The formalism of the reduced density matrix is pursued in both length and velocity gauges of the perturbation to the crystal Hamiltonian. The covariant derivative is introduced as a convenient representation of the position operator. This allow us to write compact expressions for the reduced density matrix in any order of the perturbation which simplifies the calculations of nonlinear optical responses; as an example, we compute the first and third order contributions of the monolayer graphene. Expressions obtained in both gauges share the same formal structure, allowing a comparison of the effects of truncation to a finite set of bands. This truncation breaks the equivalence between the two approaches: its proper implementation can be done directly in the expressions derived in the length gauge, but require a revision of the equations of motion of the reduced density matrix in the velocity gauge.

In this work, the difficulties inherent to perturbative calculations in the velocity gauge are addressed. In particular, it is shown how calculations of nonlinear optical responses in the independent particle approximation can be done to any order and for any finite band model. The procedure and advantages of the velocity gauge in such calculations are described. The addition of a phenomenological relaxation parameter is also discussed. As an illustration, the nonlinear optical response of monolayer graphene is numerically calculated using the velocity gauge. arXiv:1712.04924v2 [cond-mat.other]

In this paper, we developed a basis-independent perturbative method for calculating the nonlinear optical response of arbitrary non-interacting tight-binding models. Our method is based on the non-equilibrium Keldysh formalism and allows an efficient numerical implementation within the framework of the kernel polynomial method for systems which are not required to be translation-invariant. Some proof-of-concept results of the second-order optical conductivity are presented for the special case of gapped graphene with vacancies and an on-site Anderson disordered potential.

We present KITE, a general purpose open-source tight-binding software for accurate real-space simulations of electronic structure and quantum transport properties of large-scale molecular and condensed systems with tens of billions of atomic orbitals (N ∼ 10 10 ). KITE's core is written in C++, with a versatile Python-based interface, and is fully optimised for shared memory multi-node CPU architectures, thus scalable, efficient and fast. At the core of KITE is a seamless spectral expansion of lattice Green's functions, which enables large-scale calculations of generic target functions with uniform convergence and fine control over energy resolution. Several functionalities are demonstrated, ranging from simulations of local density of states and photo-emission spectroscopy of disordered materials to large-scale computations of optical conductivity tensors and real-space wave-packet propagation in the presence of magneto-static fields and spin-orbit coupling. On-the-fly calculations of real-space Green's functions are carried out with an efficient domain decomposition technique, allowing KITE to achieve nearly ideal linear scaling in its multi-threading performance. Crystalline defects and disorder, including vacancies, adsorbates and charged impurity centers, can be easily set up with KITE's intuitive interface, paving the way to user-friendly large-scale quantum simulations of equilibrium and nonequilibrium properties of molecules, disordered crystals and heterostructures subject to a variety of perturbations and external conditions. arXiv:1910.05194v1 [cond-mat.mes-hall] 11 Oct 2019 2 IntroductionComputational modelling has become an essential tool in both fundamental and applied research that has propelled the discovery of new materials and their translation into practical applications [1]. The study of condensed phases of matter has benefited from significant advances in electronic structure theory and simulation methodologies. Among these advances are: explicitly correlated wave-function-based techniques achieving sub-chemical accuracy [2], first-principles methods to tackling electronic excitations [3], charge-self-consistent atomistic models for accurate electronic structure calculations [4], and the use of machine learning as means to finding density functionals without solving the Khon-Sham equations [5,6].Semi-empirical atomistic methods are amongst the most simple and effective methods to calculate ground-and excited-state properties of materials [7][8][9][10]. The increasingly popular tight-binding approach [11] has been employed for accurate and fast calculations of total energies and electronic structure in complex materials, including semiconductors [12,13], quantum dots [14] and super-lattices [15,16], and is particularly well-suited for implementation of O(N ) (linear scaling) algorithms for efficient calculations of total energies and forces [17].Accurate tight-binding models have been devised for a plethora of model systems, ranging from metals to ionic materials [18], and shown to correctly p...

We design a random walk to explore fractal landscapes such as those describing chaotic transients in dynamical systems. We show that the random walk moves efficiently only when its step length depends on the height of the landscape via the largest Lyapunov exponent of the chaotic system. We propose a generalization of the Wang-Landau algorithm which constructs not only the density of states (transient time distribution) but also the correct step length. As a result, we obtain a flat-histogram Monte Carlo method which samples fractal landscapes in polynomial time, a dramatic improvement over the exponential scaling of traditional uniform-sampling methods. Our results are not limited by the dimensionality of the landscape and are confirmed numerically in chaotic systems with up to 30 dimensions.