Level Spacing Statistics and the Onset of Chaos

Hello! Welcome to our next lesson.

In our previous session, we established the numerical framework for solving the exciton Schrödinger equation in the intermediate magnetic field regime. We saw that by using numerical diagonalization—specifically, building and diagonalizing the Hamiltonian matrix in a suitable basis—we can compute the full energy spectrum $E_{n}$ . This provides us with the raw data needed to probe the system's dynamics.

Today, we'll learn how to interpret this data. We will move from simply observing qualitative features like avoided crossings to performing a rigorous, quantitative analysis. This lesson directly addresses the learning outcome: to analyze level spacing statistics (e.g., Wigner-Dyson distribution) to quantitatively identify the transition to chaos. We will bridge the gap between the numerically generated energy levels and the profound physical concept of quantum chaos.

The Statistical Fingerprint of Chaos

In classical mechanics, chaos is characterized by the exponential divergence of trajectories from nearby initial conditions. In quantum mechanics, where the concept of a trajectory is ill-defined, we need a different indicator. The Bohigas-Giannoni-Schmit (BGS) conjecture provides this link: the spectral statistics of a quantum system whose classical counterpart is chaotic are described by Random Matrix Theory (RMT).

What does this mean in practice? It means we can determine if a system is chaotic by analyzing the statistical distribution of the spacings between its energy levels.

Integrable (Regular) Systems: The energy levels are uncorrelated. They behave like random numbers sprinkled on an energy axis, and their spacings can be arbitrarily small.
Chaotic Systems: The interactions that cause chaos also induce correlations between energy levels. This leads to a phenomenon called level repulsion, where the probability of finding two levels very close to each other is suppressed.

The following video provides an excellent introduction to RMT and how it connects to complex physical systems. It masterfully builds the intuition for why level spacings are the key observable.

Random Matrices in Unexpected Places: Atomic Nuclei, Chaotic Billiards, Riemann Zeta #SoME2

To understand the foundational ideas of Random Matrix Theory and its surprising applicability, watch these sections from the video 'Random Matrices in Unexpected Places' by the ℏistas channel.

Please watch the following parts: 00:00 - 02:10: This introduces the core idea that complex, deterministic systems (like atomic nuclei or chaotic billiards) can exhibit statistical properties described by random matrices. 07:19 - 11:56: This section explains the central quantities of interest in RMT. Pay close attention to the focus on the matrix's spectrum (its eigenvalues), as this is precisely what we are analyzing.

Uncorrelated vs. Correlated Spectra: Poisson vs. Wigner-Dyson

Let's now formalize the two distinct statistical signatures.

1. The Poisson Distribution: The Signature of Integrability

Imagine the energy levels of an integrable system are completely independent events. The probability of finding a level in a small energy interval is constant and independent of the surrounding levels. In this case, the distribution of spacings $s$ between adjacent levels follows a Poisson distribution:

$P_{P} (s) = e^{- s}$

(after normalizing the average spacing to one). The most important feature is that $P_{P} (s = 0) = 1$ . This means that very small spacings are the most probable outcome; energy levels can be arbitrarily close and frequently cross.

2. The Wigner-Dyson Distribution: The Signature of Chaos

For a chaotic system, the eigenvalues repel each other. This repulsion dramatically alters the spacing distribution. RMT predicts a Wigner-Dyson distribution. The simplest form, which arises from considering the eigenvalues of a 2x2 symmetric random matrix, captures the essential physics:

$P (s) \propto s e^{- s^{2} / α}$

The crucial feature here is the linear factor of $s$ . This ensures that as the spacing approaches zero, the probability also goes to zero: $P (s \to 0) = 0$ . This is the mathematical signature of level repulsion.

The video you just started continues with a fantastic derivation of both of these distributions, providing deep insight into their origins.

Random Matrices in Unexpected Places: Atomic Nuclei, Chaotic Billiards, Riemann Zeta #SoME2

Let's continue with the same video to see the explicit derivation of the Poisson and Wigner-Dyson distributions.

