Universal phase transitions in Topological lattice models
|
|
- Everett Alexander
- 5 years ago
- Views:
Transcription
1 Universal phase transitions in Topological lattice models F. J. Burnell Collaborators: J. Slingerland S. H. Simon September 2, 2010
2 Overview Matter: classified by orders Symmetry Breaking (Ferromagnet) Topological (Quantum Hall) Phase transitions between different orders Landau-Ginzburg Symmetry dictates: initial and final phase; Critical behavior
3 Symmetry breaking: Symmetry groups G 1 and G 2 tell us: Particles: representations of G 1 (G 2 ) G 1 = Z 2 G 2 = 1 2 nd order phase transition? Yes, if G 2 G 1 Condensation transition: G 1 X =0 G 2 X = S i Field theory of the transition: (Ginzburg-Landau theory ) ( ) 2 L = ( t φ) 2 φ m 2 φ 2 λφ 4
4 Topological Order (TQFT) Characterize by Topological ground state degeneracy (1 on the sphere, multiple on the torus) Excitations with anyonic statistics gapless edge modes
5 Topological Order (TQFT) Characterize by Topological ground state degeneracy (1 on the sphere, multiple on the torus) Excitations with anyonic statistics gapless edge modes effective topological field theory (TQFT) gauge theory Fixes GS degeneracy, edge modes, and quasi-particle statistics σ model of topological phases
6 Why Topological Order? e.g. Quantum Hall effect Wave functions at filling 5/2 Moore-Read Halperin 331 σ xy = 5 e 2 2 h σ xy = 5 e 2 2 h Non-Abelian anyons Abelian anyons Same Hall conductivity Different edge theories and excitations
7 Topological symmetry breaking: condensation in TQFTs When can I condense a boson to get from TQFT 1 to TQFT 2? Symmetry breaking Topological order G 1, G 2 T 1, T 2 local order parameter X Higgs mechanism
8 Superconductor Hansson, Oganesyan, and Sondhi Ann. Phys. 313, 497 Start with electrons and photons TQFT 1: U(1) gauge theory with matter
9 Superconductor Hansson, Oganesyan, and Sondhi Ann. Phys. 313, 497 Start with electrons and photons TQFT 1: U(1) gauge theory with matter Condense c k c k (charge 2e ) Cooper pair
10 Superconductor Hansson, Oganesyan, and Sondhi Ann. Phys. 313, 497 Start with electrons and photons TQFT 1: U(1) gauge theory with matter Condense c k c k (charge 2e ) Cooper pair Vortices: φ = n 2e Vortex
11 Superconductor Hansson, Oganesyan, and Sondhi Ann. Phys. 313, 497 Start with electrons and photons TQFT 1: U(1) gauge theory with matter Condense c k c k (charge 2e ) Cooper pair Vortices: φ = n 2e Charge picks up Berry s phase of π when traveling around the vortex Vortex electron
12 Superconductor Hansson, Oganesyan, and Sondhi Ann. Phys. 313, 497 Start with electrons and photons TQFT 1: U(1) gauge theory with matter Condense c k c k (charge 2e ) Cooper pair Vortices: φ = n 2e Charge picks up Berry s phase of π when traveling around the vortex electron Vortex
13 Superconductor Hansson, Oganesyan, and Sondhi Ann. Phys. 313, 497 Start with electrons and photons TQFT 1: U(1) gauge theory with matter Condense c k c k (charge 2e ) Cooper pair Vortices: φ = n 2e Charge picks up Berry s phase of π when traveling around the vortex Vortex electron Ψ Ψ
14 Superconductor Hansson, Oganesyan, and Sondhi Ann. Phys. 313, 497 Start with electrons and photons TQFT 1: U(1) gauge theory with matter Condense c k c k (charge 2e ) Cooper pair Vortices: φ = n 2e Charge picks up Berry s phase of π when traveling around the vortex TQFT 2: Z 2 gauge theory Vortex electron Ψ Ψ
15 Effects of Topological Symmetry Breaking Superconductor TSB Condense a boson: Condense a boson: = c c φ 0
16 Effects of Topological Symmetry Breaking Superconductor TSB Condense a boson: Condense a boson: = c c φ 0 Vacuum: Vacuum: Ω = (u + vc k c k ) 0 Ω = (1 + φ ) 0
17 Effects of Topological Symmetry Breaking Superconductor TSB Condense a boson: Condense a boson: = c c φ 0 Vacuum: Vacuum: Ω = (u + vc k c k ) 0 Ω = (1 + φ ) 0 Mix particles and holes: c c Identify particles that mix X φ Y
18 Effects of Topological Symmetry Breaking Superconductor TSB Condense a boson: Condense a boson: = c c φ 0 Vacuum: Vacuum: Ω = (u + vc k c k ) 0 Ω = (1 + φ ) 0 Mix particles and holes: c c Confine magnetic field (quantized vortices 2e ) Identify particles that mix X φ Y Confine excitations which braid non-trivially with condensate
19 Effects of Topological Symmetry Breaking Superconductor TSB Condense a boson: Condense a boson: = c c φ 0 Vacuum: Vacuum: Ω = (u + vc k c k ) 0 Ω = (1 + φ ) 0 Mix particles and holes: c c Confine magnetic field (quantized vortices 2e ) Identify particles that mix X φ Y Confine excitations which braid non-trivially with condensate Split particles related by broken symmetry
20 Topological symmetry breaking: condensation in TQFTs Condense a boson to get from TQFT 1 to TQFT 2? Non-Abelian Yang-Mills theories More general topological orders De Wild Propitius PhD thesis (F. A. Bais) Slingerland and Bais, PRB 79,
21 Questions about Topological Symmetry breaking Physical models that realize these transitions? lattice gauge theories Other TQFT s? How are excitations in TQFT 1 and TQFT 2 related? How do some excitations split? What is the field theory of the phase transition? Universality?
