DOCSLIB.ORG

Non-Abelian Bosonization in Two and Three Spatial Dimensions and Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Non-Abelian Bosonization in Two and Three Spatial Dimensions and Applications Non-abelian bosonization in two and three spatial dimensions and applications Yen-Ta Huanga,b, Dung-Hai Leea,b aDepartment of Physics, University of California, Berkeley, California 94720, USA bMaterials Sciences Division, Lawrence Berkeley National Laboratories, Berkeley, California 94720, USA Abstract In this paper, we generalize Witten's non-abelian bosonization in (1+1)-D to two and three spatial dimensions. Our theory applies to fermions with relativistic dispersion. The bosonized theories are non-linear sigma models with level-1 Wess-Zumino-Witten terms. We apply the bosonization results to the SU(2) gauge theory of the π-flux phase, critical spin liquids in 1,2,3 spatial dimensions, and twisted bilayer graphene. Contents Introduction 6 I Bosonization 7 1 The idea 7 2 Emergent symmetries of the massless fermion theory 8 2.1 Complex class . .8 2.2 Real class . .9 3 Mass terms and mass manifolds 10 3.1 Complex class . 11 3.2 Real class . 12 4 The symmetry anomalies of the fermionic theories 13 4.1 The 't Hooft anomaly of continuous symmetry . 13 arXiv:2104.07015v2 [cond-mat.str-el] 23 Apr 2021 4.2 A heuristic way to determine the 't Hooft anomaly . 15 4.3 Discrete global symmetry anomaly . 16 Email addresses: [email protected] (Yen-Ta Huang), [email protected] (Dung-Hai Lee) Preprint submitted to Elsevier April 27, 2021 5 Breaking the emergent symmetry by the mass terms 19 5.1 Complex class . 19 5.2 Real class . 19 6 Restoring the emergent symmetries 21 7 The conditions for the effective theory being bosonic 22 8 Fermion integration 22 8.1 Complex class . 22 8.2 Real class . 25 9 Non-linear sigma models in (2 + 1)-D and (3 + 1)-D 25 9.1 Complex class in (2 + 1)-D . 26 9.2 Real class in (2 + 1)-D . 27 9.3 Complex class in (3 + 1)-D . 28 9.4 Real class in (3 + 1)-D . 28 9.5 The value of the stiffness constant and the phases of non-linear sigma models . 29 10 Non-linear sigma models as the effective theories of interacting fermion models 30 11 Global symmetries of the non-linear sigma models 32 12 The symmetry anomalies of the nonlinear sigma models 33 12.1 Gauging the non-linear sigma models and the 't Hooft anomalies 34 12.2 Discrete symmetry anomalies . 35 13 Soliton of the non-linear sigma models and the Wess-Zumino- Witten terms 37 13.1 Soliton classification . 37 13.2 Soliton charge and the conserved U(1) charge Q ......... 38 13.3 Statistics of soliton . 39 14 A summary of part I 39 II Applications 40 15 The SU(2) gauge theory of the π-flux phase of the half-filled Hubbard model 40 15.1 The \spinon" representation of the spin operator . 40 15.2 The π-flux phase mean-field theory and the SU(2) gauge fluctu- ations . 42 15.3 Antiferromagnet, Valence bond solid, and the “deconfined” quan- tum crtical point . 46 2 16 The critical spin liquid of \bipartite Mott insulators" in D = 1 + 1; 2 + 1 and 3 + 1. 48 16.1 (1+1)-D . 49 16.1.1 The analog of the π-flux phase . 49 16.1.2 The charge-SU(2) confinement . 50 16.2 (2 + 1)-D . 51 16.2.1 The analog of the π-flux phase . 51 16.2.2 The charge-SU(2) confinement . 53 16.3 (3 + 1)-D . 54 16.3.1 The analog of the π-flux phase . 54 16.3.2 The charge-SU(2) confinement . 57 16.3.3 Gapping out the U(1) mode . 58 17 Twisted bi-layer graphene 60 17.1 Charge neutral point ν =0 ..................... 61 17.2 ν = ±1; ±2; ±3 ............................ 63 Conclusions 66 Acknowledgement 67 Appendices 67 A The emergent symmetries for (2 + 1) and (3 + 1)-D 67 A.1 Complex class in (2 + 1)-D . 67 A.2 Real class in (2 + 1)-D . 68 A.3 Complex class in (3 + 1)-D . 68 A.4 Real class in (3 + 1)-D . 69 B The mass manifolds, homotopy groups and symmetry transfor- mations 70 B.1 Complex class in (2 + 1)-D . 70 B.2 Real class in (2 + 1)-D . 71 B.3 Complex class in (3 + 1)-D . 73 B.4 Real class in (3 + 1)-D . 74 C The anomalies of the fermion theories 76 C.1 The choice of discrete symmetry generators . 76 C.2 Complex class in (1 + 1)-D . 76 C.3 Real class in (1 + 1)-D . 77 C.4 Complex class in (2 + 1)-D . 78 C.5 Real class in (2 + 1)-D . 79 C.6 Complex class in (3 + 1)-D . 79 C.7 Real class in (3 + 1)-D . 