**Report: **

The workshop took place from Sept. 20-22, 2023 at the National Graphene Institute at the University of Manchester. A total of 116 scientists registered for the workshop in addition to the 15 invited speakers and the two workshop organizers (Prof. Vladimir Falko and Prof. Johannes Lischner). Out of the 116 registered attendees, 45 attended the workshop in person while the others attended the live broadcast of the talks which was delivered as a zoom webinar. The workshop featured a mix of invited speakers (15 in total) who delivered 30 minute presentations and contributed speakers (15 in total) who deliver 20 minute presentations. In addition, poster sessions were held during lunch breaks on Sept. 20 and Sept. 22. In addition to the support from Psi-k, the workshop received financial support from the Royce Institute and the CCP9 network.

On the first day of the workshop, the focus was on twisted bilayer graphene and the understanding of its properties near the magic angle of 1.1 degree. The first talk was delivered by Prof. Tim Kaxiras who not only explained the origin of the term “Twistronics” (demonstrating that ChatGPT is not to be trusted on this topic) and also introduced a multi-scale modelling approach that highlighted the importance of atomic relaxations in this system. Prof. Kaxiras also explained that it is more appropriate to think about a range of twist angles with strong correlations rather than a single magic angle. A similar message was put forward by the talk from Darryl Foo in the same session. The second session on the first day started with an invited talk by Prof. Sid Pamareswaran who presented new insights into the nature of the correlated insulator state that is observed in magic-angle twisted bilayer graphene. Based on Hartree-Fock calculations that included the effect of hetero-strain, he demonstrated that the most promising candidate for this state is a Kekule order which is also consistent with very recent scanning tunnelling microscopy experiments. In the last session of the first day, Prof. Mei-Yin Chou introduced a new way of understanding the origin of the magic angle in twisted bilayer graphene by analyzing the properties of AA-stacked (untwisted) bilayer graphene.

The second day of the workshop started with an invited talk by Prof. Steven Louie who presented a first-principles approach for studying excitons in twisted bilayer transition metal dichalcogenides. The calculations from his group revealed that one of the low-lying excitons in a WSe2/WS2 heterostructure has a charge transfer character and can therefore not be captured by standard continuum approach which treat the exciton as strongly bound. The second talk was given by Prof. Shi who reviewed experimental spectroscopy results on twisted bilayer and demonstrated the ability to control excitons in twisted multilayers by varying the layer number. He also presented results that at high illumination intensity a fascinating excitonic Mott insulator forms in these materials. In the afternoon session, Prof. Adina Luican-Mayer demonstrated that scanning tunnelling microscopy is a powerful technique to study twisted bilayers and she compared the behaviour of these systems for twist angles near 0 and near 60 degree. Prof. Thygesen highlighted in his talk that a very large number of interesting bilayers exist and showed how this large pool of materials can be searched to find candidate systems with attractive properties, such as excitonic insulator states.

On the final day of the workshop, Prof. Sarah Haigh presented experimental studies of twisted bilayer materials using scanning electron microscopy and Prof. Neil Drummond used the Quantum Monte Carlo technique to understand the energetics of different stackings in twisted systems. In the last session, Dr Dahlia Klein introduced a new tunnelling technique with high spatial resolution that exploits the properties of defects. In the last talk of the conference, Prof. Allan MacDonald presented a theory of fractional Chern insulators in twisted homobilayer TMDs.

In summary, the workshop succeeded in its goal to bring together theorists from different communities and experimentalists working in the field of twisted bilayer materials. Many discussions took place during coffee and lunch breaks. The key insight from the conference is that the field of twistronics has matured over the last few years, but also broadened significantly to include new materials and new theoretical and experimental techniques. In the foreseeable future, this will remain a highly dynamic field with many surprises yet to come!

**Programme:**

Wednesday, September 20

9.30: Arrival, registration, coffee

10.00: Welcome (Vladimir Falko)

10.10: __Session 1: Twisted graphene materials I (Chair: Nicholas Hine)__

10.10: **Tim Kaxiras** (invited): Twisted bilayer graphene revisited: where is the “magic”?

10.40: **Niels Walet** (contributed): Electronic structure inside the domain walls of twisted and strained graphene layers

11.00: **Alessandro Principi** (contributed): Interlayer electron-hole friction in tunable twisted bilayer graphene semimetal

11.20: **Angelika Knothe** (invited): Regular and chaotic electron dynamics in ballistic (twisted) bilayer graphene cavities

11.50: **Darryl Foo** (contributed): Extended magic phase in twisted graphene multilayers

12.10: Lunch & Poster session

13.30: __Session 2: Twisted graphene materials II (Chair: Dahlia Klein)__

13.30: **Sid Parameswaran** (invited): A Spiral Twist to the “Normal” State of Moiré Graphene

14.00: **Sankalpa Ghosh **(contributed): Moiré fractals in twisted graphene layers

14.20: **Irina Grigorieva **(invited): Magnetic-field induced phase transition in heterostructures based on unconventional superconductor PdBi_{2}

14.50: **Mohammed Al Ezzi** (contributed): Topological Flat Bands in Graphene Super-moiré Lattices

15.10: Coffee break

15.30: __Session 3: Twisted graphene materials III (Chair: Kristian Thygesen)__

15.30: **Artem Mishchenko** (invited): Moiré effects in thick graphitic films with surface layer aligned with hBN

16.00: **Mei-Yin Chou** (contributed): Origin of Magic Angles in Twisted Bilayer Graphene: The Magic Ring

16.20: **Maxim Trushin** (contributed): Electron pairing across a band intersection may create a highly conductive state

16.40: **Francesco Guinea** (invited): Superconductivity in graphene stacks.

