Quasiparticle

In physics, quasiparticles and collective excitations (which are closely related) are emergent phenomena that occur when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in free space. For example, as an electron travels through a semiconductor, its motion is disturbed in a complex way by its interactions with all of the other electrons and nuclei; however it approximately behaves like an electron with a different mass (effective mass) traveling unperturbed through free space. This "electron" with a different mass is called an "electron quasiparticle".^[1] In another example, the aggregate motion of electrons in the valence band of a semiconductor is the same as if the semiconductor instead contained positively charged quasiparticles called holes. Other quasiparticles or collective excitations include phonons (particles derived from the vibrations of atoms in a solid), plasmons (particles derived from plasma oscillations), and many others.

These particles are typically called "quasiparticles" if they are related to fermions (like electrons and holes), and called "collective excitations" if they are related to bosons (like phonons and plasmons),^[1] although the precise distinction is not universally agreed upon.^[2]

The quasiparticle concept is most important in condensed matter physics since it is one of the few known ways of simplifying the quantum mechanical many-body problem.

Overview

General introduction

Solids are made of only three kinds of particles: Electrons, protons, and neutrons. Quasiparticles are none of these; instead, they are an emergent phenomenon that occurs inside the solid. Therefore, while it is quite possible to have a single particle (electron or proton or neutron) floating in space, a quasiparticle can instead only exist inside the solid.

Motion in a solid is extremely complicated: Each electron and proton is pushed and pulled (by Coulomb's law) by all the other electrons and protons in the solid (which may themselves be in motion). It is these strong interactions that make it very difficult to predict and understand the behavior of solids (see many-body problem). On the other hand, the motion of a non-interacting particle is quite simple: In classical mechanics, it would move in a straight line, and in quantum mechanics, it would move in a superposition of plane waves. This is the motivation for the concept of quasiparticles: The complicated motion of the actual particles in a solid can be mathematically transformed into the much simpler motion of imagined quasiparticles, which behave more like non-interacting particles.

In summary, quasiparticles are a mathematical tool for simplifying the description of solids. They are not "real" particles inside the solid. Instead, saying "A quasiparticle is present" or "A quasiparticle is moving" is shorthand for saying "A large number of electrons and nuclei are moving in a specific coordinated way."

Relation to many-body quantum mechanics

Any system, no matter how complicated, has a ground state along with an infinite series of higher-energy excited states.

The principal motivation for quasiparticles is that it is almost impossible to directly describe every particle in a macroscopic system. For example, a barely-visible (0.1mm) grain of sand contains around 10¹⁷ atoms and 10¹⁸ electrons. Each of these attracts or repels every other by Coulomb's law. In quantum mechanics, a system is described by a wavefunction, which, if the particles are interacting (as they are in our case), depends on the position of every particle in the system. So, each particle adds three independent variables to the wavefunction, one for each coordinate needed to describe the position of that particle. Because of this, directly approaching the many-body problem of 10¹⁸ interacting electrons by straightforwardly trying to solve the appropriate Schrödinger equation is impossible in practice, since it amounts to solving a partial differential equation not just in three dimensions, but in 3x10¹⁸ dimensions – one for each component of the position of each particle.

One simplifying factor is that the system as a whole, like any quantum system, has a ground state and various excited states with higher and higher energy above the ground state. In many contexts, only the "low-lying" excited states, with energy reasonably close to the ground state, are relevant. This occurs because of the Boltzmann distribution, which implies that very-high-energy thermal fluctuations are unlikely to occur at any given temperature.

Quasiparticles and collective excitations are a type of low-lying excited state. For example, a crystal at absolute zero is in the ground state, but if one phonon is added to the crystal (in other words, if the crystal is made to vibrate slightly at a particular frequency) then the crystal is now in a low-lying excited state. The single phonon is called an elementary excitation. More generally, low-lying excited states may contain any number of elementary excitations (for example, many phonons, along with other quasiparticles and collective excitations).^[3]

When the material is characterized as having "several elementary excitations", this statement presupposes that the different excitations can be combined together. In other words, it presupposes that the excitations can coexist simultaneously and independently. This is never exactly true. For example, a solid with two identical phonons does not have exactly twice the excitation energy of a solid with just one phonon, because the crystal vibration is slightly anharmonic. However, in many materials, the elementary excitations are very close to being independent. Therefore, as a starting point, they are treated as free, independent entities, and then corrections are included via interactions between the elementary excitations, such as "phonon-phonon scattering".

Therefore, using quasiparticles / collective excitations, instead of analyzing 10¹⁸ particles, one needs to deal with only a handful of somewhat-independent elementary excitations. It is, therefore, a very effective approach to simplify the many-body problem in quantum mechanics. This approach is not useful for all systems, however: In strongly correlated materials, the elementary excitations are so far from being independent that it is not even useful as a starting point to treat them as independent.

