Forward-scattering Approximation Research Articles

Quantum state complexity meets many-body scars

Scar eigenstates in a many-body system refers to a small subset of non-thermal finite energy density eigenstates embedded into an otherwise thermal spectrum. This novel non-thermal behaviour has been seen in recent experiments simulating a one-dimensional PXP model with a kinetically-constrained local Hilbert space realised by a chain of Rydberg atoms. We probe these small sets of special eigenstates starting from particular initial states by computing the spread complexity associated to time evolution of the PXP hamiltonian. Since the scar subspace in this model is embedded only loosely, the scar states form a weakly broken representation of the Lie algebra. We demonstrate why a careful usage of the forward scattering approximation (FSA), instead of any other method, is required to extract the most appropriate set of Lanczos coefficients in this case as the consequence of this approximate symmetry. Only such a method leads to a well defined notion of a closed Krylov subspace and consequently, that of spread complexity. We show this using three separate initial states, namely |Z2⟩,|Z3⟩ and the vacuum state, due to the disparate classes of scar states hosted by these sectors. We also discuss systematic methods of remedying the imperfections in the FSA setup stemming from these approximate symmetries.

Periodically driven Rydberg chains with staggered detuning

We study the stroboscopic dynamics of a periodically driven finite Rydberg chain with staggered ($\Delta$) and time-dependent uniform ($\lambda(t)$) detuning terms using exact diagonalization (ED). We show that at intermediate drive frequencies ($\omega_D$), the presence of a finite $\Delta$ results in violation of the eigenstate thermalization hypothesis (ETH) via clustering of Floquet eigenstates. Such clustering is lost at special commensurate drive frequencies for which $\hbar \omega_d=n \Delta$ ($n \in Z$) leading to restoration of ergodicity. The violation of ETH in these driven finite-sized chains is also evident from the dynamical freezing displayed by the density-density correlation function at specific $\omega_D$. Such a correlator exhibits stable oscillations with perfect revivals when driven close to the freezing frequencies for initial all spin-down ($|0\rangle$) or Neel ($|{\mathbb Z}_2\rangle$, with up-spins on even sites) states. The amplitudes of these oscillations vanish at the freezing frequencies and reduces upon increasing $\Delta$; their frequencies, however, remains pinned to $\Delta/\hbar$ in the large $\Delta$ limit. In contrast, for the $|{\bar {\mathbb Z}_2}\rangle$ (time-reversed partner of $|{\mathbb Z}_2\rangle$) initial state, we find complete absence of such oscillations leading to freezing for a range of $\omega_D$; this range increases with $\Delta$. We also study the properties of quantum many-body scars in the Floquet spectrum of the model as a function of $\Delta$ and show the existence of novel mid-spectrum scars at large $\Delta$. We supplement our numerical results with those from an analytic Floquet Hamiltonian computed using Floquet perturbation theory (FPT) and also provide a semi-analytic computation of the quantum scar states within a forward scattering approximation (FSA).

Statistics of Green's functions on a disordered Cayley tree and the validity of forward scattering approximation

The accuracy of the forward scattering approximation for two-point Green's functions of the Anderson localization model on the Cayley tree is studied. A relationship between the moments of the Green's function and the largest eigenvalue of the linearized transfer-matrix equation is proved in the framework of the supersymmetric functional-integral method. The new large-disorder approximation for this eigenvalue is derived and its accuracy is established. Using this approximation the probability distribution of the two-point Green's function is found and compared with that in the forward scattering approximation (FSA). It is shown that FSA overestimates the role of resonances and thus the probability for the Green's function to be significantly larger than its typical value. The error of FSA increases with increasing the distance between points in a two-point Green's function.

