# Tag Archives: Topology

## *Private Communication*

Here is a correspondence that took place between Sarang and I, following his post last week about emergence and upward heritability, which was in response to a couple of my posts (here and here).

Subject: Post

Hey Sarang,

Great post — I’m glad that you wrote an article from the other side.

I do have a couple questions/comments though, as you are likely more knowledgeable on this subject than I.

To my understanding, GL theory is a coarse-grained version of BCS theory and many experimental properties of classic superconductors can be calculated using both methods. I also like this example because it is one of the few places I know of where one can derive the coarse-grained model from the underlying one (I am an experimentalist after all, so there may be many others I’m unaware of). Am I being misled in thinking that something “survived” in going from one level of the theory to another, or is this not a good example of “new principles at a different scale” that Laughlin and Pines referred to?

Anshul

Re:Post

This is a good question; I should have a succinct answer but don’t. Some thoughts:

1. I don’t see BCS as a “microscopic derivation” because a lot of coarse-grained ideas go into writing down the reduced BCS Hamiltonian — you throw out the Coulomb interaction b’se of Bohm-Pines and then you reduce the remaining interactions to Hartree-Fock and pairing channels because of Fermi surface kinematics, so what remains is an exactly solvable model. The BCS coupling is in practice a phenomenological parameter, which is backed out from the gap. (Otherwise we would not be so bad at computing Tc’s.) The symmetry-broken (i.e., non-number-conserving) BCS wavefunction violates “heritability” because particle number conservation is precisely the sort of symmetry/conservation law that is supposed to persist from one level to the next. So I see BCS as mostly a reverse-engineered microscopic justification (i.e. a way of saying, look, you can get superconductivity with just electrons) rather than an example of reasoning *from* microscopic considerations to macroscopic results.

2. More generally, when you write down a solvable, microscopically specified toy model that describes some emergent phenomenon, I do not think this counts as a *deduction* unless the decisions on what effects to include and to neglect are based on microscopic considerations. And the renormalization group tells you that such microscopically informed decisions about what effects are important/worth keeping will in general be wrong. “Relevant” and “irrelevant” are properties of the RG flow, not of the initial Hamiltonian.

3. Of course macro-stuff is made up of micro-stuff; the issue is whether the relevant conceptual architecture is inherited or distinct. I’m arguing that it is distinct whenever the thermodynamic limit is nontrivial: there are notions like fractionalized excitations and Goldstone modes that cannot be articulated without reference to the thermodynamic limit. In a finite system there is never a sharp distinction between collective and non-collective parts of the spectrum; “in practice” we know how to identify collective excitations even in relatively small systems, but when we do this we are invoking thermodynamic rather than microscopic concepts.

Re:Re:Post

Thanks for your response. I have been taking some time to think about your answers. I also have a few comments, to which you can reply if you have the time.

1. BCS is not a “microscopic derivation”, but a lot of the work leading up to the full formulation (such as the Bardeen-Pines interaction) required careful thought about how phonons could cause an attractive interaction. Ultimately, the theory is a toy, but one based on some (I would consider) meticulously thought out microscopic considerations. I would even consider the demonstration of the isotope effect a microscopic experiment in some sense. As for number conservation, Tony’s book Quantum Liquids shows that this is not necessary to formulate a theory of superconductivity, but only a trick to make calculations of experimental quantities sufficiently easier. I think to say that actual particle number is not conserved in real life would be quite unnerving.

2. This point, I perhaps don’t understand as well in light of the response given for part 1. It seems to me that the Bardeen-Pines thought process was extremely significant.

3. I personally find it quite stunning that e.g. the Aharonov-Bohm effect (a microscopic effect) and the Quantum Hall quantization (a macroscopic effect) bear such a striking resemblance. These are the kinds of phenomena I speculate (and I suppose Wilczek might also) may be upwardly heritable.

Ultimately, it may be impossible to know whether L&P are right because one cannot solve the Schroedinger equation for 10^23 particles or put it on a computer to see if one gets the right answer. I choose to believe, however, that it would because it is more “natural” (perhaps my mind is lazy). This is where I think L&P are radical. They may have reasons for saying that the Schroedinger equation does not capture some essential physics, but until it is definitively shown to be true, I don’t think I will accept it. This does not mean that I don’t think that there is different physics on different scales. Water is indisputably wet. But to get the Schroedinger equation to exhibit this macroscopic property is indeed futile, though it may be possible in principle — and hydrodynamics is undoubtedly a less unwieldy description.

Anderson himself in his More is Different article says “the concept of broken symmetry has been borrowed by particle physicists, but their use of the term is strictly an analogy, whether a deep or specious one remains to be understood.”

I guess you say perhaps specious and I say perhaps deep?

