Glossary

10.4 Berry phase

4 min read•august 21, 2024

reveals the geometric nature of quantum states in condensed matter systems. It arises from cyclic evolution in parameter space, providing insights into topological properties of materials and their electronic structure.

This concept plays a crucial role in understanding phenomena like band structure and . The mathematical framework connects quantum mechanics with differential geometry, enabling analysis of material properties and prediction of physical phenomena.

Concept of Berry phase

  • Berry phase fundamentally relates to the geometric properties of quantum states in condensed matter systems
  • Provides insights into the topological nature of materials and their electronic structure
  • Plays a crucial role in understanding various phenomena in condensed matter physics, from band structure to topological insulators

Geometric phase in quantum mechanics

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  • Arises from the cyclic evolution of a quantum system in parameter space
  • Depends only on the path taken in parameter space, not the rate of evolution
  • Manifests as an additional phase factor in the wavefunction, distinct from the dynamical phase
  • Can be observed in systems with degeneracies or near-degeneracies in their energy levels
    • Spin-1/2 particles in a rotating magnetic field accumulate a Berry phase

Adiabatic evolution of quantum states

  • Occurs when system parameters change slowly compared to the system's characteristic timescales
  • Quantum state follows the instantaneous eigenstates of the Hamiltonian
  • Adiabatic theorem ensures the system remains in its instantaneous eigenstate
  • Berry phase emerges naturally in the
    • Slowly varying magnetic field acting on a spin system demonstrates adiabatic evolution

Cyclic evolution and phase accumulation

  • Involves returning the system to its initial state after a cyclic change in parameters
  • Total phase accumulated consists of both dynamical and geometric components
  • Berry phase represents the geometric component, independent of the evolution time
  • Measured by interferometric techniques or through its effects on physical observables
  • Analogous to the Aharonov-Bohm effect for charged particles in electromagnetic fields

Mathematical formulation

  • Mathematical framework of Berry phase connects quantum mechanics with differential geometry
  • Provides a powerful tool for analyzing topological properties of condensed matter systems
  • Enables the calculation and prediction of various physical phenomena in materials

Berry connection and curvature

  • An(R)=i⟨n(R)∣∇R∣n(R)⟩A_n(R) = i\langle n(R)|\nabla_R|n(R)\rangle represents the local gauge potential
  • Fn(R)=∇R×An(R)F_n(R) = \nabla_R \times A_n(R) analogous to magnetic field in parameter space
  • Integral of Berry curvature over a closed surface yields the Berry phase
  • Berry curvature determines the topology of the band structure in crystals
    • Non-zero Berry curvature leads to anomalous velocity in semiclassical electron dynamics

Parallel transport in parameter space

  • Describes the evolution of quantum states along a path in parameter space
  • Ensures the phase relationship between nearby states is maintained
  • Geometric meaning of Berry phase as the failure of parallel transport to return to the initial state
  • Analogous to parallel transport of vectors on curved surfaces in differential geometry
    • Parallel transport of a vector on a sphere results in a rotation after a closed loop

Gauge transformations and invariance

  • Berry connection transforms under local gauge transformations of the wavefunction
  • Berry curvature and phase remain gauge-invariant quantities
  • Allows for different choices of basis states while preserving physical observables
  • Connects Berry phase to gauge theories in particle physics and field theory
    • U(1) gauge transformations in electromagnetism analogous to Berry phase gauge freedom

Berry phase in crystals

  • Berry phase concepts applied to electronic states in crystalline solids
  • Provides insights into the topological properties of band structures
  • Crucial for understanding phenomena like polarization and Hall effects in materials

Bloch waves and Brillouin zone

  • describe electronic states in periodic potentials of crystals
  • represents the fundamental domain of crystal momentum k-space
  • Berry phase can be defined for Bloch states as they evolve in k-space
  • Periodic boundary conditions in k-space lead to quantization of Berry phase
    • Graphene's honeycomb lattice exhibits Dirac points with non-trivial Berry phase

k-space topology and band structure

  • Berry curvature distribution in k-space determines the topology of energy bands
  • Band crossings and degeneracies associated with non-zero Berry curvature
  • Topological invariants () calculated from Berry curvature integrals
  • Determines the existence of topologically protected edge states
    • Quantum Hall systems exhibit chiral edge states due to non-trivial band topology

Zak phase in one-dimensional systems

  • Berry phase accumulated when k traverses the entire Brillouin zone in 1D
  • Related to the polarization of insulators and topological properties
  • Quantized in the presence of inversion or time-reversal symmetry
  • Can be measured through or bulk polarization
    • Su-Schrieffer-Heeger model of polyacetylene exhibits quantized

Key Terms to Review (19)