Thursday, September 21

__9.20: Session 1: Twisted TMDs I (Chair: Neil Drummond)__

9.20: **Steven** **Louie** (invited): Excitons and photophysics of 2D van der Waals structures

9.50: **Sufei Shi** (invited): Valley-polarized Exitonic Mott Insulator in WS2/WSe2 Moiré Superlattices

10.20: Coffee break

__10.40: Session 2: Twisted TMDs II (Chair: Sid Parameswaran)__

10.40: **Brian** **Gerardot** (invited): Optically probing correlated states in mult-orbital moiré systems

11.10: **Andres Grandos del Aguila** (contributed): Ultrafast exciton fluid flow in an atomically-thin MoS2 semiconductor

11.30: **Guang-Yu Guo** (contributed): Ab initio studies of nonlinear optical responses of 2D semiconductors

11.50: **Samuel Magorrian** (contributed): One-dimensional confinement in moiré superlattices of twisted 1T’-WTe2 bilayers

12.10: **Sergey Slizovskiy** (contributed): Kagome quantum oscillations in graphene superlattices

12.30: Lunch & Poster session

13.30: __Session 3: Twisted TMDs III (Chair: Angelika Knothe)__

13.30: **Adina** **Luican-Mayer** (invited): Scanning Tunneling Microscopy of twisted 2D semiconductors

14.00: **Andor Kormanyos** (contributed): Induced spin-orbit coupling in twisted graphene-TMDC heterobilayers

14.20: **Kristian Thygesen** (invited): Emergent properties of van der Waals bilayers revealed by computational stacking

14.50: Coffee break

15.10: __Session 4: Twisted TMDs IV (Chair: Sarah Haigh)__

15.10: **Nicholas** **Hine** (invited): Combining Large Scale DFT and Machine Learned Interatomic Potentials to Simulate Twisted Bilayers, Heterostructures and Alloys of 2D Materials

15.40: **Pierre Pantaleon-Peralta** (contributed): Designing Moiré Patterns by Strain

16.00: **Aitor Garcia-Ruiz** (contributed): FE polarization in mixed-stacking graphene tetra layers

Friday, September 22

9.20: __Session 1: (Chair: Sufei Shi)__

9.20: **Sarah Haigh** (invited): Understanding Twisted 2D material heterostructures Using Scanning Transmission Electron Microscopy

9.50: **Neil Drummond** (invited): Adhesion of Graphene to Hexagonal Boron Nitride

10.20: Coffee break

10.40: __Session 2: (Chair: Brian Gerardot) __

10.40: **Dahlia Klein** (invited): Atomic SET: a new technique for high-resolution potential imaging

11.10: **Lorenzo Sponza** (contributed): Electronic structure and optical response of twisted boron nitride bilayers

11.30: **Allan MacDonald** (invited): Magic Angles and Fractional Chern Insulators in Twisted Homobilayer TMDs

12.00: Closing (Johannes Lischner)

**Abstracts of invited talks:**

**Dahlia Klein (Weizmann Institute)**

__“Atomic SET: a new technique for high-resolution potential imaging”__

Imaging the local electrostatic potential of quantum materials plays a crucial role in understanding charge order, broken symmetries, and phase transitions. Until now, the most sensitive tool for such imaging is the scanning single electron transistor (SET), which has unearthed a wealth of information in van der Waals systems. However, it is spatially limited by the lithographically defined dimensions of a quantum dot on a tip or cantilever hovering above the sample of interest, resulting in a resolution on the order of 100 nanometers. In this talk, we introduce a new experimental approach, which we call Atomic SET, to image the electrostatic potential in 2D systems. It achieves two orders of magnitude improvement in spatial resolution and operates from room temperature down to cryogenic temperatures. This scanning charge detector is built from the same platform as the quantum twisting microscope (QTM): we assemble in situ van der Waals heterostructures by bringing 2D tip and sample surfaces into contact while simultaneously scanning the sample. This geometry overcomes the limits of previous scanning SETs, enabling resolution of about 1 nanometer. Our technique promises to open up wide-ranging opportunities for direct nanoscale visualization of electronic phenomena with unprecedented spatial resolution in a number of 2D systems, including imaging topological edge states and within moiré length scales.

**Francesco Guinea (IMDEA Institute)**

__Superconductivity in graphene stacks.__

Superconductivity has been observed in a number of twisted and untwisted graphene multilayers. The dependence of the superconducting properties on the geometry of the graphene stack will be discussed. The possibility of novel phenomena due to non trivial order parameters will also be highlighted.

**Neil Drummond (University of Lancaster)**

__Adhesion of Graphene to Hexagonal Boron Nitride
__Abstract: We investigate interlayer binding and search for metastable

structures in van der Waals heterobilayers of two-dimensional

materials with near-aligned hexagonal lattice vectors but

incommensurate lattice constants, presenting numerical data for

bilayers of hexagonal boron nitride (hBN) and graphene. Diffusion

quantum Monte Carlo methods, which provide an accurate treatment of

van der Waals interactions, are used to parametrise an adhesion

potential as a function of local lattice offset between monolayers of

hBN and graphene. Using a continuum model of the elastic energy and

the adhesion potential, we find a unique solution for the displacement

field at each small misalignment angle. Our diffusion quantum Monte

Carlo interlayer binding energies provide benchmark data for the

development of dispersion-corrected exchange-correlation functionals,

and we have used them to parameterise carbon-nitrogen and carbon-boron

interatomic pair potentials for use in atomistic modelling of van der

Waals heterostructures.

**Tim Kaxiras (Harvard University)**

__Twisted bilayer graphene revisited: where is the “magic”?__

The moiré pattern observed experimentally in twisted bilayer graphene (tBLG) clearly shows the formation of different types of domains. These domains can be explained by the atomic relaxation, both in-plane and out-of-plane, using continuum elasticity theory and the Generalized Stacking Fault Energy (GSFE) concept. Moreover, the atomic relaxation significantly affects the electronic states, leading to a pair of flat bands at the charge neutrality point which are separated by band gaps from the rest. These features appear for a small range of twist angles, that we call the “magic range”, around the twist angle of 1^{o}. We discuss how all these aspects of the system are crucial for understanding the origin of correlated states and superconductivity in tBLG. We also present a minimal model that can capture these features with 2 flat and 2 auxiliary bands and explore the implications of the model for correlated electron behavior in the context of the Hubbard model.

**Angelika Knothe (University of Regensburg)**

__Regular and chaotic electron dynamics in ballistic (twisted) bilayer graphene cavities__

The dispersion of any given material is crucial for its charge carriers’ dynamics. In many (moiré) multilayers, the low-energy dispersions lack rotational symmetry. Here, we study the influence of such symmetry-breaking and deformed Fermi lines on ballistic charge carrier dynamics. For all-electronic, gate-defined cavities in Bernal and twisted bilayer graphene, we developed a trajectory-tracing algorithm aware of the material’s electronic properties and details of the confinement. We show how the anisotropic dispersion of (twisted) bilayer graphene induces chaotic and regular dynamics depending on gate voltage, twist angle, and doping, even for a fully symmetric circular cavity. Our results demonstrate the emergence of nonstandard fermion optics solely due to anisotropic material characteristics.