22 What follows 1. Condensation and confinement in lattice Ising gauge theory (Toric code) 2. SU(2) 2 SU(2) 2 Chern-Simons theory Ising gauge theory 3. Phase transitions between more general topological phases SU(2)k SU(2) k Chern-Simons theory and the Ising phase transition
23 I. Toric Code to Vacuum Toric code (Ising gauge theory with matter) A. Kitaev, Ann. Phys. 303 ( 03) 2 Edges σ z = ±1 (electric flux) E = 0 : σ z (i) = 1 σz =1 σz = 1 Magnetic flux: σ x flips all electric fluxes from + to
24 I. Toric Code to Vacuum Toric code (Ising gauge theory with matter) A. Kitaev, Ann. Phys. 303 ( 03) 2 Edges σ z = ±1 (electric flux) E = 0 : σ z (i) = 1 σz =1 σz = 1 Magnetic flux: σ x flips all electric fluxes from + to Hamiltonian : H = V ɛ e σz (i) P ɛ m σx (i) Plaquette and vertex projectors commute Exactly solvable
25 I. Toric Code to Vacuum Ground State Loop gas Excitations Boson e (Electric source) Boson m (Vortex) Fermion em (Both!) 1 Statistics : Z 2 topological order e with m, em : 1 m with e, em : 1
26 Phase transition and confinement Confining transition : Decrease gap to creating a vortex ɛ m ɛ m (1 α) Add an E 2 term that spontaneously creates and hops vortices: H = α σ z (i) edges i References: Fradkin and Shenker PRD 19, 3682 ( 79) Trebst et. al. PRL 98, ( 07) C. Castelnovo and C. Chamon PRB ( 08 ) Vidal PRB 79, ( 09) etc.
27 Phase transition and confinement H = V ɛ e σz (i) P (1 α)ɛ m σx (i) α σ z (i) edges i Small α :deconfined ( Toric Code ) QCP α 1:confined ( Trivial )
28 Effects of Topological Symmetry Breaking Superconductor Mix particles and holes: c c Confine magnetic field (quantized vortices 2e ) TSB Identify particles that mix X φ Y Confine excitations which braid non-trivially with condensate Split particles related by broken symmetry
29 II. SU(2) 2 SU(2) 2 Toric code String net lattice Hamiltonian Levin and Wen PRB 71, ( 05) Hilbert space : Edge labels 1, σ, ψ Allowed vertices ( E = 0 ): (1, 1, 1) (1, ψ, ψ) (σ, σ, ψ) (σ, σ, 1) Allowed label flips (construct B 2 term): = Add flux (gauge theory) fusion (CFT)
30 Lattice Hamiltonian: H = V ɛ V δ E1,E 2,E 3 P ɛ m P (0) P Vertex term P V : energy penalty for vertices that are not allowed! Plaquette projector P P : flips strings (changes their electric flux) Projects onto states with no magnetic flux Commuting projectors: the model is exactly solvable Topological order: SU(2) 2 SU(2) 2 Chern-Simons theory
31 Excitations Two sectors (R and L): particles can carry both electric and magnetic flux electric : R and L magnetic : L only σ R ψ R ψ ψ Particle Type ψ R, ψ L σ R, σ L, ψ R σ L, σ R ψ L σ R σ L ψ R ψ L Statistics Fermion Chiral anyon Boson Boson
32 Condensation Boson to condense: ψ vortex product state ψ R ψ L violates plaquettes has no electric flux Z 2 flux addition rules ψ ψ
33 Phase transition Condense ψ vortices: decrease mass gap for ψ vortex Pair-create ψ vortices: H = J x i=edges ( 1)nσ(i) deconfined ( SU(2) 2 SU(2) 2 ) QCP confined ( Toric Code )
34 Questions about Topological Symmetry breaking Physical models that realize these transitions? lattice gauge theories Other TQFT s? Levin-Wen +... How are excitations in TQFT 1 and TQFT 2 related? How do some excitations split? What is the field theory of the phase transition? Universality?
35 The confined phase Identified: ψ R ψ L (ψ R ψ L ) ψ R = ψ L Confined: any excitation with an odd number of σ R,L strings σ R, σ L, ψ R σ L, σ R ψ L Split: σ R σ L σ R σ L (σ, σ) 1 (σ, σ) ψ (σ, σ)1 : No electric flux (σ, σ) ψ : electric flux ψ Flux of ψ conserved once σ eliminated!
36 The confined phase Deconfined Particle Confined particle 1 1 ψ R ψ L Condensed: ψ R ψ L 1 ψ R, ψ L Identified: ψ R ψ L σ R,L Confined σ R ψ L, ψ R σ L Confined σ R σ L Split : (σ, σ) 1, (σ, σ) ψ Net spectrum: 1, ψ, (σ, σ) 1, (σ, σ) ψ Bosons (σ, σ)1, (σ, σ) ψ with a mutual braiding phase of 1 Fermion ψ which has mutual braiding statistics of 1 with both (σ, σ) s. Final theory: Toric code! (Z 2 Topological order) e (σ, σ) ψ m (σ, σ) 1 e m ψ
37 Questions about Topological Symmetry breaking Physical models that realize these transitions? lattice gauge theories Other TQFT s? Levin-Wen +... How are excitations in TQFT 1 and TQFT 2 related? How do some excitations split? What is the field theory of the phase transition? Universality
38 Toric code and TFIM T = 0: no electric sources are created. Dual to the transverse field Ising model: +1 i i no vortex spin up σ z (i) S x (i)s x 1 vortex spin down Electric flux Domain σ x S z (i) = ±1
39 Toric Code and TFIM T = 0: no electric sources are created. Dual to the transverse field Ising model: i i j x spin up σ z (i) S x (i)s x (j) spin down Electric flux Domain wall Vortex pair creation: Ising interaction
40 Toric Code and TFIM Another picture of the phase diagram: Small α :ToricCode ( Paramagnetic ) QCP (3D Ising) α 1:Trivial ( Ferromagnetic )
41 SU(2) 2 SU(2) 2 and TFIM Objective: SU(2) 2 SU(2) 2 ( Paramagnetic ) QCP 3D Ising Toric Code (Ferromagnetic )
42 SU(2) 2 SU(2) 2 and TFIM Effective Hamiltonian for the phase transition Ising! H = H 0 + J z 2 H 0 = V P ( P (0) P m e δ(e 1, E 2, E 3 ) + P ) P(ψ) P J x ( P (0) P i=edges + P(ψ) P ( 1) nσ(i) ) Effective TFIM!
