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Flat band moir\'e graphene systems have emerged as a quintessential platform to investigate correlated phases of matter. A plethora of interaction-driven ground states have been proposed, and yet despite extensive experimental effort, there has been little direct evidence that distinguishes between the various phases, in particular near charge neutrality point. Here, we use a nanoscale scanning superconducting quantum interference device to image the local thermodynamic quantum oscillations in alternating-twist trilayer graphene at magnetic fields as low as 56 mT, which reveal ultrafine details of the density of states and of the renormalization of the single-particle band structure by Coulomb interactions. We find that the charging self-energy due to occupied electronic states, is critical in explaining the high carrier density physics. At half-filling of the conduction flat band, we observe a Stoner-like symmetry breaking, suggesting that it is the most robust mechanism in the hierarchy of phase transitions. On approaching charge neutrality, where the charging energy is negligible and exchange energy is dominant, we find the ground state to be a nematic semimetal which is favored over gapped states in the presence of heterostrain. In the revealed semimetallic phase, the flat-band Dirac cones migrate towards the mini-Brillouin zone center, spontaneously breaking the C_3 rotational symmetry. Our low-field local quantum oscillations technique presents an alluring avenue to explore the ground states of diverse strongly interacting van der Waals systems., Comment: 30 pages, 4 main text figures, 6 Extended Data figures

Quantum oscillations originating from the quantization of the electron cyclotron orbits provide ultrasensitive diagnostics of electron bands and interactions in novel materials. We report on the first direct-space nanoscale imaging of the thermodynamic magnetization oscillations due to the de Haas-van Alphen effect in moir\'e graphene. Scanning by SQUID-on-tip in Bernal bilayer graphene crystal-axis-aligned to hBN reveals abnormally large magnetization oscillations with amplitudes reaching 500 {\mu}_B/electron in weak magnetic fields, unexpectedly low frequencies, and high sensitivity to the superlattice filling fraction. The oscillations allow us to reconstruct the complex band structure in exquisite detail, revealing narrow moir\'e bands with multiple overlapping Fermi surfaces separated by unusually small momentum gaps. We identify distinct sets of oscillations that violate the textbook Onsager Fermi surface sum rule, signaling formation of exotic broad-band particle-hole superposition states induced by coherent magnetic breakdown., Comment: 30 pages, 5 main text figures, 6 supplementary figures

The scanning superconducting quantum interference device (SQUID) fabricated on the tip of a sharp quartz pipette (SQUID-on-tip) has emerged as a versatile tool for nanoscale imaging of magnetic, thermal, and transport properties of microscopic devices of quantum materials. We present the design and performance of a scanning SQUID-on-tip microscope in a top-loading probe of a cryogen-free dilution refrigerator. The microscope is enclosed in a custom-made vacuum-tight cell mounted at the bottom of the probe and is suspended by springs to suppress vibrations caused by the pulse tube cryocooler. Two capillaries allow in-situ control of helium exchange gas pressure in the cell that is required for thermal imaging. A nanoscale heater is used to create local temperature gradients in the sample, which enables quantitative characterization of the relative vibrations between the tip and the sample. The spectrum of the vibrations shows distinct resonant peaks with maximal power density of about 27 nm/Hz$^{1/2}$ in the in-plane direction. The performance of the SQUID-on-tip microscope is demonstrated by magnetic imaging of the MnBi$_2$Te$_4$ magnetic topological insulator, magnetization and current distribution imaging in a SrRuO$_3$ ferromagnetic oxide thin film, and by thermal imaging of dissipation in graphene., Comment: Submitted to Review of Scientific Instruments

