We investigate localization properties in a family of deterministic (i.e. no disorder) nearest neighbor tight binding models with quasiperiodic onsite modulation. We prove that this family is self-dual under a generalized duality transformation. The self-dual condition for this general model turns out to be a simple closed form function of the model parameters and energy. We introduce the typical density of states as an order parameter for localization in quasiperiodic systems. By direct calculations of the inverse participation ratio and the typical density of states we numerically verify that this self-dual line indeed defines a mobility edge in energy separating localized and extended states. Our model is a first example of a nearest neighbor tight binding model manifesting a mobility edge protected by a duality symmetry. We propose a realistic experimental scheme to realize our results in atomic optical lattices and photonic waveguides.Anderson localization [1] is a universal and extensively studied property of a quantum particle or a wave in a disordered medium. An interesting consequence of Anderson localization is a quantum phase transition between extended and localized states as a function of the disorder strength. In three dimensional systems with random (uncorrelated) disorder, a localization transition occurs as the strength of disorder crosses a critical value forming a sharp energy dependent mobility edge at the phase boundary separating localized/extended states below/above the mobility edge. Scaling theory [2] has shown the absence of this critical behavior in one and two dimensions where all states, at least in the absence of interaction, are known to be localized in a disordered system, pushing the mobility edge effectively to infinite energy. However, in one dimension this picture changes for a quasiperiodic system with two incommensurate (but deterministic) lattice potentials, which in a loose qualitative sense might be construed to be a highly correlated disorder, albeit perfectly well-defined with no randomness whatsoever. Aubry and Andre [3] showed that a 1D tight binding model with an onsite cosine modulation incommensurate with the underlying lattice has a self-dual symmetry and manifests an energy independent localization transition as a function of the modulation strength, i.e., all states are either localized or extended depending on the relative strength of the incommensurate modulation potential with respect to the lattice potential. [21] to show the existence of a many body localization transition in the quasiperiodic AAH model in the presence of weak interactions. The duality-driven and energy independent localization transition in the AAH model does not manifest a mobility edge, which is a hallmark of the disorder tuned localization transition in 3D. The 1D localization transition in the AAH model, defined by the self-duality point, thus has no analog in disorder-driven Anderson localization, and does not give any insight into the physics of 3D mobility edges. Recent works have ...

We numerically study the measurement-driven quantum phase transition of Haar-random quantum circuits in 1 + 1 dimensions. By analyzing the tripartite mutual information we are able to make a precise estimate of the critical measurement rate pc = 0.17(1). We extract estimates for the associated bulk critical exponents that are consistent with the values for percolation, as well as those for stabilizer circuits, but differ from previous estimates for the Haar-random case. Our estimates of the surface order parameter exponent appear different from that for stabilizer circuits or percolation, but we are unable to definitively rule out the scenario where all exponents in the three cases match. Moreover, in the Haar case the prefactor for the entanglement entropies Sn depends strongly on the Rényi index n; for stabilizer circuits and percolation this dependence is absent. Results on stabilizer circuits are used to guide our study and identify measures with weak finite-size effects. We discuss how our numerical estimates constrain theories of the transition.Nonequilibrium quantum systems can undergo various phase transitions in their dynamics; characterizing such transitions is a key open question in modern quantum statistical physics. So far, these nonequilibrium phase transitions have been studied primarily for isolated quantum systems [1, 2] and for steady states of dissipative systems [3,4]. One much-studied case is the many-body localization transition [2], which can be seen either (i) as a dynamical transition at which thermalization slows down and stops as a parameter (e.g., the disorder strength in a spin chain) is tuned or (ii) as an entanglement transition at which the many-body eigenstates of the system change from volume-law to area-law entangled. Recently, a different type of entanglement transition was discovered [5][6][7] in the steady-state entanglement of the states produced by individual quantum trajectories [8-11] of a repeatedly-measured quantum many-body system. As the system is measured at an increasing rate, this single-trajectory entanglement goes from volume-law to area-law (see Fig. 1(a)), as has been demonstrated both numerically, and analytically in certain tractable limits [6, 7,[12][13][14][15][16][17]. This measurementdriven non-equilibrium quantum phase transition can also be interpreted as a purification transition [18] that can collapse a mixed state to a pure state through a sufficiently large rate of local projective measurements.A measurement driven transition is expected to occur for any form of quantum chaotic dynamics, e.g. in both random circuit [6, 7] and Hamiltonian [19] dynamics. Current studies have mainly focused on quantum circuits, acting on an array of qudits (of local Hilbert space dimension q); these are believed to be generic models of chaotic quantum dynamics [20][21][22][23][24][25][26]. Various choices of gates have been explored numerically [6, 7, 15]. In specific limiting cases, analytic results (or large-scale simu-lations) exist. Specifically, the transition in...

