Doug's presentation was his response to the recent article in Science magazine by Phil Anderson questioning the validity of a bosonic glue for the high temperature cuprate superconductors (Science 317, 1705 (2007)). Phil's contention is that the pairing is coming from the superexchange energy, J, which is in essence instantaneous in nature. As a consequence, he contends that a proper theory must be quite different from the strong coupling theories developed in the 1960s in regards to electron-phonon mediated pairing. The implication is that J is the "elephant", and that any dynamics causing bumps and wiggles in ARPES, tunneling, and optics spectra is the "mouse".
What Doug and his colleagues like Mark Jarrell have done is to use a dynamical cluster approximation to calculate the singlet pairing vertex in the Hubbard model for values of the Hubbard U ranging from 4t to 12t, where 8t is the bandwidth of the electronic states. This vertex, Gamma, can be thought of as a sum of an irreducible part, Lambda, plus induced interaction terms due to repeated scattering in the particle-hole channel. The latter can be divided into an S=0 and S=1 part. As the temperature is lowered, Gamma develops a momentum structure with a peak at q=(pi,pi), despite the fact that Lambda is structureless (the latter defines an effective U, denoted as U-bar). This behavior of Gamma is mirrored in the S=1 part of the induced interaction (the S=0 part is depressed around q=(pi,pi) instead), indicating that it is this term where the real action lies. They then write down a gap equation, and find out that the dominant eigenvalue displays the d-wave cos(kx)-cos(ky) behavior.
Now what about the dynamics? The pairing self-energy decays as a function of Matsubara frequency out to an energy scale of t, and this is mirrored by the frequency dependence of the dynamic spin susceptibility, chi. Equating the two, he finds that Gamma is equal to (3/2) (U-bar)^2 chi(q,omega), as expected for a spin fluctuation picture for the pairing. Therefore, Doug concludes that at the level of the Hubbard model, one can indeed think in terms of a pairing glue.
Next week, we will find out what Phil has to say about all of this.
Several questions were raised after Doug's talk.
Jorge Hirsch - What are the consequences? Doug - There will be structure in the ARPES and infrared conductivity that can be related to the frequency dependence of the normal and pairing self-energies.
Chandra Varma - Is this approach equivalent to RPA? Doug - Yes, except that the dynamic spin susceptibility is quantitatively different from what RPA gives.
Gabi Kotliar - What about analytic continuation, and how is this work related to my own? Doug - We have a lower Hubbard band, coherent structure near the Fermi energy, and an upper Hubbard band, as in your work.
Peter Hirschfeld - How does U-bar vary with doping? Doug - We don't know yet.
Chandra Varma - Does your pair vertex, etc., scale with the antiferromagnetic correlation length? Doug - We don't know yet.
Thursday, August 9, 2007
Doug Scalapino, UCSB: Is There Pairing Glue in the Hubbard Model?
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THE QUESTION OF PAIRING GLUE IN THE CUPRATE SUPERCONDUCTORS: REPLY
I could not disagree more with both halves of the final sentence of this comment: there is not a crucial pairing glue in the cuprates, nor is there any need for further experimental work to settle this question. To encourage such work is to distract attention from the many fascinating and fundamental questions which still confront us in this field.
Regretfully, I will have to go into some technical detail in order to answer this criticism. The numerical analysis of Scalapino’s reference 7, and the analysis of experiments in reference 8 and 9, all have, logically, two pieces. Let me take ref 7 for definiteness. Here Scalapino’s group do two computations. The first, while difficult, is logically unimpeachable: it is a carefully designed simulation of the properties of the “t-J” model which we both agree is by far the main candidate Hamiltonian (It embodies both the “mammoth” and the “elephant” of my reference 6 comment.) Sure enough, they find d-wave superconductivity and other properties that agree with experiments. In step 2, they attempt to back out from the measured quantities a number of theoretical constructs, such as the “pairing interaction vertex”, assuming that the underlying theory is conventional Feynman-Dyson diagram theory. The interaction vertex which is derived does not look at all like the “J” term in the Hamiltonian, and in particular is small at high energy and big at low energy, where J is the same at any energy. (I should mention that the vertex is still everywhere attractive).
This is a remarkably unlikely result. It says that somehow all the low-frequency “spin fluctuations” have killed this giant interaction at the high-frequency end, but left it intact at low frequency to do its work on the pairing gap. This is not the way things work! It amounts to completely discarding my “elephant” J and replacing it with nothing but its indirect consequences. It also contradicts the simplest (correct) mean field theory of the t-J model (called because of its simplicity the “Plain Vanilla” theory.)
