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I come from a HEP background and moved to condensed matter physics. I keep seeing the word “gap” being thrown around a lot: this system has a gap, this is a gapless system, the spectrum is gapped, energy gap, band gap… etc.

They seem to be closely related to each other, quoting Wikipedia:

In solid-state physics, an energy gap is an energy range in a solid where no electron states exist, i.e. an energy range where the density of states vanishes.

whereas

In many-body physics, most commonly within condensed-matter physics, a gapped Hamiltonian is a Hamiltonian for an infinitely large many-body system where there is a finite energy gap separating the (possibly degenerate) ground space from the first excited states. A Hamiltonian that is not gapped is called gapless.

and so on. Furthermore, what is the physical significance of all these gaps?

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    $\begingroup$ Quick analogy: to produce a electron-positron pair you need to provide energy above the rest energy of the electron. To produce an conduction electron in a semiconductor you need to provide enough energy to create an electron-hole pair. $\endgroup$
    – Mauricio
    Oct 7, 2021 at 8:38

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Discrete spectra and gap in quantum mechanics

Let me first remind that discretenes of energy levels plays a very important role in quantum mechanics, e.g., it explains the spectral lines of atoms, photoelectric effect, etc. When we are dealing with systems of many particles, their energy levels often merge in a band, but the discreteness of the spectrum still shows itself in the form of a gap. Alternatively, in some systems that are not characterized by a discrete spectrum, such as atomic liquids, a gap may appear as a result of interactions.

Band gap

@PetrPovolodov has already pointed out the role of the gap in semiconductor optical spectra, which is very close to the discreteness of the energy levels in single atoms when we discuss, e.g., optical absorption.

The electric phenomena in semiconductors are also dependent on the gap - e.g., the concentration of electrons in the conduction band and holes in the valence band depends on how many valence electrons are excited across the gap at the given temperature, i.e., it follows the activation law: $$e^{\frac{E_g}{k_BT}}$$

Gapped and gapless systems

From the point of view of the theory of critical phenomena, the presence/absence of the gap characterizes different phases of a system. Thus, semiconductors/insulators differ from metals by the fact that in the former the lowest energy excitations are gapped, while in the latter the excitations are gapless. This creates a huge difference between these types of materials (insulating vs. conducting). Why some systems are insulators while others conductors is known as metal-insulator transition (although this transition is difficult to observe in a single material). Similarly, appearance of a gap often characterizes other phenomena, such as the emergence of superconductivity.

Example
An example was discussed in the comments which nicely illustrates the different meanings of a gap. Let us consider a model of multi-level sites coupled via tunneling: $$ H=\sum_\alpha\left[h_\alpha\sum_jc_{j,\alpha}^\dagger c_{j,\alpha} + \lambda_\alpha\sum_j(c_{j,\alpha}^\dagger c_{j+1,\alpha}+h.c.)\right] $$ Diagonalizing this Hamiltonian will give us something like $$ \bar{H}=\sum_{k,\alpha}\epsilon_\alpha \bar{n}_{k,\alpha},\bar{n}_{k,\alpha}=\bar{c}_{k,\alpha}^\dagger \bar{c}_{k,\alpha},\\\epsilon_{k,\alpha}=h_\alpha+2\lambda_\alpha\cos(\frac{2\pi k}{N}) $$ We see that every level has transformed into a band of width $\lambda\alpha$, centered at the original level energy $\h_\alpha$.

Band (non-)overlap: gap in the density-of-states
Depending on the values of $h\alpha$ and $\lambda_\alpha$ the bands may or may not overlap, even though the original levels were non-degenerate. Thus, if we calculate the density-of-states, it may or may not have regions where it is zero, which will affect many properties of the material, such as optical absorption, electric conductance, etc.

Band filling: excitation gap
Suppose that the bands do not overlap. They are filled with electrons up to the Fermi level, $E_F$. If the Fermi level lies between two non-overlapping bands, in the region where the density-of-states is zero, the only way to excited electrons is by transferring them from the filled band to the empty band, i.e., across the gap. In this case the material is insulator, whereas, if the Fermi level were within one of the bands, the excitations would be gapless and the material would be a metal.

Coulomb interaction
Let us now add to our Hamiltonian one-site Coulomb interaction: $$ H_C=\frac{1}{2}\sum_{\alpha,\beta\neq\alpha}\sum_j U_{\alpha\beta}n_{j,\alpha}n_{j,\beta} $$ (For coherence of discussion I neglect spin, although usually this Hamiltonian is called Hubbard model and the Coulomb interaction is between the spin opposite states.)