Adiabatic approximation: The adiabatic approximation is a method used in quantum mechanics and thermodynamics, where changes occur slowly enough that a system remains in its instantaneous eigenstate throughout the process. This concept allows for simplifying complex systems by assuming that the system's dynamics are governed primarily by its energy levels, making it easier to analyze phenomena like the Berry phase.
Angle-resolved photoemission spectroscopy: Angle-resolved photoemission spectroscopy (ARPES) is a powerful experimental technique used to study the electronic structure of materials by measuring the energy and momentum of electrons emitted from a sample when it is illuminated by ultraviolet or X-ray radiation. This method provides critical information about the band structure, Fermi surface, and other electronic properties of solids, which are essential for understanding phenomena like superconductivity, surface states, and quantum phase transitions.
Berry Connection: The Berry connection is a mathematical construct that arises in quantum mechanics and condensed matter physics, describing how the wave function of a system evolves as parameters change. It captures the geometric phase that accumulates when a system undergoes adiabatic processes and its parameters are varied along a closed path in parameter space. This concept is key to understanding phenomena such as the Berry phase, which reveals insights into the geometric properties of quantum states.
Berry Curvature: Berry curvature is a mathematical concept that describes the geometric phase acquired by a quantum system as it evolves along a closed path in parameter space. It provides insights into the topology of the parameter space and is crucial for understanding phenomena like the quantum Hall effect and topological phases of matter, connecting deeply with concepts such as the Berry phase and Chern insulators.
Berry Phase: Berry phase is a geometric phase acquired over the course of a cycle when a quantum system is subjected to adiabatic, cyclic changes in its parameters. This concept is crucial in understanding phenomena in condensed matter physics, as it connects to the geometric properties of the wavefunctions, which can influence observable physical effects such as the behavior of electrons in various materials, including topological insulators and systems experiencing the quantum Hall effect.
Bloch Waves: Bloch waves are quantum mechanical wave functions that describe the behavior of particles, such as electrons, in a periodic potential, like a crystal lattice. They arise from the periodicity of the crystal structure and show that the wave functions can be expressed as a plane wave modulated by a periodic function, reflecting the symmetry and properties of the lattice.
Brillouin zone: A Brillouin zone is a uniquely defined region in reciprocal space that contains all the distinct wave vectors for a periodic lattice. It plays a crucial role in understanding the electronic properties of solids, particularly in defining energy bands and the behavior of electrons under periodic potentials.
Chern Numbers: Chern numbers are topological invariants associated with certain classes of complex vector bundles over manifolds, often used to characterize the geometry and topology of quantum systems. They provide important insights into the properties of materials, particularly in contexts involving electronic band structures and quantum Hall effects, where they help explain phenomena like edge states and quantized conductance.
Coherent States: Coherent states are specific quantum states of a harmonic oscillator that exhibit properties closely resembling classical oscillatory motion. These states are important in quantum mechanics because they minimize the uncertainty relation and represent the quantum state that most closely follows classical trajectories, making them essential for understanding phenomena like the Berry phase.
Geometric Phase: The geometric phase is a phase factor acquired by a quantum system when it undergoes an adiabatic evolution around a closed path in parameter space. This concept is crucial in understanding how the quantum state of a system can be affected by the geometric properties of the space in which it evolves, rather than just by the dynamics of the Hamiltonian governing the system.
Interference experiments: Interference experiments are scientific procedures designed to demonstrate the wave nature of particles by showing how different wavefronts can combine to produce a resultant wave pattern. These experiments, particularly with light and electrons, reveal fundamental aspects of quantum mechanics, such as superposition and phase relationships, which are critical in understanding phenomena like the Berry phase.
Michael Berry: Michael Berry is a prominent physicist known for his contributions to theoretical physics, particularly in the area of quantum mechanics. He is most recognized for the concept of the Berry phase, which describes a geometric phase acquired over the course of a cyclic adiabatic process in a quantum system. This phase plays a crucial role in understanding phenomena in various fields such as quantum computing, optics, and condensed matter physics.
Non-abelian gauge fields: Non-abelian gauge fields are a type of field that are associated with non-abelian symmetries in gauge theories, where the commutation relations of the gauge group do not vanish. This leads to interactions that are fundamentally different from those found in abelian gauge theories, such as electromagnetism. In this context, non-abelian gauge fields give rise to interesting phenomena, including self-interactions and the concept of confinement, which play a crucial role in various physical systems.
Quantum entanglement: Quantum entanglement is a fundamental phenomenon in quantum mechanics where two or more particles become interconnected such that the state of one particle instantaneously influences the state of another, regardless of the distance separating them. This non-local connection challenges classical intuitions about separability and locality, leading to unique implications in various fields like quantum computing and quantum information theory.
Quantum Hall Effect: The quantum Hall effect is a quantum phenomenon observed in two-dimensional electron systems under low temperatures and strong magnetic fields, where the Hall conductivity becomes quantized in integer or fractional values. This effect is crucial for understanding electron behavior in low-dimensional systems and has deep connections to topological phases of matter and various advanced materials.
Shou-Cheng Zhang: Shou-Cheng Zhang is a prominent physicist known for his significant contributions to condensed matter physics, particularly in the study of topological phases of matter and their implications. His research has played a crucial role in understanding topological semimetals and the Berry phase, advancing our knowledge of how these phenomena manifest in materials and their potential applications in future technologies.
Topological Insulators: Topological insulators are materials that behave as insulators in their bulk while supporting conducting states on their surfaces or edges. This unique property arises from the topological order of the electronic band structure, which distinguishes them from ordinary insulators, allowing for robust surface states that are protected against scattering by impurities or defects.
Topological quantum computing: Topological quantum computing is a computational approach that uses the principles of topology to manipulate and store quantum information. This method relies on anyons, which are quasi-particles that exhibit non-abelian statistics, allowing for the encoding of information in a way that is inherently resistant to local disturbances and errors. By leveraging the topological properties of these anyons, topological quantum computers aim to achieve fault-tolerant computation, making them a promising avenue for robust quantum information processing.
Zak phase: The Zak phase is a geometric phase acquired by the wave function of a particle as it is adiabatically transported around a closed path in momentum space. This phase is crucial in understanding the topology of electronic states in periodic systems, particularly in relation to the band structure and the behavior of insulating materials under certain conditions.
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