Spectral instability of random Fredholm operators

How Magical Is Magic-Angle Graphene?

Twisted bilayer graphene (TBG), which has drawn much attention in recent years, arises from van der Waals materials gathering each component together via van der Waals force. It is composed of two sheets of graphene rotated relatively to each other. Moiré potential, resulting from misorientation between layers, plays an essential role in determining the band structure of TBG, which directly relies on the twist angle. Once the twist angle approaches a certain critical value, flat bands will show up, indicating the suppression of kinetic energy, which significantly enhances the importance of Coulomb interaction between electrons. As a result, correlated states like correlated insulators emerge from TBG. Surprisingly, superconductivity in TBG is also reported in many experiments, which drags researchers into thinking about the underlying mechanism. Recently, the interest in the atomic reconstruction of TBG at small twist angles comes up and reinforces further understandings of properties of TBG. In addition, twisted multilayer graphene receives more and more attention, as they could likely outperform TBG although they are more difficult to handle experimentally. In this review, we mainly introduce theoretical and experimental progress on TBG. Besides the basic knowledge of TBG, we emphasize the essential role of atomic reconstruction in both experimental and theoretical investigations. The consideration of atomic reconstruction in small-twist situations can provide us with another aspect to have an insight into physical mechanism in TBG. In addition, we cover the recent hot topic, twisted multilayer graphene. While the bilayer situation can be relatively easy to resolve, multilayer situations can be really complicated, which could foster more unique and novel properties. Therefore, in the end of the review, we look forward to future development of twisted multilayer graphene.

Let $( {\Omega ,\mathcal{B}} )$ be a measurable space, ${\rm X},Y$ be separable Hilbert spaces. Let T be a random linear operator from $\Omega \times {\rm X}$ into Y. Let $T^\dag ( \omega )$ denote the generalized inverse of $T( \omega )$, for $\omega \in \Omega $. Questions of measurability of $T^\dag ( \omega )$ are investigated in this paper, and in particular the following results are established: (i) If T is bounded, then $T^\dag $ is a random operator. (ii) If T is a closed operator with dense domain and if $T^\dag $ is bounded, then $T^\dag $ is a random operator under some mild restriction on the domains of $T( \omega )$ and $T^ * ( \omega ),\omega \in \Omega $. The results are applied to the measurability of best approximate solutions of random linear operator equations.

Moiré is More: Access to New Properties of Two-Dimensional Layered Materials

Growing twisted bilayer graphene at small angles

RANDOM LINEAR OPERATORS ONPROBABILISTIC UNITARY SPACES TranXuan Quy, DangHung Thang,and NguyenThinh Abstract. In this paper, we are concerned with abstract random linearoperators on probabilistic unitary spaces which are a generalization ofgeneralized random linear operators on a Hilbert space defined in [25].The representation theorem for abstract random bounded linear operatorsand some results on the adjoint of abstract random linear operators aregiven. 1. IntroductionLet (Ω,F,P) be a complete probability space and X,Y be Banach spaces.A mapping f : Ω × X → Y is said to be a random operator (or a randommapping) defined on X with values in Y if for each x ∈ X, the mappingω → f(ω,x) is a Y-valued random variable. Equivalently, a random operatordefined on X with values in Y is a mapping from X into the space L Y0 (Ω)of all Y -valued random variables. A random operator f : X → L Y0 (Ω) issaid to be a random linear operator if f is linear. The interest in randomoperators has been arouse not only for its own right as a random generalizationof usual deterministic operators but also for their widespread applications inother areas. Research in theory of random operators has been carried out inmany directions including random linear operators which provide a frameworkof stochastic integral, infinite random matrix (see e.g. [1], [2], [15]-[20], [23]-[26]), random fixed points of random operators and random operator equations,(e.g [3]-[14], [18], [21], [22] and references therein). As an extension of randomlinear operators, generalized random linear operators on a separable Hilbertspace were introduced and investigated in [25].In this paper, generalized random linear operators on a separable Hilbertspace are extended to abstract random linear operators on probabilistic unitaryspaces. Section 2 presents the definitions and some properties of probabilistic