Constraint-induced breaking and restoration of ergodicity in spin-1 PXP models

Eigenstate Thermalization Hypothesis(ETH) has played a pivotal role in understanding ergodicity and its breaking in isolated quantum many-body systems. Recent experiment on 51-atom Rydberg quantum simulator and subsequent theoretical analysis have shown that hardcore kinetic constraint can lead to weak ergodicity breaking. In this work, we demonstrate, using 1d spin-1 PXP chains, that miscellaneous type of ergodicity can be realized by adjusting the hardcore constraints between different components of nearest neighbor spins. This includes ETH violation due to emergent shattering of Hilbert space into exponentially many subsectors of various sizes, a novel form of non-integrability with an extensive number of local conserved quantities and strong ergodicity. We analyze these different forms of ergodicity and study their impact on the non-equilibrium dynamics of a Z2 initial state. We use forward scattering approximation (FSA) to understand the amount of Z2-oscillation present in these models. Our work shows that not only ergodicity breaking but an appropriate choice of constraints can lead to restoration of ergodicity as well.

How periodic driving stabilizes and destabilizes Anderson localization on random trees

Motivated by the link between Anderson localisation on high-dimensional graphs and many-body localisation, we study the effect of periodic driving on Anderson localisation on random trees. The time dependence is eliminated in favour of an extra dimension, resulting in an extended graph wherein the disorder is correlated along the new dimension. The extra dimension increases the number of paths between any two sites and allows for interference between their amplitudes. We study the localisation problem within the forward scattering approximation (FSA) which we adapt to this extended graph. At low frequency, this favours delocalisation as the availability of a large number of extra paths dominates. By contrast, at high frequency, it stabilises localisation compared to the static system. These lead to a regime of re-entrant localisation in the phase diagram. Analysing the statistics of path amplitudes within the FSA, we provide a detailed theoretical picture of the physical mechanisms governing the phase diagram.

Out-of-equilibrium phase diagram of the quantum random energy model

In this paper we study the out-of-equilibrium phase diagram of the quantum version of Derrida's Random Energy Model, which is the simplest model of mean-field spin glasses. We interpret its corresponding quantum dynamics in Fock space as a one-particle problem in very high dimension to which we apply different theoretical methods tailored for high-dimensional lattices: the Forward-Scattering Approximation, a mapping to the Rosenzweig-Porter model, and the cavity method. Our results indicate the existence of two transition lines and three distinct dynamical phases: a completely many-body localized phase at low energy, a fully ergodic phase at high energy, and a multifractal "bad metal" phase at intermediate energy. In the latter, eigenfunctions occupy a diverging volume, yet an exponentially vanishing fraction of the total Hilbert space. We discuss the limitations of our approximations and the relationship with previous studies.

Strong ergodicity breaking due to local constraints in a quantum system

Quantum systems that violate the eigenstate thermalisation hypothesis thereby falling outside the paradigm of conventional statistical mechanics are of both intellectual and practical interest. We show that such a breaking of ergodicity may arise purely due to local constraints on random many-body Hamiltonians. As an example, we study an ergodic quantum spin-1/2 model which acquires a localised phase upon addition of East-type constraints. We establish its phenomenology using spectral and dynamical properties obtained by exact diagonalisation. Mapping the Hamiltonian to a disordered hopping problem on the Fock space graph we find that potentially non-resonant bottlenecks in the Fock-space dynamics, caused by spatially local segments of frozen spins, lie at the root of localisation. We support this picture by introducing and solving numerically a class of random matrix models that retain the bottlenecks. Finally, we obtain analytical insight into the origins of localisation using the forward-scattering approximation. A numerical treatment of the forward-scattering approximation yields critical points which agree quantitatively with the exact diagonalisation results.

