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Register a new userNumerical Study of the Localization-Delocalization Transition for Vibrations in Amorphous Silicon
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Numerical studies of amorphous silicon in harmonic approximation show that the highest 3.5% of vibrational normal modes are localized. As vibrational frequency increases through the boundary separating localized from delocalized modes, near omega_c=70meV, (the mobility edge) there is a localization-delocalization (LD) transition, similar to a second-order thermodynamic phase transition. By a numerical study on a system with 4096 atoms, we are able to see exponential decay lengths of exact vibrational eigenstates, and test whether or not these diverge at omega_c. Results are consistent with a localization length xi which diverges above omega_c as (omega-omega_c)^{-p} where the exponent is p = 1.3 +/- 0.5. Below the mobility edge we find no evidence for a diverging correlation length. Such an asymmetry would contradict scaling ideas, and we suppose it is a finite-size artifact. If the scaling regime is narrower than our 1 meV resolution, then it cannot be seen directly on our finite system.
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We show that harmonic vibrations in amorphous silicon can be decomposed to transverse and longitudinal components in all frequency range even in the absence of the well defined wave vector ${bf q}$. For this purpose we define the transverse component of the eigenvector with given $omega$ as a component, which does not change the volumes of Voronoi cells around atoms. The longitudinal component is the remaining orthogonal component. We have found the longitudinal and transverse components of the vibrational density of states for numerical model of amorphous silicon. The vibrations are mostly transverse below 7 THz and above 15 THz. In the frequency interval in between the vibrations have a longitudinal nature. Just this sudden transformation of vibrations at 7 THz from almost transverse to almost longitudinal ones explains the prominent peak in the diffusivity of the amorphous silicon just above 7 THz.
We use a transfer-matrix method to study the localization properties of vibrations in a `mass and spring model with simple cubic lattice structure. Disorder is applied as a box-distribution to the force-constants $k$ of the springs. We obtain the reduced localization lengths $Lambda_M$ from calculated Lyapunov exponents for different system widths to roughly locate the squared critical transition frequency $omega_{text{c}}^2$. The data is finite-size scaled to acquire the squared critical transition frequency of $omega_{text{c}}^2 = 12.54 pm 0.03$ and a critical exponent of $ u = 1.55 pm 0.002$.
We propose a new approach to probing ergodicity and its breakdown in quantum many-body systems based on their response to a local perturbation. We study the distribution of matrix elements of a local operator between the systems eigenstates, finding a qualitatively different behaviour in the many-body localized (MBL) and ergodic phases. To characterize how strongly a local perturbation modifies the eigenstates, we introduce the parameter ${cal G}(L)=langle ln (V_{nm}/delta) rangle$, which represents a disorder-averaged ratio of a typical matrix element of a local operator $V$ to the energy level spacing, $delta$; this parameter is reminiscent of the Thouless conductance in the single-particle localization. We show that the parameter ${cal G}(L)$ decreases with system size $L$ in the MBL phase, and grows in the ergodic phase. We surmise that the delocalization transition occurs when ${cal G}(L)$ is independent of system size, ${cal G}(L)={cal G}_csim 1$. We illustrate our approach by studying the many-body localization transition and resolving the many-body mobility edge in a disordered 1D XXZ spin-1/2 chain using exact diagonalization and time-evolving block decimation methods. Our criterion for the MBL transition gives insights into microscopic details of transition. Its direct physical consequences, in particular logarithmically slow transport at the transition, and extensive entanglement entropy of the eigenstates, are consistent with recent renormalization group predictions.
In $e$-beam evaporated amorphous silicon ($a$-Si), the densities of two-level systems (TLS), $n_{0}$ and $overline{P}$, determined from specific heat $C$ and internal friction $Q^{-1}$ measurements, respectively, have been shown to vary by over three orders of magnitude. Here we show that $n_{0}$ and $overline{P}$ are proportional to each other with a constant of proportionality that is consistent with the measurement time dependence proposed by Black and Halperin and does not require the introduction of additional anomalous TLS. However, $n_{0}$ and $overline{P}$ depend strongly on the atomic density of the film ($n_{rm Si}$) which depends on both film thickness and growth temperature suggesting that the $a$-Si structure is heterogeneous with nanovoids or other lower density regions forming in a dense amorphous network. A review of literature data shows that this atomic density dependence is not unique to $a$-Si. These findings suggest that TLS are not intrinsic to an amorphous network but require a heterogeneous structure to form.
We analyze many-body localization (MBL) to delocalization transition in Sherrington-Kirkpatrick (SK) model of Ising spin glass (SG) in the presence of a transverse field $Gamma$. Based on energy resolved analysis, which is of relevance for a closed quantum system, we show that the quantum SK model has many-body mobility edges separating MBL phase which is non-ergodic and non-thermal from the delocalized phase which is ergodic and thermal. The range of the delocalized regime increases with increase in the strength of $Gamma$ and eventually for $Gamma$ larger than $Gamma_{CP}$ the entire many-body spectrum is delocalized. We show that the Renyi entropy is almost independent of the system size in the MBL phase, hinting towards an area law in this infinite range model while the delocalized phase shows volume law scaling of Renyi entropy. We further obtain spin glass transition curve in energy density $epsilon$-$Gamma$ plane from the collapse of eigenstate spin susceptibility. We demonstrate that in most of the parameter regime SG transition occurs close to the MBL transition indicating that the SG phase is non-ergodic and non-thermal while the paramagnetic phase is delocalized and thermal.
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