Some Comments on Physical Mathematics
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3-Fermion Topological Quantum Computation [1]
3-Fermion topological quantum computation [1] Sam Roberts1 and Dominic J. Williamson2 1Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia 2Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA I. INTRODUCTION Topological quantum computation (TQC) is currently the most promising approach to scalable, fault-tolerant quantum computation. In recent years, the focus has been on TQC with Kitaev's toric code [2], due to it's high threshold to noise [3, 4], and amenability to planar architectures with nearest neighbour interactions. To encode and manipulate quantum information in the toric code, a variety of techniques drawn from condensed matter contexts have been utilised. In particular, some of the efficient approaches for TQC with the toric code rely on creating and manipulating gapped-boundaries, symmetry defects and anyons of the underlying topological phase of matter [5{ 10]. Despite great advances, the overheads for universal fault-tolerant quantum computation remain a formidable challenge. It is therefore important to analyse the potential of TQC in a broad range of topological phases of matter, and attempt to find new computational substrates that require fewer quantum resources to execute fault-tolerant quantum computation. In this work we present an approach to TQC for more general anyon theories based on the Walker{Wang mod- els [11]. This provides a rich class of spin-lattice models in three-dimensions whose boundaries can naturally be used to topologically encode quantum information. The two-dimensional boundary phases of Walker{Wang models accommodate a richer set of possibilities than stand-alone two-dimensional topological phases realized by commuting projector codes [12, 13]. -
FIELDS MEDAL for Mathematical Efforts R
Recognizing the Real and the Potential: FIELDS MEDAL for Mathematical Efforts R Fields Medal recipients since inception Year Winners 1936 Lars Valerian Ahlfors (Harvard University) (April 18, 1907 – October 11, 1996) Jesse Douglas (Massachusetts Institute of Technology) (July 3, 1897 – September 7, 1965) 1950 Atle Selberg (Institute for Advanced Study, Princeton) (June 14, 1917 – August 6, 2007) 1954 Kunihiko Kodaira (Princeton University) (March 16, 1915 – July 26, 1997) 1962 John Willard Milnor (Princeton University) (born February 20, 1931) The Fields Medal 1966 Paul Joseph Cohen (Stanford University) (April 2, 1934 – March 23, 2007) Stephen Smale (University of California, Berkeley) (born July 15, 1930) is awarded 1970 Heisuke Hironaka (Harvard University) (born April 9, 1931) every four years 1974 David Bryant Mumford (Harvard University) (born June 11, 1937) 1978 Charles Louis Fefferman (Princeton University) (born April 18, 1949) on the occasion of the Daniel G. Quillen (Massachusetts Institute of Technology) (June 22, 1940 – April 30, 2011) International Congress 1982 William P. Thurston (Princeton University) (October 30, 1946 – August 21, 2012) Shing-Tung Yau (Institute for Advanced Study, Princeton) (born April 4, 1949) of Mathematicians 1986 Gerd Faltings (Princeton University) (born July 28, 1954) to recognize Michael Freedman (University of California, San Diego) (born April 21, 1951) 1990 Vaughan Jones (University of California, Berkeley) (born December 31, 1952) outstanding Edward Witten (Institute for Advanced Study, -
Arxiv:1705.01740V1 [Cond-Mat.Str-El] 4 May 2017 2
Physics of the Kitaev model: fractionalization, dynamical correlations, and material connections M. Hermanns1, I. Kimchi2, J. Knolle3 1Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany 2Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA and 3T.C.M. group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge, CB3 0HE, United Kingdom Quantum spin liquids have fascinated condensed matter physicists for decades because of their unusual properties such as spin fractionalization and long-range entanglement. Unlike conventional symmetry breaking the topological order underlying quantum spin liquids is hard to detect exper- imentally. Even theoretical models are scarce for which the ground state is established to be a quantum spin liquid. The Kitaev honeycomb model and its generalizations to other tri-coordinated