quasicrystals
While I was in college doing bachelor degree I was looking for a topic for my BS diploma. The topic my advisor (Artem Oganov) suggested was, while important for industry applications, world transformation and all that, did not spark any internal excitement.
I came across Dan Shechtman Nobel lecture about quasicrystals and decided that I ought to do something along these lines, mainly interested in high-dimensional approach developed by Kitaev, Kalugin and Levitov [1]. Quasicrystals are materials that are not periodic (most of the materials are) yet exhibit some sort of order. The fact that they are not periodic makes ab-initio (first principles, DFT) calculations, well, hard. First principles calculations were what Artem’s lab was doing and I decided that I should do something to make that task bearable. KKL [1], [2] papers said that quasicrystals can be represented as crystals in higher dimensional space. For example, 1D Fibonacci quasicrystal was represented as a 2D crystals. What was not clear to me, however, how to write an analog of a Schroedinger equation in that high-dimensional space so that periodic boundary conditions were there.
Turns out, if we modify the kinetic part of the Schroedinger equation in higher dimensional space for an electron in a periodic potential, we will have an equation that is easy to map on a normal Schroedinger equation of an electron in a quasiperiodic potential in a physical space. That I did and also proved (under one assumption that an equivalent of the Bloch theorem holds) validity of the mapping.
I also generalized the framework to quasicrystals with arbitrary dimension and a number of incommensurate periods.
Science journalists really liked the paper and as a result it was covered in several newspapers/journals.
One famous scientist, however, did not share enthusiasm of the journalists and accused me of plagiating KKL papers [1,2]. My impression of the discussion was, however, that he did not read the paper, but only the popular science article. That accident still puzzles me.
In fact, somewhere in the middle of the manuscript writing I discovered that the equation was written before [3]. What they were saying is that there is a relation between spectrums of two equations, and the relation is such that any solution of the higher-dimensional equation would correspond to Bloch-like solution of the normal Schrödinger equation, the inverse relation (low D mapping -> high D) has not been established. I have also (ugh, that really looks like I’m apologetic of something) had expressions for c-coefficients (or \tau, they linked to the symmetry) that they didn’t.
https://www.nature.com/articles/srep11492.pdf
[1] Kalugin, P., Kitaev, A., & Levitov, L. (1986). Electron spectrum of a one-dimensional quasicrystal. Zh. Eksp. Teor. Fiz, 91, 692-701.
[2] Kalugin, P. A., Kitaev, A. I., & Levitov, L. S. (1985). Al 0.86 Mn 0.14: a six-dimensional crystal. ZhETF Pisma Redaktsiiu, 41, 119-121.
[3] Lu, J. P., & Birman, J. L. (1987). Electronic structure of a quasiperiodic system. Physical Review B, 36(8), 4471.
https://www.gazeta.ru/science/2015/09/01_a_7731545.shtml?ysclid=lq7hopnfr527390806 (second half about the “relative” of graphene)
dynamics of an antiferromagnetic Hubbard layer in contact with the environment
I got interested in the quantum Monte-Carlo while working with Artem and consequently joined Alexey Rubtsov’s group at RQC. Long story short, I barely did any Monte-Carlo, but learned a whole new world of open quantum systems and quantum non-equilibrium. Surprisingly, the work was initially motivated by high-temperature superconductors, but because of either my stupidity or complexity of the problem we ended up doing mean-field for an antiferromagnetically ordered Hubbard system driven out of equilibrium through the quench of the interaction parameter. First for an isolated system, then for an open one.
It was known before for BCS [1,2] that because the closed system is integrable, it has dynamical regimes where persistent oscillations of the order parameter (superconducting gap) occur. Because the AFM order can be easily mapped to a BCS, same happens for the system we looked at.
For an open system, we identified three distinct dynamical regimes (slowly decaying oscillations, Landau damping, strongly damped), that are related to the ones observed in the isolated system. It was somewhat surprising (for us at least) that the system, despite being open (we had EOM equations with memory kernel) does not equilibrate to its equilibrium state after a quench. In fact, relaxation we observed numerically, was extremely slow (power law). The fact that the thermalization did not happen in some of the regimes we explained through the competition of the Landau damping [3] mechanism of relaxation and the relaxation through the bath. If the bath wins over the Landau damping (larger values of coupling to the bath), the system relaxes closer to a new equilibrium than it would for smaller values of the coupling to the environment. The fact that there is a competition between the Landau damping and relaxation through the environment was, I think, unknown – all authors who were involved in the work were surprised by this fact.
