# Semi-Poisson Statistics in Relativistic Quantum Billiards with Shapes of Rectangles

## Abstract

**:**

## 1. Introduction

## 2. Review of Characteristic Features of Neutrino Billiards

## 3. Tools Employed for the Analysis of Properties of the Eigenstates

## 4. Numerical Results for the Symmetry-Projected Eigenstates of Rectangular NBs

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Casati, G.; Valz-Gris, F.; Guarnieri, I. On the connection between quantization of nonintegrable systems and statistical theory of spectra. Lett. Nuovo Cim.
**1980**, 28, 279–282. [Google Scholar] [CrossRef] - Mehta, M.L. Random Matrices; Academic Press: London, UK, 1990. [Google Scholar]
- Bohigas, O.; Giannoni, M.J.; Schmit, C. Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws. Phys. Rev. Lett.
**1984**, 52, 1. [Google Scholar] [CrossRef] - Berry, M. Structural Stability in Physics; Pergamon Press: Berlin, Germany, 1979. [Google Scholar]
- Giannoni, M.; Voros, A.; Zinn-Justin, J. (Eds.) Chaos and Quantum Physics; Elsevier: Amsterdam, The Netherlands, 1989. [Google Scholar]
- Guhr, T.; Müller-Groeling, A.; Weidenmüller, H.A. Random-matrix theories in quantum physics: Common concepts. Phys. Rep.
**1998**, 299, 189. [Google Scholar] [CrossRef] - Haake, F.; Gnutzmann, S.; Kuś, M. Quantum Signatures of Chaos; Springer: Berlin/Heidelberg, Germany, 2018. [Google Scholar]
- Heusler, S.; Müller, S.; Altland, A.; Braun, P.; Haake, F. Periodic-orbit theory of level correlations. Phys. Rev. Lett.
**2007**, 98, 044103. [Google Scholar] [CrossRef] - Gutzwiller, M.C. Periodic Orbits and Classical Quantization Conditions. J. Math. Phys.
**1971**, 12, 343–358. [Google Scholar] [CrossRef] - Gutzwiller, M.C. Chaos in Classical and Quantum Mechanics; Springer: Berlin/Heidelberg, Germany, 1990. [Google Scholar]
- Sinai, Y.G. Dynamical systems with elastic reflections. Russ. Math. Surv.
**1970**, 25, 137. [Google Scholar] [CrossRef] - Bunimovich, L.A. On the Ergodic Properties of Nowhere Dispersing Billiards. Commun. Math. Phys.
**1979**, 65, 295. [Google Scholar] [CrossRef] - Berry, M.V. Regularity and chaos in classical mechanics, illustrated by three deformations of a circular ‘billiard’. Eur. J. Phys.
**1981**, 2, 91. [Google Scholar] [CrossRef] - Berry, M.V.; Tabor, M. Calculating the bound spectrum by path summation in actionangle variables. J. Phys. A
**1977**, 10, 371. [Google Scholar] [CrossRef] - Robnik, M.; Veble, G. On spectral statistics of classically integrable systems. J. Phys. A
**1998**, 31, 4669. [Google Scholar] [CrossRef] - Drożdż, S.; Speth, J. Near-ground-state spectral fluctuations in multidimensional separable systems. Phys. Rev. Lett.
**1991**, 67, 529–532. [Google Scholar] [CrossRef] - Gutkin, E. Billiards in polygons. Physica D
**1986**, 19, 311–333. [Google Scholar] [CrossRef] - Gutkin, E. Billiards in polygons: Survey of recent results. J. Stat. Phys.
**1996**, 83, 7–26. [Google Scholar] [CrossRef] - Bogomolny, E. Formation of superscar waves in plane polygonal billiards. J. Phys. Commun.
**2021**, 5, 055010. [Google Scholar] [CrossRef] - Casati, G.; Chirikov, B.V.; Guarneri, I. Energy-Level Statistics of Integrable Quantum Systems. Phys. Rev. Lett.
**1985**, 54, 1350–1353. [Google Scholar] [CrossRef] - Artuso, R.; Casati, G.; Guarneri, I. Numerical study on ergodic properties of triangular billiards. Phys. Rev. E
**1997**, 55, 6384–6390. [Google Scholar] [CrossRef] - Casati, G.; Prosen, T. Mixing Property of Triangular Billiards. Phys. Rev. Lett.
