The question is how is it possible to distinguish one electron forming a Cooper pair in the superconducting phase or the composite fermions having anyonic statistic behaviour in the Fractional Quantum Hall Effect o the electrons? What are "particles" in liquid He-3?
Things surely can have different names depending on what you want to emphasize, and depending on your perspective.
Let us go back to the magnon. Paul Dirac in around 1929-1930 restricted the enegy scale of a pair of electrons to just two levels, and showed that the anti-symmetrization can be re-interpreted as a transition between singlet and triplet states. The energy scale of this was named J. Then within that energy scale, the electronic Hamiltonian or Lagrangian can be transformed into an expression written purely in terms of spin operators. These obey Pauli statistics. But an approximation (e.g., Holstein's) gives Bose statistics and the excitations are called magnons if you want to think of them as particles. Alternatively, you can think of them as spin-waves.
Now, if you go to energy scales bigger than J, the two-level approximation is not enough, and you need to return to the electronic Hamiltonian. A simple form of that is the tight-binding (or LCAO) model. Here we can now have electron-hole pairs. They can be quantized into "quasi-particles" known as excitons. They can be singlet excitons and triplet excitons. The energy difference between singlet and triplet excitations is of the order of J, but the energy scale of excitons is of the order of the band-gap.
You can increase the energy scales even further. Then the tight-binding models have to be replaced by a hamiltonian which fully includes the nuclei and electrons, and use a very large Hilbert space of functions ( see sec 7.1 to 7.3 of my book for details, relation to entanglement etc). At this level, you have electrons, nuclei, and photons as the essential fundamental particles.
You can increase the energy scale even further, up to, say the nuclear energy scales, when you need to write a Lagrangian (or Hamiltonian) in terms of protons, neutrons, electrons, photons, and fields that transmit the nuclear interactions (baryon fields).
The "flow" from one type of theory to another as the energy and length scales are changed is given by the renormalization group".
At the highest energy scales we know of today, we are at the Plank scale of length and energy. That is the per-inflationary state. In the earliest state we assume that the laws of symmetry, and some laws like least action, conservation etc., hold, and these give us the capacity to construct excitations (in strings, branes or what ever). That gives you string theory or M-brane theory. The initial inflationary expansion may well be some quasi-particle of a topology or field at even higher energy scales than the Plank scale. In that sense, the whole universe is an excitation whose internal modes are the lower energy stuff that evolves along the flow of the renormalization group, and produce the particles, quasiparticles that we talk about, molecules etc that we know.
In that picture, the evolution of the whole universe is simply the decay of this quasi-particle, seen from an inside perspective! But we don't call it the decay of a quasi-particle. We have other names like "cosmic evolution".
Cooper pair electrons obey completely different statistics than individual electrons. Individual freely moving electrons is fermionic particles while Cooper electrons practically are bosons.
At the same time to break one of the Cooper pair is necessary to spend a considerable energy because such a rupture is accompanied by a change in the energies of all the other pairs, the number of which is macroscopically large.
Consider such a complex "particle" by parts does not make sense.
A similar situation occurs in liquid 3He, whose atoms have spin ½ and form a typical quantum Fermi liquid. Superfluidity of liquid helium-3 is due to the same Bose condensate that the Cooper electron pairs. Liquid helium-3 is a quantum Fermi liquid, ie it consists of particles of fermions with spin ½. In such systems, superfluidity can be observed under certain conditions when between fermions there are attractive forces that lead to the formation of bound states of fermion pairs. And the "particles" here are bound (but not chemicaly) pair of 3He atoms with different spin.
Mattuck is a wonderful book which calculates quasiparticles in many-body theory using the concepts of quantum field theory. I agree with you that it is a very interesting book for a beginner but it doesn't take into account concepts more modern as Fermi liquids or fractional quantum Hall Effect, i.e. topological field theory. My question tries to see if anybody could give a clear difference between the concepts that I have purposed for "particle"
Dear Lev,
By the way Lev (Landau was the father of the concept of quasi-particle), say you have a distinguished name for working on these issues. Let me try to answer you.
The strange for me is the concept of particle? quasi-particle? collective excitons? for the Cooper pair. The energy for mantaining correlated the two electrons is really very small for low temperatures, milli-electron volts because the electrons close to the Fermi level are very far one of the other, hundreds of nameters i.e. several times the lattice constante, correlated by a phonon . Therefore how do you call this object? What is the relation with the concept of fundamental particle that we have always in mind? What are the other exemples also mentioned in my question into this physical context? Do you follow me?
It was my impression that these concepts are very precisely defined. A particle is a singular object. A quasiparticle is fermionic and arises when you have many particles together which behave in a manner which is different from a 'bare' particle. You refer to these quasiparticles as being 'dressed'. Collective excitations are essentially the same thing, only they are the bosonic result of some many body dressing of either fermionic or bosonic constituents.
A cooper pair is a non-perturbative bound state which forms as a result of the dressing which occurs. There is nothing shocking to me regarding the meV scale. 1meV=11.6K, so this is a meaningful scale on the temperatures in question. But I think I see the mistake you are making.
You are assuming that you can have a single cooper pair, separated by a long distance, and are asking how you can call this an object. And the answer is, that your intuition is correct. Cooper pairs only ever exist as the result of many electrons forming pairs (referred to as the condensate). You therefore have many electrons which are behaving as if they are paired. However, related to your anyon remark, the electrons do not 'know' who it is they are paired with. This is a many-particle instability.
You can never measure a cooper pair. You can measure an electron, or the absence of an electron, OR that all the electrons move macroscopically in groups of two ( and from this infer cooper pairs) but you cannot probe a system and get a cooper pair out. It is not a particle.
Please rephrase the question if I have misunderstoof.
@Kai and @James,
I must disagree with you on the concept of particle as it relates to Cooper Pairs.
A particle is what we get from quantising a Hamiltonian, and therefore this is what is typically measured in experiments. It is certainly possible to measure Cooper pairs in tunneling type experiments, and to observe the 2e step in conduction, or alternatively in superconducting box type experiments (Cooper-pair box).
In such experiments it can also be possible to observe single electrons, and these are called quasi-particles, as they are not the quanta pertinent to the system Hamiltonian.
The terminology around dressed particles can be somewhat loose, unfortunately, but it is generally taken to mean an excitation that contains both a particle and a field. For me, the canonical example is to be found in cavity quantum electrodynamics, where a particle (in this case perhaps the state of an electron) is dressed by an optical field, such that the collective state is a superposition of particle-like and field-like terms.
Thank you for your contributions but my question was devoted to ask about the different concepts of particle and how they can be adapted to different scales for obtaining different results. One electron in quantum field theory is seen as a fundamental particle which has an electromagnetic classical radius with a singularity in the chage and the energy as the radius is going much smaller than the previously mentioned . And obviously this cannot be because, among other things the electrons couldn't move at all following the relation between mass and energy provided by special relativity. But this problem was solved using the mathematical methods of renormalization in Quantum Electrodynamics which the abelian part of the Quantum Field Theory; Schwinger, Feynman and Tomonaga had the Nobel Price for this wonderful work. Physically the idea was to assume a bare electric charge surrounded by high energy photons able to transform in electron-positrons pairs which produced a electric dipoles and therefore changed the dielectric constant in such a form that the new particle could have a lower finite energy. The electron is not then point-like particle as Poincare or Lorentz have discussed, but a dressed particle with high energy photons.
But if we carry the electrons to a solid and we apply many-body theory, we can get it as an isolated particle just considering it is dressed with more particles as it could be phonons in superconductivity for obtaining Bogoliubov quasiparticles (electron+phonon). By the way, experimentally is possible to see the techniques used for measuring this quasiparticle
Science, Volume 297, Issue 5584, pp. 1148-1151 (2002)
And I could follow with well stablished concepts but there are many examples which they are not simple to distinguish and even difficult to say what one is working there. Please, what do you think about this problem?
Regarding the davis/mcelroy science article. ARPES is a measure of electronic states. This is quasiparticle interference. That produces peaks where there exists a strong nesting vector between momenta at the Fermi level. It is... not a complete picture, to say the least.
cheers!
A "collective excitation" is a state of motion of a system of many "fundamental particles".
A "particle" is an irreducible representation of the symmetry group of the system.
A "quasiparticle" is a "particle" and a "collective excitation".
A "dressed particle" is a "collective excitation" associated to an individual "fundamental particle" immersed in a "medium".
"Fundamental particles" are not experimentally observable.
