skip to main content

Title: Deconstructing effective non-Hermitian dynamics in quadratic bosonic Hamiltonians

Unlike their fermionic counterparts, the dynamics of Hermitian quadratic bosonic Hamiltonians are governed by a generally non-Hermitian Bogoliubov-de Gennes effective Hamiltonian. This underlying non-Hermiticity gives rise to adynamically stableregime, whereby all observables undergo bounded evolution in time, and adynamically unstableone, whereby evolution is unbounded for at least some observables. We show that stability-to-instability transitions may be classified in terms of a suitablygeneralizedPTsymmetry, which can be broken when diagonalizability is lost at exceptional points in parameter space, but also when degenerate real eigenvalues split off the real axis while the system remains diagonalizable. By leveraging tools from Krein stability theory in indefinite inner-product spaces, we introduce an indicator of stability phase transitions, which naturally extends the notion of phase rigidity from non-Hermitian quantum mechanics to the bosonic setting. As a paradigmatic example, we fully characterize the stability phase diagram of a bosonic analogue to the Kitaev–Majorana chain under a wide class of boundary conditions. In particular, we establish a connection between phase-dependent transport properties and the onset of instability, and argue that stable regions in parameter space become of measure zero in the thermodynamic limit. Our analysis also reveals that boundary conditions that support Majorana zero modes in the fermionic Kitaev chain are precisely the same that support stability in the bosonic chain.

more » « less
Author(s) / Creator(s):
; ;
Publisher / Repository:
IOP Publishing
Date Published:
Journal Name:
New Journal of Physics
Page Range / eLocation ID:
Article No. 083004
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    Broken symmetries in topological condensed matter systems have implications for the spectrum of Fermionic excitations confined on surfaces or topological defects. The Fermionic spectrum of confined (quasi-2D)3He-A consists of branches of chiral edge states. The negative energy states are related to the ground-state angular momentum,Lz=(N/2), forN/2Cooper pairs. The power law suppression of the angular momentum,Lz(T)(N/2)[123(πT/Δ)2]for0TTc, in the fully gapped 2D chiral A-phase reflects the thermal excitation of the chiral edge Fermions. We discuss the effects of wave function overlap, and hybridization between edge states confined near opposing edge boundaries on the edge currents, ground-state angular momentum and ground-state order parameter of superfluid3He thin films. Under strong lateral confinement, the chiral A phase undergoes a sequence of phase transitions, first to a pair density wave (PDW) phase with broken translational symmetry atDc216ξ0. The PDW phase is described by a periodic array of chiral domains with alternating chirality, separated by domain walls. The period of PDW phase diverges as the confinement lengthDDc2. The PDW phase breaks time-reversal symmetry, translation invariance, but is invariant under the combination of time-reversal and translation by a one-half period of the PDW. The mass current distribution of the PDW phase reflects this combined symmetry, and originates from the spectra of edge Fermions and the chiral branches bound to the domain walls. Under sufficiently strong confinement a second-order transition occurs to the non-chiral ‘polar phase’ atDc19ξ0, in which a single p-wave orbital state of Cooper pairs is aligned along the channel.

    more » « less
  2. Abstract

    The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study this problem in the followinga prioriunstable Hamiltonian system with a time-periodic perturbationHε(p,q,I,φ,t)=h(I)+i=1n±12pi2+Vi(qi)+εH1(p,q,I,φ,t),where(p,q)Rn×Tn,(I,φ)Rd×Tdwithn,d⩾ 1,Viare Morse potentials, andɛis a small non-zero parameter. The unperturbed Hamiltonian is not necessarily convex, and the induced inner dynamics does not need to satisfy a twist condition. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbationsH1. Indeed, the set of admissibleH1isCωdense andC3open (a fortiori,Cωopen). Our perturbative technique for the genericity is valid in theCktopology for allk∈ [3, ∞) ∪ {∞,ω}.

    more » « less
  3. In this paper, we consider the linear convection-diffusion equation in one dimension with periodic boundary conditions, and analyze the stability of fully discrete methods that are defined with local discontinuous Galerkin (LDG) methods in space and several implicit-explicit (IMEX) Runge-Kutta methods in time. By using the forward temporal differences and backward temporal differences, respectively, we establish two general frameworks of the energy-method based stability analysis. From here, the fully discrete schemes being considered are shown to have monotonicity stability, i.e. theL2L^2norm of the numerical solution does not increase in time, under the time step conditionτ<#comment/>≤<#comment/>F(h/c,d/c2)\tau \le \mathcal {F}(h/c, d/c^2), with the convection coefficientcc, the diffusion coefficientdd, and the mesh sizehh. The functionF\mathcal {F}depends on the specific IMEX temporal method, the polynomial degreekkof the discrete space, and the mesh regularity parameter. Moreover, the time step condition becomesτ<#comment/>≲<#comment/>h/c\tau \lesssim h/cin the convection-dominated regime and it becomesτ<#comment/>≲<#comment/>d/c2\tau \lesssim d/c^2in the diffusion-dominated regime. The result is improved for a first order IMEX-LDG method. To complement the theoretical analysis, numerical experiments are further carried out, leading to slightly stricter time step conditions that can be used by practitioners. Uniform stability with respect to the strength of the convection and diffusion effects can especially be relevant to guide the choice of time step sizes in practice, e.g. when the convection-diffusion equations are convection-dominated in some sub-regions.

    more » « less
  4. Abstract

    CeOs4Sb12, a member of the skutterudite family, has an unusual semimetallic low-temperatureL-phase that inhabits a wedge-like area of the fieldH—temperatureTphase diagram. We have conducted measurements of electrical transport and megahertz conductivity on CeOs4Sb12single crystals under pressures of up to 3 GPa and in high magnetic fields of up to 41 T to investigate the influence of pressure on the differentHTphase boundaries. While the high-temperature valence transition between the metallicH-phase and theL-phase is shifted to higherTby pressures of the order of 1 GPa, we observed only a marginal suppression of theS-phase that is found below 1 K for pressures of up to 1.91 GPa. High-field quantum oscillations have been observed for pressures up to 3.0 GPa and the Fermi surface of the high-field side of theH-phase is found to show a surprising decrease in size with increasing pressure, implying a change in electronic structure rather than a mere contraction of lattice parameters. We evaluate the field-dependence of the effective masses for different pressures and also reflect on the sample dependence of some of the properties of CeOs4Sb12which appears to be limited to the low-field region.

    more » « less
  5. Abstract

    Chiral and helical Majorana fermions are two archetypal edge excitations in two-dimensional topological superconductors. They emerge from systems of different Altland–Zirnbauer symmetries and characterized byZandZ2topological invariants respectively. It seems improbable to tune a pair of co-propagating chiral edge modes to counter-propagate in a single system without symmetry breaking. Here, we explore the peculiar behaviors of Majorana edge modes in topological superconductors with an additional ‘mirror’ symmetry which changes the bulk topological invariant toZZtype. A theoretical toy model describing the proximity structure of a Chern insulator and apx-wave superconductor is proposed and solved analytically to illustrate a direct transition between two topologically nontrivial phases. The weak pairing phase has two chiral Majorana edge modes, while the strong pairing phase is characterized by mirror-graded Chern number and hosts a pair of counter-propagating Majorana fermions protected by the mirror symmetry. The edge theory is worked out in detail, and implications to braiding of Majorana fermions are discussed.

    more » « less