Localization in the Incommensurate Systems: A Plane Wave Study via Effective Potentials

In this paper, we apply the effective potentials in the localization landscape theory (Filoche et al., 2012, Arnold et al., 2016) to study the spectral properties of the incommensurate systems. We uniquely develop a plane wave method for the effective potentials of the incommensurate systems and utilize that, the localization of the electron density can be inferred from the effective potentials. Moreover, we show that the spectrum distribution can also be obtained from the effective potential version of Weyl's law. We perform some numerical experiments on some typical incommensurate systems, showing that the effective potential provides an alternative tool for investigating the localization and spectrum distribution of the systems.Comment: 14page

Counting eigenvalues of Schr\"odinger operators using the landscape function

We prove an upper and a lower bound on the rank of the spectral projections of the Schr\"odinger operator $-\Delta + V$ in terms of the volume of the sublevel sets of an effective potential $\frac{1}{u}$. Here, $u$ is the `landscape function' of [(David, G., Filoche, M., & Mayboroda, S. (2021) Advances in Mathematics, 390, 107946)], namely a solution of $(-\Delta + V)u = 1$ in \bbR^d. We prove the result for non-negative potentials satisfying a Kato-type and a doubling condition, in all spatial dimensions, in infinite volume, and show that no coarse graining is required. Our result yields in particular a necessary and sufficient condition for discreteness of the spectrum. In the case of polynomial potentials, we prove that the spectrum is discrete if and only if no directional derivative vanishes identically.Comment: To appear in Journal of Spectral Theor

Random Network Models and Quantum Phase Transitions in Two Dimensions

An overview of the random network model invented by Chalker and Coddington, and its generalizations, is provided. After a short introduction into the physics of the Integer Quantum Hall Effect, which historically has been the motivation for introducing the network model, the percolation model for electrons in spatial dimension 2 in a strong perpendicular magnetic field and a spatially correlated random potential is described. Based on this, the network model is established, using the concepts of percolating probability amplitude and tunneling. Its localization properties and its behavior at the critical point are discussed including a short survey on the statistics of energy levels and wave function amplitudes. Magneto-transport is reviewed with emphasis on some new results on conductance distributions. Generalizations are performed by establishing equivalent Hamiltonians. In particular, the significance of mappings to the Dirac model and the two dimensional Ising model are discussed. A description of renormalization group treatments is given. The classification of two dimensional random systems according to their symmetries is outlined. This provides access to the complete set of quantum phase transitions like the thermal Hall transition and the spin quantum Hall transition in two dimension. The supersymmetric effective field theory for the critical properties of network models is formulated. The network model is extended to higher dimensions including remarks on the chiral metal phase at the surface of a multi-layer quantum Hall system.Comment: 176 pages, final version, references correcte

The landscape law for tight binding Hamiltonians

The present paper extends the landscape theory pioneered in [FM, ADFJM2, DFM] to the tight-binding Schr\"odinger operator on $\Z^d$. In particular, we establish upper and lower bounds for the integrated density of states in terms of the counting function based upon the localization landscape

Mathematical Methods in Quantum Chemistry

The field of quantum chemistry is concerned with the modelling and simulation of the behaviour of molecular systems on the basis of the fundamental equations of quantum mechanics. Since these equations exhibit an extreme case of the curse of dimensionality (the Schrödinger equation for N electrons being a partial differential equation on R3N ), the quantum-chemical simulation of even moderate-size molecules already requires highly sophisticated model-reduction, approximation, and simulation techniques. The workshop brought together selected quantum chemists and physicists, and the growing community of mathematicians working in the area, to report and discuss recent advances on topics such as coupled-cluster theory, direct approximation schemes in full configuration-interaction (FCI) theory, interacting Green’s functions, foundations and computational aspects of densityfunctional theory (DFT), low-rank tensor methods, quantum chemistry in the presence of a strong magnetic field, and multiscale coupling of quantum simulations

Strong disorder RG approach of random systems

There is a large variety of quantum and classical systems in which the quenched disorder plays a dominant r\^ole over quantum, thermal, or stochastic fluctuations : these systems display strong spatial heterogeneities, and many averaged observables are actually governed by rare regions. A unifying approach to treat the dynamical and/or static singularities of these systems has emerged recently, following the pioneering RG idea by Ma and Dasgupta and the detailed analysis by Fisher who showed that the Ma-Dasgupta RG rules yield asymptotic exact results if the broadness of the disorder grows indefinitely at large scales. Here we report these new developments by starting with an introduction of the main ingredients of the strong disorder RG method. We describe the basic properties of infinite disorder fixed points, which are realized at critical points, and of strong disorder fixed points, which control the singular behaviors in the Griffiths-phases. We then review in detail applications of the RG method to various disordered models, either (i) quantum models, such as random spin chains, ladders and higher dimensional spin systems, or (ii) classical models, such as diffusion in a random potential, equilibrium at low temperature and coarsening dynamics of classical random spin chains, trap models, delocalization transition of a random polymer from an interface, driven lattice gases and reaction diffusion models in the presence of quenched disorder. For several one-dimensional systems, the Ma-Dasgupta RG rules yields very detailed analytical results, whereas for other, mainly higher dimensional problems, the RG rules have to be implemented numerically. If available, the strong disorder RG results are compared with another, exact or numerical calculations.Comment: review article, 195 pages, 36 figures; final version to be published in Physics Report