Dear Miguel,
Thank you for your answer, but what is a magnon of a ferromagnetic materials as Fe3O4 into your classification?
let me take this opportunity to say that the quasi particles, like collective excitations are excited states, which could be applied to them the same idea as in QFT electrons as described. When temperatures as low as several degrees became experimentally available, physicists discovered that an enormous amount of experimental data on normal metals can be described by the model where one neglects electron–electron interactions. This apparent miracle was explained by Landau who demonstrated that the interaction pattern drastically simplifies
close to the Fermi surface. Provided the system does not undergo a symmetry breaking phase transition, all interactions except forward scattering effectively vanish on the Fermi surface and low-lying excitations carry quantum numbers of electrons. Therefore they are called quasi-particles as Alexi M. Tsvelik defined in his text:
Quantum Field Theory in Condensed Matter Physics, Cambridge University Press, 2007
which is a modern version of the classic book
A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (1963, 1975). Prentice-Hall, New Jersey; Dover Publications, New York
or
D. Pines, and P. Nozières, The Theory of Quantum Liquids (1966). W.A. Benjamin, New York. Volume I: Normal Fermi Liquids (1999). Westview Press, Boulder
Nowdays it was written another excellent book:
Anthony Duncan, The Conceptual Framework of Quantum Field Theory, Oxford University Press, 2012.
@Baldomir
A magnon is a "quasiparticle", which means that whatever the "true" ground state of Fe3O4 is, the structure of the low-lying excited states of the whole system is such that it can be approximately described as a collection of weakly-coupled particles. And one of the "free particle" modes happens to be, on closed inspection, just a "collective excitation" of the spin state of the "fundamenta particles".
You can look at the symmetry-breaking phase transition in the opposite way, as a "symmetry-restoring phase transition": As energy is increased across the phase transition point, the description in terms of the low-energy quasiparticles breaks down and a different kind of elementary excitation appears. When the energy is high enough, all that's left is "fundamental particles".
Dear Miguel,
From my point of view the magnon can be seen as a collective excitation or as a quasi particle. In the first case you can consider a precession of the spins due to a wave spin moving on the ferromagnet spin lattice. On the other hand, you can think as a magnetic moment which is moving as a defect of the spin lattice and in this last case we have then a quasi particle. Perhaps the only form to define properly if it is or not a quasi particle the defect that we have imagined is going to the criterium of Landau and to observe if h/t
@Daniel
You seem to be implying that "a particle" requires that you can associate it to a lattice site. By that definition, a phonon is not a particle.
Which is fine with me as far as naming goes, but phonons do behave like a quantum gas of point Bose particles for all intents and purposes, at least at low energies and occupation numbers.
Dear Miguel,
You are right, the phonon is not a quasi particle , it is a collective excitation. And you are right with your remark You are right the phonon is not a quasi particle , it is a collective excitation, from my humble point of view. And you are right with your remark on it, there are many different of excitation states in condensed matter which have a misunderstanding name.
But the phonon hamiltonian looks like the hamiltonian of a free particle. Even magnons have an effective mass.
In the weak (fundamental-particle) coupling limit the "quasiparticle" really is an excitation localised at a lattice site. This is described for instance in Feyman's lectures Vol. III. But in the strong-coupling limit (as in phonons or magnons) an excitation localised at a lattice site is not even close to describing the "quasiparticle" eigenstates.
Dear Miguel.
Thank you for your interest.
Right, absolutely agree with you but notice how we have an undefined concept which can be used without a clear cut-off. The limits of criteria as the one given by Landau are not enough and I do not know if actually there is a good classification.
Fundamental particles, or quasi-particles, or whatever, are all excitations in their quantum fields. Thus electrons and positrons are excitations of the Dirac field. The photon is the excitation of the electromagnetic field, a phone is an excitation of an elastic field, a magnon is an excitation of a spin field, skyrmions, composite Fermions etc, are found in quantum-hall fields etc.
Many paradoxical things become clear if you recognize that particles are simply excitations in a quantum field that can pop up at some location with a probability proportional to the square of the amplitude of the field. Read my discussion of the two-slit experiment (that you can do with electrons, photons, phonons, or even buckyballs), in my book that I mention further below, to see how the usual paradoxical questions (about which path did the electron take etc) become clarified.
The only thing that distinguishes a "fundamental particle" like an electron, or a 'quasiparticle' like a phonon or a magnon or a BSC-quasiparticle (in superconducirtivity) is the energy-scale applicable to each particle (and t the boundary conditions imposed on the quantum field).
This issue of energy scales and boundary conditions imposed on the differential equations defining the fields are very important but often ignored in physics discussions. That is why I have emphasized these repeatedly in my book which also attempts to address philosophical questions associated with physics questions.
See:
"A physicist's view of matter and mind", published by World Scientific last June. See
http://www.amazon.com/A-Physicists-View-Matter-Mind/dp/9814425419
Thus, an electron (a fundamental elementary particle) can exist at huge energies, unlike what are called 'quasi-particles'. A helium atom is also a particle if you probe it with low energies (significantly smaller than its ionization potential). The energy scales of plasmons (in metals) are in the electron volt range. But phonons and BCS_quasiparticles are in the milivolt range.
Dear Dharma,
Thank you very much for your answer and the reference to your appealing book.
I am absolutely agree with you that the particles are just excited states of fields. The paradigmatic case is the photon (boson) and the electron or positron (fermions) as states in the Dirac equation. In high energy physics it seems quite clear what could be the definition of particle or field, even if we go to Yang-Mills equations and to enlarge the U(1) group of symmetry and we use the Noether's theorem for conserving them into a variational context.
The problem for me, and this is actually my question, comes when you try to distinguish these fields, quasi particles, collective excitations or composite particles. Coming back to your answer, what is a magnon for you? what is a BCS-quasiparticle that you say? Thank you in advance.
Let me use examples from condensed matter physics where representants of the mentioned types of particles exist:
Dressed particle can be represented e.g., by a polaron which is an electron in periodic lattice (Bloch electron), accompanied by its polarized neighborhood (phonons). Here, both the lattice electron and polaron are quasiparticles.
Quasiparticles: If a Hamiltonian of a system on a periodic lattice (e.g. Hamiltonian of electron in a periodic potential, vibrations of the lattice sites, etc ) can diagonalized, usually by Fourier transformation into Bloch reciprocal space, to a form of an effective particle-like Hamiltonian with an effective mass, then it is possible to define quasiparticles as respective excitations of the whole lattice : Bloch electrons, phonons, excitons, magnons , polarons, polaritons.
Collective excitations in CMP can be introduced as an alternative representation restoring translation invariance for some localized excitations (solitons, kinks, instantons) of interacting systems with specific symmetry .
Eva makes an important point: "collective excitation" includes "nonperturbative" objects which are essentially distinct from "particles" (broadly understood). The distinction between "particle", "quasiparticle" and "fundamental particle" is more philosophical than anything else as the mathematical formalism for "particles" (and fields) is the same (at least in limited energy ranges as Dharma points out: at high energy "quasiparticles" decay into "fundamental" particles).
When you break a Cooper pair you obtain what is known as Bogoliubov quasiparticle, which in principle would be the same as a polaron because we have an electron plus a phonon. What are the differences?
If you have an skyrmion as a soliton and following the interesting observation of Eva about the definitions of collective excitations as solitons; how is it possible to define it as being a fermion into de massive Thirring model (using the Bethe ansatz) and therefore as a quasiparticle?
I think that it is necessary to have a proper definition for all these excitations on solid state physics, what do you think? It seems that we have different concepts or names for calling the same object
Daniel, the Cooper pairing is pairing in the k-space, while the polaron is the dressed electron (hole) in the direct space.
It seems to me, that there is a bit of misunderstanding: the soliton as a special solution of the massive Thirring model is a localized object, \Phi= \Phi (X), but the respective Lagrangiang is translation invariant. In order to restore the invariance it is necessary to introduce an arbitrary constant coordinate X_0, \Phi(X-X_0) . If we set X_0(t) as time dependent, then by integrating the Lagrangian over X we obtain Lagrangian as function of X_0 (t) , \dot X_0(t), i.e. dynamic equation for X_0(t) implies. (The same can be done for the width of the soliton L(t), so that one obtains two coupled equations for X_0(t), L(t). These quations represent then dynamics of the soliton as a particle (not a quasiparticle ) and are called collective coordinates.
In the linear theories on a periodic latiice, the base for introducing the quasiparticles (objects of linear theories only) is the Fourier transformation into k-space, which diagonalizes respective Hamiltonians and enables definition of quasiparticles. There can remain some (small) additional terms which can be considered as interactions within a perturbation theory.
