentum). Our three systems are the cubic cell with periodic boundary conditions, whose degenerate eigenfunctions are plane waves; the abstract 3D sphere (3sphere), whose degenerate eigenfunctions are hyperspherical harmonics and which has finite volume but no boundary; and the isotropic, 3D harmonic oscillator (3DHO), whose nodal structure is largely contained within the classically allowed region, where the energy exceeds the potential such that it could describe an isotropic, harmonically trapped Bose–Einstein condensate^{22}. Further information and illustrations of these eigenfunction systems appear in Supplementary Note 1 and Supplementary Figs 1–4.
We find that vortex knots occur with high probability at sufficiently high energy in all of these random wave superpositions. The results suggest that even in lowenergy cavity modes, possibly accessible to experiments, knotted vortices will occur with some reasonable probability. The statistical details of random vortex knotting (with respect to the types of knot that occur, the length of knotted vortex curves and the eigenfunction energy) depends strongly on the wave system, and we perform an analysis and comparison of these properties.
Results
Knots in highenergy random eigenfunctions
Figure 1 shows the nodal/vortex structure of a typical chaotic eigenfunction in each model system: in Fig. 1a, a cube with periodic boundaries; in Fig. 1b, the 3sphere; and in Fig. 1c, the 3DHO. In all three cases the random modes are labelled by energy E_{N} with principal quantum number N (further details of how the model systems are generated and how the vortices are located is given in Supplementary Notes 1 and 2). In each of the Figs 1d–g, a single vortex line from these eigenfunctions is shown alongside a simpler projection of the same knot. Each vortex line resembles a random walk^{12,23}.
Our analysis of these knots uses the standard conventions of mathematical knot theory^{24}, in which knots in closed curves can be factorized into prime knots, which are tabulated according to their minimum number of crossings (examples are shown in Supplementary Figure 5). Both prime knots (for example, as shown in Fig. 1d,f–g), and composites of prime knots (for example, as shown in Fig. 1e), are identified in the curve data by a combination of several schemes. After simplifying the longest curves by a geometric relaxation method^{23}, several topological invariants are computed for each curve, which distinguish knots of different types. These are, the absolute value of the Alexander polynomial^{25} Δ(t), evaluated for t at certain roots of unity; the hyperbolic volume^{26}; and the second and thirdorder Vassiliev invariants^{27,28}. Although other knot invariants can have more discriminatory power than these separately^{24,29}, they tend to be significantly more demanding in computer time; in practice the combination of this particular set of invariants discriminates almost all tabulated knots^{24,29} (up to at least 14 crossings; the methods are described further in Supplementary Note 3). Each of these invariants encodes different topological and geometric information about each knot^{24}. When comparing the knottedness of different vortex loops, we follow previous studies^{25,27} in using the (positive integervalued) knot determinant Δ(−1) as the primary quantitative measure of their knotting complexity.
We have identified ∼50,000 knots from around 10^{9} curves (of different lengths), computed from about 10^{6} random eigenfunctions (at different energies). Some of these knotted curves are relatively short, such as that in Fig. 1d, although the majority of vortex knots are much longer, such as that in Fig. 1e, whose full spatial extent spreads over several periodic cells. At large lengthscales, each curve approximates a Brownian random walk^{12,23}; for closed curves, the radius of gyration scales with the square root of the total length. Most of the vortex knots occur in closed loops; in the 3DHO, however, most vortex lines stretch to spatial infinity as almoststraight lines in the classically forbidden region. Most of the knots in the 3DHO are in these open curves, such as that represented in Fig. 1g. In the periodic cube, many of the longer lines wrap around the cell a nonzero number of times (that is, they have nontrivial homology, so in a tiling of the 3D cells they would be infinitely long and periodic^{12}). Since knots in such a periodic space have not been adequately classified mathematically^{24}, we only consider knots in closed loops with trivial homology in this system.
Links, which are configurations of two or more vortex curves that are topologically entangled with one another, also occur frequently. In fact, we find links to be more common than knots, consistent with previous investigations of random optical fields at smaller lengthscales in which links are found to be common but knots were not detected^{30}. The restriction to knotting in the present study was chosen as it allows comparison with extensive studies of the knotting probability of random curves (as random walks)^{25}, whereas the study of random linking is not so well developed. Furthermore, linking of open curves (in the 3DHO) is not well defined. For these reasons, our analysis is limited to the statistics of vortex knotting.
Probabilities and complexities of knottedness
As one might expect, the random eigenfunction statistics show that longer knotted vortex curves display a greater complexity of knot types, as measured by the knot determinant Δ(−1). This is represented in Fig. 2 for each of the three systems at fixed energy. These histograms indicate that the distribution of log Δ(−1) with respect to vortex length L apparently scales according to a power law. Longer curves are also more likely to be knotted; the probability of a given vortex loop being unknotted decreases exponentially according to L (Fig. 2 insets), as previously studied for random walks modelling polymers^{25,27,31}. The value of the unknotting exponents is different for each of the three systems, given in the caption.
