Quasiparticle

	Quasiparticle
Classification:	List of quasi-particles

Quasi-particles (from Lat. Quas (i) "like", "something like") - a concept in quantum mechanics , the introduction of which allows you to significantly simplify the description of complex quantum systems with interaction, such as solids and quantum liquids.

For example, an extremely complex description of the motion of electrons in semiconductors can be simplified by introducing a quasiparticle called conduction electron , differing from an electron by mass and moving in free space. Phonons are used to describe the oscillations of atoms in the lattice sites in the theory of condensed matter, and magnons are used to describe the propagation of elementary magnetic excitations in a system of interacting spins .

Content

Introduction

The idea of using quasiparticles was first proposed by L. D. Landau in the theory of Fermi-liquid to describe liquid helium-3 , later it was used in the theory of condensed matter. It is impossible to describe the state of such systems directly by solving the Schrödinger equation with about ¹⁰²³ interacting particles. It is possible to circumvent this difficulty by reducing the problem of the interaction of particles to a simpler problem with noninteracting quasiparticles.

Quasi-particles in a Fermi liquid

The introduction of quasiparticles for a Fermi liquid is made by a smooth transition from the excited state of an ideal system (without interaction between particles), obtained from the ground one, with the distribution function $n_{0}({\vec {p}})$ ${\ displaystyle n_ {0} ({\ vec {p}})}$ by adding a particle with momentum ${\vec {p}}$ ${\ displaystyle {\ vec {p}}}$ , adiabatic inclusion of interaction between particles. With such an inclusion, an excited state of a real Fermi liquid arises with the same momentum, since it persists in the collision of particles. As the interaction is activated, the added particle draws the particles surrounding it into motion, forming a perturbation. Such a disturbance is called a quasi-particle. The state of the system thus obtained corresponds to the real ground state plus a quasi-particle with momentum ${\vec {p}}$ ${\ displaystyle {\ vec {p}}}$ and the energy corresponding to this disturbance. In such a transition, the role of gas particles (in the absence of interaction) passes to elementary excitations (quasi-particles), the number of which coincides with the number of particles and which, like particles, obey the Fermi-Dirac statistics .

Quasi-particles in solids

Phonon as a quasi-particle

Describing the state of solids by directly solving the Schrödinger equation for all particles is practically impossible because of the large number of variables and the complexity of taking into account the interaction between particles. Such a description can be simplified by introducing quasiparticles — elementary excitations relative to a certain ground state. Often, taking into account only the lowest energy excitations relative to this state is sufficient to describe the system, since, according to the Boltzmann distribution , states with large energies are less likely to be given. Let us consider an example of the use of quasiparticles for the description of atomic oscillations in lattice sites.

An example of low-energy excitations is a crystal lattice at absolute zero , when an elementary perturbation of a certain frequency, that is, a phonon, is added to the ground state in which there are no oscillations in the lattice. It happens that the state of the system is characterized by several elementary excitations, and these excitations, in turn, can exist independently of each other, in which case this state is interpreted by a system of noninteracting phonons. However, it is not always possible to describe the state by noninteracting quasiparticles due to the anharmonic oscillation in the crystal. However, in many cases, elementary excitations can be considered as independent. Thus, it can be approximated that the crystal energy associated with the oscillation of atoms in the lattice sites is equal to the sum of the energy of a certain ground state and the energies of all phonons.

Quantization of oscillations by the example of a phonon

Consider a scalar model of the crystal lattice, according to which atoms oscillate along one direction. Using the basis of plane waves, we write the expression for the displacements of atoms in a node:

u_{n}(t)=\sum _{\vec {k}}Q_{\vec {k}}(t)\phi _{\vec {k}}({\vec {r}}_{n}),

{\ displaystyle u_ {n} (t) = \ sum _ {\ vec {k}} Q _ {\ vec {k}} (t) \ phi _ {\ vec {k}} ({\ vec {r}} _ {n}),}

\phi _{\vec {k}}({\vec {r}}_{n})={\frac {1}{\sqrt {N}}}e^{i{\vec {k}}{\vec {r}}_{n}}.