**Sid Parameswaran (Oxford)**

__A Spiral Twist to the “Normal” State of Moiré Graphene
__Abstract: Intense experimental efforts have uncovered a wide range of interaction-driven phenomena in magic-angle twisted bilayer graphene (TBG), including the observation of quantised topological responses, correlated insulating states, and most famously, gate-tunable superconductivity. These phenomena challenge a weak-coupling treatment yet the strong-coupling limit, despite admitting an elegant hidden symmetry structure (which I will review), is also in tension with experiments: for instance, it fails to fully explain the sequence of insulating and semimetallic states at commensurate fillings and their associated Landau fan diagrams. In this talk, I will argue that even very small strain (ubiquitous in experimental samples) drives the system to an intermediate coupling regime, where the normal-state phase diagram is dominated by a novel translation symmetry-breaking order dubbed the incommensurate Kekulé spiral (IKS). This order manifests as a Kekulé distortion on the microscopic graphene scale whose phase rotates on the moiré scale: its existence hence relies essentially on the multiscale nature of moiré materials. I will describe the properties and origin of the IKS state and discuss recent scanning tunnelling experiments in twisted bilayer (arXiv:2303.00024) and trilayer (arXiv:2304.10586) graphene that directly visualised IKS order.

This talk is based on Phys. Rev. X 11, 041063 (2021), Phys. Rev. Lett. 128,156401 (2022), arXiv:2303.13602, and ongoing work.

**Brian Gerardot (Heriot-Watt University)**

__Optically probing correlated states in mult-orbital moiré systems__

The interplay of charge, spin, lattice, and orbital degrees of freedom leads to a wide range of emergent phenomena in strongly correlated systems. In heterobilayer transition metal dichalcogenide moiré systems, recent observations of Mott insulators and generalized Wigner crystals are well described by triangular lattice single-orbital Hubbard models based on K-valley derived moiré bands. Richer phase diagrams, mapped onto multi-orbital Hubbard models, are possible with hexagonal lattices in G-valley derived moiré bands and additional layer degrees of freedom. Here I will first discuss how optically probing the behaviour of exciton-polarons can be used to distinguish the layer and valley degrees of freedom of the correlated states. Then I will present results on tunable interaction between strongly correlated hole states hosted by G- and K-derived moiré bands in a monolayer MoSe / natural WSe bilayer device.

**Nicholas Hine (Warwick University)**

__Combining Large Scale DFT and Machine Learned Interatomic Potentials to Simulate Twisted Bilayers, Heterostructures and Alloys of 2D Materials__

The electronic and vibrational properties of 2D materials whose minimal models are large, such as disordered alloys, twisted bilayers and misaligned heterostructures, are challenging for conventional approaches to Density Functional Theory (DFT) due to their unfavourable scaling with system size. This work uses two complementary approaches to large scale simulation, firstly Linear-Scaling DFT, using the ONETEP code [1], and secondly construction of Machine-Learned Interatomic Potentials. We demonstrate MLIP surrogate models for 2D systems whose accuracy closely matches that of DFT, with efficient use of training data, constructed using equivariant neural networks such as MACE [2]. These are then used to study alloy systems, and they enable us to fully characterize the vibrational properties and predicted Raman spectra of ternary and quaternary alloys of form Mo_{1-x}W_{x}S_{2-2y}Se_{2y}. MLIPs are also being applied to vibrational spectroscopy in twisted bilayer TMDs, where ab initio evaluation of the Hessian is unfeasibly expensive. Finally, for electronic properties including band structure, MLIPs can be applied to geometry pre-relaxation of twisted and heterostructured systems, enabling DFT calculations with accurate corrugation at the large system sizes required to minimize strain. Applications of spectral function unfolding of the results of LS-DFT to TMD heterostructures involving graphene and hBN will also be discussed.

[1] www.onetep.org and J.C.A. Prentice *et al*, *J. Chem. Phys*. **152**, 174111 (2020).

[2] I. Batatia *et al*., *Advances in Neural Information Processing Systems* **35**, 11423 (2022).

**Kristian Thygesen (DTU)**

__Emergent properties of van der Waals bilayers revealed by computational stacking____ __

The field of 2D materials has evolved with tremendous pace over the past decade and is currently impacting many contemporary subfields of physics including spintronics, valleytronics, unconventional superconductivity, multiferroics, and quantum light sources. Van der Waals (vdW) stacking of 2D materials offers unique opportunities for creating designer structures with novel properties absent in the constituent monolayers with the emergence of superconducting phases in twisted bilayer graphene being a particularly striking example. Unfortunately, experimental advancements beyond the proof-of-principle level are impeded by the vast size of the configuration space defined by layer combinations and stacking orders. To improve on this situation, we use an automated density functional theory (DFT) based workflow to stack all known monolayers in all possible (non-twisted) stacking configurations. We validate the approach by comparison to experimentally observed stacking orders and compute a range of electronic, magnetic, and vibrational properties for more than 2000 homobilayers. We identify bilayers that support two or more (meta)stable stackings with different magnetic or electrical properties making them candidates for the emerging field of slidetronics. If time permits, I will discuss electrical tuning of excitons in transition metal dichalcogenide homobilayers and predictions of exciton insulators and superfluidity in Janus bilayers.

References:

https://arxiv.org/abs/2304.01148

**Sarah Haigh (University of Manchester)**

__Understanding Twisted 2D material heterostructures Using Scanning Transmission Electron Microscopy__

Scanning transmission electron microscopy (STEM) is a powerful approach to investigate local atomic reconstruction for suspended samples. We have used advanced STEM methods to reveal the unusual lattice reconstruction that occurs at the interfaces in twisted transition metal dicholcogenide (TMD) bilayers for both parallel and anti-parallel stacking configurations [1]. Complementary scanning tunnelling microscopy (STM) measurements show that such reconstruction creates strong piezoelectric textures, which can be engineered by the application of applied field in the electron microscope [2]. For such investigations, scanning electron microscopy (SEM) is a more versatile approach than STM or STEM as it allows study of more realistic device architectures, with encapsulation, back gating and on conventional support. We will present recent work showing how to optimise imaging of twist domains in such devices. Furthermore. we will discuss how twisted interfaces can lead to an unusual potential landscape resulting in ultra-fast ion exchange behaviour for few layer clays and micas [3].

