Entanglement Entropy in 2+1 Chern-Simons Theory
|
|
- Jodie Jacobs
- 6 years ago
- Views:
Transcription
1 Entanglement Entropy in 2+1 Chern-Simons Theory Shiying Dong UIUC With: Eduardo Fradkin, Rob Leigh, Sean Nowling arxiv: hep-th/ /27/2008 Great Lakes String University of Wisconsin-Madison
2 Motivation Candidate of black hole entropy 2+1 Gravity, BTZ Order parameter of topological states Fractional quantum Hall effect p+ip superconductors Quantum computing with anyons
3 Definition For a system consisting of two subsystems A, B, from any pure state φ, the density matrix ρ = φ φ, define the reduced density matrix on A by ρ A = tr B ρ, and the entanglement entropy iss A = tr(ρ A ln ρ A ). For a pure state, S A = S B. The entanglement entropy should depend on the common features of A and B.
4 Scale Dependence It depends on the interface length scale L, the correlation length ξ, and the ultraviolet cutoff ε. Area law: When the interface is rotational symmetric, the leading term is proportional to the area of the interface. In general, for spatial dimension d, S A = g d 1 ( L ɛ )d 1 + g d 2 ( L ɛ )d g 0 ln L ɛ + S 0. H.Casini and M. Huerta 06
5 Universal Terms In odd d dimensions, or even d dimensions with non-smooth interfaces, the entanglement entropy has a logarithmic divergent term, which is universal. Otherwise, there is a universal constant term. In particular, d=1, S A = β ln L ɛ δ, d=2, S A = αl γ. H.Casini and M. Huerta 06
6 Calculation Define Z n = tr(ρ n A ), for integer n. There is an unambiguous analytic continuation to real n 1. S A = lim n 1 n Z n. In practice we usually have to normalize it, S A = lim n 1 n Z n Z 1 n. P. Calabrese and J. Cardy 04
7 2D Free Boson CFT ρ = lim β e βh ρ A = tr B ρ
8 Z n = tr(ρ A n ) e.g., n=3 u v n-sheeted w
9 T (w) = ( dz dw )2 T (z) + c 12 Where z = ( w u w v ) 1 n = c(1 1/n2 ) 24 = T (w)φ n(u)φ n (v) Φ n (u)φ n (v) ±n = c 24 (1 1 n 2 ). S A = c 3 {z, w} (v u) 2 (w u) 2 (w v) 2 And so ln( u v ɛ ). P. Calabrese and J. Cardy 04
10 2D Massive Free Boson A R m 2 ln ZB n = 1 2 B R ξ=1/m R. The Green function is defined on the n-sheeted complex plane. m 2 ln ZB n = 1 2m 2 ( 1 12n + n 2 (mr 1 2 )2 ), S A = 1 6 ln 1 mɛ. d 2 rg n ( r, r) P. Calabrese and J. Cardy 04
11 2D Massive Free Fermion m ln ZF n = d 2 r trs n ( r, r). m 2 ln ZF n = 1 2m 2 ( 1 24n n 2 (mr 1 2 )2 ), S A = 1 12 ln 1 mɛ. The linear divergence is cancelled between the bosons and fermions. m 2 ln(zb n Zn F ) = 1 24m 2 n ( ),
12 Summary for 2D The logarithmic term in the entanglement entropy of 2D free QFT is universal. It is proportional to the conformal anomaly of the system. It is also proportional to the number of interfaces between A and B subsystems.
13 2+1 Chern-Simons The Hilbert space on a 2d closed surface is spanned by the conformal blocks of the WZW CFT living on that surface. The wavefunctions can be written as the partition function of the gauged WZW model, ψ J(A z )=exp[ ik Tr( 2π JA)] [Dg]exp[ikS + (g, A z, J)] E. Witten 89, 92, Elitzur, Moore, Schwimmer and Serberg, 89
14 We define a state by doing path integral on a 3D manifold enclosed by the surface. And the density matrix has two manifolds with opposite orientations. Trace over B means to identify the boundary value of the Chern-Simons fields on the two B surfaces, and sum over them properly. This means ρ A = tr B ρ is generated by gluing the two manifolds along their B surfaces. To calculate Z n = tr(ρ n A ), we need to glue n pieces of the ρ A manifolds, and study the CS partition function of the final manifold.