lattices are chief counterexamples | they are exactly solvable, harbor a variety of quantum spin liquid phases, and are also relevant for certain transition metal compounds including the polymorphs of (Na,Li)2IrO3 Iridates and RuCl3. In this review, we give an overview of the rich physics of the Kitaev model, including 2D and 3D fractionalization as well as dynamical correlations and behavior at finite temperatures. We discuss the different materials, and argue how the Kitaev model physics can be relevant even though most materials show magnetic ordering at low temperatures. arXiv:1705.01740v1 [cond-mat.str-el] 4 May 2017 2 CONTENTS I. Introduction 2 II. Kitaev quantum spin liquids 3 A. The Kitaev model 3 B. Classifying Kitaev quantum spin liquids by projective symmetries 4 C. Confinement and finite temperature 5 III. Symmetry and chemistry of the Kitaev exchange 6 IV. -
Inflation, Large Branes, and the Shape of Space
Inflation, Large Branes, and the Shape of Space Brett McInnes National University of Singapore email: [email protected] ABSTRACT Linde has recently argued that compact flat or negatively curved spatial sections should, in many circumstances, be considered typical in Inflationary cosmologies. We suggest that the “large brane instability” of Seiberg and Witten eliminates the negative candidates in the context of string theory. That leaves the flat, compact, three-dimensional manifolds — Conway’s platycosms. We show that deep theorems of Schoen, Yau, Gromov and Lawson imply that, even in this case, Seiberg-Witten instability can be avoided only with difficulty. Using a specific cosmological model of the Maldacena-Maoz type, we explain how to do this, and we also show how the list of platycosmic candidates can be reduced to three. This leads to an extension of the basic idea: the conformal compactification of the entire Euclidean spacetime also has the topology of a flat, compact, four-dimensional space. arXiv:hep-th/0410115v2 19 Oct 2004 1. Nearly Flat or Really Flat? Linde has recently argued [1] that, at least in some circumstances, we should regard cosmological models with flat or negatively curved compact spatial sections as the norm from an Inflationary point of view. Here we wish to argue that cosmic holography, in the novel form proposed by Maldacena and Maoz [2], gives a deep new interpretation of this idea, and also sharpens it very considerably to exclude the negative case. This focuses our attention on cosmological models with flat, compact spatial sections. Current observations [3] show that the spatial sections of our Universe [as defined by observers for whom local isotropy obtains] are fairly close to being flat: the total density parameter Ω satisfies Ω = 1.02 0.02 at 95% confidence level, if we allow the imposition ± of a reasonable prior [4] on the Hubble parameter. -
Christopher Elliott
Christopher Elliott Contact Details Full Name Christopher James Elliott Date of Birth 20th Sep, 1987 Nationality British Contact Address University of Massachusetts Department of Mathematics and Statistics 710 N Pleasant St Amherst, MA 01003 USA E-mail Address [email protected] Work Experience 2019– Visiting Assistant Professor, University of Massachusetts, Amherst 2016–2019 Postdoctoral Fellow, Institut des Hautes Etudes´ Scientifiques Education 2010–2016 PhD, Northwestern University Advisors: Kevin Costello and David Nadler Thesis Title: Gauge Theoretic Aspects of the Geometric Langlands Correspondence. 2009–2010 MMath (Mathematics Tripos: Part III), University of Cambridge, With Distinction. Part III Essay: D-Modules and Hodge Theory 2006–2009 BA (hons) (Mathematics), University of Cambridge, 1st Class. Research Visits 2014–2018 Perimeter Institute (7 visits, each 1-3 weeks) Oct–Nov 2017 MPIM, Bonn Oct 2017 Hausdorff Institute, Bonn Nov 2016 MPIM, Bonn Research Interests I’m interested in mathematical aspects and applications of quantum field theory. In particular • The construction and classification of (not necessarily topological) twists of classical and quantum field theories, especially using techniques of derived algebraic geometry and homotopical algebra. • The connection between structures appearing in various versions of the geometric Langlands correspon- dence and twists of four-, five- and six-dimensional supersymmetric gauge theories. • The theory of factorization algebras as a model for perturbative quantum field theory, possibly with bound- ary conditions and defects. Publications and Preprints 1. Quantum Geometric Langlands Categories from N = 4 Super Yang–Mills Theory (joint with Philsang Yoo), arXiv:2008.10988 2. Spontaneous Symmetry Breaking: a View from Derived Geometry (joint with Owen Gwilliam), Journal of Geometry and Physics, Vol 162, 2021, arXiv:2008.02302 1 3. -
