Bare and dressed

This is a physics post… oh, come on, you do not really believe that I am using a title including the word bare in order to fool search engines and increase the number of hits, do you ? I mean, really, how stupid would it be to try and disseminate throughout a post keywords so as to attract pathetic sex-obsessed readers always looking for nudity, or porn, or otherwise explicit, graphical descriptions of acts of sexual nature performed by consenting adults

So, anyway, in two recent remarkable posts, Doug Natelson explains in eminently simple and intuitive terms the concept of quasi-particle, one of the most powerful and frequently utilized in condensed matter physics. Doug illustrates this idea by describing the polaron, i.e., a quasi-particle consisting of an electron moving in a crystalline solid, instantaneously surrounded by a “cloud” of slightly displaced lattice ions, which are attracted to the electron on account of their positive electric charge (that of the electron being negative).

A commonly used terminology is that, inside the crystal, an electron becomes “dressed” (as opposed to being “bare”, i.e., in vacuo) by its interaction with the ions, the “dress” being the local polarization associated to the displaced ions in the nearest vicinity of the traveling electron. This dress is continuously, dynamically changing, as the electron moves through the lattice; ions abandon the cloud as an electron moves past them, to be replaced by other ions located along the way. As it turns out, this dress that the electron now carries with it affects not only some of its properties as an individual particle (i.e., its energy, mass etc.), but also its interaction with other (dressed) electrons. In other words, two electrons inside a crystal do not interact the same way they would in vacuo. Indeed, they experience a “renormalized” (or, again, dressed) interaction, which may at times look very different than the one in vacuo, leading to completely new phenomena (I know, is condensed matter physics cool or what ?).

For example, while two electrons repel each other in vacuo, as they have the same electric charge and like charges repel, they can effectively attract inside some crystals, e.g., those of a Superconductor, where this attraction leads to the formation of pairs of bound electrons.
How does this metamorphosis of the interaction occur ? In many different ways, depending on the background in which the original particles move; in turn, this means that the set of all possible effective interactions is virtually unlimited.

The formal description of the interaction between dressed particles can be quite involved, but the basic idea is simple, and can be illustrated fairly intuitively using examples drawn from actual physical systems (as opposed to more or less ridiculous analogies with human behavior [0]).
One of my favorite playgrounds for this type of exercise, is antiferromagnets. An antiferromagnetic solid is one in which ions feature a magnetic moment, i.e., they can be regarded as tiny, identical bar magnets. Magnetic moments associated to ions at adjacent lattice sites interact, in such a way that if the bar magnet of a given ion points, say, “North”, those of the nearest ions will point “South”, and vice versa — i.e., magnetic moments of neighboring ions tend to be anti-parallel [1].

Schematic representation of a one-dimensional antiferromagnet

Fig. 1. Schematic representation of a one-dimensional antiferromagnet

Fig. 1 represents schematically the configuration of lowest energy of a one-dimensional antiferromagnet. The bar magnet at each site points in the direction opposite to that of the magnets sitting on the immediate left and right (“nearest neighbors”).
Let us put it more quantitatively, by associating an energy –J to each bond (i.e., a pair of nearest-neighboring sites) for which magnetic moments are anti-parallel, and +J to bonds where they are parallel; here, J is a number whose magnitude expresses the strength of the coupling between moments [2]. In the configuration shown above, there are only anti-parallel bonds, which means that the system picks up a contribution –J from each. No other configuration has lower energy.
“Flipping” a single bar magnet (i.e., turning it upside down) increases the energy by 4J, as for two bonds (those involving sites at the left and right of the one whose bar magnet is flipped) the energy increases by 2J (it goes from –J to +J), as the corresponding magnetic moments go from being anti-parallel to parallel.

OK, let us make things a bit more interesting, now.
In some cases, it is possible to dope an antiferromagnetic crystal (i.e., introduce a fraction of foreign impurity atoms). As a result of this operation, some of the original ions will either acquire or lose an electron, in turn 1) losing their magnetic moment (completely or partially) 2) gaining extra electric charge (either positive or negative depending on whether electrons are lost or captured). In other words, some sites will look magnetically “empty”, while the ions will have different electric charge than that of the other ions.
Let us for definiteness assume that ions in the “undoped” lattice are all positively charged, i.e., they have been stripped of one or more electrons. Consider the case where doping causes some of them to lose an extra electron, therefore becoming even more positively charged than the other ions, with the concurrent vanishing of their magnetic moment [3]. Let us call these ions holes, with reference to the missing magnetic moment.
The situation now looks as illustrated in Fig. 2 (holes are represented as white circles:

Holes in a one-dimensional antiferromagnets. Broken bonds are shown by blue stumps of line.

Fig. 2. Holes in a one-dimensional antiferromagnets. Broken bonds are shown by blue stumps of line.


Let us now ask ourselves the question: do holes attract or repel ? Equivalently phrased, will the lowest energy configuration be one with holes all clustered together or one with holes far apart ?
The answer would seem obvious. Because holes have a strong positive charge, they surely must repel, and indeed they would in vacuo. However, the situation is complicated here by the presence of the antiferromagnetic background of the underlying ionic lattice. Let us compute first the energy “cost” of introducing a hole; because the number of antiferromagnetic bonds that are “broken” on injecting a single hole (in one dimension) is two (shown by blue stumps of line in Fig. 2), the cost in terms of energy is 2J (since the hole has no magnetic moment, the energy of its interaction with the magnetic moment of any other ion is zero). If a second hole is introduced, and if it is far away from the first one (as in the above figure), then four antiferromagnetic bonds are broken, at a total energy cost of 4J.
Holes in a one-dimensional antiferromagnet. Configuration with holes on adjacent lattice sites.