{\ displaystyle \ phi _ {\ vec {k}} ({\ vec {r}} _ {n}) = {\ frac {1} {\ sqrt {N}}} e ^ {i {\ vec {k }} {\ vec {r}} _ {n}}.}

In this form $Q_{\vec {k}}$ ${\ displaystyle Q _ {\ vec {k}}}$ called generalized coordinates. Then the Lagrangian of the system:

L=\sum _{n}{\frac {m{\dot {u}}_{n}^{2}}{2}}-{\frac {1}{2}}\sum _{n,n^{'}}A({\vec {r}}_{n}-{\vec {r}}_{n^{'}})u_{n}u_{n^{'}}

{\ displaystyle L = \ sum _ {n} {\ frac {m {\ dot {u}} _ {n} ^ {2}} {2}} - {\ frac {1} {2}} \ sum _ {n, n ^ {'}} A ({\ vec {r}} _ {n} - {\ vec {r}} _ {n ^ {'}}) u_ {n} u_ {n ^ {'} }}

put it in terms $Q_{\vec {k}}$ ${\ displaystyle Q _ {\ vec {k}}}$ as:

L={\frac {m}{2}}\sum {\vec {k}}({\dot {Q}}_{\vec {k}}^{*}{\dot {Q}}_{\vec {k}}-\omega _{\vec {k}}^{2}Q_{\vec {k}}^{*}Q_{\vec {k}}).

{\ displaystyle L = {\ frac {m} {2}} \ sum {\ vec {k}} ({\ dot {Q}} _ {\ vec {k}} ^ {*} {\ dot {Q} } _ {\ vec {k}} - \ omega _ {\ vec {k}} ^ {2} Q _ {\ vec {k}} ^ {*} Q _ {\ vec {k}}).}

Hence the canonical impulse and the Hamiltonian are expressed:

P_{\vec {k}}={\frac {\delta L}{\delta {\dot {Q}}_{\vec {k}}}}=m{\dot {Q}}_{\vec {k}}^{*},

{\ displaystyle P _ {\ vec {k}} = {\ frac {\ delta L} {\ delta {\ dot {Q}} _ {\ vec {k}}}} = m {\ dot {Q}} _ {\ vec {k}} ^ {*},}

H=\sum _{\vec {\vec {k}}}P_{\vec {k}}{\dot {Q}}_{\vec {k}}-L={\frac {1}{2m}}\sum _{\vec {k}}(P_{\vec {k}}P_{\vec {k}}^{*}+m^{2}\omega _{\vec {k}}^{2}Q_{\vec {k}}^{*}Q_{\vec {k}}).

{\ displaystyle H = \ sum _ {\ vec {\ vec {k}}} P _ {\ vec {k}} {\ dot {Q}} _ {\ vec {k}} - L = {\ frac {1 } {2m}} \ sum _ {\ vec {k}} (P _ {\ vec {k}} P _ {\ vec {k}} ^ {*} + m ^ {2} \ omega _ {\ vec {k }} ^ {2} Q _ {\ vec {k}} ^ {*} Q _ {\ vec {k}}).}

The action quantization is performed by the requirement of operator commutation rules for the generalized coordinate and momentum ( $\hbar =1$ ${\ displaystyle \ hbar = 1}$ ):

[Q_{\vec {k}},P_{{\vec {k}}^{'}}]=i\delta _{{\vec {k}},{\vec {k}}^{'}},

{\ displaystyle [Q _ {\ vec {k}}, P _ {{\ \ vec {k}} ^ {'}}] = i \ delta _ {{\ vec {k}}, {\ vec {k}} ^ {'}},}

[Q_{\vec {k}},Q_{{\vec {k}}^{'}}]=[P_{\vec {k}},P_{{\vec {k}}^{'}}]=0.

{\ displaystyle [Q _ {\ vec {k}}, Q _ {{\ \ vec {k}} ^ {'}}] = [P _ {\ vec {k}}, P _ {{\ vec {k}} ^ { '}}] = 0.}

To go to the phonon representation, the secondary quantization language is used, defining the birth operators $a_{\vec {k}}^{+}$ ${\ displaystyle a _ {\ vec {k}} ^ {+}}$ and destruction $a_{\vec {k}}$ ${\ displaystyle a _ {\ vec {k}}}$ quantum phonon field:

[a_{\vec {k}},a_{{\vec {k}}^{'}}^{+}]=i\delta _{{\vec {k}},{\vec {k}}^{'}}\,\,\,\,\,\,\,[a_{\vec {k}},a_{{\vec {k}}^{'}}]=0.