If I found a result which so blatantly made no sense, I am afraid that I would have abandoned the method rather than attacking those of us who seem to have found the right answers by doing things otherwise. It is very attractive to abandon the methodological assumptions, because the same set of assumptions, applied in the normal state, have had zero success in describing its unique and very anomalous properties.
Recently I have found a set of methods which do give correct answers for the normal state. I am only beginning to apply them to the superconducting state but I can already see that they will tend to correct the variation of the vertex with energy, possibly leaving it almost constant. What is conclusive is that they show that conventional diagram theory cannot deal correctly with the t-J model; and that a “glue” is unnecessary.
(i) P W Anderson, et al, J Phys Cond Matter
(ii) P W Anderson, Nature Physics 2, 626 (2006)
(iii)P A Casey et al, cond-mat
Prof. Scalapino is 100% correct, and Prof. Anderson is 50% correct.
The solution lies in work done by Huygens 400 years ago, with the two clocks and their synchronized pendulums. Coupled oscillators, which synchronized because of two things: proximity and frequency.
As applied to this problem, Anderson is right that the mammoth and the elephant are important. They serve to position the relevant oscillators properly, so that they can connect with each other. But Scalapino is right that the low frequency is critical to the actual coupling. See Art Winfree.
Following up on my previous post, think about Huygens' wall clocks in 1657. Their oscillating pendulums coupled based on several factors:
a. proximity
b. Same orientation (both flat on the same wall)
c. Similar low frequencies (close enough to couple)
d. Communication between the two oscillators: they could talk to each other and hear each other.
This example contains all the elements that need to be analyzed in understanding superconductors--new or old. Low temp superconductors or high temp superconductors. Specifics may vary, but surely the general attributes will be the same. These four elements are the common denominator. And the same four apply to other phases of matter, and their transitions. Pressure and temperature are simply different words for position and frequency.
Prof. Anderson's elements probably create proper position and proper orientation. The (relatively) low frequency that is of interest to Prof. Scalapino probably allows the relevant oscillating parts to couple, provided that they can communicate with each other.
We know that Cooper pairs in BCS superconductors can communicate with each other over quite a considerable distance. Many factors can affect the range over which oscillators can communicate with each other. See, for example, the research on quantum magnets done by Stanford researchers in 2002.
This analytical framework comes from coupled oscillator theory, as developed by Art Winfree in the 1960's. Think Malaysian fireflies, heart cells, or the gaits of a horse--trot, canter, gallop. Physicists don't use this framework, but it is quite helpful. In evaluating the debate between Scalapino and Anderson, it is useful to reflect on how each critical element that they cite relates to the four factors (proximity, orientation, oscillator frequency and oscillator communication).
A good summary of Winfree's work can be found on the Cornell web site of Steve Strogatz--a 1993 Scientific American article by Strogatz and Stewart (Dec. '93).
Of course, the oscillators in question must also be capable, if they are coupled, of producing the desired order either directly or indirectly.
Many of the new quantum phases of matter can be analyzed with this same framework. Typically, positioning and orientation are established by optical or magnetic controls; and then extremely low temperatures reduce oscillators to frequencies low enough for the oscillators to couple.
In this, my third post, I will explain specifically how to apply coupled oscillator theory to high temp superconductivity.
Please begin by reviewing my post immediately above. Done? Then follow me.
Let me describe precisely what I expect to find. I am looking for oscillators. The oscillators in question will be facets of electrons. Electrons have many different types of oscillations. There are inherent oscillations, such as orbit or spin; and there are special oscillations, such as hopping, or the regular flipping of poles, that might arise only under specific conditions. For my purpose, the only requirement is that these oscillators must be limit cycle oscillators. Once again, see Art Winfree's work from 1965 to 1970 for a refresher, or simply read his obituary in Siam (January 2003) written by Prof. Strogatz of Cornell.
The oscillators that I expect to find will be of several different types or sets. The various sets will have numerous interconnections and some kind of regular structure. The sets will have a propensity to synchronize, and each set is likely to synchronize under different conditions. Synchronization will depend on factors cited by Winfree, including proximity relative to each other and frequency of oscillation. It is likely that some sets will be able to synchronize only if other sets have done so first.