Each of the model systems studied has a finite spatial extent, namely the side length of the cube and the diameters of the 3sphere and classically allowed region of the 3DHO. In the latter two cases, this finite size imposes a longlength cutoff S to the Brownian scaling of long vortex lines. In the 3sphere and 3DHO, loops whose radius of gyration exceeds S are confined, and the Brownian and confined regimes have different knotting probability exponents. Both knotting probability and complexity increase distinctly with length in the confined regime, where long curves are restricted to volumes much smaller than the corresponding Brownian radius of gyration would allow.
In contrast, the cube’s periodic boundary conditions allow the vortex lines to have extent greater than the cube’s side length, although the Brownian scaling of loop sizes gives way at some S to the lines with nontrivial homology, which do not contribute to the knot count. Figure 2b,c shows that the minimal complexity of knots in the confined regime also scales with L; long vortices are always knotted in these systems, and the knot is usually very complex. By contrast, under periodic boundary conditions (Fig. 2a) even the longest vortex loops can be topologically trivial. In all of these cases, the scaling depends weakly on the energy of the eigenfunction, at least in the ranges of energy considered in the simulations.
Lowenergy knots and the impact of eigenfunction symmetry
The total vortex length in random eigenfunctions is proportional to their energy E_{N} (see Supplementary Note 1). It is therefore natural to expect knotted vortex lines to occur more frequently in higherenergy eigenfunctions. This probability of knotting with energy is shown for eigenfunctions of the 3sphere and 3DHO in Fig. 3. In the 3sphere, where all vortex loops must be closed, this probability reaches 99% when N≈16, which is surprisingly small to guarantee such a degree of topological complexity (as illustrated in the insets of Fig. 3).
The energies simulated in the 3DHO are insufficient to guarantee that a random eigenfunction will contain a knotted vortex (Fig. 3a), as the distribution of vortex lengths in the classically allowed region prefers a greater number of shorter curves. The knotting probability of a 3DHO eigenfunction also strongly depends on whether the principal quantum number N is odd or even, since 3DHO energy eigenstates, and hence the zeroes (that is, the vortex tangle), must be paritysymmetric with respect to spatial inversion through the origin; in fact, the nodal system of vortex curves is required to be strongly negatively amphicheiral^{32} (tangents at parityopposite vortex points must be parallel). Furthermore, when N is odd, exactly one open vortex line must pass through the origin, such as the lowenergy example in Fig. 4a. Since strongly negatively amphicheiral knots must be open (see Supplementary Note 4 and Supplementary Fig. 6), this line can take such a configuration and is commonly knotted, contributing to a larger knotting probability when N is odd (other knots can occur when N is odd or even, but only in antipodal pairs, as shown in Fig. 4b). This symmetry constraint applies to all energies considered in the simulations.
Eigenfunctions of the 3sphere must also be symmetric under inversion between antipodal points, but here the curve system must be strongly positively amphicheiral (opposite vortex points have antiparallel tangents). The simplest prime knot with this symmetry is in fact the 10crossing knot 10_{99} (ref. 32), and an example of a knotted vortex of this type is shown in Fig. 4c.
Discussion
On the basis of computer experiments, we have shown that knotted vortex lines are common in random quantum wavefunctions, even at comparatively low energies. These results complement previous rigorous mathematical studies, where small knots (of any type) have been proven to occur with nonzero probability in random eigenfunctions of the 3DHO^{33} and hydrogen atom^{34} at high energies. These knots would be quite different from those studied in the present work, as they occupy limited volumes and, in the 3DHO, occur with exponentially small probabilities. As such, those knots may be thought of as highly structured superoscillatory phenomena^{35}, known to be very rare. On the other hand, the knotted vortices emphasized here can extend throughout the entire wavefunction, and are rather common except at the lowest energies; in fact, the very smallest knots we find in the 3DHO are an order of magnitude larger than the bounds on the size of superoscillatory examples. Preliminary numerical studies of vortex knots in random eigenfunctions of hydrogen indicate similar behaviour to the 3DHO described here, where eigenfunctions containing knotted vortices are typically beyond a certain energy (and comparable mode count), although unlike the 3DHO and 3sphere, the nodal sets are not constrained by an amphicheiral symmetry.
Although our computer experiments have been limited to modes of fixed energy, we do not anticipate major differences in systems of random waves with different power spectra, such as a turbulent Bose–Einstein condensate^{22} with characteristic power law spectrum, especially when sizes of system are similar to or greater than those considered here.
