22,328 research outputs found
Hodge theory and deformations of SKT manifolds
We use tools from generalized complex geometry to develop the theory of SKT
(a.k.a. pluriclosed Hermitian) manifolds and more generally manifolds with
special holonomy with respect to a metric connection with closed skew-symmetric
torsion. We develop Hodge theory on such manifolds showing how the reduction of
the holonomy group causes a decomposition of the twisted cohomology. For SKT
manifolds this decomposition is accompanied by an identity between different
Laplacian operators and forces the collapse of a spectral sequence at the first
page. Further we study the deformation theory of SKT structures, identifying
the space where the obstructions live. We illustrate our theory with examples
based on Calabi--Eckmann manifolds, instantons, Hopf surfaces and Lie groups.Comment: 46 pages, 9 figures; v5: Added theorem 5.16 and expanded example 5.17
to show that the only Calabi-Eckman manifolds to admit SKT structures are S^1
x S^1, S^1 x S^3 and S^3 x S^
Non-equilibrium physics of Rydberg lattices in the presence of noise and dissipative processes
We study the non-equilibrium dynamics of driven spin lattices in the presence
of decoherence caused by either laser phase noise or strong decay. In the first
case, we discriminate between correlated and uncorrelated noise and explore
their effect on the mean density of Rydberg states and the full counting
statistics (FCS). We find that while the mean density is almost identical in
both cases, the FCS differ considerably. The main method employed is the
Langevin equation (LE) but for the sake of efficiency in certain regimes, we
use a Markovian master equation and Monte Carlo rate equations, respectively.
In the second case, we consider dissipative systems with more general power-law
interactions. We determine the phase diagram in the steady state and analyse
its generation dynamics using Monte Carlo rate equations. In contrast to
nearest-neighbour models, there is no transition to long-range-ordered phases
for realistic interactions and resonant driving. Yet, for finite laser
detunings, we show that Rydberg lattices can undergo a dissipative phase
transition to a long-range-ordered antiferromagnetic (AF) phase. We identify
the advantages of Monte Carlo rate equations over mean field (MF) predictions
Fibrations and stable generalized complex structures
A generalized complex structure is called stable if its defining
anticanonical section vanishes transversally, on a codimension-two submanifold.
Alternatively, it is a zero elliptic residue symplectic structure in the
elliptic tangent bundle associated to this submanifold. We develop
Gompf-Thurston symplectic techniques adapted to Lie algebroids, and use these
to construct stable generalized complex structures out of log-symplectic
structures. In particular we introduce the notion of a boundary Lefschetz
fibration for this purpose and describe how they can be obtained from genus one
Lefschetz fibrations over the disk.Comment: 35 pages, 2 figure
Spectral Methods from Tensor Networks
A tensor network is a diagram that specifies a way to "multiply" a collection
of tensors together to produce another tensor (or matrix). Many existing
algorithms for tensor problems (such as tensor decomposition and tensor PCA),
although they are not presented this way, can be viewed as spectral methods on
matrices built from simple tensor networks. In this work we leverage the full
power of this abstraction to design new algorithms for certain continuous
tensor decomposition problems.
An important and challenging family of tensor problems comes from orbit
recovery, a class of inference problems involving group actions (inspired by
applications such as cryo-electron microscopy). Orbit recovery problems over
finite groups can often be solved via standard tensor methods. However, for
infinite groups, no general algorithms are known. We give a new spectral
algorithm based on tensor networks for one such problem: continuous
multi-reference alignment over the infinite group SO(2). Our algorithm extends
to the more general heterogeneous case.Comment: 30 pages, 8 figure
- …