[1] A. Weston et al Nature Nanotechnology, 15 592–597 (2020). [2] A. Weston et al Nature Nanotechnology 17, 390–395, (2022). [3] Y. Zou et al Nature Materials, 20, 1677–1682 (2021)

**Steven Louie (UC Berkeley)**

__Excitons and photophysics of 2D van der Waals structures__

Recent experiments revealed signatures of novel exciton states and intriguing fast pump-probe optical responses in 2D van der Waals structures. The nature of many of these phenomena remains to be fully understood. Here, we present results on the photophysics of these systems based on an *ab initio* interacting *n*-particle Green’s function approach. We show that there is a rich diversity of excitons in transition metal dichalcogenide (TMD) moiré superlattices, including unforeseen novel intralayer charge-transfer moiré excitons. In pump-probe calculations, we discovered a self-driven exciton-Floquet effect in 2D materials, wherein prominent satellite bands and renormalization of the quasiparticle bands are induced by excitons, analogously to the optical Floquet effect driven by photons. We demonstrated a new many-body mechanism (direct coupling of intralayer with interlayer excitons) in the ultrafast optical response of TMD heterobilayers. Moreover, we showed strong excitonic physics in 2D materials can greatly enhances their nonlinear optical responses (e.g., shift currents and SHG). This has led to our discovery of a striking phenomenon of formation of light-induced shift current vortex crystals in TMD moiré systems – *i.e*., 2D periodic arrays of moiré-scale current vortices and associated magnetic fields with remarkable intensity under laboratory laser setup.

Our studies are made possible with the development of two new methods that allow for the *ab initio*calculations of excitonic physics and optical response of systems with thousands of atoms in the unit cell and in the time domain.

**Sufei Shi (Carnegie Mellon University)**

__Valley-polarized Exitonic Mott Insulator in WS2/WSe2 Moiré Superlattice
__The strongly enhanced electron-electron interactions in semiconducting moiré superlattices formed by transition metal dichalcogenide heterobilayers have led to a plethora of intriguing fermionic correlated states. Meanwhile, interlayer excitons in a type-II aligned heterobilayer moiré superlattice, with electrons and holes separated in different layers, inherit this enhanced interaction and suggest that tunable correlated bosonic quasiparticles with a valley degree of freedom could be realized. Here, we determine the spatial extent of interlayer excitons and the band hierarchy of correlated states that arises from the strong repulsion between interlayer excitons and correlated electrons in a WS2/WSe2 moiré superlattice. We also find evidence that an excitonic Mott insulator state emerges when one interlayer exciton occupies one moiré cell. Further, the valley polarization of the excitonic Mott insulator state is enhanced by nearly one order of magnitude. Our study demonstrates that the WS2/WSe2 moiré superlattice is a promising platform for engineering and exploring new correlated states of fermion, bosons, and a mixture of both.

**I.V. Grigorieva (University of Manchester)**

__Magnetic-field induced phase transition in heterostructures based on unconventional superconductor PdBi _{2}__

Centrosymmetric superconductor b-PdBi_{2} has attracted much attention recently due to its topologically nontrivial band structure and predicted unconventional superconductivity. However, most studies so far – typically conducted in zero magnetic field – only detect a single s-wave gap. I will review our recent studies where we used tunnelling spectroscopy on PdBi_{2}-based van der Waals heterostructures to demonstrate a magnetic-field driven transition from s-wave superconductivity in low magnetic field to unconventional pairing and a nodal gap in parallel magnetic fields above ~0.2T. Our theory shows that the transition is associated with local breaking of the inversion symmetry and the associated reconstruction of the electronic bands in magnetic field.

**Allan MacDonald (UT Austin)**

__Magic Angles and Fractional Chern Insulators in Twisted Homobilayer TMDs__

Recent experiments [1, 2] have reported the first observations of fractional Chern insulators

(FCIs), exotic states of matter that display a fractional quantum Hall effect in the absence of a magnetic field. The FCIs were discovered in the hole fluids of AA-stacked K-valley transition metal dichalcogenide (TMD) twisted homobilayers. Earlier theoretical work had hinted that FCI states might appear in systems of this type by showing that their moire minibands could carry Chern numbers [3], that the moire band width could mysteriously vanish [4] near a magic twist angle, and that the bands have almost ideal quantum geometry [5] when flat. I will explain [6] the appearance of magic angle flat bands and fractional Chern insulators in these systems by mapping their continuum model to a Landau level problem and strategies to use DFT to help optimize FCI properties.

[1] J. Cai *et al.,* Signatures of fractional quantum anomalous hall states in twisted MoTe_{2}, Nature 10.1038/s41586-023-06289-w (2023).

[2] Y. Zeng *et al.*, Integer and fractional chern insulators in twisted bilayer MoTe_{2} (2023), arXiv:2305.00973.

[3] F. Wu *et al.* Topological insulators in twisted transition metal dichalcogenide homobilayers, Phys. Rev. Lett. 122, 086402 (2019).

[4] T. Devakul *et al.*, Magic in twisted transition metal dichalcogenide bilayers, Nature

Communications 12, 6730 (2021).

[5] N. Morales-Duran *et al. *Pressure–enhanced fractional chern insulators in moire transition

metal dichalcogenides along a magic line (2023), arXiv:2304.06669.

[6] N. Morales-Duran *et al., *Magic Angles and Fractional Chern Insulators in Twisted Homobilayer TMDs, (2023) arXiv: 2308.03143.

