In the high-transition-temperature (high-T(c)) superconductors the pseudogap phase becomes predominant when the density of doped holes is reduced. Within this phase it has been unclear which electronic symmetries (if any) are broken, what the identity of any associated order parameter might be, and which microscopic electronic degrees of freedom are active. Here we report the determination of a quantitative order parameter representing intra-unit-cell nematicity: the breaking of rotational symmetry by the electronic structure within each CuO(2) unit cell. We analyse spectroscopic-imaging scanning tunnelling microscope images of the intra-unit-cell states in underdoped Bi(2)Sr(2)CaCu(2)O(8 +) (delta) and, using two independent evaluation techniques, find evidence for electronic nematicity of the states close to the pseudogap energy. Moreover, we demonstrate directly that these phenomena arise from electronic differences at the two oxygen sites within each unit cell. If the characteristics of the pseudogap seen here and by other techniques all have the same microscopic origin, this phase involves weak magnetic states at the O sites that break 90 degrees -rotational symmetry within every CuO(2) unit cell.

In the stripe-ordered state of a strongly-correlated two-dimensional electronic system, under a set of special circumstances, the superconducting condensate, like the magnetic order, can occur at a non-zero wave-vector corresponding to a spatial period double that of the charge order. In this case, the Josephson coupling between near neighbor planes, especially in a crystal with the special structure of La2−xBaxCuO4, vanishes identically. We propose that this is the underlying cause of the dynamical decoupling of the layers recently observed in transport measurements at x = 1/8.High-temperature superconductivity (HTSC) was first discovered [1] in La 2−x Ba x CuO 4 . A sharp anomaly [2] in T c (x) occurs at x = 1/8 which is now known to be indicative [3,4] of the existence of stripe order and of its strong interplay with HTSC. Recently, a remarkable dynamical layer decoupling has been observed [5] associated with the superconducting (SC) fluctuations below the spinstripe ordering transition temperature, T spin = 42K.While T c (x), as determined by the onset of a bulk Meissner effect, reaches values up to T c (x = 0.1) = 33 K for x somewhat smaller and larger than x = 1/8, T c (x) drops to the range 2-4 K for x = 1/8. However, in other respects, superconductivity appears to be optimized for x = 1/8. The d-wave gap determined by ARPES has recently been shown [6] to be largest for x = 1/8. Moreover, strong SC fluctuations produce an order of magnitude drop [5] in the in-plane resistivity, ρ ab , at T ≈ T spin , which is considerably higher than the highest bulk SC.The fluctuation conductivity reveals heretofore unprecedented characteristics (as described schematically in Fig. 1): 1) ρ ab drops rapidly with decreasing temperature from T spin down to T KT ≈ 16K, at which point it becomes unmeasurably small. In the range T spin > T > T KT , the temperature dependence of ρ ab is qualitatively of the Kosterlitz-Thouless form, as if the SC fluctuations were strictly confined to a single copperoxide plane. 2) By contrast, the resistivity perpendicular to the copper-oxide planes, ρ c , increases with decreasing temperatures from T ⋆ > ∼ 300 K, down to T ⋆⋆ ≈ 35 K. For T < T ⋆⋆ , ρ c decreases with decreasing temperature, but it only becomes vanishingly small below T 3D ≈ 10 K. Within experimental error, for T KT > T > T 3D , the resistivity ratio, ρ c /ρ ab , is infinite! 3) The full set of usual characteristics of the SC state, the Meissner effect and perfect conductivity, ρ ab = ρ c = 0, is only observed below T c = 4K. Thus, for T 3D > T > T c , a peculiar form of fragile 3D superconductivity exists.The above listed results are new, so an extrinsic explanation of some aspects of the data is possible. Here we assume that the measured properties do reflect the bulk behavior of La 2−x Ba x CuO 4 . We show that there is a straightforward way in which stripe or-

