Flat bands and high-order Van Hove singularities

We show that an interacting electronic system with a single, ordinary or extended Van Hove point that crosses the Fermi energy is unstable against triplet superconductivity. For extended Van Hove points we find that the transition temperature is uniquely determined by the exponent that governs the divergence of the electronic density of states at the Fermi level. Enhancing this exponent should drastically increase Tc. The Cooper pair wave function of the triplet state is found to be highly non-local, with a non-monotonic momentum dependence and a steep slope near the nodes of the gap. These findings go beyond of what a logarithmic renormalization group treatment would yield. They demonstrate that the supermetal state, discussed in the context of extended Van Hove points, is ultimately unstable against superconductivity.

Using scanning-tunneling-microscopy and spectroscopy on heterostructures of twisted bilayer graphene on hexagonal Boron-Nitride, we show that the ensuing super-moiré structures display a rich landscape of moiré-crystals and quasicrystals. We reveal a phase-diagram comprised of commensurate moiré-crystals embedded in swaths of moiré quasicrystals. The 1:1 commensurate moiré crystal, which is expected to be a Chern insulator exhibiting orbital magnetism, should only exist at one point on this phase-diagram, implying that it ought to be practically undetectable. Surprisingly we find that commensurate moiré crystals exist over a wide range of parameters, providing evidence of an unexpected relaxation mechanism favoring self-alignment. The remainder of the phase-diagram is comprised of moiré quasicrystals with emergent rotational symmetries of the electronic wave function. These moiré quasicrystals provide a synthetic platform for creating and exploring such rarely found in nature structures.

I will discuss the analysis of Fermi surface instabilities close to Van Hove singularities (high order and otherwise) using parquet RG. Particular emphasis will be given to hexagonal lattice systems. I will also discuss possible applications to Kagome metals.

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.

Resonant magnetoresistance measurements on graphene FETs under “ohmic” low current conditions have been used to study a wealth of novel physics. At high currents (up to 100 A$m^{-1}$), electrons are driven far from equilibrium with the atomic lattice vibrations so that their kinetic energy can exceed the thermal energy of the phonons. Here, we report three non-equilibrium phenomena in monolayer graphene at high currents [1]: (i) a “Doppler-like” shift and splitting of the frequencies of the transverse acoustic (TA) phonons, emitted resonantly when the electrons undergo inter-Landau level (LL) transitions [2] and revealed by shifts of peaks in the differential resistance, $r_y$; (ii) an intra-LL Mach effect with the emission of TA phonons when the electrons approach supersonic speed (∼1.4 x $10^4$ ms$^{-1}$), revealed by a strong and broad magnetic field-independent peak in $r_y$ at $I∼1$ mA and (iii) the onset of elastic inter-LL transitions and an associated resonance maximum in $r_y$ at a critical carrier drift velocity (see Fig 1b). The third phenomenon is analogous to a “superfluid” Landau velocity and the formation of magneto-excitons involving a quantum jump, $h/m$, in the electron LL circulation. All three quantum phenomena can be unified in a single resonance equation. They offer avenues for research on out-of-equilibrium phenomena in other two-dimensional systems. [1] M.T. Greenaway et al., Nature Communications 12, 6392 (2021) [2] P. Kumaravadivel et al., Nature Communications 10, 3334 (2019)

We develop the transfer-learning accelerated large-scale simulation, and investigate the lattice relaxation and moiré flat bands in twisted MoTe2 for a wide range of angles. Built on the realistic continuum models with multiple Chern bands, our multi-band exact diagonalization analysis reveals significant band-mixing effects and the strong competition between charge density wave orders and integer (fractional) quantum anomalous Hall states.

In the realm of condensed matter physics, the fractional quantum Hall effect stands as a singular experimental manifestation of topological order, characterized by the presence of anyons—quasiparticles that bear fractional charge and exhibit exchange statistics diverging from conventional fermions and bosons. This phenomenon, observed over four decades ago, was still missing the direct observation of similar topological orders arising purely from band structure—without the application of strong magnetic fields. In 2023 within the span of a few months, several pioneering experiments have illuminated this once theoretical domain. Studies on twisted homobilayer MoTe2 and pentalayer rhombohedral graphene placed on hBN have finally unveiled the existence of fractional Chern insulators (FCIs), the zero-magnetic field analog of fractional quantum Hall states. The journey to this point, preceded by over a decade of theoretical frameworks and predictions surrounding FCIs, yet the experimental revelations have proved to be richer and more surprising than expected. In this talk, we will present how the combination of ab-initio and quantum many-body calculations can help us capture the different features observed in experiments. We will discuss the potential future for this exciting booming field, including the possible observation of fractional topological insulators, a yet-never observed topological ordered phase preserving the time reversal symmetry.

