Study Guide:Mathematical Mesoscopic Physics/Mesoscopic Physics Glossary

This is an alphabetical list of terms in mesoscopic physics and in mathematics that may be useful for someone studying (the mathematical aspects of) mesoscopic physics. The definitions/explanations are kept brief, though some terms might refer to a more extended entry on a separate page, or contain appropriate links to Wikipedia and other resources. The purpose of this list is to have an easily accessible (easily printable), thematically focussed overview. You might want to print this and use it as a cheat sheet during talks outside of your specialty.

A note on the mathematics entries: Since this page is meant as a quick reference or a survival kit, it is in many cases neither necessary nor desirable to provide a formal and complete mathematical definition. It may be preferable to come up with an illuminating example, explain what a concept is good for and which subdiscipline of mathematics contains the prerequisites for grasping the technical definition of a term. If you would like to explain a concept in more depth, please consider adding to the specialized study guides on our Mathematics for Mesoscopic Physics page.

Glossary

• Boltzmann distribution - In thermodynamic equilibrium, the probability of finding the system in an energy eigenstate with energy $E_{n}$ is proportional to $exp(-E_{n}/k_{B}T)$ . (Read more: "Boltzmann Distribution")
• Charging energy - The energy (or energy scale) needed to add one additional electron to a quantum dot, due to the Coulomb repulsion between electrons. Often, one may describe this interaction effectively by assigning a capacitance $C$ to the dot (depending on its shape etc.). Then the charging energy scale would be $e^{2}/C$ , where $e$ is the electron charge.
• Conductance - The conductance G of an electrical sample is the ratio of the electrical current and the voltage that is applied to produce it, i.e. the inverse of the resistance: $G=I/V$ . In mesoscopic circuits, it is often useful to introduce the 'conductance quantum' $G_{Q}=e^{2}/h$ .
• Coulomb blockade - Electron-electron interactions on a quantum dot (a finite region containing an integer number N of electrons) cause an energy contribution of the form $e^{2}(N-N_{0})^{2}/2C$ , with a reference value $N_{0}$ determined by a gate voltage. This energy cost prevents the tunneling of additional electrons onto the dot at low temperatures (Coulomb blockade), and gives rise to very small conductance through the dot. The Coulomb blockade is only absent for special (half-integer) values of $N_{0}$ .
• Decoherence is the loss of a sharp relation between quantum-mechanical phases (e.g., of a wave function at two distinct points in space), due to coupling of the system to an 'environment' of inaccessible degrees of freedom. To find the expectation value of a system observable, one thus has to trace out these d.o.f.s from the (pure) density matrix of the total system: The accessible part no longer evolves unitarily (like a closed system), and phase relations are quickly delocalized into the environment. This destroys superpositions of macroscopically different states, and is hence responsible for all classical aspects of our universe.
• Disorder potential - A random potential $V(x)$ (i.e. a realization of a stochastic scalar field), representing the spatially fluctuating potential energy of a particle. In real nanoscale devices, such a potential is produced by the random location of charged impurity atoms and other defects. The propagation of electron waves through such a disorder potential is one of the main subjects of mesoscopic physics.
• elastic scattering - A process in which a particle is scattered from a fixed potential such that its final energy remains equal to its initial energy. This is the typical process for electron motion in a disorder potential, where electrons only change their direction of flight but not their energy during scattering. (To be contrasted with inelastic scattering)
• Fermi distribution - the equilibrium distribution of fermions over energy $\epsilon$ , $f(\epsilon )=1/(\exp[(\epsilon -\mu )/k_{B}T]+1)$ , depending on temperature $T$ and chemical potential $\mu$ . (Read more: "Fermi Dirac statistics" (Wikipedia))
• Fermi energy - Due to the Pauli principal (anti symmetry of the multiparticle wavefunction) the multiparticle ground state of a System of $N$ fermions can not be the $N$ fold occupiet single particle ground state (as for bosons, see Bose-Einstein condesation), but rather the $N$ lowest levels will be populated. (The Fermi distribution converges to the $\theta$ heaviside function for $T\to 0$ ) This state is called the fermi see and the highest occupied energy is the Fermi energy $\epsilon _{F}$ .
• Fermi surface - Manifold of momenta ${\vec {k}}$ at the fermi energy, i.e. $=E_{k}^{-1}(\epsilon _{F})$ where $\epsilon _{F}$ is the Fermi energy and $E_{k}:\mathbb {R} ^{d}\to \mathbb {R}$ the dispersion relation mapping momentum to energy. For e.g. free fermions $E_{k}({\vec {k}})={\frac {1}{2m}}k^{2}$ the Fermi surface is a sphere.
• Fibre bundle - is a map $\pi :E\rightarrow B$ (sometimes called a "projection") between two manifolds E (the "total space") and B (the "base space"), such that locally  the inverse image $\pi ^{-1}(U)$ of an open set U around x is a direct product as a topological space. For examples, both the Moebius band and the cylinder are line bundles over the circle.
• Holomorphic - A function defined on an open subset of $\mathbb {C}$ (or more generally of $\mathbb {C} ^{n}$ ) which can be locally expressed by convergent power series. For example, the exponential function is holomorphic, but the function $z\mapsto \mathrm {Re} (z)$ is not.
• Inelastic scattering - A process in which a particle is scattered from another particle or by a time-dependent potential, such that its final energy is different from its initial energy (only the overall energy is still conserved).
• Involuton - An automorphism $i:G\rightarrow G$ (i.e. a bijective map to itself which preserves all the algebraic structures) of an algebraic object G (e.g. a group, a ring, a Lie algebra,...) such that $i^{2}=1$ , i.e. a map which is its own inverse. For example, take the complex conjugate $x+iy\mapsto x-iy$ in $\mathbb {C}$ .
• Lie group - A group which is also a smooth manifold, and the group structures are differentiable in the manifold structure. Prominent examples are the matrix Lie group $GL_{n}$ and its various Lie subgroups $SL_{n}$ , $SO_{n}$ , $SU_{n}$ , $SP_{n}$ ,... as well as the Lorentz group $SO(3,1)$ and it's extension by affine translations, the Poincaré group.
• Luttinger Liquid - In one dimension, the low-energy excitations of interacting fermions are simply density waves propagating through the system. There is a close analogy between one dimensional fermion systems and classical (Euler)liquids. This is why these systems are often called "Tomonaga-Luttinger" liquids. It turns out that it is possible to diagonalize the Hamiltonian in terms of these (bosonic) density waves and that one can re-express the fermionic operators by the bosonic ones. This powerful technique is known as "bosonization" and allows for the simple calculation of fermionic Green's functions etc. .
• Manifold - A topological Space equipped with a differentiable structure. Locally diffeomorphic to $\mathbb {R} ^{n}$ via charts. Examples are $\mathbb {R} ^{n}$ , the spheres $S^{n}$ , Lie Groups, Symmetric Spaces,... (Read more: "Manifold" (extended Wikipedia article))
• Partition Function - Describes a system in thermal equilibrium. Given some macroscopic constraints it is defined as the total number ${\mathcal {Z}}$ of accessible states (compatible with these constraints). It can thus be thought of as the statistical weight of the macroscopic state/constraints.
• projective space - A projective space is a compactification of a vector space, i.e. it is the result of adding, in a natural way, "points at infinity" to make a vector space compact. Mathematically, it is constructed as the quotient space of a vector space by the group of dilations. For example, the real projective plane $\mathbb {RP} ^{2}$ , which is a compactification of the real plane $\mathbb {R} ^{2}$ , is constructed as the quotient of $\mathbb {R} ^{3}$ by the scaling action of $\mathbb {R}$ . In other words, the two triplets (1,1,1) and (2,2,2) represent the same point on the real projective plane.
• Reservoir - Term used for a big piece of material (e.g. a wire) that is being connected to a quantum dot, such that electrons can tunnel from the dot into the reservoir and vice versa.
• Symplectic manifold - A smooth manifold equipped with a closed, non-degenerate, skew-symmetric $2$ -form $\omega$ . Here a skew-symmetric $2$ -form means a section of the vector bundle $\Lambda ^{2}T^{\ast }M\rightarrow M$ , i.e. $\omega \in \Gamma (M,\Lambda ^{2}T^{\ast }M)$ , where $\Lambda ^{2}T^{\ast }M$ denotes the second exterior power of the cotangent bundle of $M$ . In other words, for any point $p\in M$ we have a skew-symmetric bilinear form $\omega (p)$ on the tangent space $TM$ . Non-degeneracy of $\omega$ means non-degeneracy of $\omega (p)$ for any $p\in M$ . Furthermore $\omega$ is required to be closed, that is the exterior derivative $d\omega$ vanishes. Examples are the $2$ -sphere or any Kähler manifold.
• Thouless energy - Particles in disordered systems (larger than the so-called mean free path) move by diffusion. The Thouless time, $t_{D}\,\sim ~\,L^{2}/D$ is the time it takes a particle to propagate once across a system of finite extension $L$ ($D$ is the diffusion constant.) The Thouless energy $E_{C}\equiv \hbar /t_{D}$ is the energy scale associated to the Thouless time by the quantum uncertainty principle.
• Tunnel coupling - A part of the Hamiltonian that describes the possibility for electrons to tunnel through a barrier, e.g. from some wire into a quantum dot.

