When reading the physics literature, we often see the term "condensate".

Some examples:

  • in the string net model (Wen, Levin), one will say the string "condensate".
  • in QCD, people talk about the quark-gluon condensate.
  • In stat mech, one talks about the bose einstein condensate.
  • People will talk about 'monopole condensation' to describe confinement transition.

My understanding is that in some cases the ground state acquires an interesting non-zero expectation value of "something" after the phase transition, but even then it's not clear.

What do condensed matter physicists mean when they say something "condenses"? (a mathematical definition in terms of operators would help).

If there are other examples of condensation, I'd be interested to learn more.

  • $\begingroup$ I do not have any knowledge of string-net condensation.. but usually the mean value of the relevant field acquires a non zero value. $\endgroup$
    – Quillo
    Dec 5 '21 at 10:35
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    $\begingroup$ Yes, that's the standard way to talk about an order parameter. So some questions immediately come to mind: does a symmetry have to be broken for something to condense? (QCD seems to say no). What if there's no local order param? (2d XY model). Do we still say there's condensation? Is forming a condensate synonymous to 2nd order phase transition? $\endgroup$ Dec 5 '21 at 19:17
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    $\begingroup$ The answer to your question may be simpler than you think. The original meaning of the word is "to thicken," to reduce, or especially to create liquid out of vapor. So mathematicians and physicists in several disparate fields have found reason to use this word metaphorically to describe something in their work. $\endgroup$
    – RC_23
    Dec 7 '21 at 6:12
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    $\begingroup$ There is a funny anecdote from Landau & Lifshitz, Vol3 (stat mech), $62 p.181: "The steady increase of particles in the state with ε = 0 is called Bose-Einstein condensation. It should be emphasised that this refers only to "condensation" in momentum space; no condensation actually occurs in the gas, of course." So the authors felt the need to clarify the term "condense". Thinking of BEC as condensation in momentum space is a really helpful picture in my opinion. $\endgroup$ Dec 7 '21 at 6:52

I may be able to offer some hints. I am also looking forward to more answers (Ryan please!) because this is definitely a complicated subject, with contributions from people coming from very different backgrounds (hep-th, cond-mat, pure math, etc.).

"Condense" may mean different things to different people. I believe the oldest precise definition was that an operator condenses if it picks up a vacuum expectation value. The physical interpretation is that the operator acquires a macroscopic value, i.e., it becomes something like a constant background, a spacetime filling object.

Note that vevs typically imply that a symmetry is spontaneously broken (and not the other way around!). Indeed, if the operator is charged under a given symmetry, then its condensate breaks the symmetry. Of course there are situations where an operator that is not charged under any symmetry condenses, and in those cases the vev does not break any symmetries, and thus there is no spontaneous symmetry breaking. The simplest example is, I think, the Schwinger model, where the basic meson condenses but there are no symmetries to begin with, so nothing is broken. Of course, Schwinger is gapped and has a unique, trivial vacuum state.

In QCD, the object that typically condenses is the quark bilinear $\psi_L\psi_R$. If this operator has a non-zero expectation value, it breaks the chiral symmetry $SU(N)_L\times SU(N)_R$ down to the diagonal subgroup. This breaking means that there are Goldstone bosons, the pions. There are other objects that could condense beyond the quark meson (e.g., with enough supersymmetry it is known that certain monopoles condense; I believe they break the magnetic $U(1)$ one-form symmetry and hence the Goldstone photon becomes massive and is gapped out, hence the theory becomes confining. But don't quote me on that).

More generally, condensing means that you allow the object to have a macroscopic value. In cond-mat, I believe, this is often achieved, roughly speaking, by dropping certain terms from the Hamiltonian, i.e., those terms that energetically penalize the excitation of the object you want to condense. If the object to condense does not have a term in the Hamiltonian, then it will cost no energy to produce it, and the vacuum state will have a non-zero excitation number for it, i.e., the object has acquired a macroscopic value.

A different notion of condensing is that of gauging. If you gauge a symmetry generated by that operator, then effectively you allow that operator to proliferate, because gauging mean that you identify any two objects that differ by the operator you gauged. Thus, the gauged operator is identified with the vacuum. This is reminiscent of the previous paragraph, since the Hamiltonian must be gauge invariant, i.e., it cannot depend on the object you gauged. (Note that in cond-mat, gauging is often enforced energetically, i.e., you impose the Gauss law by penalizing configurations that violate it).

This last notion is the one used for strings. Condensing a string means gauging the one-form symmetry it generates. Gauging means summing over bundles, i.e., you introduce a mesh of strings, which reproduces the fact that the condensed object acquires a macroscopic value. When you gauge a string, you sum over all possible insertions, and therefore the string appears everywhere: the vacuum consists of a spacetime filling mesh of strings.

