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Mark H Fischer

Publications and source records attributed to Mark H Fischer.

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Introduction to Machine Learning for the Sciences

This is an introductory machine-learning course specifically developed with STEM students in mind. Our goal is to provide the interested reader with the basics to employ machine learning in their own projects and to familiarize themself with the terminology as a foundation for further reading of the relevant literature. In these lecture notes, we discuss supervised, unsupervised, and reinforcement learning. The notes start with an exposition of machine learning methods without neural networks, such as principle component analysis, t-SNE, clustering, as well as linear regression and linear classifiers. We continue with an introduction to both basic and advanced neural-network structures such as dense feed-forward and conventional neural networks, recurrent neural networks, restricted Boltzmann machines, (variational) autoencoders, generative adversarial networks. Questions of interpretability are discussed for latent-space representations and using the examples of dreaming and adversarial attacks. The final section is dedicated to reinforcement learning, where we introduce basic notions of value functions and policy learning.

physics.comp-ph

Superconductivity and Local Inversion-Symmetry Breaking

Inversion and time reversal are essential symmetries for the structure of Cooper pairs in superconductors. The loss of one or both leads to modifications to this structure and can change the properties of the superconducting phases in profound ways. Lacking inversion, superconductivity in noncentrosymmetric materials has become an important topic, in particular, in the context of topological superconductivity as well as unusual magnetic and magneto-electric properties. Recently, crystal structures with local, but not global inversion-symmetry breaking have attracted attention, as superconductivity can exhibit phenomena not naively expected in centrosymmetric materials. After introducing the concept of locally noncentrosymmetric crystals and different material realizations, we discuss consequences of such local symmetry breaking on the classification, the expected and, in parts, already observed phenomenology of unconventional superconductivity, and possible topological superconducting phases.

cond-mat.supr-con

Flat bands with fragile topology through superlattice engineering on single-layer graphene

'Magic'-angle twisted bilayer graphene has received a lot of interest due to its flat bands with potentially non-trivial topology that lead to intricate correlated phases. A spectrum with flat bands, however, does not require a twist between multiple sheets of van der Waals materials, but rather can be realized with the application of an appropriate periodic potential. Here, we propose the imposition of a tailored periodic potential onto a single graphene layer through local perturbations that could be created via lithography or adatom manipulation, which also results in an energy spectrum featuring flat bands. Our first-principle calculations for an appropriate decoration of graphene with adatoms indeed show the presence of flat bands in the spectrum. Furthermore, we reveal the topological nature of the flat bands through a symmetry-indicator analysis. This non-trivial topology manifests itself in corner-localized states with a filling anomaly as we show using a tight-binding model. Our proposal of a single decorated graphene sheet provides a new versatile route to study correlated phases in topologically non-trivial, flat band structures.

cond-mat.mes-hall

Atomic limit and inversion-symmetry indicators for topological superconductors

Symmetry indicators have proven to be extremely helpful in identifying topologically non-trivial crystalline insulators using symmetry-group representations of their Bloch states. An extension of this approach to superconducting systems requires defining an appropriate atomic limit for Bogoliubov-de-Gennes Hamiltonians. Here, we introduce such a notion of atomic limit and derive a $\mathbb{Z}_{2^d}$-valued symmetry indicator for inversion-symmetric superconductors of $d$ dimensions. This indicator allows for a refined topological classification including higher-order phases for systems in the superconducting symmetry classes D and DIII. We further elucidate the bulk-boundary correspondence of these phases using Dirac surface theories. Requiring only the normal-state band structure and the superconducting order-parameter symmetry as an input, this indicator is well suited for a search of topological superconductors using first-principles calculations.

cond-mat.supr-con

Deviation from the Fermi-Liquid Transport Behavior in the Vicinity of a Van Hove Singularity

Recent experiments revealed non-Fermi-liquid resistivity in the unconventional superconductor Sr$_{2}$RuO$_{4}$ when strain pushes one of the Fermi surfaces close to a van Hove singularity. The origin of this behavior and whether it can be understood from a picture of well defined quasiparticles is unclear. We employ a Boltzmann transport analysis beyond the single relaxation-time approximation based on a single band which undergoes a Lifshitz transition, where the Fermi surface crosses a van Hove singularity, either due to uni-axial or epitaxial strain. First analytically investigating impurity scattering, we clarify the role of the diverging density of states together with the locally flat band at the point of the Lifshitz transition. Additionally including electron-electron scattering numerically, we find good qualitative agreement with resistivity measurements on uni-axially strained Sr$_{2}$RuO$_{4}$, including the temperature scaling and the temperature dependence of the resistivity peak. Our results imply that even close to the Lifshitz transition, a description starting from well-defined quasiparticles holds. To test the validity of Boltzmann transport theory near a van Hove singularity, we provide further experimentally accessible parameters, such as thermal transport, the Seebeck coefficient, and Hall resistivity and compare different strain scenarios.