Dear Eva thank you very much for your contribution,
I am not sure if I have understood you properly. You are right that the Cooper pair is defined in k-space, but this feature cannot avoid to know if we are speaking or not of certain kind of excitation. k and r may be straightforwardly related by Fourier transformations assuming that we have spatial periodicity.
It is also very difficult to follow your worry on the spatial traslation symmetry condition: this is obviously a necessary condition if we want to have a defined moment or k, and also necessary for solving a sine-Gordon equation. But what was trying to say is that you can have a topological term that you can add to a chiral Lagrangian, whose integral depends upon the homotopy class. This provides you with topological sectors in the quantized general model of Thirring. Therefore what is important is that you have non-local (no dependent of coordinates) sectors as skyrmions. What are they particles or quasi particles? How do you relate your soliton with this topological last one?For me it is quite difficult to say if I follow the usual definitions.
Essentially (much of) the difference is due to history, what people knew at some moment in time. and what properties of matter they want(ed) to highlight.
An electron is a ``fundamental'' particle in the sense that, up to now, it behaves like a point charge and experiments at the LHC can test and have tested, whether it's possible to observe any ``substructure''-the answer is No.
The term ``dressed particle'' is, sometimes, used to distinguish a particle without interactions from the same particle when interactions are taken into account. One sometimes says that the interactions ``dress'' the particle. People sometimes talk about ``bare'' charge, or mass to stress that these quantities are attributed to the ``free'' particle, when interactions with other particles can be neglected.
In a superconductor the electrons can form (Cooper) pairs, that condense in the superconducting phase. But it is possible to perform measurements, when not all are condensed, of course, that show that these objects, of charge twice that of an electron, possess some structure: they have a size, given by the coherence length, in real space. Also they're spin-0 objects (in ``s-wave'' superconductors), since the two electrons are in the singlet state, for example; in ``p-wave'' binding they would be in the spin-1 state. Since such a pair is a bound state, its mass is greater than twice the mass of the electron. Here the mass is the ``effective mass'' of the electron, when it moves in the medium.
Quasi-particles is a term that is used to describe the particle that corresponds to an excitation of a medium, when we want to stress that it has a finite lifetime--so the term makes sense for times much less than this lifetime.
A collective excitation, as the name suggests, is an excitation of a medium that corresponds to the coherent motion of more than one particles-and, by convention, more than two.
The electron is also a dressed particle. Its observed mass is assumed to be the result of 'regularization' of the infinities in dealing with the base mass in quantum electrodynamics. The main difference between fundamental particles and quasi-particles or dressed particles is simply the energy range of the associated Lagrangians and the boundary conditions put on the fields. They enter into the integrals and cutoffs used in the regularization of the divergent integrals of field theories. I have discussed Fermi-liquid quasiparticles in Ch. 8 of my book, but the issue of keeping track of energy scales and boundary conditions is, in m view, very important for clear conceptualization of the foundations of physics, as I emphasize through out chapters 1, 2, of my book (physicist's view of matter and mind) and in many other chapters. It is the boundary conditions that produce quantization.
Of all the particles, the photon is the most difficult one to understand with its zero mass and impossibility of localization. Willis Lamb and others have tried to even do away with the concept of a photon! However, I have discussed its conceptual basis in Sec 6.3.2 on wards in my book, and argued that every observable photon is actually a polariton in the sense defined there.
There has been quite a bit of discussion here lately, but I want to clarify one simple point raised by @Daniel: namely what is a magnon? Answer: a (composite) boson.
Ultimately the reason for responding here is that much of the discussions above seem to be concentrating on classifying excitations, which I believe to be more a distraction from the fact that the phenomenology that is associated with the excitations is more important, and of special interest to some of my research is in highlighting the connections and applying results from one branch of physics into another.
It is certainly the case that a magnon is a particle: it behaves like a particle, it can be quantised like a particle, and can even be guided at the single particle level (to show also its wave-like properties). However it is a composite boson, comprised of a collective excitation of many spins. There is no real reason to presuppose strongly or weakly coupled spins in this case.
Because it is a collective excitation, rather than an intrinsic particle, there are different channels for decoherence and loss (based on the lattice), but in many cases this is not so significant, although of course the fact that relatively low energy might be required to excite magnons (and indeed other collective excitations like phonons) means that low temperature operation is often required to demonstrate effects.
As an example of two-magnon effects which shows scattering akin to particle-particle interactions (and I apologise for referencing work I have been involved with, but there are many good references as well in this paper), I refer you to Longo et al., Phys. Lett. A 377, 1242 (2013). There is also a fairly clear (IMO) description of how to use the Holstein-Primakoff transformation to derive the magnonic creation/annihilation operators in a spin chain, which is of course background, but we needed to do it for the interactions between bound and scattering magnons in our problem.
Daniel, for completeness, I would like to distinguish between different languages:
The procedure I presented above is related to the sine-Gordon Lagrangian, known to be equivalent to the massive Thirring model for Q=0, with the use of the Rice's Ansatz: M.J.Rice, Phys.Rev.B 28, 3587 (1983)., also R. Rajaraman, Solitons and Instantons, 1982.
Dear Nicolis, Dharma and Greentree,
First of all thank you very much for your answers which are full of knowledge and good physical sense, but unfortunatelly they do not solve the problem: how can we distinguish these terms which nowdays are well accepted in condensed matter and in physics?.
It seems that Nicolis say that they have only a historical origin and we do not need to be worry if their name corresponds to an actual physical concept. I must say that this is not my humble opinion and think that the concepts are clear but no their application in the literature. My best example is the magnon that Greentre call a (composite) boson. Frankly I do not understand this name: it is true that it is a boson but I do not follow composite in this context. In fact a magnon is always considered as a collective excitation due to a perturbation of electron's spins but with could be also seen as a quasiparticle carrying mass and having a finite lifetime.
Dharma refers that the details are in his book, but I think that we only need to know the solution to a narrow question without going to further deeper readings, obviously this does not prevents to do it in another moment.
Dear Eva,
Thank you very much but I am more interested to know how is possible that we can have unique name for the soliton when we can interpret it in so different physical behavior. One a very local object acting as a quasiparticle and another as a topological excitation, do you follow me? The subject is very huge and we could go to the very fashionable interesting literature on the subject of skyrmions but the question is to solve how we can distinguish in condensed matter all these excited states and if the names are univoque.
To distinguish them we need a well-defined setup. It *is* possible to distinguish a Cooper pair from an electron in a superconductor: we measure charge, spin and mass.
In a magnetic material the magnetization varies in time and space. It's possible to describe this as a wave of given wavenumber and frequency, which corresponds to one mode. It is also possible to make wave packets, superpositions of many modes. Such wavepackets don't have well defined wavenumber or frequency, but well-defined position in real space: they can be considered as particles and are called magnons. Within this approximation we may associate other properties to them, namely statistics-and they turn out to be bosons, sometimes.
As magnetic excitations, they involve the spin of constituents of matter, namely electrons. So if we probe their structure we see that they are not of the same sort as the electrons of the medium. When we want to stress the wave-like properties we then talk about spin waves-when we want to stress the particle-like properties, we talk about magnons.
What's interesting is that we *can* see, experimentally, that magnons or spin waves make sense as such beyond some scale: they *do* involve many electrons of the material. When we can start probing individual electrons, the description in terms of magnons/spin waves is no longer appropriate.
What I'd like to stress is that what things are *called* is convention: what they *do* is physics. As long as it's clear how to calculate with them and how to measure them, what they're called is of historical interest.
It is precisely experiments such as the Hall effect which allow us to see that the "proper" description of certain systems is not in terms of "fundamental" particles. Just like the classical Hall effect can show current carriers to be positively charged (i.e., holes), so the fractional quantum Hall effect shows that the current carriers have fractional charge. The genius of Laughlin was to exhibit a kind of collective state that can have fractional charge.
This precisely shows the ``metaphysical'' nature of many such terms, when carried beyond their scope-which, of course, is realized a posteriori and highlights that it is the system that controls what terms make sense.
In the fractional quantum Hall effect it turns out that the objects that are useful to be described as particles carry fractional charge. What's, however, interesting is that it is possible to realize transitions, from such a state to others where the particles carry integer charge (in units of the electron's charge). incidentally, while the Laughlin state is, of course, the correct description, how it arises in terms of the dynamics of these systems is, still, not a settled issue.