We show that the emergence of the axial anomaly is a universal phenomenon for a generic three dimensional metal in the presence of parallel electric (E) and magnetic (B) fields. In contrast to the expectations of the classical theory of magnetotransport, this intrinsically quantum mechanical phenomenon gives rise to the longitudinal magnetoresistance for any three dimensional metal. However, the emergence of the axial anomaly does not guarantee the existence of negative longitudinal magnetoresistance. We show this through an explicit calculation of the longitudinal magnetoconductivity in the quantum limit using the Boltzmann equation, for both short-range neutral and long-range ionic impurity scattering processes. We demonstrate that the ionic scattering contributes a large positive magnetoconductivity ∝ B 2 in the quantum limit, which can cause a strong negative magnetoresistance for any three dimensional or quasi-two dimensional metal. In contrast, the finite range neutral impurities and zero range point impurities can lead to both positive and negative longitudinal magnetoresistance depending on the underlying band structure. In the presence of both neutral and ionic impurities, the longitudinal magnetoresistance of a generic metal in the quantum limit initially becomes negative, and ultimately becomes positive after passing through a minimum. We discuss in detail the qualitative agreement between our theory and recent observations of negative longitudinal magnetoresistance in Weyl semimetals TaAs and TaP, Dirac semimetals Na3Bi, Bi1−xSbx, and ZrTe5, and quasi-two dimensional metals PdCoO2, α-(BEDT-TTF)2I3 which do not possess any bulk three dimensional Dirac or Weyl quasiparticles.

We study the quantum phase diagram of a three dimensional noninteracting Dirac semimetal in the presence of either quenched axial or scalar potential disorder, by calculating the average and the typical density of states as well as the inverse participation ratio using numerically exact methods. We show that as a function of the disorder strength a half-filled (i.e., undoped) Dirac semimetal displays three distinct ground states, namely an incompressible semimetal, a compressible diffusive metal, and a localized Anderson insulator, in stark contrast to a conventional dirty metal that only supports the latter two phases. We establish the existence of two distinct quantum critical points, which respectively govern the semimetal-metal and the metal-insulator quantum phase transitions and also reveal their underlying multifractal nature. Away from half-filling the (doped) system behaves as a diffusive metal that can undergo Anderson localization only, which is shown by determining the mobility edge and the phase diagram in terms of energy and disorder.

We investigate many-body localization in the presence of a single-particle mobility edge. By considering an interacting deterministic model with an incommensurate potential in one dimension we find that the single-particle mobility edge in the noninteracting system leads to a many-body mobility edge in the corresponding interacting system for certain parameter regimes. Using exact diagonalization, we probe the mobility edge via energy resolved entanglement entropy (EE) and study the energy resolved applicability (or failure) of the eigenstate thermalization hypothesis (ETH). Our numerical results indicate that the transition separating area and volume law scaling of the EE does not coincide with the non-thermal to thermal transition. Consequently, there exists an extended non-ergodic phase for an intermediate energy window where the many-body eigenstates violate the ETH while manifesting volume law EE scaling. We also establish that the model possesses an infinite temperature many-body localization transition despite the existence of a single-particle mobility edge. We propose a practical scheme to test our predictions in atomic optical lattice experiments which can directly probe the effects of the mobility edge.Thermalization, a commonplace phenomenon in various physical settings, can naturally fail in isolated disordered quantum interacting systems, making standard concepts of quantum statistical mechanics invalid. The fundamental theoretical underpinning of thermalization in quantum systems has been postulated in the form of the eigenstate thermalization hypothesis (ETH) [1,2]. Recently, it has been shown using perturbative arguments that the presence of interaction and disorder in a closed quantum system could lead to many-body localization (MBL) [3] with such an interacting quantum MBL state being non-thermal. A hallmark of MBL is its violation of the ETH [2], where a local subsystem fails to thermalize with its environment [4]. MBL has now been established nonperturbatively in lattice models with finite energy density, where numerical evidence points towards the existence of MBL all the way to infinite temperature [5,6]. Further numerical work [7][8][9] and a rigorous mathematical proof [10] for the existence of the MBL phase have mounted compelling evidence for the existence of such a 'finite-temperature' MBL phase which eventually gives way to an extended phase at strong enough interaction. Although much of the MBL work has focused on the interacting one dimensional (1d) fermionic Anderson model with random disorder [11] (and closely related spin models), it turns out that MBL also exists without any disorder [8,12,13] for the Aubry-Andre-Azbel-Harper (AAAH) model [14][15][16], which is a non-random 1d model with a quasiperiodic onsite potential. We emphasize that neither 1d Anderson model nor AAAH model manifests a single-particle mobility edge (SPME).In the absence of a SPME, interactions act on the Fock space of Slater determinants of either completely localized or delocalized single-particle eigenst...