80 3 D Fermion integration 80 D.1 Integrating out real versus complex fermions . 81 D.2 Integrating out complex fermions . 82 D.2.1 The stiffness term . 84 D.2.2 The WZW (topological) term . 85 D.3 The fermion integration results for sufficiently large n so that the WZW term is stabilized . 86 D.3.1 Complex class in (1 + 1)-D . 86 D.3.2 Real class in (1 + 1)-D . 88 D.3.3 Complex class in (2 + 1)-D . 89 D.3.4 Real class in (2 + 1)-D . 90 D.3.5 Complex class in (3 + 1)-D . 91 D.3.6 Real class in (3 + 1)-D . 93 D.4 The less relevant real terms originate from Eq.(118) and Eq.(119) 94 D.4.1 (1 + 1)-D . 94 D.4.2 (2 + 1)-D . 95 D.4.3 (3 + 1)-D . 96 E Emergent symmetries of the nonlinear sigma models 100 E.1 Complex class in (2 + 1)-D . 100 E.2 Complex class in (3 + 1)-D . 102 F Anomalies of the nonlinear sigma models 104 F.1 The ('t Hooft) anomalies associated with continuous symmetries 104 F.1.1 Complex class in (1 + 1)-D . 104 F.1.2 Real class in (1 + 1)-D . 105 F.1.3 Complex class in (2 + 1)-D . 106 F.1.4 Real class in (2 + 1)-D . 109 F.1.5 Complex class in (3 + 1)-D . 110 F.1.6 Real class in (3 + 1)-D . 110 F.2 Anomalies with respect to the discrete groups . 112 G Soliton's statistics 112 G.1 Complex class in (2 + 1)-D . 113 G.2 Real class in (2 + 1)-D . 114 G.2.1 How to test whether a proposed Z2 soliton is trivial or not 115 G.2.2 The Berry phase of a self-rotating Z2 soliton . 119 G.3 Complex class in (3 + 1)-D . 119 G.4 Real class in (3 + 1)-D . 121 H Bosonization for small flavor number 122 H.1 Complex class in 2 + 1 D with n =2................ 122 H.2 Mass manifold enlargement . 123 H.3 The derivation of the Hopf term . 123 H.4 Gauging small n non-linear sigma models . 125 4 I Massless fermions as the boundary of bulk topological insula- tors/superconductors 126 I.1 The Z classification . 126 I.2 Construction of the bulk SPT . 127 J The decoupling of the charge-SU(2) gauge field from the low energy non-linear sigma model after confinement 128 K The WZW term in the (3 + 1)-D real class non-linear sigma model 129 5 Introduction Bosonization in (1 + 1)-D has been a very useful theoretical tool. It allows one to map a theory, where the fundamental degrees of freedom are fermionic, to a theory with bosonic degrees of freedom. Often, things that can be seen easily in one picture are difficult to see in the other. The best-known bosonization is the abelian bosonization [1, 2, 3, 4], where fermions are solitons in the Bose field. A shortcoming of the abelian bosonization, when fermions have flavor (e.g., spin) degrees of freedom, is that the flavor symmetries are hidden. This problem was solved by Witten's non-abelian bosonization [5]. In this paper we generalize Witten's non-abelian bosonization to (2 + 1) and (3 + 1) space-time dimensions. The limitation of our theory is that it only applies to fermions with relativistic dispersion. (However, we do not restrict the Fermi velocity to be the speed of light.) In the absence of a mass gap, such theories have Dirac-like dispersion rela- tion. In one space dimension, massless fermions are generically relativistic at low energies. In two and three space dimensions, relativistic massless fermions have been discovered in many experimental condensed matter systems. Examples in- clude graphene and twisted bilayer graphene, Dirac and Weyl semi-metal,...etc. Moreover, relativistic massless fermions can appear in the mean-field theory of strongly correlated systems. Such theory serves as the starting point of a more rigorous treatment. For example, the \spinon π-flux phase" mean-field theory sets the stage for a gauge theory description of the Mott insulating state of the half-filled Hubbard model. Another important area where relativistic massless fermions appear is at the boundary of topological insulators or superconductors, which are simple ex- amples of symmetry-protected topological (SPT) phases. The classification of topological insulator/superconductor[6, 7] can be viewed as asking how many copies of the massless fermion theories on the boundary are required to couple together before a symmetry-allowed mass term emerges. The paper contains two major parts: I. bosonization and II. applications. Each of them contains several sections, namely, 14 sections in part I and 3 sections in part II.