It is probable, however, that on the level beyond mean-field coupling of the collective mode to the environment will lead to the full thermalization. For weak coupling to the environment that should happen, however, at much longer times. But to be honest, we never checked. Could be a decent question for a MS diploma perhaps.
Another thing one would do it to include fluctuations of the phase of the order parameter in the dynamics, but that could be computationally expensive.
While it was me who performed all the calculations, and Alexey who came with an idea for the project, I benefited a lot from working with a then postdoc at RQC – Pedro Ribeiro, whose push for perfection – he suggested, for example, writing everything in pseudo-spin notation –made both the manuscript and the science behind it by order of magnitude better than it was initially.
https://link.aps.org/accepted/10.1103/PhysRevB.95.024309
[1] Yuzbashyan, E. A., Tsyplyatyev, O., & Altshuler, B. L. (2006). Relaxation and persistent oscillations of the order parameter in fermionic condensates. Physical review letters, 96(9), 097005.
[2] Barankov, R. A., Levitov, L. S., & Spivak, B. Z. (2004). Collective Rabi oscillations and solitons in a time-dependent BCS pairing problem. Physical review letters, 93(16), 160401.
[3] Landau L. D., Zh. Eksp. Teor. Fiz. 16, 574 (1946).
higher harmonics for Mott insulator
One of the perks of working at RQC were generously funded trips abroad. My first trip from the quantum center as a visiting student was to Berlin, where I spent a month or so working with Misha Ivanov and Olga Smirnova. They were experts in high-harmonic generation – when you shine at something with a laser at frequency N, something will often reemit it back at the same frequency N (and often below it, if some relaxation process are present). However, if the laser is superpowerful, the thing will reemit not only light at frequency N, but also a noticeable amount of light at frequencies 2N, 3N and so on – that is higher harmonics generation [1] . “Something” is often a gas of weakly interacting atoms.
Alexey, was, in turn, an expert in strongly correlated quantum matter.
So the scientific question was generated analogously to a meme about a pineapple: what happens if we shine with a super powerful laser to a 1D Mott insulator [2]? In that project, Misha’s postdoc was doing exact time-evolution numerically, and I was teaching everyone (mainly Rui) about delights of condensed matter physics. Everyone, in turn, were destroying my ignorance about the high-harmonics generation and related math.
Besides that, I made an attempt to do Keldysh technique for a strongly driven Mott insulator, but realized after a while that trying to reach that regime perturbatively in interaction parameter was not the cleverest idea. I did then mean-field numerically for 1D AFM state in Hubbard model driven with an electrical field. In both cases (MF and ED), order parameter goes to zero soon after the laser pulse and an insulator becomes a conductor. My perspective was that it is an effective heating. Temperature raises because of the energy injected by the laser pulse and hence the Mott insulator melts.
The high-harmonics picture was very different from mean-field results – most likely, because interactions between elementary excitation (doublons and holons for charge excitations) appear to be important for it, since a transition insulator-conductor happens at a finite excitation density.
Rui found Takashi Oka’s [3] analytical paper for the Mott insulator case about breakdown of Mott insulator. The phenomena Taskashi aptly described by Dykhne formula was very similar to a high-energy phenomena of creation particle-antiparticle pairs out of vacuum [2]. The vacuum in this case is an insulating state, and particles-antiparticles are the pairs of elementary excitations.
The resulting paper was published in Nature and, according to Rui, got experimentalists very excited.
I, in turn, felt a bit burnt out after my trip to Berlin and went for a short vacation to Caucasus.
There is still a question whether the regime where the breakdown through the holon-doublon production wins over a simple heating is observable – a simulation with a few atomic sites possibly does not capture thermalization mechanism that well and neither do the mean-field calculation. In the hindsight, what was happening in the mean-field simulation is simply destruction of the ground state by generation a whole lot of single-particle excitations. Similar to heating, but not quite that.
Overall, it turned out that despite the fact the research idea was on the surface, I learned a lot from that trip through diving into an unknown area and Olga’s and Misha’s willingness to share their skills and knowledge.
https://www.nature.com/articles/s41566-018-0129-0
[1] Vampa, G., McDonald, C. R., Orlando, G., Klug, D. D., Corkum, P. B., & Brabec, T. (2014). Theoretical analysis of high-harmonic generation in solids. Physical review letters, 113(7), 073901.
[2] Schwinger, J. (1951). On gauge invariance and vacuum polarization. Physical Review, 82(5), 664.