Understand that we are not exactly looking for paired electrons. We are looking for oscillations that emanate from electrons and couple or synchronize, usually at a transition temperature, or when nudged together by pressure. We want to find something that causes the electrons to hold hands so that they are united in some manner. That synchrony or coherence will cause both of the gaps that we find in high temp superconductors (the superconducting gap and the pseudogap). Since there are two gaps, it is likely that we are looking for at least two different sets of coupled oscillators, or the same oscillators behaving differently in different environments.
while we are at it, whatever we find should be capable of explaining all of the other interesting behaviors of the cuprates.
Now that I have set the stage, let's put it all together, starting with the cuprates themselves and especially the doping. Doping really means holes. Why do holes help these cuprates display more interesting behavior, and why does the number of holes have to be just right to produce the most interesting behavior? The answer is that there is an oscillating relationship between the particles and the holes, and that relationship is a repetitive or limit cycle oscillation. These oscillations become synchronized when the doping is just right. If there are too many holes, the moving particles aren't forced to synchronize, because there are plenty of holes to go around. If there are too few holes, the problem is the reverse; in that case, the oscillators, now fewer in number, can't communicate effectively with each other for two reasons. The oscillators are too few and far apart, and the particles that are not able to dance (because there is no place to go) block whatever weak signals might exist.
So the holes give rise to a dance, once we have the right number of holes, and and the dancers synchronize under certain conditions. That synchrony then puts certain other oscillators (such as flipping poles) in proximity to each other in a highly regular order in space and time. Thus, a first set of coupled oscillators gives rise to preliminary organization in a second set of oscillators of a different nature. That second set of oscillators eventually synchronizes when new conditions, such as lower temperature, make that possible.
A good example hereis the Millennium Bridge in London. The wobbling problem with that bridge arose because there were lots of oscillators (people, or more precisely the legs of people), all in a confined space, and accordingly organized by that confinement; and then they all responded in the same way, in time and space, to a wobble in the bridge. (Their response made the problem worse, unfortunately.) Credit here goes to Brian Josephson, incidentally, for elucidating this matter.
I'm going to continue this discussion in another post, which will follow immediately.
This post is a continuation of my previous post immediately above.
Note that the cuprates provide a great structure for these oscillations, in addition to providing the holes that create the oscillations in the first place. The cuprates offer a highly regular but also complex structure, with lots of potential neighborly connections in all directions. This ensures that the proximity and the orientation of all the oscillations will be ripe for synchrony. So the cuprates offer all the conditions that Art Winfree considered significant in his theory of coupled oscillators, as described in my previous post.
Now let's identify some of the candidate oscillations more specifically. First, let's listen to Anderson. His beloved J is "determined by the virtual hopping of an electron of a given spin to an adjacent site containing an electron with an opposite spin." (That quotation comes from Scalapino's e letter to Science Magazine, dated December 5, 2007.) Hopping, virtual or otherwise, is an oscillation. The change from spin to opposite spin in a continuing cycle is also an oscillation. That's good: two kinds of oscillation for the price of one. Once the hopping becomes synchronized, all of the spins will be in the right position for eventual synchrony themselves.
Now let's try Scalapino. His spin fluctuations are also oscillations. Or as he puts it in the Aspen material above, he studies interaction terms due to repeated scattering in the particle-hole channel. He divides that into an S = 1 part and an S = O part, and finds a peak for the former and a depression for the latter. I'm not sure how to count all these oscillations, but I see maybe four oscillations for the price of one. Note also that these oscillations are related to the holes.
Scalapino's mistake is that he thinks there is some pairing glue, which to many people means some new attractive force. The organization that arises in coupled oscillators is not a new force; it is simply organization in response to a traditional force, where the organization is necessitated by the particular conditions.
So we have plenty of limit cycle oscillators in the cuprates. Aside from the ones cited by Anderson or Scalapino, the literature on high temp superconductors is full of interesting oscillations. The ingredients are the holes; the rich variety of oscillations that are triggered by the holes; the proximity of those oscillations due to the structure of the cuprates; and the interconnections among the various sets of oscillations in time and space. These oscillations all tend to synchronize in their various sets--different sets at different times--based on proximity and frequency, just as Winfree's theory predicts. Superconductivity is just one manifestation of this coordination. Superconductivity occurs when the last little class of oscillators synchronizes. That final synchrony is made possible by various other oscillators that have already synchronized, and in doing so, have properly positioned the last little class of oscillators in relation to each other, again in both space and time. Lower temperature then lowers the frequency of those oscillations to a point where they can synchronize.