Load more

Recommended publications
  • Bosonization and Mirror Symmetry Arxiv:1608.05077V1 [Hep-Th]
    Bosonization and Mirror Symmetry Shamit Kachru1, Michael Mulligan2,1, Gonzalo Torroba3, and Huajia Wang4 1Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA 2Department of Physics and Astronomy, University of California, Riverside, CA 92511, USA 3Centro At´omico Bariloche and CONICET, R8402AGP Bariloche, ARG 4Department of Physics, University of Illinois, Urbana-Champaign, IL 61801, USA Abstract We study bosonization in 2+1 dimensions using mirror symmetry, a duality that relates pairs of supersymmetric theories. Upon breaking supersymmetry in a controlled way, we dynamically obtain the bosonization duality that equates the theory of a free Dirac fermion to QED3 with a single scalar boson. This duality may be used to demonstrate the bosonization duality relating an O(2)-symmetric Wilson-Fisher fixed point to QED3 with a single Dirac fermion, Peskin-Dasgupta-Halperin duality, and the recently conjectured duality relating the theory of a free Dirac fermion to fermionic QED3 with a single flavor. Chern-Simons and BF couplings for both dynamical and background gauge fields play a central role in our approach. In the course of our study, we describe a \chiral" mirror pair that may be viewed as the minimal supersymmetric generalization of the two bosonization dualities. arXiv:1608.05077v1 [hep-th] 17 Aug 2016 Contents 1 Introduction 1 2 Mirror symmetry and its deformations 4 2.1 Superfields and lagrangians . .5 2.2 = 4 mirror symmetry . .7 N 2.3 Deformations by U(1)A and U(1)R backgrounds . 10 2.4 General mirror duality . 11 3 Chiral mirror symmetry 12 3.1 Chiral theory A .
    [Show full text]
  • Physics 215C: Particles and Fields Spring 2019
    Physics 215C: Particles and Fields Spring 2019 Lecturer: McGreevy These lecture notes live here. Please email corrections to mcgreevy at physics dot ucsd dot edu. Last updated: 2021/04/20, 14:28:40 1 Contents 0.1 Introductory remarks for the third quarter................4 0.2 Sources and acknowledgement.......................7 0.3 Conventions.................................8 1 Anomalies9 2 Effective field theory 20 2.1 A parable on integrating out degrees of freedom............. 20 2.2 Introduction to effective field theory.................... 25 2.3 The color of the sky............................. 30 2.4 Fermi theory of Weak Interactions..................... 32 2.5 Loops in EFT................................ 33 2.6 The Standard Model as an EFT...................... 39 2.7 Superconductors.............................. 42 2.8 Effective field theory of Fermi surfaces.................. 47 3 Geometric and topological terms in field theory actions 58 3.1 Coherent state path integrals for bosons................. 58 3.2 Coherent state path integral for fermions................. 66 3.3 Path integrals for spin systems....................... 73 3.4 Topological terms from integrating out fermions............. 86 3.5 Pions..................................... 89 4 Field theory of spin systems 99 4.1 Transverse-Field Ising Model........................ 99 4.2 Ferromagnets and antiferromagnets..................... 131 4.3 The beta function for 2d non-linear sigma models............ 136 4.4 CP1 representation and large-N ...................... 138 5 Duality 148 5.1 XY transition from superfluid to Mott insulator, and T-duality..... 148 6 Conformal field theory 158 6.1 The stress tensor and conformal invariance (abstract CFT)....... 160 6.2 Radial quantization............................. 166 6.3 Back to general dimensions......................... 172 7 Duality, part 2 179 7.1 (2+1)-d XY is dual to (2+1)d electrodynamics.............