43 SU(2) 2 SU(2) 2 and TFIM ψ-vortex Ising spin on the plaquette (Exact mapping = non-local) Plaquette terms: ( ) P (0) P + P(ψ) P 1 ( ) P (0) P P(ψ) P S z ψ ψψ Creation terms : Ising interaction S x (i)s x (j) σ domain wall (Ising gauge field) =( 1) n σ = i j S x (i)s x (j)
44 Questions about Topological Symmetry breaking Physical models that realize these transitions? lattice gauge theories Other TQFT s? Levin-Wen +... How are excitations in TQFT 1 and TQFT 2 related? How do some excitations split? What is the field theory of the phase transition? Universality Ising
45 Universality Why Ising transition? ψ L ψ R Ising spin (ψl ψ R ) 2 1 ψl ψ R carries no electric flux Pair-create ψ vortices without creating other excitations H commutes with n σ, n ψ Other theories with similar transitions? TFIM SU(2)k SU(2) k Transverse-field Potts Order q simple current
46 Conclusions Phase transitions between different topological orders Mechanism: Yang-Mills theory: Higgs General TQFT: Topological Symmetry Breaking Toy models realizing these Topological lattice Hamiltonians + perturbation Field Theory & Universality Condense Ising-like particle: TFIM transition +1 no vortex spin up i σz(i) i Sx Condense Potts-like particle: Transverse field Potts transition 1 vortex spin down Electric flux Do
47 Universality and Generalizations What is the field theory describing topological phase transitions? Symmetry breaking: Critical theory insensitive to microscopics; depends only on symmetry breaking Here: Critical theory is indifferent to many features of the topological order Condense a boson which behaves like an Ising spin Transition: TFIM Generalizations: Condense a boson which behaves like a q-state Potts spin (order q simple current) Effective theory of phase transitions: transverse field Potts model
48 Condensation in SU(2) 3 SU(2) 3 Lattice Hamiltonian (Levin-Wen type) Excitations: spins 0, 1/2 L,R, 1 L,R, 3/2 L,R Condense φ 3/2 L 3/2 R Boson with 2 states per site Carries no electric flux P (0) + P (φ) = Preserves spin on each edge mod 1 P (0) P (φ) = Interchanges 1/2-int int. spins Modify the Hamiltonian: H = ɛ V P V 1 [ɛ m (P (0) + P (φ)) 2 V P ( +J z P (0) P (φ))] + J x ( 1) 2s Effective Ising model of phase transition! e
49 Condensation in SU(2) 3 SU(2) 3 Uncondensed Condensed (0, 0), (0, 1), (1, 0), (1, 1) (0, 0), (0, 1), (1, 0), (1, 1) (3/2, 3/2), (3/2, 1/2), (1/2, 3/2), (1/2, 1/2) Identified with above (0, 1/2), (0, 3/2), (1, 1/2), (1, 3/2) Confined (1/2, 0, (3/2, 0), (1/2, 1), 3/2, 1) Confined Final theory:doubled Fibbonacci Generalizes to any odd k ( SO(3) k SO(3) k )
50 Condensation in SU(2) k SU(2) k (k even) Start: doubled Chern-Simons theory particles spin 0, 1/2,...k/2 in non-interacting R and L sectors Realizeable via a lattice Hamiltonian of commuting projectors (string net) R L symmetric boson: k/2r k/2 L, which violates only plaquettes in the lattice model Phase transition: condense k/2 R k/2 L Effective description of ground-states + condensing boson: TFIM Dual effective description: Ising gauge theory Phase transition: Ising type
51 Condensation in SU(2) k SU(2) k (k even) End: a new topological theory... Confinement of particles of net 1/2-integer spin Particle types: spin j L i R, j/2 L i/2 R, j = 0, 1, k/2 1, i = 0, 1,...k/2, i > j Splitting of k/4l k/4 R due to an emergent Z 2 symmetry in confined phase Features of the final theory k = 4: D 3 gauge theory k > 4...? In general, it is not a gauge theory...
Criticality in topologically ordered systems: a case study
Criticality in topologically ordered systems: a case study Fiona Burnell Schulz & FJB 16 FJB 17? Phases and phase transitions ~ 194 s: Landau theory (Liquids vs crystals; magnets; etc.) Local order parameter
More informationPhase transitions in topological lattice models via topological symmetry breaking
AER hase transitions in topological lattice models via topological symmetry breaking To cite this article: F J Burnell et al 2012 New J. hys. 14 015004 View the article online for updates and enhancements.