Progress in the understanding of quantum critical properties of itinerant electrons has been hindered by the lack of effective models which are amenable to controlled analytical and numerically exact calculations. Here we establish that the disorder driven semimetal to metal quantum phase transition of three dimensional massless Dirac fermions could serve as a paradigmatic toy model for studying itinerant quantum criticality, which is solved in this work by exact numerical and approximate field theoretic calculations. As a result, we establish the robust existence of a non-Gaussian universality class, and also construct the relevant low energy effective field theory that could guide the understanding of quantum critical scaling for many strange metals. Using the kernel polynomial method (KPM), we provide numerical results for the calculated dynamical exponent (z) and correlation length exponent (ν) for the disorder-driven semimetal (SM) to diffusive metal (DM) quantum phase transition at the Dirac point for several types of disorder, establishing its universal nature and obtaining the numerical scaling functions in agreement with our field theoretical analysis.

We study the many-body localization aspects of single-particle mobility edges in fermionic systems. We investigate incommensurate lattices and random disorder Anderson models. Many-body localization and quantum nonergodic properties are studied by comparing entanglement and thermal entropy, and by calculating the scaling of subsystem particle number fluctuations, respectively. We establish a nonergodic extended phase as a generic intermediate phase (between purely ergodic extended and nonergodic localized phases) for the many-body localization transition of non-interacting fermions where the entanglement entropy manifests a volume law (hence, 'extended'), but there are large fluctuations in the subsystem particle numbers (hence, 'nonergodic'). Based on the numerical results, we expect such an intermediate phase scenario may continue to hold even for the manybody localization in the presence of interactions as well. We find for many-body fermionic states in non-interacting one dimensional Aubry-André and three dimensional Anderson models that the entanglement entropy density and the normalized particle-number fluctuation have discontinuous jumps at the localization transition where the entanglement entropy is sub-thermal but obeys the "volume law". In the vicinity of the localization transition we find that both the entanglement entropy and the particle number fluctuations obey a single parameter scaling based on the diverging localization length. We argue using numerical and theoretical results that such a critical scaling behavior should persist for the interacting many-body localization problem with important observable consequences. Our work provides persuasive evidence in favor of there being two transitions in many-body systems with single-particle mobility edges, the first one indicating a transition from the purely localized nonergodic many-body localized phase to a nonergodic extended many-body metallic phase, and the second one being a transition eventually to the usual ergodic many-body extended phase.

We numerically study the effect of short-ranged potential disorder on massless noninteracting threedimensional Dirac and Weyl fermions, with a focus on the question of the proposed (and extensively theoretically studied) quantum critical point separating semimetal and diffusive-metal phases. We determine the properties of the eigenstates of the disordered Dirac Hamiltonian (H) and exactly calculate the density of states (DOS) near zero energy, using a combination of Lanczos on H 2 and the kernel polynomial method on H. We establish the existence of two distinct types of low-energy eigenstates contributing to the disordered density of states in the weak-disorder semimetal regime. These are (i) typical eigenstates that are well described by linearly dispersing perturbatively dressed Dirac states and (ii) nonperturbative rare eigenstates that are weakly dispersive and quasilocalized in the real-space regions with the largest (and rarest) local random potential. Using twisted boundary conditions, we are able to systematically find and study these two (essentially independent) types of eigenstates. We find that the Dirac states contribute low-energy peaks in the finite-size DOS that arise from the clean eigenstates which shift and broaden in the presence of disorder. On the other hand, we establish that the rare quasilocalized eigenstates contribute a nonzero background DOS which is only weakly energy dependent near zero energy and is exponentially small at weak disorder. We also find that the expected semimetal to diffusive-metal quantum critical point is converted to an avoided quantum criticality that is "rounded out" by nonperturbative effects, with no signs of any singular behavior in the DOS at the energy of the clean Dirac point. However, the crossover effects of the avoided (or hidden) criticality manifest themselves in a so-called quantum critical fan region away from the Dirac energy. We discuss the implications of our results for disordered Dirac and Weyl semimetals, and reconcile the large body of existing numerical work showing quantum criticality with the existence of these nonperturbative effects.