**1999**, 83, 4729–4732. [Google Scholar] [CrossRef] - Lozej, Č.; Casati, G.; Prosen, T. Quantum chaos in triangular billiards. Phys. Rev. Res.
**2022**, 4, 013138. [Google Scholar] [CrossRef] - Marklof, J. Spectral Form Factors of Rectangle Billiards. Comm. Math. Phys.
**1998**, 199, 169. [Google Scholar] [CrossRef] - Zemlyakov, A.N.; Katok, A.B. Topological transitivity of billiards in polygons. Mat. Notes
**1975**, 18, 291. [Google Scholar] [CrossRef] - Richens, P.J.; Berry, M.V. Pseudointegrable systems in classical and quantum mechanics. Physica D
**1981**, 2, 495. [Google Scholar] [CrossRef] - Mirbach, B.; Korsch, H.J. Longlived states and irregular dynamics in inelastic collisions: Analysis of a polygon billiard model. Nonlinearity
**1989**, 2, 327. [Google Scholar] [CrossRef] - Życzkowski, K. Classical and quantum billiards-integrable, nonintegrable and pseudo-integrable. Act. Phys. Pol. B
**1992**, 49, 245–270. [Google Scholar] - Biswas, D.; Jain, S.R. Quantum description of a pseudointegrable system: The π/3-rhombus billiard. Phys. Rev. A
**1990**, 42, 3170–3185. [Google Scholar] [CrossRef] [PubMed] - Shudo, A.; Shimizu, Y. Extensive numerical study of spectral statistics for rational and irrational polygonal billiards. Phys. Rev. E
**1993**, 47, 54–62. [Google Scholar] [CrossRef] - Shudo, A.; Shimizu, Y.; Šeba, P.; Stein, J.; Stöckmann, H.J.; Życzkowski, K. Statistical properties of spectra of pseudointegrable systems. Phys. Rev. E
**1994**, 49, 3748–3756. [Google Scholar] [CrossRef] - Hasselblatt, B.; Katok, A. (Eds.) Handbook of Dynamical Systems; Elsevier: Amsterdam, The Netherlands, 2002; Volume 1A. [Google Scholar]
- Bogomolny, E.B.; Gerland, U.; Schmit, C. Models of intermediate spectral statistics. Phys. Rev. E
**1999**, 59, R1315–R1318. [Google Scholar] [CrossRef] - Gorin, T. Generic spectral properties of right triangle billiards. J. Phys. A Math. Gen.
**2001**, 34, 8281. [Google Scholar] [CrossRef] - Bäcker, A.; Schubert, R. Chaotic eigenfunctions in momentum space. J. Phys. A Math. Gen.
**1999**, 32, 4795. [Google Scholar] [CrossRef] - Husimi, K. Some formal properties of the density matrix. Proc. Phys. Math. Soc. Jpn.
**1940**, 22, 264. [Google Scholar] - Bäcker, A.; Fürstberger, S.; Schubert, R. Poincaré Husimi representation of eigenstates in quantum billiards. Phys. Rev. E
**2004**, 70, 036204. [Google Scholar] [CrossRef] [PubMed] - Berry, M.V.; Mondragon, R.J. Neutrino Billiards: Time-Reversal Symmetry-Breaking Without Magnetic Fields. Proc. R. Soc. London A
**1987**, 412, 53. [Google Scholar] - Weyl, H. Elektron und Gravitation. I. Z. Physik
**1929**, 56, 330. [Google Scholar] [CrossRef] - Bolte, J.; Keppeler, S. A Semiclassical Approach to the Dirac Equation. Ann. Phys.
**1999**, 274, 125–162. [Google Scholar] [CrossRef] - Dietz, B. Circular and Elliptical Neutrino Billiards: A Semiclassical Approach. Act. Phys. Pol. A
**2019**, 136, 770. [Google Scholar] [CrossRef] - Dietz, B.; Li, Z.Y. Semiclassical quantization of neutrino billiards. Phys. Rev. E
**2020**, 102, 042214. [Google Scholar] [CrossRef] - Baym, G. Lectures on Quantum Mechanics; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Dresselhaus, M.; Dresselhaus, G.; Eklund, P. Science of Fullerenes and Carbon Nanotubes; Academic Press: San Diego, CA, USA, 1996. [Google Scholar] [CrossRef]
- Reich, S.; Maultzsch, J.; Thomsen, C.; Ordejón, P. Tight-binding description of graphene. Phys. Rev. B