More informationTopology driven quantum phase transitions
Topology driven quantum phase transitions Dresden July 2009 Simon Trebst Microsoft Station Q UC Santa Barbara Charlotte Gils Alexei Kitaev Andreas Ludwig Matthias Troyer Zhenghan Wang Topological quantum
More informationKitaev honeycomb lattice model: from A to B and beyond
Kitaev honeycomb lattice model: from A to B and beyond Jiri Vala Department of Mathematical Physics National University of Ireland at Maynooth Postdoc: PhD students: Collaborators: Graham Kells Ahmet Bolukbasi
More informationDefects in topologically ordered states. Xiao-Liang Qi Stanford University Mag Lab, Tallahassee, 01/09/2014
Defects in topologically ordered states Xiao-Liang Qi Stanford University Mag Lab, Tallahassee, 01/09/2014 References Maissam Barkeshli & XLQ, PRX, 2, 031013 (2012) Maissam Barkeshli, Chaoming Jian, XLQ,
More informationNon-abelian statistics
Non-abelian statistics Paul Fendley Non-abelian statistics are just plain interesting. They probably occur in the ν = 5/2 FQHE, and people are constructing time-reversal-invariant models which realize
More informationEffective Field Theories of Topological Insulators
Effective Field Theories of Topological Insulators Eduardo Fradkin University of Illinois at Urbana-Champaign Workshop on Field Theoretic Computer Simulations for Particle Physics and Condensed Matter
More informationMutual Chern-Simons Landau-Ginzburg theory for continuous quantum phase transition of Z2 topological order
Mutual Chern-Simons Landau-Ginzburg theory for continuous quantum phase transition of Z topological order The MIT Faculty has made this article openly available. Please share how this access benefits you.
More informationClassification of Symmetry Protected Topological Phases in Interacting Systems
Classification of Symmetry Protected Topological Phases in Interacting Systems Zhengcheng Gu (PI) Collaborators: Prof. Xiao-Gang ang Wen (PI/ PI/MIT) Prof. M. Levin (U. of Chicago) Dr. Xie Chen(UC Berkeley)
More informationTopological order from quantum loops and nets
Topological order from quantum loops and nets Paul Fendley It has proved to be quite tricky to T -invariant spin models whose quasiparticles are non-abelian anyons. 1 Here I ll describe the simplest (so
More informationBraid Group, Gauge Invariance and Topological Order
Braid Group, Gauge Invariance and Topological Order Yong-Shi Wu Department of Physics University of Utah Topological Quantum Computing IPAM, UCLA; March 2, 2007 Outline Motivation: Topological Matter (Phases)
More informationRealizing non-abelian statistics in quantum loop models
Realizing non-abelian statistics in quantum loop models Paul Fendley Experimental and theoretical successes have made us take a close look at quantum physics in two spatial dimensions. We have now found
More informationSPIN-LIQUIDS ON THE KAGOME LATTICE: CHIRAL TOPOLOGICAL, AND GAPLESS NON-FERMI-LIQUID PHASE
SPIN-LIQUIDS ON THE KAGOME LATTICE: CHIRAL TOPOLOGICAL, AND GAPLESS NON-FERMI-LIQUID PHASE ANDREAS W.W. LUDWIG (UC-Santa Barbara) work done in collaboration with: Bela Bauer (Microsoft Station-Q, Santa
More informationAnyons and topological quantum computing
Anyons and topological quantum computing Statistical Physics PhD Course Quantum statistical physics and Field theory 05/10/2012 Plan of the seminar Why anyons? Anyons: definitions fusion rules, F and R
More informationTopological Quantum Computation from non-abelian anyons
Topological Quantum Computation from non-abelian anyons Paul Fendley Experimental and theoretical successes have made us take a close look at quantum physics in two spatial dimensions. We have now found
More informationSymmetric Surfaces of Topological Superconductor
Symmetric Surfaces of Topological Superconductor Sharmistha Sahoo Zhao Zhang Jeffrey Teo Outline Introduction Brief description of time reversal symmetric topological superconductor. Coupled wire model
More information(Effective) Field Theory and Emergence in Condensed Matter
(Effective) Field Theory and Emergence in Condensed Matter T. Senthil (MIT) Effective field theory in condensed matter physics Microscopic models (e.g, Hubbard/t-J, lattice spin Hamiltonians, etc) `Low
More informationJiannis K. Pachos. Introduction. Berlin, September 2013
Jiannis K. Pachos Introduction Berlin, September 203 Introduction Quantum Computation is the quest for:» neat quantum evolutions» new quantum algorithms Why? 2D Topological Quantum Systems: How? ) Continuum
More informationGeneralized Global Symmetries
Generalized Global Symmetries Anton Kapustin Simons Center for Geometry and Physics, Stony Brook April 9, 2015 Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries
More informationTopological Phases in One Dimension
Topological Phases in One Dimension Lukasz Fidkowski and Alexei Kitaev arxiv:1008.4138 Topological phases in 2 dimensions: - Integer quantum Hall effect - quantized σ xy - robust chiral edge modes - Fractional
More informationConfinement-deconfinement transitions in Z 2 gauge theories, and deconfined criticality
HARVARD Confinement-deconfinement transitions in Z 2 gauge theories, and deconfined criticality Indian Institute of Science Education and Research, Pune Subir Sachdev November 15, 2017 Talk online: sachdev.physics.harvard.edu
More informationBoundary Degeneracy of Topological Order
Boundary Degeneracy of Topological Order Juven Wang (MIT/Perimeter Inst.) - and Xiao-Gang Wen Mar 15, 2013 @ PI arxiv.org/abs/1212.4863 Lattice model: Toric Code and String-net Flux Insertion What is?
More informationBoulder School 2016 Xie Chen 07/28/16-08/02/16
Boulder School 2016 Xie Chen 07/28/16-08/02/16 Symmetry Fractionalization 1 Introduction This lecture is based on review article Symmetry Fractionalization in Two Dimensional Topological Phases, arxiv:
More informationInteger quantum Hall effect for bosons: A physical realization
Integer quantum Hall effect for bosons: A physical realization T. Senthil (MIT) and Michael Levin (UMCP). (arxiv:1206.1604) Thanks: Xie Chen, Zhengchen Liu, Zhengcheng Gu, Xiao-gang Wen, and Ashvin Vishwanath.