** **

**Abstracts of contributed talks**

**Aitor Garcia-Ruiz (University of Manchester)**

__Rhombohedral graphite with a twin boundary defect: Flat bands, ferroelectricity, and spectroscopic signatures __

Materials featuring flat bands, like twisted bilayer graphene or rhombohedral graphite, often exhibit rich correlated physics, due to their high electron density at the Fermi level. In this presentation, I will analyse one such material: rhombohedral graphite with a twin boundary. In this material, two rhombohedral graphite films are stacked on top of each other with a different stacking orientation, leaving an ABA- stacked trilayer buried inside the structure, as shown in Fig.1 (a). Its band structure features two pairs of nearly flat bands localised at the surfaces and the twin boundary, highlighted in green and red colors in Fig.1 (b), respectively and I present an effective model to describe these bands [1]. I will discuss several spectroscopic techniques, such as optical absorption and ARPES, which can be used to characterise these materials, with emphasis on the smallest member of this family: ABCB tetralayer graphene [2]. Finally, I demonstrate that this class of materials features an asymmetric charge density redistribution, inherited by the lack of inversion symmetry [3], and we propose how the internal electric fields that they generate can be exploited to construct ferroelectric devices using marginally twisted graphene multilayers.

- [1] A. Garcia-Ruiz, S. Slizovskiy, and V. I. Fal’ko, Adv. Mat. Int.,
**7**, (2023), - [2] A. McEllistrim, A. Garcia-Ruiz, Z. Woodwin, V. I. Fal’ko, Phys. Rev. B,
**107**,155147 (2023). - [3] A. Garcia-Ruiz, V. V. Enaldiev, A. McEllistrim, and V. I. Fal’ko, Nano Lett.
**23**, 10, 4120 ( 2023).

**Sergey Slizovskiy (University of Manchester)**

__Kagome quantum oscillations in graphene superlattices__

Periodic systems feature the Hofstadter butterfly spectrum produced by

Brown-Zak minibands of electrons formed when the magnetic flux through

the lattice unit cell is commensurate with a flux quantum. In

experiments these Brown–Zak minibands lead to 1/B periodic oscillations

in the magnetoresistance.

Quantum oscillations, such as the Shubnikov – de Haas effect and the

Aharonov-Bohm effect, are also characteristic for electronic systems

with closed orbits in real space and reciprocal space. Here we show the

intricate relation between these two phenomena by tracing quantum

magneto-oscillations to Lifshitz transitions in graphene superlattices,

realized in three different material systems. The oscillations persist

even at relatively low fields and very much above liquid-helium

temperatures and occur close to a Lifshitz transition. The oscillations

originate from Aharonov–Bohm interference on cyclotron trajectories

that form a Kagome-shaped network characteristic for Lifshitz

transitions. In contrast to Shubnikov – de Haas oscillations, the Kagome

oscillations are robust against thermal smearing. At high magnetic

fields, our description is a reinterpretation of Brown-Zak oscillations

which were attributed to the ballistic propagation of Brown-Zak fermions

in zero effective magnetic field [Kumar et al, Science 357, 181 (2017)

and Proc.Natl.Acad.Sci.115, 5135 (2018)]. At low magnetic fields the

oscillations can be detected even when the Hofstadter butterfly spectrum

is smeared out by electron scattering and interband matrix elements of

the velocity operator are dominating the conductivity [J. Vucicevic and

R. šZitko, PRL 12, 196601 (2021) and PRB 104, 205101 (2021)]. We argue

that Kagome quantum oscillations are generic for two-dimensional

crystals with discrete rotational symmetries close to Lifshitz transitions.

Reference: https://arxiv.org/abs/2303.06403

**Alessandro Principi (University of Manchester)**

__Interlayer electron-hole friction in tunable twisted bilayer graphene semimetal__

Charge-neutral conducting systems represent a class of materials with unusual properties governed by electron-hole (e-h) interactions. Depending on the quasiparticles’ statistics, band structure, and device geometry these semimetallic phases of matter can feature unconventional responses to external fields that often defy simple interpretations in terms of single-particle physics. Here we show that small-angle twisted bilayer graphene (SA-TBG) offers a highly-tunable system in which to explore interactions-limited electron conduction. In particular, the transition from a non-degenerate charge-neutral Dirac fluid to a compensated two-component e-h Fermi liquid where spatially separated electrons and holes experience strong mutual friction. This friction results in a resistivity that scales as the temperature square. By employing a dual-gated device architecture, a resistivity that accurately follows the e-h drag prediction is found. Our results provide a textbook illustration of a smooth transition between different interaction-limited transport regimes and clarify the conduction mechanisms in charge-neutral SA-TBG.

**Mohammed Al Ezzi (National University of Singapore)**

__Topological Flat Bands in Graphene Super-moiré Lattices
__Moiré-pattern based potentials have emerged as a driving force behind the exploration of exotic physics in various condensed matter systems. While these potentials have induced correlated phenomena in almost all commonly studied 2D materials, monolayer graphene has remained an exception. In this letter, we present a theoretical study demonstrating that a single layer of graphene, when placed between two bulk boron nitride crystal substrates with the right twist angles that create a commensurate super-moiré potential, can support a topological ultra-flat band. This configuration, which can be easily fabricated using existing experimental capabilities [1], offers one of the simplest platforms for the realization of topological flat bands. Notably, the flat band originates from the second hole band and exhibits robustness against Hartree-Fock corrections. As a result, we anticipate that this system will provide a fertile ground for the investigation of strongly correlated physics. Furthermore, our continuum model demonstrates that similar topological flat bands also arise in graphene bilayers and trilayers when subjected to similar sandwiching by boron nitride substrates. These findings unveil new possibilities for the engineering of moiré-induced topological flat bands in graphene-based systems, opening up avenues for further exploration and potential applications in condensed matter physics.

[1] Junxiong Hu*, Junyou Tan*, Mohammed M Al Ezzi*, Udvas Chattopadhyay, Jian Gou, Yuntian Zheng, Zihao Wang, Jiayu Chen, Reshmi Thottathil, Jiangbo Luo, et al. “Controlled alignment of supermoiré lattice in double-aligned graphene heterostructures”. Nature Communications 14, 4142 (2023).

**Andres Granados del Aguila (National University of Singapore)**

__Ultrafast exciton fluid flow in an atomically-thin MoS2 semiconductor __

Excitons (coupled electron-hole pairs) in semiconductors may form collective states with spectacular non-linear properties. We have experimentally realized a collective state of short-lived excitons in a direct-bandgap, atomically-thin MoS2 semiconductor whose propagation resembles that of a classical liquid. The signatures are the anomalous transport of the exciton density that propagates uniformly through the MoS2 monolayer regardless of crystallographic defects and geometrical constraints. The exciton fluid flows over ultra-long distances (at least 60 micrometers) with a speed of ∼1.8×10^7 m/s (∼6% the speed of light). The collective phase emerges above a critical laser power, in the absence of free charges and below a critical temperature (usually Tc ~150 K) approaching room-temperature in hBN-encapsulated devices. Our theoretical simulations suggests

that momentum is conserved and local-equilibrium is achieved among excitons, which are

compatible features with a fluid dynamics description of the exciton transport. Our findings have implications for ultrafast exciton-mediated optical switches, exciton-valley Hall

devices and on-chip exciton circuitry.