**2002**, 66, 035412. [Google Scholar] [CrossRef] - Novoselov, K.S.; Geim, A.K.; Morozov, S.V.; Jiang, D.; Zhang, Y.; Dubonos, S.V.; Grigorieva, I.V.; Firsov, A.A. Electric Field Effect in Atomically Thin Carbon Films. Science
**2004**, 306, 666–669. [Google Scholar] [CrossRef] - Geim, A.; Novoselov, K. The rise of graphene. Nat. Mater.
**2007**, 6, 183. [Google Scholar] [CrossRef] - Beenakker, C.W.J. Colloquium: Andreev reflection and Klein tunneling in graphene. Rev. Mod. Phys.
**2008**, 80, 1337. [Google Scholar] [CrossRef] - Castro Neto, A.H.; Guinea, F.; Peres, N.M.R.; Novoselov, K.S.; Geim, A.K. The electronic properties of graphene. Rev. Mod. Phys.
**2009**, 81, 109. [Google Scholar] [CrossRef] - Ponomarenko, L.A.; Schedin, F.; Katsnelson, M.I.; Yang, R.; Hill, E.W.; Novoselov, K.S.; Geim, A.K. Chaotic Dirac Billiard in Graphene Quantum Dots. Science
**2008**, 320, 5874. [Google Scholar] [CrossRef] [PubMed] - Güttinger, J.; Stampfer, C.; Hellmüller, S.; Molitor, F.; Ihn, T.; Ensslin, K. Charge detection in graphene quantum dots. Appl. Phys. Lett.
**2008**, 93, 212102. [Google Scholar] [CrossRef] - Güttinger, J.; Frey, T.; Stampfer, C.; Ihn, T.; Ensslin, K. Spin States in Graphene Quantum Dots. Phys. Rev. Lett.
**2010**, 105, 116801. [Google Scholar] [CrossRef] - Wallace, P.R. The Band Theory of Graphite. Phys. Rev.
**1947**, 71, 622–634. [Google Scholar] [CrossRef] - Polini, M.; Guinea, F.; Lewenstein, M.; Manoharan, H.C.; Pellegrini, V. Artificial graphene as a tunable Dirac material. Nat. Nanotechnol.
**2013**, 8, 625. [Google Scholar] [CrossRef] [PubMed] - Akhmerov, A.R.; Beenakker, C.W.J. Detection of Valley Polarization in Graphene by a Superconducting Contact. Phys. Rev. Lett.
**2007**, 98, 157003. [Google Scholar] [CrossRef] - Akhmerov, A.R.; Beenakker, C.W.J. Boundary conditions for Dirac fermions on a terminated honeycomb lattice. Phys. Rev. B
**2008**, 77, 085423. [Google Scholar] [CrossRef] - Wurm, J.; Richter, K.; Adagideli, İ. Edge effects in graphene nanostructures: From multiple reflection expansion to density of states. Phys. Rev. B
**2011**, 84, 075468. [Google Scholar] [CrossRef] - Bittner, S.; Dietz, B.; Miski-Oglu, M.; Richter, A. Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard. Phys. Rev. B
**2012**, 85, 064301. [Google Scholar] [CrossRef] - Dietz, B.; Iachello, F.; Miski-Oglu, M.; Pietralla, N.; Richter, A.; von Smekal, L.; Wambach, J. Lifshitz and excited-state quantum phase transitions in microwave Dirac billiards. Phys. Rev. B
**2013**, 88, 104101. [Google Scholar] [CrossRef] - Dietz, B.; Klaus, T.; Miski-Oglu, M.; Richter, A. Spectral properties of superconducting microwave photonic crystals modeling Dirac billiards. Phys. Rev. B
**2015**, 91, 035411. [Google Scholar] [CrossRef] - Dietz, B.; Klaus, T.; Miski-Oglu, M.; Richter, A.; Wunderle, M.; Bouazza, C. Spectral Properties of Dirac Billiards at the van Hove Singularities. Phys. Rev. Lett.
**2016**, 116, 023901. [Google Scholar] [CrossRef] [PubMed] - Zhang, W.; Zhang, X.; Che, J.; Miski-Oglu, M.; Dietz, B. Properties of the eigenmodes and quantum-chaotic scattering in a superconducting microwave Dirac billiard with threefold rotational symmetry. arXiv
**2023**, arXiv:2302.10094v1. [Google Scholar] [CrossRef] - Libisch, F.; Stampfer, C.; Burgdörfer, J. Graphene quantum dots: Beyond a Dirac billiard. Phys. Rev. B
**2009**, 79, 115423. [Google Scholar] [CrossRef] - Wurm, J.; Rycerz, A.; Adagideli, İ.; Wimmer, M.; Richter, K.; Baranger, H.U. Symmetry Classes in Graphene Quantum Dots: Universal Spectral Statistics, Weak Localization, and Conductance Fluctuations. Phys. Rev. Lett.