More informationField Theory Description of Topological States of Matter. Andrea Cappelli INFN, Florence (w. E. Randellini, J. Sisti)
Field Theory Description of Topological States of Matter Andrea Cappelli INFN, Florence (w. E. Randellini, J. Sisti) Topological States of Matter System with bulk gap but non-trivial at energies below
More informationSymmetry Protected Topological Phases of Matter
Symmetry Protected Topological Phases of Matter T. Senthil (MIT) Review: T. Senthil, Annual Reviews of Condensed Matter Physics, 2015 Topological insulators 1.0 Free electron band theory: distinct insulating
More informationNonabelian hierarchies
Nonabelian hierarchies collaborators: Yoran Tournois, UzK Maria Hermanns, UzK Hans Hansson, SU Steve H. Simon, Oxford Susanne Viefers, UiO Quantum Hall hierarchies, arxiv:1601.01697 Outline Haldane-Halperin
More informationTopological Field Theory and Conformal Quantum Critical Points
Topological Field Theory and Conformal Quantum Critical Points One might expect that the quasiparticles over a Fermi sea have quantum numbers (charge, spin) of an electron. This is not always true! Charge
More informationMatrix product states for the fractional quantum Hall effect
Matrix product states for the fractional quantum Hall effect Roger Mong (California Institute of Technology) University of Virginia Feb 24, 2014 Collaborators Michael Zaletel UC Berkeley (Stanford/Station
More informationAn origin of light and electrons a unification of gauge interaction and Fermi statistics
An origin of light and electrons a unification of gauge interaction and Fermi statistics Michael Levin and Xiao-Gang Wen http://dao.mit.edu/ wen Artificial light and quantum orders... PRB 68 115413 (2003)
More informationCritical Spin-liquid Phases in Spin-1/2 Triangular Antiferromagnets. In collaboration with: Olexei Motrunich & Jason Alicea
Critical Spin-liquid Phases in Spin-1/2 Triangular Antiferromagnets In collaboration with: Olexei Motrunich & Jason Alicea I. Background Outline Avoiding conventional symmetry-breaking in s=1/2 AF Topological
More informationVortex States in a Non-Abelian Magnetic Field
Vortex States in a Non-Abelian Magnetic Field Predrag Nikolić George Mason University Institute for Quantum Matter @ Johns Hopkins University SESAPS November 10, 2016 Acknowledgments Collin Broholm IQM
More informationConformal Field Theory of Composite Fermions in the QHE
Conformal Field Theory of Composite Fermions in the QHE Andrea Cappelli (INFN and Physics Dept., Florence) Outline Introduction: wave functions, edge excitations and CFT CFT for Jain wfs: Hansson et al.
More informationTopological Insulators in 3D and Bosonization
Topological Insulators in 3D and Bosonization Andrea Cappelli, INFN Florence (w. E. Randellini, J. Sisti) Outline Topological states of matter: bulk and edge Fermions and bosons on the (1+1)-dimensional
More informationAfter first studying an example of a topological phase and its underlying structures, we study effective field theories for 2D topological phases.
1 Boulder notes by Victor V Albert I CHETAN NAYAK After first studying an example of a topological phase and its underlying structures, we study effective field theories for D topological phases I1 Example
More informationShunsuke Furukawa Condensed Matter Theory Lab., RIKEN. Gregoire Misguich Vincent Pasquier Service de Physique Theorique, CEA Saclay, France
Shunsuke Furukawa Condensed Matter Theory Lab., RIKEN in collaboration with Gregoire Misguich Vincent Pasquier Service de Physique Theorique, CEA Saclay, France : ground state of the total system Reduced
More informationMagnets, 1D quantum system, and quantum Phase transitions
134 Phys620.nb 10 Magnets, 1D quantum system, and quantum Phase transitions In 1D, fermions can be mapped into bosons, and vice versa. 10.1. magnetization and frustrated magnets (in any dimensions) Consider
More informationLecture 2: Deconfined quantum criticality
Lecture 2: Deconfined quantum criticality T. Senthil (MIT) General theoretical questions Fate of Landau-Ginzburg-Wilson ideas at quantum phase transitions? (More precise) Could Landau order parameters
More informationChiral spin liquids. Bela Bauer
Chiral spin liquids Bela Bauer Based on work with: Lukasz Cinco & Guifre Vidal (Perimeter Institute) Andreas Ludwig & Brendan Keller (UCSB) Simon Trebst (U Cologne) Michele Dolfi (ETH Zurich) Nature Communications
More informationIntoduction to topological order and topologial quantum computation. Arnau Riera, Grup QIC, Dept. ECM, UB 16 de maig de 2009
Intoduction to topological order and topologial quantum computation Arnau Riera, Grup QIC, Dept. ECM, UB 16 de maig de 2009 Outline 1. Introduction: phase transitions and order. 2. The Landau symmetry
More informationFrom Luttinger Liquid to Non-Abelian Quantum Hall States
From Luttinger Liquid to Non-Abelian Quantum Hall States Jeffrey Teo and C.L. Kane KITP workshop, Nov 11 arxiv:1111.2617v1 Outline Introduction to FQHE Bulk-edge correspondence Abelian Quantum Hall States
