Haldane on the second quantum revolution

This week I attended a great public lecture by Duncan Haldane: "Quantum Mechanics After One Hundred Years, and the 'Second Quantum Revolution' Today"

Starting from the discovery of quantum mechanics, he explained how the concept of quantum entanglement is fueling today's second quantum revolution and its connection to his Nobel Prize-winning work.

Haldane remarked that his work on quantum spin chains was controversial. He had theorists accosting him at conferences arguing he was wrong. These kinds of disputes among theorists are best settled by experiment. Undoubtedly, Haldane would not have received his Nobel Prize if his predictions had not been validated by experiments. How can you motivate some experimental group to be interested in your theory? If it generates controversy!

Similarly, experiments often are the drive for fresh theoretical advances. For example, the experimental discovery of the quantum Hall and fractional quantum Hall effects came before the theoretical predictions or understanding. 

A good example of this is a second important work by Haldane also cited in his prize: quantum Hall effects in absence of Landau levels. This now-seminal work went largely unnoticed for a decade, because the model based on a two-dimensional honeycomb lattice seemed unfeasible to realize in an experiment. Later, the unanticipated work experimental isolation of graphene drove theorists to this fresh area. Haldane's early theory work was recognised as the foundation for the discovery of time reversal-symmetric topological insulators and the whole "zoo" of topological materials that followed.

Haldane also emphasized the importance of luck in making ground-breaking discoveries. von Klitzing was not the first person to attempt quantum Hall measurements, but previous attempts had used a different experimental setup: varying current with a fixed magnetic field. Imperfections in the current source led to fluctuations in the measured resistivity, which seemed to be consistent with previous approximate theoretical calculations based on perturbation theory. von Klitzing's approach of measuring resistivity as a function of magnetic field strength, with current kept fixed, led to unexpectedly precise quantization which needed new theory to explain. 

Haldane's take-home message was thus: anyone can win a Nobel Prize, but you need luck and the perseverance to defend your work if it is challenged.

An earlier iteration of this talk is available here. A more detailed write-up is available here. 

   

IBS-APCTP Conference on Advances in The Physics of Topological and Correlated Matter

Last month I attended the IBS-APCTP Conference on Advances in The Physics of Topological and Correlated Matter. It was great to be attending a conference in-person after so long! Slides from many of the talks are available here. Summaries of some of the talks from the notes that I took:

------

Prof. Changyoung Kim (IBS/SNU) spoke about tunable anomalous Hall conductivity in correlated electron systems. The anomalous Hall effect is a Hall effect that occurs in the absence of an external magnetic field in materials exhibiting a spontaneous magnetization of the electron spins. An extreme example of this is half-metals, where the energy bands are completely filled (insulating) for one spin and partially-filled (conducting) for the other spin. 

One way to potentially enhance the anomalous Hall conductivity is to dope the material to tune the Fermi level of the partially-filled band to a region of high Berry curvature, but in some cases this kind of perturbation picture will fail due to the doping-inducing a reconstruction of the band structure to form a distinct phase. It is important to distinguish the intrinsic anomalous Hall conductivity from (potentially much larger) extrinsic (disorder-induced) contributions.

Another approach applicable to thin film materials is substrate-induced magnetization, which can lead to interesting thickness-dependent material properties including the magnitude and even sign of the anomalous Hall conductivity changing as single layers are added or removed. These changes can be resolved using spin-resolved ARPES.

------

Prof. Masatoshi Sato (Kyoto University) talked about Majorana fermions in topological superconductors, particularly differences between spinless and spinful Majorana fermions; the latter require additional symmetries for protection. The symmetry requirements will likely make it quite extremely challenging to carry out robust braiding operations required for topological quantum computation. Another application of Majorana particles outside of quantum computing is to use them to probe the properties of the underlying superconductor, since they share the same symmetries as the underlying superconducting gap function.

------

Prof. Takashi Oka (University of Tokyo) covered Floquet engineering of topological bands and novel topological nonlinear optical effects, including Hall response dependent on a nonlinear gauge-invariant curvature term, related to theoretical analysis published in Nature Physics at the end of last year. Floquet engineering of topological bands has of course been a very influential idea since the late 2000s (Oka and Aoki's 2009 paper has amassed more than 1000 citations!), with a variety of linear (and nonlinear) phenomena occurring sensitive to the period and strength of the periodic Floquet driving.

------

Prof. Gil-Ho Lee (POSTECH) discussed his group's experimental work on Floquet engineering of graphene using microwaves. The effective strength of the Floquet-induced band structure modifications scales as \( E / \omega^2 \), where \( E \) is the (normalized) field strength of the drive and \( \omega \) is its frequency. By using a lower frequency microwave drive (instead of more conventional optical pumping), the required field strength is reduced, making it practical to study continuous wave driving while maintaining a sufficiently low temperature. The catch, however, is that the lower drive frequency is accompanied by smaller Floquet band gaps that require scanning tunneling microscopy to resolve. Ongoing work aims to observe Floquet topological bands induced by polarized microwave driving, and to improve the coupling between the microwave antenna and the graphene sheet to probe higher Floquet drive strengths.

------

Dr. Andrew Pierce (Harvard -> Cornell?) talked about thermodynamic measurements of topological states in magic angle graphene, the subject of his PhD studies and three (!) Nature Physics papers published in the last year ([1], [2], [3]). He mentioned that bilayer graphene is an extremely fickle system - every sample is different and it is very hard to reproduce experimental results. Trilayer graphene, on the other hand, can offer a more reliable platform for exploring moire phenomena including twist-induced superconductivity. 

------

Dr. Robert-Jan Slager (Cambridge University) talked about unconventional multi-gap topological phases. In the conventional topological band theory we usually consider a single band gap whose topological properties (and the existence or absence of robust edge states) is determined by topological invariants of all the bands below the gap, leading to the well-known periodic table of topological insulators. Interestingly, the topological band theory can be generalized to novel multi-gap topological phases which are trivial under the conventional topological band theory, involving partitions of the band structure into three or more sectors. One example of non-trivial multi-gap topological invariant is the Euler class. These ideas are now being generalized to Floquet systems.

------

Prof. Aris Alexandradinata (UC Santa Cruz) presented a topological principle in photovoltaics. Bulk photovoltaic effects known as shift currents are attracting interest as a potential route towards making more efficient solar cells. A nonzero shift current requires a material with broken inversion symmetry. Beyond this necessary condition it is not clear what is the best way to maximize the strength of the shift current. New classes of topological phases may provide a route to finding materials with stronger shift currents. He currently as a few PhD and postdoctoral research fellow positions open!

------

Prof. Emil Bergholtz (Stockholm University) talked about fractional Chern insulators and quantum geometry in moire materials. A long-standing goal in the field of topological materials has been the search for fractional Chern insulators, which are hoped to exhibit high temperature fractional quantum Hall states. The "simple" picture of a fractional Chern insulators is that they are lattice systems exhibiting nearly flat bands with nonzero Chern numbers that form a lattice analogue of the Landau levels underlying the conventional quantum Hall and fractional quantum Hall effects. In practice, however, the physics is more complicated. Even if the energy dispersion is relatively flat, other properties of the band including its quantum metric and Berry curvature can be strongly non-uniform, disrupting the analogy with continuum Landau levels and destroying the fragile fractional quantum Hall states. This seems to also be the case with the nearly flat topological bands arising in twisted bilayer graphene: although the single particle band structure is flat, interactions combined with the non-trivial momentum-dependence of the quantum metric lead to self-consistent (Hartree-Fock) bands with strong dispersion, inhibiting the formation of fractional quantum Hall states. He also has some postdoc openings!