On the role of the second-order derivative term in the calculation of convergent beam diffraction patterns

The simulation of (scanning) transmission electron microscopy images and diffraction patterns is most often performed using the forward-scattering approximation where the second-order derivative term in z is assumed to be small with respect to the first-order derivative term in the modified Schrödinger equation. This assumption is very good at high incident electron energies, but breaks down at low energies. In order to study the differences between first- and second-order methods, convergent beam electron diffraction patterns were simulated for silicon at the [111] zone-axis orientation at 20 keV and compared using electron intensity difference maps and integrated intensity profiles. The geometrical differences in the calculated diffraction patterns could be explained by an Ewald surface analysis. Furthermore, it was found that solutions based on the second-order derivative equation contained small amplitude oscillations that need to be resolved in order to ensure numerical integration stability. This required the use of very small integration steps resulting in significantly increased computation time compared to the first-order differential equation solution. Lastly, the efficiency of the numerical integration technique is discussed.

Clustering of Nonergodic Eigenstates in Quantum Spin Glasses.

The two primary categories for eigenstate phases of matter at a finite temperature are many-body localization (MBL) and the eigenstate thermalization hypothesis (ETH). We show that, in the paradigmatic quantum p-spin models of the spin-glass theory, eigenstates violate the ETH yet are not MBL either. A mobility edge, which we locate using the forward-scattering approximation and replica techniques, separates the nonergodic phase at a small transverse field from an ergodic phase at a large transverse field. The nonergodic phase is also bounded from above in temperature, by a transition in configuration-space statistics reminiscent of the clustering transition in the spin-glass theory. We show that the nonergodic eigenstates are organized in clusters which exhibit distinct magnetization patterns, as characterized by an eigenstate variant of the Edwards-Anderson order parameter.

The many-body localized phase of the quantum random energy model

The random energy model (REM) provides a solvable mean-field description of the equilibrium spin glass transition. Its quantum sibling (the QREM), obtained by adding a transverse field to the REM, has similar properties and shows a spin glass phase for sufficiently small transverse field and temperature. In a recent work, some of us have shown that the QREM further exhibits a many-body localization - delocalization (MBLD) transition when viewed as a closed quantum system, evolving according to the quantum dynamics. This phase encloses the familiar equilibrium spin-glass phase. In this paper we study in detail the MBLD transition within the forward-scattering approximation and replica techniques. The predictions for the transition line are in good agreement with the exact diagonalization numerics. We also observe that the structure of the eigenstates at the MBLD critical point changes continuously with the energy density, raising the possibility of a family of critical theories for the MBLD transition.

Many-body mobility edge in a mean-field quantum spin glass.

The quantum random energy model provides a mean-field description of the equilibrium spin glass transition. We show that it further exhibits a many-body localization-delocalization (MBLD) transition when viewed as a closed quantum system. The mean-field structure of the model allows an analytically tractable description of the MBLD transition using the forward-scattering approximation and replica techniques. The predictions are in good agreement with the numerics. The MBLD transition lies at energy density significantly above the equilibrium spin glass transition, indicating that the closed system dynamics freezes well outside of the traditional glass phase. We also observe that the structure of the eigenstates at the MBLD critical point changes continuously with the energy density, raising the possibility of a family of critical theories for the MBLD transition.

TH‐E‐BRE‐02: A Forward Scattering Approximation to Dose Calculation Using the Linear Boltzmann Transport Equation

Purpose:To investigate the use of the linear Boltzmann transport equation as a dose calculation tool which can account for interface effects, while still having faster computation times than Monte Carlo methods. In particular, we introduce a forward scattering approximation, in hopes of improving calculation time without a significant hindrance to accuracy.Methods:Two coupled Boltzmann transport equations were constructed, one representing the fluence of photons within the medium, and the other, the fluence of electrons. We neglect the scattering term within the electron transport equation, resulting in an extreme forward scattering approximation to reduce computational complexity. These equations were then solved using a numerical technique for solving partial differential equations, known as a finite difference scheme, where the fluence at each discrete point in space is calculated based on the fluence at the previous point in the particle's path. Using this scheme, it is possible to develop a solution to the Boltzmann transport equations by beginning with boundary conditions and iterating across the entire medium. The fluence of electrons can then be used to find the dose at any point within the medium.Results:Comparisons with Monte Carlo simulations indicate that even simplistic techniques for solving the linear Boltzmann transport equation yield expected interface effects, which many popular dose calculation algorithms are not capable of predicting. Implementation of a forward scattering approximation does not appear to drastically reduce the accuracy of this algorithm.Conclusion:Optimized implementations of this algorithm have been shown to be very accurate when compared with Monte Carlo simulations, even in build up regions where many models fail. Use of a forward scattering approximation could potentially give a reasonably accurate dose distribution in a shorter amount of time for situations where a completely accurate dose distribution is not required, such as in certain optimization algorithms.