These coordinated oscillations, in several different shapes and sizes, all taken together, create high temperature superconductivity.
Wait...I'm not done yet. I will continue this discussion in another post to follow later today.
Oh, I just remembered. I promised to explain how BCS theory (low-temp superconductivity) is similar to my theory outlined above for high-temp superconductivity. (See my second post above.) I forgot about that, so here goes:
BCS theory runs as follows:
oscillations organize electrons and their spins precisely
My theory about high temp superconductivity runs as follows:
oscillations organize electrons and their spins precisely
There's a certain similarity between the two theories, don't you think?
If you think my summary of BCS theory is inaccurate, look up the definition of phonon. And reacquaint yourself with the Meissner effect. The Wikipedia articles will suffice in both cases.
Here is the specific explanation of high temperature superconductivity.
There are three principal mechanisms--one at the level of the atoms, and two at the level of the electrons. All three mechanisms essentially stem from Art Winfree's law of coupled oscillators, but at the same time they are not far removed from BCS theory.
Let me start with the electron level, since that is where most of you think I will trip up.
Think of an arbitrary set of 24 electrons and holes, in the copper and oxygen plane. This number is useful because it is a highly composite number. Assuming a 1 in 6 ratio of holes, then there will be 4 holes in this set.
The majority of electrons in this set will pair by spin with another electron--spin up, spin down. Nothing fancy there.
But four electrons will pair up with holes. (For reasons I won't bother to explain right now, these electrons will all be coppers, and the holes will be "found," you might say,in the oxygens.) They will pair by charge, with the holes faking status as a positive charge. This will have the effect of making the orbits of those four electrons quite definitive and orderly (unlike the electrons paired with other electrons, which will have coordinated spin, but not precisely coordinated orbits.)
Continuing with the four electron--hole pairs, the one-half spin of the electron in the pair will now be free to look for an additional dance partner. So it will coordinate its spin with another electron that has a free dance card for spin dances. Naturally, that will mean it is free to align by spin with another such electron--one of the three other electrons in this arbitrary set of 24 that is not already spin-paired.
The result is a Cooper pair. Each electron will be paired by charge/orbit (with a hole) and paired by spin (with another electron). So it will be highly regular and orderly. Traditional "old" Cooper pairs have these same two characteristics: they are paired by orbit and they are paired by spin. But "old" Cooper pairs have just one dance partner--another copper electron. These new Cooper pairs have two partners--their Cooper partner, which is a spin relationship with another copper electron, and their hole partner, which is a charge (and orbit) relationship. And then the orbits of Cooper partner A and partner B will themselves pair exactly (and in fact that may already have been dictated by other regularities in the overall system).
This reasoning is based on Art Winfree's coupled oscillator theory, but at the same time it is fully consistent with physics. Art Winfree would describe these orbit and spin relationships as exactly antisynchronous two oscillator systems. That's the formula humans use to walk, for example.
Now let me turn to the atomic level, and more specifically the behavior of the lattice. It is all well and good to explain some individual "one off" pairing at the electron level, which I believe I have done above, but that is not enough. A full solution has to show that the lattice itself can fully synchronize, at least in the critical copper oxide plane. All the copper and oxygen atoms in the plane have to be highly coordinated in a form of mass synchrony.
Again I turn to Winfree. I see this as a five oscillator system--actually two five oscillator systems. First, there is a five oscillator system comprised of four copper atoms and another atom above (lanthanum, strontium, etc.) Second, there is another, conceptually identical five oscillator system comprised of four oxygen atoms and a copper atom slightly above them. This is the Perovskite structural essence--itself a frequently noted characteristic of the HTS superconductors.
Now, amazingly, there is a good article on the web that describes the physics of a horse and rider system. The four legs of the horse are four oscillators, and the rider is the fifth oscillator. It's by four authors out of Florida Atlantic University. The underlying principles are based on Winfree, whom they cite. Google it now, please. One of the authors is Kelso.
They describe the synchrony that emerges from such a system. And I submit that their horse and rider system is an apt description of the Perovskite structure of cuprate superconductors.