  • $\begingroup$ Thanks for the insight. The "gauging" of condensed matter physics always confused me because in that case, gauge transformations are really physical transformations, and the gauge can be broken up to some energy gap. This is not what gauging means in HEP where the gauge cannot be broken (it's built in to the dynamical variables of the theory). I assume it's in that sense that you are "gauging" the strings in the last example? $\endgroup$ Dec 7 '21 at 4:03

Condense could probably mean :-

  1. to become denser or more compact or concentrated.

  2. Change of state of matter.

Condensate could mean :-

  1. The liquid phase produced by the condensation of steam or any other gas.

  2. The product of a chemical condensation reaction, other than water.

  3. Natural-gas condensate, in the natural gas industry.

Now there is more to it in deep physics like :-

  1. Bose-Einstein condensate is a state of matter that occurs when a set of atoms is cooled almost to absolute zero in which a statistical description of the positions of the atoms implies that they physically overlap each other and in effect form a single atom. enter image description here

  2. Fermionic condensate is basically a pairing of electrons (which are fermions) to produce a condensate is a crucial feature of superconductivity, so a fermionic condensate would give us crucial insights into the mechanisms behind superconductivity, as well as superfluidity.enter image description here

  1. Gluon condensate, a non-perturbative property of the QCD vacuum.

  2. Color-glass condensate (CGC) is a type of matter theorized to exist in atomic nuclei when they collide at near the speed of light.

  • $\begingroup$ @physicsdude , please verify its answer if it's correct $\endgroup$ Dec 7 '21 at 9:38
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    $\begingroup$ I do not see how this answers the question - the asker is clearly aware of the difference concrete usages of the word "condensation", they're asking for the underlying principle that makes is meaningful to refer to all these varied phenomena with the same word. You've just given more examples for the usage of "condensation". $\endgroup$
    – ACuriousMind
    Dec 7 '21 at 9:45
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    $\begingroup$ Hi, the answer I think is correct but the original question wanted some insight into what is shared between those different usage of the term if any. I'm aware of the usage of the term in various applications you highlighted, and what it means but only at the textbook level. $\endgroup$ Dec 9 '21 at 5:30

Q1:"What do condensed matter physicists mean when they say something "condenses"?"

Word "Condensation" defined as;
"Condensation is the change of the physical state of matter from the gas phase into the liquid phase, and is the reverse of vaporization."
is used by advanced physics mostly to describe both solids and liquids due to the lack of solid (pun intended) physical definition of what is the difference between solid and liquid. As both of these phases are physically very same compared to Gas which behavior can be very accurately be described with Kinetic Gas theory.

This concludes the fact, that though it as been impossible to write a simple universal description to define the physical difference between solid and liquid, it's very simple to describe condensed matter;

If molecules form some stable structure, where applying force to any single molecule of that structure, will immediately (with the speed of light) distribute part of that force to all other molecules of that structure, then they are condensed.

As a thought experiment the "non condensed" can be described as easily;

If molecules have their own unique randomly varying properties, they are not condensed.

This simple difference produces one remarkably observable aspect; a surface. Condensed matter has a clear surface which interacts with outer forces, it creates a recognizable object.

One of these forces are gravity, which also here becomes quite problematic, if you think of it; "How can regular gravity interact to non condensed matter?" -as there only are objects in scales of quantum gravity available. (Key word; Planck mass)

Q2 a mathematical definition in terms of operators would help.

A2: It should be noted, that all properties of a non condensed matter can be basically be described as a particle velocity $v$ and mean free path $l$. This simplifies the whole physics of this topic. As; $$pV=nk_BT$$

Velocity $v$ form Temperature $T$; $$v_{rms}=\sqrt{\frac{3k_BT}{m}}$$

Mean free path $l$ form Pressure $p$; $$l=\frac{k_BT}{\sqrt{2}\pi d^2p}$$ $$p=\rho R_dT$$ so $$l=\frac{k_B}{\sqrt{2}\pi d^2\rho R_d}$$

With these equations you can calculate any molecule, Temperature and Pressure combination and notice, that all matter actually condensates at the same point which can be exactly defined through a constant mean free path $l$; below this length matter is condensed and above it's Kinetic (gas alike) IF we accept here a new defintion for "Condensed" being the answer on the question;
Is the particle completely surrounded by another particles with permanent electromagnetic interaction, including attraction. Only Condensed matter has stresses and tensions and viscosity. The molecules are able to pull each other.
As this definition makes the viscous gas being "condensed" too and basically causes the lower atmosphere of planets being "condensed" up to the straight lower edge of Cumulus clouds. Funnily enough these clouds are present on all planets which has some gas surroundings, and regardless of their molecule structure, they all fit in to this definition. This certain length can also be derived straight forward from the speed of light $c$.

Those who have more interest to this topic, can observe nature by them selves, or maybe read this badly written old paper of mine.


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