Magnetoresistance and localization in bosonic insulators

We study the strong localization of hard-core bosons. Using a locator expansion we find that in the insulator, unlike for typical fermion problems, nearly all low-energy scattering paths come with positive amplitudes and hence interfere constructively. As a consequence, the localization length of bosonic excitations shrinks when the constructive interference is suppressed by a magnetic field, entailing an exponentially large positive magnetoresistance, opposite to and significantly stronger than the analogous effect in fermions. Within the forward-scattering approximation, we find that the lowest-energy excitations are the most delocalized. A similar analysis applied to random field Ising models suggests that the ordering transition is due to a delocalization initiated at zero energy rather than due to the closure of a mobility gap in the paramagnet.

Polaron in Bose–Einstein–Condensation System

We consider the motion of an impurity in a Bose–Einstein condensate system at T = 0 K with the contact interactions for boson-boson and boson-impurity. Under the forward-scattering approximation, we obtain a Fröhlich-like Hamiltonian for this system, which means that a polaron can be formed. The effective mass, the phonon number and the energy to form a polaron are obtained. We also discuss the validity of the forward-scattering approximation for this system.

Finite-frequency effects in global surface-wave tomography

SUMMARY We compare traditional ray-theoretical surface-wave tomography with finite-frequency tomography, using 3-D Born sensitivity kernels for long-period, fundamental-mode dispersion measurements. The 3-D kernels preserve sidelobes beyond the first Fresnel zone, and fully account for the directional dependence of surface-wave scattering, and the effects of timedomain tapering and seismic source radiation. Tomographic inversions of Love and Rayleigh phase-delay measurements and synthetic checkerboard tests show that (1) small-scale S-wave velocity anomalies are better resolved using finite-frequency sensitivity kernels, especially in the lowermost upper mantle; (2) the resulting upper-mantle heterogeneities are generally stronger in amplitude than those recovered using ray theory and (3) finite-frequency tomographic models fit long-period dispersion data better than ray-theoretical models of comparable roughness. We also examine the reliability of 2-D, phase-velocity sensitivity kernels in global surface-wave tomography, and show that phase-velocity kernels based upon a forwardscattering approximation or previously adopted geometrical simplifications do not reliably account for finite-frequency wave-propagation effects. 3-D sensitivity kernels with full consideration of directional-dependent seismic scattering are the preferred method of inverting long-period dispersion data. Finally, we derive 2-D boundary sensitivity kernels for lateral variations in crustal thickness, and show that finite-frequency crustal effects are not negligible in long-period surface-wave dispersion studies, especially for paths along continent‐ocean boundaries. Unfortunately, we also show that, in global studies, linear perturbation theory is not sufficiently accurate to make reliable crustal corrections, due to the large difference in thickness between oceanic and continental crust.

