The easiest way of understanding SDW, CDW, is to consider first what happens in 1D metals. The response function for charge and for spin diverges for q = 2k_F. Adding a interaction between the chains in the RPA framework will lead to a finite temperature symmetry breaking, with a charge modulation \Delta\rho cos(2k_Fr+\phi) accompanied by a lattice distortion (the famous Peirls transition) or a spin modulation, depending which electron-electron interaction or electron-phonon interaction is dominant. In 2D, you can have partial nesting of the Fermi Surface, leading to the same phenoma . But strong electron electron interaction can also impose some charge ordering (the famous stripes inn High Tc)  Presently the question to know wether the nesting or the correlation is responsible for CDW in High Tc is still debated, and even 2D systems very weel known like NbSe2, there is a controversy.

Hi Berthier, thank you very much for your reply. However, I think your reply is beyond my knowledge scope. Could you explain some more? What do the SDW and CDW physical models look like? What the effects of the presence of SDW and CDW on physical properties, such as electric and thermal conductivities, magnetism and even electronic structure?

Usually, the electonic density or the spin density, is supposed to be invariant by a lattice translation. A CDW introduces a new periodicity which can be commensurate  (the wave number is simple fraction) or incommensurate (the periodicity never matches that of the lattice, all the cells see a different charge or spin density, and the wave vector is a irrationnal number). The onset of a charge (spin) density wave opens a gap at the Fermi surface, which can be complete (metal-insulator transition) or partial. More information later.

Hi Berthier, you mentioned the phrase - Fermi surface nesting. Could you give some more introduction about Fermi surface nesting. How does it look like? In general, what kind of materials can have a Fermi surface nesting. Does it influence the stability of a phase, such as magnetic phase? 

Following up on the other terms that you asked about, one should note that “spin” is often used as a catch-all shorthand for magnetic moments that may be mixtures of spin and orbital moments. The precise meaning of many of these terms depends on the context in which they are used.  With that in mind:

 “Spin fluctuation” can refer to any motion (or dynamic behavior)  of spins either singly or collectively, and the fluctuations could arise from thermal energy, or quantum zero point motion, etc.

“Spin excitation” usually refers to a magnetic excitation out of a ground state – this can be an excitation associated with a single magnetic ion or a collective excitation from a many-particle ground state.

“Spin gap” refers to an energy gap in the spin excitation spectrum.

“Spin wave” usually refers to a collective magnetic excitation from a magnetically ordered ground state – the magnetic analogue of a lattice wave, (i.e. phonon).

Switching away from magnetism, “charge order” usually refers to a situation where the same atomic species might occur in a material with different valences, and the atoms with one or the other integer valences are arranged in an ordered pattern.  To illustrate by  example, say one has a chemical formula of an insulating crystalline material A2[X] where  A is a specific ion and the [X] represents something that has a net charge equal to an odd number of electrons.  Let’s use 3 electrons as an example.  Then the nominal charge on the “A” species is 1.5.  If the electrons are localized on the A ion then overall charge neutrality implies that half of the A ions are in a state with valence +1, and the other half have valence +2.  If the crystal forms such  that there are two crystallographically inequivalent  sites for A ions, and one always occurs with valence +1, and the other with valence +2, the system is charge ordered.  (One could imagine an alternative in which the valence on a given A atom varies randomly from site to site.

One should emphasize once again that context is important, and the papers that you are reading might be using these terms somewhat differently than what is written above.

Here are references of some papers of my own on CDW in transition metals dichalcogehides, blue bronzes and more recent ones in High T_c. Inside you will find basic references;

Claude Berthier, Denis Jérome, Philippe Molinié, Jean Rouxel

Charge Density Waves in layered structure: a N.M.R. study on a 2H-NbSe2 single crystal Solid State Comm., 19, 131-5 (1976)

Claude Berthier, Philippe Molinié, Denis Jérome

Evidence for a connection between charge density waves and the pressure enhancement of superconductivity in 2H-NbSe2 Solid State Comm., 18, 1393-5 (1976)

Claude Berthier,  Denis Jérome, Philippe Molinié

         NMR study on a 2H-NbSe2 single crystal : a microscopic investigation of the charge density waves state  J. Phys.C.: Solid State physics 11, 797-814 (1978)

Pierre Ségransan, Andras Janossy, Claude Berthier, Jacques Marcus, Patrick. Butaud

N.M.R. evidence of the Frölich mode in Rb0.30MoO3  Phys. Rev. Lett., 56 (1986) 1854-7.

Andras Janossy, Claude Berthier, Pierre Ségransan, Patrick Butaud

  Measurement of the Sliding Charge Density Wave Phase velocity in Rb0.3MoO3    Phys. Rev. Lett., 59 (1987) 2348-51

Katsuaki Kodama, Masashi Takigawa, Mladen Horvati, Claude Berthier, H. Kageyama, Yutaka Ueda, Shin Miyahara, Federico Becca and Frédéric Mila

 Observation of a magnetic superstructure in the 1/8 magnetization plateau phase of SrCu2(BO3)2 Science, 298, 395-9 (2002)

G. Koutroulakis, Vesna  F. Mitrović, Mladen  Horvatić, Claude Berthier, Gérard Lapertot, Jacques Flouquet

Field Dependence of the Ground State in the Exotic Superconductor CeCoIn5: a Nuclear Magnetic Resonance Investigation, Phys. Rev. Lett. 101, 047004 (2008)

Ko-ichi Hiraki, M.  Kitahara, T. Takahashi, Hadrien Mayaffre,  Mladen  Horvatić, Claude Berthier, Shinya Uji, H. Tanaka, B. Zhou, A. Kobayashi, and H. Kobayashi

Evidence of Charge Disproportionationin l-Type BETS Based Organic Superconductors

N. Hardy, Ruixing Liang, Doug. A. Bonn & Marc-Henri Julien

Magnetic-field-induced charge-stripe order in the high-temperature superconductor YBa2Cu3Oy

Nature 477, 191 (2011)

Masashi Takigawa, Mladen Horvatić, Takeshi Waki, Steffen Krämer, Claude Berthier, Florence Lévy-Bertrand, Ilya Sheikin, Hiroshi Kageyama, Yutaka Ueda, Frederic Mila

Incomplete devil's staircase in the magnetization curve of SrCu2(BO3)2

            PRL 110, 067210 (2013).

T. Wu, Hadrien Mayaffre, Steffen Krämer, Mladen  Horvatić, Claude Berthier, W.N. Hardy, R. Liang, D.A. Bonn, Marc-Henri Julien

Short-range charge order reveals the role of disorder in the pseudogap state of high-Tc superconductors

Nat. Commun. 6 6438 (2015). http://dx.doi.org/10.1038/ncomms7438

Tao Wu, Hadrien Mayaffre, Steffen Steffen Krämer, Mladen Morvatić, Claude Berthier, Philip L. Kuhns, Arneil P. Reyes, Ruixing Liang, WN Hardy, DA Bonn & Marc-Henri Julien

Emergence of charge order from the vortex state of a high-temperature superconductor

Nature Communications 4, 2113 (2013

Y. Ihara, Mingee Jeong, Hadrien Mayaffre, Claude Berthier, Mladen Horvatić, H. Seki, and A. Kawamoto

13C NMR study of the charge ordered state near the superconducting transition in the organic superconductor ''-(BEDT-TTF)4[(H3O)Ga(C2O4)3]‧C6H5NO2

Phys. Rev. B 90, 121106(R) (2014)

Hi Stephen, thank you very much for your comprehensive explanation. These phrases become more clear for me now.

Berthier, thank you very much again.