**Darryl Foo (National University of Singapore)**

__Extended magic phase in twisted graphene multilayers__

Theoretical and experimental studies have verified the existence of “magic angles” in twisted bilayer graphene, where the twist between layers gives rise to flat bands and consequently highly correlated phases. Narrow bands can also exist in multilayers with alternating twist angles, and recent theoretical work suggests that they can also be found in trilayers with twist angles between neighboring layers in the same direction.

We show here that flat bands exist in a variety of multilayers where the ratio between twist angles is close to coprime integers. We generalize previous analyses, and, using the chiral limit for interlayer coupling, give examples of many combinations of twist angles in stacks made up of three and four layers which lead to flat bands. The technique we use can be extended to systems with many layers. Our results suggest that flat bands can exist in graphene multilayers with angle disorder, that is, narrow samples of turbostatic graphite.

**Andor Kormanys (****Eötvös Loránd University, Budapest****)**

__Induced spin-orbit coupling in twisted graphene-TMDC heterobilayers__

It has been demonstrated using several different experimental techniques that a strong spin-orbit coupling (SOC) can be induced in graphene when it is placed on top of semiconducting transition metal dichalcogenides (TMDCs). We have developed a theoretical approach to predict the dependence of the induced SOC on the twist angle between the graphene and TMDC layer. We find that the strength of the induced SOC can be significantly tuned by the twist angle[1].

We have also shown [2] that the induced Rashba type SOC depends on a quantum phase which had not received much attention previously. We discuss how this quantum phase affects the spin-polarization of the graphene bands and its potential effect on spin-to-charge conversion measurements. Extending our model to the case of graphene encapsulated by two TMDC layers, we show that in twisted trilayers this quantum phase can lead to interference effects in the induced Rashba spin-orbit coupling.

[1] A. David, P. Rakyta, A. Kormányos, and G. Burkard, Phys. Rev B 100, 085412 (2019).

[2] Cs. G Péterfalvi, A. David, P. Rakyta, G. Burkard, and A. Kormányos, Physical Review Research 4, L022049 (2022)

**Maxim Trushin (National University of Singapore)**

__ELECTRON PAIRING ACROSS A BAND INTERSECTION MAY CREATE A HIGHLY CONDUCTIVE STATE__

Twisted stacks of 2D materials provide an exciting opportunity to explore interacting electrons in strongly correlated regimes not accessible so far. A few different correlated electronic phase states can emerge in the same sample depending on temperature, electron density, and other parameters tunable externally [1]. Here, we report on a new electron correlated state that may emerge due to the cross-band superconducting pairing in a 2D Dirac semimetal as soon as the Dirac velocity falls down below a certain threshold [2]. The state can be seen as a conventional Fermi sea, where the empty and filled states are separated by a layer of paired electrons, playing a role of the Fermi level. The thickness of the correlated electron layer is given by a self-consistent order parameter deduced from a mean-field equation. Even though no superconducting gap opens, the electrons turn out to be immune to elastic scattering by, e.g., intrinsic disorder. Hence, the electrical resistivity is expected to demonstrate a sudden drop once the electrons transition to the state predicted. This is what has been observed recently in twisted double-bilayer graphene at some specific filling factors corresponding to high density of states at the Fermi level [3].

[1] Shen, C., Chu, Y., Wu, Q. et al. Correlated states in twisted double bilayer graphene. Nat. Phys. 16, 520-525 (2020).

[2] M. Trushin, L. Peng, G. Sharma, G. Vignale, S. Adam, arXiv:2207.05974.

[3] He, M., Li, Y., Cai, J. et al. Symmetry breaking in twisted double bilayer graphene. Nat. Phys. 17, 26-30 (2021).

**Mei-Yin Chou (Academia Sinica, Taiwan)**

__Origin of Magic Angles in Twisted Bilayer Graphene: The Magic Ring__

The unexpected discovery of superconductivity and strong electron correlation in twisted bilayer graphene (TBG), a system containing only sp electrons, is considered as one of the most intriguing developments in two-dimensional materials in recent years. The key feature is the emergent flat energy bands near the Fermi level, a favorable condition for novel many-body phases, at the so-called “magic angles”. The physical origin of these interesting flat bands has been elusive to date, hindering the construction of an effective theory for the unconventional electron correlation. In this work, we have identified the importance of charge accumulation in the AA region of the moiré supercell and the most critical role of the Fermi ring in AA-stacked bilayer graphene. We show that the magic angles can be predicted by the moiré periodicity determined by the size of this Fermi ring. The resonant criterion in momentum space makes it possible to coherently combine states on the Fermi ring through scattering by the moiré potential, leading to flat bands near the Fermi level. We thus establish the physical origin of the magic angles in TBG and identify the characteristics of one-particle states associated with the flat bands for further many-body investigations.

**Sankalpa Ghosh (****INDIAN INSTITUTE OF TECHNOLOGY, DELHI****)**

__Moire fractals in twisted graphene layers__

Twisted bilayer graphene (TBLG) subject to a sequence of commensurate external periodic potentials reveals the formation of moir\’e fractals that share striking similarities with the central place theory (CPT) of economic geography, thus uncovering a remarkable connection between twistronics and the geometry of economic zones. The moir\'{e} fractals arise from the self-similarity of the hierarchy of Brillouin zones (BZ) so formed, forming a nested subband structure within the bandwidth of the original moir\'{e} bands. The fractal generators for TBLG under these external potentials are derived and we explore their impact on the hierarchy of the BZ edges. Furthermore, we uncover parallels between the modification of the BZ hierarchy and magnetic BZ formation in the Hofstadter butterfly, allowing us to construct an incommensurability measure for moir\'{e} fractals as a function of the twist angle. The resulting band structure hierarchy bolsters correlation effects, pushing more bands within the same energy window for both commensurate and incommensurate TBLG.

