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Register a new userThe almost mobility edge in the almost Mathieu equation
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Harpers equation (aka the almost Mathieu equation) famously describes the quantum dynamics of an electron on a one dimensional lattice in the presence of an incommensurate potential with magnitude $V$ and wave number $Q$. It has been proven that all states are delocalized if $V$ is less than a critical value $V_c=2t$ and localized if $V> V_c$. Here, we show that this result (while correct) is highly misleading, at least in the small $Q$ limit. In particular, for $V<V_c$ there is an abrupt crossover akin to a mobility edge at an energy $E_c$; states with energy $|E|<E_c$ are robustly delocalized, but those in the tails of the density of states, with $|E|>E_c$, form a set of narrow bands with exponentially small bandwidths $ sim t exp[-(2pialpha/Q)]$ (where $alpha$ is an energy dependent number of order 1) separated by band-gaps $ sim t Q$. Thus, the states with $|E|> E_c$ are almost localized in that they have an exponentially large effective mass and are easily localized by small perturbations. We establish this both using exact numerical solution of the problem, and by exploiting the well known fact that the same eigenvalue problem arises in the Hofstadter problem of an electron moving on a 2D lattice in the presence of a magnetic field, $B=Q/2pi$. From the 2D perspective, the almost localized states are simply the Landau levels associated with semiclassical precession around closed contours of constant quasiparticle energy; that they are not truly localized reflects an extremely subtle form of magnetic breakdown.
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We consider the spectrum of the almost Mathieu operator $H_alpha$ with frequency $alpha$ and in the case of the critical coupling. Let an irrational $alpha$ be such that $|alpha-p_n/q_n|<c q_n^{-varkappa}$, where $p_n/q_n$, $n=1,2,dots$ are the convergents to $alpha$, and $c$, $varkappa$ are positive absolute constants, $varkappa<56$. Assuming certain conditions on the parity of the coefficients of the continued fraction of $alpha$, we show that the central gaps of $H_{p_n/q_n}$, $n=1,2,dots$, are inherited as spectral gaps of $H_alpha$ of length at least $cq_n^{-varkappa/2}$, $c>0$.
Results are presented for the dynamics of an almost strong edge mode which is the quasi-stable Majorana edge mode occurring in non-integrable spin chains. The dynamics of the edge mode is studied using exact diagonalization, and compared with time-evolution with respect to an effective semi-infinite model in Krylov space obtained from the recursion method. The effective Krylov Hamiltonian is found to resemble a spatially inhomogeneous SSH model where the hopping amplitude increases linearly with distance into the bulk, typical of thermalizing systems, but also has a staggered or dimerized structure superimposed on it. The non-perturbatively long lifetime of the edge mode is shown to be due to this staggered structure which diminishes the effectiveness of the linearly growing hopping amplitude. On taking the continuum limit of the Krylov Hamiltonian, the edge mode is found to be equivalent to the quasi-stable mode of a Dirac Hamiltonian on a half line, with a mass which is non-zero over a finite distance, before terminating into a gapless metallic bulk. The analytic estimates are found to be in good agreement with the numerically obtained lifetimes of the edge mode.
In this paper, we study a one-dimensional tight-binding model with tunable incommensurate potentials. Through the analysis of the inverse participation rate, we uncover that the wave functions corresponding to the energies of the system exhibit different properties. There exists a critical energy under which the wave functions corresponding to all energies are extended. On the contrary, the wave functions corresponding to all energies above the critical energy are localized. However, we are surprised to find that the critical energy is a constant independent of the potentials. We use the self-dual relation to solve the critical energy, namely the mobility edge, and then we verify the analytical results again by analyzing the spatial distributions of the wave functions. Finally, we give a brief discussion on the possible experimental observation of the invariable mobility edge in the system of ultracold atoms in optical lattices.
Thermalization of random-field Heisenberg spin chain is probed by time evolution of density correlation functions. Studying the impacts of average energies of initial product states on dynamics of the system, we provide arguments in favor of the existence of a mobility edge in the large system-size limit.
Many interesting experimental systems, such as cavity QED or central spin models, involve global coupling to a single harmonic mode. Out-of-equilibrium, it remains unclear under what conditions localized phases survive such global coupling. We study energy-dependent localization in the disordered Ising model with transverse and longitudinal fields coupled globally to a $d$-level system (qudit). Strikingly, we discover an inverted mobility edge, where high energy states are localized while low energy states are delocalized. Our results are supported by shift-and-invert eigenstate targeting and Krylov time evolution up to $L=13$ and $18$ respectively. We argue for a critical energy of the localization phase transition which scales as $E_c propto L^{1/2}$, consistent with finite size numerics. We also show evidence for a reentrant MBL phase at even lower energies despite the presence of strong effects of the central mode in this regime. Similar results should occur in the central spin-$S$ problem at large $S$ and in certain models of cavity QED.
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