Topological phases, non-equilibrium dynamics and parallels of black hole phenomena in condensed matter

This dissertation deals with two broad topics - Majorana modes in Kitaev chain and parallels of black hole phenomena in the quantum Hall effect. Majorana modes in topological superconductors are of fundamental importance as realizations of real solutions to the Dirac equation and for their anyonic exchange statistics. They are realised as zero energy edge modes in one-dimensional topological superconductors, modeled by the Kitaev chain Hamiltonian. Here an extensive study is made on the wavefunction features of these Majorana modes. It is shown that the Majorana wavefunction has two distinct features- a decaying envelope and underlying oscillations. The latter becomes important when one considers the coupling between the Majorana modes in a finite-sized chain. The coupled Majorana modes form a non-local Dirac fermionic state which determines the ground state fermion parity. The dependance of the fermion parity on the parameters of the system is purely determined by the oscillatory part of the Majorana wavefunctions. Using transfer matrix method, one can uncover a new boundary in the phase diagram, termed as `circle of oscillations', across which the oscillations in the wavefunction and the ground-state fermionic parity cease to exist. This is closely related to the circle that appears in the context of transverse field XY spin chain, within which the spin-spin correlations have oscillations. For a finite sized system, the circle is further split into mutliple ellipses called `parity sectors'. The parity oscillations have a scaling behaviour i.e oscillations for different superconducting gaps can be scaled to collapse to a single plot. Making use of results from random matrix theory for class D systems, one can also predict the robustness of certain features of fermion parity switches in the presence of disorder and comment on the critical properties of the MBS wavefunctions and level crossings near zero energy. These results could provide directions for making measurements on zero-bias conductance oscillations and the parameter range of operations for robust parity switches in realistic disordered system. On the front of non-equilibrium dynamics, the effect of Majorana modes on the dynamical evolution of the ground state under time variation of a Hamiltonian parameter is studied. The key result is the failure of the ground state to evolve into opposite parity sectors under the dynamical tuning of the system within the topological phase. This dramatic lack of adiabaticity is termed as ‘parity blocking’. A real-space time-dependent formalism is also developed using Pfaffian correlations, where simple momentum space methods fail. This formalism can be used for calculating the non-equilibrium quantities, such as adiabatic fidelity and the residual energy in a system with open boundaries. The consideration of Majorana modes in non-equilibrium dynamics lead to deviation from Kibble-Zurek physics and non-analyticities in adiabatic fidelity even within the topological phase. The second part of the thesis deals with uncovering structural parallels of black hole phenomena such as the Hawking-Unruh effect and quasinormal modes in quantum Hall systems. The Hawking-Unruh effect is the emergence of a thermal state when a vacuum of a quantum field theory on a given spacetime is restricted to a submanifold bounded by an event horizon. The thermal state manifests as Hawking radiation in the context of a black hole spacetime with an event horizon. The Unruh effect is a simpler example where a family of accelerating observers in Minkowski spacetime are confined by the lightcone structure and the Minkowski vacuum looks as a thermal state to them. The key element in understanding the Hawking-Unruh effect is the Rindler Hamiltonian or the boost. The boost acts as the generator of time translation for the quantum states in the Rindler wedge giving rise to thermality. In this thesis it is shown that due to an exact isomorphism between the Lorentz algebra in Minkowksi spacetime and the algebra of area preserving transformations in the lowest Landau level of quantum hall effect, an applied saddle potential acts as an equivalent to the Rindler Hamiltonian giving rise to a parallel of Hawking-Unurh effect. In the lowest Landau level, the saddle potential is reduced to the problem of scattering off an inverted harmonic oscillator(IHO) and the tunneling probability assumes the form of a thermal distribution. The IHO also has scattering resonances which are poles of the scattering matrix in the complex energy plane. The scattering resonances are states with time-decaying behavior and have purely incoming/outgoing probability current. These states are identified as quasinormal modes analogous to those occurring the scattering off an effective potential in black hole spacetimes. The quasinormal decay is an unexplored effect in quantum Hall systems and provides a new class of time-dependent probe of quantum Hall physics. The parallels between the relativistic symmetry generators and the potentials applied in the lowest Landau level also open up an avenue for studying Lorentz Kinematics and symplectic phase space dynamics in the lowest Landau level. These parallels open up new avenues of exploration in the quantum Hall effect