15 A Simplest Example: S 2 ρ = φ φ ρ A
16 Tr(ρ A n ) = Z n Z 1 n = Z(S3 ) Z(S 3 ) n S A = lim n 1 = ln S 00 = ln D n Z(S3 ) 1 n = ln Z(S 3 ) E. Witten 89 D = i d 2 i = i ( S 0i S 00 ) 2 = 1 S 00 quantum dimension modular S matrix
17 Remarks In 2+1 theory, we have S A = αl γ in general. Since Chern-Simons theory is topological, there is no scale dependence, only the topological piece survives. If we move away the topological phase, we can still calculate the topological entropy by computing D γ 0 = S A + S B + S C S AB S AC S BC + S ABC. A C B A. Kitaev and J. Preskill 06
18 S 2 With Two Interfaces ρ = B2 A B1 φ B1* A* B2* φ b2 b1 b1 b2 A b1 A* ρ A A A* b1 b2 = A b2 A*
19 Now the final manifold is the connected sum of two S 3 s along n S 2 s, Tr(ρ A n ) = Z n Z 1 n = Z(S3, S 3, n) Z(S 3 ) n = Z(S3 ) 2 Z(S 3 ) 2n. The entanglement entropy is doubled, S A = lim n 1 n Z(S3 ) 2(1 n) = 2 ln D. In general, S 2 with I interfaces gives us S A = I ln D.
20 Useful Facts The key to generalize the operation is that, if any three manifold is a connected sum of two submanifolds, with their interface supporting only one state, we can cut it into two pieces. On S 2 with two punctures, the Hilbert space is one dimensional if they are a conjugate pair, zero otherwise. A single link inside S 3 has expectation value S 0 j.
21 General Manifolds and States Finding all the conformal blocks Squeeze all the interfaces Cut and glue around each interface The ears will cancel after the normalization
22 An Example: 2-Tori A b B Ψ = {i,j,k} φ {i,j,k} {i, j, k} j i k A b B
23 D 2 A b B k1 S 2 i1 j k2 i2 2n j Z n = {{i,k},j} S 0 j n φ {it,j,k t }φ ψ A ψ A {it,j} ψ B ψ B {kt,j} {i t,j,k t+1 } (S j 0 ) 2 t=1 S 3
24 General Result Normalize the basis states Z n = n (S j 0 ) 1 n Ψ Ψ = ψ A ψ A ψ B ψ B S 0 j =1 (S 0 j ) 1 n tr(ρ j ) n {{i,k},j} t=1 φ {it,j,k t }φ {i t,j,k t+1 } = j S A = I ln S 0 0 {j i }( # of interfaces I d ji )tr i=1 quantum dimensions all possible configurations around interfaces projected density matrix ( ρ{ji } I i=1 d ln ρ ) {j i } I j i i=1 d. j i
25 Summary for Chern-Simons The entanglement entropy has a vacuum contribution, which is proportional to the number of interfaces. The nontrivial part comes from the sewing law of CFT. The total entropy is a sum of the traditional entanglement entropy from all the sewing channels. There is a microscopic degeneracy for all the states, associated with the quantum dimensions of the states defined on loops.
