From graphene to borophene

In the beginning there was graphene, and graphene had some very remarkable properties which have attracted enormous interest over the years. For the unfamiliar, graphene is a two-dimensional sheet of carbon atoms arranged in a honeycomb lattice structure. The tight binding energy band structure has the peculiar property that its conduction and valence bands touch at the corners of the Brillouin zone. These corners are known as Dirac points because the low energy electronic degrees of freedom are governed by an effective Dirac equation,

$$ i \partial_t \psi = v_F (\boldsymbol{p} \cdot \boldsymbol{\hat{\sigma}} ) \psi,$$

where $\boldsymbol{\hat{\sigma}}$ are the Pauli matices, $\boldsymbol{p}$ is the in-plane momentum, and $v_F$ is the Fermi velocity, which acts as an effective speed of light.

 

The role of spin is in graphene's effective Dirac equation is played by the sublattice degree of freedom – the underlying honeycomb has two sublattices, termed A and B, that are inequivalent. This is known as a pseudospin. Because of this Dirac equation description, electrons in graphene can emulate a variety of interesting phenomena from high energy physics, such as the Klein paradox.

This interesting Dirac physics is not specific to graphene, but emerges in any periodic potential with a honeycomb lattice structure, including photonic systems. In this case, the electronic wavefunction is replaced by the optical field envelope, and the effective potential can be controlled by modulating the local refractive index. For example, in the case of semiconductor microcavities, the potential modulation can be created by selective etching of the cavity to form a honeycomb structure. The resulting photonic band structure can be observed experimentally by measuring the energy-resolved photoluminescence spectrum from the cavity, which reveals a neat Dirac cone structure where the two energy bands cross.



One of the interesting properties of a Dirac cone is that it has an emergent rotational symmetry. Even though the potential is inhomogeneous and breaks the continuous rotational symmetry, the energy eigenvalues in the vicinity of the Dirac cone are invariant under rotations. This in-plane rotational symmetry leads to a conserved total angular momentum J, which is the sum of the usual orbital angular momentum, and a pseudospin angular momentum associated with the sublattice degree of freedom. Rewriting the effective Dirac Hamiltonian in terms of pseudospin raising and lowering operators $\sigma_{\pm} = \hat{\sigma}_x \pm i \hat{\sigma}_y$,

$$ i \partial_t \psi = v_F ( e^{-i \varphi} \hat{\sigma}_+ +e^{i \varphi} \hat{\sigma_-} ) \psi,$$

we see that a flip of the pseudospin must be accompanied by a change in the orbital angular momentum (corresponding to the angular phase winding terms $\exp(\pm i \varphi)$. Therefore, if a Dirac cone is excited with a spin up state, one can measure a phase vortex in the spin down field component. This pseudospin-mediated vortex generation has been observed using photonic waveguide lattices.

Around 2011, multiple groups proposed various generalizations of graphene to higher-order conical intersections (reviewed here). The effective Hamiltonian at a higher order conical intersection can be obtained by replacing the spin ½ Pauli operators in the Dirac equation with spin s matrices. The resulting band structures similarly have intersecting conical bands with energies determined by the spin projection parallel to the momentum. 

 

There is a qualitative difference between intersections with integer and half-integer pseudospin. For the integer s case, the spin projection can vanish, corresponding to a flat band with zero energy for all momenta. The first studies of higher order conical intersections however were limited to tight binding models that seemed quite difficult to implement in practice, requiring for example laser-assisted hopping or fine-tuned multilayer structures.

Around the same time, different groups realized that the s=1 conical intersection could be observed using a relatively simple square lattice structure known as the Lieb lattice, which is obtained by removing one quarter of the sites from an ordinary square lattice. Starting from a tight binding model, one can show that the band structure close to the Brillouin zone corners is described by a spin-1 variant of the Dirac equation. This band structure is interesting because one has conical bands with a vanishing wave effective mass intersecting a flat band with an infinite wave effective mass.



One of the important consequences of the flat band is that waves in this band do not propagate, they remain localized. This nondiffracting property of flat band states was observed in 2015 by two groups (papers here and here). A second interesting observable difference of the Lieb lattice is that different pseudospin states have can have differing dynamics, dependent on the magnitude of the initial pseudospin. Initial states with pseudospin plus or minus one partially excite the flat band, leading to a splitting of a beam between a rapidly-expanding conical diffraction component, and a residual flat band component. In these conical diffraction experiments, it's also possible to observe a pseudospin-mediated generation of charge two phase vortices.

What about higher values of the pseudospin s? It gets more challenging. Usually we design conical intersections within the framework of a weak coupling (tight binding) approximation in which coupling between second and most distant neighbours is assumed to be zero. This requires a lattice that is sufficiently deep lattice or has a large separation between the sites. But minimizing second neighbour coupling in this manner also makes the nearest neighbour coupling weaker, reducing the overall energy bandwidth and making it more difficult to resolve the different bands at the conical intersection. This problem naturally gets worse the more intersecting bands one has. So while there have been many studies of Dirac cones and Lieb lattices, the extension of these ideas to higher pseudospin systems is more challenging. Feasible proposals are scarce and most have required complicated fine-tuned models that are difficult to implement.

