Numerical results suggest that the quantum Hall effect at ν = 5/2 is described by the Pfaffian or anti-Pfaffian state in the absence of disorder and Landau level mixing. Those states are incompatible with the observed transport properties of GaAs heterostructures, where disorder and Landau level mixing are strong. We show that the recent proposal of a PH-Pfaffian topological order by Son is consistent with all experiments. The absence of the particle-hole symmetry at ν = 5/2 is not an obstacle to the existence of the PH-Pfaffian order since the order is robust to symmetry breaking.One of the most interesting features of topological insulators and superconductors is their surface behavior. A great variety of gapless and topologically-ordered gapped surface states have been proposed [1]. Such states are anomalous, that is, they can only exist on the surface of a 3D bulk system and not in a stand-alone film. Finding experimental realizations of exotic surface states proved difficult and most of them have remained theoretical proposals. Thus, it came as a surprise when Son [2] argued that one such exotic state [3], made of Dirac composite fermions with the particle-hole symmetry (PHS), has long been observed experimentally in a two-dimensional system: the electron gas in the quantum Hall effect (QHE) with the filling factor ν = 1/2.At first sight, Son's idea violates the fermion doubling theorem [4]. However, the theorem does not apply to interacting systems such as the one Son considered. Besides, in contrast to the conditions of the doubling theorem, the action of PHS is nonlocal in QHE since it involves filling a Landau level. Interestingly, the picture of composite Dirac fermions sheds light on the geometrical resonance experiments [5] which the classic theory [6] of the 1/2 state could not explain. In this paper we show that a closely related idea [2] provides a natural explanation of the observed phenomenology on the enigmatic QHE plateau at ν = 5/2 in GaAs.Cooper pairing of Dirac composite fermions in the s channel results in a fractional QHE state dubbed PHPfaffian [2,7], where "PH" stands for "particle-hole". We argue that the PH-Pfaffian topological order is present on the QHE plateau at ν = 5/2. This might seem unlikely because the particle-hole symmetry is violated by the Landau level mixing (LLM) in the observed states of the second Landau level in GaAs. Besides, numerics [11][12][13][14][15][16] supports the Pfaffian [8] and anti-Pfaffian [9,10] states at ν = 5/2 even in the presence of PHS. At the same time, the existing numerical work, which always neglects strong disorder and typically neglects strong LLM, is not yet in the position to explain the physics of the 5/2 state, as is evidenced by a large discrepancy between numerical and experimental energy gaps [17]. Note that a recent attempt to incorporate LLM [16] into simulations led to the manifestly wrong conclusion that the 5/2 state does not exist at realistic LLM. Indeed, as discussed in Ref. [16], the existing perturbative methods are justifi...

Because of the bulk gap, low energy physics in the quantum Hall effect is confined to the edges of the 2D electron liquid. The velocities of edge modes are key parameters of edge physics. They were determined in several quantum Hall systems from time-resolved measurements and high-frequency ac transport. We propose a way to extract edge velocities from dc transport in a point contact geometry defined by narrow gates. The width of the gates assumes two different sizes at small and large distances from the point contact. The Coulomb interaction across the gates depends on the gate width and affects the conductance of the contact. The conductance exhibits two different temperature dependencies at high and low temperatures. The transition between the two regimes is determined by the edge velocity. An interesting feature of the low-temperature I−V curve is current oscillations as a function of the voltage. The oscillations emerge due to charge reflection from the interface of the regions defined by the narrow and wide sections of the gates.

We present egglog, a fixpoint reasoning system that unifies Datalog and equality saturation (EqSat). Like Datalog, egglog supports efficient incremental execution, cooperating analyses, and lattice-based reasoning. Like EqSat, egglog supports term rewriting, efficient congruence closure, and extraction of optimized terms. We identify two recent applications -- a unification-based pointer analysis in Datalog and an EqSat-based floating-point term rewriter -- that have been hampered by features missing from Datalog but found in EqSat or vice-versa. We evaluate our system by reimplementing those projects in egglog. The resulting systems in egglog are faster, simpler, and fix bugs found in the original systems.

Formal Methods for the Informal Engineer (FMIE) was a workshop held at the Broad Institute of MIT and Harvard in 2021 to explore the potential role of verified software in the biomedical software ecosystem. The motivation for organizing FMIE was the recognition that the life sciences and medicine are undergoing a transition from being passive consumers of software and AI/ML technologies to fundamental drivers of new platforms, including those which will need to be mission and safety-critical. Drawing on conversations leading up to and during the workshop, we make five concrete recommendations to help software leaders organically incorporate tools, techniques, and perspectives from formal methods into their project planning and development trajectories.