A Second-Order Distributed Trotter-Suzuki Solver with a Hybrid Kernel

The Trotter-Suzuki approximation leads to an efficient algorithm for solving the time-dependent Schr\"odinger equation. Using existing highly optimized CPU and GPU kernels, we developed a distributed version of the algorithm that runs efficiently on a cluster. Our implementation also improves single node performance, and is able to use multiple GPUs within a node. The scaling is close to linear using the CPU kernels, whereas the efficiency of GPU kernels improve with larger matrices. We also introduce a hybrid kernel that simultaneously uses multicore CPUs and GPUs in a distributed system. This kernel is shown to be efficient when the matrix size would not fit in the GPU memory. Larger quantum systems scale especially well with a high number nodes. The code is available under an open source license.Comment: 11 pages, 10 figure

Finding Exponential Product Formulas of Higher Orders

In the present article, we review a continual effort on generalization of the Trotter formula to higher-order exponential product formulas. The exponential product formula is a good and useful approximant, particularly because it conserves important symmetries of the system dynamics. We focuse on two algorithms of constructing higher-order exponential product formulas. The first is the fractal decomposition, where we construct higher-order formulas recursively. The second is to make use of the quantum analysis, where we compute higher-order correction terms directly. As interludes, we also have described the decomposition of symplectic integrators, the approximation of time-ordered exponentials, and the perturbational composition.Comment: 22 pages, 9 figures. To be published in the conference proceedings ''Quantum Annealing and Other Optimization Methods," eds. B.K.Chakrabarti and A.Das (Springer, Heidelberg

Intriguing sets of vertices of regular graphs

Intriguing and tight sets of vertices of point-line geometries have recently been studied in the literature. In this paper, we indicate a more general framework for dealing with these notions. Indeed, we show that some of the results obtained earlier can be generalized to larger classes of graphs. We also give some connections and relations with other notions and results from algebraic graph theory. One of the main tools in our study will be the Bose-Mesner algebra associated with the graph

Slow quench dynamics of the Kitaev model: anisotropic critical point and effect of disorder

We study the non-equilibrium slow dynamics for the Kitaev model both in the presence and the absence of disorder. For the case without disorder, we demonstrate, via an exact solution, that the model provides an example of a system with an anisotropic critical point and exhibits unusual scaling of defect density $n$ and residual energy $Q$ for a slow linear quench. We provide a general expression for the scaling of $n$ ($Q$) generated during a slow power-law dynamics, characterized by a rate $\tau^{-1}$ and exponent $\alpha$, from a gapped phase to an anisotropic quantum critical point in $d$ dimensions, for which the energy gap $\Delta_{\vec k} \sim k_i^z$ for $m$ momentum components ($i=1..m$) and $\sim k_i^{z'}$ for the rest $d-m$ components ($i=m+1..d$) with $z\le z'$: $n \sim \tau^{-[m + (d-m)z/z']\nu \alpha/(z\nu \alpha +1)}$ ($Q \sim \tau^{-[(m+z)+ (d-m)z/z']\nu \alpha/(z\nu \alpha +1)}$). These general expressions reproduce both the corresponding results for the Kitaev model as a special case for $d=z'=2$ and $m=z=\nu=1$ and the well-known scaling laws of $n$ and $Q$ for isotropic critical points for $z=z'$. We also present an exact computation of all non-zero, independent, multispin correlation functions of the Kitaev model for such a quench and discuss their spatial dependence. For the disordered Kitaev model, where the disorder is introduced via random choice of the link variables $D_n$ in the model's Fermionic representation, we find that $n \sim \tau^{-1/2}$ and $Q\sim \tau^{-1}$ ($Q\sim \tau^{-1/2}$) for a slow linear quench ending in the gapless (gapped) phase. We provide a qualitative explanation of such scaling.Comment: 10 pages, 11 Figs. v

