
Periodic modulations of electron density in materials, often persisting beyond order-disorder transitions.Charge density waves are periodic modulations of electron density in certain crystalline materials, where the electrons organize into a standing wave pattern rather than being uniformly distributed. This phenomenon occurs in low-dimensional metals, such as layered compounds, and represents a collective quantum state of electrons. Charge density waves are often accompanied by a periodic distortion of the crystal lattice, and they can persist even when the material undergoes an order-disorder transition, making them a rich subject for studying quantum many-body physics.
Charge density waves were theoretically predicted by Rudolf Peierls in the 1930s, who showed that a one-dimensional metal is unstable against lattice distortion. The first experimental observations were made in the 1970s in layered transition-metal dichalcogenides like niobium diselenide.
In a charge density wave, the electron gas becomes unstable against a periodic modulation of its density, typically due to strong electron-phonon coupling or Fermi surface nesting. The electrons and the lattice distort together, creating a new ground state with a lower energy. This state has an energy gap at the Fermi level, turning the material into a semiconductor or insulator. The wave can be pinned by impurities or move under an applied electric field, leading to nonlinear transport phenomena.
Charge density waves are studied for potential applications in high-frequency oscillators, memory devices, and as model systems for understanding superconductivity and other quantum phenomena.
In some materials, charge density waves can move through the crystal like a 'sliding' state, carrying current without dissipation, similar to superconductivity but at higher temperatures.
| \rho_{\rm GJ} | variable | Goldreich-Julian charge density — charge density coulomb per cubic metre |
| \mathbf{\Omega} | variable | Angular velocity vector of the neutron star — angular velocity radian per second |
| \mathbf{B} | variable | Magnetic field vector — magnetic field tesla |
| c | constant | Universal constant, speed of light in vacuum — speed of light = 299792458 metre per second |
| 2\pi | constant | Mathematical constant, 2π — two pi = 6.283 |
| \cdot | operator | Scalar product of two vectors — dot product |
| n | variable | number density — number density per metre cubed |
| r | variable | radial distance — radius metre |
| β | variable | power-law index — power law index |
| ∝ | operator | is proportional to — proportionality |
| p(θ) | variable | Probability density in new variable θ — probability density |
| p_u(φ) | variable | Probability density in original variable φ — probability density |
| φ | variable | Original variable |
| θ | variable | New variable |
| det(∂φ/∂θ) | operator | Scales the density to conserve probability mass — jacobian determinant |
| \omega_p | variable | plasma angular frequency — angular frequency radian per second |
| n | variable | electron number density — number density per cubic metre |
| e | variable | elementary charge — electric charge coulomb |
| m | variable | electron mass — mass kilogram |
| \epsilon_0 | constant | permittivity of free space — vacuum permittivity = 8.8541878128e-12 farad per metre |
| \sqrt{} | operator | takes the square root — square root |
| S_I | variable | power spectral density of current noise — power spectral density ampere squared per hertz |
| e | variable | elementary charge — elementary charge coulomb |
| I | variable | average electric current — electric current ampere |
| J^{\mu\nu\alpha} | variable | Angular momentum density tensor — angular momentum density kilogram per metre per second |
| x^{\mu} | variable | Spacetime coordinate — position metre |
| x^{\nu} | variable | Spacetime coordinate — position metre |
| T^{\nu\alpha} | variable | Stress-energy tensor component — stress energy tensor joule per cubic metre |
| T^{\mu\alpha} | variable | Stress-energy tensor component — stress energy tensor joule per cubic metre |