Dear Nicolis,
For being more concrete, if I write that the magnon is a quasiparticle, what do you think about?
Of course you can-but what matters is what you do with the equations that describe its properties and how you plan your experiments. These need to be given prominence, not the name. Of course terms have history and that's what I tried to highlight in my previous message. But history isn't physics. A bound state of a charm quark and a charm antiquark is called the J/psi: one group calls it J, another calls it psi. But everyone understands the physics involved. Gell-Mann called some objects ``quarks'', Zweig called them ``aces''-which term is more widely used is sociology, what they express is physics. The ``top'' quark for some time was called ``truth''. And so on. What matters is consistency of terminology: if you use ``quasiparticle'' in more than one ways, you should explain the reason. If you use it in a way that isn't the same with what other people do, it's useful to spell out your definitions. But once you write the equations and describe your apparatus, who cares what it's called?
Things surely can have different names depending on what you want to emphasize, and depending on your perspective.
Let us go back to the magnon. Paul Dirac in around 1929-1930 restricted the enegy scale of a pair of electrons to just two levels, and showed that the anti-symmetrization can be re-interpreted as a transition between singlet and triplet states. The energy scale of this was named J. Then within that energy scale, the electronic Hamiltonian or Lagrangian can be transformed into an expression written purely in terms of spin operators. These obey Pauli statistics. But an approximation (e.g., Holstein's) gives Bose statistics and the excitations are called magnons if you want to think of them as particles. Alternatively, you can think of them as spin-waves.
Now, if you go to energy scales bigger than J, the two-level approximation is not enough, and you need to return to the electronic Hamiltonian. A simple form of that is the tight-binding (or LCAO) model. Here we can now have electron-hole pairs. They can be quantized into "quasi-particles" known as excitons. They can be singlet excitons and triplet excitons. The energy difference between singlet and triplet excitations is of the order of J, but the energy scale of excitons is of the order of the band-gap.
You can increase the energy scales even further. Then the tight-binding models have to be replaced by a hamiltonian which fully includes the nuclei and electrons, and use a very large Hilbert space of functions ( see sec 7.1 to 7.3 of my book for details, relation to entanglement etc). At this level, you have electrons, nuclei, and photons as the essential fundamental particles.
You can increase the energy scale even further, up to, say the nuclear energy scales, when you need to write a Lagrangian (or Hamiltonian) in terms of protons, neutrons, electrons, photons, and fields that transmit the nuclear interactions (baryon fields).
The "flow" from one type of theory to another as the energy and length scales are changed is given by the renormalization group".
At the highest energy scales we know of today, we are at the Plank scale of length and energy. That is the per-inflationary state. In the earliest state we assume that the laws of symmetry, and some laws like least action, conservation etc., hold, and these give us the capacity to construct excitations (in strings, branes or what ever). That gives you string theory or M-brane theory. The initial inflationary expansion may well be some quasi-particle of a topology or field at even higher energy scales than the Plank scale. In that sense, the whole universe is an excitation whose internal modes are the lower energy stuff that evolves along the flow of the renormalization group, and produce the particles, quasiparticles that we talk about, molecules etc that we know.
In that picture, the evolution of the whole universe is simply the decay of this quasi-particle, seen from an inside perspective! But we don't call it the decay of a quasi-particle. We have other names like "cosmic evolution".
@Daniel,
The term composite boson is only to distinguish the excitation from an elementary boson. So a photon travelling in a vacuum is an elementary boson. The magnon is a composite boson because it cannot exist without the spins that it is an excitation of. The same goes for phonons, which are composite bosons of the vibrational energy.
As @Stam says, what matters here is what you want to do with the particles and the experiments that you want to perform. And additionally, there can be different decay properties and pathways for a composite particle compared to an elementary particle, so some details of the comparisons between elementary and composite may be important.
I would not refer to a magnon as a quasi-particle. I note that Wikipedia suggests: "Usually, an elementary excitation [referring to collective excitations that we have been discussing] is called a "quasiparticle" if it is a fermion and a "collective excitation" if it is a boson.[1] However, the precise distinction is not universally agreed.[2]" (http://en.wikipedia.org/wiki/Quasiparticle). The issue for me is that the particles such as composite bosons really are particles, inasmuch as they behave like particles, they result from exactly the same quantisation physics as particles, and they can be measured as particles. Furthermore, quasiparticle has a clear meaning, at least in the context of superconductors, which has the potential to cause confusion.
Magnons are quasiparticles obtained as a result of quantization of spin waves in ferromagnetic crystals.
For this definiton as well as those analogical for other quasiparticles in question as excitations of linear theories (Bloch electrons, phonons, excitons etc ), see e.g. the textbook by Ashcroft, Mermin; Solid State Physics.
I am quite agree with you, Eva; the usual form of seeing the magnon is as a quasiparticle. In fact it is taken the same picture as the phonons for the calculations and I think in pages 330-336 of the 8th edition of C.Kittel where you can see even the scattering of this quasi particle with a neutron.
From my humble point of view there is a confusion in the literature on this issue and I am not in agreement with the above people who seems to define freely the words. In the theory of fundamental particles everybody can distinguish univocally what is a baryon or a meson as parts of hadrons, and to perfectly distinguish them, for instance, of the leptons. Thanks to that it was clearly possible to find the interaction of quarks by means of gluons in the different views as coulors or flavors depending of the symmetry used in the theory. The meson J/psi has zero value for the flavor, i.e. they have the same number of charm of quarks and antiquarks. This is a meson with a mass of 3096,9 MeV/c^2 and a lifetime of 7,2 10 ^(-21) seconds. That is, we have very well defined the physics and the particles with their names for particle physics and J/psi is a name univoque to a meson, not the case that we have in condensed matter that it seems that we have the SAME physical object with different not NAMES but classifications. Notice that quasiparticle is a classification of many different objects as phonon, polaron, plasmon, hole...
Daniel, yes, the particle physics obviously uses different terminology related to fundamental patricles than the solid state physics:Here the base for the description of excitations is the periodic lattice (nowadays also in quantum optics - the optical lattice). Among their characteristics the wave vector is always taken as a "quantum number" when classifying the Bloch band states.
There is also to be mentioned an ambiguity in the terminology: sometimes there is used as an alternative to the term "quasiparticle ", the term "collective excitation". Logically it is correct, because all the lattice "shares" the excitation (therefore the wave vector), but lateron the term collective excitation has been used rather for nonlinear (nonperturbative) excitations, solitons, etc.
Magnaons are quasiparticles, but with respect to the electronic Hailtonian, and the magnon is a particles with respect to the restricted quantum-field of the spin Hamiltonian. The electron is a particle with respect to the vacuum with the electromagnetic and Dirac fields integrated out. If they are kept, the electron bare mass does NOT agree with the observed mass. So, with respect to the more detailed hamiltonian, the electron is a quasiparticle with a bare mass renormalized to an effective mass that we observe. Similarly, the g-factor of the electron has got modified etc, etc. These renormalizations and construction of the observed excitations from the bare excitations was what got Dyson, Schwinger, Feynmann and Tomanaga etc their Nobel prize.
So, what is a quasiparticle or a particle has to be specified with with respect to the defining hamilonian (i.e., its enegy scale). If you just say "quasi-particle", that is OK in a community where everybody knows what hamiltonian you have in the back of your mind.
I have said this enough times, and so I will drop out of this discussion.
Thank you dear Dharma,
What is the hamiltonian for the skyrmion as a topological exctation. What is the hamiltonian for the magnon as collective excitation? How could you give their mass at rest or lifetimes? Thanks in advance.
Perhaps my last questions might carry you to misunderstand my opinion or what I am trying to look for with my initial question. Notice that my question would be nonsense if we could easily translate the excited states in, say, quasi-particles or in collective states.
The question is that there are even situations where these concepts are not possible to define, one paradigmatic example is given by a chain of interacting electrons in one 1-dimensional metal. In this case is not possible to define quasi-particles o collective excitations as you seem to reduce the possibilities of the excited states. It was necessary to create the concept of Luttinger liquid for this case and the charge and spin could be separated objects moving at different velocities. One interesting application are the electrons moving on the edge of a Fractional Hall Effect system or on a Topological Insulator.
Indeed: in the system you mention, what happens is that, as soon as you introduce interactions, you cannot express the excitations as local functionals of the original fields. So you can ask whether there exist *other* fields, in terms of which the excitations can be expressed locally-and you find, of course, that these new fields are *non-local* expressions of the original ones.
