26 research outputs found
Special Holonomy and Two-Dimensional Supersymmetric Sigma-Models
Two-dimensional sigma-models describing superstrings propagating on manifolds
of special holonomy are characterized by symmetries related to covariantly
constant forms that these manifolds hold, which are generally non-linear and
close in a field dependent sense. The thesis explores various aspects of the
special holonomy symmetries.Comment: PhD thesis, 149 pages; King's College London, 200
Two-Dimensional Supersymmetric Sigma Models on Almost-Product Manifolds and Non-Geometry
We show that the superconformal symmetries of the (1,1) sigma model decompose
into a set of more refined symmetries when the target space admits projectors
, and the orthogonal complements , covariantly constant with
respect to the two natural torsionful connections that arise
in the sigma model. Surprisingly the new symmetries still close to form copies
of the superconformal algebra, even when the projectors are not integrable, so
one is able to define a superconformal theory not associated with a particular
geometry, but rather with non-integrable projectors living on a larger
manifold. We show that this notion of non-geometry encompasses the locally
non-geometric examples that arise in the T-duality inspired doubled
formulations, with the benefit that it is more generally applicable, as it does
not depend on the existence of isometries, or invariant structures beyond
and . We derive the conditions for (2,2) supersymmetry in
the projective sense, thus extending the relation between (2,2) theories and
bi-Hermitian target spaces to the non-geometric setting. In the bosonic
subsector we propose a BRST type approach to defining the physical degrees of
freedom in the non-geometric scenario.Comment: 27 pages; version published in Classical and Quantum Gravity
(excluding the BV appendix
Covariantly constant forms on torsionful geometries from world-sheet and spacetime perspectives
The symmetries of two-dimensional supersymmetric sigma models on target
spaces with covariantly constant forms associated to special holonomy groups
are analysed. It is shown that each pair of such forms gives rise to a new one,
called a Nijenhuis form, and that there may be further reductions of the
structure group. In many cases of interest there are also covariantly constant
one-forms which also give rise to symmetries. These geometries are of interest
in the context of heterotic supergravity solutions and the associated
reductions are studied from a spacetime point of view via the Killing spinor
equations.Comment: 33 pages, minor modifications, version published in JHE
Time Evolution and Deterministic Optimisation of Correlator Product States
We study a restricted class of correlator product states (CPS) for a
spin-half chain in which each spin is contained in just two overlapping
plaquettes. This class is also a restriction upon matrix product states (MPS)
with local dimension ( being the size of the overlapping regions of
plaquettes) equal to the bond dimension. We investigate the trade-off between
gains in efficiency due to this restriction against losses in fidelity. The
time-dependent variational principle formulated for these states is numerically
very stable. Moreover, it shows significant gains in efficiency compared to the
naively related matrix product states - the evolution or optimisation scales as
for the correlator product states versus for the unrestricted
matrix product state. However, much of this advantage is offset by a
significant reduction in fidelity. Correlator product states break the local
Hilbert space symmetry by the explicit selection of a local basis. We
investigate this dependence in detail and formulate the broad principles under
which correlator product states may be a useful tool. In particular, we find
that scaling with overlap/bond order may be more stable with correlator product
states allowing a more efficient extraction of critical exponents - we present
an example in which the use of correlator product states is several orders of
magnitude quicker than matrix product states.Comment: 19 pages, 14 figure
Compact Neural Networks based on the Multiscale Entanglement Renormalization Ansatz
This paper demonstrates a method for tensorizing neural networks based upon
an efficient way of approximating scale invariant quantum states, the
Multi-scale Entanglement Renormalization Ansatz (MERA). We employ MERA as a
replacement for the fully connected layers in a convolutional neural network
and test this implementation on the CIFAR-10 and CIFAR-100 datasets. The
proposed method outperforms factorization using tensor trains, providing
greater compression for the same level of accuracy and greater accuracy for the
same level of compression. We demonstrate MERA layers with 14000 times fewer
parameters and a reduction in accuracy of less than 1% compared to the
equivalent fully connected layers, scaling like O(N).Comment: 8 pages, 2 figure
Particles, holes and solitons: a matrix product state approach
We introduce a variational method for calculating dispersion relations of
translation invariant (1+1)-dimensional quantum field theories. The method is
based on continuous matrix product states and can be implemented efficiently.
We study the critical Lieb-Liniger model as a benchmark and excelent agreement
with the exact solution is found. Additionally, we observe solitonic signatures
of Lieb's Type II excitation. In addition, a non-integrable model is introduced
where a U(1)-symmetry breaking term is added to the Lieb-Liniger Hamiltonian.
For this model we find evidence of a non-trivial bound-state excitation in the
dispersion relation
Quantum Gross-Pitaevskii Equation
We introduce a non-commutative generalization of the Gross-Pitaevskii
equation for one-dimensional quantum gasses and quantum liquids. This
generalization is obtained by applying the time-dependent variational principle
to the variational manifold of continuous matrix product states. This allows
for a full quantum description of many body system ---including entanglement
and correlations--- and thus extends significantly beyond the usual mean-field
description of the Gross-Pitaevskii equation, which is known to fail for
(quasi) one-dimensional systems. By linearizing around a stationary solution,
we furthermore derive an associated generalization of the Bogoliubov -- de
Gennes equations. This framework is applied to compute the steady state
response amplitude to a periodic perturbation of the potential.Comment: 4.{\epsilon} pages + references and 4 pages supplementary material
(small revisions + extended discussion of periodic potential example
Conformal data from finite entanglement scaling
In this paper, we apply the formalism of translation invariant (continuous) matrix product states in the thermodynamic limit to (1+1)-dimensional critical models. Finite bond dimension bounds the entanglement entropy and introduces an effective finite correlation length, so that the state is perturbed away from criticality. The assumption that the scaling hypothesis holds for this kind of perturbation is known in the literature as finite entanglement scaling. We provide further evidence for the validity of finite entanglement scaling and based on this formulate a scaling algorithm to estimate the central charge and critical exponents of the conformally invariant field theories describing the critical models under investigation. The algorithm is applied to three exemplary models; the cMPS version to the nonrelativistic Lieb-Liniger model and the relativistic massless boson, and MPS version to the one-dimensional quantum Ising model at the critical point. Another new aspect to our approach is that we directly use the (c)MPS induced correlation length rather than the bond dimension as scaling parameter. This choice is motivated by several theoretical arguments as well as by the remarkable accuracy of our results
Conformal data from finite entanglement scaling
In this paper, we apply the formalism of translation invariant (continuous) matrix product states in the
thermodynamic limit to (1 + 1)-dimensional critical models. Finite bond dimension bounds the entanglement
entropy and introduces an effective finite correlation length, so that the state is perturbed away from criticality.
The assumption that the scaling hypothesis holds for this kind of perturbation is known in the literature as finite
entanglement scaling. We provide further evidence for the validity of finite entanglement scaling and based on this
formulate a scaling algorithm to estimate the central charge and critical exponents of the conformally invariant
field theories describing the critical models under investigation. The algorithm is applied to three exemplary
models; the cMPS version to the nonrelativistic Lieb-Liniger model and the relativistic massless boson, and MPS
version to the one-dimensional quantum Ising model at the critical point. Another new aspect to our approach is
that we directly use the (c)MPS induced correlation length rather than the bond dimension as scaling parameter.
This choice is motivated by several theoretical arguments as well as by the remarkable accuracy of our results.Peer ReviewedPostprint (author’s final draft