2023/09/08 by Michael Berry, M V Berry
Physics and Astronomy · #Physics of Superconductivity and Magnetism #Quantum and electron transport phenomena #Quantum many-body systems
paper · doi:10.1088/1361-6404/acf81e
openalex publication_date 2023/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23
Abstract For a quantum spin driven cyclically by a slowly-rotated magnetic field, geometric phases are well understood. If the cycle takes a long time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> </mml:math> the leading-order (dynamical) phase is proportional to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>T</mml:mi> </mml:math> and the geometric phase is the contribution independent of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>T</mml:mi> <mml:mo>.</mml:mo> </mml:math> The dynamical and geometric phases are the first two terms of a series in slowness <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>T</mml:mi> <mml:mo>.</mml:mo> </mml:math> Here it is shown with an exactly solvable example that the corrections are of two types: smooth, proportional to powers of slowness, and oscillatory: essential singularities in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> </mml:math> in the form of trigonometric functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>T</mml:mi> </mml:math> divided by powers of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>T</mml:mi> <mml:mo>.</mml:mo> </mml:math> The calculations are elementary and therefore suitable for presentation in graduate quantum theory courses.