Daniel: The low dimensional systems, D=1,2 are of course very different, because the concept of mean field as a basis for the 3D quasiparticle desciption does not apply.
Therefore, the nonlinear interaction terms which can be in 3D included into the Hartree-Fock mean field+ small interactions, remain explicit and the system persists to be nonlinear. Then the concepts of either the Luttinger or massive Thirring models were created. The separation of the interaction term in the sine-Gordon equation on effectively noninteracting bosons is then a matter of the value of the coefficient \beta in the cos (\beta \Phi). The role of excitations in, e.g. massive Thirring model, or the equivalent sine-Gordon model play nonlinear exitations as the solitons, breathers etc. (Luther, Poeschel, Emery, Solyom, ...)
These are the concepts as used in 1D interacting electron gas, or in the theory of Peierls instability (Su-Schriefer-Heeger model). .
Regarding the edge of a Fractional Hall effect system or the topological insulator , I would appreciate if You send me the references.
Thank You.
Dear Nicolis,
Thank you for your answer. Why do you have this problem in one dimension and no in three? Please, could you explain a little bit deepler the sentence "you cannot express the excitations as local functionals of the original fields"
Dear Eva,
Thank you very much by your contribution. Although there some points in your answer that I need to analyse deeper, I must say that I follow you. But I am not agree that the main problem is that you cannot have a mean field approach in 1 or 2-dimensions for having topological solutions there. In any case I write you to references on the subject that you have asked me:
M. M. Vazifeh and M. Franz, Spin response of electrons on the surface of a topological insulator, Phys. Rev. B 86, 045451 (2012)
M. M. Vazifeh and M. Franz, Quantization and 2π periodicity of the axion action in topological insulators,Phys. Rev. B 82, 233103 (2010)
It's possible to see this in many ways-and each person has their favorite, depending on their background and their interests. One way, I think, to see this is how it is possible (or not) to define locality in each case. So look at the propagator of a local field and how it behaves at large space-like distances where you expect it to describe the propagation of a physical particle. In one dimension, you'll find, in fact, that it *increases* with the distance, in two dimensions it increases, but logarithmically and only in three dimensions and above does it decrease. You deduce from this result that correlations increase with distance in one and two dimensions, therefore, local means the whole system, i.e. there isn't any notion of locality for fields with these properties.
That's the, rough, ``intuitive''', idea. I'll try to look up some references that are more precise.
Excitations in quantum-hall systems, (e.g., fractional quantum Hall effect) are very interesting because you cannot do mean-field theory. However, using exact diagonalization of small numbers of electrons, or QMC, or classical-maps of the quantum problem (as in Laughlin's ansatz), we can construct the excitations. Now, at each energy scale there is a specific Hamiltonian. The excitations at that level can interact via residual interactions to form composite particles which are the quasiparticles at a higher level etc. The resulting hierarchy of particles (or quasi-particles when viewed from a different level) is known as the Haldane Hierarchy.
(The hierarchy can be mapped onto a hierarchy of classical fluids, as we showed in a paper by MacDonald, Aers and Myself [ Physical Review B 31 (8) 5529.] where reference to the Haldane Hierarchy of quasi-particles may be found. Each classical fluid is made up of a set of particles which correspond to the quasiparticles of a parent quantum system.
Stam Nicolis: Strictly mathematically, the dependences of the interaction behaviour on the dimensions You mentioned is certainly correct, if the interaction would be confined in 1D.. However, real physical structures often behave differently from those ideal mathematical models. There use to exist components of interaction in other dimensions, so that in reality we are left with quasi-1D, or quasi-2D structures. .
Examples: A charged impurity in 1D chain of atoms (molecules) exhibits 3D-like Coulomb field.
A typical example of quasi-1D behaviour is the Peierls instability in 1D polymers, which should not exist according to the Mermin-Wagner theorem in 1D, however the interactions responsible for the symmetry breaking are not confined in 1D.
Similar behaviour is known also in 2D Coulomb plasma .
Dear Daniel, thank You very much for the references. I am not sure what kind of topological excitations You mean.
Generally, it is nothing surprising that mean field theories do not work in D=1,2, simply because there is not enough neighbour particles for creating the mean field for a given particle. For example, if we take 1D Ising model, according to the exact Ising solution, there does not exist phase transition (T_c=0) except if there act infinite-range interactions, while the mean field theory allows for the phase transition
independently of the dimension.
Dear Eva,
I'm afraid that we have a misunderstood between us. In 1-D Ising model for a mean field approach is given by
Z = e^{-\beta J m^2 N z /2} \left[2 \cosh\left(\frac{h+m J z}{k_BT} \right)\right]^{N}
( sorry for the formula but I cannot copy it better and it is only for fixing the idea)
where Z is the partition function, h external magnetic field, T temperature, m magnetic moment taken as the mean value of the spins, z the coordination number, J is the coupling constant between two spins which only can take values +1 (up) or -1 (down) because we are in an Ising model, with cubic symmetry.
You are right that we have a singularity for T=0, which separates the ordered ferromagnetic phase T0 taking the critical temperature at cero. Notice that we have, e.g. the same problem for the Coumbic potencial in the center of the souce charge or so on. This is not a topological problem as I was thinkin when I wrote you. The topological problem is related with the homotopy classes that you can define in two dimensions or in three for a given hamiltonian containing terms as Chern numbers or so on.
In any case now I understand your previous answer
Dear Daniel, thank You very much. I shall look at the topological terms in 2D.
Dear Nicolis,
I do not fully understand your explanations and perhaps we are speaking about different things. Sorry.
Your last answer was based in the propagator or the Green function, but if you do that then you are assuming a perturbative method (great advantage of having quasiparticles) which cannot give us, in any case, topological solutions. That is to say, why do you employ a method which you know that is not the good one? Why are not going directly to topological excitations or models?
It would be very nice if you could send the promised reference because it is possible that I had misunderstand your explanation. Thank you in advance.
Dear Eva,
let me try to help with the topological solutions that I am speaking on.
The first topological particle was introduced in 1932 by Dirac for justifying the quantization of the electric charge, for instance that so different particles as the electron or the proton were having exactly the same electric charge but with different sign. He introduced the magnetic monopole as a long string where never could touch another particle, i.e. the density of probability on the string will be always zero. Some years later, Wu and Yang could prove that the string was not necessary at all if it was used a Principal Fiber Bundle where the fiber was U(1), unitary group of dimension one, base the space-time and the physical configuration containing the magnetic monopole would be the non trivial topological one. The topological solution allowed to have locally the divergence of B equal zero although we had a magnetic source particle. Mathematically we enlarged the Gauss theorem for the integrals surrounding the magnetic monopole.. Later on Schwinger solved the relativistic invariance of the quantum theory of the magnetic and electric charges introducing a more general model for the string showing that the commutaton relations for the density of the energy-momentum tensor of the electromagnetic fields do not contradice the relativistic invariance if a quantization condition for both electric and magnetic charges is satisfied. In both cases we have that first homotopy pi_1( sphere dimension 1) =Z integers and notice that locally we have SO(2) equivalent U(1), i.e. el grupo ortogonal special which keep invariant the metric tensor in two dimensions is equivalent to the electromagnetic group which has as Noether current the electric (or magnetic) charge.
Finally t'Hooft and Polyakov generalized the group of symmetry to SU(2) into a model with a soliton without singularities and which is nowadays the solution found in great unified theories. The above group is equivalent to one sphere of two dimensions and the homotopy group which make it non trivial is then of two dimensions two. This is the most clear physical case of topological particle that I know.
Dear Daniel,
thank You for Your explanations. The last mentioned SU(2) symmetry of some condensed matter models lead to application of the field-theoretical concepts.
Therefore our different starting points in the discussion.
Dear Dharma-Wardana,
I was reading your Phys.Rev.B and it is very interesting indeed, specially the figure 3 but you use only one hamiltonian and with a Hartree-Fock approach, isn't? You compare your results with the Haldane Hierarchy assuming you have quasielectrons and quasiholes. This is extremelly interesting for me but I cannot see the correspondence between exciton-hamiltonian that I can follow of your answers to my initial question. In any case, could we try to write a table where the my excitations are classified without any doubt?. Perhaps you could also enlarge the number of concepts for determining them as it could be the hamiltonian, although I do not think it would be necessary
statistics "particle"
Screening electron fermion fundamental particle
Bloch electron fermion quasi particle
Wannier electron fermion quasi particle
Phonon boson collective excitation
Polaron fermion quasi particle
Plasmon boson collective excitation
Bogolibov electron fermion quasi particle
In a simplest conceptual frame work, a particle which can exist in vacuum is a fundamental particle, while same particle (say electron in a conductor) is a quasi- particle, and phonon is a collective excitation; perhaps it is difficult to clearly distinguish between quasi-particle and dressed particle. However, if there is a distinction it will be a useful information.