Vortices are the hallmarks of hydrodynamic flow. Recent studies indicate that strongly-interacting electrons in ultrapure conductors can display signatures of hydrodynamic behavior including negative nonlocal resistance, Poiseuille flow in narrow channels, and a violation of the Wiedemann-Franz law. Here we provide the first visualization of whirlpools in an electron fluid. By utilizing a nanoscale scanning superconducting quantum interference device on a tip (SQUID-on-tip) we image the current distribution in a circular chamber connected through a small aperture to an adjacent narrow current carrying strip in high-purity type-II Weyl semimetal WTe2. In this geometry, the Gurzhi momentum diffusion length and the size of the aperture determine the vortex stability phase diagram. We find that the vortices are present only for small apertures, whereas the flow is laminar (non-vortical) for larger apertures, consistent with the theoretical analysis of the hydrodynamic regime and in contrast to the expectations of ballistic transport in WTe2 at low temperatures. Moreover, near the vortical-to-laminar transition, we observe a single vortex in the chamber splitting into two vortices, a behavior that can occur only in the hydrodynamic regime and cannot be sustained by ballistic transport. These findings suggest a novel mechanism of hydrodynamic flow: instead of the commonly considered electron-electron scattering at the bulk, which becomes extremely weak at low temperatures, the spatial diffusion of charge carriers' momenta is enabled by small-angle scattering at the planar surfaces of thin pure crystals. This surface-induced para-hydrodynamics opens new avenues for exploring and utilizing electron fluidics in high-mobility electron systems., Comment: Main text: 15 pages, 5 figures. Method: 18 pages, 9 Extended Data figures. Supplementary videos: 2

Charge carriers in magic angle graphene come in eight flavors described by a combination of their spin, valley, and sublattice polarizations. When the inversion and time reversal symmetries are broken by the substrate or by strong interactions, the degeneracy of the flavors can be lifted and their corresponding bands can be filled sequentially. Due to their non-trivial band topology and Berry curvature, each of the bands is classified by a topological Chern number, leading to the quantum anomalous Hall and Chern insulator states at integer fillings $\nu$ of the bands. It has been recently predicted, however, that depending on the local atomic-scale arrangements of the graphene and the encapsulating hBN lattices, rather than being a global topological invariant, the Chern number C may become position dependent, altering transport and magnetic properties of the itinerant electrons. Using a SQUID-on-tip, we directly image the nanoscale Berry-curvature-induced equilibrium orbital magnetism, the polarity of which is governed by the local Chern number, and detect its two constituent components associated with the drift and the self-rotation of the electronic wave packets. At $\nu=1$, we observe local zero-field valley-polarized Chern insulators forming a mosaic of microscopic patches of C=-1, 0, or 1, governed by the local sublattice polarization, consistent with predictions. Upon further filling, we find a first-order phase transition due to recondensation of electrons from valley K to K', which leads to irreversible flips of the local Chern number and the magnetization, and to the formation of valley domain walls giving rise to hysteretic global anomalous Hall resistance. The findings shed new light on the structure and dynamics of topological phases and call for exploration of the controllable formation of flavor domain walls and their utilization in twistronic devices., Comment: 29 pages, 5 main text figures and 8 supplementary figures

In a Type II superconductor, the vortex core behaves like a normal metal. Consequently, the single-particle density of states in the vortex core of a conventional Type II superconductor remains either flat or (for very clean single crystals) exhibits a peak at zero bias due to the formation of Caroli-de Gennes-Matricon bound state inside the core. Here we report an unusual observation from scanning tunneling spectroscopy measurements in a weakly pinned thin film of the conventional s-wave superconductor a-MoGe, namely, that a soft gap in the local density of states continues to exist even at the center of the vortex core. We ascribe this observation to rapid fluctuation of vortices about their mean position that blurs the boundary between the gapless normal core and the gapped superconducting region outside. Analyzing the data as a function of magnetic field we show that the variation of fluctuation amplitude as a function of magnetic field is consistent with quantum zero-point motion of vortices., Comment: arXiv admin note: substantial text overlap with arXiv:1905.01045

The work reported in my doctoral thesis is an experimental study of vortex dynamics and phase transitions in thin films of type II superconductors using scanning tunneling spectroscopy, low frequency ac susceptibility measurements and complimentary transport measurements. Chapter I and II cover basics of superconductivity and methodologies used in the thesis. Chapter III discusses the effect of periodic pinning centers on the geometry and dynamics of vortex lattice in NbN thin films. Consequent study of dynamic transition of vortex Mott-like to vortex metal-like state is described. In Chapter IV, effect of strong disorder on vortex lattice in NbN thin films is studied. Here we show magnetic field induced granularity gives rise to pseudogap phase which is utilized to explain superconductor to insulator-like transition in stronger disorder. Chapter V contains the study of 2-dimensional vortex lattice melting in a-MoGe thin films. We discuss the observation of hexatic vortex fluid phase and the BKTHNY two-step melting in this scenario. Effect of sample thickness on this phases is also described. In Chapter VI, we study the possibility of quantum fluctuation of vortices in weakly pinned a-MoGe thin films. Effect of pinning on this fluctuation and a possibility of a quantum to thermal crossover is also explored., Comment: 149 pages, A thesis submitted to the Tata Institute of Fundamental Research, Mumbai for the PhD degree in Physics

