2,091 research outputs found
Differentiable Programming Tensor Networks
Differentiable programming is a fresh programming paradigm which composes
parameterized algorithmic components and trains them using automatic
differentiation (AD). The concept emerges from deep learning but is not only
limited to training neural networks. We present theory and practice of
programming tensor network algorithms in a fully differentiable way. By
formulating the tensor network algorithm as a computation graph, one can
compute higher order derivatives of the program accurately and efficiently
using AD. We present essential techniques to differentiate through the tensor
networks contractions, including stable AD for tensor decomposition and
efficient backpropagation through fixed point iterations. As a demonstration,
we compute the specific heat of the Ising model directly by taking the second
order derivative of the free energy obtained in the tensor renormalization
group calculation. Next, we perform gradient based variational optimization of
infinite projected entangled pair states for quantum antiferromagnetic
Heisenberg model and obtain start-of-the-art variational energy and
magnetization with moderate efforts. Differentiable programming removes
laborious human efforts in deriving and implementing analytical gradients for
tensor network programs, which opens the door to more innovations in tensor
network algorithms and applications.Comment: Typos corrected, discussion and refs added; revised version accepted
for publication in PRX. Source code available at
https://github.com/wangleiphy/tensorgra
Color-Kinematics Duality for QCD Amplitudes
We show that color-kinematics duality is present in tree-level amplitudes of
quantum chromodynamics with massive flavored quarks. Starting with the color
structure of QCD, we work out a new color decomposition for n-point tree
amplitudes in a reduced basis of primitive amplitudes. These primitives, with k
quark-antiquark pairs and (n-2k) gluons, are taken in the (n-2)!/k! Melia
basis, and are independent under the color-algebra Kleiss-Kuijf relations. This
generalizes the color decomposition of Del Duca, Dixon, and Maltoni to an
arbitrary number of quarks. The color coefficients in the new decomposition are
given by compact expressions valid for arbitrary gauge group and
representation. Considering the kinematic structure, we show through explicit
calculations that color-kinematics duality holds for amplitudes with general
configurations of gluons and massive quarks. The new (massive) amplitude
relations that follow from the duality can be mapped to a well-defined subset
of the familiar BCJ relations for gluons. They restrict the amplitude basis
further down to (n-3)!(2k-2)/k! primitives, for two or more quark lines. We
give a decomposition of the full amplitude in that basis. The presented results
provide strong evidence that QCD obeys the color-kinematics duality, at least
at tree level. The results are also applicable to supersymmetric and
D-dimensional extensions of QCD.Comment: 33 pages + refs, 7 figures, 4 tables; v3 minor corrections, journal
versio
Normal Coordinates and Primitive Elements in the Hopf Algebra of Renormalization
We introduce normal coordinates on the infinite dimensional group
introduced by Connes and Kreimer in their analysis of the Hopf algebra of
rooted trees. We study the primitive elements of the algebra and show that they
are generated by a simple application of the inverse Poincar\'e lemma, given a
closed left invariant 1-form on . For the special case of the ladder
primitives, we find a second description that relates them to the Hopf algebra
of functionals on power series with the usual product. Either approach shows
that the ladder primitives are given by the Schur polynomials. The relevance of
the lower central series of the dual Lie algebra in the process of
renormalization is also discussed, leading to a natural concept of
-primitiveness, which is shown to be equivalent to the one already in the
literature.Comment: Latex, 24 pages. Submitted to Commun. Math. Phy
Kranc: a Mathematica application to generate numerical codes for tensorial evolution equations
We present a suite of Mathematica-based computer-algebra packages, termed
"Kranc", which comprise a toolbox to convert (tensorial) systems of partial
differential evolution equations to parallelized C or Fortran code. Kranc can
be used as a "rapid prototyping" system for physicists or mathematicians
handling very complicated systems of partial differential equations, but
through integration into the Cactus computational toolkit we can also produce
efficient parallelized production codes. Our work is motivated by the field of
numerical relativity, where Kranc is used as a research tool by the authors. In
this paper we describe the design and implementation of both the Mathematica
packages and the resulting code, we discuss some example applications, and
provide results on the performance of an example numerical code for the
Einstein equations.Comment: 24 pages, 1 figure. Corresponds to journal versio
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