Quasiparticle

Prerequisites: Canonical Quantum Mechanics. Full path (17 pages)

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Intuitive A story about quasiparticles | Gravity and Levity

Why is it interesting?
Intuitive
Concrete
Abstract

Why is it interesting?

The similarities between the vacuum of space and low-temperature phases of matter are legendary in physics. Not only are phases static, uniform quantum states, but their most subtle internal motions are physically indistinguishable from elementary particles very generally. This is one of the most astonishing facts in science, and something students always find upsetting and difficult to believe. But they eventually become convinced after looking at enough experiments, for the evidence is plenti and consistent. In fact, the more one studies the mathematical descriptions of cold phases, the more accustomed one gets to using the parallel terminologies of matter and space interchangeably. Thus instead of a phase of matter we speak of a vacuum. Instead of particles we speak of excitations. Instead of collective motions we speak of quasiparticles. The prefix "quasi" turns out to be a vestige of the historical battles over the physical meaning of these objects and conveys no meaning. In private conversations one drops the pretense and refers to the objects as particle.

page 112 in "A different Universe" by Robert Laughlin

Intuitive

A solid contains something like a hundred thousand billion billion electrons and nuclei per cubic centimetre, all interacting with each other. Describing every particle is hopeless. The useful observation is that the low-energy excitations of such a system often behave like a dilute gas of weakly interacting particles. Each of these excitations has a definite energy and momentum, and often a charge and a spin. These particle-like excitations are called quasiparticles.

Some quasiparticles are dressed versions of real particles. An electron moving through a metal pushes the other electrons and the ions around it, and this cloud moves along with it. The electron together with its cloud behaves like a particle with a different mass, the effective mass. A similar picture is a person walking through a crowd: the people who make room and close in behind make the walker effectively heavier. Other quasiparticles have no counterpart among the original particles at all. A phonon is a quantum of a sound wave in a crystal, a collective vibration of all atoms. A hole is the absence of an electron in an otherwise filled band, and it moves like a particle with positive charge.

Quasiparticles are usually not stable forever. They decay or scatter after some time, and the concept is useful when this lifetime is long compared to the time scale set by their energy. Much of solid state physics, for example electrical conduction in metals, is described in terms of quasiparticles.

Concrete

Phonons. Take a chain of atoms of mass $m$ and spacing $a$ , connected by springs with constant $K$ , with displacements $u_n$ :

$H = \sum_n \left[\frac{p_n^2}{2m} + \frac{K}{2}(u_{n+1}-u_n)^2\right] .$

The plane wave ansatz $u_n \propto e^{i(kna - \omega t)}$ solves the equations of motion if

$\omega(k) = 2\sqrt{\frac{K}{m}}\left|\sin\frac{ka}{2}\right| .$

After quantization, in terms of normal mode operators $b_k$ with $[b_k, b_{k'}^\dagger] = \delta_{kk'}$ ,

$H = \sum_k \hbar\omega(k)\left(b_k^\dagger b_k + \frac12\right) .$

The interacting chain of atoms has become a set of independent harmonic oscillators. The state $b_k^\dagger|0\rangle$ is one phonon with energy $\hbar\omega(k)$ and crystal momentum $\hbar k$ , defined modulo $2\pi\hbar/a$ . Phonons are bosons, and anharmonic terms in the potential make them interact and decay.

Electrons in a band. An electron in a periodic potential has energies $\varepsilon_n(k)$ that form bands. Near a band minimum at $k_0$ ,

$\varepsilon(k) \approx \varepsilon(k_0) + \frac{\hbar^2(k-k_0)^2}{2m^*}, \qquad \frac{1}{m^*} = \frac{1}{\hbar^2}\frac{d^2\varepsilon}{dk^2}\bigg|_{k_0} ,$

so the electron moves like a free particle with effective mass $m^*$ , which can be much smaller or larger than the electron mass. Near the maximum of an almost filled band, $d^2\varepsilon/dk^2 < 0$ . The missing electron, the hole, then responds to fields like a particle with charge $+e$ and positive mass $-m^*$ .