{\ displaystyle [a _ {\ vec {k}}, a _ {{\ \ vec {k}} ^ {'}} ^ {+}] = i \ delta _ {{\ \ vec {k}}, {\ vec { k}} ^ {'}} \, \, \, \, \, \, \, [a _ {\ vec {k}}, a _ {{\ vec {k}} ^ {'}}] = 0. }

By direct calculation, you can verify that the required switching rules are fulfilled for the operators:

Q_{\vec {k}}={\frac {1}{\sqrt {2m\omega _{\vec {k}}}}}(a_{\vec {k}}^{+}e^{i\omega t}+a_{-{\vec {k}}}e^{-i\omega _{\vec {k}}t}),

{\ displaystyle Q _ {\ vec {k}} = {\ frac {1} {\ sqrt {2m \ omega _ {\ vec {k}}}}} (a _ {\ vec {k}} ^ {+} e ^ {i \ omega t} + a _ {- {\ vec {k}}} e ^ {- i \ omega _ {\ vec {k}} t}),}

P_{\vec {k}}=i{\sqrt {\frac {\omega _{\vec {k}}m}{2}}}(a_{-{\vec {k}}}^{+}e^{i\omega t}-a_{\vec {k}}e^{-i\omega _{\vec {k}}t}).

{\ displaystyle P _ {\ vec {k}} = i {\ sqrt {\ frac {\ omega _ {\ vec {k}} m} {2}}} (a _ {- {\ vec {k}}} ^ {+} e ^ {i \ omega t} -a _ {\ vec {k}} e ^ {- i \ omega _ {\ vec {k}} t}).}

Replacing the sign of complex conjugation $Q_{\vec {k}}^{*}$ ${\ displaystyle Q _ {\ vec {k}} ^ {*}}$ on $Q_{\vec {k}}^{+}$ ${\ displaystyle Q _ {\ vec {k}} ^ {+}}$ and taking into account that energy is an even function of a quasimomentum, $\omega _{\vec {k}}=\omega _{-{\vec {k}}}$ ${\ displaystyle \ omega _ {\ vec {k}} = \ omega _ {- {\ vec {k}}}}$ (from homogeneity), we obtain expressions for the kinetic and potential parts of the Hamiltonian:

K={\frac {1}{2m}}\sum _{\vec {k}}P_{\vec {k}}P_{-{\vec {k}}}=-{\frac {1}{4}}\sum _{\vec {k}}\omega _{\vec {k}}(a_{-{\vec {k}}}^{+}-a_{\vec {k}})(a_{\vec {k}}^{+}-a_{-{\vec {k}}}),

{\ displaystyle K = {\ frac {1} {2m}} \ sum _ {\ vec {k}} P _ {\ vec {k}} P _ {- {\ vec {k}}} = - {\ frac { 1} {4}} \ sum _ {\ vec {k}} \ omega _ {\ vec {k}} (a _ {- {\ vec {k}}} ^ {+} - a _ {\ vec {k} }) (a _ {\ vec {k}} ^ {+} - a _ {- {\ vec {k}}}),}

H={\frac {m\omega _{\vec {k}}^{2}}{2}}\sum _{\vec {k}}Q_{\vec {k}}Q_{-{\vec {k}}}={\frac {1}{4}}\sum _{\vec {k}}\omega _{\vec {k}}(a_{-{\vec {k}}}^{+}-a_{\vec {k}})(a_{\vec {k}}^{+}-a_{-{\vec {k}}}).

{\ displaystyle H = {\ frac {m \ omega _ {\ vec {k}} ^ {2}} {2}} \ sum _ {\ vec {k}} Q _ {\ vec {k}} Q _ {- {\ vec {k}}} = {\ frac {1} {4}} \ sum _ {\ vec {k}} \ omega _ {\ vec {k}} (a _ {- {\ vec {k}} } ^ {+} - a _ {\ vec {k}}) (a _ {\ vec {k}} ^ {+} - a _ {- {\ vec {k}}}).}

Then the Hamiltonian takes the form:

H=\sum _{\vec {k}}\omega _{\vec {k}}(a_{\vec {k}}^{+}a_{\vec {k}}+{\frac {1}{2}}).