(English) The goal of this thesis is to probe the infrared optical response of twisted bilayer graphene (TBG) using Fourier transform infrared spectroscopy (FTIR). First, I used a commercial FTIR to measure the TBG in the mid-infrared range at room temperature. I improved the device fabrication technique and fabricated the TBG devices with a large area and simultaneously a low inhomogeneity. I observe that the TBG has abundant optical absorption features originating from the interband transitions that are uniquely determined by the twist angle. Then, I want to probe the interband transition of the TBG that lies in the terahertz range, which evolves the flat band of the TBG that hosts strongly correlated effects. I built a homemade FTIR that works in both the mid-infrared and terahertz range. I wired the cryostat carefully and achieved an electrical noise level approaching the Johnson noise limit. By guiding the light from the FITR into the cryostat, I successfully measured the exciton states in the Bernal bilayer graphene device over a broad spectral range, demonstrating that the system is ready for future experimental study of TBG. (Català) L'objectiu d'aquesta tesi és sondar la resposta òptica infraroja del grafè de bicapa retorçada (TBG) mitjançant l'espectroscòpia infraroja de transformada de Fourier (FTIR). Primer, vaig utilitzar un FTIR comercial per mesurar el TBG en el rang d'infraroig mitjà a temperatura ambient. Vaig millorar la tècnica de fabricació del dispositiu i vaig fabricar els dispositius TBG amb una gran àrea i alhora una baixa deshomogeneïtat. Observo que el TBG té abundants característiques d'absorció òptica originades a partir de les transicions entre bandes que es determinen únicament per l'angle de gir. Aleshores, vull investigar la transició entre bandes del TBG que es troba en el rang de terahertzs, que evoluciona la banda plana del TBG que allotja efectes fortament correlacionats. Vaig construir un FTIR casolà que funciona tant en l'infraroig mitjà com en el rang de terahertz. Vaig connectar el criostat amb cura i vaig aconseguir un nivell de soroll elèctric que s'acostava al límit de soroll de Johnson. En guiar la llum del FITR al criostat, vaig mesurar amb èxit els estats d'excitons al dispositiu de grafè de bicapa Bernal en un ampli rang espectral, demostrant que el sistema està preparat per a futurs estudis experimentals de TBG. (Español) El objetivo de esta tesis es investigar la respuesta óptica infrarroja del grafeno bicapa retorcido (TBG) mediante espectroscopia infrarroja por transformada de Fourier (FTIR). Primero, utilicé un FTIR comercial para medir el TBG en el rango del infrarrojo medio a temperatura ambiente. Mejoré la técnica de fabricación del dispositivo y fabriqué los dispositivos TBG con un área grande y, al mismo tiempo, una baja inhomogeneidad. Observé que el TBG tiene abundantes características de absorción óptica que se originan a partir de las transiciones entre bandas que están determinadas únicamente por el ángulo de torsión. Luego, quiero investigar la transición entre bandas del TBG que se encuentra en el rango de terahercios, que desarrolla la banda plana del TBG que alberga efectos fuertemente correlacionados. Construí un FTIR casero que funciona tanto en el rango del infrarrojo medio como en el de los terahercios. Conecté el criostato con cuidado y logré un nivel de ruido eléctrico que se acerca al límite de ruido de Johnson. Al guiar la luz desde el FITR hacia el criostato, medí con éxito los estados de excitón en el dispositivo de grafeno bicapa de Bernal en un amplio rango espectral, lo que demuestra que el sistema está listo para futuros estudios experimentales de TBG.

Moiré superlattices of 2D materials with a small twist angle are thought to exhibit appreciable flexoelectric effect, though unambiguous confirmation of their flexoelectricity is challenging due to artifacts associated with commonly used piezoresponse force microscopy (PFM). For example, unexpectedly small phase contrast (≈8°) between opposite flexoelectric polarizations is reported in twisted bilayer graphene (tBG), though theoretically predicted value is 180°. Here a methodology is developed to extract intrinsic moiré flexoelectricity using twisted double bilayer graphene (tDBG) as a model system, probed by lateral PFM. For small twist angle samples, it is found that a vectorial decomposition is essential to recover the small intrinsic flexoelectric response at domain walls from a large background signal. The obtained threefold symmetry of commensurate domains with significant flexoelectric response at domain walls is fully consistent with the theoretical calculations. Incommensurate domains in tDBG with relatively large twist angles can also be observed by this technique. A general strategy is provided here for unraveling intrinsic flexoelectricity in van der Waals moiré superlattices while providing insights into engineered symmetry breaking in centrosymmetric materials.

Controlling the balance between piezoelectric and flexoelectric effects is crucial for tailoring the electromechanical responses of a material. In twisted graphene, it is found that the electromechanical response near the domain walls (DWs) is dominated by either the flexoelectric effect as in twisted bilayer graphene (tBLG) or the piezoelectric effect as in twisted monolayer–bilayer graphene (tMBG). The codominance of both effects in a single system is rare. Here, utilizing lateral piezoresponse force microscopy (LPFM), we show that piezoelectric and flexoelectric effects can coexist and are equally important in twisted double bilayer graphene (tDBG), termed as the piezo-flexoelectric effect. Unlike tBLG and tMBG, distinctive two-step LPFM spatial profiles are captured across the moiré DWs of tDBG. By decomposing the LPFM signal into axisymmetric and antisymmetric components, we find that the angular dependence of both components satisfies sinusoidal relations. Quantitatively, the in-plane piezoelectric coefficient of DWs in tDBG is determined to be 0.15 pm/V by dual AC resonance tracking (DART) LPFM measurement. The conclusion is further supported by continuum mechanics simulations. Our results demonstrate that the stacking configuration serves as a powerful tuning knob for modulating the electromechanical responses of twisted van der Waals materials.