    [Show full text]
  • Effective Field Theories for Generalized Spin Ladders
    Effective Field Theories for Generalized Spin Ladders Vom Fachbereich Physik der Universitat¨ Hannover zur Erlangung des Grades Doktor der Naturwissenschaften – Dr. rer. nat. – genehmigte Dissertation von M. Sc. Teimuraz Vekua geboren am 23.05.1976 in Tbilissi, Georgien 2003 Referent: Prof. Dr. H.-J. Mikeska Korreferent: Prof. Dr. O. Lechtenfeld Tag der Promotion: 15. Juli 2003 Effektive Feldtheorien fur¨ verallgemeinerte Spinleitern Kurzzusammenfassung Thema der vorliegenden Arbeit ist die Untersuchung des Phasendiagramms verallgemei- nerter Spinleitern mit effektiver feldtheoretischer Beschreibung. Zwei verschiedene Spin Systeme werden untersucht. Im ersten Teil der Doktorarbeit wird eine antiferromagneti- sche SU(2)-symmetrische Spinleiter mit zusatzlicher¨ zyklischer Vier-Spin-Wechselwirkung (Ringaustausch) untersucht. Das Hauptziel dieses Teiles der Doktorarbeit ist die Erklarung¨ des Frustrationseffektes produziert durch Ringaustausch auf dem Grundzustandsphasendia- gramm und des Verhaltens des Modelles bei endlichen Temperaturen. Die Bosonisierung-Fermionisierungsmethode wird verwendet fur¨ die Beschreibung der nie- derenergetischen Anregungen der Spinleiter mit den massiven, schwach wechselwirkenden Majorana Fermionen. Diese Methode ermoglicht¨ auch einen Teil des Grundzustandsphasen- diagramms fur¨ genugend¨ starke Frustration zu bekommen. Der gewohnliche¨ Sprossensinglett- Grundzustand der Spinleiter geht durch einen Quantenphasenubergang¨ in eine spontan dime- risierte Phase mit zunehmendem Ringaustausch uber.¨ Eine weitere
    [Show full text]
  • Fermions in Two Dimensions and Exactly Solvable Models
    Doctoral Thesis Fermions in two dimensions and exactly solvable models Jonas de Woul Mathematical Physics, Department of Theoretical Physics School of Engineering Sciences Royal Institute of Technology, SE-106 91 Stockholm, Sweden Stockholm, Sweden 2011 Typeset in LATEX Akademisk avhandling f¨or avl¨aggande av Teknologie Doktorsexamen (TeknD) inom ¨amnesomr˚adet Teoretisk Fysik. Scientific thesis for the degree of Doctor of Philosophy (PhD) in the subject area of Theoretical Physics. ISBN 978-91-7501-174-5 TRITA-FYS-2011:56 ISSN 0280-316X ISRN KTH/FYS/--11:56--SE c Jonas de Woul, November 2011 Printed in Sweden by Universitetsservice US AB, Stockholm November 2011 Abstract This Ph.D. thesis in mathematical physics concerns systems of interacting fermions with strong correlations. For these systems the physical properties can only be described in terms of the collective behavior of the fermions. Moreover, they are often characterized by a close competition between fermion localization versus de- localization, which can result in complex and exotic physical phenomena. Strongly correlated fermion systems are usually modelled by many-body Hamil- tonians for which the kinetic- and interaction energy have the same order of mag- nitude. This makes them challenging to study as the application of conventional computational methods, like mean field- or perturbation theory, often gives un- reliable results. Of particular interest are Hubbard-type models, which provide minimal descriptions of strongly correlated fermions. The research of this thesis focuses on such models defined on two-dimensional square lattices. One motivation for this is the so-called high-Tc problem of the cuprate superconductors. A main hypothesis is that there exists an underlying Fermi surface with nearly flat parts, i.e.