What's Inside
Newsletter A publication of the Controlled Release Society Volume 32 • Number 1 • 2015 What’s Inside 42nd CRS Annual Meeting & Exposition pH-Responsive Fluorescence Polymer Probe for Tumor pH Targeting In Situ-Gelling Hydrogels for Ophthalmic Drug Delivery Using a Microinjection Device Interview with Paolo Colombo Patent Watch Robert Langer Awarded the Queen Elizabeth Prize for Engineering Newsletter Charles Frey Vol. 32 • No. 1 • 2015 Editor Table of Contents From the Editor .................................................................................................................. 2 From the President ............................................................................................................ 3 Interview Steven Giannos An Interview with Paolo Colombo from University of Parma .............................................. 4 Editor 42nd CRS Annual Meeting & Exposition .......................................................................... 6 What’s on Board Access the Future of Delivery Science and Technology with Key CRS Resources .............. 9 Scientifically Speaking pH-Responsive Fluorescence Polymer Probe for Tumor pH Targeting ............................. 10 Arlene McDowell Editor In Situ-Gelling Hydrogels for Ophthalmic Drug Delivery Using a Microinjection Device ........................................................................................................ 12 Patent Watch ................................................................................................................... 14 Special -
Anomalies, Dualities, and Topology of D = 6 N = 1 Superstring Vacua
RU-96-16 NSF-ITP-96-21 CALT-68-2057 IASSNS-HEP-96/53 hep-th/9605184 Anomalies, Dualities, and Topology of D =6 N =1 Superstring Vacua Micha Berkooz,1 Robert G. Leigh,1 Joseph Polchinski,2 John H. Schwarz,3 Nathan Seiberg,1 and Edward Witten4 1Department of Physics, Rutgers University, Piscataway, NJ 08855-0849 2Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030 3California Institute of Technology, Pasadena, CA 91125 4Institute for Advanced Study, Princeton, NJ 08540 Abstract We consider various aspects of compactifications of the Type I/heterotic Spin(32)/Z2 theory on K3. One family of such compactifications in- cludes the standard embedding of the spin connection in the gauge group, and is on the same moduli space as the compactification of arXiv:hep-th/9605184v1 25 May 1996 the heterotic E8 × E8 theory on K3 with instanton numbers (8,16). Another class, which includes an orbifold of the Type I theory re- cently constructed by Gimon and Polchinski and whose field theory limit involves some topological novelties, is on the moduli space of the heterotic E8 × E8 theory on K3 with instanton numbers (12,12). These connections between Spin(32)/Z2 and E8 × E8 models can be demonstrated by T duality, and permit a better understanding of non- perturbative gauge fields in the (12,12) model. In the transformation between Spin(32)/Z2 and E8 × E8 models, the strong/weak coupling duality of the (12,12) E8 × E8 model is mapped to T duality in the Type I theory. The gauge and gravitational anomalies in the Type I theory are canceled by an extension of the Green-Schwarz mechanism. -
Kitaev Materials