More informationNon-Abelian Statistics. in the Fractional Quantum Hall States * X. G. Wen. School of Natural Sciences. Institute of Advanced Study
IASSNS-HEP-90/70 Sep. 1990 Non-Abelian Statistics in the Fractional Quantum Hall States * X. G. Wen School of Natural Sciences Institute of Advanced Study Princeton, NJ 08540 ABSTRACT: The Fractional Quantum
More informationSymmetry Protected Topological Phases
CalSWARM 2016 Xie Chen 06/21/16 Symmetry Protected Topological Phases 1 Introduction In this lecture note, I will give a brief introduction to symmetry protected topological (SPT) phases in 1D, 2D, and
More informationTHE UNIVERSITY OF CHICAGO TOPOLOGICAL PHASES AND EXACTLY SOLUBLE LATTICE MODELS A DISSERTATION SUBMITTED TO
THE UNIVERSITY OF CHICAGO TOPOLOGICAL PHASES AND EXACTLY SOLUBLE LATTICE MODELS A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE OF DOCTOR OF
More informationTopological Insulators
Topological Insulators Aira Furusai (Condensed Matter Theory Lab.) = topological insulators (3d and 2d) Outline Introduction: band theory Example of topological insulators: integer quantum Hall effect
More informationDeconfined Quantum Critical Points
Deconfined Quantum Critical Points Outline: with T. Senthil, Bangalore A. Vishwanath, UCB S. Sachdev, Yale L. Balents, UCSB conventional quantum critical points Landau paradigm Seeking a new paradigm -
More informationUnderstanding Topological Order with PEPS. David Pérez-García Autrans Summer School 2016
Understanding Topological Order with PEPS David Pérez-García Autrans Summer School 2016 Outlook 1. An introduc
More informationTopological order of a two-dimensional toric code
University of Ljubljana Faculty of Mathematics and Physics Seminar I a, 1st year, 2nd cycle Topological order of a two-dimensional toric code Author: Lenart Zadnik Advisor: Doc. Dr. Marko Žnidarič June
More informationDeconfined Quantum Critical Points
Deconfined Quantum Critical Points Leon Balents T. Senthil, MIT A. Vishwanath, UCB S. Sachdev, Yale M.P.A. Fisher, UCSB Outline Introduction: what is a DQCP Disordered and VBS ground states and gauge theory
More informationFrom Majorana Fermions to Topological Order
From Majorana Fermions to Topological Order Arxiv: 1201.3757, to appear in PRL. B.M. Terhal, F. Hassler, D.P. DiVincenzo IQI, RWTH Aachen We are looking for PhD students or postdocs for theoretical research
More informationFermionic topological quantum states as tensor networks
order in topological quantum states as Jens Eisert, Freie Universität Berlin Joint work with Carolin Wille and Oliver Buerschaper Symmetry, topology, and quantum phases of matter: From to physical realizations,
More informationFractional Quantum Hall States with Conformal Field Theories
Fractional Quantum Hall States with Conformal Field Theories Lei Su Department of Physics, University of Chicago Abstract: Fractional quantum Hall (FQH states are topological phases with anyonic excitations
More informationField Theory Description of Topological States of Matter
Field Theory Description of Topological States of Matter Andrea Cappelli, INFN Florence (w. E. Randellini, J. Sisti) Outline Topological states of matter Quantum Hall effect: bulk and edge Effective field
More informationarxiv: v1 [cond-mat.str-el] 22 Oct 2015
arxiv:1510.06627v1 [cond-mat.str-el] 22 Oct 2015 Generating domain walls between topologically ordered phases using quantum double models Miguel Jorge Bernabé Ferreira, a Pramod Padmanabhan, a,b Paulo
More informationIntroduction to Topological Error Correction and Computation. James R. Wootton Universität Basel
Introduction to Topological Error Correction and Computation James R. Wootton Universität Basel Overview Part 1: Topological Quantum Computation Abelian and non-abelian anyons Quantum gates with Abelian
More informationSuperinsulator: a new topological state of matter
Superinsulator: a new topological state of matter M. Cristina Diamantini Nips laboratory, INFN and Department of Physics and Geology University of Perugia Coll: Igor Lukyanchuk, University of Picardie
More informationNon-Abelian Anyons in the Quantum Hall Effect
Non-Abelian Anyons in the Quantum Hall Effect Andrea Cappelli (INFN and Physics Dept., Florence) with L. Georgiev (Sofia), G. Zemba (Buenos Aires), G. Viola (Florence) Outline Incompressible Hall fluids:
More informationTopological quantum computation
NUI MAYNOOTH Topological quantum computation Jiri Vala Department of Mathematical Physics National University of Ireland at Maynooth Tutorial Presentation, Symposium on Quantum Technologies, University
More informationParamagnetic phases of Kagome lattice quantum Ising models p.1/16
Paramagnetic phases of Kagome lattice quantum Ising models Predrag Nikolić In collaboration with T. Senthil Massachusetts Institute of Technology Paramagnetic phases of Kagome lattice quantum Ising models
More informationΨ({z i }) = i<j(z i z j ) m e P i z i 2 /4, q = ± e m.