    [Show full text]
  • Bosonization I: the Fermion–Boson Dictionary
    17 Bosonization I: The Fermion–Boson Dictionary “Bosonization” refers to the possibility of describing a theory of relativistic Dirac fermions obeying standard anticommutation rules by a boson field theory. While this may be possible in all dimensions, it has so far proved most useful in d = 1, where the bosonic version of the given fermionic theory is local and simple, and often simpler than the Fermi theory. This chapter should be viewed as a stepping stone toward a more thorough approach, for which references are given at the end. In this chapter I will set up the bosonization machine, explaining its basic logic and the dictionary for transcribing a fermionic theory to a bosonic theory. The next chapter will be devoted to applications. To my knowledge, bosonization, as described here, was first carried out by Lieb and Mattis [1] in their exact solution of the Luttinger model [2]. Later, Luther and Peschel [3] showed how to use it to find asymptotic (low momentum and energy) correlation functions for more generic interacting Fermi systems. It was independently discovered in particle physics by Coleman [4], and further developed by Mandelstam [5]. Much of what I know and use is inspired by the work of Luther and Peschel. 17.1 Preamble Before getting into any details, I would first like to answer two questions. First, if bosonization applies only to relativistic Dirac fermions, why is it of any interest to condensed matter theory where relativity is not essential? Second, what is the magic by which bosonization helps us tame interacting field theories? As for the first question, there are two ways in which Dirac fermions enter condensed matter physics.
    [Show full text]
  • A Schema for Duality, Illustrated by Bosonization
    A Schema for Duality, Illustrated by Bosonization Sebastian De Haro and Jeremy Butterfield Trinity College, Cambridge, CB2 1TQ, United Kingdom Department of History and Philosophy of Science, University of Cambridge Free School Lane, Cambridge CB2 3RH, United Kingdom [email protected], [email protected] Friday 22 Dec 2017 Abstract In this paper we present a schema for describing dualities between physical theories (Sections 2 and 3), and illustrate it in detail with the example of bosonization: a boson- fermion duality in two-dimensional quantum field theory (Sections 4 and 5). The schema develops proposals in De Haro (2016, 2016a): these proposals include construals of notions related to duality, like representation, model, symmetry and inter- pretation. The aim of the schema is to give a more precise criterion for duality than has so far been considered. The bosonization example, or boson-fermion duality, has the feature of being simple yet rich enough to illustrate the most relevant aspects of our schema, which also apply to more sophisticated dualities. The richness of the example consists, mainly, in its concern with two non-trivial quantum field theories: including massive Thirring-sine-Gordon duality, and non-abelian bosonization. This prompts two comparisons with the recent philosophical literature on dualities:| (a) Unlike the standard cases of duality in quantum field theory and string theory, where only specific simplifying limits of the theories are explicitly known, the boson- fermion duality is known to hold exactly. This exactness can be exhibited explicitly. (b) The bosonization example illustrates both the cases of isomorphic and non-isomorphic models: which we believe the literature on dualities has not so far discussed.
    [Show full text]
  • Bosonization Vs. Supersymmetry
    University of Kentucky UKnowledge University of Kentucky Doctoral Dissertations Graduate School 2006 BOSONIZATION VS. SUPERSYMMETRY Herbert Morales University of Kentucky, [email protected] Right click to open a feedback form in a new tab to let us know how this document benefits ou.y Recommended Citation Morales, Herbert, "BOSONIZATION VS. SUPERSYMMETRY" (2006). University of Kentucky Doctoral Dissertations. 429. https://uknowledge.uky.edu/gradschool_diss/429 This Dissertation is brought to you for free and open access by the Graduate School at UKnowledge. It has been accepted for inclusion in University of Kentucky Doctoral Dissertations by an authorized administrator of UKnowledge. For more information, please contact [email protected]. ABSTRACT OF DISSERTATION Herbert Morales The Graduate School University of Kentucky 2005 BOSONIZATION VS. SUPERSYMMETRY ABSTRACT OF DISSERTATION A dissertation submitted in partial ful¯llment of the requirements for the degree of Doctor of Philosophy in the College of Arts and Sciences at the University of Kentucky By Herbert Morales Lexington, Kentucky Director: Dr. Alfred Shapere, Professor of Physics and Astronomy Lexington, Kentucky 2005 Copyright c Herbert Morales 2005 ° ABSTRACT OF DISSERTATION BOSONIZATION VS. SUPERSYMMETRY We study the conjectured equivalence between the O(3) Gross-Neveu model and the supersymmetric sine-Gordon model under a naive application of the bosonization rules. We start with a review of the equivalence between sine-Gordon model and the massive Thirring model. We study the models by perturbation theory and then determine the equivalence. We ¯nd that the dependence of the identi¯cations on the couplings can change according to the de¯nition of the vector current.