**2009**, 102, 056806. [Google Scholar] [CrossRef] - Dietz, B. Relativistic quantum billiards with threefold rotational symmetry: Exact, symmetry-projected solutions for the equilateral neutrino billiard. Act. Phys. Pol. A
**2021**, 140, 473. [Google Scholar] [CrossRef] - Yu, P.; Zhang, W.; Dietz, B.; Huang, L. Quantum signatures of chaos in relativistic quantum billiards with shapes of circle- and ellipse-sectors. J. Phys. Math. Theor.
**2022**, 55, 224015. [Google Scholar] [CrossRef] - McIsaac, P. Symmetry-Induced Modal Characteristics of Uniform Waveguides - I: Summary of Results. IEEE Trans. Microw. Theory Tech.
**1975**, 23, 421–429. [Google Scholar] [CrossRef] - Sieber, M. Semiclassical transition from an elliptical to an oval billiard. J. Phys. A
**1997**, 30, 4563–4596. [Google Scholar] [CrossRef] - Waalkens, H.; Wiersig, J.; Dullin, H.R. Elliptic Quantum Billiard. Ann. Phys.
**1997**, 260, 50–90. [Google Scholar] [CrossRef] - Gaddah, W.A. Exact solutions to the Dirac equation for equilateral triangular billiard systems. J. Phys. A Math. Theor.
**2018**, 51, 385304. [Google Scholar] [CrossRef] - Zhang, W.; Dietz, B. Microwave photonic crystals, graphene, and honeycomb-kagome billiards with threefold symmetry: Comparison with nonrelativistic and relativistic quantum billiards. Phys. Rev. B
**2021**, 104, 064310. [Google Scholar] [CrossRef] - Gaddah, W.A. Discrete symmetry approach to exact bound-state solutions for a regular hexagon Dirac billiard. Phys. Script.
**2021**, 96, 065207. [Google Scholar] [CrossRef] - Greiner, W.; Schäfer, A. (Eds.) Quantum Chromodynamics; Springer: New York, NY, USA, 1994. [Google Scholar]
- Alberto, P.; Fiolhais, C.; Gil, V.M.S. Relativistic particle in a box. Eur. J. Phys.
**1996**, 17, 19. [Google Scholar] [CrossRef] - Alonso, V.; Vincenzo, S.D. General boundary conditions for a Dirac particle in a box and their non-relativistic limits. J. Phys. A Math. Gen.
**1997**, 30, 8573. [Google Scholar] [CrossRef] - Alonso, V.; Vincenzo, S.D.; Mondino, L. On the boundary conditions for the Dirac equation. Eur. J. Phys.
**1997**, 18, 315. [Google Scholar] [CrossRef] - Alberto, P.; Das, S.; Vagenas, E.C. Relativistic particle in a three-dimensional box. Phys. Lett. A
**2011**, 375, 1436–1440. [Google Scholar] [CrossRef] - Yusupov, J.; Otajanov, D.; Eshniyazov, V.; Matrasulov, D. Classical and quantum dynamics of a kicked relativistic particle in a box. Phys. Lett. A
**2018**, 382, 633–638. [Google Scholar] [CrossRef] - Dietz, B. Unidirectionality and Husimi functions in constant-width neutrino billiards. J. Phys. A Math. Theor.
**2022**, 55, 474003. [Google Scholar] [CrossRef] - Bäcker, A. Numerical Aspects of Eigenvalue and Eigenfunction Computations for Chaotic Quantum Systems. In The Mathematical Aspects of Quantum Maps; Esposti, M.D., Graffi, S., Eds.; Springer: Berlin/Heidelberg, Germany, 2003; pp. 91–144. [Google Scholar] [CrossRef]
- Yu, P.; Dietz, B.; Xu, H.Y.; Ying, L.; Huang, L.; Lai, Y.C. Kac’s isospectrality question revisited in neutrino billiards. Phys. Rev. E
