# Quasi-particles: how typical are they?

1. Jan 12, 2015

### Couchyam

It seems fascinating to me that quasi-particles appear everywhere, and can appear at multiple scales within the same system. Why is this? Is there a good intuitive reason for why entities behaving like free particles should exist at various scales in a system that is otherwise strongly interacting? Are there any large' quantum systems that have no quasi-particles at any scale? If so, how would we find them experimentally (or perhaps more ambitiously, predict their existence)?

2. Jan 14, 2015

### ZapperZ

Staff Emeritus
No one appears to address this, and from my reading, what you're asking is a bit vague.

What exactly do you mean by "quasiparticles" here? It is a bit confusing because you are claiming that it is "everywhere", whereas the term "quasiparticles" has a very specific and narrow definition within the Landau's Fermi Liquid theory. For example, if you look at the single-particle spectral function, one can argue that there are Landau's quasiparticles only in the weak-coupling limit, i.e. when the single-particle spectral function is "sharp", to put it crudely. This is why people argue on whether the Fermi liquid model is valid for the optimally-doped and underdoped cuprate superconductors. Their spectral function peak is "broad" and the concept of "quasiparticles" may not be valid in that doping range. So it is not true that they can exist at ANY scale.

Furthermore, there are no "quasiparticles" in Luttinger liquid system, i.e. low dimensional conductor that has even a weak interaction. The spin-charge separation that is often exhibited by the violation of the Wiedermann-Franz law is a clear signature that these are not Landau's quasiparticles.

Zz.

3. Jan 14, 2015

### Couchyam

My question above is extremely vague, and I appreciate your thoughtful answer. A more appropriate question (though unfortunately still vague) might be How typical are particle-like excitations'. I will try to avoid using wrong terminology, but I don't have much experience in condensed matter so I may make mistakes.

It seems like particle-like excitations in effective field theory are analogous to order parameter fields in statistical mechanics with Gaussian behavior (i.e Gaussian distributed with some kernel $K(x-y)$). An order parameter field can have Gaussian statistics on one length scale, but then evolve under renormalization group flow to a completely different kind of statistics on another scale. In this sense, the order parameter field exhibits statistical mechanical quasiparticles' at one scale, which may dissolve as the system is viewed from larger or smaller scales. Of course, this works for any order parameter field with Gaussian behavior on some scale, so I haven't figured out yet what the right way to approach typicallness' is in this setting. Maybe it could have something to do with how robust the Gaussian behavior is under renormalization group flow (eliminating contrived examples), and the number' of independent robust Gaussian order parameter fields in the system.

The quantum systems are more interesting to me because there is an underlying Hilbert space with unitary time evolution, and it seems like only special types of coarse-graining processes would lead to near unitary time evolution in the resulting density matrix. If the time evolution is roughly unitary then a pure-state density matrix of the subsystem can be described by a wave function, which I am calling a collective excitation' (or a group of collective excitations).

Some questions:
If a quantum effective field theory encodes both the effective Hamiltonian and Hilbert space for unitary subsystem dynamics (or the most unitary possible'), then could a sensible definition for the typicalness' of a collective excitation be the range of length or energy scales over which the kinetic term for the excitation field is a relevant operator?
Also, in the example of a doped superconductor, does the doping have the effect of breaking translation invariance in the system?

The ideas written above are still very vague, so please let me know if anything needs clarification or is probably wrong. Thanks!

4. Jan 14, 2015

### ZapperZ

Staff Emeritus
If we believe in QFT, then I'd say that one can model any kind of interaction with "particle-like excitations". Now, you may argue that there are systems in which perturbation method fails and thus, such particle interaction is not accurate, but I don't think that that is what you are asking here.

I'm a bit confused here. The existence of order parameter has more to do with the "ordering" or the symmetry length scale, rather than the emergence of particle-like interactions. So I don't quite understand the connection between the two.

Zz.

5. Jan 14, 2015

### Couchyam

Yes, the lack of particle-like excitations at a given energy scale could be related to the failure of perturbation theory, in the following sense. Fixed points of RG transformations correspond to regions in parameter space where the model is stable under scale changes (either zooming out or zooming in). The quadratic part of the Lagrangian (which encodes the free particle behavior) is then robust' under scale changes, in the sense mentioned in my post above. Perturbation theory may fail at these points because of large coupling, but for weak coupling at least it should be possible to do meaningful calculations over a wide range of energies. Between RG fixed points, the form of the Lagrangian density or shape of the probability distribution changes relatively rapidly. Tentatively, I would say that the quadratic part of the effective Lagrangian at these scales can only correspond to a particle-like excitation in a weaker sense than near a fixed point. It would be difficult to make sensible predictions in a wide range of energy scales using an effective theory renormalized at an intermediate scale (between fixed points) because of how rapidly the parameters in the Lagrangian change: accurately predicting changes in the operators that appear in the Lagrangian density would require infinite sums in perturbation theory (the labels perturbative' and nonperturbative' are somewhat loose, but I would say that perturbation theory fails' in a certain sense). This would relate the failure of perturbation theory to the lack of sensible quasiparticles.

Once the existence of the order parameter(s) is established, new order parameters can be defined as functionals of the original. For example, in the definition of the renormalization group it is conventional to consider a set of order parameters defined at each scale (where the scale is the 1-dimensional parameter in RG transformations). At a given scale, the order parameter may have a probability measure $\mu(\phi)$. By analogy with probability distributions in finite dimensions, it seems reasonable that there could be functionals, say $\Psi\{\phi\}$, such that the induced probability distribution on $\Psi$ is Gaussian, or near Gaussian. Then, I would say that $\Psi$ is a Gaussian (or near Gaussian) order parameter'. In quantum field theory, this would correspond to a field redefinition (e.g. dressing' an electron with phonons or photons), leading to a Lagrangian density that can be treated perturbatively.

Thanks!

6. Jan 14, 2015

### ZapperZ

Staff Emeritus
So, doesn't this kinda answered your question? Look for a system or the regime where perturbation theory fails, and you now have a situation where you can't use any form of quasiparticles or particle-like interactions.

Zz.

7. Jan 14, 2015

### Couchyam

More or less, but I think failure of perturbation theory is a little subjective since it depends on how the expansion is made. An alternative approach is to look for RG trajectories that avoid approaching fixed points as much as possible. The models or effective field theories associated with these RG trajectories would then correspond to systems that have no distinctive particle-like excitations at any energy scale. I'm not sure how `stable' this definition is with respect to field redefinitions however: the question is then whether it is possible to move or create an RG fixed point near the trajectory by simply redefining fields.

Of course, actually computing the RG flow far away from a fixed point would be prohibitively difficult if not impossible. I guess from this perspective, the reason that particle-like excitations appear so often at multiple scales is that most RG trajectories would pass by multiple unstable fixed points.

Last edited: Jan 14, 2015
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