On a spectral flow formula for the homological index

We study unbounded Hermitian operators with dense domain in Hilbert space. As is known, the obstruction for a Hermitian operator to be selfadjoint or to have selfadjoint extensions is measured by a pair of deficiency indices, and associated deficiency spaces; but in practical problems, the direct computation of these indices can be difficult. Instead, in this paper we identify additional structures that throw light on the problem. We will attack the problem of computing deficiency spaces for a single Hermitian operator with dense domain in a Hilbert space which occurs in a duality relation with a second Hermitian operator, often in the same Hilbert space.

The common form for degenerate boundary conditions for the operator \(D^4\) (\(D^n\)) is found. It is shown that the matrix for coefficients of degenerate boundary conditions has a two diagonal form and the elements for one of the diagonal are units. Operator \(D^4\) whose spectrum fills the entire complex plane are studied, too. Earlier, examples of eigenvalue problems for the differential operator of even order with common boundary conditions (not containing a spectral parameter) whose spectrum fills the entire complex plane were given. However, in connection with this, another question arises whether there are other examples of such operators. In this paper we show that such examples exist. Moreover, all eigenvalue boundary problems for the operator \(D^4\) whose spectrum fills the entire complex plane are described. It is proved that the characteristic determinant is identically equal to zero if and only if the matrix of coefficients of boundary conditions has a two diagonal form. The elements of this matrix for one of the diagonal are units, and the elements of the other diagonal are 1, \(-1\) and an arbitrary constant.

The main aim of this paper is to introduce and study the subclass of not necessarily continuous, normal random operators, establishing connections with other subclasses of random operators, as well as with the existing concept of random projection operator-valued measure. Hence, after recalling some basic facts regarding random operators on a complex separable Hilbert space, theorems about transforming the class of not necessarily continuous decomposable random operators into the class of purely contractive random operators are proved. These are applied to obtain integral representations for not necessarily continuous normal or self-adjoint random operators on a Hilbert space with respect to the corresponding random projection operator-valued measures.

This article explores the generalized Cauchy–Riemann system on the entire complex plane. The coefficient for the conjugation of the desired function belongs to the Hölder space and, for |z| > 1 , equals eimϕ , where m is an integer. For m ≤ 0 , the system was shown to have no nonzero solutions that grow no faster than a polynomial. For m ≥ 0 , the complete set of regular solutions, i.e., those without singularities in the finite part of the plane, was constructed. The obtained solutions were expressed as series of Bessel functions of an imaginary argument. From the resulting set, the solutions bounded on the entire plane were distinguished, and the dimension of the real linear space of these solutions, which equals m , was determined.

Two-dimensional (2D) materials exhibit enhanced physical, chemical, electronic, and optical properties compared to their bulk counterparts, making them highly promising for next generation nanoelectronics. Among these materials, graphene has attracted significant attention due to its exceptional physical and electronic characteristics.<br/>In recent years, the focus was on moiré superlattice formed by stacking two monolayers of graphene with a relative rotation, the so-called twisted bilayer graphene (TBG). At small twist angles, interlayer interactions in TBG can cause strong correlation effects, showing correlated insulating, superconducting, orbital ferromagnetic, and Chern insulating behaviors. At larger twist angles, TBG exhibits quantum Hall states, further enriching its electronic phase diagram. <br/>Hexagonal boron nitride (h-BN) is often used in TBG devices to provide additional control over the graphene layers. Its honeycomb lattice is structurally like that of graphene. The heteroatomic composition and difference lattice constant lead to the fundamental symmetry differences, which significantly affects the electronic properties of TBG—whether the h-BN is aligned or misaligned with the graphene layers.<br/>As a summary, this thesis primarily focuses on the effect of the h-BN substrate on TBG with different twist angles. The TBGs are fabricated on h-BN as field effect transistor devices.<br/>We employed low-temperature electronic transport to investigate the symmetry-breaking effects induced by the h-BN substrate in the TBG/h-BN trilayer. By carefully tuning two twist angles in three layers, we observed topologically non-trivial flat bands in TBGs near the magic angle and symmetry-broken quantum Hall states in TBG with larger twist angles.