**2020**, 101, 032215. [Google Scholar] [CrossRef] [PubMed] - Yu, P.; Dietz, B.; Huang, L. Quantizing neutrino billiards: An expanded boundary integral method. New J. Phys.
**2019**, 21, 073039. [Google Scholar] [CrossRef] - Leyvraz, F.; Schmit, C.; Seligman, T.H. Anomalous spectral statistics in a symmetrical billiard. J. Phys. A
**1996**, 29, L575. [Google Scholar] [CrossRef] - Keating, J.P.; Robbins, J.M. Discrete symmetries and spectral statistics. J. Phys. A
**1997**, 30, L177. [Google Scholar] [CrossRef] - Robbins, J.M. Discrete symmetries in periodic-orbit theory. Phys. Rev. A
**1989**, 40, 2128–2136. [Google Scholar] [CrossRef] [PubMed] - Joyner, C.H.; Müller, S.; Sieber, M. Semiclassical approach to discrete symmetries in quantum chaos. J. Phys. A
**2012**, 45, 205102. [Google Scholar] [CrossRef] - Weyl, H. Über die Abhängigkeit der Eigenschwingungen einer Membran und deren Begrenzung. J. Reine Angew. Math.
**1912**, 141, 1. [Google Scholar] [CrossRef] - Oganesyan, V.; Huse, D.A. Localization of interacting fermions at high temperature. Phys. Rev. B
**2007**, 75, 155111. [Google Scholar] [CrossRef] - Atas, Y.Y.; Bogomolny, E.; Giraud, O.; Roux, G. Distribution of the Ratio of Consecutive Level Spacings in Random Matrix Ensembles. Phys. Rev. Lett.
**2013**, 110, 084101. [Google Scholar] [CrossRef] - Atas, Y.; Bogomolny, E.; Giraud, O.; Vivo, P.; Vivo, E. Joint probability densities of level spacing ratios in random matrices. J. Phys. A
**2013**, 46, 355204. [Google Scholar] [CrossRef] - McLachlan, N. (Ed.) Theory and Application of Mathieu Functions; Oxford University Press: London, UK, 1947. [Google Scholar]
- Morse, P.; Feshbach, H. (Eds.) Methods of Theoretical Physics; MacGraw-Hill: New York, NY, USA, 1953. [Google Scholar]
- Abramowitz, M.; Stegun, I.A. (Eds.) Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables; Dover: New York, NY, USA, 2013. [Google Scholar]
- Dietz, B.; Smilansky, U. A scattering approach to the quantization of billiards—The inside—Outside duality. Chaos
**1993**, 3, 581. [Google Scholar] [CrossRef] [PubMed] - Elkies, N.D.; McMullen, C.T. Gaps in n mod 1 and ergodic theory. Duke Math. J.
**2004**, 123, 95–139. [Google Scholar] [CrossRef] - El-Baz, D.; Marklof, J.; Vinogradov, I. The two-point correlation function of the fractional parts of sqrt(n) is Poisson. Proc. Am. Math. Soc.
**2015**, 143, 2815–2828. [Google Scholar] [CrossRef] - Wang, J.; Casati, G.; Prosen, T. Nonergodicity and localization of invariant measure for two colliding masses. Phys. Rev. E
**2014**, 89, 042918. [Google Scholar] [CrossRef] [PubMed] - Huang, J.; Zhao, H. Ultraslow diffusion and weak ergodicity breaking in right triangular billiards. Phys. Rev. E
**2017**, 95, 032209. [Google Scholar] [CrossRef] [PubMed] - Bäcker, A.; Schubert, R.; Stifter, P. On the number of bouncing ball modes in billiards. J. Phys. Math. Gen.
**1997**, 30, 6783. [Google Scholar] [CrossRef] - Bogomolny, E.; Schmit, C. Structure of Wave Functions of Pseudointegrable Billiards. Phys. Rev. Lett.
**2004**, 92, 244102. [Google Scholar] [CrossRef] - Dietz, B.; Heine, A.; Heuveline, V.; Richter, A. Test of a numerical approach to the quantization of billiards. Phys. Rev. E
**2005**, 71, 026703. [Google Scholar] [CrossRef] - Sieber, M.; Smilansky, U.; Creagh, S.C.; Littlejohn, R.G. Non-generic spectral statistics in the quantized stadium billiard. J. Phys. A
**1993**, 26, 6217. [Google Scholar] [CrossRef]