This is a key part of the lesson. Please watch these sections carefully: 18:12 - 21:06: Follow the derivation of the exponential (Poissonian) distribution for the spacings of uncorrelated eigenvalues. 21:06 - 22:54: This part provides an intuitive argument for why correlated eigenvalues (from a chaotic system) should behave differently, predicting a linear dependence for small spacings. 22:54 - 26:33: Here, the Wigner-Dyson distribution is explicitly derived for a 2x2 real symmetric random matrix. This beautifully demonstrates how level repulsion, P(s) \propto s, emerges from the structure of the problem. 26:33 - 28:09: This section directly compares the Poissonian and Wigner-Dyson distributions and summarizes the concept of level repulsion.

The RMT Symmetry Classes: GOE, GUE, and GSE

The simple $s \cdot e^{- s^{2}}$ form is just the beginning. The precise shape of the Wigner-Dyson distribution depends on the fundamental symmetries of the Hamiltonian, leading to different "ensembles" in RMT. Your background in advanced quantum mechanics will make the symmetry arguments here feel familiar.

Gaussian Orthogonal Ensemble (GOE): Applies to systems that are invariant under time reversal. The Hamiltonian can be represented by a real symmetric matrix. The level spacing distribution exhibits linear repulsion:
$P_{GO E} (s) = \frac{π}{2} s e^{- π s^{2} /4} (\propto s for small s)$
Gaussian Unitary Ensemble (GUE): Applies to systems where time-reversal symmetry is broken (e.g., by an external magnetic field). The Hamiltonian is a complex Hermitian matrix. This ensemble shows stronger, quadratic level repulsion:
$P_{G U E} (s) = \frac{32}{π ^{2}} s^{2} e^{- 4 s^{2} / π} (\propto s^{2} for small s)$
Gaussian Symplectic Ensemble (GSE): A more specialized case for systems with time-reversal symmetry but with half-integer spin and no rotational symmetry. Repulsion is even stronger: $P_{GS E} (s) \propto s^{4}$ .

These distributions provide concrete theoretical curves against which we can compare our numerical data.

Introduction to the computational analysis of static ...

These lecture notes provide a concise summary of the different random matrix ensembles and their corresponding spacing distributions.

Please read Section IV.A.1, 'Level spacing distributions and symmetries'. It clearly defines the GOE, GUE, and GSE, gives their explicit formulas, and states the physical conditions under which each applies. This formalizes the concepts from the video.

The Practical Workflow: Unfolding the Spectrum

There is a critical practical step before we can make a comparison. The theoretical RMT distributions assume a spectrum where the average level spacing is constant and equal to one. However, the density of states of a real physical system is almost never uniform; it changes with energy.

To compare our calculated spectrum to RMT, we must first "unfold" it. This is a rescaling procedure that transforms the original energy levels $E_{n}$ into a new set of levels $ϵ_{n}$ that have a uniform average spacing of one.

The procedure is as follows:

Calculate the cumulative level density, $N (E)$ : This function gives the number of energy levels with energy less than or equal to $E$ . For our discrete spectrum, this is a staircase function.
Fit a smooth function: We fit a smooth, monotonically increasing function, $N ˉ (E)$ , to the staircase. This function represents the average cumulative density.
Rescale the levels: The new, unfolded energy levels are given by $ϵ_{n} = N ˉ (E_{n})$ . By construction, the spectrum ${ϵ_{n}}$ now has a mean density of 1 everywhere.
Calculate spacings and histogram: We compute the nearest-neighbor spacings of the unfolded spectrum, $s_{n} = ϵ_{n + 1} - ϵ_{n}$ , and create a histogram of these values. This histogram can now be directly compared to the theoretical $P_{GO E} (s)$ or $P_{G U E} (s)$ .

Introduction to the computational analysis of static ...

The same lecture notes provide an excellent, practical guide on how to perform this crucial unfolding step.

Please read Section IV.B, 'Unfolding the spectrum'. It explains why this step is necessary and provides a clear recipe for carrying it out.

Test your understanding!

You numerically calculate 1000 energy levels for a system between E=0 and E=10. You find that the levels are much denser at low energy than at high energy. You naively calculate the spacings $s_{n} = E_{n + 1} - E_{n}$ and plot their distribution. Why would this histogram be misleading if you compare it directly to a Wigner-Dyson distribution?