Ref:1. Moiré fractals in twisted graphene layers: https://arxiv.org/abs/2306.04580

( D. Aggarwal, R. Narula and SG)

2. A primer on twistronics: a massless Dirac fermion’s journey to moiré patterns and flat bands in twisted bilayer graphene: https://iopscience.iop.org/article/10.1088/1361-648X/acb984

( D. Aggarwal, R. Narula and SG)

**Niels Walet (University of Manchester)**

__Electronic srructure inside the domain walls of twisted and strained graphene layers__

In this talk I will describe recent work on the electrionic structure inside the domain walls of twisted and strained graphene, concentrating on the effect of relaxation on the local density of states. This is important for understanding the nature of these dominant features at low energy in large domains.

**Lorenzo Sponza (CNRS, ONERA)**

__Electronic structure and optical response of twisted boron nitride bilayers__

Bilayers of 2D materials have recently emerged as minimal systems where to tune the exceptional properties of their constituent layers. By changing the alignment of one layer with respect to the other (twist angle), it is possible to modify the interactions between the two sheets and the local chemical environment.

In this context, boron nitride (BN) twisted bilayers are studied because the twist angle determines the formation and the flatness of several groups of degenerate bands [1, 2, 3]. This leads to a modulation of its optical properties [2, 4] and controls the emergence of correlated phases [1]. Moreover, the presence of two atomic species gives access to different stackings, hence providing an additional degree of freedom with respect to graphene bilayers.

In our work [5] that in boron nitride twisted bilayers, for a given moiré periodicity, there are five different stackings which preserve the monolayer hexagonal symmetry (i.e. the invariance upon rotations of 120°) and not only two as always discussed in literature. We introduce some definitions and a nomenclature that identify unambiguously the twist angle and the stacking sequence of any hexagonal bilayer with order-3 rotation symmetry. Moreover, we employ density functional theories and tight-binding models to study the evolution of the band structure and optical response as a function of the twist angle for the five stacking sequences.

Support by the EU Horizon 2020 research and innovation program is gratefully acknowledged.

[1] L. Xian, Dante E. Kennes et al., Nano Letters 19, 4934-4940 (2019)

[2] X. Zhao, Y. Yang, D.-B. Zhang, and S.-H. Wei, Phys. Rev. Lett. 124, 086401 (2020)

[3] H Ochoa and A. Asenjo-Garcia, Phys. Rev. Lett. 125, 037402 (2020)

[4] H. Y. Lee, M. M. Al Ezzi, N. Raghuvanshi et al. Nano Letters 21, 2832-2839 (2021)

[5] S. Latil, H. Amara, L. Sponza, SciPost Phys. 14, 053 (2023)

**Pierre Anthony Pantaleon-Peralta (****Instituto Madrileño de Estudios Avanzados en Nanociencia, IMDEA****)**

__Designing Moire Patterns by Strain__

Experiments conducted on twisted two-dimensional materials have shown the presence of a

plethora of moiré patterns with different forms and shapes. We find that the lattice structure

of these systems is strongly dependent on the strain magnitude and direction, resulting in the different observed moiré patterns. Specifically, for moiré systems composed of honeycomb lattices, we found that the shape of the primitive unit cell depends on both strain and twist. By modifying

the strain direction, we can describe a family of primitive unit cells with different geometries, providing the opportunity to design moiré systems with varied configurations.

**Guang-Yu Guo (Department of Physics, National Taiwan University, Taipei 10617, Taiwan)**

__Ab initio studies of nonlinear optical responses of 2D semiconductors__

Nonlinear optical (NLO) properties, such as bulk photovoltaic effect (BPVE), second harmonic generation (SHG) and third-order nonlinear photovoltaic Hall effect (THE), of 2D semiconductors have received great attention in recent years because of their promising applications in, e.g., optoelectronics, high efficient solar energy harvesting and high sensitive THz radiation detection. Interestingly, NLO responses of solids have also been shown recently to be a powerful probe of geometric structure of quantum states in the materials [1-2]. In this talk, I will report the main results and interesting findings of our recent ab initio investigations on the SHG and BPVE in PT-symmetric antiferromagnetic CrI3 bilayer [3], few-layer helical chainlike selenium and tellurium [4], few-layer pentagonal transition metal dichalcogenide semiconductors PdS2 and PdSe2 [5] as well as THE in quantum spin Hall insulator gemanene and Chern insulator (LaOsO3)2 bilayer [2].

The speaker wants to thank many collaborators especially V. K. Gudelli, M. Cheng, Z.-Z. Zhu, J. Ahn, N. Nagaosa and A. Vishwanath. He also acknowledges the support from the National Science and Technology Council as well as NCTS, Taiwan.

[1] J. Ahn, G.-Y. Guo and N. Nagaosa, Phys. Rev. X 10, 041041 (2020).

[2] J. Ahn, G.-Y. Guo, N. Nagaosa and A. Vishwanath, Nature Phys. 18, 290 (2022).

[3] V. K. Gudelli and G.-Y. Guo, Chin. J. Phys. 68, 896 (2020).

[4] M. Cheng, Z.-Z. Zhu and G.-Y. Guo, Phys. Rev. B 103, 245415 (2021)

[5] V. K. Gudelli and G.-Y. Guo, New J. Phys. 23, 093028 (2021)

List of registered attendees

Md. Sakib | Hasan khan | Khulna University of engineering & technology |

Ankita | Jaiswal | IIT(ISM) Dhanbad India |

Priyanka | Kumari | Indian Institute of Technology, Jodhpur |

Jean Baptiste | Fankam Fankam | University of the Witwatersrand, Johannesburg, South Africa |