Distinction between quasiparticles and collective excitations

Usually, an elementary excitation is called a "quasiparticle" if it is a fermion and a "collective excitation" if it is a boson.^[1] However, the precise distinction is not universally agreed upon.^[2]

There is a difference in the way that quasiparticles and collective excitations are intuitively envisioned.^[2] A quasiparticle is usually thought of as being like a dressed particle: It is built around a real particle at its "core", but the behavior of the particle is affected by the environment. A standard example is the "electron quasiparticle": A real electron particle, in a crystal, behaves as if it had a different mass. On the other hand, a collective excitation is usually imagined to be a reflection of the aggregate behavior of the system, with no single real particle at its "core". A standard example is the phonon, which characterizes the vibrational motion of every atom in the crystal.

However, these two visualizations leave some ambiguity. For example, a magnon in a ferromagnet can be considered in one of two perfectly equivalent ways: (a) as a mobile defect (a misdirected spin) in a perfect alignment of magnetic moments or (b) as a quantum of a collective spin wave that involves the precession of many spins. In the first case, the magnon is envisioned as a quasiparticle, in the second case, as a collective excitation. However, both (a) and (b) are equivalent and correct descriptions. As this example shows, the intuitive distinction between a quasiparticle and a collective excitation is not particularly important or fundamental.

The problems arising from the collective nature of quasiparticles have also been discussed within the philosophy of science, notably in relation to the identity conditions of quasiparticles and whether they should be considered "real" by the standards of, for example, entity realism.^[4][5]

Effect on bulk properties

By investigating the properties of individual quasiparticles, it is possible to obtain a great deal of information about low-energy systems, including the flow properties and heat capacity.

In the heat capacity example, a crystal can store energy by forming phonons, and/or forming excitons, and/or forming plasmons, etc. Each of these is a separate contribution to the overall heat capacity.

History

The idea of quasiparticles originated in Lev Landau's theory of Fermi liquids, which was originally invented for studying liquid helium-3. For these systems a strong similarity exists between the notion of quasiparticle and dressed particles in quantum field theory. The dynamics of Landau's theory is defined by a kinetic equation of the mean-field type. A similar equation, the Vlasov equation, is valid for a plasma in the so-called plasma approximation. In the plasma approximation, charged particles are considered to be moving in the electromagnetic field collectively generated by all other particles, and hard collisions between the charged particles are neglected. When a kinetic equation of the mean-field type is a valid first-order description of a system, second-order corrections determine the entropy production, and generally take the form of a Boltzmann-type collision term, in which figure only "far collisions" between virtual particles. In other words, every type of mean-field kinetic equation, and in fact every mean-field theory, involves a quasiparticle concept.

Examples of quasiparticles and collective excitations

This section contains examples of quasiparticles and collective excitations. The first subsection below contains common ones that occur in a wide variety of materials under ordinary conditions; the second subsection contains examples that arise only in special contexts.

More common examples

• In solids, an electron quasiparticle is an electron as affected by the other forces and interactions in the solid. The electron quasiparticle has the same charge and spin as a "normal" (elementary particle) electron, and like a normal electron, it is a fermion. However, its mass can differ substantially from that of a normal electron; see the article effective mass.^[1] Its electric field is also modified, as a result of electric field screening. In many other respects, especially in metals under ordinary conditions, these so-called Landau quasiparticles closely resemble familiar electrons; as Crommie's "quantum corral" showed, an STM can clearly image their interference upon scattering.
• A hole is a quasiparticle consisting of the lack of an electron in a state; it is most commonly used in the context of empty states in the valence band of a semiconductor.^[1] A hole has the opposite charge of an electron.
• A phonon is a collective excitation associated with the vibration of atoms in a rigid crystal structure. It is a quantum of a sound wave.
• A magnon is a collective excitation^[1] associated with the electrons' spin structure in a crystal lattice. It is a quantum of a spin wave.
• A roton is a collective excitation associated with the rotation of a fluid (often a superfluid). It is a quantum of a vortex.
• In materials, a photon quasiparticle is a photon as affected by its interactions with the material. In particular, the photon quasiparticle has a modified relation between wavelength and energy (dispersion relation), as described by the material's index of refraction. It may also be termed a polariton, especially near a resonance of the material. For example, an exciton-polariton is a superposition of an exciton and a photon; a phonon-polariton is a superposition of a phonon and a photon.
• A plasmon is a collective excitation, which is the quantum of plasma oscillations (wherein all the electrons simultaneously oscillate with respect to all the ions).
• A polaron is a quasiparticle which comes about when an electron interacts with the polarization of its surrounding ions.
• An exciton is an electron and hole bound together.
• A plasmariton is a coupled optical phonon and dressed photon consisting of a plasmon and photon.