**Figure 1.**(

**Left**) Spectral properties of the symmetry-projected eigenstates of the ellipse NB with $\u03f5=0.65$ (red histogram and circles: $l=0$; green histogram and triangles: $l=1$), and the semi-ellipse NB with $\u03f5=0.65$ (violet histogram and diamonds), $\u03f5=0.5$ (orange histogram and plus) and $\u03f5=0.1$ (turquoise histogram and stars). (

**Right**) Ratio distributions for (

**a**) the symmetry-projected eigenstates of the ellipse NB with $l=0$, (

**b**) $l=1$, (

**c**) all eigenvalues of the ellipse NB (red) and the corresponding ellipse QB (turquoise) and (

**d**) the semi-ellipse NB for $\u03f5=0.5$ (green) and $\u03f5=0.1$ (red). They are compared to the GOE (black solid line), Poisson (black dashed line) and semi-Poisson (black dashed–dotted lines) statistics.

**Figure 2.**(

**Left**) Spectral properties of the ${R}_{1}$, i.e., square NB (violet histogram and diamonds) and its symmetry-projected eigenstates (red histogram and circles: $l=0$; green histogram and triangles: $l=1$; orange histogram and plus: $l=2$; turquoise histogram and stars: $l=3$). (

**Right**) Ratio distributions for (

**a**) the symmetry-projected eigenstates of the ${R}_{1}$ NB with $l=0$ (red) and $l=2$ (green), (

**b**) $l=1$ (red) and $l=3$ (green), (

**c**) all eigenvalues of the square NB (red) and the corresponding square QB (turquoise) and (

**d**) the ${45}^{\circ}$-triangle, i.e., ${T}_{1}$ NB (red). They are compared to the GOE (black solid line), Poisson (black dashed line) and semi-Poisson (black dashed–dotted lines) statistics.

**Figure 3.**(

**Left**) Comparison of the length spectra of, from bottom to top, the full square NB (black line), for $l=0$ (blue line), $l=1$ (violet line), $l=2$ (green), $l=3$ (red), and the square QB (maroon). (

**Right**) Comparison of the length spectra of, from bottom to top, the rectangular ${R}_{3}$ NB (black line), for $l=0$ (red line), $l=1$ (green line), and the corresponding QB (blue).

**Figure 4.**(

**a**) From left to right, the momentum distribution in the $({q}_{x},{q}_{y})$ plane, the real parts of the spinor components ${\psi}_{1}\left(\mathit{r}\right)$ and ${\psi}_{2}\left(\mathit{r}\right)$ function in the $(x,y)$ plane, the local current in the $(x,y)$ plane and the Husimi functions in the Birkhoff coordinate plane $(s,p)$, where $s=0$ at the center of the lower horizontal side and increases in counterclockwise direction, for the symmetry-projected eigenstates states of the square NB with $l=0$ and, from top to bottom, numbers $n=49,51,68,75$. (

**b**) Same as left for $l=1$.

**Figure 5.**(

**Left**) Spectral properties of the symmetry-projected eigenstates of the rectangular ${R}_{2}$ NB (red histogram and circles: $l=0$; green histogram and triangles: $l=1$), the ${R}_{2}$ QB (violet histogram, lines and dots), and the ${R}_{2}$ NB (turquoise histogram and diamonds). (

**Right**) Spectral properties of the symmetry-projected eigenstates of the ${R}_{3}$ NB (red histogram and circles: $l=0$; green histogram and triangles: $l=1$), the ${R}_{3}$ QB (violet histogram, lines and dots), the ${R}_{3}$ NB (turquoise histogram and diamonds) and the triangular ${T}_{3}$ (magenta histogram and squares) and ${T}_{4}$ NB (brown histogram and dots). They are compared to the GOE (black solid line), Poisson (black dashed line) and semi-Poisson (black dashed–dotted lines) statistics.

