Quantum Oscillations in the Zeroth Landau Level: Serpentine Landau Fan and the Chiral Anomaly.

We identify an unusual mechanism for quantum oscillations in nodal semimetals, driven by a single pair of Landau levels periodically closing their gap at the Fermi energy as a magnetic field is varied. These "zero Landau level" quantum oscillations (ZQOs) appear in the nodal limit where the zero-field Fermi volume vanishes and have distinctive periodicity and temperature dependence. We link the Landau spectrum of a two-dimensional (2D) nodal semimetal to the Rabi model, and show by exact solution that, across the entire Landau fan, pairs of opposite-parity Landau levels are intertwined in a "serpentine" manner. We propose 2D surfaces of topological crystalline insulators as natural settings for ZQOs. In certain 3D nodal semimetals, ZQOs lead to oscillations of anomaly physics. We propose a transport measurement capable of observing such oscillations, which we demonstrate numerically.

Recursive contact tracing in Reed-Frost epidemic models.

We introduce a Reed-Frost epidemic model with recursive contact tracing and asymptomatic transmission. This generalizes the branching-process model introduced by the authors in a previous work (Bulchandani et al 2021Phys. Biol.18045004) to finite populations and general contact networks. We simulate the model numerically for two representative examples, the complete graph and the square lattice. On both networks, we observe clear signatures of a contact-tracing phase transition from an 'epidemic phase' to an 'immune phase' as contact-network coverage is increased. We verify that away from the singular line of perfect tracing, the finite-size scaling of the contact-tracing phase transition on each network lies in the corresponding percolation universality class. Finally, we use the model to quantify the efficacy of recursive contact-tracing in regimes where epidemic spread is not contained.

Ergodic and nonergodic many-body dynamics in strongly nonlinear lattices.

The study of nonlinear oscillator chains in classical many-body dynamics has a storied history going back to the seminal work of Fermi et al. [Los Alamos Scientific Laboratory Report No. LA-1940, 1955 (unpublished)]. We introduce a family of such systems which consist of chains of N harmonically coupled particles with the nonlinearity introduced by confining the motion of each individual particle to a box or stadium with hard walls. The stadia are arranged on a one-dimensional lattice but they individually do not have to be one dimensional, thus permitting the introduction of chaos already at the lattice scale. For the most part we study the case where the motion is entirely one dimensional. We find that the system exhibits a mixed phase space for any finite value of N. Computations of Lyapunov spectra at randomly picked phase space locations and a direct comparison between Hamiltonian evolution and phase space averages indicate that the regular regions of phase space are not significant at large system sizes. While the continuum limit of our model is itself a singular limit of the integrable sinh Gordon theory, we do not see any evidence for the kind of nonergodicity famously seen in the work of Fermi et al. Finally, we examine the chain with particles confined to two-dimensional stadia where the individual stadium is already chaotic and find a much more chaotic phase space at small system sizes.

Digital herd immunity and COVID-19.

A population can be immune to epidemics even if not all of its individual members are immune to the disease, so long as sufficiently many are immune-this is the traditional notion of herd immunity. In the smartphone era a population can be immune to epidemicseven if not a single one of its members is immune to the disease-a notion we call 'digital herd immunity', which is similarly an emergent characteristic of the population. This immunity arises because contact-tracing protocols based on smartphone capabilities can lead to highly efficient quarantining of infected population members and thus the extinguishing of nascent epidemics. When the disease characteristics are favorable and smartphone usage is high enough, the population is in this immune phase. As usage decreases there is a novel 'contact-tracing phase transition' to an epidemic phase. We present and study a simple branching-process model for COVID-19 and show that digital immunity is possible regardless of the proportion of non-symptomatic transmission.

Distinct critical behaviors from the same state in quantum spin and population dynamics perspectives.