26 Thank you.
STUDIES ON TWO TOPICS IN THEORETICAL PHYSICS: ENTANGLEMENT ENTROPY AND HOLOGRAPHIC SUPERCONDUCTORS SHIYING DONG DISSERTATION
c 2010 Shiying Dong STUDIES ON TWO TOPICS IN THEORETICAL PHYSICS: ENTANGLEMENT ENTROPY AND HOLOGRAPHIC SUPERCONDUCTORS BY SHIYING DONG DISSERTATION Submitted in partial fulfillment of the requirements
More informationEntanglement Entropy for Disjoint Intervals in AdS/CFT
Entanglement Entropy for Disjoint Intervals in AdS/CFT Thomas Faulkner Institute for Advanced Study based on arxiv:1303.7221 (see also T.Hartman arxiv:1303.6955) Entanglement Entropy : Definitions Vacuum
More informationAnomalies and SPT phases
Anomalies and SPT phases Kazuya Yonekura, Kavli IPMU Review (mainly of [1508.04715] by Witten) [1607.01873] KY [1610.07010][1611.01601] collaboration with Yuji Tachikawa Introduction What is the most general
More informationHolographic Entanglement Beyond Classical Gravity
Holographic Entanglement Beyond Classical Gravity Xi Dong Stanford University August 2, 2013 Based on arxiv:1306.4682 with Taylor Barrella, Sean Hartnoll, and Victoria Martin See also [Faulkner (1303.7221)]
More informationMutual Information in Conformal Field Theories in Higher Dimensions
Mutual Information in Conformal Field Theories in Higher Dimensions John Cardy University of Oxford Conference on Mathematical Statistical Physics Kyoto 2013 arxiv:1304.7985; J. Phys. : Math. Theor. 46
More informationHigher Spin AdS/CFT at One Loop
Higher Spin AdS/CFT at One Loop Simone Giombi Higher Spin Theories Workshop Penn State U., Aug. 28 2015 Based mainly on: SG, I. Klebanov, arxiv: 1308.2337 SG, I. Klebanov, B. Safdi, arxiv: 1401.0825 SG,
More informationQuantum Information and Entanglement in Holographic Theories
Quantum Information and Entanglement in Holographic Theories Matthew Headrick randeis University Contents 1 asic notions 2 1.1 Entanglement entropy & mutual information............................ 2 1.2
More informationAnomalies and SPT phases
Anomalies and SPT phases Kazuya Yonekura, Kavli IPMU Based on A review of [1508.04715] by Witten [1607.01873] KY [1609.?????] Yuji Tachikawa and KY Introduction One of the motivations: What is the most
More informationEntanglement in Quantum Field Theory
Entanglement in Quantum Field Theory John Cardy University of Oxford DAMTP, December 2013 Outline Quantum entanglement in general and its quantification Path integral approach Entanglement entropy in 1+1-dimensional
More informationUniversal terms in the Entanglement Entropy of Scale Invariant 2+1 Dimensional Quantum Field Theories
Universal terms in the Entanglement Entropy of Scale Invariant 2+1 Dimensional Quantum Field Theories Eduardo Fradkin Department of Physics and Institute for Condensed Matter Theory, University of Illinois
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 informationString theory effects on 5D black strings
String theory effects on 5D black strings Alejandra Castro University of Michigan Work in collaboration with J. Davis, P. Kraus and F. Larsen hep-th/0702072, hep-th/0703087, 0705.1847[hep-th], 0801.1863
More informationEntanglement in Quantum Field Theory
Entanglement in Quantum Field Theory John Cardy University of Oxford Landau Institute, June 2008 in collaboration with P. Calabrese; O. Castro-Alvaredo and B. Doyon Outline entanglement entropy as a measure
More informationLecture 8: 1-loop closed string vacuum amplitude