Kitaev Materials Simon Trebst Institute for Theoretical Physics University of Cologne, 50937 Cologne, Germany Contents 1 Spin-orbit entangled Mott insulators 2 1.1 Bond-directional interactions . 4 1.2 Kitaev model . 6 2 Honeycomb Kitaev materials 9 2.1 Na2IrO3 ...................................... 9 2.2 ↵-Li2IrO3 ..................................... 10 2.3 ↵-RuCl3 ...................................... 11 3 Triangular Kitaev materials 15 3.1 Ba3IrxTi3 xO9 ................................... 15 − 3.2 Other materials . 17 4 Three-dimensional Kitaev materials 17 4.1 Conceptual overview . 18 4.2 β-Li2IrO3 and γ-Li2IrO3 ............................. 22 4.3 Other materials . 23 5 Outlook 24 arXiv:1701.07056v1 [cond-mat.str-el] 24 Jan 2017 Lecture Notes of the 48th IFF Spring School “Topological Matter – Topological Insulators, Skyrmions and Majoranas” (Forschungszentrum Julich,¨ 2017). All rights reserved. 2 Simon Trebst 1 Spin-orbit entangled Mott insulators Transition-metal oxides with partially filled 4d and 5d shells exhibit an intricate interplay of electronic, spin, and orbital degrees of freedom arising from a largely accidental balance of electronic correlations, spin-orbit entanglement, and crystal-field effects [1]. With different ma- terials exhibiting slight tilts towards one of the three effects, a remarkably broad variety of novel forms of quantum matter can be explored. On the theoretical side, topology is found to play a crucial role in these systems – an observation which, in the weakly correlated regime, has lead to the discovery of the topological band insulator [2, 3] and subsequently its metallic cousin, the Weyl semi-metal [4, 5]. Upon increasing electronic correlations, Mott insulators with unusual local moments such as spin-orbit entangled degrees of freedom can form and whose collective behavior gives rise to unconventional types of magnetism including the formation of quadrupo- lar correlations or the emergence of so-called spin liquid states. -
Doing Physics with Math ...And Vice Versa
Doing physics with math ...and vice versa Alexander Tabler Thessaloniki, December 19, 2019 Thessaloniki, December 19, 2019 0 / 14 Outline 1 Physics vs. Mathematics 2 Case study: a question about shapes 3 Enter physics & quantum field theory 4 Summary Thessaloniki, December 19, 2019 1 / 14 1. Physics vs. Mathematics Thessaloniki, December 19, 2019 1 / 14 Common origins, divergent futures At some point, Mathematics and Physics were the same discipline: Natural philosophy. Calculus: mathematics or physics? \Isaac Newton" - Godfrey Kneller - commons.curid=37337 Thessaloniki, December 19, 2019 2 / 14 Today, Math=Physics6 Mathematics: - Definition. Theorem. Proof. (Bourbaki-style mathematics) - Logical rigor, \waterproof" - \Proof"-driven (Theoretical) Physics: - Less rigor - Computation-oriented (\shut up and calculate") - Experiment-driven ...in principle Thessaloniki, December 19, 2019 3 / 14 An early 20th century divorce Freeman Dyson, 1972: \The marriage between mathematics and physics has recently ended in divorce"1 F. Dyson - by ioerror - commons.curid=71877726 1 Dyson - \Missed Opportunities", Bull. Amer. Math. Soc., Volume 78, Number 5 (1972), 635-652. Thessaloniki, December 19, 2019 4 / 14 A fruitful relation However, strong bonds still remain: - Modern theoretical physics uses more abstract mathematics - There is a deep relationship between advances in both fields - This relationship is mutually beneficial https://www.ias.edu/ideas/power-mirror-symmetry Thessaloniki, December 19, 2019 5 / 14 2. Case study: a question about shapes Thessaloniki, -
String Theory for Pedestrians
String Theory for Pedestrians Johanna Knapp ITP, TU Wien Atominstitut, March 4, 2016 Outline Fundamental Interactions What is String Theory? Modern String Theory String Phenomenology Gauge/Gravity Duality Physical Mathematics Conclusions Fundamental Interactions What is String Theory? Modern String Theory Conclusions Fundamental Interactions • Four fundamental forces of nature • Electromagnetic Interactions: Photons • Strong Interactions: Gluons • Weak Interactions: W- and Z-bosons • Gravity: Graviton (?) • Successful theoretical models • Quantum Field Theory • Einstein Gravity • Tested by high-precision experiments Fundamental Interactions What is String Theory? Modern String Theory Conclusions Gravity • General Relativity 1 R − g R = 8πGT µν 2 µν µν • Matter $ Curvature of spacetime • No Quantum Theory of Gravity! • Black Holes • Big Bang Fundamental Interactions What is String Theory? Modern String Theory Conclusions Particle Interactions • Strong, weak and electromagnetic interactions can be described by the Standard Model of Particle Physics • Quantum Field Theory • Gauge Theory Fundamental Interactions What is String Theory? Modern String Theory Conclusions Unification of Forces • Can we find a unified quantum theory of all fundamental interactions? • Reconciliation of two different theoretical