The rapid transmission of the highly contagious novel coronavirus has been represented through several data-guided approaches across targeted geographies, in an attempt to understand when the pandemic will be under control and imposed lockdown measures can be relaxed. However, these epidemiological models predominantly based on training data employing number of cases and fatalities are limited in that they do not account for the spatiotemporal population dynamics that principally contributes to the disease spread. Here, a stochastic cellular automata enabled predictive model is presented that is able to accurate describe the effect of demography-dependent population dynamics on disease transmission. Using the spread of coronavirus in the state of New York as a case study, results from the computational framework remarkably agree with the actual count for infected cases and deaths as reported across organizations. The predictions suggest that an extended lockdown in some form, for up to 180 days, can significantly reduce the risk of a second wave of the outbreak. In addition, increased availability of medical testing is able to reduce the number of infected patients, even when less stringent social distancing guidelines and imposed. Equipping this stochastic approach with demographic factors such as age ratio, pre-existing health conditions, robustifies the model to predict the transmittivity of future outbreaks before they transform into an epidemic., Comment: 14 pages, 6 figures

We investigate the evolution of superconductivity with decreasing film thickness in ultrathin amorphous MoGe (a-MoGe) films using a combination of sub-Kelvin scanning tunneling spectroscopy, magnetic penetration depth measurements and magneto-transport measurements. We observe that superconductivity is strongly affected by quantum and classical phase fluctuations for thickness below 5 nm. The superfluid density is strongly suppressed by quantum phase fluctuations at low temperatures and evolves towards a linear-T dependence at higher temperatures. This is associated with a rapid decrease in the superconducting transition temperature, Tc, and the emergence of a pronounced pseudogap above Tc. These observations suggest that at strong disorder the destruction of superconductivity follows a Bosonic route where the global superconducting state is destroyed by phase fluctuations even though the pairing amplitude remains finite.

We investigate the magnetic field evolution of the superconducting state in a strongly disordered NbN thin film which exhibits a magnetic field tuned superconductor to insulator-like transition, employing low temperature scanning tunneling spectroscopy (STS). Transport measurements of the sample reveals a characteristic magnetic field, which separates the low field state where resistance decreases with decreasing temperature, i.e. dR/dT > 0 and a high-field state where dR/dT < 0. However, STS imaging of the superconducting state reveals a smooth evolution across this field and the presence of a robust pseudogap on both sides of this characteristic field. Our results suggest that the superconductor-insulator transition might be a percolative transition driven by the shrinking of superconducting fraction with magnetic field., Comment: 22 pages, 8 figures including supplementary material

Recently, detailed real space imaging using scanning tunneling spectroscopy of the vortex lattice in a weakly pinned a-MoGe thin film revealed that the vortex lattice melts in two steps with temperature or magnetic field, going first from a vortex solid to a hexatic vortex fluid and then from a hexatic vortex fluid to an isotropic vortex liquid. In this paper, we show that the resistance in the vortex fluid states is extremely sensitive to radio-frequency electromagnetic perturbation. In the presence of very low-amplitude excitation above few kilohertz, the resistance increases by several orders of magnitude. On the other hand when the superconductor is well shielded from external electromagnetic radiation, the dissipation in the sample is very small and the resistance is below our detection limit., Comment: pdf file with figures

Quantum fluids refer to a class of systems that remain in fluid state down to absolute zero temperature. In this letter, using a combination of magnetotransport and scanning tunneling spectroscopy down to 300 mK, we show that vortices in a very weakly pinned a-MoGe thin film can form a quantum vortex fluid. Under the application of a magnetic field perpendicular to the plane of the film, the vortex state transforms from a vortex solid to a hexatic vortex fluid and eventually to an isotropic vortex liquid. The fact that the two latter states remain fluid down to absolute zero temperature is evidenced from the electrical resistance which saturates to a finite value at low temperatures. Furthermore, scanning tunneling spectroscopy measurements reveal a soft gap at the center of each vortex, which arises from large zero point fluctuation of vortices., Comment: In view of more recent findings, the data in this (unpublished) preprint has been reanalyzed: Kindly see arXiv:1909.06573 and arXiv:2102.12996 for more rigorous analysis with more extensive data

