Effect of the Soliton Width on Nonequilibrium Exchange Phases of Anyons.

Unlike bosons and fermions, quasiparticles in two-dimensional quantum systems, known as anyons, exhibit statistical exchange phases that range between 0 and π. In fractional quantum Hall states, these anyons, possessing a fraction of the electron charge, traverse along chiral edge channels. This movement facilitates the creation of anyon colliders, where coupling different edge channels through a quantum point contact enables the observation of two-particle interference effects. Such configurations are instrumental in deducing the anyonic exchange phase via current cross-correlations. Prior theoretical models represented dilute anyon beams as discrete steps in the boson fields. However, our study reveals that incorporating the finite width of the soliton shape is crucial for accurately interpreting recent experiments, especially for collider experiments involving anyons with exchange phases θ>π/2, where prior theories fall short.

Boundary between noise and information applied to filtering neural network weight matrices.

Deep neural networks have been successfully applied to a broad range of problems where overparametrization yields weight matrices which are partially random. A comparison of weight matrix singular vectors to the Porter-Thomas distribution suggests that there is a boundary between randomness and learned information in the singular value spectrum. Inspired by this finding, we introduce an algorithm for noise filtering, which both removes small singular values and reduces the magnitude of large singular values to counteract the effect of level repulsion between the noise and the information part of the spectrum. For networks trained in the presence of label noise, we indeed find that the generalization performance improves significantly due to noise filtering.

Machine Learning Optimization of Majorana Hybrid Nanowires.

As the complexity of quantum systems such as quantum bit arrays increases, efforts to automate expensive tuning are increasingly worthwhile. We investigate machine learning based tuning of gate arrays using the covariance matrix adaptation evolution strategy algorithm for the case study of Majorana wires with strong disorder. We find that the algorithm is able to efficiently improve the topological signatures, learn intrinsic disorder profiles, and completely eliminate disorder effects. For example, with only 20 gates, it is possible to fully recover Majorana zero modes destroyed by disorder by optimizing gate voltages.

Random matrix analysis of deep neural network weight matrices.

Neural networks have been used successfully in a variety of fields, which has led to a great deal of interest in developing a theoretical understanding of how they store the information needed to perform a particular task. We study the weight matrices of trained deep neural networks using methods from random matrix theory (RMT) and show that the statistics of most of the singular values follow universal RMT predictions. This suggests that they are random and do not contain system specific information, which we investigate further by comparing the statistics of eigenvector entries to the universal Porter-Thomas distribution. We find that for most eigenvectors the hypothesis of randomness cannot be rejected, and that only eigenvectors belonging to the largest singular values deviate from the RMT prediction, indicating that they may encode learned information. In addition, a comparison with RMT predictions also allows to distinguish networks trained in different learning regimes-from lazy to rich learning. We analyze the spectral distribution of the large singular values using the Hill estimator and find that the distribution cannot in general be characterized by a tail index, i.e., is not of power-law type.

Soft mode in the dynamics of over-realizable online learning for soft committee machines.

Overparametrized deep neural networks trained by stochastic gradient descent are successful in performing many tasks of practical relevance. One aspect of overparametrization is the possibility that the student network has a larger expressivity than the data generating process. In the context of a student-teacher scenario, this corresponds to the so-called over-realizable case, where the student network has a larger number of hidden units than the teacher. For online learning of a two-layer soft committee machine in the over-realizable case, we present evidence that the approach to perfect learning occurs in a power-law fashion rather than exponentially as in the realizable case. All student nodes learn and replicate one of the teacher nodes if teacher and student outputs are suitably rescaled and if the numbers of student and teacher hidden units are commensurate.

Bulk-Boundary Correspondence for Non-Hermitian Hamiltonians via Green Functions.

Genuinely non-Hermitian topological phases can be realized in open systems with sufficiently strong gain and loss; in such phases, the Hamiltonian cannot be deformed into a gapped Hermitian Hamiltonian without energy bands touching each other. Comparing Green functions for periodic and open boundary conditions we find that, in general, there is no correspondence between topological invariants computed for periodic boundary conditions, and boundary eigenstates observed for open boundary conditions. Instead, we find that the non-Hermitian winding number in one dimension signals a topological phase transition in the bulk: It implies spatial growth of the bulk Green function.

Fractional Coulomb blockade for quasi-particle tunneling between edge channels.

In the fractional quantum Hall effect, the elementary excitations are quasi-particles with fractional charges as predicted by theory and demonstrated by noise and interference experiments. We observe Coulomb blockade of fractional charges in the measured magneto-conductance of a 1.4-micron-wide quantum dot. Interaction-driven edge reconstruction separates the dot into concentric compressible regions with fractionally charged excitations and incompressible regions acting as tunnel barriers for quasi-particles. Our data show the formation of incompressible regions of filling factors 2/3 and 1/3. Comparing data at fractional filling factors to filling factor 2, we extract the fractional quasi-particle charge e */e = 0.32 ± 0.03 and 0.35 ± 0.05. Our investigations extend and complement quantum Hall Fabry-Pérot interference experiments investigating the nature of anyonic fractional quasi-particles.