    [Show full text]
  • Jhep08(2019)117
    Published for SISSA by Springer Received: June 14, 2019 Accepted: August 10, 2019 Published: August 22, 2019 JHEP08(2019)117 One-dimensional bosonization and the SYK model Jeff Murugana;b and Horatiu Nastasec aThe Laboratory for Quantum Gravity & Strings, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7700, South Africa bKavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, U.S.A. cInstituto de F´ısica Te´orica, UNESP-Universidade Estadual Paulista, R. Dr. Bento T. Ferraz 271, Bl. II, Sao Paulo 01140-070, SP, Brazil E-mail: [email protected], [email protected] Abstract: We explore the possibility of extending the familiar bosonization of two di- mensions to (0 + 1)-dimensional systems with a large number of degrees of freedom. As an application of this technique, we consider a class of SYK-type models, and argue that the corresponding action on the gravity dual of the SYK model is given by an STS sequence of dualities. Keywords: AdS-CFT Correspondence, Field Theories in Lower Dimensions ArXiv ePrint: 1812.11929 Open Access, c The Authors. https://doi.org/10.1007/JHEP08(2019)117 Article funded by SCOAP3. Contents 1 Introduction1 2 Bosonization as a duality in 1-dimension4 2.1 Review of bosonization duality in 1+1 dimensions4 2.2 Duality in (0+1)-dimensions6 2.3 Kaluza-Klein compactification and dimensional reduction7 JHEP08(2019)117 2.4 Luttinger liquid bosonization and the SYK model8 3 Action on SYK-like models and gravity dual 10 3.1 Jackiw-Teitelboim gravity from near-horizon extremal BTZ 11 3.2 The action of bosonization on the gravity dual 13 4 Conclusions 14 1 Introduction From a value-for-effort perspective, bosonization has to be one of the most versatile tools in low-dimensional quantum field theory.
    [Show full text]
  • Non Chiral Bosonization of a Luttinger Liquid
    NON CHIRAL BOSONIZATION OF A LUTTINGER LIQUID A thesis submitted for the degree of Doctor of Philosophy Joy Prakash Das Under the guidance of Prof. Girish. S. Setlur Department of Physics Indian Institute of Technology Guwahati Guwahati 781 039, Assam, India TH-2101_146121021 TH-2101_146121021 NON CHIRAL BOSONIZATION OF A LUTTINGER LIQUID A thesis submitted for the degree of Doctor of Philosophy Joy Prakash Das Roll No. 146121021 Under the guidance of Prof. Girish. S. Setlur Department of Physics Indian Institute of Technology Guwahati Guwahati 781 039, Assam, India TH-2101_146121021 January 2019 TH-2101_146121021 In matters of science, a thousand proclamations by so-called experts are outweighed by the humble reasoning of a single individual. - Galileo Galilei TH-2101_146121021 TH-2101_146121021 Dedicated to my family TH-2101_146121021 TH-2101_146121021 Declaration The work in this thesis entitled \Non chiral bosonization of a Luttinger liquid" has been carried out by me under the supervision of Prof. Girish S. Setlur, Department of Physics, Indian Institute of Technology Guwahati. No part of this thesis has been submitted elsewhere for award of any other degree or qualification. The research works have been carried out in the period from January, 2015 to December, 2018. In keeping with general practice of reporting scientific observations, due acknowledg- ments have been made wherever the work described is based on the findings of other investi- gations. Place: IIT Guwahati Joy Prakash Das Date: Roll No. 146121021 TH-2101_146121021 i TH-2101_146121021 ii Certificate This is to certify that the research work contained in this thesis entitled \Non chiral bosonization of a Luttinger liquid" by Mr.