It is an old question whether an original, and intuitively compelling, picture of the current flowing in a narrow channel along the sample edge is the physically correct one. Motivated by recent experiments {\it locally} imaging the quantized current flow in a Chern insulating (Bi, Sb)$_2$Te$_3$ heterostructure, we theoretically demonstrate the possibility of a broad `edge state' meandering away from the sample boundary deep into the sample bulk. Further, we show that varying experimental parameters permits continuously tuning between narrow edge states and meandering channels all the way to incompressible bulk transport. This accounts for various features observed in, and differing between, experiments. Overall, this underscores the robustness of topological condensed matter physics, but it also unveils a phenomenological richness hidden by topological censorship.

We present the time-dependent Quantum Geometric Tensor (tQGT) as a comprehensive tool for capturing the geometric character of insulators observable within linear response. We show that tQGT describes the zero-point motion of bound electrons and acts as a generating function for generalized sum rules of electronic conductivity. Therefore, tQGT enables a systematic and basis-independent framework to compute the instantaneous response of insulators, including optical mass, orbital angular momentum, and the dielectric constant in low-energy effective theories. It allows for a consistent approximation across these quantities upon restricting the number of occupied and unoccupied states in an effective low-energy description of an infinite quantum system. We outline how lattice interference can generate quantum geometry in periodic systems. We examine spectral weight transfer from small frequencies to high frequencies by creating geometrically frustrated flat bands.

The interplay between spontaneous symmetry breaking and topology can result in exotic quantum states of matter. A celebrated example is the quantum anomalous Hall (QAH) effect, which exhibits an integer quantum Hall effect at zero magnetic field due to topologically nontrivial bands and intrinsic magnetism. In the presence of strong electron-electron interactions, fractional-QAH (FQAH) effect at zero magnetic field can emerge, which is a lattice analog of fractional quantum Hall effect without Landau level formation. In this talk, I will first review our experimental observation of FQAH and associated effects in twisted MoTe2 bilayer, using combined magneto-optical and -transport measurements. I will then present an optical probe of putative zero-field composite Fermi liquid state. Lastly, I will discuss the latest progress in the zero-field Fractional Chern insulator.

I will discuss how the topological structure of the flat bands in moiré graphene systems can stabilize a host of complex charge-ordered states with intervalley coherence that can be frutifully viewed as "Berry-frustrated" exciton condensates. Apart from identifying a host of potential new correlated insulating states in a range of moiré graphene multilayers, this theory also sheds new light on the emergence of the "Kekulé spiral" charge orders that theoretically proposed [1-3] in twisted bilayer and symmetric trilayer graphene and recently observed via scanning tunneling microscopy experiments [4,5]. [1] Y.H. Kwan, G. Wagner, T. Soejima, M.P. Zaletel, S.H. Simon, S.A. Parameswaran, and N. Bultinck, Phys. Rev. X 11, 041063 (2021). [2] G. Wagner, Y.H. Kwan, S.H. Simon, N. Bultinck, and S.A. Parameswaran, Phys. Rev. Lett. 128, 156401 (2022). [3] Z. Wang, Y.H. Kwan, G. Wagner, N. Bultinck, S.H. Simon, and S.A. Parameswaran, arXiv:2310.16094 (2023). [4] K.P. Nuckolls et al, Nature 620, 525(2023). [5] H. Kim et al, Nature 623, 942 (2023).

Both Landau levels in a magnetic field and flat bands in moire materials give rise to strong correlation among electrons. These interaction can stabilize a wide range of electronic states, mostly studies using transport measurements. Our work focuses on high-resolution scanning tunneling microscopy measurements to directly visualizing electronic states of the correlated states. I will describe a wide range of experiments probing various broken symmetry states and their melting into exotic superconductors and other correlated electronic states we have done in the past few years.