There is a deep connection between the ground states of transverse-field spin systems and the late-time distributions of evolving viral populations-within simple models, both are obtained from the principal eigenvector of the same matrix. However, that vector is the wave-function amplitude in the quantum spin model, whereas it is the probability itself in the population model. We show that this seemingly minor difference has significant consequences: Phase transitions that are discontinuous in the spin system become continuous when viewed through the population perspective, and transitions that are continuous become governed by new critical exponents. We introduce a more general class of models that encompasses both cases and that can be solved exactly in a mean-field limit. Numerical results are also presented for a number of one-dimensional chains with power-law interactions. We see that well-worn spin models of quantum statistical mechanics can contain unexpected new physics and insights when treated as population-dynamical models and beyond, motivating further studies.

Interacting multi-channel topological boundary modes in a quantum Hall valley system.

Symmetry and topology are central to understanding quantum Hall ferromagnets (QHFMs), two-dimensional electronic phases with spontaneously broken spin or pseudospin symmetry whose wavefunctions also have topological properties1,2. Domain walls between distinct broken-symmetry QHFM phases are predicted to host gapless one-dimensional modes-that is, quantum channels that emerge because of a topological change in the underlying electronic wavefunctions at such interfaces. Although various QHFMs have been identified in different materials3-8, interacting electronic modes at these domain walls have not been probed. Here we use a scanning tunnelling microscope to directly visualize the spontaneous formation of boundary modes at domain walls between QHFM phases with different valley polarization (that is, the occupation of equal-energy but quantum mechanically distinct valleys in the electronic structure) on the surface of bismuth. Spectroscopy shows that these modes occur within a topological energy gap, which closes and reopens as the valley polarization switches across the domain wall. By changing the valley flavour and the number of modes at the domain wall, we can realize different regimes in which the valley-polarized channels are either metallic or develop a spectroscopic gap. This behaviour is a consequence of Coulomb interactions constrained by the valley flavour, which determines whether electrons in the topological modes can backscatter, making these channels a unique class of interacting one-dimensional quantum wires. QHFM domain walls can be realized in different classes of two-dimensional materials, providing the opportunity to explore a rich phase space of interactions in these quantum wires.

Fast Preparation of Critical Ground States Using Superluminal Fronts.

We propose a spatiotemporal quench protocol that allows for the fast preparation of ground states of gapless models with Lorentz invariance. Assuming the system initially resides in the ground state of a corresponding massive model, we show that a superluminally moving "front" that locally quenches the mass, leaves behind it (in space) a state arbitrarily close to the ground state of the gapless model. Importantly, our protocol takes time O(L) to produce the ground state of a system of size ∼L^{d} (d spatial dimensions), while a fully adiabatic protocol requires time ∼O(L^{2}) to produce a state with exponential accuracy in L. The physics of the dynamical problem can be understood in terms of relativistic rarefaction of excitations generated by the mass front. We provide proof of concept by solving the proposed quench exactly for a system of free bosons in arbitrary dimensions, and for free fermions in d=1. We discuss the role of interactions and UV effects on the free-theory idealization, before numerically illustrating the usefulness of the approach via simulations on the quantum Heisenberg spin chain.

Obtaining highly excited eigenstates of the localized XX chain via DMRG-X.

We benchmark a variant of the recently introduced density matrix renormalization group (DMRG)-X algorithm against exact results for the localized random field XX chain. We find that the eigenstates obtained via DMRG-X exhibit a highly accurate l-bit description for system sizes much bigger than the direct, many-body, exact diagonalization in the spin variables is able to access. We take advantage of the underlying free fermion description of the XX model to accurately test the strengths and limitations of this algorithm for large system sizes. We discuss the theoretical constraints on the performance of the algorithm from the entanglement properties of the eigenstates, and its actual performance at different values of disorder. A small but significant improvement to the algorithm is also presented, which helps significantly with convergence. We find that, at high entanglement, DMRG-X shows a bias towards eigenstates with low entanglement, but can be improved with increased bond dimension. This result suggests that one must be careful when applying the algorithm for interacting many-body localized spin models near a transition.This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.

Obtaining Highly Excited Eigenstates of Many-Body Localized Hamiltonians by the Density Matrix Renormalization Group Approach.