Fractionalization of charge and statistics in graphene and related structures M. Franz University of British Columbia franz@physics.ubc.ca January 5, 2008 In collaboration with: C. Weeks, G. Rosenberg,
More informationUniversal Topological Phase of 2D Stabilizer Codes
H. Bombin, G. Duclos-Cianci, D. Poulin arxiv:1103.4606 H. Bombin arxiv:1107.2707 Universal Topological Phase of 2D Stabilizer Codes Héctor Bombín Perimeter Institute collaborators David Poulin Guillaume
More informationComposite Dirac liquids
Composite Dirac liquids Composite Fermi liquid non-interacting 3D TI surface Interactions Composite Dirac liquid ~ Jason Alicea, Caltech David Mross, Andrew Essin, & JA, Physical Review X 5, 011011 (2015)
More informationProximity-induced magnetization dynamics, interaction effects, and phase transitions on a topological surface
Proximity-induced magnetization dynamics, interaction effects, and phase transitions on a topological surface Ilya Eremin Theoretische Physik III, Ruhr-Uni Bochum Work done in collaboration with: F. Nogueira
More informationarxiv: v1 [cond-mat.str-el] 6 May 2010
MIT-CTP/4147 Correlated Topological Insulators and the Fractional Magnetoelectric Effect B. Swingle, M. Barkeshli, J. McGreevy, and T. Senthil Department of Physics, Massachusetts Institute of Technology,
More informationarxiv:cond-mat/ v2 [cond-mat.mes-hall] 27 Sep 2006
arxiv:cond-mat/0607743v2 [cond-mat.mes-hall] 27 Sep 2006 Topological degeneracy of non-abelian states for dummies Masaki Oshikawa a Yong Baek Kim b,c Kirill Shtengel d Chetan Nayak e,f Sumanta Tewari g
More informationTopological Order and the Kitaev Model
Topological Order and the Kitaev Model PGF5295-Teoria Quântica de Muitos Corpos em Matéria Condensada Juan Pablo Ibieta Jimenez January 8, 2016 Abstract In this notes we intend to discuss the concept of
More informationVortices and vortex states of Rashba spin-orbit coupled condensates
Vortices and vortex states of Rashba spin-orbit coupled condensates Predrag Nikolić George Mason University Institute for Quantum Matter @ Johns Hopkins University March 5, 2014 P.N, T.Duric, Z.Tesanovic,
More informationA Superconducting Quantum Simulator for Topological Order and the Toric Code. Michael J. Hartmann Heriot-Watt University, Edinburgh qlightcrete 2016
A Superconducting Quantum Simulator for Topological Order and the Toric Code Michael J. Hartmann Heriot-Watt University, Edinburgh qlightcrete 2016 Topological Order (in 2D) A 2-dimensional physical system
More informationQuantum Spin Liquids and Majorana Metals
Quantum Spin Liquids and Majorana Metals Maria Hermanns University of Cologne M.H., S. Trebst, PRB 89, 235102 (2014) M.H., K. O Brien, S. Trebst, PRL 114, 157202 (2015) M.H., S. Trebst, A. Rosch, arxiv:1506.01379
More informationWiring Topological Phases
1 Wiring Topological Phases Quantum Condensed Matter Journal Club Adhip Agarwala Department of Physics Indian Institute of Science adhip@physics.iisc.ernet.in February 4, 2016 So you are interested in
More informationChern-Simons Theory and Its Applications. The 10 th Summer Institute for Theoretical Physics Ki-Myeong Lee
Chern-Simons Theory and Its Applications The 10 th Summer Institute for Theoretical Physics Ki-Myeong Lee Maxwell Theory Maxwell Theory: Gauge Transformation and Invariance Gauss Law Charge Degrees of
More informationIs the composite fermion a Dirac particle?
Is the composite fermion a Dirac particle? Dam T. Son (University of Chicago) Cold atoms meet QFT, 2015 Ref.: 1502.03446 Plan Plan Composite fermion: quasiparticle of Fractional Quantum Hall Effect (FQHE)
More informationFinite Temperature Quantum Memory and Haah s Code
Finite Temperature Quantum Memory and Haah s Code S.M. Kravec 1 1 Department of Physics, University of California at San Diego, La Jolla, CA 92093 This paper addresses the question of whether realizations
More informationDimer model implementations of quantum loop gases. C. Herdman, J. DuBois, J. Korsbakken, K. B. Whaley UC Berkeley
Dimer model implementations of quantum loop gases C. Herdman, J. DuBois, J. Korsbakken, K. B. Whaley UC Berkeley Outline d-isotopic quantum loop gases and dimer model implementations generalized RK points
More informationTopological Quantum Computation A very basic introduction
Topological Quantum Computation A very basic introduction Alessandra Di Pierro alessandra.dipierro@univr.it Dipartimento di Informatica Università di Verona PhD Course on Quantum Computing Part I 1 Introduction
More informationExchange statistics. Basic concepts. University of Oxford April, Jon Magne Leinaas Department of Physics University of Oslo
University of Oxford 12-15 April, 2016 Exchange statistics Basic concepts Jon Magne Leinaas Department of Physics University of Oslo Outline * configuration space with identifications * from permutations
More informationInti Sodemann (MIT) Séptima Escuela de Física Matemática, Universidad de Los Andes, Bogotá, Mayo 25, 2015
Inti Sodemann (MIT) Séptima Escuela de Física Matemática, Universidad de Los Andes, Bogotá, Mayo 25, 2015 Contents Why are the fractional quantum Hall liquids amazing! Abelian quantum Hall liquids: Laughlin
More information2-Group Global Symmetry
2-Group Global Symmetry Clay Córdova School of Natural Sciences Institute for Advanced Study April 14, 2018 References Based on Exploring 2-Group Global Symmetry in collaboration with Dumitrescu and Intriligator
More informationWhich Spin Liquid Is It?