The exchange interaction can lead to correlated states in low dimensional systems such as the graphene family. Regions of large density of states are especially prone to correltaion effects, an example that will be discussed is the recently identified exchange driven quantum anomalous Hall (QAH) nu=2 state that exhibits quantized charge Hall conductance close to zero magnetic field as well as spin, valley and spin-valley anomalous quantum Hall effects and out-of-plane ferroelectricity in suspended bilayer graphene [1]. In the case that bilayers are encapsulated in h-BN, a large displacement field can be applied allowing the opening of a gap in the density of states with a concomitant van-Hove-singularity close to the band edges. We will discuss our recent measurements [2] in such device structures that indicate that close to the band edges novel states appear that are distinct from Stoner [3,4] and other single particle physics. For example, one identified state is consistent with a Chern insulating state at finite density in the valence band. [1] F. R. Geisenhof, F. Winterer, A. M. Seiler, J. Lenz, T. Xu, F. Zhang and R. T. Weitz, "Quantum anomalous Hall octet driven by orbital magnetism in bilayer graphene", Nature 598, 53 (2021) [2] A. M. Seiler, F. R. Geisenhof, F. Winterer, K. Watanabe, T. Taniguchi, T. Xu, F. Zhang and R. T. Weitz, "Quantum cascade of new correlated phases in trigonally warped bilayer graphene", Nature 608, (2022) 298 [3] H. Zhou, Y. Saito, L. Cohen, W. Huynh, C. L. Patterson, F. Yang, T. Taniguchi, K. Watanabe, A. F. Young, “Isospin magnetism and spin-triplet superconductivity in Bernal bilayer graphene”, Science 375 (2022), 774 [4] S. C. de la Barrera, S. Aronson, Z. Zheng, K. Watanabe, T. Taniguchi, Q. Ma, P. Jarillo-Herrero, R. Ashoori, “Cascade of isospin phase transitions in Bernal bilayer graphene at zero magnetic field” Nature Physics 18, (2022) 771

Among the variety of correlated states found in twisted bilayer graphene (TBG), a strong reorganization of the density of states, up to tenths of meV, including oscillations of the remote bands energies in Scanning Tunneling Microscope experiments and sawtooth peaks in the inverse compressibility were detected and named cascades. These cascades happen in a much larger energy, twist angle and temperature range than other correlated phenomena in TBG, pointing to a hierarchy of phenomena [1]. Many proposals to explain the cascades involved symmetry breaking. Using Dynamical Mean Field Theory (DMFT) calculations we showed [2] that the cascades emerge already in the normal state, in which no symmetry breaking order is present, due to the formation of local moments and heavy quasiparticles. Besides explaining the signatures in STM and compressibility measurements, we predicted a strong momentum differentiation in the incoherent spectral weight, associated to the fragile topology of TBG. After reviewing the phenomenology of the cascades and our first results, in the talk I will discuss more recent calculations in which we both address another set of experimental results and make new predictions for future measurements. [1] Wong et al, Nature 582, 198 (2020), Zondiner et al, Nature 582, 203 (2020). Polski et al, arXiv:2205.05225 [2] A. Datta, M.J. Calderón, A. Camjayi, and E. Bascones, Nature Communications 14, 5036 (2023)

At high-order Van Hove points the dispersion is exceptionally flat and the density of states has a power-law divergence. In my talk, I will discuss our analysis of the competition between different ordering tendencies using unbiased renormalization group approaches, both a parquet RG as well as a functional RG with high momentum resolution. In the simple parquet RG, we find that for purely repulsive interactions, the two key competitors are ferromagnetism and chiral superconductivity. For a small attractive exchange interaction, we find a new type of spin Pomeranchuk order, in which the spin order parameter winds around the Fermi surface. The supermetal state, predicted for a single high-order Van Hove point, is an unstable fixed point in our case. Using the functional RG, we also study full band models and the effect of (non-local) Coulomb interactions as relevant for intercalated graphene and other materials. We explore the competing correlated states and the phase diagram for various models in the vicinity of the high-order Van Hove points.

We analyze quantum fluctuation effects at the onset of incommensurate $2k_F$ charge- or spin-density wave order in two-dimensional metals, for a system where the ordering wave vector ${\bf Q}$ connects a pair of hot spots on the Fermi surface with a vanishing Fermi surface curvature. We first compute the order parameter susceptibility and the fermion self-energy in random phase approximation (RPA). Logarithmic divergences are subsequently treated by a renormalization group analysis. The coupling between the order parameter fluctuations and the fermions vanishes logarithmically in the low-energy limit. As a consequence, the logarithmic divergences found in RPA do not sum up to anomalous power laws. Instead, only logarithmic corrections to Fermi liquid behavior are obtained. In particular, the quasiparticle weight and the Fermi velocity vanish logarithmically at the hot spots. L. Debbeler and W. Metzner, Phys. Rev. B 107, 165152 (2023). L. Debbeler and W. Metzner, arXiv:2403.00007.