[3] Oka, T. (2012). Nonlinear doublon production in a Mott insulator: Landau-Dykhne method applied to an integrable model. Physical Review B, 86(7), 075148.
https://mbi-berlin.de/research/highlights/details/from-insulator-to-conductor-in-a-flash
high-temperature exciton insulator in double bilayer graphene
Upon my arrival to the University of Texas at Austin I first took a project from Andrew Potter about a single driven spin connected to a bath but that did not result in anything publishable I think. After a while I took a project from Allan MacDonald.
Allan MacDonald had an argument with Konstantin Efetov about the critical temperature of the exciton phase transition in graphene bilayers. Allan’s postdocs did mean-field calculation with a Coulomb interaction for a graphene bilayer. Their estimate of the BKT-temperature was extremely high, around room temperature [1].
Efetov’s paper [2], in contrast had the instability analysis with screened interaction. Because screening (in RPA approximation) is proportional to the number of species, and bilayer has 4 low-energy degrees of freedom, the estimate of the interaction was rather small and, consequently, using MacMillan formula (instability in exciton condensates is also logarithmic, like in BCS), one could find that the critical temperature is vanishingly small.
Allan wanted me to do the same estimate as Efetov did, but for a double bilayer (two AB bilayers on top of another).
Long story short, instability analysis showed that things are even worse in the double bilayer because of the increase in the number of degrees of freedom (the power in the exponent in the MacMillan formula gets multiplied by the number of them). After realizing that we switched to something else.
After finishing the calculation, I talked (personally) to another high-caliber scientist at a different place about that problem. He seemed to be puzzled by the research idea, suggested we could do something else, but we never agreed on what – it is hard to agree on anything over emails – and so I delved into calculation of the tunneling currents in TMDs.
[1] Min, H., Bistritzer, R., Su, J. J., & MacDonald, A. H. (2008). Room-temperature superfluidity in graphene bilayers. Physical Review B, 78(12), 121401.
[2] Kharitonov, M. Y., & Efetov, K. B. (2008). Electron screening and excitonic condensation in double-layer graphene systems. Physical Review B, 78(24), 241401.
tunnelling enhancement through exciton fluctuations in the double bilayer
As a project, Allan suggested me looking at the Emmanuel Tutuc’s [1] paper about enhanced tunneling in the double bilayer graphene. One of the possible interpretation was that the drastic enhancement in the tunneling conductivity (derivative of the tunneling current with respect to the bias) was because of the formation of the exciton condensate between the bilayers. Tutuc’s group looked at the various regimes, while I looked at the one where the valence band of one bilayer has an energy overlap with the conductance band of another bilayer – Dmitry Efimkin, who was my office mate at the time, looked at a different regime before me – at a regime where instead the energy overlap happens between alike bands. In his work, he managed to explain the enhancement simply through the fluctuations.
I haven’t published the result yet – basically, someone has stolen my laptop with it (and a dog chewed up my homework).
[1] Burg, G. W., Prasad, N., Kim, K., Taniguchi, T., Watanabe, K., MacDonald, A. H., ... & Tutuc, E. (2018). Strongly enhanced tunneling at total charge neutrality in double-bilayer graphene-WSe 2 heterostructures. Physical review letters, 120(17), 177702.
moire excitons
The next project was about moire excitons – excitons are still bound pairs of electrons and holes, yet they are situated in a long-period lattice. The question was how their energies behave with doping.
Turns out, the question is computationally hard as there are many bands and interaction vertices have non-diagonal elements.
Our main finding was that the energy shift is of order E_{1s}(a_B/a_m)^2 – still unpublished for reasons other than the laptop went missing.
coexistence of the superconductivity and interlayer coherence
There could be superconductivity in 2D layers. There could be coherence between different layers. Can the two things coexist?
That was roughly the question Allan had in summer of 2020. It was mainly inspired by the superconductivity in moire graphene bilayers [1].
First, one should agree what is coherence. In my definition, that state corresponds to an order parameter of the form
Coherence is pseudo-magnetism (instead of spin degree of freedom we have layers) in the x direction. If interaction between the layers of the system is sufficiently strong (a criteria analogous to what we have in the Stoner magnetism), we can have both s-wave superconductivity and interlayer coherence.
We also found that to have both order parameters non-zero, one would need to have interlayer superconductivity as well —
I met a lot of opposition from my advisor while doing that work – issues about interlayer superconductivity and massive phases are somewhat non-trivial. Mean-field equations, for example, were derived via three different approaches in the summer of 2020.
However, that interaction forced me to make many of the explanations in the article simpler, and a result, the final work better.