Which Spin Liquid Is It? Some results concerning the character and stability of various spin liquid phases, and Some speculations concerning candidate spin-liquid phases as the explanation of the peculiar
More informationThe Toric-Boson model and quantum memory at finite temperature
The Toric-Boson model and quantum memory at finite temperature A.H., C. Castelnovo, C. Chamon Phys. Rev. B 79, 245122 (2009) Overview Classical information can be stored for arbitrarily long times because
More informationKai Sun. University of Michigan, Ann Arbor. Collaborators: Krishna Kumar and Eduardo Fradkin (UIUC)
Kai Sun University of Michigan, Ann Arbor Collaborators: Krishna Kumar and Eduardo Fradkin (UIUC) Outline How to construct a discretized Chern-Simons gauge theory A necessary and sufficient condition for
More informationBosonization of lattice fermions in higher dimensions
Bosonization of lattice fermions in higher dimensions Anton Kapustin California Institute of Technology January 15, 2019 Anton Kapustin (California Institute of Technology) Bosonization of lattice fermions
More informationAnyon Physics. Andrea Cappelli (INFN and Physics Dept., Florence)
Anyon Physics Andrea Cappelli (INFN and Physics Dept., Florence) Outline Anyons & topology in 2+ dimensions Chern-Simons gauge theory: Aharonov-Bohm phases Quantum Hall effect: bulk & edge excitations
More informationQuantum Monte Carlo study of a Z 2 gauge theory containing phases with and without a Luttinger volume Fermi surface
Quantum Monte Carlo study of a Z 2 gauge theory containing phases with and without a Luttinger volume Fermi surface V44.00011 APS March Meeting, Los Angeles Fakher Assaad, Snir Gazit, Subir Sachdev, Ashvin
More informationSymmetry protected topological phases in quantum spin systems
10sor network workshop @Kashiwanoha Future Center May 14 (Thu.), 2015 Symmetry protected topological phases in quantum spin systems NIMS U. Tokyo Shintaro Takayoshi Collaboration with A. Tanaka (NIMS)
More informationSmall and large Fermi surfaces in metals with local moments
Small and large Fermi surfaces in metals with local moments T. Senthil (MIT) Subir Sachdev Matthias Vojta (Augsburg) cond-mat/0209144 Transparencies online at http://pantheon.yale.edu/~subir Luttinger
More informationBerry s phase in Hall Effects and Topological Insulators
Lecture 6 Berry s phase in Hall Effects and Topological Insulators Given the analogs between Berry s phase and vector potentials, it is not surprising that Berry s phase can be important in the Hall effect.
More informationThink Globally, Act Locally
Think Globally, Act Locally Nathan Seiberg Institute for Advanced Study Quantum Fields beyond Perturbation Theory, KITP 2014 Ofer Aharony, NS, Yuji Tachikawa, arxiv:1305.0318 Anton Kapustin, Ryan Thorngren,
More informationSimulations of Quantum Dimer Models
Simulations of Quantum Dimer Models Didier Poilblanc Laboratoire de Physique Théorique CNRS & Université de Toulouse 1 A wide range of applications Disordered frustrated quantum magnets Correlated fermions
More informationTransition in Kitaev model
Transition in Kitaev model Razieh Mohseninia November, 2013 1 Phase Transition Landau symmetry-breaking theory(1930-1980) different phases different symmetry Liquid solid Paramagnetic ferromagnetic 2 Topological
More informationFractional quantum Hall effect and duality. Dam T. Son (University of Chicago) Canterbury Tales of hot QFTs, Oxford July 11, 2017
Fractional quantum Hall effect and duality Dam T. Son (University of Chicago) Canterbury Tales of hot QFTs, Oxford July 11, 2017 Plan Plan General prologue: Fractional Quantum Hall Effect (FQHE) Plan General
More informationGolden chain of strongly interacting Rydberg atoms
Golden chain of strongly interacting Rydberg atoms Hosho Katsura (Gakushuin Univ.) Acknowledgment: Igor Lesanovsky (MUARC/Nottingham Univ. I. Lesanovsky & H.K., [arxiv:1204.0903] Outline 1. Introduction
More informationarxiv: v3 [cond-mat.str-el] 19 Dec 2016
Symmetry enriched string-nets: Exactly solvable models for SET phases Chris Heinrich, Fiona Burnell, 2 Lukasz Fidkowski, 3 and Michael Levin Department of Physics, James Frank Institute, University of
More informationHigh-Temperature Criticality in Strongly Constrained Quantum Systems
High-Temperature Criticality in Strongly Constrained Quantum Systems Claudio Chamon Collaborators: Claudio Castelnovo - BU Christopher Mudry - PSI, Switzerland Pierre Pujol - ENS Lyon, France PRB 2006
More informationQuantum magnetism and the theory of strongly correlated electrons
Quantum magnetism and the theory of strongly correlated electrons Johannes Reuther Freie Universität Berlin Helmholtz Zentrum Berlin? Berlin, April 16, 2015 Johannes Reuther Quantum magnetism () Berlin,
More informationEntanglement, fidelity, and topological entropy in a quantum phase transition to topological order
Entanglement, fidelity, and topological entropy in a quantum phase transition to topological order A. Hamma, 1 W. Zhang, 2 S. Haas, 2 and D. A. Lidar 1,2,3 1 Department of Chemistry, Center for Quantum
More informationLIBERATION ON THE WALLS IN GAUGE THEORIES AND ANTI-FERROMAGNETS
LIBERATION ON THE WALLS IN GAUGE THEORIES AND ANTI-FERROMAGNETS Tin Sulejmanpasic North Carolina State University Erich Poppitz, Mohamed Anber, TS Phys.Rev. D92 (2015) 2, 021701 and with Anders Sandvik,
More informationTopological Electromagnetic and Thermal Responses of Time-Reversal Invariant Superconductors and Chiral-Symmetric band insulators
Topological Electromagnetic and Thermal Responses of Time-Reversal Invariant Superconductors and Chiral-Symmetric band insulators Satoshi Fujimoto Dept. Phys., Kyoto University Collaborator: Ken Shiozaki
More informationThe Role Of Magnetic Monopoles In Quark Confinement (Field Decomposition Approach)
The Role Of Magnetic Monopoles In Quark Confinement (Field Decomposition Approach) IPM school and workshop on recent developments in Particle Physics (IPP11) 2011, Tehran, Iran Sedigheh Deldar, University
More informationVacuum degeneracy of chiral spin states in compactified. space. X.G. Wen
Vacuum degeneracy of chiral spin states in compactified space X.G. Wen Institute for Theoretical Physics University of California Santa Barbara, California 93106 ABSTRACT: A chiral spin state is not only
More information