A quasiparticle (as a quantum of an excitation field in a periodic crystal with a tensor effective mass ) can be dressed, e.g. polaron is a dressed Bloch electron (quasiparticle). In polar crystals it is "dressed" due to interaction with phonons The dressing is reflected in the increase of its effective mass due to the interaction but the characteristics of a quasiparticle remain. The modification of the effective mass is caused by a polarization of the electron neighbourhood which provides a potential well where the electron is selflocalized .
Phonon is also a quasiparticle. The quasiparticles are also collective excitations, because all the crystal shares respective excitation. The terminology is not so strict. In any case, quasiparticles are quanta of the excitation fields in periodic lattices, e.g. phonons are quanta of atomic (molecular) oscillation fields in periodic lattices.
Similarly, excitons nad magnons are quanta of the optical excitation and spin waves in crystals, respectively
Perhaps it would be a good idea try to define what is a quasi-particle and a collective excitation before going to a classification.
These terms are quite confused in the literature and in different applications. For instance, quasi particles in BCS superconductivity are spinful objects with no well defined charge, in contrast with singlet Cooper pairs, which haven't spin but their electric charge 2e it is. In fact, the quasi particles in this context are just combinations of electrons and holes through their second quatized associated operators for obtaining excited states very close to the BCS fundamental state . Where is here the dressed concept that Landau had for transforming the Fermi liquid in the Fermi gas? For me is a very different concept of quasi particle in superconductivity than metallic electrons, for instance. What do you think about?
Dear Jain,
The electrons in "vacuum" are also dressed particles. The renormalization is obtained thanks to be able to dress the bare electron with high energy photons which at the same time are able to transform in pairs electron-positron. Vacuum is not nothing, of course.
The same picture was translated to solid state physics by Landau when tried to get Fermi gas of particles close to the Fermi level when it has another system with interaction among them. This is always the same problem of many-body theory, try to avoid the interaction. For instance the density funcional theory (DFT) which makes it for the fundamental state of an electronic system.
As a clear example of real and quasi particle, we can quote photon and phonon, respectively. Perhaps it will be helpful if address a real particle such as electron, or even an atom in a fluid as dressed particle. In any case the use of different terms is confusing but it is not difficult to understand the context and meaning.
The Kondo effect is one of the exemples in solid state physics that does not need to introduce quasiparticles. You need only a two-orbital molecule described by an Anderson-type Hamiltonian for explaining the Kondo temperature of diluted magnetic impurity in a electron gas. See for example
Phys.Rev.B 149, 491 (1966)
Dear Jain,
I agree only partially with you. Notice the dificulty to distinguish the dispersion curves of polaritons (quasiparticles) between photons or phonons within the longitudinal and transversal optical modes. Where they follow the Lyddane-Sachs-Teller relation. Phys.Rev. 132, 563 (1963)
For clearness and to avoid misunderstandings, allow me to remind, that the quasiparticles in condensed matter physics are characterized by the effective mass which is the tensor {m_{i,j}}, i,j,=x,y,z depending on the crystal structure. Only for the special case of cubic symmetry, it becomes a scalar, m_{i,j}=m\delta_{i,j}. Similarly, the wave vector k is a quasiwave vector, because it is a quasicontinuous (quantized so that the energy spectrum E_{nk} shows very small level spacings ),
k_i= (2\pi/N_ia_i ).n_i, -N_i
Dear Eva,
I am sorry to be in disagreement with your previos answer for trying to characterize the quasi particles. Let me to put some points clear:
1- I never use the tensor expresion of the mass for the quasi-particles because it is taken effective mass calculated on the bands in the first Brillouin zone. It is taken locally as one scalar due to the impossibility to know the real motion of the quasi particles and the crystaline symmetries are also taking into account within the bands used. Finally, the quasi- particle concept tries to renomalize the mass of the fundamental particle interacting in a medium, using the propagator associated to the Hamiltonian, assuming perturbation methods for avoiding the real interaction dressing them with colllective excitations .
2- But the quasi particles are, over all, determined by the lifetime which is assumed to tend to infinity in the Fermi level. That is to say they are almost absolutely stable particles there. The lifetime stems from the collition with other quasi-particles. When the lifetime of the quasi-particle is infinite, the state with such a particle is one eigenstate of the whole system that they are defined. Therefore we can know quite well if they are a physical object or not.
These are the two features which define what a quasi-paricle is and they are quite general without being necessary to have them in a crystal with discrete as it can happens with a Fermi liquid in general.
@Daniel:
The Hall effect and quantum Hall effect allow the experimental measurement of the charge and number density of current carriers, while effective mass can be determined using cyclotron resonance and other methods. Even the anisotropy ot the effective mass is potentially observable by these means.
Given that, and knowing nothing of the microscopic structure of materials, a physicist equipped with 1930's quantum theory would formulate an effective field theory in which the building blocks would be what we'd call "Landau quasiparticles". The effective field theory would cease to work at high enough energies for experiments to reveal the "underlying" structure of the material - for instance, the photoelectric effect would reveal the nature of the "fundamental" "current carrier"; X-ray or neutron diffraction would reveal the crystal structure, Raman spectroscopy whould probe the phonon spectrum, and so on and so forth.
I don't think it's very productive to insist on strict definitions of the terms once one understands how an "effective field theory" works.
Dear Miguel,
I am not agree with you. The definitions that we have in the literature and even the ones given a long this discussion are not univoque. I remember when very early you have defined them in the basic and important aspects.
It is true that we are now in a cycle where the amount of knowledge more important on the subject is going into details, sometimes a little far. But I must say that I have learned things and linked new ideas.
In any case, my hope is to have anybody who could give us a new scope of this issue and distinguish those concepts better. Notice that we haven't touch yet the most important ones which are related with the topology and how we can compare them. I have found many mistakes in the literature with these names which obviously I am not going to mention, but which show the confusion of the subject in physics.
Dear Daniel,
when the energy of the quasiparticle (Bloch electron) approaches the Fermi energy, it vanishes in the Fermi sea, because it is defined as excitation above the Fermi energy.
The term "quasi" in condensed matter physics stresses the difference between the particle outside the crystal and inside it, nothing more.
Dear Eva,
The energy of the quasiparticle Epp is linear close to the nivel of Fermi Ef
Eqp= Ef+ Vf( p-pf)
where the slope is the velocity of Fermi
Vf= (partial derivative of Eqp respect to p) in p=pf
and the effective mass mf is given by
mf= pf/ Vf
which is calculate using the bands of energy if we are in a solid and no the tensorial mass because we do not distinguish between Ki and Kj of different directions in the crystal. Obviously this can be done if the anisotropy of the medium is very high, but in any case this is outside of the concept of quasiparticle and belongs to every particle which could move within it.
Generally, the effective mass is defined as the inverse of the second derivative of the energy dispersion law by k , \delta^2 E(k)/ dk^2 in the point of a minimum of the band (for a cubic structure, otherwise the definition is modificated because of its tensor character). Only in the case of quadratic dispersion law it is a constant.
If the dispersion law is linear, then the electron is effectively a massless particle. (This is the case of electron in 1D metal or of the Dirac electrons in 2D graphene ).
Dear Eva,
In the vicinity of the Fermi surface, the energy of the quasi-particle defined by Landau, is linear in the desplacement moment p-pf. This is for answering you that it is not necessary to take into account the general definition of tensorial mass Mij
( 1/M)ij = (2pi/h)(2pi/h)(second derivative of energy of the quasi particle respecto Ki and Kj)
(sorry for writing so badly de equation. I hope that it is possible to follow the idea).
This property of the energy of the quasi-particles that their energy on the surface of Fermi is exactly the Fermi energy and also that the lifetime is infinite.
Dear Daniel,
naturally the dispersion law in this case is linear near the vicinity of the Fermi energy. But, in this case, the electron is the Dirac fermion and massless. Remarkable physics which follows from these properties is now widely discussed in the literature on graphene since its discovery ~2005.
There is no contradiction in the definition of the effective masses for metals and insulators, the important point is the difference of occupation of energy bands by electrons in metals and semiconductors (insulators):
As to the energy band of a metal, energy is periodic as usually but, because of its usual half-filling for a metal respective energy concides with the Fermi energy.