Experiments are finally revealing intricate facts about graphene which go beyond the ideal picture of relativistic Dirac fermions in pristine two dimensional (2D) space, two years after its first isolation [1,2]. While observations of rippling [3,4,5] added another dimension to the richness of the physics of graphene, scanning single electron transistor images displayed prevalent charge inhomogeneity [6]. The importance of understanding these non-ideal aspects cannot be overstated both from the fundamental research interest since graphene is a unique arena for their interplay, and from the device applications interest since the quality control is a key to applications. We investigate the membrane aspect of graphene and its impact on the electronic properties. We show that curvature generates spatially varying electrochemical potential. Further we show that the charge inhomogeneity in turn stabilizes ripple formation.Since its unexpected isolation [1], free standing graphene, a single atomic layer of carbon atoms forming a 2D honeycomb lattice, has risen as an intriguing and promising metamaterial. Not only the charming notion of manipulating relativistic fermions on a table top but also the potential of the graphene-based electronics, is propelling the current enthusiasm [2,7]. However, for the graphene-based electronics it is vital to understand the interplay and connection among observed real material aspects.The observations of ripples in suspended graphene using transmission electron microscopy (TEM) [3] and in graphene on SiO 2 substrate using scanning tunneling microscopy (STM) [4,5] summon the membrane aspect of graphene to the foreground. Statistical mechanics of membranes has long been an important branch of soft condensed matter physics [8], with its application to biological systems being one of its driving forces. However, it has been irrelevant for the study of conventional 2D electronic systems which are buried in the semiconductor heterojunction structure. Graphene being a single layer of carbon atoms that can be gated, it forms the first example of an electronic membrane that is subject to direct probes.Electron-hole puddles in graphene imaged by scanning single electron transistor (SET) [6] suggest that such charge inhomogeneity should play an important role in limiting the transport characteristics of graphene. One cause of such charge inhomogeneity could be remote charged impurities in the substrate as it was noted in Refs. [9,10]. By building an effective theory based on microscopic calculations, we find the corrugations to generate inhomogeneous electrochemical potential directly on the graphene membrane; thus identify corrugations as another cause of charge inhomogeneity. We predict control over irregular corrugations to improve the transport properties of graphene greatly. At present, the technology for stabilizing perfectly flat graphene is not available. One possibility is to epitaxially grow monolayer graphene in a controlled manner. While current epitaxial growth technique only yields ...

Even though the rare-earth tritellurides are tetragonal materials with a quasi two dimensional (2D) band structure, they have a "hidden" 1D character. The resultant near-perfect nesting of the Fermi surface leads to the formation of a charge density wave (CDW) state. We show that for this band structure, there are two possible ordered phases: A bidirectional "checkerboard" state would occur if the CDW transition temperature were sufficiently low, whereas a unidirectional "striped" state, consistent with what is observed in experiment, is favored when the transition temperature is higher. This result may also give some insight into why, in more strongly correlated systems, such as the cuprates and nickelates, the observed charge ordered states are generally stripes as opposed to checkerboards.