frameworks • Quantum Theory of Gravity • Why is unification a good idea? • It worked before: electromagnetism, electroweak interactions • Evidence of unification of gauge couplings • Open problems of fundamental physics should be solved. Fundamental Interactions What is String Theory? Modern String Theory Conclusions History of the Universe • Physics of the early universe Fundamental Interactions What is String Theory? Modern String Theory Conclusions Energy Budget of the Universe • Only about 5% of the matter in the universe is understood. • What is dark matter? Extra particles? Fundamental Interactions What is String Theory? Modern String Theory Conclusions String Theory Fact Sheet • String theory is a quantum theory that unifies the fundamental forces of nature. -
SHELDON LEE GLASHOW Lyman Laboratory of Physics Harvard University Cambridge, Mass., USA
TOWARDS A UNIFIED THEORY - THREADS IN A TAPESTRY Nobel Lecture, 8 December, 1979 by SHELDON LEE GLASHOW Lyman Laboratory of Physics Harvard University Cambridge, Mass., USA INTRODUCTION In 1956, when I began doing theoretical physics, the study of elementary particles was like a patchwork quilt. Electrodynamics, weak interactions, and strong interactions were clearly separate disciplines, separately taught and separately studied. There was no coherent theory that described them all. Developments such as the observation of parity violation, the successes of quantum electrodynamics, the discovery of hadron resonances and the appearance of strangeness were well-defined parts of the picture, but they could not be easily fitted together. Things have changed. Today we have what has been called a “standard theory” of elementary particle physics in which strong, weak, and electro- magnetic interactions all arise from a local symmetry principle. It is, in a sense, a complete and apparently correct theory, offering a qualitative description of all particle phenomena and precise quantitative predictions in many instances. There is no experimental data that contradicts the theory. In principle, if not yet in practice, all experimental data can be expressed in terms of a small number of “fundamental” masses and cou- pling constants. The theory we now have is an integral work of art: the patchwork quilt has become a tapestry. Tapestries are made by many artisans working together. The contribu- tions of separate workers cannot be discerned in the completed work, and the loose and false threads have been covered over. So it is in our picture of particle physics. Part of the picture is the unification of weak and electromagnetic interactions and the prediction of neutral currents, now being celebrated by the award of the Nobel Prize. -
2005 March Meeting Gears up for Showtime in the City of Angels
NEWS See Pullout Insert Inside March 2005 Volume 14, No. 3 A Publication of The American Physical Society http://www.aps.org/apsnews 2005 March Meeting Gears Up for Reborn Nicholson Medal Showtime in the City of Angels Stresses Mentorship Established in memory of Dwight The latest research relevance to the design R. Nicholson of the University of results on the spin Hall and creation of next- Iowa, who died tragically in 1991, effect, new chemistry generation nano-electro- and first given in 1994, the APS with superatoms, and mechanical systems Nicholson Medal has been reborn several sessions cel- (NEMS). Moses Chan this year as an award for human ebrating all things (Pennsylvania State Uni- outreach. According to the infor- Einstein are among the versity) will talk about mation contained on the Medal’s expected highlights at evidence of Bose-Einstein web site (http://www.aps.org/praw/ the 2005 APS March condensation in solid he- nicholso/index.cfm), the Nicholson meeting, to be held later lium, while Stanford Medal for Human Outreach shall Photo Credit: Courtesy of the Los Angeles Convention Center this month in Los University’s Zhixun Shen be awarded to a physicist who ei- Angeles, California. The will discuss how photo- ther through teaching, research, or Photo from Iowa University Relations. conference is the largest physics Boltzmann, and Ehrenfest, but also emission spectroscopy has science-related activities, Dwight R. Nicholson. meeting of the year, featuring some Emmy Noether, one of the rare emerged as a leading tool to push