Show answer

The Wigner-Dyson distribution assumes a uniform mean level spacing. Your raw spectrum violates this assumption. The high density of states at low energy will contribute a large number of very small spacings to your histogram, while the sparse high-energy region will contribute a few large spacings. This will skew your distribution, making it look more like a Poisson distribution (peaked at small $s$ ) than it actually is. You must unfold the spectrum first to remove the effect of the non-uniform density of states before you can reliably test for level repulsion.

Case Study: Magnetoexcitons in Cu₂O

Let's now apply this machinery to our system of interest. Your work on excitons in high magnetic fields involves precisely these concepts.

For a hydrogen atom in a magnetic field, time-reversal symmetry is broken. However, a more subtle antiunitary symmetry often remains (a combination of time reversal and a spatial reflection), which is why its chaotic spectrum typically follows GOE statistics.

For an exciton in a semiconductor like Cu₂O, the situation is different. The underlying cubic crystal lattice introduces additional terms in the Hamiltonian. As the following paper demonstrates, the combination of the external magnetic field and the cubic crystal potential can break all antiunitary symmetries, even the subtle ones that survive in the hydrogen atom.

This leads to a clear prediction: the level spacing statistics of magnetoexcitons in Cu₂O should transition to GUE statistics, a hallmark of profound quantum chaos.

arXiv:1609.04278v2 [cond-mat.mes-hall] 23 Dec 2016

This research paper is a direct application of everything we have discussed. It shows how level spacing analysis is used in cutting-edge research to understand magnetoexcitons.

Please read the following key parts of this paper: Introduction (Section idx=0): Focus on the first paragraph which introduces the BGS conjecture and the distinction between Poisson, GOE, and GUE statistics. Analytical argument (Section idx=2): You don't need to follow every step of Eq. (9), but grasp the main argument: the cubic part of the Hamiltonian (H_c) is not invariant under the residual symmetry operation that the hydrogen-like part is, thus breaking all antiunitary symmetries. Numerical analysis and Results (Sections idx=3, 4): Read from the paragraph beginning 'In our numerical calculations...' to the end of the paper. Pay close attention to: The mention of unfolding the spectra. The decision to plot the cumulative distribution function F(s) = \int_0^s P(x)dx, which is better for smaller datasets than a histogram. The explicit formulas given for the Poisson, GOE, and GUE distributions (Eqs. 12-14). Figure 1, which is the key result: it shows the numerical data (red dots) for the exciton spectrum perfectly matching the theoretical GUE curve (green solid line) as the cubic term (\delta') becomes significant.

This paper beautifully demonstrates the entire process: a theoretical prediction based on symmetry, numerical calculation of the spectrum, proper statistical analysis (unfolding), and comparison with RMT to confirm the presence and nature of quantum chaos.

Conclusion

Today, we have equipped ourselves with the quantitative tools to diagnose quantum chaos. By analyzing the statistical distribution of energy level spacings from the spectra we learned to compute in the last lesson, we can move beyond qualitative observation to robustly classify the dynamics of our system.

Key Takeaways:

Quantum chaos manifests as level repulsion in the energy spectrum.
The level spacing statistics of integrable systems follow a Poisson distribution, with no level repulsion ( $P (0) > 0$ ).
The statistics of chaotic systems follow a Wigner-Dyson distribution, which shows level repulsion ( $P (s \to 0) = 0$ ).
The specific form of the distribution (GOE for $P \propto s$ , GUE for $P \propto s^{2}$ ) depends on the system's time-reversal and spatial symmetries.
A crucial practical step is unfolding the spectrum to ensure a uniform mean level density before analysis.
For magnetoexcitons in cubic semiconductors like Cu₂O, the interplay of the magnetic field and crystal structure can break all antiunitary symmetries, leading to GUE statistics.

Preview of the Next Lesson

We have now seen that in the intermediate field regime, the Hamiltonian mixes basis states extensively, leading to chaotic spectra. What is the physical consequence of this state mixing? In the next lesson, we will evaluate the field-induced mixing of ortho- and para-exciton states and the resulting relaxation of optical selection rules. We will see how this chaos allows for previously "forbidden" transitions to become visible, a direct and measurable consequence of the physics we've studied today.

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