Lecture 8: 1-loop closed string vacuum amplitude José D. Edelstein University of Santiago de Compostela STRING THEORY Santiago de Compostela, March 5, 2013 José D. Edelstein (USC) Lecture 8: 1-loop vacuum
More informationRényi Entropy in AdS 3 /CFT 2
Rényi Entropy in AdS 3 /CFT 2 Bin Chen R Peking University String 2016 August 1-5, Beijing (a) Jia-ju Zhang (b) Jiang Long (c) Jie-qiang Wu Based on the following works: B.C., J.-j. Zhang, arxiv:1309.5453
More informationAnomalies and Entanglement Entropy
Anomalies and Entanglement Entropy Tatsuma Nishioka (University of Tokyo) based on a work with A. Yarom (Technion) (to appear) T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 1 / 35 Roles of entanglement entropy
More informationThe boundary state from open string fields. Yuji Okawa University of Tokyo, Komaba. March 9, 2009 at Nagoya
The boundary state from open string fields Yuji Okawa University of Tokyo, Komaba March 9, 2009 at Nagoya Based on arxiv:0810.1737 in collaboration with Kiermaier and Zwiebach (MIT) 1 1. Introduction Quantum
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 informationBlack Hole Entropy and Gauge/Gravity Duality
Tatsuma Nishioka (Kyoto,IPMU) based on PRD 77:064005,2008 with T. Azeyanagi and T. Takayanagi JHEP 0904:019,2009 with T. Hartman, K. Murata and A. Strominger JHEP 0905:077,2009 with G. Compere and K. Murata
More informationEntanglement Entropy in Extended Quantum Systems
Entanglement Entropy in Extended Quantum Systems John Cardy University of Oxford STATPHYS 23 Genoa Outline A. Universal properties of entanglement entropy near quantum critical points B. Behaviour of entanglement
More informationChern-Simons gauge theory The Chern-Simons (CS) gauge theory in three dimensions is defined by the action,
Lecture A3 Chern-Simons gauge theory The Chern-Simons (CS) gauge theory in three dimensions is defined by the action, S CS = k tr (AdA+ 3 ) 4π A3, = k ( ǫ µνρ tr A µ ( ν A ρ ρ A ν )+ ) 8π 3 A µ[a ν,a ρ
More informationTopological Entanglement Entropy from the Holographic Partition Function
Journal of Statistical Physics, Vol. 126, No. 6, March 2007 ( C 2007 ) DOI: 10.1007/s10955-006-9275-8 Topological Entanglement Entropy from the Holographic Partition Function Paul Fendley, 1 Matthew P.
More informationOne Loop Tests of Higher Spin AdS/CFT
One Loop Tests of Higher Spin AdS/CFT Simone Giombi UNC-Chapel Hill, Jan. 30 2014 Based on 1308.2337 with I. Klebanov and 1401.0825 with I. Klebanov and B. Safdi Massless higher spins Consistent interactions
More informationOne-loop Partition Function in AdS 3 /CFT 2
One-loop Partition Function in AdS 3 /CFT 2 Bin Chen R ITP-PKU 1st East Asia Joint Workshop on Fields and Strings, May 28-30, 2016, USTC, Hefei Based on the work with Jie-qiang Wu, arxiv:1509.02062 Outline
More informationMany-body topological invariants for topological superconductors (and insulators)
Many-body topological invariants for topological superconductors (and insulators) Shinsei Ryu The University of Chicago July 5, 2017 Outline Motivations: the Kitaev chain with interactions The kitaev chain
More informationScale and Conformal Invariance in d = 4
Scale and Conformal Invariance in d = 4 Joseph Polchinski work with Markus Luty & Riccardo Rattazzi arxiv:1204xxxx N = 4 Super Yang-Mills Theory, 35 Years After Caltech, March 31, 2102 Overview: Scale
More informationHolographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary Holographic entanglement entropy for time dependent states and disconnected regions Durham University INT08: From Strings to Things, April 3, 2008 VH, M.