    [Show full text]
  • One-Dimensional Bosonization and the SYK Model
    One-dimensional bosonization and the SYK model Jeff Murugana,b∗ and Horatiu Nastasec† aThe Laboratory for Quantum Gravity & Strings, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7700, South Africa bKavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106 cInstituto de F´ısica Te´orica, UNESP-Universidade Estadual Paulista R. Dr. Bento T. Ferraz 271, Bl. II, Sao Paulo 01140-070, SP, Brazil Abstract We explore the possibility of extending the familiar bosonization of two dimen- sions to (0 + 1)- dimensional systems with a large number of degrees of freedom. As an application of this technique, we consider a class of SYK-type models, and argue that the corresponding action on the gravity dual of the SYK model is given by an STS sequence of dualities. arXiv:1812.11929v2 [hep-th] 7 Aug 2019 ∗E-mail address: [email protected] †E-mail address: [email protected] 1 Introduction From a value-for-effort perspective, bosonization has to be one of the most versatile tools in low-dimensional quantum field theory. Originally discovered in the context of particle physics by Coleman [1] as a peculiar relation between the massive Thirring model and sine-Gordon theory, it was then sharpened and made precise by Mandelstam through an operator relation that directly maps the fermionic operators of the Thirring model to bosonic operators [2]. By now there is an extensive literature on the technique and since it is well beyond the purview of this note to review all the developments, we refer the reader to the excellent collection of original seminal works with commentary in [3].
    [Show full text]
  • 4 an Introduction to Bosonization D
    4 An Introduction to Bosonization D. S´en´echal ABSTRACT After general comments on the relevance of field theory to condensed matter systems, the continuum description of interacting elec- trons in 1D is summarized. The bosonization procedure is then introduced heuristically, but the precise quantum equivalence between fermion and boson is also presented. Then the exact solution of the Tomonaga–Lut- tinger model is carried out. Two other applications of bosonization are then sketched. We end with a quick introduction to non-Abelian bosonization. 1 Quantum Field Theory in Condensed Matter Before embarking on a technical description of what bosonization is and how it helps understanding the behavior of interacting electrons in one di- mension, let us make some general comments on the usefulness of quantum field theory in the study of condensed matter systems. First of all, what is a quantum field theory? It has taken over half a century, from the late 1920s to the 1980s, for a satisfactory answer to this fundamental question to creep into the general folklore of theoretical physics, the most significant progress being the introduction of the renormalization group (RG). A field theory is by definition a physical model defined on the continuum as opposed to a lattice. However, quantum fluctuations being at work on all length scales, a purely continuum theory makes no sense, and a momentum cutoff Λ (or another physically equivalent procedure) has to be introduced in order to enable meaningful calculations. Such a regularization of the field theory invariably introduces a length scale Λ−1, which is part of the definition of the theory as one of its parameters, along with various coupling constants, masses, and so on.
    [Show full text]
  • 3D Bosonization: Flavor-Violated, Bordered, Quivered, and Stacked
    3d Bosonization: Flavor-Violated, Bordered, Quivered, and Stacked Kyle Aitken A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2021 Reading Committee: Laurence Yaffe, Chair Lukasz Fidkowski Andreas Karch Program Authorized to Offer Degree: Physics University of Washington Abstract 3d Bosonization: Flavor-Violated, Bordered, Quivered, and Stacked Kyle Aitken Chair of the Supervisory Committee: Professor Laurence Yaffe Physics In this work we discuss several aspects of the 3d bosonization dualities, which are a conjec- tured equivalence between Chern-Simons matter theories in 2 + 1-dimensions. We propose a new phrase diagram for the most general 3d bosonization duality when the number of matter particles is large. We discuss how the most general form the dualities can be ren- dered consistent on a manifold with boundaries. Novel quiver dualities, both bosonic and fermionic, are constructed which serve as a generalization of the well-known particle-vortex duality and which also have application to domain walls in 3 + 1-dimensional QCD. We argue the rank-one dualities can be connected to the rank-two dualities via an orbifolding procedure. Finally, we show how such 2 + 1-dimensional dualities can be related to known 3 + 1-dimensional electromagnetic duality. TABLE OF CONTENTS Page List of Figures . iii List of Tables . vii Chapter 1: Introduction to 3d Bosonization . .1 1.1 Four Systems with Dualities . .1 1.2 3d Bosonization . .6 1.3 The Master Bosonization Duality . 15 1.4 Outline . 21 Chapter 2: Flavor-Violated Dualities . 23 2.1 Review of QCD3 Symmetry-Breaking .
    [Show full text]