292 research outputs found
Studies On Fractionalization And Topology In Strongly Correlated Systems From Zero To Two Dimensions
The interplay among symmetry, topology and condensed matter systems has deepened our understandings of matter and lead to tremendous recent progresses in finding new topological phases of matter such as topological insulators, superconductors and semi-metals. Most examples of the aforementioned topological materials are free fermion systems, in this thesis, however, we focus on their strongly correlated counterparts where electron-electron interactions play a major role. With interactions, exotic topological phases and quantum critical points with fractionalized quantum degrees of freedom emerge. In the first part of this thesis, we study the problem of resonant tunneling through a quantum dot in a spinful Luttinger liquid. It provides the simplest example of a (0+1)d system with symmetry-protected phase transitions. We show that the problem is equivalent to a two channel SU(3) Kondo problem and can be mapped to a quantum Brownian motion model on a Kagome lattice. Utilizing boundary conformal field theory, we find the universal peak conductance and compute the scaling behavior of the resonance line-shape.
For the second part, we present a model of interacting Majorana fermions that describes a superconducting phase with a topological order characterized by the Fibonacci topological field theory. Our theory is based on a SO(7)1=SO(7)1/(G2)1 x (G2)1 coset construction and implemented by a solvable two-dimensional network model. In addition, we predict a closely related \u27\u27anti-Fibonacci\u27\u27 phase, whose topological order is characterized by the tricritical Ising model. Finally, we propose an interferometer that generalizes the Z2 Majorana interferometer and directly probes the Fibonacci non-Abelian statistics.
For the third part, we argue that a correlated fluid of electrons and holes can exhibit fractional quantum Hall effects at zero magnetic field. We first show that a Chern insulator can be realized as a free fermion model with p-wave(m=1) excitonic pairing. Its ground state wavefunction is then worked out and generalized to m\u3e1. We give several pieces of evidence that this conjectured wavefunction correctly describes a topological phase, dubbed \u27\u27fractional excitonic insulator\u27\u27, within the same universality class as the corresponding Laughlin state at filling 1/m. We present physical arguments that gapless states with higher angular momentum pairing between energy bands are conducive to forming the fractional excitonic insulator in the presence of repulsive interactions. Without interactions, these gapless states appear at topological phase transitions which separate the trivial insulator from a Chern insulator with higher Chern number. Since the nonvanishing density of states at these higher angular momentum band inversion transitions can give rise to interesting many-body effects, we introduce a series of minimal lattice models realizations in two dimensions. We also study the effect of rotational symmetry broken electron-hole exciton condensation in our lattice models using mean field theory
Aspects of Defect Topology in Smectic Liquid Crystals
We study the topology of smectic defects in two and three dimensions. We give
a topological classification of smectic point defects and disclination lines in
three dimensions. In addition we describe the combination rules for smectic
point defects in two and three dimensions, showing how the broken translational
symmetry of the smectic confers a path dependence on the result of defect
addition.Comment: 19 pages, 13 figure
Relation Networks for Object Detection
Although it is well believed for years that modeling relations between
objects would help object recognition, there has not been evidence that the
idea is working in the deep learning era. All state-of-the-art object detection
systems still rely on recognizing object instances individually, without
exploiting their relations during learning.
This work proposes an object relation module. It processes a set of objects
simultaneously through interaction between their appearance feature and
geometry, thus allowing modeling of their relations. It is lightweight and
in-place. It does not require additional supervision and is easy to embed in
existing networks. It is shown effective on improving object recognition and
duplicate removal steps in the modern object detection pipeline. It verifies
the efficacy of modeling object relations in CNN based detection. It gives rise
to the first fully end-to-end object detector
Homotopy Classification of loops of Clifford unitaries
Clifford quantum circuits are elementary invertible transformations of
quantum systems that map Pauli operators to Pauli operators. We study periodic
one-parameter families of Clifford circuits, called loops of Clifford circuits,
acting on -dimensional lattices of prime -dimensional qudits. We
propose to use the notion of algebraic homotopy to identify topologically
equivalent loops. We calculate homotopy classes of such loops for any odd
and , and . Our main tool is the Hermitian K-theory,
particularly a generalization of the Maslov index from symplectic geometry. We
observe that the homotopy classes of loops of Clifford circuits in
-dimensions coincide with the quotient of the group of Clifford
Quantum Cellular Automata modulo shallow circuits and lattice translations in
-dimensions.Comment: 25 page
Double-tower Solutions for Higher Order Prescribed Curvature Problem
We consider the following higher order prescribed curvature problem on \begin{equation*} D^m \tilde u=\widetilde{K}(y) \tilde
u^{m^{*}-1} \quad \mbox{on} \ {\mathbb {S}}^N, \qquad \tilde u >0 \quad
\mbox{in} \ {\mathbb {S}}^N. \end{equation*} where is a
radial function, and is order differential
operator given by \begin{equation*}
D^m=\prod_{i=1}^m\left(-\Delta_g+\frac{1}{4}(N-2i)(N+2i-2)\right),
\end{equation*} where is the Riemannian metric. We prove
the existence of infinitely many double-tower type solutions, which are
invariant under some non-trivial sub-groups of and their energy can be
made arbitrarily large.Comment: 34 pages, 0 figures. arXiv admin note: substantial text overlap with
arXiv:2205.14482 by other author
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