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A quantum phase transition (QPT) occurs between two competing phases of matter at zero temperature, driven by quantum fluctuations. Though the presence of these fluctuations is well established, they have not been locally imaged in space and their local dynamics has not been studied so far. We use a scanning superconducting quantum interference device to image quantum fluctuations in the vicinity of the QPT from a superconductor to an insulator. We find fluctuations of the diamagnetic response in both space and time that survive well below the transition temperature, demonstrating their quantum nature. The fluctuations appear as telegraph-like noise with a range of characteristic times and a non-monotonic temperature dependence, revealing unexpected quantum granularity. The lateral dimension of these fluctuations grows towards criticality, offering a new measurable length scale. Our results provide physical insight about the reorganization of phases across a QPT, with implications for any theoretical description. This paves a new route for future quantum information applications.

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The Berezinskii-Kosterlitz-Thouless transition is a very specific phase transition where all thermodynamic quantities are smooth. Therefore, it is difficult to determine the critical temperature in a precise way. In this paper we demonstrate how neural networks can be used to perform this task. In particular, we study how the accuracy of the transition identification depends on the way the neural networks are trained. We apply our approach to three different systems: (i) the classical XY model, (ii) the phase-fermion model, where classical and quantum degrees of freedom are coupled and (iii) the quantum XY model. ordered phase is not described by a local order parameter [1][2][3][4]. Instead, its formation is connected with suppression of non-local topological defects which are difficult to identify.In this paper, we demonstrate the application of machine learning approaches to identify topological transitions in a few different types of two-dimensional classical and quantum systems. In particular, we study the classical XY (c-XY) model, the phase-fermion (PF) [21] where the interaction is only between quantum and classical degrees of freedom, and the fully quantum XY (q-XY) model.The first of these models has already been thoroughly analyzed in [17]. The authors have shown there that treating spin configurations as raw images in the case of a feed-forward network does not lead to the correct value of the critical temperature. Instead, they propose to preprocess the spin configurations into vorticity and then use the results to train two kinds of artificial neural networks (ANN): a one-layer feed-forward network and a deep convolutional network. In both cases, the results are scaled with the system size towards the correct value of the critical temperature, but the one-layer network performed poorly for large systems. In the present approach, we do not have convolutional layers, but we use a deep feed-forward network composed of four fully-connected layers (we learned that the choice of particular meta-parameters is not crucial for the network performance). What is important is that, instead of using raw spin configurations where each spin is represented by a number from 0 to 2π (which gives rather inaccurate results, as demonstrated in [17]), we represent the configurations as vectors of sines and cosines of the spin angles, which reflects the system's symmetry.

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