In fact, the copper atoms again are subject to dual constraints. They have to behave precisely as the legs of a horse in the one case; and they also have to behave as a rider in respect to the four oxygens. The oxygens have only a single role, as the legs of the (second) horse. The result is that all coppers and all oxygens in the plane will be synchronized perfectly as far as the eye can see (if the eye could see at this level).
For the coppers, the relevant image is that of a Centaur. I will let you, dear reader, puzzle over that image.
So I have every reason to believe that the lattice will produce exactly the type of proximity and coordinated position of the atoms that will lead to synchrony at the atomic level within the copper oxygen plane; and that synchrony will in turn support, reinforce and magnify the effects that I describe above in relation to the electrons.
So there you have it. A complete theoretical explanation of high temperature superconductivity. Please excuse all the formulas and odd symbols.
Now let me explain the reasoning behind the theory I expressed in the previous post.
The result that produces HTS superconductivity must look something like the Cooper pairs of low temp superconductors, in the sense that two copper electrons are exactly correlated in two ways: by orbital angular momentum and by spin. And there must be some form of coherence or synchrony in the system as a whole, involving links among thousands or millions of these pairs.
But to get that result, I decided there must be several mechanisms--not just the one lattice-charge mechanism of BCS theory. So I looked for three different mechanisms: one to coordinate spin, one to coordinate orbital angular momentum (not to zero, but to the next best level of coordination), and one to create order or synchrony within the lattice, or at least within the relevant portion of the lattice--the copper oxide plane.
This is simply a divide and conquer approach, which is frequently successful in solving any type of puzzle. In addition, this multiple mechanism approach allowed me to look for coupling effects that might occur at higher temperatures. Any direct electron to electron pairing can only be achieved with a lot of anesthesia (meaning very low temperature in this case). Multiple indirect couplings must be at work.
As to the contribution from the lattice itself, I assumed that the perovskite structure was a critical clue. The horse and rider model described in my previous post is a perfect match for perovskites. I assumed also that the behavior of the lattice must be complex and dynamic, because observational techniques have not disclosed any clear and simple lattice effects. And some behaviors, such as spin flipping or other transitory phenomena, suggest a dynamic lattice with some type of regular variation. Dynamic action within the lattice would also help the holes sort themselves into the ideal positions, and might even help move the superconducting current through the system.
I assumed also that Winfree's laws of coupled oscillators would give me a pattern to consider--probably either a four oscillator pattern or a five oscillator pattern. (More on Winfree, and my faith in his patterns, below.) A four or five oscillator pattern in the Winfree system can combine in several different ways. I wanted some Winfree model with multiple combinations since the HTS cuprates show a wide range of interesting phases: antiferromagnetic, underdoped pseudogap, superconductivity, and a host of transport problems and other oddities thereafter. I was pretty sure that it would not be a two oscillator system or a three oscillator system since the effects stemming from such simple patterns would be too limited or basic, and any such simple patterns would have been obvious at some point over 22 years of intense observation going back to 1986.
As to the electrons, I wanted one or more mechanisms to coordinate orbital angular momentum and another mechanism to coordinate spins.
As to orbital angular momentum of the electrons, the fact that holes can pair with electrons is salient. That pairing will be by charge (negative and positive), and that might help control orbit, particularly since the electron in question would be in the D shell. A copper electron in the D shell is more docile than the relatively untamed electron in the S shell, simply because it is more cabin'd, cribbed, confined and bound in. That's like the difference between a trained quarterhorse and a wild mustang.
In addition,a hole-electron pair has a reduced or mitigated negative charge; so it is easier for the electron in that pair to coordinate in some way with an electron in another hole-electron pair. Less inherent repulsion between those two partially neutralized electrons means one less obstacle to any type of pairing between them.
So coherent orbital angular momentum might be produced by several factors: the well defined lattice, the motion of the lattice, the d shell, the pairing with the hole, and the pairing by spin described in the following paragraph.
That left spin as the final element. There, it would be quite natural for an electron-hole pair with a spin of one-half to coordinate with another electron-hole pair with a spin of one-half. Once that pairing becomes effective, the same two electrons may coordinate their orbital angular momentum as well, if that coordination has not already been achieved through the other mechanisms I described above.
Now let me address quantum mechanics and Art Winfree. My theory is based heavily on Art Winfree's notion that limit cycle oscillators have a tendency to couple in various ways when proximity and frequency of oscillation are just right. How can I justify relying so much on that "law" and simply ignore quantum mechanics?