Fig. 3. Holes in a one-dimensional antiferromagnet. Configuration shown is that with holes on adjacent lattice sites.


Now, if the second hole is inserted right next to the first, as shown in Fig. 3, then only three antiferromagnetic bonds are broken, i.e., the energy cost is lower by J than placing holes far away from each other.
Of course, placing them near each other costs energy too, because they repel due to their electric charge — let us call U the electrostatic energy cost in vacuo of placing two holes side by side. For simplicity, we assume that such energy cost drops to zero if holes are further away than nearest neighbors.
So, recapitulating:
1) If holes are far away, the energy is 4J above that of the antiferromagnetically ordered “undoped” system.
2) If holes are near each other, the energy is 3J + U above that of the undoped system.

If the value of J is greater than U, then 3J + U is less than 4J, i.e., in the presence of two holes the system attains its lowest energy by placing holes near each other. That is, the antiferromagnetic background results into an effective attraction between objects that would normally repel [4]. Here too, the effect is that of “dressing” the “bare” interaction between holes, in some cases profoundly changing its character (from repulsive to attractive) [5].

One of the most fascinating aspects of the idea of effective interactions, is that albeit it originates within condensed matter (or, more properly, many-body) physics, it is actually not restricted to that realm — indeed, it is far more general than that. For, essentially every interaction is “effective”, i.e. can be thought of as arising from fundamental forces through a process that involves the two particles as players, but is also mediated by their individual interactions with the background, whichever that is. In a sense, particles always move as if over a “mattress”, which they “depress” locally, in one way or another. An attraction arises when two particles decide to sit at the bottom of the same depression of the mattress, as in the above example of the doped antiferromagnet.
This is true even if the background is vacuum. For example, the interaction between an electron and its positively charged anti-particle (positron) in vacuo is not merely electrostatic, but also involves the creation and annihilation of particles of light (photons) which are continuously emitted and absorbed by the two particles as they interact with the background electromagnetic field [6].
In fact, the notions of “dressing” of particles into “quasiparticles”, and the emergence of “effective” interactions, are arguably among the most fundamental, general and powerful in physics.

Notes
[0] My PhD advisor used to rehash quite often (as in, every day) the following analogy, of which there exist many variations, some not suitable for a family oriented blog: a smoker is someone from whom many of us prefer to stay away, for reasons not having anything to do with the person, whom we may find quite likable when (s)he is not smoking. These reasons are deemed valid by authorities, and as a result smoking in public is severely restricted. Smokers typically find themselves in the same place, either outside a building, or on the porch, or in a suitably designated smoking area, i.e., they spend time together, not because they necessarily like each other, but because of the “repulsion” that all of them suffer from the rest of society each time they light a cigarette. Thus, smokers attract each other effectively, as a result of their individual interaction with society at large.

[1] The microscopic origin of this peculiar interaction is interesting and well understood, but need not concern us here. Incidentally, there are also materials, called ferromagnets, in which parallel orientation is favored.

[2] We are making here the simplifying assumption that moments can have only two orientations, namely “up” and “down”. One may wonder whether more general orientations are possible, and the answer is yes, although this is one of those issues that quantum mechanics makes a bit tricky. Suffice to say that the basic ideas expounded here remain valid, and are not significantly altered, if one assumes an arbitrary orientation of local moments, as long as the interaction favors anti-parallelism.

[3] Magnetic moments typically arise from the presence of an unpaired electron in an atomic shell.

[4] Even if J is less than U, i.e., if no effective attraction arises, the repulsion that holes experience when close to each other is weakened compared to what it would be in vacuo. The corresponding energy difference in this case is U-J instead of U.

[5] In a richer (and more realistic) model, a hole can “trade place” with a nearest-neighboring ion, which retains its moment orientation upon “hopping” to the site previously occupied by the hole. This physical mechanism has the effect of “scrambling” the local antiferromagnetic order, and the hole becomes a quasi-hole, namely an object consisting of the original “bare” hole, “dressed” by a cloud of (almost) randomly oriented magnetic moments, in a background in which they are otherwise anti-ferromagnetically aligned. This is very similar to the polaron example, actually, except that instead of enhancing locally the polarization like electrons do, holes depress antiferromagnetism.
The resulting interaction between quasi-holes is more complicated than in the simple example shown above; one of the important differences is that it is dynamic, not static, i.e., it depends on time.

[6] There is really no such thing as vacuum, actually. Indeed, fluctuations of the electromagnetic field, as a result of which photons continually appear out of nowhere (interacting with charged particles that are present) can never be turned off, even at zero temperature.

4 Responses to “Bare and dressed”

  1. Cath@VWXYNot? Says:

    Oh, are you a physicist?

  2. Massimo (formerly known as Okham) Says:

    No, I fix washers for a living…

  3. Cath@VWXYNot? Says:

    You’re not very good at it though, IIRC.

    There’s a discussion about the impact of the budget on Canadian science over at the Nature Network, if you’re interested. All biologists so far though!

  4. Cherish Says:

    Doing my best Bevus and Butthead impression:

    He said sex!

    *snort, snort*

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