Wishlist

This is a collection of terms someone thought should be included. Feel free to add anything else here that comes to your mind (either physical or mathematical concepts that are potentially useful for mesoscopic physics or its mathematical aspects). If you feel you can give a brief definition, please choose 'edit this page' and move the term up to the glossary (inserting it according to alphabetical order). Preferably (and if possible) include a commonly encountered example, especially for mathematical concepts. Note: In the glossary, we do not distinguish between mathematical and physical concepts.

Just attending some talk and jotting down the words that are still unclear to you may be enough to get ideas for extending this list.

Mathematics terms

If you are a mathematician, you might help us by defining one of the following terms (be brief and give an example, if possible). Please move the definitions up to the glossary, in the form shown there!

fibration, ray, character, form, wedge product, orbit, quotient space, symmetric space, monodromy, Hessian, group representation, cone, super Lie Algebra, endomorphism, irreducible representation, C* algebra

Physics terms

If you are a physicist, try to define one of these:

metallic, gate, electrode, depleted, wave guide, transverse modes, channels, evanescent waves, tunnel contact, quantum point contact, lead, relaxation, chemical potential, scattering states, second quantization, S-matrix, Landauer-Büttiker formula, quasiparticle, coherent propagation, dephasing, quantum dot, Coulomb blockade, Two-terminal conductance, linear response, Kubo formula, fluctuation-dissipation theorem, occupation number, density of states, retarded Green's function, single-particle states, lifetime, decay rate, weak localization, strong localization, Anderson model, universality, renormalization group

Footnotes

1. That is to say, for every point x on B there exists an open set U around x in B such that ...