Anshul

Re:Re:Re:Post

1-2. I guess I think the isotope effect experiment is precisely the opposite of a deduction *from* microscale *to* macroscale. The approach in that kind of expt is to identify the relevant terms in the microscopic Hamiltonian by varying them separately and see if the “answer” changes. In other words what it actually is is a deduction *from* a macroscale fact (Tc depends on isotopic mass) *to* a microscale conclusion about what terms in the microscopic Hamiltonian are truly important. Similarly, a key ingredient in Bardeen-Pines is the previous (Bohm-Pines) result that the Coulomb interaction gets sufficiently renormalized by the Debye frequency that it can actually lose to the electron-phonon coupling. Again, there is nothing deductively “microscopic” about this result: you have to include renormalization due to electron-hole pairs at all energies above the Debye frequency in order to see it, so the infrared behavior of the theory is already smuggled in! Not that there is anything wrong with this, but I do not think it can be correctly interpreted as a deduction from small scales to large scales. The really hard part of the BCS problem, after all, was writing down the BCS Hamiltonian, and the data used to do this came almost entirely from (a) macroscale experiments and (b) theoretical considerations which included infrared/emergent physics. If you want to make deductions from solving the Hamiltonian at the scale of a few atoms it is hopeless because at that scale what you have is just this huge Coulomb interaction and nothing else, and there is no obvious path from there toward a paired state.

The number conserving v. of BCS is basically like the Schrodinger’s cat ground state (all up + all down) in the Ising model. The fact that using the usual variable-number wavefunction is even *possible* undermines the notion that microscopic symmetries/conservation laws tell you anything useful about macroscopic physics.

3. I think part of the problem is that if you define “upward heritability” vaguely enough anything can upwardly heritable. I read Wilczek as saying: given some set of specific, robust facts about the microscopic physics (e.g. symmetries, AB periodicities), you can derive strong constraints on the kinds of macroscopic physics that are possible. I think these constraints are actually weak to the point of barely existing [1] (because the microscopic symmetries can be broken and new symmetries can emerge; the AB periodicity can change…). Perhaps you want to argue that it’s surprising the macroscopic system can even be described using the language of symmetries and AB periodicity at all? I suppose I’m not surprised by that — clearly some symmetries are preserved under composition, the trouble for small -> large deduction is that a priori you don’t know which ones they will be.

I agree that if you solved the Sch. Eq. exactly for a huge system with precisely defined parameters you would get the right answer. However, (a) even slight imprecision in parameters or slight approximation can give you a wildly wrong answer, and (b) even wildly imprecise calculations can give you a qualitatively correct answer. Therefore, if you run a simulation starting from measured parameters with any uncertainty, in the thermodynamic limit you will have no reason (in principle) to trust your simulation. There are phenomena very like chaos that happen under coarse-graining.

[1] I should mention one sense in which Wilczek is right that there is a constraint. Whenever the original symmetry is global and continuous, the constraint says: “either the system will respect this symmetry in the thermodynamic limit or there will be gapless modes.” However this is not a very useful constraint, as there is no way even in principle to count the gapless modes.

Re:Re:Re:Re:Post

This most recent email makes a lot of sense to me.

I should just point out that Laughlin and Pines said that even in principle, one could not obtain certain effects from the Schrodinger equation, that there did not exist a deductive link at all. This is the statement that I found quite jarring. I can totally get on board with what you said below though, that makes a lot more sense.

Thanks again, I actually learned something from this conversation.

Re:Re:Re:Re:Re:Post

Thanks — yes it was useful having to write this stuff out. Again I don’t know that I agree with L&P — I read their paper a long time ago and was bothered by it because it seemed to be saying some crazy things among the many correct things.

## Quantized Vortices in Superfluid 4He

There are a couple nice old PRLs  from the Packard Group at Berkeley demonstrating the existence of quantized vortices and the vortex pattern in superfluid 4He. See here (pdf!) and here (paywall). Just as a little bit of background for those who are unfamiliar: below 2.17K helium undergoes a liquid to liquid phase transition from a “normal” liquid to a superfluid. The superfluid is characterized by several properties including zero viscosity (similar to electrons in a superconductor), second sound (effectively, temperature waves), and quantized vorticity.

The observation of this latter property was captured vividly in the series of images taken by the Packard Group shown below. To induce vortex formation, the authors rotated the bucket in which the superfluid had been placed. Because of the zero viscosity, the superfluid remained still until a critical velocity was reached where a single vortex formed in the center of the bucket. As the angular velocity was increased, more and more vortices started to form and the authors show the pattern formed by these vortices in the presence of 1-11 vortices.

Interestingly, there are two different stable configurations for which 6 vortices can form as shown in the figure. I happen to know that Richard Feynman, who had done a lot of the prior theoretical work on vortices in superfluid 4He, sent a personal letter to the authors of these papers to thank them for their elegant experiment.

## Theory of Everything – Laughlin and Pines

I recently re-visited a paper written in 2000 by Laughlin and Pines entitled The Theory of Everything (pdf!). The main claim in the paper is that what we call the theory of everything in condensed matter (the Hamiltonian below) does not capture “higher organizing principles”. Condensed Concepts blog has a nice summary of the article.