Partial Equilibration of the Anti-Pfaffian Edge due to Majorana Disorder.

We consider electrical and thermal equilibration of the edge modes of the anti-Pfaffian quantum Hall state at ν=5/2 due to tunneling of the Majorana edge mode to trapped Majorana zero modes in the bulk. Such tunneling breaks translational invariance and allows scattering between Majorana and other edge modes in such a way that there is a parametric difference between the length scales for equilibration of charge and heat transport between integer and Bose mode, on the one hand, and for thermal equilibration of the Majorana edge mode, on the other hand. We discuss a parameter regime in which this mechanism could explain the recent observation of quantized heat transport [M. Banerjee et al., Nature (London) 559, 205 (2018)NATUAS0028-083610.1038/s41586-018-0184-1].

Flux Superperiods and Periodicity Transitions in Quantum Hall Interferometers.

For strongly screened Coulomb interactions, quantum Hall interferometers can operate in a novel regime: the intrinsic energy gap can be larger than the charging energy, and addition of flux quanta can occur without adding quasiparticles. We show that flux superperiods are possible and reconcile their appearance with the Byers-Yang theorem. We explain that the observation of anyonic statistical phases is possible by tuning to the transition from a regime with constant chemical potential to a regime with constant particle density, where a flux superperiod changes to a periodicity with one flux quantum at a critical magnetic field strength.

Voigt Exceptional Points in an Anisotropic ZnO-Based Planar Microcavity: Square-Root Topology, Polarization Vortices, and Circularity.

Voigt points represent propagation directions in anisotropic crystals along which optical modes degenerate, leading to a single circularly polarized eigenmode. They are a particular class of exceptional points. Here, we report the fabrication and characterization of a dielectric, anisotropic optical microcavity based on nonpolar ZnO that implements a non-Hermitian system and mimics the behavior of Voigt points in natural crystals. We prove the exceptional-point nature by monitoring the complex-square-root topology of the mode eigenenergies (real and imaginary parts) around the Voigt points. Polarization state analysis shows that these artificially engineered Voigt points behave as vortex cores for the linear polarization and sustain chiral modes. Our findings apply to any planar microcavity with broken cylindrical symmetry and, thus, pave the way for exploiting exceptional points in widespread optoelectronic devices such as vertical cavity surface emitting lasers and resonant cavity light emitting diodes.

Nonlocal Polarization Feedback in a Fractional Quantum Hall Ferromagnet.

In a quantum Hall ferromagnet, the spin polarization of the two-dimensional electron system can be dynamically transferred to nuclear spins in its vicinity through the hyperfine interaction. The resulting nuclear field typically acts back locally, modifying the local electronic Zeeman energy. Here we report a nonlocal effect arising from the interplay between nuclear polarization and the spatial structure of electronic domains in a ν=2/3 fractional quantum Hall state. In our experiments, we use a quantum point contact to locally control and probe the domain structure of different spin configurations emerging at the spin phase transition. Feedback between nuclear and electronic degrees of freedom gives rise to memristive behavior, where electronic transport through the quantum point contact depends on the history of current flow. We propose a model for this effect which suggests a novel route to studying edge states in fractional quantum Hall systems and may account for so-far unexplained oscillatory electronic-transport features observed in previous studies.

Current Correlations from a Mesoscopic Anyon Collider.

Fermions and bosons are fundamental realizations of exchange statistics, which governs the probability for two particles being close to each other spatially. Anyons in the fractional quantum Hall effect are an example for exchange statistics intermediate between bosons and fermions. We analyze a mesoscopic setup in which two dilute beams of anyons collide with each other, and relate the correlations of current fluctuations to the probability of particles excluding each other spatially. While current correlations for fermions vanish, negative correlations for anyons are a clear signature of a reduced spatial exclusion as compared to fermions.

Critical flow and dissipation in a quasi-one-dimensional superfluid.

In one of the most celebrated examples of the theory of universal critical phenomena, the phase transition to the superfluid state of (4)He belongs to the same three-dimensional (3D) O(2) universality class as the onset of ferromagnetism in a lattice of classical spins with XY symmetry. Below the transition, the superfluid density ρs and superfluid velocity v s increase as a power law of temperature described by a universal critical exponent that is constrained to be identical by scale invariance. As the dimensionality is reduced toward 1D, it is expected that enhanced thermal and quantum fluctuations preclude long-range order, thereby inhibiting superfluidity. We have measured the flow rate of liquid helium and deduced its superfluid velocity in a capillary flow experiment occurring in single 30-nm-long nanopores with radii ranging down from 20 to 3 nm. As the pore size is reduced toward the 1D limit, we observe the following: (i) a suppression of the pressure dependence of the superfluid velocity; (ii) a temperature dependence of v s that surprisingly can be well-fitted by a power law with a single exponent over a broad range of temperatures; and (iii) decreasing critical velocities as a function of decreasing radius for channel sizes below R ≃ 20 nm, in stark contrast with what is observed in micrometer-sized channels. We interpret these deviations from bulk behavior as signaling the crossover to a quasi-1D state, whereby the size of a critical topological defect is cut off by the channel radius.

