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

Cameron Beetar, Jeff Murugan

TL;DR

This work investigates how magnetic fields reshape quantum information scrambling in single-particle systems by analyzing thermal OTOCs in magnetic billiards. Using a spectral Hashimoto-type construction, the authors compute $C_T(t)$ for canonical position-momentum operators and extract Lyapunov-like exponents $λ_L(T,B)$ across temperature and field strength, revealing a magnetic-county crossover from chaotic-like spreading to magnetic rigidity. A contrasting guiding-center OTOC built from $X,Y$ coordinates exhibits no exponential growth, underscoring the operator- and basis-dependence of scrambling diagnostics in Landau-quantized systems. Together, these results establish a controlled framework linking geometry, magnetic fields, spectral structure, and scrambling, with potential implications for quantum control and transport in mesoscopic and topological settings.

Abstract

We revisit thermal out-of-time-order correlators (OTOCs) in single-particle quantum systems, focusing on magnetic billiards. Using the stadium billiard as a testbed, we compute the thermal OTOC $C_T(t) = -\langle [x(t), p]^2 \rangle_β$ and extract Lyapunov-like exponents $λ_L$ that quantify early-time growth. We map out $λ_L(T, B)$, revealing a crossover from quantum chaos to magnetic rigidity. In parallel, we compute an alternative OTOC built from guiding-center operators, which exhibits qualitatively distinct dynamics and no exponential growth. Our results offer a controlled framework for probing scrambling, temperature dependence, and the interplay of geometry and magnetic fields in quantum systems.

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

TL;DR

This work investigates how magnetic fields reshape quantum information scrambling in single-particle systems by analyzing thermal OTOCs in magnetic billiards. Using a spectral Hashimoto-type construction, the authors compute for canonical position-momentum operators and extract Lyapunov-like exponents across temperature and field strength, revealing a magnetic-county crossover from chaotic-like spreading to magnetic rigidity. A contrasting guiding-center OTOC built from coordinates exhibits no exponential growth, underscoring the operator- and basis-dependence of scrambling diagnostics in Landau-quantized systems. Together, these results establish a controlled framework linking geometry, magnetic fields, spectral structure, and scrambling, with potential implications for quantum control and transport in mesoscopic and topological settings.

Abstract

We revisit thermal out-of-time-order correlators (OTOCs) in single-particle quantum systems, focusing on magnetic billiards. Using the stadium billiard as a testbed, we compute the thermal OTOC and extract Lyapunov-like exponents that quantify early-time growth. We map out , revealing a crossover from quantum chaos to magnetic rigidity. In parallel, we compute an alternative OTOC built from guiding-center operators, which exhibits qualitatively distinct dynamics and no exponential growth. Our results offer a controlled framework for probing scrambling, temperature dependence, and the interplay of geometry and magnetic fields in quantum systems.
Paper Structure (16 sections, 68 equations, 12 figures)

This paper contains 16 sections, 68 equations, 12 figures.

Table of Contents

  1. Introduction
  2. OTOCs in quantum mechanics
  3. A 2-dimensional harmonic oscillator
  4. The Morse Potential
  5. OTOCs for a quantum particle in a magnetic field
  6. Worked example: circular magnetic billiard
  7. From circular to general billiards
  8. Thermal OTOCs for the magnetic stadium billiard
  9. Temperature and $B$-field dependence of the OTOC growth
  10. Guiding-center OTOCs in a transverse magnetic field
  11. Guiding-center coordinates
  12. Guiding-Center OTOCs and bulk–edge separation
  13. Conclusions
  14. Comments on Numerics
  15. Numerics for Section \ref{['sec:qm_otoc']}
  16. ...and 1 more sections

Figures (12)

  • Figure 1: Microcanonical (left) and thermal (right) OTOC's for $T=0.01$, $a=0.5$, $m=1$, and varying $D_e$ values.
  • Figure 2: Microcanonical (left) and thermal (right) OTOC's for $T=1.0$, $D_e=16$, $m=1$, and varying $a$ values.
  • Figure 3: Microcanonical (left) and thermal (right) OTOC's for $a=0.5$, $D_e=10$, $m=1$, and varying $T$ values.
  • Figure 4: Microcanonical (left) and thermal (right) OTOC's for a charged particle in a circular magnetic billiard $B=0, 1, 2$ and $4$.
  • Figure 5: Thermal OTOC's for a charged particle in a stadium magnetic billiard with magnetic field strengths $B=0, 1, 2$ and $4$.
  • ...and 7 more figures