Because we can measure quantities like $e^2/h$ and $h/2e$ in quantum hall experiments and superconducting rings respectively, it must be that the theory of everything does not capture some essential physics that emerges only on a different scale. In their words:

These things [e.g. that we can measure $e^2/h$] are clearly true, yet they cannot be deduced by direct calculation from the Theory of Everything, for exact results cannot be predicted by approximate calculations. This point is still not understood by many professional physicists, who find it easier to believe that a deductive link exists and has only to be discovered than to face the truth that there is no link. But it is true nonetheless. Experiments of this kind work because there are higher organizing principles in nature that make them work.

If I am perfectly honest, I am one of those people that “believes that a deductive link exists”. Let me take the example of the BCS Hamiltonian. I do think that it is reasonable to start with the theory of everything, make a series of approximations, and arrive at the BCS Hamiltonian. From BCS, one can then derive the Ginzburg-Landau (GL) equations as shown by G’orkov (pdf!). Not only that, one can obtain the Josephson effect (where one can measure $h/2e$) by using either a BCS or a GL approach.

The reason I bring this example up is because, I would rather believe that a deductive link does exist and that even though approximations have been made, that there is some topological property that has survives to each “higher” level. Said another way, in going from the TOE to BCS to GL, one keeps some fundamental topological characteristics in tact.

It is totally possible that what I am saying is gobbledygook. But I do think that the Laughlin-Pines viewpoint is speculative, radical, and has perhaps taken the Anderson “more is different” perspective too far. It is a thought-provoking article partly because of weight that the authors’ names carry and partly because of the self-belief of the article’s tone, but I am a little more conservative in my scientific outlook. The TOE may not always be useful, but I don’t think that means that “no deductive link exists” either.

I’m curious to know whether you see things like Laughlin and Pines.

## Just a Little Thought on TKNN

Thouless, Kohmoto, Nightingale and den Nijs in 1982 wrote a landmark paper relating the Hall conductivity to the Chern number (also now known as the TKNN invariant).

It is well-known that the Quantized Hall Effect (QHE) is an extremely robust phenomenon of topological origin (pdf!). One can think of the Hall conductivity as measuring the total Chern number of the occupied Landau levels. What baffles me about the TKNN result is that despite the robustness and topological character of the QHE, the authors are able to use linear response theory.

This must mean that second and higher order responses must somehow be exponentially suppressed and that the response is perfectly linear. I have not come across a proof of this in the literature, though it may very well be an (boneheaded!) oversight on my part. This line of questioning also applies to the Quantum Spin Hall Effect and the Quantum Anomalous Hall Effect.

## It’s Not Just a (Berry) Phase

Just a few years after Berry’s breakthrough paper (pdf!) on the phase now named after him, it was recognized by Zak (pdf!) that this effect could play a role in a solid state setting. Zak realized that in the Bloch Hamiltonian, the crystal momentum, k, could be treated as a “parameter” similar to how other “parameters” had been treated in Berry’s original work. More concretely:

$H(\textbf{k}) = e^{-i\textbf{k}\cdot r}He^{i\textbf{k}\cdot r}$

While phenomena that are now considered prototypical Berry phase effects were understood before the Berry paper, it was Berry who unified many concepts under a general framework. For instance, the Aharonov-Bohm effect, the Dirac monopole problem and the Integer Quantum Hall Effect were all understood separately, but the Berry phase concept presented a huge leap forward in consolidating these seemingly disparate phenomena.

It should be mentioned that the three cases mentioned above are all instances where the Berry phase is topological and results in a robust quantization through a topological invariant (known as the Chern number). However, the Berry phase does not have to be topological in nature and can have “geometrical” consequences. Many of these “geometrical” consequences are discussed in a great review article by Xiao, Chang and Niu.

One particular “geometrical” consequence stands out: the semiclassical equations of motion for electrons in a solid, as detailed in textbooks such as Ashcroft and Mermin, are incomplete! The application of an electric field in crystals with either broken time-reversal symmetry or broken inversion symmetry may, under certain circumstances, exhibit a transverse velocity associated with a non-zero Berry curvature (i.e. an application of an electric field in the x-direction will result in electrons travelling in the y-direction!). This experimental paper by NP Ong’s group demonstrates this effect (pdf!), and this paper by Karplus and Luttinger (paywall!) theoretically explains the transverse velocity in ferromagnets, again before Berry’s seminal paper.

I’ll also mention briefly that the seemingly mundane phenomenon of electrical polarization in solids is another case that cannot be explained without Berry-esque concepts, but I’ll leave that topic for another day.

It truly is stunning how many effects can be accounted for under the umbrella of Berry/Zak ideas. I’ll even restrain myself from mentioning topological insulators and Chern insulators (paywall) / the quantum anomalous hall effect (paywall).