Enhanced Bulk-Edge Coulomb Coupling in Fractional Fabry-Perot Interferometers.

Recent experiments use Fabry-Perot (FP) interferometry to claim that the ν=5/2 quantum Hall state exhibits non-Abelian topological order. We note that the experiments appear inconsistent with a model neglecting bulk-edge Coulomb coupling and Majorana tunneling, so we reexamine the theory of FP devices. Even a moderate Coulomb coupling may strongly affect some fractional plateaus, but very weakly affect others, allowing us to model the data over a wide range of plateaus. While experiments are consistent with the ν=5/2 state harboring Moore-Read topological order, they may have measured Coulomb effects rather than an "even-odd effect" due to non-Abelian braiding.

Transmission phase lapses through a quantum dot in a strong magnetic field.

The phase of the transmission amplitude through a mesoscopic system contains information about the system's quantum mechanical state and excitations thereof. In the absence of an external magnetic field, abrupt phase lapses occur between transmission resonances of quantum dots and can be related to the signs of tunneling matrix elements. They are smeared at finite temperatures. By contrast, we show here that in the presence of a strong magnetic field, phase lapses represent a genuine interaction effect and may occur also on resonance. We identify a relevant physical regime where these phase lapses are robust against finite temperature broadening.

Shot-noise signatures of charge fractionalization in the ν=2 quantum Hall edge.

We investigate the effect of interactions on shot noise in ν=2 quantum Hall edges, where a repulsive coupling between copropagating edge modes is expected to give rise to charge fractionalization. Using the method of nonequilibrium bosonization, we find that even asymptotically the edge distribution function depends in a sensitive way on the interaction strength between the edge modes. We compute shot noise and the Fano factor from the asymptotic distribution function, and from comparison with a reference model of fractionalized excitations, we find that the Fano factor can be close to the value of the fractionalized charge.

Modulation of Majorana-induced current cross-correlations by quantum dots.

We study charge transport through a topological superconductor with a pair of Majorana end states coupled to leads via quantum dots with resonant levels. The nonlocality of the Majorana bound states opens the possibility of crossed Andreev reflection with nonlocal shot noise due to the injection of an electron into one end of the superconductor followed by the emission of a hole at the other end. In the space of energies of the two resonant quantum dot levels, we find a four peaked cloverlike pattern for the strength of noise due to crossed Andreev reflection, distinct from the single ellipsoidal peak found in the absence of Majorana bound states.

Superfluid stiffness of a driven dissipative condensate with disorder.

Observations of macroscopic quantum coherence in driven systems, e.g. polariton condensates, have strongly stimulated experimental as well as theoretical efforts during the last decade. We address the question of whether a driven quantum condensate is a superfluid, allowing for the effects of disorder and its nonequilibrium nature. We predict that for spatial dimensions d<4 the superfluid stiffness vanishes once the condensate exceeds a critical size, and treat in detail the case d=2. Thus a nonequilibrium condensate is not a superfluid in the thermodynamic limit, even for weak disorder, although superfluid behavior would persist in small systems.

Influence of topological excitations on shapiro steps and microwave dynamical conductance in bilayer exciton condensates.

The quantum Hall state at total filling factor ν(T)=1 in bilayer systems realizes an exciton condensate and exhibits a zero-bias tunneling anomaly, similar to the Josephson effect in the presence of fluctuations. In contrast to conventional Josephson junctions, no Fraunhofer diffraction pattern has been observed, due to disorder induced topological defects, so-called merons. We consider interlayer tunneling in the presence of microwave radiation, and predict Shapiro steps in the tunneling current-voltage characteristic despite the presence of merons. Moreover, the Josephson oscillations can also be observed as resonant features in the microwave dynamical conductance.

Dephasing by a zero-temperature detector and the Friedel sum rule.

Detecting the passage of an interfering particle through one of the interferometer's arms, known as "which path" measurement, gives rise to interference visibility degradation (dephasing). Here, we consider a detector at equilibrium. At finite temperature, dephasing is caused by thermal fluctuations of the detector. More interestingly, in the zero-temperature limit, equilibrium quantum fluctuations of the detector give rise to dephasing of the out-of-equilibrium interferometer. This dephasing is a manifestation of an orthogonality catastrophe, which differs qualitatively from Anderson's. Its magnitude is directly related to the Friedel sum rule.