Lifetime and the Green's function. In an interacting system the probability amplitude for an added electron with momentum $\hbar k$ to stay in that state is described by the single-particle Green's function. With $\hbar = 1$ and energies $\xi_k$ measured from the chemical potential, a quasiparticle shows up as

$G^R(k,\omega) \approx \frac{Z_k}{\omega - \xi_k + i\Gamma_k} + \text{(smooth background)} ,$

with $0 < Z_k \leq 1$ the weight of the bare electron in the quasiparticle and $1/\Gamma_k$ (up to a factor 2 depending on whether one means amplitude or probability) the lifetime. The quasiparticle picture makes sense if $\Gamma_k \ll |\xi_k|$ . In a Fermi liquid, phase space restrictions from the Pauli principle give $\Gamma_k \propto \xi_k^2$ at zero temperature, so quasiparticles close to the Fermi surface are long-lived.

Abstract

The precise notion is spectral. For a many-body system with ground state $|0\rangle$ and an operator $c_k^\dagger$ that adds a particle of momentum $k$ , the retarded Green's function has the Lehmann representation in terms of exact eigenstates, and the spectral function

$A(k,\omega) = -\frac{1}{\pi}\operatorname{Im} G^R(k,\omega)$

is non-negative and normalized, $\int d\omega\, A(k,\omega) = 1$ for fermions. In a finite system it is a sum of delta functions. In the thermodynamic limit a quasiparticle is a peak of $A$ whose width is small compared to its distance from the chemical potential, or equivalently a pole of the analytic continuation of $G^R$ into the lower half plane close to the real axis. Writing Dyson's equation $G^R = (\omega - \epsilon^0_k - \Sigma^R(k,\omega))^{-1}$ with self-energy $\Sigma$ , the quasiparticle energy solves $E_k = \epsilon^0_k + \operatorname{Re}\Sigma^R(k,E_k)$ , and

$Z_k = \left(1 - \frac{\partial\operatorname{Re}\Sigma^R}{\partial\omega}\bigg|_{E_k}\right)^{-1}, \qquad \Gamma_k = -Z_k\operatorname{Im}\Sigma^R(k,E_k) .$

Landau's Fermi liquid theory assumes adiabatic continuity: the low-lying states of the interacting system are in one-to-one correspondence with those of the free Fermi gas, labelled by occupation numbers $n_k$ of quasiparticles. The energy is a functional

$E[n] = E_0 + \sum_k \epsilon_k\,\delta n_k + \frac12\sum_{k,k'} f_{kk'}\,\delta n_k\,\delta n_{k'} + \ldots ,$

where the Landau parameters $f_{kk'}$ encode the residual interactions. The theory breaks down where $Z\to0$ , for example in one dimension, where the low-energy excitations are collective (Luttinger liquids).

Bosonic collective modes appear as poles in two-point functions of composite operators, for instance the density-density response for plasmons and phonons or the spin-spin response for magnons. When a continuous symmetry is spontaneously broken, Goldstone's theorem guarantees gapless modes; phonons in a crystal (broken translations) and magnons in a magnet (broken spin rotations) are examples, with $\omega\propto k$ for antiferromagnets and $\omega\propto k^2$ for ferromagnets. Quasiparticles can also carry quantum numbers that no constituent has: in the fractional quantum Hall state at filling $\nu = 1/3$ the elementary excitations have charge $e/3$ and fractional (anyonic) exchange statistics.

In relativistic quantum field theory the elementary particles are themselves excitations above the vacuum, poles of the propagator with residue $Z$ (Källén and Lehmann). The difference to condensed matter is the symmetry of the ground state: the vacuum is Lorentz invariant, while a material picks a rest frame, so quasiparticle dispersions need not have the form $\omega^2 = k^2 + m^2$ .

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