My answer to that is simple. Art Winfree's law of coupled oscillators is quite similar to quantum mechanics. Both say that only certain combinations are possible, and they must be exact. The oscillations that Winfree tracks are conceptually identical to waves, wave functions and the motion of particles in physics. I believe the result I propose is fully consistent with quantum mechanics.
Winfree's theory struck me many years ago as a tool that might explain phase transitions generally. This particular effort, which uses his theory to explain HTS, is just a special, though particularly interesting, case. If my broader view is correct, then Ockham's Razor also points in favor of this proposed explanation, since the explanation for HTS should, in a perfect world, be based on a theory that would help explain all other phase transitions.
Time for another installment in this series, like a Dickens novel.
I did a poor job in my last couple of posts in discussing D wave symmetry. Low temp superconducting is S wave, whereas high temp is D wave. The former has two lobes, while the latter has four lobes. Four lobes signify a complex process, with two interactions rather than one. That is consistent with my explanation in the preceding two posts, since I describe two separate pairings--one being a charge-orbit pairing between Electron A and Hole A; and another being a spin up spin down pairing between Electron A and Electron B. (Electron B's orbit is in turn controlled by a relationship with Hole B.) That will produce D wave symmetry rather than S wave symmetry, because each Cooper electron is engaged in two separate relationships.
Let me also address the lattice. As noted above, I believe that a complex lattice dynamic, which can be modeled as a horse and rider system, emerges in the perovskites due primarily to their structure--which duplicates a horse and rider system, in an abstract way. Just as the rider and the four legs of the horse are perfectly coordinated, so two are the five key atoms in the perovskite. Conventional wisdom for HT superconductors is that there is no lattice effect, because there is no isotope effect, unlike the isotope effect observed in low temp (BCS)superconductors. While that may be true, the more important fact is that the cuprate superconductors all have a similar lattice structure--the perovskite structure. I submit that this similarity in structure suggests a lattice effect in high temp superconductors, much as varied results from isotopes suggested a lattice effect in BCS theory. While the cuprate superconductors are varied, one thing they have in common is general lattice structure in the portion of the lattice that is proximate to the copper oxide plane. Isotope variations are not likely to produce an effect in HT superconductors because they do not change the lattice structure in any significant way. It's the rhythm of the entire five piece band that counts in HT superconductors, not the individual beat (phonon) of a single instrument.
So my approach is compatible with D wave symmetry. It is compatible with BCS theory, broadly construed, in that the lattice plays a role, and in the fact that Cooper pairs are, once again, two copper electrons paired by spin and total momentum--although they are paired not directly by one force (lattice vibrations or phonons), but indirectly as the result of about three separate influences.
I believe my theory also does the best job of explaining how Cooper pairs might emerge at the high temperatures of HT superconductivity. There must be multiple influences to produce this result, and those influences must come from every quarter: the lattice, the holes, regular dynamic movement, charge, spin, and the synchronies that Art Winfree found among countless groups of limit cycle, weakly coupled oscillators. Many hands make light work.
Now it is time for some definitive quantitative and technical data to prove my theory.
I refer to a 2007 article entitled "A determination of the pairing interaction in the High Tc cuprate superconductor ..." by Little, Holcomb, Ghiringelli et al. Read it now, on the web.
The authors state:
"Our conclusion is that the high superconducting transition temperature of the optimally doped cuprates is the result of the combined contributions of a moderately strong phonon interaction, and an excitonic-like interaction from the d-to-d excitations of the Cu d-shell."
This conclusion is exactly in line with my theory, in every respect.
First, their conclusion confirms that there is a contribution from the lattice and a contribution from the level of the electrons, and that both are needed to explain the high transition temperature. This is exactly consistent with my theory--as I said above, many hands make light work.
Second, they conclude that the contribution from the level of the electrons involves copper electrons, as I did.
Third, the electronic interaction is based in the d-shell, as I posited.
Fourth, they say that the contribution from the electronic level is "an excitonic-like interaction..." An "exciton" is a bound state of an electron and an imaginary particle called an electron hole. That definition of an exciton comes from Wikipedia. In other words, their data support my notion that a copper electron pairs with a hole. (I go on to say that the same copper electron then pairs again, by spin, with another electron that is also in an electron-hole relationship.)
Here's a confession: I did not know about this article until this morning. Yet the conclusion and the data support my theory in multiple respects. That's either a stunning coincidence or, far more likely, a near proof of my theory.
