On computing non-equilibrium dynamics following a quench

Computing the non-equilibrium dynamics that follows a quantum quench is difficult, even in exactly solvable models. Results are often predicated on the ability to compute overlaps between the initial state and eigenstates of the Hamiltonian that governs time evolution. Except for a handful of known cases, it is generically not possible to find these overlaps analytically. Here we develop a numerical approach to preferentially generate the states with high overlaps for a quantum quench starting from the ground state or an excited state of an initial Hamiltonian. We use these preferentially generated states, in combination with a "high overlap states truncation scheme" and a modification of the numerical renormalization group, to compute non-equilibrium dynamics following a quench in the Lieb-Liniger model. The method is non-perturbative, works for reasonable numbers of particles, and applies to both continuum and lattice systems. It can also be easily extended to more complicated scenarios, including those with integrability breaking.

Interpolating data on the Cubed Sphere with Spherical Harmonics

<p>The Cubed Sphere is a grid commonly used in numerical simulation in climatology. In this talk we present recent progress<br>on the algebraic and geometrical properties of this highly symmetrical grid.<br>First, an analysis of the symmetry group of the Cubed Sphere will be presented: this group <br>is identified as the group of the Cube, [1]. Furthermore, we show how to construct a discrete Spherical Harmonics (SH) basis associated to <br>the Cubed Sphere. This basis displays a truncation scheme relating the zonal and longitudinal <br>mode numbers reminiscent of the rhomboidal truncation on the Lon-Lat grid.<br>The new analysis allows to derive new quadrature rules of  interest for applications in any kind of spherical modelling. In addition,<br>we will comment on applications in mathematical climatology and meteorology, [2].</p><p>[1] J.-B. Bellet, Symmetry group of the equiangular Cubed Sphere, preprint, IECL, Univ. Lorraine, 2020, submitted</p><p>[2] J.-B. Bellet, M. Brachet and J.-P. Croisille, Spherical Harmonics on The Cubed Sphere, IECL, Univ. Lorraine, 2021, Preprint.</p>

Equation of motion truncation scheme based on partial orthogonalization

We introduce a general scheme to consistently truncate equations of motion for Green’s functions. Our scheme is guaranteed to generate physical Green’s functions with real excitation energies and positive spectral weights. There are free parameters in our scheme akin to mean field parameters that may be determined to get as good an approximation to the physics as possible. As a test case we apply our scheme to a two-pole approximation for the 2D Hubbard model. At half-filling we find an insulating solution with several interesting properties: it has low expectation value of the energy and it gives upper and lower Hubbard bands with the full non-interacting bandwidth in the large U limit. Away from half-filling, in particular in the intermediate interaction regime, our scheme allows for several different phases with different number of Fermi surfaces and topologies. Graphic abstract

A Numerically Exact Method for Dissipative Dynamics of Qubits

We propose a numerical technique suitable for simulating the dynamics of reduced density matrix of a qubit interacting with its environment. Our approach, based on a combination of short-iterative Lanczos method (SIL) and a flexible truncation scheme, allows to include in the physical description multiple-excitation processes, beyond weak coupling and Markov approximations. We perform numerical simulations of two different model Hamiltonians, that are relevant in the field of adiabatic quantum computation (AQC), and we show that our technique is able to recover the correct thermodynamic behavior of the qubit-bath system, from weak to intermediate coupling regime.

Efficient Model-Assisted Probability of Detection and Sensitivity Analysis for Ultrasonic Testing Simulations Using Stochastic Metamodeling

Model-assisted probability of detection (MAPOD) and sensitivity analysis (SA) are important for quantifying the inspection capability of nondestructive testing (NDT) systems. To improve the computational efficiency, this work proposes the use of polynomial chaos expansions (PCEs), integrated with least-angle regression (LARS), a basis-adaptive technique, and a hyperbolic truncation scheme, in lieu of the direct use of the physics-based measurement model in the MAPOD and SA calculations. The proposed method is demonstrated on three ultrasonic testing cases and compared with Monte Carlo sampling (MCS) of the physics model, MCS-based kriging, and the ordinary least-squares (OLS)-based PCE method. The results show that the probability of detection (POD) metrics of interests can be controlled within 1% accuracy relative to using the physics model directly. Comparison with metamodels shows that the LARS-based PCE method can provide up to an order of magnitude improvement in the computational efficiency.

Irregular parameter dependence of numerical results in tensor renormalization group analysis

We study the parameter dependence of numerical results obtained by the tensor renormalization group. We often observe irregular behavior as the parameters are varied with the method. Using the two-dimensional Ising model we explicitly show that the sharp cutoff used in the truncated singular value decomposition causes this unwanted behavior when the level crossing happens between singular values below and above the truncation order as the parameters are varied. We also test a smooth cutoff, instead of the sharp one, as a truncation scheme and discuss its effects.