What I still don’t understand, however, is that on mean-field level, there is a mild enhancement of the superconducting order parameter for large interlayer repulsion – order parameters are coupled – was not able to tell anything remotely rigorous.
https://arxiv.org/abs/2204.00438
[1] Cao, Y., Fatemi, V., Fang, S., Watanabe, K., Taniguchi, T., Kaxiras, E., & Jarillo-Herrero, P. (2018). Unconventional superconductivity in magic-angle graphene superlattices. Nature, 556(7699), 43-50.
[2] Andreev, A. F. (1965). Thermal conductivity of the intermediate state of superconductors II. Sov. Phys. JETP, 20, 1490.
PIP-phase in ABC-graphene
My officemate Huang Chunli pointed me to an article with observation of the superconductivity in the rhombohedral graphene trilayer [1]. They looked at quantum oscillations at different doping fractions at large displacement field (electric field perpendicular to the plane of the graphene layers).
As a function of doping, they observed a number of the phase transitions in that and their previous paper [2]. Superconducting phase there goes along the boundary between the normal phase and so-called PIP-phase (partially isospin polarized phase). It also seemed vaguely reminiscent to both what was happening in the high-temperature superconductors [3] and in moire materials [4], so I thought it should be worthy to explore the PIP-phase.
At the time, there was a work by Michael Zaletel where they interpreted the PIP phase as a half-metallic phase with coherence between different valleys [4]. The phase they had effectively was happening at zero transferred momentum (or K-K’ if one do not wish to call valley a pseudospin).
The first transition was happening not long after opening of a second Fermi surface: at the same hole doping, there were two Fermi surfaces, one is hole-like and another is electron-like. Such shape of the Fermi surface can be seen in the quantum oscillations picture.
I did not encounter many PIP phases before neither did I see any systems with the annular Fermi surface, so it seemed reasonable that the annular Fermi surface is one of the main factors contributing to the formation of the PIP phase – a conductive phase with resistance of order of 1.5-2 larger than the normal phase.
But then there got to be something that distinguishes between the PIP phase and the valley polarized phases – that role was given to the anisotropy of the Fermi surface. I took then a very simple one-band model that has both features: Fermi surface anisotropy and annular Fermi surface.
Turns out, the instability in the electron-hole channel between different valleys is the strongest at the finite momentum in agreement with some other works [5,6]. The momentum at which the instability is the strongest, corresponds to the difference between the two Fermi surface.
The state that forms as a result I called “partial excitonic condensation” for the following reason: part of the order parameter consists of electron-hole pairs residing on different Fermi surface (and different valleys), and part of the order parameter acquires its value through electron-hole pairs between the same Fermi surfaces (and different valleys). Admittedly, it is somewhat similar to the charge density wave, yet charge density wave systems normally do not have the annular Fermi surface.
Electron-hole pairs residing on different Fermi surface are more similar to an exciton condensate, because their effective masses have an opposite sign. The instability then is of Cooper type. Electron-hole pairs residing on the same Fermi surface have effective masses of the same sign and hence the state is more similar to pseudospin magnetism. Thus, only part of the particles contributing to the order parameter are in the exciton condensate and hence the name. This picture also allows to understand why resistance in this phase is higher.
We published a preprint, then I delved into an idea of having an analytical LG-style calculation of the conductance – that is what I am currently doing for unforgivably long time.
https://arxiv.org/abs/2303.17350
[1] Zhou, H., Xie, T., Taniguchi, T., Watanabe, K., & Young, A. F. (2021). Superconductivity in rhombohedral trilayer graphene. Nature, 598(7881), 434-438.
[2] Zhou, H., Xie, T., Ghazaryan, A., Holder, T., Ehrets, J. R., Spanton, E. M., ... & Young, A. F. (2021). Half-and quarter-metals in rhombohedral trilayer graphene. Nature, 598(7881), 429-433.
[3] Prokof’ev, D. D., Volkov, M. P., & Boikov, Y. A. (2003). Pseudogap and its temperature dependence in YBCO from the data of resistance measurements. Physics of the Solid State, 45, 1223-1232.
[4] Chatterjee, S., Wang, T., Berg, E., & Zaletel, M. P. (2022). Inter-valley coherent order and isospin fluctuation mediated superconductivity in rhombohedral trilayer graphene. Nature communications, 13(1), 6013.
[5] You, Y. Z., & Vishwanath, A. (2022). Kohn-luttinger superconductivity and intervalley coherence in rhombohedral trilayer graphene. Physical Review B, 105(13), 134524.
[6] Lu, D. C., Wang, T., Chatterjee, S., & You, Y. Z. (2022). Correlated metals and unconventional superconductivity in rhombohedral trilayer graphene: a renormalization group analysis. Physical Review B, 106(15), 155115.