I'm pleased to note that two of the authors, W A Little and M J Holcomb, are from Stanford.
Now I am going to take a different tack, in which I stick with my general theory but apply it in a different and more direct way.
In this new approach, I begin by reiterating the views I expressed above about the lattice, its importance, and its oscillations (think phonons, if you prefer).
As I said, I suspect that the lattice oscillations are highly organized in patterns much like the legs of a horse, in which the horse's gait is a trot. This would apply to any group of four copper atoms, and any group of four oxygen atoms. See the discussion in my preceding posts.
If that is correct, then this "horse trotting" model means that two copper atoms (upper left and lower right, or lower left and upper right, just as a horse's legs move in a trot) will move together. One such paired motion will result in phonons with a given orientation; and the other such paired motion will result in phonons with a diagonally opposed orientation. The same holds true for any group of four oxygen atoms.
In other words, the portion of the lattice that is the copper oxygen plane will have two highly organized phonon waves which will serve as warp and weft to organize the electrons in that plane in a highly symmetrical fashion.
More specifically, this "warp and weft" organization from the lattice phonons could produce the hallmark D wave symmetry. As I noted in an earlier post, D wave symmetry suggests that two mechanisms organize the four lobes of the D wave (not one, three or four mechanisms); and it further suggests that those two mechanisms must be perfectly synchronized with each other.
One virtue of this alternate approach, among several virtues, is that it hews closely to the original BCS theory (in which lattice vibration is critical), while recasting it in Art Winfree terms, which is in my judgment a more fundamental, and more accurate, way to understand all phases of matter and all phase transitions.
Let me suggest another variation. Once again, I start with Art Winfree and his theory of coupled oscillators, as expressed in my first few posts. I remain confident that one or more synchronies create superconductivity, and that those synchronies are consistent with Winfree's theory of coupled oscillators.
I also repeat my analysis of Cooper pair formation, as set forth four posts above and three posts above.
The variation i propose is that there is a second step, after the formation of the Cooper pairs. In the second step, two sets of Cooper pairs become connected as coupled oscillators. Thus a synchronized pair in turn synchronizes (at a lower temperature) with another such synchronized pair.
This is consistent with my early posts. Position and orientation of the oscillating units are critical; and so is temperature, which determines the frequency of the oscillations. Once a set of oscillators synchronizes, that might lead to a second form of synchrony at an even lower temperature. These are the core ingredients in Winfree's theory of coupled oscillators.
This two step theory fits rather well with the data. The first synchrony (pair formation) might be the pseudogap phase. The second step (pairing between two sets of pairs) might be the superconducting phase.
This might also fit nicely with d wave symmetry. There would be four lobes to a two by two pairing. Those lobes might be anisotropic.
When I refer to Cooper pairs, I should explain that such pairs are not likely to behave exactly the same as they do in BCS superconductors, where they are less constrained. As a result, they may not superconduct until they reach a second stage in which one Cooper pair is synchronized and connected with another Cooper pair. This is the primary reason why I have pondered a four way pairing (two by two), as opposed to the two way pairing I suggested in a prior post.
The 45 degree vulnerability to magnetic force seen in the pseudogap phase would be compatible with this theoretical model, since the unit cell is shifted at a 45 degree angle.
This model is also consistent with the abstract model of a horse walking at a trot, which I mention above on several occasions. The horse walking model, abstracted into Winfree coupled oscillator theory, involves two by two pairing in a dynamic format. A dynamic but consistent model would be maddeningly difficult to observe, but would still present the spatial precision necessary for superconductivity, which under any theoretical perspective must be based on some precise order.
Finally, this model might be a better fit also with the other, non-superconducting phases of the cuprates.
What about superinsulators, which are described as the exact opposite of superconductors? In 2008, researchers at Argonne National Laboratory discovered that titanium nitride in a thin film format turns into a superinsulator when cooled near absolute zero and bathed in a magnetic field. There is a sudden transition at which point the electrical resistance is infinite. Titanium nitride is a superconductor (old, low temp variety) under other conditions.
This discovery (made well after my first posts on this site) supports my basic point, that Art Winfree's coupled oscillator model explains superconductivity and other phases of matter and their transitions.
The microscopic explanation for superinsulators is still under discussion. The Argonne team published an explanation in 2008; there was a well qualified criticism within a month or two thereafter; and then the Argonne team published a more detailed explanation in Nature in December 2009.