The eigenstates of many-body localized (MBL) Hamiltonians exhibit low entanglement. We adapt the highly successful density-matrix renormalization group method, which is usually used to find modestly entangled ground states of local Hamiltonians, to find individual highly excited eigenstates of MBL Hamiltonians. The adaptation builds on the distinctive spatial structure of such eigenstates. We benchmark our method against the well-studied random field Heisenberg model in one dimension. At moderate to large disorder, the method successfully obtains excited eigenstates with high accuracy, thereby enabling a study of MBL systems at much larger system sizes than those accessible to exact-diagonalization methods.

Phase Structure of Driven Quantum Systems.

Clean and interacting periodically driven systems are believed to exhibit a single, trivial "infinite-temperature" Floquet-ergodic phase. In contrast, here we show that their disordered Floquet many-body localized counterparts can exhibit distinct ordered phases delineated by sharp transitions. Some of these are analogs of equilibrium states with broken symmetries and topological order, while others-genuinely new to the Floquet problem-are characterized by order and nontrivial periodic dynamics. We illustrate these ideas in driven spin chains with Ising symmetry.

Eigenstate thermalization and representative states on subsystems.

We consider a quantum system A∪B made up of degrees of freedom that can be partitioned into spatially disjoint regions A and B. When the full system is in a pure state in which regions A and B are entangled, the quantum mechanics of region A described without reference to its complement is traditionally assumed to require a reduced density matrix on A. While this is certainly true as an exact matter, we argue that under many interesting circumstances expectation values of typical operators anywhere inside A can be computed from a suitable pure state on A alone, with a controlled error. We use insights from quantum statistical mechanics-specifically the eigenstate thermalization hypothesis (ETH)-to argue for the existence of such "representative states."

How universal is the entanglement spectrum?

It is now commonly believed that the ground state entanglement spectrum (ES) exhibits universal features characteristic of a given phase. In this Letter, we show that this belief is false in general. Most significantly, we show that the entanglement Hamiltonian can undergo quantum phase transitions in which its ground state and low-energy spectrum exhibit singular changes, even when the physical system remains in the same phase. For broken symmetry problems, this implies that the low-energy ES and the Rényi entropies can mislead entirely, while for quantum Hall systems, the ES has much less universal content than assumed to date.

Kibble-Zurek scaling and string-net coarsening in topologically ordered systems.

We consider the non-equilibrium dynamics of topologically ordered systems driven across a continuous phase transition into proximate phases with no, or reduced, topological order. This dynamics exhibits scaling in the spirit of Kibble and Zurek but now without the presence of symmetry breaking and a local order parameter. The late stages of the process are seen to exhibit a slow, coarsening dynamics for the string-net that underlies the physics of the topological phase, a potentially interesting signature of topological order. We illustrate these phenomena in the context of particular phase transitions out of the Abelian Z2 topologically ordered phase of the toric code/Z2 gauge theory, and the non-Abelian SU(2)k ordered phases of the relevant Levin-Wen models.

Coulomb phase diagnostics as a function of temperature, interaction range, and disorder.

The emergent gauge field characteristic of the Coulomb phase of spin ice betrays its existence via pinch points in the spin structure factor S in reciprocal space which takes the form of a transverse projector P at low temperature: S(q) ~ P ~ q perpendicular(2)/q(2). We develop a theory which establishes the fate of the pinch points at low and high temperature, for hard and soft spins, for short- and long-ranged (dipolar) interactions, as well as in the presence of disorder. We find that their detailed shape can be used to read off the relative sizes of entropic and magnetic Coulomb interactions of monopoles in spin ice, and we resolve the question of why pinch points have been experimentally observed for Ho(1.7)Y(0.3)Ti(2)O(7) even at high temperature in the presence of strong disorder.

Magnetic Coulomb fields of monopoles in spin ice and their signatures in the internal field distribution.

Fractionalization-the breaking up of an apparently indivisible microscopic degree of freedom-is one of the most counterintuitive phenomena in many-body physics. Here we study its most fundamental manifestation in spin ice, the only known fractionalized magnetic compound in 3D: we directly visualize the 1/r(2) magnetic Coulomb field of monopoles that emerge as the atomic magnetic dipoles fractionalize. We analyze the internal magnetic field distribution, relevant for local experimental probes. In particular, we present new zero-field NMR measurements that exhibit excellent agreement with the calculated line shapes, noting that this experimental technique can in principle measure directly the monopole density in spin ice. The distribution of field strengths is captured by a simple analytical form that exhibits a low density of low-field sites-in apparent disagreement with reported muon spin rotation results. Counterintuitively, the density of low-field locations decreases as the local ferromagnetic correlations imposed by the ice rules weaken.