At the point of the nesting, the dispersion law is exactly linear.
For semiconductors or insulators, the conduction band is occupied usually close to the band minimum edge, so that the effective mass is given by the definition given in Your and mine above contributions where the dispersion law is quadratic.
.
Dear Eva,
I am absolutely agree with your last commentary. What I had not understood was your previous one where you wrote about the tensor mass related with the quasi-particles:
For clearness and to avoid misunderstandings, allow me to remind, that the quasiparticles in condensed matter physics are characterized by the effective mass which is the tensor {m_{i,j}}, i,j,=x,y,z depending on the crystal structure.
I think that it was out of our discussion of distinguishing the excitation states in condensed matter by the assigning of a "univoque" name and universally accepted as it could be quasiparticle or collective excitation.
I am agree with you that the linear dispersion law is extensively used in graphene although as a consequence of having Dirac relativistic electronic behaviour for the massless electrons instead of the non relativistic Schrödinger context.
Please, could we go a step further trying to distinguish these excitations or perhaps it happens that we have we arrived to the end of this discussion? In any case this end it was not by knowing the solution to the question, but for lack of new ideas. From my humble point of view we have had interesting purposes, where the two extremes are :
Greentree have spoken of composite boson or composite fermión, whitout more distintions, for defining all the excited states in condensed matter. That is to say, it seems that the coined terms as quasi particle or collective excitation are not necessary,
Nicolis avoids to enter in these names as defined physical features. It seems that we are justified to employ them as we wish.
It's obviously true that the structure of the dispersion relations is physically relevant. What one *calls* these excitations is not so relevant-that they obey linear of quadratic dispersion relations is. It is the properties of graphene that lead to linear dispersion relations, not the other way around. Of course, once we have identified this dispersion relation in grapheme, we may ask, what features of the material are really relevant to its appearance and, whether we could find them in other materials-and how does it come about that, ``typically'' we come across quadratic dispersion relations.
So we realize what are some *intrinsic* features that are physically interesting-and what are, just, of historical significance.
I always assume that behind a name is a concept to distinguish it. Obviously the name is nothing without a physical correspondence and what is worse: if we have several concepts or names for the same physical concept we have confusion. As a mathematical could say we have no functions and even we cannot define them for calling the events.
@Daniel, I think you right but I feel that this discussion can not help in solving the problem of multiple names for common event or concept or a single name for conceptually different events or physical entities. It can be solved only when the issue is discussed and debated in a physical society meeting to reach suitable decision. The world of physics is suffering from many such problems and even more serious problems. For example there are many ideas or concepts which clearly do not help in understanding several intriguing experimental phenomena in terms of conventional ideas but the world of physicists is not ready to give an alternative concept or idea a chance of being discussed and debated. While as physicists our common goal is to establish the truth of nature, however, it appears that believers of conventional ideas in many such cases are simply protecting their own interests. I may be wrong but it appears the problem at hand is not good enough to find its solution at this forum.
Dear Jain,
Thank you for your answer, but I am not agree with you. This is a forum, just a forum of physicists, and we can discuss the problems as you have told. But I think that when you write or ask in a forum, you are so responsable as if you were in a commitee. Here we have more than 500 persons ( a great proportion of physicists that they are waiting for a useful discussion) who has entered for reading our opinions and I think that they deserve a clear answer, which could be that we do not know to distinguish among all these concepts but we employ them in books and in scientific papers. Your complaint is already a good contribution.
I am sure that there are people who could give us more insight on this issue or at least we have to tell clearly what is the state of the knowledge of all that we have tried to answer it. This could also help.
It is indeed well established in CMP that the form of the dispersion law is a crucial characteristics of excitations which are quanta of respective excitation fields of crystals . Their name - quasiparticles for them is widely used and the concrete name (Bloch electron, B. hole, polaron, exciton phonon, magnon) tells an information on the concrete dispersion law . If the dispersion law is quadratic, there is effective mass (tensor). From this point of view, there is not quite clear why this concept in not satisfactory? All these excitations can be called collective from known reasons. What is wrong there? As for CM physicists, I do not believe that they consider this concept doubtfull and try to modify it for a better? Usually such sorrows have linguists, who are not satisfied with some words used in commom language. ...
Dear Eva,
Could you write a correspondence among the excited states that you mentioned ("composite particles") with their dispersion laws? Thank you, this could be very helpful indeed.
Dear Daniel,
This task is more than technically possible within this correspondence, so that
I attach a file with more comprehensive classification of the excitations in conventional 3D models in CMP.(It can be easily completed within literature).
I use only the standard terminology as is usual in CMP. Some speculations of new classification of these excitations are not productive and bring only confusions.
Dear Eva,
I am very sorry to have to say that you are absolutely wrong and your equations are no quasiparticles at all in the form that you present them. The concept of quasiparticle was introduced for avoiding the electron-electron interaction in a Fermi liquid to try to get a Fermi gas( by Landau using the adiabatic principle). You need to dress the particle with collective excitations and do not write the hamiltonian assuming the particle interaction as is done in Kittel (notice that he never speaks about quasiparticles), only in an small paragraph and outside of this concept. Ascroft and Mermin, do it in pages 350-351 for mentioning the Fermi liquid and to say that they are not going to enter in the subject.
The proper books for the subject are:
L.D. Landau and E.M. Lifshitz,Statistical Physics,part 2 (Pergamon, Oxford, 1980).
P. Nozieres,Theory of interacting Fermi systems(Bejamin, New York 1964).
D. Pines and P. Nozieres,The theory of quantum liquids, Vol. 1 Normal Fermi liquids
(Bejamin, NewYork 1966).
G. Baym and C. Pethick,Landau Fermi-liquid theory
(Wiley, Ney York 1991)
I attach you a paper a good paper devoted to this subject
Daniel, the Fermi liquid theory is another subject than that handled in CMP. I do not mind anything against QP in Landau liquids. The wave vectors there are continuous However, in CMP, the basic medium for the excitations is the periodic lattice, while in the theory of Fermi-liquids this is not the case.
I can also recommend plenty of textbooks on Condensed Matter Physics, partly given in the file attached. The Bloch formalism and related quasipartices are standard and inevitable part of the CMP.
It does not mean that You are wrong, only Your attention is turned to another direction, where the concept of quasiparticles is different because the system is different.
I would be glad if You could admit importance of the concept of quasiparticles in CMP, which, I believe, is in consideration also in Your Institution.
For a quick insight look at the Table "Quasiparticle ZOO " at
http://www.tf.uni-kiel.de/matwis/amat/semitech_en/kap_9/advanced/t9_1_1.html
Dear Eva,
If you want tell me that there are quasiparticles in solids, I agree. Obviously there are but they are not treated as you have done in your notes where the concept of particle is assumed (see also the Kittel that you have mentioned). They are dressed changing their effective mass (tensor of order zero= scalar) for renormalizing them and also with a lifetime for solving a many-body problem. Thus we have the same physical object but considered in a very different form that you have writen. Please go to the serious reference of Ascroft, page 350-351 or read the paper that I have attached you.
I would say, that there exists a unifying view for quasiparticles in both fields , as follows:
A problem of many interacting particles (difficult to solve) is converted to one-particle problem (they can also weakly interact). These are quasiparticles.
Dear Eva,
We need to take care because we have written a lot of things and if everything is OK as it happens in many no scientific "chats", then I can understand the people which says that the question is finished; even when we can see that nobody has answered it : how to distinguish them? The definition is clear in the literature but what is not clear (at least for me) is its application to the physical objects that we call with those names.
I am sure that there is people who can answer this question with enough accuracy!
The authors of the methods I described are Bloch (1929)-Bloch formalism, Frenkel (1931)-Frenkel excitons, Wannier (1937)- Mott (1938), Wannier-Mott excitons, Brillouin (Brillouin zones), Bohm and Pines (1951, 1952), Landau, Pekar (1948), Feynman and many others. Lateron also the authors of Green's function methods in CMP. contributed by the definition of quasiparticles via singularities of respective Green functions
I would like to finish this discussion by a conclusion, that
Quasipartcles are the concept which is able to convert interacting many body systems (hardly solvable) onto one-body systems of quasiparticles.