cond-mat.str-el

Superconductivity without inversion and time-reversal symmetries

The traditional symmetries that protect superconductivity are time-reversal and inversion. Here, we examine the minimal symmetries protecting superconductivity in two dimensions and find that time-reversal symmetry and inversion symmetry are not required, and having a combination of either symmetry with a mirror operation on the basal plane is sufficient. We classify superconducting states stabilized by these two symmetries, when time-reversal and inversion symmetries are not present, and provide realistic minimal models as examples. Interestingly, several experimentally realized systems, such as transition metal dichalcogenides and the two-dimensional Rashba system belong to this category, when subject to an applied magnetic field.

cond-mat.supr-con

Topological superconductivity in monolayer transition metal dichalcogenides

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 due to 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 favors two topological superconducting states: inter-pocket paired state with Chern number 2 and intra-pocket 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.

cond-mat.supr-con

Dynamics of a Many-Body-Localized System Coupled to a Bath

Coupling a many-body-localized system to a dissipative bath necessarily leads to delocalization. Here, we investigate the nature of the ensuing relaxation dynamics and the information it holds on the many-body-localized state. We formulate the relevant Lindblad equation in terms of the local integrals of motion of the underlying localized Hamiltonian. This allows to map the quantum evolution deep in the localized state to tractable classical rate equations. We consider two different types of dissipation relevant to systems of ultra-cold atoms: dephasing due to inelastic scattering on the lattice lasers and particle loss. Only the latter mechanism shows a pronounced effects of interactions on the relaxation of observables.

cond-mat.str-el

Spin-Torque Generation in Topological-Insulator-Based Heterostructures

Heterostructures utilizing topological insulators exhibit a remarkable spin-torque efficiency. However, the exact origin of the strong torque, in particular whether it stems from the spin-momentum locking of the topological surface states or rather from spin-Hall physics of the topological-insulator bulk remains unclear. Here, we explore a mechanism of spin-torque generation purely based on the topological surface states. We consider topological-insulator-based bilayers involving ferromagnetic metal (TI/FM) and magnetically doped topological insulators (TI/mdTI), respectively. By ascribing the key theoretical differences between the two setups to location and number of active surface states, we describe both setups within the same framework of spin diffusion of the non-equilibrium spin density of the topological surface states. For the TI/FM bilayer, we find large spin-torque efficiencies of roughly equal magnitude for both in-plane and out-of-plane spin torques. For the TI/mdTI bilayer, we elucidate the dominance of the spin-transfer-like torque. However, we cannot explain the orders of magnitude enhancement reported. Nevertheless, our model gives an intuitive picture of spin-torque generation in topological-insulator-based bilayers and provides theoretical constraints on spin-torque generation due to topological surface states.

cond-mat.mes-hall

Fluctuation and strain effects in a chiral $p$-wave superconductor

For a tetragonal material, order parameters of $p_x$ and $p_y$ symmetry are related by rotation and hence have the same $T_{\rm c}$ at a mean-field level. This degeneracy can be lifted by a symmetry-breaking field, like (uniaxial) in-plane strain, such that at $T_{\rm c}$, the order parameter is only of $p_x$ or $p_y$ symmetry. Only at a lower temperature also the respective other order parameter condenses to form a chiral $p$-wave state. At the mean-field level, the derivative of $T_{\rm c}$ with strain is discontinuous at zero strain. We analyze consequences of (thermal) fluctuations on the strain-temperature phase diagram within a Ginzburg-Landau approach. We find that the order-parameter fluctuations can drive the transition to be weakly first order, rounding off this discontinuity. We discuss the possibility of a second-order transition into a non-superconducting time-reversal-symmetry-breaking phase and consequences for the spin-triplet superconductor Sr$_2$RuO$_4$.

cond-mat.supr-con

Gap Symmetry and Stability Analysis in the Multi-Orbital Fe-Based Superconductors

The iron-based superconductors allow for a zoo of possible order parameters due to their orbital degrees of freedom. These order parameters are often written in an orbital basis, as this allows to distinguish their different symmetry. Unlike in standard single-band superconductors, where electrons of the same band - and hence, energy - are paired, a general order parameter in such a multi-orbital system also contains pairing of electrons belonging to different bands. As this corresponds to pairing of electrons of different energy, such order parameters are energetically less stable. Here, we present a simple criterion for a stability analysis of the orbital part of the gap function based on the basic principle that electrons of equal energy are paired in the superconducting state. This not only allows to find the most stable states, but also to identify the terms in a (tight-binding) Hamiltonian suppressing any given superconducting state. Our approach thus allows to identify the minimal Hamiltonian necessary to compare competing instabilities.

cond-mat.supr-con

Spin-Orbit Coupling in LaAlO$_3$/SrTiO$_3$ interfaces: Magnetism and Orbital Ordering

The combination of Rashba spin-orbit coupling and electron correlations can induce unusual phenomena in the metallic interface between SrTiO$_3$ and LaAlO$_3$. We consider effects of Rashba spin-orbit coupling at this interface in the context of the recent observation of anisotropic magnetism. Firstly, we show how Rashba spin-orbit coupling in a system near a band-edge can account for the observed magnetic anisotropy. Secondly, we investigate the coupling between in-plane magnetic-moment anisotropy and nematicity in the form of an orbital imbalance between d$_{xz}$ / d$_{yz}$ orbitals. We estimate this coupling to be substantial in the low electron density regime. Such an orbital ordering can affect magneto transport.

cond-mat.str-el