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2D Materials exhibit a wide range of novel physics, much of it driven by behaviour such as flat bands and van Hove singularities emerging when layered materials are combined as twisted multi-layer systems or heterostructures. However, the electronic and vibrational properties of 2D materials whose minimal models are large are challenging for conventional modelling approaches due to their unfavourable scaling with system size. Our 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 can perform geometry and vibrational spectroscopy calculations in situations where ab initio evaluation of the Hessian is unfeasibly expensive. For electronic properties, MLIPs can be applied to geometry pre-relaxation of twisted and heterostructured systems, enabling LS-DFT calculations with accurate corrugation at the large system sizes required to minimize strain. Applications of spectral function unfolding for heterostructures involving TMDs, hBN and graphene and hBN will 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).

Recently, there has been a surge of interest in topological quantum materials exhibiting nontrivial topological states, marking a dynamic frontier in condensed matter physics. Theoretical predictions and experimental observations have unveiled a spectrum of intriguing topological states, including topological insulators, Dirac and Weyl semimetals. Within the realm of topological quantum materials, those featuring a kagome lattice have recently garnered significant attention. This lattice not only gives rise to geometrically frustrated magnetism but also hosts a nontrivial topological electronic structure, showcasing Dirac points, van Hove singularities, and flat bands. The unique structure of the kagome lattice, coupled with multiple spin, charge, and orbit degrees of freedom, creates a fertile ground for exploring the interplay between frustrated magnetism, nontrivial topology, and correlation effects. This interplay results in a multitude of quantum states, offering a platform for investigating emergent electronic orders and their correlations. These materials can be broadly categorized into magnetic and non-magnetic Kagome materials. Magnetic Kagome materials, such as Mn3Sn , Co3Sn2S2 , REMn6Sn6 and FeGe etc. primarily involve 3d transition metal-based Kagome systems. The interplay between magnetism and the topological band structure significantly influences the electronic response, leading, for example, to the enhancement of the Berry curvature and the emergence of a large intrinsic anomalous Hall effect due to the presence of massive Dirac or Weyl fermions. Additionally, the frustrated structure of Kagome materials allows them to host topologically protected Skyrmion lattices or noncoplanar spin textures, resulting in a topological Hall effect arising from real-space Berry phases.

The Kagome band structure hosts several features of interest that primarily include a flat band and a Dirac node with two saddle points flanking it. Even though both the saddle points induce logarithmically divergent van Hove singularities in the density of states, they differ in the way they mix the electronic orbitals from the three unit cell sites. This affects the type of instabilities they may induce. We investigate a Kagome metal using scanning tunneling spectroscopy. We identify a particular distortion on its Kagome surface termination that considerably flattens the M-type saddle point and turns it into a higher order van Hove singularity pinned to the Fermi energy. We capture the process of its generation both in ab initio calculation and with a minimal tight binding model. The strong interactions result in a d-wave Pomeranchuck instability. We visualize it both in real and in reciprocal space as a cascade of wavefunction distributions that spontaneously break the rotational symmetry imposed by the underlying distorted Kagome lattice without generating any additional Bragg peaks. The evolution of the wave function across the Fermi energy further suggests the spontaneously deformed Fermi surface gives rise to charge pumping. The type of Kagome distortion that generates the higher order van Hove singularity may be common to other Kagome materials.

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Partially flat bands and higher order Van Hove singularities are predicted to generate a number of unusual electronic states and lattice instabilities in metals built from kagome networks. Two recent experimental examples of this have been reported in the AV3Sb5 and RV6sn6 classes of kagome metals, both of which have a series of saddle points near their Fermi levels. In this talk, I will discuss the nature of charge correlations that form in these compounds and how they map to models of intertwined lattice and charge instabilities in these materials. I will focus mainly on the nature of short-range, diffuse correlations that appear once the primary charge instabilities in these materials are suppressed in these compounds and insights that can be gained from them.

