35 research outputs found
Two-dimensional Valence Bond Solid (AKLT) states from electrons
Two-dimensional AKLT model on a honeycomb lattice has been shown to be a
universal resource for quantum computation. In this valence bond solid,
however, the spin interactions involve higher powers of the Heisenberg coupling
, making these states seemingly unrealistic on
bipartite lattices, where one expects a simple antiferromagnetic order. We show
that those interactions can be generated by orbital physics in multiorbital
Mott insulators. We focus on electrons on the honeycomb lattice and
propose a physical realization of the spin- AKLT state. We find a phase
transition from the AKLT to the Neel state on increasing Hund's rule coupling,
which is confirmed by density matrix renormalization group (DMRG) simulations.
An experimental signature of the AKLT state consists of protected, free
spins-1/2 on lattice vacancies, which may be detected in the spin
susceptibility
Optimal Renormalization Group Transformation from Information Theory
Recently a novel real-space RG algorithm was introduced, identifying the
relevant degrees of freedom of a system by maximizing an information-theoretic
quantity, the real-space mutual information (RSMI), with machine learning
methods. Motivated by this, we investigate the information theoretic properties
of coarse-graining procedures, for both translationally invariant and
disordered systems. We prove that a perfect RSMI coarse-graining does not
increase the range of interactions in the renormalized Hamiltonian, and, for
disordered systems, suppresses generation of correlations in the renormalized
disorder distribution, being in this sense optimal. We empirically verify decay
of those measures of complexity, as a function of information retained by the
RG, on the examples of arbitrary coarse-grainings of the clean and random Ising
chain. The results establish a direct and quantifiable connection between
properties of RG viewed as a compression scheme, and those of physical objects
i.e. Hamiltonians and disorder distributions. We also study the effect of
constraints on the number and type of coarse-grained degrees of freedom on a
generic RG procedure.Comment: Updated manuscript with new results on disordered system
Relevance in the Renormalization Group and in Information Theory
The analysis of complex physical systems hinges on the ability to extract the
relevant degrees of freedom from among the many others. Though much hope is
placed in machine learning, it also brings challenges, chief of which is
interpretability. It is often unclear what relation, if any, the architecture-
and training-dependent learned "relevant" features bear to standard objects of
physical theory. Here we report on theoretical results which may help to
systematically address this issue: we establish equivalence between the
information-theoretic notion of relevance defined in the Information Bottleneck
(IB) formalism of compression theory, and the field-theoretic relevance of the
Renormalization Group. We show analytically that for statistical physical
systems described by a field theory the "relevant" degrees of freedom found
using IB compression indeed correspond to operators with the lowest scaling
dimensions. We confirm our field theoretic predictions numerically. We study
dependence of the IB solutions on the physical symmetries of the data. Our
findings provide a dictionary connecting two distinct theoretical toolboxes,
and an example of constructively incorporating physical interpretability in
applications of deep learning in physics
Relevance in the Renormalization Group and in Information Theory
The analysis of complex physical systems hinges on the ability to extract the relevant degrees of freedom from among the many others. Though much hope is placed in machine learning, it also brings challenges, chief of which is interpretability. It is often unclear what relation, if any, the architecture- and training-dependent learned “relevant” features bear to standard objects of physical theory. Here we report on theoretical results which may help to systematically address this issue: we establish equivalence between the field-theoretic relevance of the renormalization group, and an information-theoretic notion of relevance we define using the information bottleneck (IB) formalism of compression theory. We show analytically that for statistical physical systems described by a field theory the relevant degrees of freedom found using IB compression indeed correspond to operators with the lowest scaling dimensions. We confirm our field theoretic predictions numerically. We study dependence of the IB solutions on the physical symmetries of the data. Our findings provide a dictionary connecting two distinct theoretical toolboxes, and an example of constructively incorporating physical interpretability in applications of deep learning in physics
Exactly soluble models for fractional topological insulators in 2 and 3 dimensions
We construct exactly soluble lattice models for fractionalized, time reversal
invariant electronic insulators in 2 and 3 dimensions. The low energy physics
of these models is exactly equivalent to a non-interacting topological
insulator built out of fractionally charged fermionic quasiparticles. We show
that some of our models have protected edge modes (in 2D) and surface modes (in
3D), and are thus fractionalized analogues of topological insulators. We also
find that some of the 2D models do not have protected edge modes -- that is,
the edge modes can be gapped out by appropriate time reversal invariant, charge
conserving perturbations. (A similar state of affairs may also exist in 3D). We
show that all of our models are topologically ordered, exhibiting fractional
statistics as well as ground state degeneracy on a torus. In the 3D case, we
find that the models exhibit a fractional magnetoelectric effect.Comment: 29 pages, 9 figure