One way to overcome these problems is to consider conical intersections that are protected by permutation symmetries. The idea is to associate a conical intersection with a permutation symmetry matrix. By re-interpreting the symmetry as an adjacency matrix of a graph, one can embed the degeneracy into a periodic lattice. Detuning the wavevector away from p=0 breaks the symmetry, lifting the degeneracy, which produces a conical intersection in the dispersion relation. This approach can be used to systematically create conical intersections of a desired order. More importantly, the resulting lattices typically involve symmetric and relatively close-packed structures, giving rise to larger bandwidths!

For the case of a five-fold degeneracy corresponding to pseudospin 2, the permutation symmetry approach generates a lattice known as chiral borophene. It can be obtained by considering as a unit cell a filled hexagon, removing one of the corners, and rotating the remaining sites. This gives rise to a lattice with broken mirror symmetry, which has two inequivalent chiral variants. The tight binding band structure has 5 intersecting bands at p=0, with a 6th band separated by a large gap.



The effective Hamiltonian describing the band structure close to $p=0$ is a little more complicated than the usual Dirac Hamiltonian,

$$ i \partial_t \psi = c_0 \boldsymbol{p} \cdot \boldsymbol{\hat{S}} + c_1 \boldsymbol{p} \cdot \{ \boldsymbol{\hat{S}}, \hat{S}_z^2 \} - c_2 \hat{1}, $$

with a second term proportional to an anticommutator of the spin-2 matrices. The reason for this is basically that the spin-2 matrices allow for more non-trivial terms that respect the rotational symmetry. The effect of this additional term is to control the relative opening angle between the pairs of conical bands. When the lattice is excited by a state with pseudospin 2, the conservation of total angular momentum means that phase vortices with charge up to 4 can be generated by post-selecting on different output pseudospin states, as shown in the simulation results published here.

The middle band in chiral borophene is not very flat, and actually has considerable dispersion over the Brillouin zone, even in the nearest neighbour tight binding model. Flat dispersion only occurs along high symmetry lines, which corresponds to the existence of non-diffracting line states. Similar to the case of the Lieb lattice, these nondiffracting states can be excited by considering an input with a staggered phase profile. The diffraction of such states is strongly suppressed compared to a similar input with a flat phase profile, as you can see in these experimental results.

The pseudospin-2 occurring in latices such as chiral borophene opens up many interesting properties for wave manipulation and nonlinear optics, including the possibility for cascaded wave mixing between the partially flat and conical bands, generation of high charge vortices, and the introduction of strain or other perturbations to open up topological band gaps, and possible analogies with the physics and propagation of gravitons (which also have spin 2). It will also be interesting to see whether this lattice can be realized as a two-dimensional electronic material.

The localization landscape

Localization of waves due to destructive interference in disordered media - Anderson localization - has been a subject of intense investigation for more than 50 years. The original paper has now been cited more than 15,000 times according to Google Scholar. Being a property of linear, non-interacting Hamiltonians subjected to a random potential, one might think that everything there is to know about this problem would have already been studied to death a long time ago, with current work focusing on understanding peculiarities arising under special circumstances.

Delightfully, this is not the case! Back in February, this preprint caught my attention due to its keywords of many-body localization and persistent homology. Specifically, the authors have used persistent homology to characterize the shape of the "localization landscape" of a many-body Hamiltonian. 

What is the localization landscape?

The localization landscape was first proposed by Filoche and Mayboroda in an article in PNAS which, despite its broad applicability to understanding wave transport in a variety of settings (condensed matter, photonics, acoustics), did not attract as much interest as other arguably more specialized areas such as non-Hermitian systems or topological edge states. In short, the localization landscape u of a Hamiltonian $\mathcal{H}$ satisfying certain properties is the solution of the linear equation

$$\mathcal{H} u = \mathcal{1},$$

where $\mathcal{1}$ is a vector with all elements equal to 1. u corresponds to the steady-state response of the medium to a uniformly-distributed source.

Left: The localization landscape of a disordered two-dimensional wave medium. Right: Five lowest energy eigenstates of the Hamiltonian, which are localized to distinct peaks of the landscape bounded by lines of small $u$ (red). For more details, read the paper!

Remarkably, it can be shown that $u$ bounds the spatial extend of the low energy eigenmodes of $\mathcal{H}$. Thus, instead of solving the eigenvalue problem and plotting its low energy modes individually to find out where they are localized, plotting $u$ alone is enough to find all the effective low-energy valleys in a medium. Moreover, as a solution to a linear system of equations, $u$ is easier to obtain than the eigenstates themselves. A more rigorous discussion (including the properties that must be satisfied by $\mathcal{H}$ can be found in the PNAS article, which is a pleasant and accessible read.