Topological superconductors have become an important research topic, stimulated by their potential for quantum and superconducting applications. Since the topology only protects against perturbations smaller than the superconducting gap, it is of paramount importance to find topological superconductors with large gaps and high transition temperatures. One possible path to high-temperature topological superconductivity is to consider materials where the Fermi level lies close to van Hove singularities (VHSs), which enhance the density of states. Motivated by this, we theoretically analyze the conditions under which topological superconductivity emerges close to van Hove fillings on the square and kagome lattices. For the kagome lattice we investigate how the different sublattice textures of the m-type and p-type VHSs influence the propensity for topological superconductivity. For both lattices we find a variety of different topological superconducting states and discuss their properties, including Majorana edge and corner modes.

We examine the ground-state phase diagram of interacting two-component fermions in a time-reversal invariant pair of flat bands arising in a Hofstadter model where each component experiences oppositely directed magnetic flux. Based on a density-matrix renormalisation group analysis, we examine the stability of fractional topological insulator states, composed of pairs of equal-filling fractional Chern insulator states, to inter-species interactions. Our results confirm the stability of these FTI states up to a critical value of inter-species interaction strength of typically $0.5$ - $0.7$ of the intra-species interaction, where we find transitions to a topologically trivial state. We also examine the stability of the FTI phase towards spontaneous time-reversal breaking towards states with different densities of particles for each species, but find that the $\nu=2/3+2/3$ state maintains time-reversal symmetry for the examined parameters. We further propose the helical entanglement spectrum as a diagnostic for the presence of the $\nu=1/3+1/3$ FTI phase and demonstrate that it yields the edge counting resulting from a $U(1)\times U(1)$ chiral boson theory.

We start with exploring the effect of a superlattice on the massless Dirac fermions and show how twisted van de waals hetero-structures, particularly twisted graphene layers, and twisted hBN-graphene layers can be described in terms of them. Then we show how twisted bilayer graphene (TBLG) subject to a sequence of commensurate external periodic potentials reveals the formation of moiré 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 fractal generators for TBLG under these external potentials are derived, and we explore their impact on the hierarchy of the BZ edges. We show that several conventional super-moiré systems consisting of even dissimilar layers may be considered as ideal moiré fractals subject to a weak periodic perturbation, thus enabling the quantitative determination of the number of in-gap states that form between the two lowest bands of the base moiré structures. The resulting band structure hierarchy bolsters correlation effects, pushing more bands within the same energy window for both commensurate and incommensurate TBLG. Furthermore, we illustrate that the density distribution of the wave function of these moiré fractals possesses distinct fractal characteristics. We also discuss the effect of strain-induced deformations on the moir\'{e} structures and the moir\'{e} fractals. (The abstract is based on the first two papers in the list of publications) and another manuscript under preparation all co-authored with Deepanshu Aggarwal and Rohit Narula.

The search for new types of exotic topological orders has recently rekindled the interest in Fermi surface reconstructions. Of particular interest are Electronic Topological (Lifshitz) transitions where the number of Fermi surface sheets changes abruptly under the influence of external parameters like chemical doping, pressure, or magnetic field. Lifshitz transitions are generally associated with the presence of critical points in the electronic band structure, i. e., maxima, minima, or saddle points whose presence follows directly from lattice periodicity. As their separation from the chemical potential is of the order of the bandwidth, the critical points hardly affect the low temperature behavior of “conventional” metals. In heavy-fermion materials, however, the widths of the quasi-particle bands are strongly reduced by electronic correlations and, consequently, magnetic fields can drive Lifshitz transitions. The characteristic anomalies in the equilibrium and transport properties provide a method to test the quasi-particle dispersion away from the Fermi surface. The values of the field at which the transitions occur reflects the microscopic mechanism leading to the formation of the heavy quasi-particles. Here we demonstrate that the magnetic field-dependent anomalies in the Seebeck coefficient provide detailed information not only on the critical points, i. e., their character and position relative to the chemical potential but also on the effective mass tensor, i. e.,the quasi-particle dispersion in the vicinity of the critical points. For lanthanide-based HFS, the theoretical analysis is based on Renormalized Band (RB) structure calculations assuming that the heavy quasi-particles result from a Kondo effect. For U-based HFS, on the other hand, we adopt the fully microscopic model which emphasizes the role of intra-atomic Hund's rule-type correlations for appearance of heavy quasi-particle masses. The calculations reproduce the observed positions of the anomalies surprisingly well.