Knotting of vortices in chaotic complex 3D eigenfunctions is a complex counterpart to Chladni’s original observation of the structure of cavity modes, and we anticipate that generic 3D complex cavity modes^{1,2} will contain knotted vortices. Our results lead us to expect that almost every random quantum eigenfunction at sufficiently high energy contains at least one knotted vortex line and, at least in the systems considered here, this is common even at the relatively low energies currently accessible to numerical experiment. Knotted nodal lines with characteristic complexity scalings may therefore be a complex, 3D counterpart to the nodal statistics proposed to signify quantum chaos^{36} for realvalued chaotic eigenfunctions in two dimensions. On the basis of the mode counts^{2} of the energies in Fig. 3, we expect knotted vortices to occur with 50% probability from somewhere between the 500th (3sphere) to 3,000th (3DHO) mode of a chaotic cavity.
Methods
Sampling random eigenfunctions
We generate random eigenfunctions of model systems via standard random superpositions of their degenerate eigenfunctions. Further details and comparisons of these systems, especially their eigenfunctions, are included in Supplementary Note 2.
Tracking vortices (nodal lines)
Within each random eigenfunction, phase vortices are numerically tracked by sampling the wavefunction on a cubic lattice (initial voxel size ∼0.1λ) and checking around each numerical grid plaquette for the circulation of phase indicating the passage of a vortex line through the face of the grid cell. The phase is only guaranteed to circulate in this way when the amplitude is zero but the complex scalar gradient is nonzero (that is, the vortices occur as transverse intersections of real and imaginary nodal surfaces), but this is generically the case in random eigenfunctions; the algorithm would otherwise fail to converge on the vortex line but, as anticipated, we have never observed this to happen. This core procedure does not always fully capture the geometry of the vortex curve (especially where two vortex lines approach closely), and so the above procedure is augmented with a recursive resampling of the numerical grid, at successively higher resolutions, in any location where the vortex line tracking is ambiguous; we call this the RRCG algorithm (see Supplementary Note 2). Since vortex lines must be continuous, the result of the algorithm is a piecewiselinear representation of each vortex in the entire largescale tangle. The vortex topology is retained, and geometry well recovered, even with a relatively poor initial sampling resolution. The algorithm generalizes to each of periodic boundary conditions, the 3sphere and the 3DHO via an appropriate choice of lattice boundary conditions, and in the case of the 3sphere by sampling on multiple Cartesian lattices and joining appropriate boundaries to reproduce the geometry of the sphere.
Detecting and classifying knots and links
The topology of vortices is investigated using the standard knot invariants described in the main text, whose values depend only on the topology of a single curve and from which the knot type can be inferred. It is computationally expensive to use some invariants known to be more powerful (for example, the Jones or HOMFLYPT polynomials), or to identify the exact knot type of every curve. When necessary (Figs 1 and 4) we are able to do so unambiguously using the Alexander polynomial (evaluated at certain roots of unity), hyperbolic volume and Vassiliev invariants of degree 2 and 3. We further use the knot determinant to measure the topological complexity of vortex curves even without identifying their specific knot types.
Data availability
All relevant data are available from the corresponding author on request.
Additional information
How to cite this article: Taylor, A. J. & Dennis, M. R. Vortex knots in tangled quantum eigenfunctions. Nat. Commun. 7:12346 doi: 10.1038/ncomms12346 (2016).
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Acknowledgements
We thank Michael Berry for valuable discussion and for originally proposing the problem investigated here, and Andrea Aiello, Stu Whittington, Daniel PeraltaSalas and Dmitry Jakobson for useful suggestions. This research was funded by a Leverhulme Trust Research Programme Grant. This work was carried out using the computational facilities of the Advanced Computing Research Centre, University of Bristol. We are grateful to the KITP for hospitality during the early part of this work. Alexander Taylor was partially funded by the Engineering and Physical Sciences Research Council and Mark Dennis by the the Royal Society of London.
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A.J.T. created the numerical routines for simulating eigenfunction modes and performing topological analysis. Other contributions were shared equally.
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Supplementary Information
Supplementary Figures 16, Supplementary Notes 14 and Supplementary References (PDF 9477 kb)
Supplementary Movie 1
A figureeight vortex knot in a random eigenfunction of the threedimensional harmonic oscillator, viewed from different directions. (MOV 3953 kb)
Supplementary Movie 2
Two mirrored trefoil vortex knots in a random eigenfunction of the threedimensional harmonic oscillator, viewed from different directions. (MOV 4013 kb)
Supplementary Movie 3
The knot 10_99 formed from vortices in a random eigenfunction of the the 3sphere, viewed from different directions. (MOV 18061 kb)
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Taylor, A., Dennis, M. Vortex knots in tangled quantum eigenfunctions. Nat Commun 7, 12346 (2016). https://doi.org/10.1038/ncomms12346
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DOI: https://doi.org/10.1038/ncomms12346
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