Braids, posets and orthoschemes

In this article we study the curvature properties of the order complex of a graded poset under a metric that we call the ``orthoscheme metric''. In addition to other results, we characterize which rank 4 posets have CAT(0) orthoscheme complexes and by applying this theorem to standard posets and complexes associated with four-generator Artin groups, we are able to show that the 5-string braid group is the fundamental group of a compact nonpositively curved space.Comment: 33 pages, 16 figure

A degree and forbidden subgraph condition for a k-contractible edge

An edge in a k-conected graph is said to be k-contractible if the contraction of it results in a k-connected graph. We say that k-connected graph G satis es “ degree-sum conditon ”if Σx2V (W)degG(x) 3k +2 holds for any connected subgraph W of G with 　｜W｜= 3. Let k be an integer such that k 5. We prove that if a k-connected graph with no K1+C4 satis es degree-sum condition, then it has a k-contractible edge.電気通信大学201

Chern-Simons theory of magnetization plateaus on the kagome lattice

Frustrated spin systems on Kagome lattices have long been considered to be a promising candidate for realizing exotic spin liquid phases. Recently, there has been a lot of renewed interest in these systems with the discovery of experimental materials such as Volborthite and Herbertsmithite that have Kagome like structures. In this thesis I will focus on studying frustrated spin systems on the Kagome lattice using a spin-1/2 antiferromagnetic XXZ Heisenberg model in the presence of an external magnetic field as well as other perturbations. Such a system is expected to give rise to magnetization platueaus which can exhibit topological characteristics in certain regimes. We will first develop a flux-attachment transformation that maps the Heisenberg spins (hard-core bosons) onto a problem of fermions coupled to a Chern-Simons gauge field. This mapping relies on being able to define a consistent Chern-Simons term on the lattice. Using this newly developed mapping we analyse the phases/magnetization plateaus that arise at the mean-field level and also consider the effects of adding fluctuations to various mean-fi eld states. Along the way, we show how to discretize an abelian Chern-Simons gauge theory on generic 2D planar lattices that satisfy certain conditions. We find that as long as there exists a one-to-one correspondence between the vertices and plaquettes defined on the graph, one can write down a discretized lattice version of the abelian Chern-Simons gauge theory. Using the newly developed flux attachment transformation, we show the existence of chiral spin liquid states for various magnetization plateaus for certain range of parameters in the XXZ Heisenberg model in the presence of an external magnetic field. Speci cally, in the regime of XY anisotropy the ground states at the 1/3 and 2/3 plateau are equivalent to a bosonic fractional quantum Hall Laughlin state with filling fraction 1/2 and that the 5/9 plateau is equivalent to the first bosonic Jain daughter state at filling fraction 2/3. Next, we also consider the effects of several perturbations: a) a chirality term, b) a Dzyaloshinskii-Moriya term, and c) a ring-exchange type term on the bowties of the kagome lattice, and inquire if they can also support chiral spin liquids as ground states. We find that the chirality term leads to a chiral spin liquid even in the absence of an uniform magnetic field, with an effective spin Hall conductance of 1/2 in the regime of XY anisotropy. The Dzyaloshinkii-Moriya term also leads a similar chiral spin liquid but only when this term is not too strong. An external magnetic field when combined with some of the above perturbations also has the possibility of giving rise to additional plateaus which also behave like chiral spin liquids in the XY regime. Under the in influence of a ring-exchange term we find that provided its coupling constant is large enough, it may trigger a phase transition into a chiral spin liquid by the spontaneous breaking of time-reversal invariance. Finally, we also present some numerical results based on some exact diagonalization studies. Here, we specifically focus on the 2/3-magnetization plateau which we previously argued should be a chiral spin liquid with a spin hall conductance of 1/2 . Such a topological state has a non-trivial ground state degeneracy and it excitations are described by semionic quasiparticles. In the numerical analysis, we analyse the ground state degeneracy structure on various Kagome clusters of different sizes. We compute modular matrices from the resultant minimally entangled states as well as the Chern numbers of various eigenstates all of which provide strong evidence that the 2/3-magnetization plateau very closely resembles a chiral spin liquid state with the expected characteristics