Despite rapidly growing interest in harnessing machine learning in the study of quantum manybody systems, training neural networks to identify quantum phases is a nontrivial challenge. The key challenge is in efficiently extracting essential information from the many-body Hamiltonian or wave function and turning the information into an image that can be fed into a neural network. When targeting topological phases, this task becomes particularly challenging as topological phases are defined in terms of non-local properties. Here we introduce quantum loop topography (QLT): a procedure of constructing a multi-dimensional image from the "sample" Hamiltonian or wave function by evaluating two-point operators that form loops at independent Monte Carlo steps. The loop configuration is guided by characteristic response for defining the phase, which is Hall conductivity for the cases at hand. Feeding QLT to a fully-connected neural network with a single hidden layer, we demonstrate that the architecture can be effectively trained to distinguish Chern insulator and fractional Chern insulator from trivial insulators with high fidelity. In addition to establishing the first case of obtaining a phase diagram with topological quantum phase transition with machine learning, the perspective of bridging traditional condensed matter theory with machine learning will be broadly valuable.Introduction-Machine learning techniques have been enabling neural networks to successfully recognize and interpret big data sets of images and speeches [1]. Through supervised trainings with a large number of data sets, neural networks 'learn' to recognize key features of a universal class. Very recently, rapid and promising development has been made from this perspective on numerical studies of condensed matter systems, including dynamical systems[2-6], systems undergoing phase transitions [7][8][9][10][11][12][13], as well as quantum many-body systems. Also established is the theory connection to renormalization group [14,15]. Exciting successes in application of machine learning to symmetry broken phases [7][8][9][10] may be attributed to the locality of the defining property of the target phases: the order parameter field. The snap-shots of order parameter configuration form images that can be readily fed into neural networks that have been developed to recognize patterns in images.Unfortunately many novel states cannot be numerically detected through a local order parameter. For one, all topological phases are intrinsically defined in terms of non-local topological properties. Not only many-body localized states of growing interest [16] fit into this category, even a superconducting state fits in here since the superconducting order parameter explicitly breaks particle number conservation [17]. In order for neural networks to learn to recognize and identify such phases, we need to supply them with "images" that contain relevant non-local information. Clearly information based on single site is insufficient. One approach to detecting topologic...

Despite much interest in engineering new topological surface (edge) states using structural defects, such topological surface states have not been observed yet. We show that recently imaged tilt boundaries in gated multilayer graphene should support topologically protected gapless edge states. We approach the problem from two perspectives: the microscopic perspective of a tight-binding model and an ab initio calculation on a bilayer, and the symmetry-protected topological (SPT) state perspective for a general multilayer. Hence, we establish the tilt-boundary edge states as the first concrete example of the edge states of symmetry-protected Z-type SPT, protected by no-valley mixing, electron-number conservation, and time-reversal T symmetries. Further, we discuss possible phase transitions between distinct SPTs upon symmetry changes. Combined with a recently imaged tilt-boundary network, our findings may explain the long-standing puzzle of subgap conductance in gated bilayer graphene. This proposal can be tested through future transport experiments on tilt boundaries. In particular, the tilt boundaries offer an opportunity for the in situ imaging of topological edge transport.

We study the coexisting smectic modulations and intra-unit-cell nematicity in the pseudogap states of underdoped Bi(2)Sr(2)CaCu(2)O(8+δ). By visualizing their spatial components separately, we identified 2π topological defects throughout the phase-fluctuating smectic states. Imaging the locations of large numbers of these topological defects simultaneously with the fluctuations in the intra-unit-cell nematicity revealed strong empirical evidence for a coupling between them. From these observations, we propose a Ginzburg-Landau functional describing this coupling and demonstrate how it can explain the coexistence of the smectic and intra-unit-cell broken symmetries and also correctly predict their interplay at the atomic scale. This theoretical perspective can lead to unraveling the complexities of the phase diagram of cuprate high-critical-temperature superconductors.

Theoretically, it has been known that breaking spin degeneracy and effectively realizing spinless fermions is a promising path to topological superconductors. Yet, topological superconductors are rare to date. Here we propose to realize spinless fermions by splitting the spin degeneracy in momentum space. Specifically, we identify monolayer hole-doped transition metal dichalcogenide (TMD)s as candidates for topological superconductors out of such momentum-space-split spinless fermions. Although electron-doped TMDs have recently been found superconducting, the observed superconductivity is unlikely topological because of the near spin degeneracy. Meanwhile, hole-doped TMDs with momentum-space-split spinless fermions remain unexplored. Employing a renormalization group analysis, we propose that the unusual spin-valley locking in hole-doped TMDs together with repulsive interactions selectively favours two topological superconducting states: interpocket paired state with Chern number 2 and intrapocket paired state with finite pair momentum. A confirmation of our predictions will open up possibilities for manipulating topological superconductors on the device-friendly platform of monolayer TMDs.