More informationQuantum Entanglement and the Geometry of Spacetime
Quantum Entanglement and the Geometry of Spacetime Matthew Headrick Brandeis University UMass-Boston Physics Colloquium October 26, 2017 It from Qubit Simons Foundation Entropy and area Bekenstein-Hawking
More informationLattice study of quantum entanglement in SU(3) Yang-Mills theory at zero and finite temperatures
Lattice study of quantum entanglement in SU(3) Yang-Mills theory at zero and finite temperatures Yoshiyuki Nakagawa Graduate School of Science and Technology, Niigata University, Igarashi-2, Nishi-ku,
More informationQuantum entanglement, it s entropy, and why we calculate it
Quantum entanglement, it s entropy, and why we calculate it Piotr Witkowski Max Planck Institute for Physics 14.7 2016 Munich 1 What is entanglement? 2 Quantifying entanglement the entropy 3 The (very)
More informationMany-body topological invariants for topological superconductors (and insulators)
Many-body topological invariants for topological superconductors (and insulators) Shinsei Ryu The University of Chicago June 6, 2017 Outline Motivations: the Kitaev chain with interactions The kitaev chain
More informationAnyonic Quantum Computing
Anyonic Quantum Computing 1. TQFTs as effective theories of anyons 2. Anyonic models of quantum computing (anyon=particle=quasi-particle) Topological quantum computation: 1984 Jones discovered his knot
More informationUniversal phase transitions in Topological lattice models
Universal phase transitions in Topological lattice models F. J. Burnell Collaborators: J. Slingerland S. H. Simon September 2, 2010 Overview Matter: classified by orders Symmetry Breaking (Ferromagnet)
More informationAdS/CFT Correspondence and Entanglement Entropy
AdS/CFT Correspondence and Entanglement Entropy Tadashi Takayanagi (Kyoto U.) Based on hep-th/0603001 [Phys.Rev.Lett.96(2006)181602] hep-th/0605073 [JHEP 0608(2006)045] with Shinsei Ryu (KITP) hep-th/0608213
More informationProperties of entropy in holographic theories
Properties of entropy in holographic theories Matthew Headrick randeis University Contents 0 Definitions 1 Properties of entropy Entanglement entropy in QFT 3 Ryu-Takayanagi formula 6 Monogamy 8 5 SS of
More informationAspects of Renormalized Entanglement Entropy
Aspects of Renormalized Entanglement Entropy Tatsuma Nishioka (U. Tokyo) based on 1207.3360 with Klebanov, Pufu and Safdi 1401.6764 1508.00979 with Banerjee and Nakaguchi T. Nishioka (Tokyo) Oct 15, 2015
More informationIntroduction to the Ryu-Takayanagi Formula
Introduction to the Ryu-Takayanagi Formula PHYS 48300 String Theory-1, Masaya Fukami {13 March 2018} 1 Introduction The idea of holography has played central roles in recent developments of string theory.
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 informationBoundaries, Interfaces and Dualities
Boundaries, Interfaces and Dualities Dualities I Complementary weakly coupled descriptions in a space of exactly marginal couplings T1 T5 T2 T4 T3 Dualities II IR free effective description of asymptotically
More informationKnot Homology from Refined Chern-Simons Theory
Knot Homology from Refined Chern-Simons Theory Mina Aganagic UC Berkeley Based on work with Shamil Shakirov arxiv: 1105.5117 1 the knot invariant Witten explained in 88 that J(K, q) constructed by Jones
More informationContact interactions in string theory and a reformulation of QED
Contact interactions in string theory and a reformulation of QED James Edwards QFT Seminar November 2014 Based on arxiv:1409.4948 [hep-th] and arxiv:1410.3288 [hep-th] Outline Introduction Worldline formalism
More informationEntanglement in quantum phase transition
Entanglement in quantum phase transition Huihuo Zheng Department of Physics, University of Illinois at Urbana-Champaign Urbana, IL 61801-3080, USA (Dated: May 14, 2012) Abstract A quantum phase transition
More informationRefined Chern-Simons Theory, Topological Strings and Knot Homology
Refined Chern-Simons Theory, Topological Strings and Knot Homology Based on work with Shamil Shakirov, and followup work with Kevin Scheaffer arxiv: 1105.5117 arxiv: 1202.4456 Chern-Simons theory played
More informationEntanglement in Topological Phases
Entanglement in Topological Phases Dylan Liu August 31, 2012 Abstract In this report, the research conducted on entanglement in topological phases is detailed and summarized. This includes background developed
More informationRG Flows, Entanglement & Holography Workshop. Michigan Center for Theore0cal Physics September 17 21, 2012
RG Flows, Entanglement & Holography Workshop Michigan Center for Theore0cal Physics September 17 21, 2012 Intersec0ons in Theore0cal Physics: Par$cle Physics Sta$s$cal Mechanics Renormalization Group Flows
More informationD.Blanco, H.C., L.Y.Hung, R. Myers (2013)