Group actions on injective spaces and Helly graphs

These are lecture notes for a minicourse on group actions on injective spaces and Helly graphs, given at the CRM Montreal in June 2023. We review the basics of injective metric spaces and Helly graphs, emphasizing the parallel between the two theories. We also describe various elementary properties of groups actions on such spaces. We present several constructions of injective metric spaces and Helly graphs with interesting actions of many groups of geometric nature. We also list a few exercises and open questions at the end.Comment: Comments are welcome! v2: some references adde

Pseudometrics, The Complex of Ultrametrics, and Iterated Cycle Structures

Every set X, finite of cardinality n say, carries a set M(X) of all possible pseudometrics. It is well known that M(X) forms a convex polyhedral cone whose faces correspond to triangle inequalities. Every point in a convex cone can be expressed as a conical sum of its extreme rays, hence the interest around discovering and classifying such rays. We shall give examples of extreme rays for M(X) exhibiting all integral edge lengths up to half the cardinality of X. By intersecting the cone with the unit cube we obtain the convex polytope of bounded-by-one pseudometrics BM(X). Analogous to extreme rays, every point in a convex polytope arises as a convex combination of extreme points. Extreme rays of BM(X) give rise to very special extreme points of ̄BM(X) as we may normalize a nonzero pseudometric to make its largest distance 1. We shall give a simple and complete characterization of extremeness for metrics with only edge lengths equal to 1/2 and 1. Then we shall use this characterization to give a decomposition result for the upper half of BM(X). BM(X) contains the set of bounded-by-1 pseudoultrametrics, U(X). Ultrametrics satisfy a stronger version of the triangle inequality, and have an interesting structure expressed in terms of partition chains. We will describe the topology of U(X) and its subset of scaled ultrametrics, SU(X), up to homotopy equivalence. Every permutation on a set X can be written as a product of disjoint cycles that cover X. In this way, a permutation generalizes a partition. An iterated cycle structure (ICS) will then be the associated generalization of a partition chain. Analogously, we will compute the “Euler-characteristic” of the set of iterated cycle structures