However, there is general agreement that a superinsulator is perfectly organized, probably in Cooper pairs, but with the Cooper pairs synchronized in some manner that is the exact opposite of the manner in which superconducting Cooper pairs are organized.
This is the essence of Art Winfree's coupled oscillator theory. Oscillators are frequently perfectly organized one way, or perfectly organized exactly the opposite way. Synchronous or anti-synchronous--exactly so, either way. Zero percent resistance or 100% resistance. (These are the choices that Art Winfree identified for two oscillator systems; other choices of organization are available for three oscillator systems, four, etc.)
In a thin film, the chances are that the elements are akin to a two oscillator system, in which one organizational result or the exact opposite result are the only two choices. In thicker materials, more than two organizational choices are available to the materials.
My conclusion is that superinsulators prove that Art Winfree's coupled oscillator theory is the proper tool to explain low temp superconductivity. BCS theory is just a subset of Art Winfree's theory about how limit cycle oscillators organize themselves.
In addition, I maintain that high temp superconductors and low temp superconductors must share the same basic organizing principle. Ockham's razor. Art Winfree's theory is a basic principle that works for both, and it is the only such principle of which I am aware. (And physicists over the past twenty years have failed to find any satisfactory explanatory principle that covers low temp and high temp superconductivity.)
I will leave it to others to decide whether I have the details right in the way I have applied the Winfree theory to superconductivity in the preceding posts. Perhaps I have not applied Winfree's theory exactly right. But I am confident that Winfree's theory is the underlying explanation for all phases of matter--old, new, super, quantum, and others yet to be discovered. The discovery of superinsulators is an important confirmation that Winfree's law of coupled oscillators is the right theory.
Let me also hazard a guess as to the details. I want to focus on two things: the extremely low temperature and the magnetic field. Both factors directly affect oscillations. Extremely low temperature has the effect of lowering the frequency of oscillation, which in turn usually increases the range of the waves that emerge from the oscillation. This probably expands the range of communication between oscillators and changes the dimensions of the organized system. As to the other ingredient--a magnetic field--this of course acts on the spin of the electrons, which is one of the oscillations that is organized in Cooper pairs. So the details of the experiment producing the superinsulator are directly relevant to oscillations--the organization of which is the concept on which my theory is based.
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Newly released observations by Riken, the research institute in Japan, confirm my theory. On April 22, Riken reported finding a mechanism for superconductivity in an iron based superconductor (a pnictide).
The release shows a picture suggesting an S +- wave structure that is unique to a material with two types of electrons.
The picture strongly suggest that two sets of waves (oscillations) emanating from different points have intersected at a precise angle, leading to a set of combined oscillations that are synchronized and perfectly organized. In other words, a complex seiche, to borrow a nautical term.
This is consistent with other recent research showing that with pnictides the angle in the structure must be exactly so for superconductivity to emerge.
These two observations support each other, and directly confirm that coupled and synchronized oscillations lead to superconductivity in the pnictides.
The Riken announcement allows me to tie together all three forms of superconductivity (BCS, cuprate, pnictide)using the coupled oscillator theory I have described in multiple posts on this site.
In each case, Art Winfree's law of coupled oscillators, applied to physics rather than biology, is the operative mechanism. In BCS theory, Cooper pairs are direct coupled pairs of electrons, coupled in antisynchronous fashion...antisynchronous as to orbit and as to spin. In the pnictides, Riken shows us that two sets of oscillations couple and synchronize if they intersect at a specific angle. The oscillations appear to be magnetic waves emanating from two different types of electrons. The result is a 2 way pairing, also exactly antisynchronous (the up dots versus the down dots, in the Riken picture). In the cuprates, there is a four way pairing (2 by 2). Each two way pairing is exactly antisynchronous, and the 2 by 2 interaction is also exactly antisynchronous.
Ockham's Razor says that if one underlying theory can explain several phenomena, it is likely to be correct. The difficulty so far is that theorists have failed to focus on the common denominator: coupled and synchronized oscillations. The precise nature of the oscillations varies among the three types of superconductors, but Art Winfree showed that limit cycle oscillators have a tendency to synchronize, regardless of the nature of the particular oscillations.
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This example contains all the elements that need to be analyzed in understanding superconductors--new or old. Low temp superconductors or high temp superconductors where the real action lies. They then write down a gap equation, and find out that the dominant eigenvalue displays the d-wave cos(kx)-cos(ky) behavior.
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