Dear Eva,
In all that I know Bloch, Wienner and Frenkel never treated the problem of quasiparticles and less with respect to the collective excitations. Feynman only superficially for the superconductivity and statistics, better reference for that point is
A.A. Abrikosov, L.P. Gorkov and I.E.Dzyaloshinski, Methods of Quantum Field Therory in Statistical Physics, Dover Publications, 1963
These are the classic references but if you want to enter in the excellent research that nowadays is done on the subject, perhaps you could go to the page of Piers Coleman. This is a problem of many-body theory and all that I have tried to tell you is that your notes where outside of this scope. I have been obliged to explain it for avoiding to increase the entropy on the issue.
Daniel,
I hope You change Your meaning, when You look at the literature on CMP. It is well konwn that
Wannier (not Wienner) introduced the concept of excitons with large radius,
Wannier excitons, in Phys. Rev. 52, 191 (1937;)
Frenkel introduced concept of excitons with small radius, Frenkel, Phys.Rev.37, 17 (1931), 37, 1276 (1937).
Feynman used his Feynman path integral method for treatment the polaron in polar crystals, R. Feyman, Phys. Rev. 97, 660 (1955).
See also R. Feynman Statistical Mechanics, W.A. Benjamin, Inc, 1972.
Feynman, Hibbs, Path Integrals.
Kuper, Whitefield, Polarons and Excitons, Edinbourgh, London, 1963.
Good insight into the theory of quasiparticles in CMT is
H. Haken, Quantenfeldtheorie des Festkoerpers, Teubner, Stuttgart, 1973 (or English translation Fieldtheoretical Methods in Solid State Physics) . There are given another important references on the quasiparticles which I do not include in this short contribution..
I know the book You recommended, there are many - Haken's book is one of that kind, very intelligible for reading.
Certainly there are many other types of quasiparticles in CMP, e.g. the Cooper pairs.
What I want to stress is, that the quasipartcles in CMP are introduced by diagonalization of Hamiltonians in k-representation, i.e., in k-representation the problems converted to one-particle Hamiltonians. This representation has been performed by Fourier transformation of Hamiltonians. That is possible because of the lattice periodicity. and absence of nonlinear terms for interactions in Hamiltonians . With higher-order terms the FT is, of course, inappropriate.
The concept of quasiparticles has been used also in many body systems without the periodicity provided by the lattices, e.g. in the models You mentioned, the method of diagonalizations are different but finally the resulting one-particle (quasiparticle) Hamiltonians were obtained. It is not so basically important what kind of diagonalization has been used, the methods are often different according to the models where they are used, but the aims can be same or similar.
Dear Eva,
A Cooper pair is not at all a quasiparticle, something which has the two electrons at one hundred of nanometers cannot be dressed to have a well defined propagator with effective mass and lifetime finites, even being bosons.
You are right, I have written Wannier badly, sorry, but notice that one thing is to have defined one excitation of a field and another very different is to have a dressed one with the surround collective excitations of the medium to avoid the interaction among them. Thus reducing the number of "particles" to calculate two as maximum.
The examples of the books of Kittel and Ashcroft for your notes were well for the first purpose, but not for the quasiparticles that I am asking for distinguishing. In any case I am very grateful to your contribution, although I think that we needed to come back to the initial question and to try to solve it.
I hope that somebody is going to help us. Thank you.
Daniel, from Your last contribution it is evident, that there is difference between the meaning of "dressing" in Your problems than that in condensed matter physics.
The effective mass of Bloch electrons (holes, excitons ) is different from the mass of the electron (hole, electron-hole pair ) in vacuum, because it is modified by the lattice.. This is not called '''dressing' in CMP. The term dressing is used for polarons (Bloch electron dressed by phonons). polaritons (dressed by phonons and photons), excitons (Bloch electron and Bloch hole pairs) .
Regarding the Cooper pairs, the situation is not quite clear, sometimes the Bogolyubov electrons are considered as quasiparticles.
Dear Eva,
The Cooper pair of superconductor phase is never a quasiparticle, while the Bogoliubov electrons always are in all that I know. In any case, this excitation is one that I have clearly understood in my picture of what is a quasiparticle. But I must say that I am not sure of all these definitions which I think they are a little bit confused in the literature. In fact this is the origin of my question.
I absolutely agree with you that the term "dressed" in CM is not often used, even when the term quasiparticle is employed. This creates more confusion to these definitions because it is the only way that I can understand what is a quasiparticle: a particle dressed by collective excitations. And also this picture justify the change of mass and the new lifetime for this object in analogy with QFT where a bar particle is used to be dressed and to be renormalized. But the physics is full of mistakes into these subjects, even they call group of renormalization, what is not at all a group. But it worked quite well in QED ( Ward-Takahashi identities) solving problems of infinities as point-like electric charge (Feynman, Schwinger and Tomonaga) and which in fact has involved a change of scales. And these concepts were extensively employed in CM by Kadanoff and Wilson.
Anyway I do not understand why these concepts, which are the same, they have not unified names in Physics and also a whole univoque definition.
So
could any body sum up the conclusions of such a lengthy discussion to help people to identify different particles as real particles, quasi-particles, dressed particles and collective motions. In case the discussion is still inconclusive, then it is a different matter but summing up after some such time would be greatly appreciated.
Some many-body Hamiltonians with interactions making the problem unsolvable are transformable to a noninteracting (or weakly interacting) one-particle system where the original interactions are.effectively included into new parameters (e.g. effective mass) . These new effective particles are quasiparticles.
Collective excitations are excitations shared by many degrees of freedom including also nonlinear nonperturbative excitations (solitons).
Dressed particles consist of a (quasi)particle + additional component of excitations which modifies the parameters (e.g.,effective mass) of the (quasi)particle.
I'll try to summarize the main contributions to this question which is still open, as I'll show in another answer.
- Lev Matyushkin started saying that the Cooper pair is a non sense to consider it as an individual "particle". The Cooper pair is a new "realitiy" very far of the electrons which form it.
- Berna Uyanık recommend us to read the Mattuck.
- Kai Zhao thinks that a Cooper pair is a quasiparticle
- James LeBlanc, for him the Cooper pair is defined into a many-particle instability within a bosonic phase made with electrons interacting with phonons. They are unable to be measured.
- Andrew Greentree, for him a particle corresponds to a quantization of a Hamiltonian and the Cooper pair is possible of being measured. The electrons into the superconductive phase are quasiparticles.
- James LeBlanc reply that the Cooper pairs are only possible to be measured with ARPES which detect picks associated to the nestings in the Fermi surface only.
- Miguel Carrion Alvarez introduce a definition of collective excitations, fundamental particles, quasiparticles and dressed particles. In the second answer he tries to apply his definitions to a magnon which he defines only as a quasiparticle. I show him that it is also possible to consider it as a collective excitation: no univoque definion exists then. After that we go also to the phonon which is not a quasiparticle too against his intuition.
I think that the confusion of what is the difference of these concepts in the literature is clear and other people continue the discussion who I'll write in another answer for avoiding to make difficult to distinguish clearly everybody.
- M. Dharma-Wardana says: Fundamental particles, or quasi-particles, or whatever, are all excitations in their quantum fields. It seems that he want to enter in the real question of condensed matter. And even he repeats that the phonon is a quasiparticle when it was deeply discussed previously that it is a collective excitation.
- Stam Nicolis starts saying: It's possible to see this in many ways-and each person has their favorite, depending on their background and their interests. It seems that he thinks that these concepts are without a serious physical meaning.
- Eva Majerníková, has discussed quite a lot these concepts but restricted them to the periodic traslational lattices and put an interesting list where tries to summarize the differences asked for those excited states. I deeply thank the details where she enters, although can create confusion because she pretends solve the issue with some formalism which is not the good one and this obligues me to respond her with another detailed written as an attached file.
- Yatendra Jain does the contribution of thinking about the vacuum where only the fundamental particles exist, from the condensed matter where many people call the excitation with names quite freely.
This is for the moment where the question is and I afraid that we need more opinions which could clearly distinguish these concepts. From my humble point of view, it is not possible to call in free form in so serious science as physics at those objects behind the names and I am sorry if I have forgotten any important contribution or detail, if that is the case please go ahead and tell us your opinion again.
Perhaps the best way to know if we understand these concepts if we apply them. Let us start with the most important particle in CM.
Could be one ELECTRON one quasiparticle in a given physical phase? Could it be on collective excitation? When could it be a dressed particle?
Apparently I see three different physical situations: One where electron has no interacting environment and in this case it is simple particle. In second case it moves in an interacting environment without dragging other constituent particles, it is then a quasi-particle, third it moves and drags other particles of the physical environment, - it is then a dressed particle (Example is electron in liquid helium where it occupies a self created cavity known as electron bubble) .
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