Bijoy | Nharangatt | S N Bose National center for Basic Sciences , India |

Mwanaidi | Namisi | University of Nairobi |

Ridha | Eddhib | University of west Bohemia |

Gautam | Gurung | Trinity College, University of Oxford |

Khalil Ur | Rahman | Riphah International University Islamabad |

Nilanthy | Balakrishnan | Keele University |

Arti | Kashyap | IIT Mandi, HP, India |

Bijal | Mehta | Sardar Vallabhbhai National Institute of Technology, Surat |

Nishan | Nepal | Tribhuvan University |

Yonas | Tirffe | Ethiopian Defense University |

Sushant Kumar | Behera | Indian Institute of Science, Bangalore-560012, India |

Zeinab | Heidari Pebdani | Helmholtz zentrum Hereon |

Athira | S J | Shiv Nadar University, Delhi NCR, India |

Celal | Yelgel | RTE University |

Subrata | Rakshit | Faculty of physics, University of Warsaw |

Rafael | Gonzalez | Universidad del Norte, Barranquilla, Colombia |

Snehal | Paladiya | Gujarat Technological University |

Shivam Jani | Shivam | S.N. Bose National Centre for Basic Sciences, Kolkata |

Guang-Yu | Guo | (1) Department of Physics, National Taiwan University, Taipei 10617, Taiwan (2) Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan |

Selin | Kilic | Queen Mary University of London |

Pierre Anthony | Pantaleon-Peralta | Instituto Madrileño de Estudios Avanzados en Nanociencia, IMDEA |

Szymon | Bartus | University College London (UCL) |

Hua | Men | Brunel University London |

Konstantinos | TERMENTZIDIS | CNRS |

Lorenzo | Sponza | CNRS, ONERA |

Jean Baptiste | Fankam Fankam | University of the Witwatersrand |

Christopher Tat Shun | Cheung | Imperial College London |

Niels | Walet | University of Manchester |

Nilanthy | Balakrishnan | Keele University |

Petros Panagis | Filippatos | University of Nottingham |

SANKALPA | GHOSH | INDIAN INSTITUTE OF TECHNOLOGY, DELHI |

Adair | Nicolson | UCL |

Mei-Yin | Chou | Academia Sinica, Taiwan |

Steffi | Woo | Oak Ridge National Laboratory |

Khalil Ur | Rahman | Govt Degree college shewa swabi |

Mahmut | Okyay | UCR |

NARENDER | KUMAR | |

Sundheep | Radhakrishnan | |

Carlos | Garcia Cervera | University of California, Santa Barbara |

Anass | Sibari | CIC energiGUNE |

Saul | Herrera | IMDEA Nanoscience and Universidad Nacional Autónoma de México |

Shivam | Jani | S.N.Bose National Centre for Basic Sciences, Kolkata |

Subhasis | Roy | University of Calcutta |

Subrata | Rakshit | Faculty of physics, University of Warsaw |

Katherine | Inzani | University of Nottingham |

Diyan Unmu | Dzujah | TU Dresden |

Ruiqi | Zhang | Binghamton university |

Bijoy | Nharangatt | S N Bose National Center for Basic Sciences |

Maxim | Trushin | National University of Singapore |

Andor | Kormányos | Eötvös Loránd University, Budapest |

Khaled | DINE | Faculty of Technology, University of Dr Moulay Tahar – Saida |

Mukesh | Singh | Indian Institute of Technology Bombay |

Stefano | Dal Forno | Nanyang Technological University |

Khushboo | Dange | Indian Institute of Technology Bombay, India |

Darryl | Foo | Centre for Advanced 2D Materials |

Zhen | Zhan | IMDEA Nanoscience |

Andres | Granados del Aguila | Institute for Functional Intelligent Materials, National University of Singapore. |

Mohammed | Al Ezzi | National University of Singapore |

Tim | Verhagen | FZU – Institute of Physics of the Czech Academy of Sciences |

Gautam | Rai | Universität Hamburg, Germany |

Samuel | Magorrian | University of Warwick |

Azar | Ostovan | University of California, Santa Barbara |

Azar | Ostovan | University of California, Santa Barbara |

Milad | Asgarpour Khansary | University of British Columbia |

Shambhu | Bhandari Sharma | University College London |

Ziwei | Wang | University of Oxford |

Priya | Mahadevan | S.N.Bose National Centre for Basic Sciences |

Sunit | Das | Indian Institute of Technology Kanpur |

Madhurita | Das | S.N.Bose National Centre for Basic Sciences |

Yedija | Teweng | Kanazawa university |

Sumanti | Patra | University of Hamburg |

Zachary | Goodwin | Harvard University |

Raj Kumar | Paudel | National Central University |

Medha | Dandu | Lawrence Berkeley National Laboratory |

Darius-Alexandru | Deaconu | The University of Manchester |

Alexandra-Daria | Dumitriu-Iovanescu | The University of Manchester |

Yuchi | He | University of Oxford |

Yuchi | He | University of Oxford |

Arkajyoti | Maity | Max Planck Institute for the Physics of Complex Systems |

Na | Xin | The University of Manchester |

Emad | Badradeen | University of Arkansas at Pine Bluff |

Sarah | Haigh | University of Manchester |

ABSIKE | Hanan | IAp/ Um6P |

absike | hanan | IAP/um6p |

Sushant Kumar | Behera | Indian Institute of Science |

Sergey | Slizovskiy | UoM |

Aitor | Garcia-Ruiz Fuentes | National Graphene Institute |

Mikhail | Kaliteevski | National Graphene Institute |

Adrian | Ceferino | IMDEA Nanociencia, Madrid |

Amit | Singh | University of Manchester |

Igor | Rozhansky | University of Manchester |

Patrick | Sarsfield | University of Manchester, National Graphene Institute |

Alessandro | Principi | University of Manchester |

Na | Xin | University of Manchester |

Arash | Mostofi | Imperial College London |

Guillermo | Parra | IMDEA Nanociencia |

Amy | Carl | University of Manchester |

James | McHugh | National Graphene Institute/University of Manchester |

Abhishek | Khedkar | KCL |

Kanta | Ogawa | Imperial College London |

Wojciech | Jankowski | University of Cambridge |

Ivan | Vera Marun | University of Manchester |

Isaac | Soltero Ochoa | National Graphene Institute – University of Manchester |

Alejandro | Jimeno-Pozo | IMDEA Nanoscience |

Sofia | Konyzheva | The University of Manchester |

Pascal | Pochet | Univ. Grenoble-Alpes & CEA |

Isaac | Soltero Ochoa | National Graphene Institute – University of Manchester |

Chi-Ruei | Pan | Academia Sinica |

Wei-En | Tseng | Institute of Atomic and Molecular Sciences,Academia Sinica, Taipei, Taiwan |

Tai ting | Lee | NTU Physics |

Wei-Chen | Wang | School of Physics, Georgia Institute of Technology |

Rutvij | Gholap | University of Manchester |