**Figure 6.**(

**Left**) Spectral properties of the triangular ${T}_{3}$ NB (red histogram and circles) and ${T}_{3}$ QB (green histogram and triangles) and the triangular ${T}_{4}$ NB (orange histogram and dots) and ${T}_{4}$ QB (violet histogram and diamonds). They are compared to the GOE (black solid line), Poisson (black dashed–dotted–dotted line), semi-Poisson (black dashed–dotted lines) and quarter-Poisson (blue dashed lines) statistics. (

**Right**) Ratio distributions for (

**a**) the $l=0$ states of the ${R}_{2}$ (green histogram) and ${R}_{3}$ (red histogram) NBs, (

**b**) same as (

**a**) for the states with $l=1$, (

**c**) the rectangular ${R}_{2}$ (green histogram) and ${R}_{3}$ (red histogram) NBs and for the rectangular ${R}_{3}$ QB (turquoise histogram), (

**d**) the ${T}_{5}$ (green histogram), ${T}_{3}$ (red histogram) and ${T}_{4}$ (brown histogram) NBs. They are compared to the GOE (black solid line), Poisson (black dashed line), semi-Poisson (black dashed–dotted lines) and quarter-Poisson (blue dashed lines) statistics.

**Figure 7.**(

**Left**) Spectral properties of the triangular ${T}_{2}$ QB and NB before ((violet histogram and stars) and (red histogram and dots)) and after extraction of contributions from orbits that lead to scarred wave functions ((orange dashed-line histogram and crosses) and (green histogram and squares)). They are compared to the GOE (black solid line), Poisson (black dashed–dotted–dotted line), semi-Poisson (black dashed–dotted lines) and GUE (black dashed lines) statistics. (

**Right**) Ratio distributions $P\left(r\right)$ (upper panels) and integrated ratio distributions $I\left(r\right)$ (lower panels) for the ${T}_{2}$ QB (

**a**,

**c**) and NB (

**b**,

**d**). They are compared to the GOE (black solid line), Poisson (black dashed–dotted–dotted line), semi-Poisson (black dashed–dotted lines) and GUE (black dashed lines) statistics.

**Figure 8.**(

**Left**) Fluctuating part of the integrated spectral density (black) and the contributions of bouncing-ball orbits that are extracted to obtain the results shown in Figure 7 (red) for the ${T}_{2}$ NB. (

**Right**) Comparison of the length spectra of the ${T}_{2}$ NB (upper part) and the ${T}_{2}$ QB (lower part).

**Figure 9.**(

**a**) Same as in Figure 4 for the symmetry-projected eigenstates of the rectangular ${R}_{3}$ NB with $l=0$ and, from top to bottom, numbers $n=57$–59. (

**b**) Same as (

**a**) for $l=1$.

**Figure 10.**(

**a**) Same as Figure 4 for the triangular ${T}_{3}$ NB with, from top to bottom, numbers $n=255,288,264$. Here, the Borkhoff coordinates are chosen such that $s=0$ at the left corner of the triangle, and it increases in the counterclockwise direction. (

**b**) From left to right, momentum distributions, intensity distributions and Husimi functions for the corresponding QB with, from top to bottom, numbers $n=266,259,267$.

**Figure 11.**First row: Local current for the eigenstates of the ${T}_{3}$ NB for, from left to right, numbers $n=$1460 (

**a**), 1453 (

**b**), 1471 (

**c**). Second row: intensity distribution for the eigenstates of the ${T}_{3}$ QB for, from left to right, numbers $n=$1459 (

**d**), 1463 (

**e**), 1469 (

**f**).

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Dietz, B.
Semi-Poisson Statistics in Relativistic Quantum Billiards with Shapes of Rectangles. *Entropy* **2023**, *25*, 762.
https://doi.org/10.3390/e25050762

**AMA Style**

Dietz B.
Semi-Poisson Statistics in Relativistic Quantum Billiards with Shapes of Rectangles. *Entropy*. 2023; 25(5):762.
https://doi.org/10.3390/e25050762

**Chicago/Turabian Style**

Dietz, Barbara.
2023. "Semi-Poisson Statistics in Relativistic Quantum Billiards with Shapes of Rectangles" *Entropy* 25, no. 5: 762.
https://doi.org/10.3390/e25050762