We analyze the transition into the most favorable ordered state for a system of 2D fermions with spin and valley degrees of freedom. We show that for short-ranged interactions and a range of rotationally invariant dispersions, the ordering transition is highly unconventional: the associated susceptibility diverges (or almost diverges) at the transition, yet immediately below it the system jumps discontinuously into a fully polarized state. We analyze the dispersion of the longitudinal and transverse collective modes in different parameter regimes above and below the transition. Additionally, we consider ordering in a system with full SU(4) symmetry and show that there is a cascade of discontinuous transitions into a set of states, which includes a quarter-metal, a half-metal and a three-quarter metal. We compare our results with the data for biased bilayer and tri-layer graphene. With Z. Raines (UMN) and L. Glazman (Yale).

Sr$_2$RuO$_4$ hosts a clean, two-dimensional Fermi liquid that is highly tunable with uniaxial pressure. Notably, a Van Hove singularity, which lies 14 meV above the Fermi level at ambient pressure, can be brought to the Fermi level by applying pressure along the <100> direction of the crystal. Compressed further, the Fermi surface undergoes a Lifshitz transition, and a magnetically ordered phase emerges at even higher compression. Together, these features make Sr$_2$RuO$_4$ an attractive system in which to study the physics of Lifshitz transitions, but the experimental conditions necessary to access the transition are incompatible with traditional thermodynamic measurements. With our ongoing development of thermodynamic probes compatible with uniaxial pressure tuning, it has become possible to study the low-temperature phase diagram of Sr$_2$RuO$_4$ across the Lifshitz transition in detail. I will discuss some of our recent results, including measurements of the stress-strain relation that reveal a strong coupling between the conduction electron system and the lattice at the Lifshitz transition, as well as measurements of the elastocaloric effect.

Through application of uniaxial stress, a Lifshitz point that lies close to the Fermi level in unstressed Sr$_2$RuO$_4$ can be tuned to the Fermi level. Experimental data has provided information on how this tuning affects the electronic structure in the rest of k-space. Through angle-resolved photoemission data, it is seen that Fermi velocities far from the Lifshitz point are not strongly affected. Through Hall effect data, there is evidence that when Sr$_2$RuO$_4$ is tuned well beyond this Lifshitz transition, the rate of electron-electron scattering on one Fermi surface falls dramatically. Penetration depth data indicate that tuning to the Lifshitz transition magnifies the superconducting gap throughout the Brillouin zone. In this talk, these data will be discussed.

The phenomenology and radical changes seen in materials properties traversing a quantum phase transition has captivated condensed matter research over the past decades. Strong electronic correlations lead to novel ground states, including magnetic order, nematicity and unconventional superconductivity. In many cases these are driven or accompanied by Lifshitz transitions. I will present spectroscopic imaging of the electronic structure of strontium ruthenates performed at temperatures down to 100mK[1] and in vector-magnetic fields, and discuss the implications for the low energy electronic structure. Notably, for several of the strontium ruthenates the surface provides a platform to study the properties of the electronic structure under conditions not accessible in the bulk.[2,3] Comparison of the experimental data with continuum local density of states calculations enables us to identify microscopic models that explain some of the macroscopic properties of the materials.[4] I will further discuss how the electronic instabilities in the surface layer are affected by the subtle changes in the electronic structure. [1] C.A. Marques et al., Atomic-scale imaging of emergent order at a magnetic-field-induced Lifshitz transition, Sci. Adv. 8, eabo7757 (2022). [2] A. Kreisel et al., Quasiparticle Interference of the van-Hove singularity in Sr2RuO4, npj Quantum Materials 6, 100 (2021). [3] C.A. Marques et al., Magnetic-Field Tunable Intertwined Checkerboard Charge Order and Nematicity in the Surface Layer of Sr2RuO4. Adv. Mat. 33, 2100593 (2021). [4] M. Naritsuka et al., Compass-like manipulation of electronic nematicity in Sr3Ru2O7, Proc. Nat. Acad. Sci. 120, e2308972120 (2023).

In addition to the usual Van Hove singularities, there are higher-order Van Hove singularities (HOVHS) with DOS divergence according to the power law. They affect different types of ordering in quantum materials. We find that the spin wave density and charge density of the phase formation can be enhanced by the presence of a singularity depending on the strength of certain interactions, with the critical temperature increasing by orders of magnitude. We discuss the application of our findings to various experimental systems such as $\mathrm{Sr_3 Ru_2 O_7}$.

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