The hexatic fluid refers to a phase in between a solid and a liquid which has short range positional order but quasi-long range orientational order. In the celebrated theory of Berezinskii, Kosterlitz and Thouless and subsequently refined by Halperin, Nelson and Young, it was predicted that a 2-dimensional hexagonal solid can melt in two steps: first, through a transformation from a solid to a hexatic fluid which retains quasi long range orientational order and then from a hexatic fluid to an isotropic liquid. In this paper, using a combination of real space imaging and transport measurements we show that the 2-dimensional vortex lattice in a-MoGe thin film follows this sequence of melting as the magnetic field is increased. Identifying the signatures of various transitions on the bulk transport properties of the superconductor, we construct a vortex phase diagram for a two dimensional superconductor., Comment: New Data added in this version

When a magnetic field is applied, the mixed state of a conventional Type II superconductor gets destroyed at the upper critical field Hc2, where the normal vortex cores overlap with each other. Here, we show that in the presence weak and homogeneous disorder the destruction of superconductivity with field follows a different route. Starting with a weakly disordered NbN thin film ( Tc ~ 9K ), we show that under the application of magnetic field the superconducting state becomes increasingly granular, where lines of vortices separate the superconducting islands. Consequently, phase fluctuations between these islands give rise to a field induced pseudogap phase, which has a gap in the electronic density of states but where the global zero resistance state is destroyed., Comment: New data added in this version

We show that under an a.c. magnetic field excitation the vortex lattice in a superconductor with periodic array of holes can undergo a transition from a Mott-like state where each vortex is localized in a hole, to a metal-like state where the vortices get delocalized. The vortex dynamics is studied through the magnetic shielding response which is measured using a low frequency two-coil mutual inductance technique on a disordered superconducting NbN film having periodic array of holes. We observe that the shielding response of the vortex state is strongly dependent on the amplitude of the a.c. magnetic excitation. At low amplitude the shielding response varies smoothly with excitation amplitude, corresponding to elastic deformation of the vortex lattice. However, above a threshold value of excitation the response shows a series of sharp jumps, signaling the onset of the Mott to metal transition. Quantitative analysis reveals that this is a collective phenomenon which depends on the filling fraction of vortices in the antidot lattice., Comment: Significantly modified version

Disorder induced melting, where the increase in positional entropy created by random pinning sites drives the order-disorder transition in a periodic solid, provides an alternate route to the more conventional thermal melting. Here, using real space imaging of the vortex lattice through scanning tunneling spectroscopy, we show that in the presence of weak pinning, the vortex lattice in a type II superconductor disorders through two distinct topological transitions. Across each transition, we separately identify metastable states formed through superheating of the low temperature state or supercooling of the high temperature state. Comparing crystals with different levels of pinning we conclude that the two-step melting is fundamentally associated with the presence of random pinning which generates topological defects in the ordered vortex lattice., Comment: Expanded and with new data added

To generalize the notion of recurrent manifold, there are various recurrent like conditions in the literature. In this paper we present a recurrent like structure, namely, \textit{super generalized recurrent manifold}, which generalizes both the hyper generalized recurrent manifold and weakly generalized recurrent manifold. The main object of the present paper is to study the geometric properties of super generalized recurrent manifold. Finally to ensure the existence of such structure we present a proper example by a suitable metric., Comment: It contains 15 pages and we invite all the interest researchers for their valuable comments

The present paper deals with the proper existence of a generalized class of recurrent manifolds, namely, hyper-generalized recurrent manifolds. We have established the proper existence of various generalized notions of recurrent manifolds. For this purpose we have presented a metric and computed its curvature properties and finally we have obtained the existence of a new class of semi-Riemannian manifolds which are non-recurrent but hyper-generalized recurrent, Ricci recurrent, conharmonically recurrent, manifold of recurrent curvature 2-forms and semisymmetric; not weakly symmetric but weakly Ricci symmetric, conformally weakly symmetric and conharmonically weakly symmetric; non-Einstein but Ricci simple; do not satisfy $P\cdot P =0$ but fulfills the condition $P \cdot P = -\frac{1}{3}Q(S, P)$, $P$ being the projective curvature tensor., Comment: It contains 20 pages and interested researchers are invited for their valuable comments