Interest is now growing in localization landscapes, thanks to recent generalizations that can bound the spatial extent of modes residing in the middle of the energy spectrum, eigenvectors of real symmetric matrices and modes of many-body interacting quantum systems. The latter remarkably shows that useful landscape functions are not limited to low-dimensional wave media, but can also bound the spreading of wavefunctions in the high-dimensional Fock space of many-body quantum systems.

A natural question that arises from these recent works is whether the localization landscape might have some potential applications in the context of quantum computing and noisy intermediate-scale quantum (NISQ) processors. For example, finding the ground state of a generic many-body Hamiltonian is a computationally challenging problem, and methods for NISQ devices such as the variational quantum eigensolver may fail to converge due to the presence of vanishing gradients or sub-optimal local minima. 

Can the localization landscape offer an easier, more hardware-efficient way to sample from low-energy solutions of a hard-to-solve Hamiltonian? Watch this space to find out!

Detecting dark solitons using persistent homology - Part 2

Previously I posted about work-in-progress on using persistent homology to analyze BEC images to identify dark solitons. We recently finished this project and the manuscript is now available on arXiv. The approach we ended up using differs from that outlined in my original post.

Our original method was based on applying persistent homology to point clouds formed by the coordinates of low intensity pixels. When applied to point clouds, persistent homology can reliably distinguish isolated minima (clusters) from the lines (loops) forming the dark solitons. However, the limitation of this method is that it requires some arbitrary choice of cutoff intensity to identify the low density pixels. Ideally when building models for machine learning we want to minimise the use of hyper-parameters that need to be optimised. For example, identifying optimal neural network hyper-parameters (number of layers, nodes, and their connectivity) is one of the most time-consuming parts of deep learning.

Therefore, in the final manuscript we instead used the "lower star image filtration" approach, which can directly compute persistent topological features of image data. This method avoids hyper-parameters and can achieve reasonable accuracy quite quickly.

On the machine learning side, we tried several different methods including support vector machines and logistic regression, which had comparable performance once we were able to identify the relevant features to use. These methods perform very well and using much less training data compared to neural networks. One important issue I overlooked in my original post was how to properly evaluate the performance of the trained model, given that the training data set had large imbalances between the three image classes. Luckily, scikit-learn includes a variety of suitable metrics for imbalanced supervised learning problems.

All in all, this project taught me a lot about persistent homology (the power of point summaries of persistence diagrams), machine learning (the power of simple models such as logistic regression if one can identify the right data features), and Jupyter notebooks (the power of being able to revise all the manuscript figures in a few seconds without having to muck around with touching them up in Inkscape).

I've now spent a year and finished two manuscripts exploring topological data analysis. My next goal is to employ these methods to solve some important and timely problem and publish the results in Physical Review Letters (yes, a generic goal that every physicist has!). The reason being that I think we need high impact publications in order to convince other physicists that these methods are really powerful and worth learning more about. I have a few promising ideas on problems to attack, so watch this space!

Creative research takes time

Nowadays funding agencies prioritise research with immediate applications. Grants for early career researchers provide funding for only a few years, requiring funded projects to have a quick and clear path to fruition. However, in basic research the eventual applications are often not those that were first envisaged.

Perhaps my favourite research project was our study of topological effects in "leaky" optical systems, published earlier this year in Nature Physics. See here for a popular summary. I am proud of this work not merely because the results were eventually published in a glossy journal, but rather because it was a pure, curiosity-driven project involving short bursts of inspiration and progress separated by many months.

This project originated from a grant application we prepared in August 2018. The host institution had expertise in photonic crystal fibers, so we were interested in whether the photonic crystal fiber platform could be used to observe interesting topological phenomena. However, photonic crystal fibers are leaky wave systems which continuously radiate energy into their environment, so it was not clear whether our experience in designing topological states for bound modes would be useful in this setting.

We started working on the idea by implementing well-known approaches for numerically computing leaky modes with the help of a tutorial article, studying some simple one-dimensional examples to gain some intuition for the platform. 

By April 2019 we had developed a tight binding-like formalism allowing us to design and study topological wave effects in leaky wave systems, but we did not know what our formalism could be useful for apart from computing the decay rates of leaky topological edge states. 

After a few conferences and many discussions with other researchers, in October 2019 we finally stumbled upon the neat idea of using radiation losses to controllably fill energy bands and measure their bulk topological invariants. 

We finished a draft manuscript by the end of November 2019 and shared with some collaborators. Their feedback was that the idea was interesting, but it wasn't clear how easy it would be to experimentally test our theory. 

We spent a few more months carrying out numerical simulations of possible designs and polishing the manuscript, finally submitting it in May 2020. Our collaborators' impression was echoed by the referees, who requested more detailed and explicit simulations of our designs in a revised manuscript, which was finally accepted in December 2020.

Take home messages:

-Find inspiration for new lines of research in old reviews and tutorials, not the latest papers published in high impact journals.

-If you get stuck don't be afraid to put the project aside for a few months and work on other directions.

-Creative research takes time - 2.5 years between the initial idea and eventual publication in this case (a theory project). Experimental projects can take even longer.