D.Blanco, H.C., L.Y.Hung, R. Myers (2013) Renormalization group flow in the space of QFT Change in the physics with scale through the change of coupling constants with the RG flow. At fixed points there
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 informationHolographic Entanglement Entropy, Fractional Quantum Hall Effect and Lifshitz-like Fixed Point
Journal of Physics: Conference Series OPEN ACCESS Holographic Entanglement Entropy, Fractional Quantum Hall Effect and Lifshitz-like Fixed Point To cite this article: Tadashi Takayanagi 2013 J. Phys.:
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 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 informationOn a holographic quantum quench with a finite size effect
On a holographic quantum quench with a finite size effect Tomonori Ugajin (U. Tokyo KITP) Based on work in progress with G.Mandal, R.Sinha Holographic Vistas on gravity and strings YITP, 2014 Introduction
More informationDisentangling Topological Insulators by Tensor Networks
Disentangling Topological Insulators by Tensor Networks Shinsei Ryu Univ. of Illinois, Urbana-Champaign Collaborators: Ali Mollabashi (IPM Tehran) Masahiro Nozaki (Kyoto) Tadashi Takayanagi (Kyoto) Xueda
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 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 informationThermalization and Revivals after a Quantum Quench in a Finite System
after a Quantum Quench in a Finite System John Cardy University of Oxford EPSRC Workshop Nottingham May 2014 arxiv:1403.3040, to appear in PRL (Global) Quantum Quench prepare an extended system (in thermodynamic
More informationMatrix Product States
Matrix Product States Ian McCulloch University of Queensland Centre for Engineered Quantum Systems 28 August 2017 Hilbert space (Hilbert) space is big. Really big. You just won t believe how vastly, hugely,
More informationSphere Partition Functions, Topology, the Zamolodchikov Metric
Sphere Partition Functions, Topology, the Zamolodchikov Metric, and Extremal Correlators Weizmann Institute of Science Efrat Gerchkovitz, Jaume Gomis, ZK [1405.7271] Jaume Gomis, Po-Shen Hsin, ZK, Adam
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 informationZ2 topological phase in quantum antiferromagnets. Masaki Oshikawa. ISSP, University of Tokyo
Z2 topological phase in quantum antiferromagnets Masaki Oshikawa ISSP, University of Tokyo RVB spin liquid 4 spins on a square: Groundstate is exactly + ) singlet pair a.k.a. valence bond So, the groundstate
More informationSPACETIME FROM ENTANGLEMENT - journal club notes -
SPACETIME FROM ENTANGLEMENT - journal club notes - Chris Heinrich 1 Outline 1. Introduction Big picture: Want a quantum theory of gravity Best understanding of quantum gravity so far arises through AdS/CFT
More informationThe TT Deformation of Quantum Field Theory
The TT Deformation of Quantum Field Theory John Cardy University of California, Berkeley All Souls College, Oxford ICMP, Montreal, July 2018 Introduction all QFTs that we use in physics are in some sense
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 informationGeometry and Physics. Amer Iqbal. March 4, 2010
March 4, 2010 Many uses of Mathematics in Physics The language of the physical world is mathematics. Quantitative understanding of the world around us requires the precise language of mathematics. Symmetries
More informationarxiv: v1 [hep-th] 26 Sep 2017
Eigenstate entanglement in the Sachdev-Ye-Kitaev model arxiv:709.0960v [hep-th] 6 Sep 07 Yichen Huang ( 黄溢辰 ) Institute for Quantum Information and Matter, California Institute of Technology Pasadena,
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 informationHolographic Entanglement Entropy. (with H. Casini, M. Huerta, J. Hung, M. Smolkin & A. Yale) (arxiv: , arxiv: )
v Holographic Entanglement Entropy (with H. Casini, M. Huerta, J. Hung, M. Smolkin & A. Yale) (arxiv:1102.0440, arxiv:1110.1084) Entanglement Entropy what is entanglement entropy? general tool; divide
More informationHolographic entanglement entropy
Holographic entanglement entropy Mohsen Alishahiha School of physics, Institute for Research in Fundamental Sciences (IPM) 21th Spring Physics Conference, 1393 1 Plan of the talk Entanglement entropy Holography
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 informationQuantum Fields in Curved Spacetime
Quantum Fields in Curved Spacetime Lecture 3 Finn Larsen Michigan Center for Theoretical Physics Yerevan, August 22, 2016. Recap AdS 3 is an instructive application of quantum fields in curved space. The
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 informationCounterterms, critical gravity and holography
Counterterms, critical gravity and holography Aninda Sinha, Indian Institute of Science, Bangalore, India, Perimeter Institute, Waterloo, Canada 14-Feb-12 1 of 27 Based on arxiv:1101.4746 Phys.Rev.Lett.