{\ displaystyle H = \ sum _ {\ vec {k}} \ omega _ {\ vec {k}} (a _ {\ vec {k}} ^ {+} a _ {\ vec {k}} + {\ frac {1} {2}}).}

Otherwise, you can rewrite:

H=\sum _{\vec {k}}E_{\vec {k}}(n_{\vec {k}}+{\frac {1}{2}}),

{\ displaystyle H = \ sum _ {\ vec {k}} E _ {\ vec {k}} (n _ {\ vec {k}} + {\ frac {1} {2}}),}

Where

n_{\vec {k}}=a_{\vec {k}}^{+}a_{\vec {k}}

{\ displaystyle n _ {\ vec {k}} = a _ {\ vec {k}} ^ {+} a _ {\ vec {k}}}

- operator of the number of particles, phonons,

E_{\vec {k}}=\omega _{\vec {k}}

{\ displaystyle E _ {\ vec {k}} = \ omega _ {\ vec {k}}}

- phonon energy with momentum

{\vec {k}}.

{\ displaystyle {\ vec {k}}.}

Such a description of oscillations in a crystal is called a harmonic approximation. It corresponds only to the consideration of quadratic terms in displacements in the Hamiltonian.

Quasiparticles in a ferromagnet, magnons

In the case of a ferromagnet , with absolute zero temperature, all the spins are aligned along one direction. This arrangement of spins corresponds to the ground state. If one of the spins is deflected from the given direction and the system is given to itself, a wave will begin to propagate. The energy of this wave will be equal to the excitation energy of the crystal, associated with a change in the orientation of the spin of an atom. This energy can be considered as the energy of a certain particle, which is called a magnon.

If the energy of a ferromagnet associated with the spin deviation is small, then it can be represented as the sum of the energies of individual propagating spin waves or, to put it differently, as the sum of the energies of the magnons.

Magnons, like phonons, obey Bose-Einstein statistics.

Properties

Quasiparticles are characterized by the vector ${\vec {p}}$ ${\ displaystyle {\ vec {p}}}$ whose properties resemble an impulse, it is called a quasi-impulse.
The energy of a quasiparticle, in contrast to the energy of an ordinary particle, has a different dependence on momentum.
Quasiparticles can interact with each other, as well as with ordinary particles.
May have charge and / or spin.
Quasi-particles with integer spin values obey Bose-Einstein statistics, with half-integer Fermi-Dirac .

Comparing Quasi-Particles with Ordinary Particles

There are a number of similarities and differences between quasiparticles and ordinary elementary particles . In many field theories (in particular, in conformal field theory ), there are generally no differences between particles and quasiparticles.

Similarities

Like an ordinary particle, a quasiparticle can be more or less localized in space and retain its localization in the process of movement.
Quasiparticles may collide and / or otherwise interact. In the collision of low-energy quasiparticles, mechanical laws of conservation of quasi-momentum and energy are fulfilled. Quasiparticles can also interact with ordinary particles (for example, with photons ).
For quasiparticles with a quadratic dispersion law (that is, energy is proportional to the square of the pulse), we can introduce the concept of effective mass . The behavior of such a quasiparticle will be very similar to the behavior of ordinary particles.

Differences

Unlike ordinary particles, which exist by themselves, including in empty space, quasi-particles cannot exist outside the environment, the oscillations of which they are.
In collisions, for many quasiparticles, the law of conservation of the quasimomentum is satisfied with an accuracy of the reciprocal lattice vector .
The law of dispersion of ordinary particles is a given, which cannot be changed. The law of dispersion of quasiparticles arises dynamically, and therefore can have the most intricate appearance.
Quasiparticles can have a fractional electric charge or a magnetic charge.

Other quasi-particles

Conduction electron - has the same charge and spin as the "normal" electron, but differs in mass.
A hole is an unfilled valence bond that manifests itself as a positive charge, in absolute value equal to the charge of an electron.
Roton - collective excitement associated with the vortex motion in the liquid.
A polaron is a quasi-particle corresponding to the polarization associated with the electron motion due to the interaction of the electron with the crystal lattice.
Plasmon - is a collective oscillation of electrons in a plasma.

Literature

Soloviev V.G. Theory of the atomic nucleus: Quasiparticles and phonons. - Energoatomizdat, 1989. - 304 p. - ISBN 5-283-03914-5 .
Kaganov M.I. "Quasi-particle". What it is?. - Knowledge, 1971. - 75 p. - 12 500 copies

Links

Quasiparticles