Quantum adiabatic algorithm and scaling of gaps at first-order quantum phase transitions.

Motivated by the quantum adiabatic algorithm (QAA), we consider the scaling of the Hamiltonian gap at quantum first-order transitions, generally expected to be exponentially small in the size of the system. However, we show that a quantum antiferromagnetic Ising chain in a staggered field can exhibit a first-order transition with only an algebraically small gap. In addition, we construct a simple classical translationally invariant one-dimensional Hamiltonian containing nearest-neighbor interactions only, which exhibits an exponential gap at a thermodynamic quantum first-order transition of essentially topological origin. This establishes that (i) the QAA can be successful even across first-order transitions but also that (ii) it can fail on exceedingly simple problems readily solved by inspection, or by classical annealing.

Microscopic model of quasiparticle wave packets in superfluids, superconductors, and paired Hall states.

We study the structure of Bogoliubov quasiparticles, bogolons, the fermionic excitations of paired superfluids that arise from fermion (BCS) pairing, including neutral superfluids, superconductors, and paired quantum Hall states. The naive construction of a stationary quasiparticle in which the deformation of the pair field is neglected leads to a contradiction: it carries a net electrical current even though it does not move. However, treating the pair field self-consistently resolves this problem: in a neutral superfluid, a dipolar current pattern is associated with the quasiparticle for which the total current vanishes. When Maxwell electrodynamics is included, as appropriate to a superconductor, this pattern is confined over a penetration depth. For paired quantum Hall states of composite fermions, the Maxwell term is replaced by a Chern-Simons term, which leads to a dipolar charge distribution and consequently to a dipolar current pattern.

Nature of the spin liquid state of the Hubbard model on a honeycomb lattice.

Recent numerical work [Z. Y. Meng et al., Nature (London) 464, 847 (2010)] indicates the existence of a spin liquid (SL) phase that intervenes between the antiferromagnetic and semimetallic phases of the half filled Hubbard model on a honeycomb lattice. To better understand the nature of this exotic phase, we study the quantum J(1)-J(2) spin model on the honeycomb lattice, which provides an effective description of the Mott insulating region of the Hubbard model. Employing the variational Monte Carlo approach, we analyze the phase diagram of the model. We find three phases-antiferromagnetic, an unusual Z(2) SL state, and a dimerized state with spontaneously broken rotational symmetry. We identify the Z(2) SL state as the likely candidate for the SL phase of the Hubbard model.

Weakly coupled Pfaffian as a type I quantum Hall liquid.

The Pfaffian phase in the proximity of a half-filled Landau level is understood to be a p+ip superconductor of composite fermions. We consider the properties of this paired quantum Hall phase when the pairing energy is small, i.e., in the weak-coupling, BCS limit, where the coherence length is much larger than the charge screening length. We find that, as in a type I superconductor, vortices attract so that, upon varying the magnetic field from its magic value at ν=5/2, the system exhibits Coulomb frustrated phase separation. We propose that the weakly and strongly coupled Pfaffians exemplify a general dichotomy between type I and type II quantum Hall fluids.

Thermal quenches in spin ice.

We study the diffusion-annihilation process which occurs when spin ice is quenched from a high temperature paramagnetic phase deep into the spin-ice regime, where the excitations--magnetic monopoles--are sparse. We find that due to the Coulomb interaction between the monopoles, a dynamical arrest occurs, in which nonuniversal lattice-scale constraints impede the complete decay of charge fluctuations. This phenomenon is outside the reach of conventional mean-field theory for a two-component Coulomb liquid. We identify the relevant time scales for the dynamical arrest and propose an experiment for detecting monopoles and their dynamics in spin ice based on this nonequilibrium phenomenon.