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 informationFrustration and Area law
Frustration and Area law When the frustration goes odd S. M. Giampaolo Institut Ruder Bošković, Zagreb, Croatia Workshop: Exactly Solvable Quantum Chains Natal 18-29 June 2018 Coauthors F. Franchini Institut
More informationTopological insulator part II: Berry Phase and Topological index
Phys60.nb 11 3 Topological insulator part II: Berry Phase and Topological index 3.1. Last chapter Topological insulator: an insulator in the bulk and a metal near the boundary (surface or edge) Quantum
More informationCovariant Prescription of Holographic Entanglement Entropy in AdS 3 and BTZ Black Hole
Master Thesis Covariant Prescription of Holographic Entanglement Entropy in AdS 3 and BTZ Black Hole Mario Benites High Energy Physics, Department of Theoretical Physics, School of Engineering Sciences
More informationConformal Blocks, Entanglement Entropy & Heavy States
Conformal Blocks, Entanglement Entropy & Heavy States Pinaki Banerjee The Institute of Mathematical Sciences, Chennai April 25, 2016 arxiv : 1601.06794 Higher-point conformal blocks & entanglement entropy
More information31st Jerusalem Winter School in Theoretical Physics: Problem Set 2
31st Jerusalem Winter School in Theoretical Physics: Problem Set Contents Frank Verstraete: Quantum Information and Quantum Matter : 3 : Solution to Problem 9 7 Daniel Harlow: Black Holes and Quantum Information
More informationHolographic Entanglement Entropy for Surface Operators and Defects
Holographic Entanglement Entropy for Surface Operators and Defects Michael Gutperle UCLA) UCSB, January 14th 016 Based on arxiv:1407.569, 1506.0005, 151.04953 with Simon Gentle and Chrysostomos Marasinou
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 informationFermionic partial transpose and non-local order parameters for SPT phases of fermions
Fermionic partial transpose and non-local order parameters for SPT phases of fermions Ken Shiozaki RIKEN Corroborators: Hassan Shapourian Shinsei Ryu Kiyonori Gomi University of Chicago University of Chicago
More informationQuantum Quench in Conformal Field Theory from a General Short-Ranged State
from a General Short-Ranged State John Cardy University of Oxford GGI, Florence, May 2012 (Global) Quantum Quench prepare an extended system at time t = 0 in a (translationally invariant) pure state ψ
More informationAnomalies, Gauss laws, and Page charges in M-theory. Gregory Moore. Strings 2004, Paris. Related works: Witten , , ,
Anomalies, Gauss laws, and Page charges in M-theory Gregory Moore Strings 2004, Paris Related works: Witten 9609122,9610234,9812012,9912086 Diaconescu, Moore, and Witten 00 Diaconescu, Freed, and Moore
More informationds/cft Contents Lecturer: Prof. Juan Maldacena Transcriber: Alexander Chen August 7, Lecture Lecture 2 5
ds/cft Lecturer: Prof. Juan Maldacena Transcriber: Alexander Chen August 7, 2011 Contents 1 Lecture 1 2 2 Lecture 2 5 1 ds/cft Lecture 1 1 Lecture 1 We will first review calculation of quantum field theory
More informationRegularization Physics 230A, Spring 2007, Hitoshi Murayama
Regularization Physics 3A, Spring 7, Hitoshi Murayama Introduction In quantum field theories, we encounter many apparent divergences. Of course all physical quantities are finite, and therefore divergences
More informationExact Solutions of 2d Supersymmetric gauge theories
Exact Solutions of 2d Supersymmetric gauge theories Abhijit Gadde, IAS w. Sergei Gukov and Pavel Putrov UV to IR Physics at long distances can be strikingly different from the physics at short distances