Inverse methods in quantum many-body physics

The interactions of many quantum particles can give rise to fascinating emergent behavior and exotic phases of matter with no classical analogues. Examples include phases with topological properties, which can occur at low temperatures in frustrated magnets and certain superconductors, and many-body localized (MBL) phases that do not obey the laws of thermodynamics, which can occur in interacting disordered magnets. Traditionally, such quantum phases of matter have been studied using a "forward" approach, where a model for the phase is solved to understand the phase's properties. In this thesis, we explore an alternative "inverse" approach to the problem, where we find models from properties, and show how inverse methods and related tools can be used to efficiently study topological and MBL physics in a new way. In Chapter 1, we introduce the theoretical background necessary for understanding this thesis. First, we discuss the typical forward approach used to study quantum physics and some of its limitations. We introduce the alternative inverse approach that we take in this thesis and give some background on how methods for solving inverse problems have been highly successful in areas such as machine learning and classical physics. Next, we describe two types of topological phases of matter, quantum spin liquids with Wilson loops and topological superconductors with Majorana zero modes (MZMs). These phases have exotic properties, such as long-ranged entanglement and anyonic quasiparticles, that make them interesting to study and potentially useful in emerging technologies such as quantum computing. Finally, we provide an overview of the phenomenon of many-body localization, the failure of many quantum particles to thermalize -- equilibrate with their surroundings -- in the presence of strong interactions and disorder. We introduce the concept of thermalization and discuss how MBL systems defy thermalization. We also explain the various key signatures of MBL physics, such as low-entanglement of eigenstates and the existence of local integrals of motion known as local bits or l-bits. In Chapter 2, we discuss the main numerical techniques that we used to study quantum many-body systems. First, we discuss the exact but computationally expensive exact diagonalization (ED) method, which can be used to study small systems with few quantum spins. Next, we discuss the variational Monte Carlo (VMC) method, which can be used to compute properties of certain classes of variational wave functions by sampling a Markov chain. Then, we explain techniques for performing calculations with tensor networks, a class of quantum states defined through the contraction of many tensors. Finally, in addition to the state-based methods we just described, we also introduce operator-based methods that we be essential for our inverse approach and our study of MBL. In Chapter 3, we introduce the eigenstate-to-Hamiltonian construction (EHC) inverse method that finds Hamiltonians with desired eigenstates. We benchmark the method with many different input states in one and two-dimensions. In each case, we find that the EHC method can find many different Hamiltonians with the target state as an eigenstate, and in many cases a ground state. We show how EHC can be used to find new Hamiltonians with interesting ground states, find Hamiltonians with degenerate ground states, and expand the ground state phase diagrams of previously studied Hamiltonians. In Chapter 4, we introduce the symmetric Hamiltonian construction (SHC) inverse method that finds Hamiltonians with desired symmetries. We use SHC to study quantum spin liquids and topological superconductors. In particular, by providing Wilson loops as input to SHC, we find new types of spin liquid Hamiltonians with properties not seen in previously studied models and, by providing MZMs as input to SHC, we find a large class of superconductor Hamiltonians with tunable MZM physics. In Chapter 5, we develop a tool that allows us to study MBL physics in higher dimensions than was previously possible. While MBL has been clearly observed in one spatial dimension, it is a key open question whether MBL survives in two or three dimensions. Because of the numerical difficulty of studying two and three dimensional quantum systems, this problem has been largely unexplored. We develop an algorithm for finding approximate l-bits, local integral of motions and a key signature of MBL physics, in arbitrary dimensions. Using this algorithm, we observe a sharp change in the properties of l-bits versus disorder strength for four different models in one, two, and three dimensional spin systems. This provides the first evidence for the existence of a thermal to MBL transition in three dimensions. In Chapter 6, we present a method for constructing a large family of Hamiltonians with magnetically ordered "spiral colored" ground states. We demonstrate how these Hamiltonian and states can be arranged into many different geometrical patterns. We also show that with slight modification these Hamiltonians can be made to realize quantum many-body scars, a type of anomalous high-energy excited state that does not exhibit thermal properties as is typical for quantum systems that thermalize. In Chapter 7, we summarize our work and provide an outlook on paths forward

Advanced Concepts in Particle and Field Theory

Uniting the usually distinct areas of particle physics and quantum field theory, gravity and general relativity, this expansive and comprehensive textbook of fundamental and theoretical physics describes the quest to consolidate the elementary particles that are the basic building blocks of nature. Designed for advanced undergraduates and graduate students and abounding in worked examples and detailed derivations, as well as historical anecdotes and philosophical and methodological perspectives, this textbook provides students with a unified understanding of all matter at the fundamental level. Topics range from gauge principles, particle decay and scattering cross-sections, the Higgs mechanism and mass generation, to spacetime geometries and supersymmetry. By combining historically separate areas of study and presenting them in a logically consistent manner, students will appreciate the underlying similarities and conceptual connections across these fields. This title, first published in 2015, has been reissued as an Open Access publication

Advanced Concepts in Particle and Field Theory

Uniting the usually distinct areas of particle physics and quantum field theory, gravity and general relativity, this expansive and comprehensive textbook of fundamental and theoretical physics describes the quest to consolidate the elementary particles that are the basic building blocks of nature. Designed for advanced undergraduates and graduate students and abounding in worked examples and detailed derivations, as well as historical anecdotes and philosophical and methodological perspectives, this textbook provides students with a unified understanding of all matter at the fundamental level. Topics range from gauge principles, particle decay and scattering cross-sections, the Higgs mechanism and mass generation, to spacetime geometries and supersymmetry. By combining historically separate areas of study and presenting them in a logically consistent manner, students will appreciate the underlying similarities and conceptual connections across these fields. This title, first published in 2015, has been reissued as an Open Access publication