Fast and stable method for simulating quantum electron dynamics

A fast and stable method is formulated to compute the time evolution of a wavefunction by numerically solving the time-dependent Schr{\"o}dinger equation. This method is a real space/real time evolution method implemented by several computational techniques such as Suzuki's exponential product, Cayley's form, the finite differential method and an operator named adhesive operator. This method conserves the norm of the wavefunction, manages periodic conditions and adaptive mesh refinement technique, and is suitable for vector- and parallel-type supercomputers. Applying this method to some simple electron dynamics, we confirmed the efficiency and accuracy of the method for simulating fast time-dependent quantum phenomena.Comment: 10 pages, 35 eps figure

Non-planar double-box, massive and massless pentabox Feynman integrals in negative dimensional approach

Negative dimensional integration method (NDIM) is a technique which can be applied, with success, in usual covariant gauge calculations. We consider three two-loop diagrams: the scalar massless non-planar double-box with six propagators and the scalar pentabox in two cases, where six virtual particles have the same mass and in the case where all of them are massless. Our results are given in terms hypergeometric functions of Mandelstam variables and for arbitrary exponents of propagators and dimension $D$ as well.Comment: Latex, 12 pages, 2 figures, uses axodraw (included

Quantum walks with an anisotropic coin I: spectral theory

We perform the spectral analysis of the evolution operator U of quantum walks with an anisotropic coin, which include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. In particular, we determine the essential spectrum of U, we show the existence of locally U-smooth operators, we prove the discreteness of the eigenvalues of U outside the thresholds, and we prove the absence of singular continuous spectrum for U. Our analysis is based on new commutator methods for unitary operators in a two-Hilbert spaces setting, which are of independent interest.Comment: 26 page

Submodels of Nonlinear Grassmann Sigma Models in Any Dimension and Conserved Currents, Exact Solutions

In the preceding paper(hep-th/9806084), we constructed submodels of nonlinear Grassmann sigma models in any dimension and, moreover, an infinite number of conserved currents and a wide class of exact solutions. In this paper, we first construct almost all conserved currents for the submodels and all ones for the one of ${\bf C}P^1$-model. We next review the Smirnov and Sobolev construction for the equations of ${\bf C}P^1$-submodel and extend the equations, the S-S construction and conserved currents to the higher order ones.Comment: 13 pages, AMSLaTex; an new section and an appendix adde

H-T phase diagram and the nature of Vortex-glass phase in a quasi two-dimensional superconductor: Sn metal layer sandwiched between graphene sheets

The magnetic properties of a quasi two-dimensional superconductor, Sn-metal graphite (MG), are studied using DC and AC magnetic susceptibility. Sn-MG has a unique layered structure where Sn metal layer is sandwiched between adjacent graphene sheets. This compound undergoes a superconducting transition at $T_{c}$ = 3.75 K at $H$ = 0. The $H$-$T$ diagram of Sn-MG is similar to that of a quasi two-dimensional superconductors. The phase boundaries of vortex liquid, vortex glass, and vortex lattice phase merge into a multicritical point located at $T^{*}$ = 3.4 K and $H^{*}$ = 40 Oe. There are two irreversibility lines denoted by $H_{gl}$ (de Almeida-Thouless type) and $H_{gl^{\prime}}$ (Gabay-Toulouse type), intersecting at $T_{0}^{\prime}$= 2.5 K and $H_{0}^{\prime}$ = 160 Oe. The nature of slow dynamic and nonlinearity of the vortex glass phase is studied.Comment: 24 pages, 13 figures; Physica C (2003), in pres

Vortex Structure in Abelian-Projected Lattice Gauge Theory

We report on a breakdown of both monopole dominance and positivity in abelian-projected lattice Yang-Mills theory. The breakdown is associated with observables involving two units of the abelian charge. We find that the projected lattice has at most a global $Z_2$ symmetry in the confined phase, rather than the global U(1) symmetry that might be expected in a dual superconductor or monopole Coulomb gas picture. Implications for monopole and center vortex theories of confinement are discussed.Comment: LATTICE99(confine), 3 pages, 2 figure