More informationHolography and the (Exact) Renormalization Group
Holography and the (Exact) Renormalization Group Rob Leigh University of Illinois ICMT: March 2014 Rob Leigh (UIUC) HRG ICMT: March 2014 1 / 21 Introduction An appealing aspect of holography is its interpretation
More informationMP 472 Quantum Information and Computation
MP 472 Quantum Information and Computation http://www.thphys.may.ie/staff/jvala/mp472.htm Outline Open quantum systems The density operator ensemble of quantum states general properties the reduced density
More informationUnified Description of (Some) Unitary and Nonunitary FQH States
Unified Description of (Some) Unitary and Nonunitary FQH States B. Andrei Bernevig Princeton Center for Theoretical Physics UIUC, October, 2008 Colaboration with: F.D.M. Haldane Other parts in collaboration
More informationA Violation Of The Area Law For Fermionic Entanglement Entropy
A Violation Of The Area Law For Fermionic Entanglement Entropy or: How much can you say in a phone call? Robert Helling 1 Hajo Leschke 2 and Wolfgang Spitzer 2 1 Arnold Sommerfeld Center Ludwig-Maximilians-Universität
More information1 Quantum field theory and Green s function
1 Quantum field theory and Green s function Condensed matter physics studies systems with large numbers of identical particles (e.g. electrons, phonons, photons) at finite temperature. Quantum field theory
More informationFermionic partial transpose fermionic entanglement and fermionic SPT phases
Fermionic partial transpose fermionic entanglement and fermionic SPT phases Shinsei Ryu University of Chicago November 7, 2017 Outline 1. Bosonic case (Haldane chain) What is partial tranpose? Why it is
More informationEntanglement entropy and the F theorem
Entanglement entropy and the F theorem Mathematical Sciences and research centre, Southampton June 9, 2016 H RESEARH ENT Introduction This talk will be about: 1. Entanglement entropy 2. The F theorem for
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 informationAlgebraic Theory of Entanglement
Algebraic Theory of (arxiv: 1205.2882) 1 (in collaboration with T.R. Govindarajan, A. Queiroz and A.F. Reyes-Lega) 1 Physics Department, Syracuse University, Syracuse, N.Y. and The Institute of Mathematical
More informationOverview: Entanglement Entropy
Overview: Entanglement Entropy Matthew Headrick Brandeis University January 27, 2014 Quantum Fields beyond Perturbation Theory KITP 0 Intro & disclaimer Over past 10 years, explosion of activity in entanglement
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 informationNUMBER THEORY IN STRING THEORY: SKEW, MOCK, FALSE, AND QUANTUM
NUMBER THEORY IN STRING THEORY: SKEW, MOCK, FALSE, AND QUANTUM Sarah M. Harrison based on work with Cheng, Duncan, Harvey, Kachru, Rayhaun ( 17) and Cheng, Chun, Ferrari, Gukov (to appear) INTRODUCTION
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 informationABJM Baryon Stability at Finite t Hooft Coupling
ABJM Baryon Stability at Finite t Hooft Coupling Yolanda Lozano (U. Oviedo) Santiago de Compostela, October 2011 - Motivation: Study the stability of non-singlet baryon vertex-like configurations in ABJM
More informationHolography for 3D Einstein gravity. with a conformal scalar field
Holography for 3D Einstein gravity with a conformal scalar field Farhang Loran Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran. Abstract: We review AdS 3 /CFT 2 correspondence
More information