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Symmetry protected topological (SPT) phases are fundamental quantum many-body states of matter beyond Landau's paradigm. Here, we introduce the concept of quantum restored SPTs (QRSPTs), where the protecting symmetry is spontaneously broken at each instance in time, but restored after time average over quantum fluctuations, so that topological features re-emerge. To illustrate the concept, we study a one-dimensional fermionic Su-Schrieffer-Heeger model with fluctuating superconducting order. We solve this problem in several limiting cases using a variety of analytical methods and compare them to numerical (density matrix renormalization group) simulations which are valid throughout the parameter regime. We thereby map out the phase diagram and identify a QRSPT phase with topological features which are reminiscent from (but not identical to) the topology of the underlying free fermion system. The QRSPT paradigm thereby stimulates a new perspective for the constructive design of novel topological quantum many-body phases., Comment: 7 pages, 6 pages supplement, 10 figures

Motivated by the importance of studying topological superconductors beyond the mean-field approximation, we here investigate mesoscopic islands of time reversal invariant topological superconductors (TRITOPS). We characterize the spectrum in the presence of strong order parameter fluctuations in the presence of an arbitrary number of Kramers pairs of Majorana edge states and study the effect of coupling the Coulomb blockaded island to external leads. In the case of an odd fermionic parity on the island, we derive an unconventional Kondo Hamiltonian in which metallic leads couple to both topological Majorana degrees of freedom (which keep track of the parity in different leads) and the overall spin-1/2 in the island. For the simplest case of a single wire (two pairs of Majorana edge states), we demonstrate that anisotropies are irrelevant in the weak coupling renormalization group flow. This permits us to solve the Kondo problem in the vicinity of a Toulouse-like point using Abelian Bosonization. We demonstrate a residual ground state entropy of $\ln(2)$, which is protected by spin-rotation symmetry, but reduced to $\ln(\sqrt{2})$ (as in the spinless topological Kondo effect) by symmetry breaking perturbations. In the symmetric case, we further demonstrate the simultaneous presence of both Fermi liquid and non-Fermi liquid like thermodynamics (depending on the observable) and derive charge and spin transport signatures of the Coulomb blockaded island., Comment: 10+13 pages, 4+1 figures

The interplay of topological electronic band structures and strong interparticle interactions provides a promising path towards the constructive design of robust, long-range entangled many-body systems. As a prototype for such systems, we here study an exactly integrable, local model for a fractionalized topological insulator. Using a controlled perturbation theory about this limit, we demonstrate the existence of topological bands of zeros in the exact fermionic Green's function and show that {in this model} they do affect the topological invariant of the system, but not the quantized transport response. Close to (but prior to) the Higgs transition signaling the breakdown of fractionalization, the topological bands of zeros acquire a finite ``lifetime''. We also discuss the appearance of edge states and edge zeros at real space domain walls separating different phases of the system. This model provides a fertile ground for controlled studies of the phenomenology of Green's function zeros and the underlying exactly solvable lattice gauge theory illustrates the synergetic cross-pollination between solid-state theory, high-energy physics and quantum information science., Comment: 5 papges, 9 pages supplement, 5 figures

We use a combination of numerical matrix product states (MPS) and analytical approaches to investigate the phase diagram of the two-leg $t$-$J$ ladder in the region of low to intermediate fillings. We choose the same coupling strength along the leg- and rung-directions, but study the effect of adding a nearest-neighbor repulsion $V$. We observe a rich phase diagram and analytically identify a Lifshitz-like band filling transition, which can be associated to a numerically observed crossover from s-wave to d-wave like superconducting quasi-long range order (QLRO). Due to the strong interactions, the Lifshitz transition is smeared into a crossover region which separates two distinct Luttinger theories with unequal physical meaning of the Luttinger parameter. Our numerically exact MPS results spotlight deviations from standard Luttinger theory in this crossover region and is consistent with Luttinger theory sufficiently far away from the Lifshitz transition. At very low fillings, studying the Friedel-like oscillations of the local density identifies a precursor region to a Wigner crystal at small values of the magnetic exchange interaction $J/t$. We discuss analytically how tuning parameters at these fillings modifies the phase diagram, and find good agreement with MPS results., Comment: 14 pages, 4 appendices, 17 figures