The effect of steep sea-waves on polarization ratio at low grazing angles

A backscattering numerical model based on a two-scale representation of surface roughness is developed for one-dimensional (1D) sea surfaces at low grazing angles. The effect of the large-scale roughness component is accounted for by a numerical solution of the integral equation for surface field obtained in the forward-scattering approximation. The presence of the small-scale roughness responsible for backscattering is treated by the small-perturbation theory. The numerical simulations accomplished support the viewpoint that the significant difference between experimentally observed and numerically calculated values of polarization ratio for low grazing angles is most likely due to inadequate modeling of surface roughness. It is demonstrated that adding a few relatively minor steep-wave-like features to the surface with the standard Pierson-Moskowitz spectrum will change the average polarization ratio dramatically, bringing its theoretical values from about -20 dB to experimentally observed values of a few negative dB. Half of this increase is due to steepening of the front faces of the undulating waves. However, the other 10 dB of increase is due to diffraction effects, which enhance the scattering coefficient for the HH-polarization on the front faces of the steep waves.

Correlation Transfer: Index of Refraction and Anisotropy Effects

The derivation of a correlation transfer equation is outlined. This correlation transfer equation comes from multiple-scattering theory for the electric field fluctuations of light propagating through suspensions of diffusing particles. The correlation equation is very similar to the radiative intensity equation. Thus, radiative solution techniques are applied to obtain results for isotropic and anisotropic one-dimensional media. For anisotropic scattering, a forward-scattering approximation is used along with the preaveraging method for handling the single-scattering correlation function. Transmitted and backscattered results are presented. Theoretical results are also presented for detection in directions other than direct backscattering or direct transmission, and results are presented for the effect of refractive index. These predictions are compared with another published theory and with experimental data for an expanded 514.5-nm laser beam scattering from suspensions of 0.107-, 0.304-, and 0.497-μm diameter polystyrene latex spheres. For intermediate optical thicknesses, preaveraging theory yields good agreement with the experimental data presented if the effects of refractive index are included in the theory. For small optical thickness, polarization effects are important and need to be included in the future to yield acceptably accurate predictions.

Correlation transfer - Index of refraction and anisotropy effects

The derivation of a correlation transfer equation is outlined. This correlation transfer equation comes from multiple-scattering theory for the electric e eld e uctuations of light propagating through suspensions of diffusing particles. The correlation equation is very similar to the radiative intensity equation. Thus, radiative solution techniques are applied to obtain results for isotropic and anisotropic one-dimensional media. For anisotropic scattering, a forward-scattering approximation is used along with the preaveraging method for handling the single-scattering correlation function. Transmitted and backscattered results are presented. Theoretical results are also presented for detection in directions other than direct backscattering or direct transmission, and results are presented for the effect of refractive index. These predictions are compared with another published theory and with experimental data for an expanded 514.5-nm laser beam scattering from suspensions of 0.107-, 0.304-, and 0.497- m diameter polystyrene latex spheres. For intermediate optical thicknesses, preaveraging theory yields good agreement with the experimental data presented if the effects of refractive index are included in the theory. For small optical thickness, polarization effects are important and need to be included in the future to yield acceptably accurate predictions.

Imaging through the atmosphere: practical instrumentation-based theory and verification of aerosol modulation transfer function: comment

Simple order-of-magnitude expressions derived from the forward-scattering small-angle approximation show that the scattering modulation transfer function (MTF) contributes insignificantly to the overall atmospheric MTF in most practical imaging applications except for precipitation, ice-crystal clouds, and some rare advection-fog or high-wind desert storm events with particles in the size range of tens of micrometers.

Quasi-isoplanatic radiation transfer through randomly moving particles

An iterative solution of an improved forward-scattering approximation for the radiative-transfer equation is derived. The narrow-angle approximation is useful for optical and acoustic beam propagation in the ocean and the atmosphere, where particle sizes are much larger than the wavelength. The improved narrow-angle approximation is useful for a cluster of anisotropic scatterers, and when the cluster is subjected to winds or currents. The solution of the derived equation is sought in terms of uniformly convergent iteration series via the two-scale embedding technique. The lowest iterative of the technique provides an approximate solution that differs from the commonly used successive substitution technique because it takes into account some of the multiple scattering terms. An example of wave propagation through randomly moving particles, useful for remote sensing, is considered. >