More on OTOCs and Chaos in Quantum Mechanics -- Magnetic Fields

Technical Summary: More on OTOCs and Chaos in Quantum Mechanics - Magnetic Fields

Problem Statement
Out-of-time-order correlators (OTOCs) have emerged as a primary diagnostic for quantum chaos and information scrambling, particularly in many-body systems where they exhibit exponential growth characterized by a Lyapunov exponent. However, in single-particle quantum systems, previous work (specifically Hashimoto et al.) established that OTOCs typically display power-law growth followed by saturation, rather than exponential divergence, even in classically chaotic geometries like the stadium billiard. This behavior arises because the finite-dimensional Hilbert space of single-particle systems suppresses indefinite operator growth, and unitary evolution replaces classical trajectory divergence.

A significant gap in the literature concerns how transverse magnetic fields modify this behavior. Magnetic fields fundamentally alter the classical phase space by introducing cyclotron motion, potentially suppressing ergodicity and inducing Landau quantization. The central problem addressed in this work is to systematically determine how a transverse magnetic field affects quantum information scrambling in single-particle systems. Specifically, the authors investigate whether magnetic confinement suppresses, enhances, or qualitatively reshapes the growth of OTOCs, and how this interplay depends on temperature and the strength of the magnetic field.

Methodology
The authors extend the spectral construction of thermal OTOCs introduced by Hashimoto et al. to magnetized quantum systems. The methodology involves the following steps:

1. Spectral Construction: The thermal OTOC is defined as $C_T(t) = -\langle [x(t), p(0)]^2 \rangle_\beta$. Using the energy eigenbasis $\{|n\rangle\}$ with eigenvalues $E_n$, this is computed by summing over microcanonical OTOCs $c_n(t)$ weighted by the Boltzmann factor $e^{-\beta E_n}$. The microcanonical OTOC is expressed purely in terms of position operator matrix elements $x_{nm}$ and energy differences, avoiding explicit wavefunction derivatives:
   $$c_n(t) = \sum_m |b_{nm}(t)|^2, \quad b_{nm}(t) = \frac{1}{2} \sum_k x_{nk} x_{km} \left[ (E_k - E_m)e^{i(E_n - E_k)t} - (E_n - E_k)e^{i(E_k - E_m)t} \right].$$

2. Numerical Implementation:

   • Integrable Benchmarks: The formalism is first validated using exactly solvable systems: a 2D harmonic oscillator and a particle in a Morse potential. These serve to test the spectral representation and explore the effects of anharmonicity and finite bound states.
   • Magnetic Billiards: The authors apply the method to a charged particle in a 2D stadium billiard (a classically chaotic system) subjected to a transverse magnetic field $B$. The Hamiltonian includes the vector potential in the symmetric gauge.
   • Eigenstate Computation: For the circular magnetic billiard, analytic solutions involving confluent hypergeometric functions are used. For the non-separable stadium geometry, the authors employ boundary methods (de Aguiar's approach) or finite-element/finite-difference discretization to compute the eigenpairs $\{E_\alpha, \psi_\alpha\}$ and the position matrix elements $x_{\alpha\beta}$.

3. Extraction of Exponents: The early-time growth of the thermal OTOC is fitted to an exponential form $C_T(t) \sim A e^{2\lambda_L t} + C_0$ to extract a Lyapunov-like exponent $\lambda_L(T, B)$. The authors caution that this is an effective short-time indicator rather than a true asymptotic Lyapunov exponent, given the absence of many-body interactions.

4. Guiding-Center Analysis: A second diagnostic is introduced using guiding-center coordinates $(X, Y)$, which represent the center of cyclotron motion. The authors compute the OTOC $C_R(t) = -\langle [X(t), Y(0)]^2 \rangle_\beta$. This formulation isolates the slow drift dynamics from the fast cyclotron motion and is sensitive to the non-commutative geometry induced by the magnetic field.

Key Results

• Canonical OTOCs ($x, p$):

  • Magnetic Field Dependence: In the absence of a magnetic field ($B=0$), the stadium billiard OTOC shows initial growth followed by saturation, consistent with previous findings. As $B$ increases, the nature of the growth changes. At intermediate fields, the OTOC exhibits structured, quasi-periodic fluctuations. At large fields ($B \gtrsim 3.5$), the dynamics are dominated by pronounced oscillations with timescales set by the inverse cyclotron frequency $\omega_c^{-1}$, signaling a crossover from chaotic to magnetically rigid dynamics.
  • Temperature Dependence: At low temperatures, the thermal ensemble is dominated by low-lying, delocalized states, resulting in slow, sublinear OTOC growth. As temperature increases, higher-energy states sensitive to the chaotic boundary contribute, leading to faster growth and a clear early-time exponential regime.
  • The $\lambda_L(T, B)$ Surface: The extracted exponent $\lambda_L$ reveals a non-monotonic structure. At fixed temperature, $\lambda_L$ increases with temperature (due to the population of chaotic eigenstates) but decreases monotonically with increasing magnetic field strength. High magnetic fields suppress $\lambda_L$ toward zero, indicating the restoration of regular dynamics due to magnetic rigidity and the confinement of trajectories to cyclotron orbits.

• Guiding-Center OTOCs ($X, Y$):

  • Distinct Dynamics: In contrast to the canonical OTOC, the guiding-center OTOC $C_R(t)$ exhibits no exponential growth. For a free particle or in the bulk, $C_R(t)$ is constant. In the presence of boundaries or potentials, it grows linearly in time, with a rate proportional to the mixed second derivative of the potential $\langle \partial_x \partial_y V \rangle$.
  • Bulk-Edge Separation: The guiding-center OTOC cleanly separates bulk and edge physics. Low-temperature thermal averages are dominated by bulk states (constant OTOC), while high temperatures populate edge states where the potential gradient induces linear growth. This highlights that the "chaos" observed in canonical operators is filtered out when observing the guiding-center degrees of freedom.

Significance and Claims
The paper claims to provide a controlled framework for probing the interplay between quantum chaos, Landau physics, and thermal scrambling in single-particle systems. The primary contributions are:

1. Extension of OTOC Formalism: The work successfully extends the Hashimoto et al. construction to magnetized systems, demonstrating that the spectral method remains robust even with the complexity of Landau-level mixing and boundary conditions.
2. Magnetic Control of Scrambling: The authors demonstrate that a transverse magnetic field acts as a tunable parameter that can suppress quantum scrambling. The construction of the $\lambda_L(T, B)$ surface characterizes a smooth crossover from a chaotic, high-temperature regime to a magnetically regularized phase.
3. Operator Dependence of Chaos: A crucial finding is that the diagnosis of chaos is operator-dependent. While canonical operators $(x, p)$ show signatures of exponential growth (at finite $T$ and moderate $B$), guiding-center operators $(X, Y)$ show only linear growth. This suggests that "chaos" in these systems is not an absolute property of the Hamiltonian but depends on which degrees of freedom are being probed. The guiding-center formulation effectively integrates out the fast cyclotron motion, filtering out short-time chaotic sensitivity.
4. Physical Interpretation: The results offer a physical picture where magnetic rigidity constrains phase-space exploration, suppressing the boundary-induced chaos typical of stadium billiards. The transition is described as a competition between thermal activation (which populates chaotic states) and magnetic confinement (which restricts motion to cyclotron orbits).

The authors conclude that while single-particle systems do not exhibit true many-body exponential scrambling, the introduction of magnetic fields and the analysis of specific operator sets allow for a nuanced understanding of how quantum information dynamics can be engineered or suppressed in mesoscopic systems.