More specialized examples

• Composite fermions arise in a two-dimensional system subject to a large magnetic field, most famously those systems that exhibit the fractional quantum Hall effect.^[6] These quasiparticles are quite unlike normal particles in two ways. First, their charge can be less than the electron charge e. In fact, they have been observed with charges of e/3, e/4, e/5, and e/7.^[7] Second, they can be anyons, an exotic type of particle that is neither a fermion nor boson.^[8]
• Stoner excitations in ferromagnetic metals
• Bogoliubov quasiparticles in superconductors. Superconductivity is carried by Cooper pairs—usually described as pairs of electrons—that move through the crystal lattice without resistance. A broken Cooper pair is called a Bogoliubov quasiparticle.^[9] It differs from the conventional quasiparticle in metal because it combines the properties of a negatively charged electron and a positively charged hole (an electron void). Physical objects like impurity atoms, from which quasiparticles scatter in an ordinary metal, only weakly affect the energy of a Cooper pair in a conventional superconductor. In conventional superconductors, interference between Bogoliubov quasiparticles is tough for an STM to see. Because of their complex global electronic structures, however, high-Tc cuprate superconductors are another matter. Thus Davis and his colleagues were able to resolve distinctive patterns of quasiparticle interference in Bi-2212.^[10]
• A Majorana fermion is a particle which equals its own antiparticle, and can emerge as a quasiparticle in certain superconductors, or in a quantum spin liquid.^[11]
• Magnetic monopoles arise in condensed matter systems such as spin ice and carry an effective magnetic charge as well as being endowed with other typical quasiparticle properties such as an effective mass. They may be formed through spin flips in frustrated pyrochlore ferromagnets and interact through a Coulomb potential.
• Skyrmions
• Spinon is represented by quasiparticle produced as a result of electron spin-charge separation, and can form both quantum spin liquid and strongly correlated quantum spin liquid in some minerals like Herbertsmithite.^[12]

See also

• Fractionalization
• List of quasiparticles
• Mean field theory
• Pseudoparticle

References

1. E. Kaxiras, Atomic and Electronic Structure of Solids, ISBN 0-521-52339-7, pages 65–69.
2. A guide to Feynman diagrams in the many-body problem, by Richard D. Mattuck, p10. "As we have seen, the quasiparticle consists of the original real, individual particle, plus a cloud of disturbed neighbors. It behaves very much like an individual particle, except that it has an effective mass and a lifetime. But there also exist other kinds of fictitious particles in many-body systems, i.e. 'collective excitations'. These do not center around individual particles, but instead involve collective, wavelike motion of all the particles in the system simultaneously."
3. Principles of Nanophotonics by Motoichi Ohtsu, p205 google books link
4. A. Gelfert, 'Manipulative Success and the Unreal', International Studies in the Philosophy of Science Vol. 17, 2003, 245–263
5. B. Falkenburg, Particle Metaphysics (The Frontiers Collection), Berlin: Springer 2007, esp. pp. 243–46
6. Physics Today Article
7. Cosmos magazine June 2008 Archived 9 June 2008 at the Wayback Machine.
8. Nature article
9. "Josephson Junctions". Science and Technology Review. Lawrence Livermore National Laboratory.
10. J. E. Hoffman; McElroy, K; Lee, DH; Lang, KM; Eisaki, H; Uchida, S; Davis, JC; et al. (2002). "Imaging Quasiparticle Interference in Bi2Sr2CaCu2O8+". Science. 297 (5584): 1148–51. arXiv:cond-mat/0209276. Bibcode:2002Sci...297.1148H. doi:10.1126/science.1072640. PMID 12142440.
11. Banerjee, A.; Bridges, C. A.; Yan, J.-Q.; et al. (4 April 2016). "Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet". Nature Materials. doi:10.1038/nmat4604. (subscription required (help)).
12. Shaginyan, V. R.; et al. (2012). "Identification of Strongly Correlated Spin Liquid in Herbertsmithite". EPL. 97 (5): 56001. arXiv:1111.0179. Bibcode:2012EL.....9756001S. doi:10.1209/0295-5075/97/56001.

Further reading

• L. D. Landau, Soviet Phys. JETP. 3:920 (1957)
• L. D. Landau, Soviet Phys. JETP. 5:101 (1957)
• A. A. Abrikosov, L. P. Gor'kov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (1963, 1975). Prentice-Hall, New Jersey; Dover Publications, New York.
• D. Pines, and P. Nozières, The Theory of Quantum Liquids (1966). W.A. Benjamin, New York. Volume I: Normal Fermi Liquids (1999). Westview Press, Boulder.
• J. W. Negele, and H. Orland, Quantum Many-Particle Systems (1998). Westview Press, Boulder
• Amusia, M., Popov, K., Shaginyan, V., Stephanovich, V. (2014). Theory of Heavy-Fermion Compounds - Theory of Strongly Correlated Fermi-Systems. Springer. ISBN 978-3-319-10825-4.

External links

• PhysOrg.com – Scientists find new 'quasiparticles'
• Curious 'quasiparticles' baffle physicists by Jacqui Hayes, Cosmos 6 June 2008. Accessed June 2008