The emergence of explainability methods has enabled a better comprehension of how deep neural networks operate through concepts that are easily understood and implemented by the end user. While most explainability methods have been designed for traditional deep learning, some have been further developed for geometric deep learning, in which data are predominantly represented as graphs. These representations are regularly derived from medical imaging data, particularly in the field of neuroimaging, in which graphs are used to represent brain structural and functional wiring patterns (brain connectomes) and cortical surface models are used to represent the anatomical structure of the brain. Although explainability techniques have been developed for identifying important vertices (brain areas) and features for graph classification, these methods are still lacking for more complex tasks, such as surface-based modality transfer (or vertex-wise regression). Here, we address the need for surface-based explainability approaches by developing a framework for cortical surface-based deep learning, providing a transparent system for modality transfer tasks. First, we adapted a perturbation-based approach for use with surface data. Then, we applied our perturbation-based method to investigate the key features and vertices used by a geometric deep learning model developed to predict brain function from anatomy directly on a cortical surface model. We show that our explainability framework is not only able to identify important features and their spatial location but that it is also reliable and valid.

Deep learning based Quantitative Susceptibility Mapping (QSM) has shown great potential in recent years, obtaining similar results to established non-learning approaches. Many current deep learning approaches are not data consistent, require in vivo training data or solve the QSM problem in consecutive steps resulting in the propagation of errors. Here we aim to overcome these limitations and developed a framework to solve the QSM processing steps jointly. We developed a new hybrid training data generation method that enables the end-to-end training for solving background field correction and dipole inversion in a data-consistent fashion using a variational network that combines the QSM model term and a learned regularizer. We demonstrate that NeXtQSM overcomes the limitations of previous deep learning methods. NeXtQSM offers a new deep learning based pipeline for computing quantitative susceptibility maps that integrates each processing step into the training and provides results that are robust and fast.

Whether it be in a man-made machine or a biological system, form and function are often directly related. In the latter, however, this particular relationship is often unclear due to the intricate nature of biology. Here we developed a geometric deep learning model capable of exploiting the actual structure of the cortex to learn the complex relationship between brain function and anatomy from structural and functional MRI data. Our model was not only able to predict the functional organization of human visual cortex from anatomical properties alone, but it was also able to predict nuanced variations across individuals.

Quantitative susceptibility mapping (QSM) has gained broad interests in the field by extracting biological tissue properties, predominantly myelin, iron and calcium from magnetic resonance imaging (MRI) phase measurements in vivo. Thereby, QSM can reveal pathological changes of these key components in a variety of diseases. QSM requires multiple processing steps such as phase unwrapping, background field removal and field-to-source-inversion. Current state of the art techniques utilize iterative optimization procedures to solve the inversion and background field correction, which are computationally expensive and require a careful choice of regularization parameters. With the recent success of deep learning using convolutional neural networks for solving ill-posed reconstruction problems, the QSM community also adapted these techniques and demonstrated that the QSM processing steps can be solved by efficient feed forward multiplications not requiring iterative optimization nor the choice of regularization parameters. Here, we review the current status of deep learning based approaches for processing QSM, highlighting limitations and potential pitfalls, and discuss the future directions the field may take to exploit the latest advances in deep learning for QSM.

Purpose: The purpose of this study is to demonstrate a method for Specific Absorption Rate (SAR) reduction for T2-FLAIR MRI sequences at 7T by predicting the required adiabatic pulse power and scaling the amplitude in a slice-wise fashion. Methods: We used a TR-FOCI adiabatic pulse for spin inversion in a T2-FLAIR sequence to improve B1+ homogeneity and calculate the pulse power required for adiabaticity slice-by-slice to minimize the SAR. Drawing on the implicit B1+ inhomogeneity present in a standard localizer scan, 3D AutoAlign localizers and SA2RAGE B1+ maps were acquired in eight volunteers. A convolutional neural network (CNN) was then trained to predict the B1+ profile from the localizers and scale factors for the pulse power for each slice were calculated. The ability to predict the B1+ profile as well as how the derived pulse scale factors affected the FLAIR inversion efficiency were assessed in transverse, sagittal, and coronal orientations. Results: The predicted B1+ maps matched the measured B1+ maps with a mean difference of 4.45% across all slices. The acquisition in the transverse orientation was shown to be most effective for this method and delivered a 40% reduction in SAR along with 1min and 30-sec reduction in scan time (28%) without degradation of image quality. Conclusion: We propose a SAR reduction technique based on the prediction of B1+ profiles from standard localizer scans using a CNN and show that scaling the inversion pulse power slice-by-slice for FLAIR sequences at 7T reduces SAR and scan time without compromising image quality.