Now, even if the Fermi level lies within a band, and even if the bands overlap, the excitation of an electron may require finite energy - this is particularly obvious, if the Coulomb interaction is greater than the band width: $$U_{\alpha\beta}>2\lambda_\alpha,2\lambda_\beta $$ (The reality is somewhat trickier in one-dimension, since we have to accountf or the Luttinger liquid effects, but the discussion applies to higher dimensions as well.)

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  • $\begingroup$ Let me see if I understood this right. So in a nutshell it is the observation that when going from, say atomic physics's spectral lines, to a large (formally infinite I presume) collection of bodies (like atoms) one expects the collective spectra to change. Indeed one can observe a variety of situations: collective systems with a continuous spectrum, discrete spectrums, continuous spectrums with jumps (gap!) and systems that transition between theses possibilities depending on some parameters. $\endgroup$ Oct 7, 2021 at 9:38
  • $\begingroup$ There are two different things in this answer: 1) a gap between bands that arise when joining many levels - like in tight-binding picture, essentially a gap in one-particle spectrum. 2) a gap that arises due to interactions, e.g., in a Hubbard model, where due to strong Coulomb repulsion the excitations require finite energy. $\endgroup$ Oct 7, 2021 at 9:47
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    $\begingroup$ @FriendlyLagrangian it could be said so, although this stretches a bit the terminology. Take $H=\sum_\alpha[h_\alpha\sum_n c_{n,\alpha}^\dagger c_{n,\alpha} + \lambda_\alpha\sum_n c_{n,\alpha}^\dagger c_{n+1,\alpha} + h.c.)$ - depending on the values of $h\alpha$ and $\lambda_\alpha$ you may or may not overlapping bads - this could be already called absence/presence of gaps in some contexts. $\endgroup$ Oct 7, 2021 at 9:57
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    $\begingroup$ Then, even if the bands do not overlap, once you start feeling them with electrons, it is usually what happens near the Fermi level that determines the behavior - so depending on whether the last band is completely filled or not, you may call your excitations gapped or gapless. Finally, if you add on-site coulomb interaction, $H_C=\sum_nU_{\alpha\beta}\sum_{\alpha,\beta}c_{n,\alpha}^\dagger c_{n,\alpha}c_{n,\beta}^\dagger c_{n,\beta}$ the gap may open, even if you didn't have it. $\endgroup$ Oct 7, 2021 at 9:59
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    $\begingroup$ Like most Hamiltonians in condensed matter many-body physics - whenever one talks about thermodynamic/statistical physics description, such interactions are implied. Think about an ideal gas - there are collisions between atoms - otherwise the thermodynamic equilibrium would never establish itself. But we neglect this collisions - there is good physical reasoning behind that, but no equations. $\endgroup$ Oct 8, 2021 at 13:29
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Crudely speaking...

When you have an isolated atom, the allowed energy states are discrete, so an electron has to gain or lose a certain minimum amount of energy to move between the levels. When you bring atoms together to form a solid, their allowed energy states change somewhat. Where you had a single energy state in each atom when they were isolated from each other, when you bring n atoms together, instead of having n instances of the same discreet energy state around each atom, you get a set of n distinct states spreading all over the solid that are closely-spaced in energy- a co-called band of states. It is as if each energy level in the isolated atom spreads to become a wide band of allowable but closely spaced energy levels in the solid. The key point is that since each band has an energy width, the energy separation between each successive band is less than the corresponding energy separation between the single states in an isolated atom. This means that where, in the isolated atom, an electron needed a certain amount of energy to jump between two successive energy levels, an electron needs less energy to jump between the corresponding bands. Indeed, depending on the nature of the solid, the energy bands can be so broad that the gap between them vanishes, so it becomes very easy for an electron to be promoted to a higher band.

You can easily imagine, therefore, that the electronic properties of a solid depend upon the width of the bands and gaps, and whether or not the bands are fully occupied (the Fermi exclusion principle prevents a band from being occupied by more electrons than there are states). If a band is not fully occupied, an electron can readily be promoted to one of the higher closely-spaced energy levels in the band. If a band is full, and there is a large energy gap to the next band with unoccupied states, then more energy is required to promote an electron to a higher state.

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An energy gap is important. For example, in semiconductors, there is an energy gap it means that in principle you have an energy limitation. For example if there is an electron in the semiconductor, it can "annihilate" with a free space (called hole) with a photon creation. So, the gap in the system means that the energy is higher than the gap energy.

Inversely is true, if you illuminate a semiconductor with a light it can absorb a photon with a higher than band gap energy. Which is known to be the photo effect

Remarks:

  1. You get a gap between electron and hole dispersions while considering for a semiconductor crystall pot=ential and solving Schrödinger equation

  2. This is true for an ideal semiconductor. There might be complications for a real one

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