Loop optimization for tensor network renormalization
|
|
- Maurice Hines
- 5 years ago
- Views:
Transcription
1 Yukawa Institute for Theoretical Physics, Kyoto University, Japan June, 6 Loop optimization for tensor network renormalization Shuo Yang! Perimeter Institute for Theoretical Physics, Waterloo, Canada Zheng-Cheng Gu (CUHK, PI) Xiao-Gang Wen (MIT, PI) arxiv:5.98
2 Overview tensor network + renormalization group = tensor network renormalization! Partition function of classical statistical system!!! Euclidean path integral of D quantum system!!! Physical properties of D quantum system e h Z = X { } exp( X hiji i j)
3 Overview tensor network + renormalization group = tensor network renormalization boundary MPS fully isotropic coarse-graining
4 Overview tensor network + renormalization group = tensor network renormalization How the physics change with scale scale transformation Non-perturbative approach, suitable for strongly correlated systems Reproduce long-range physics
5 Overview tensor network + renormalization group = tensor network renormalization How the physics change with scale scale transformation Non-perturbative approach, suitable for strongly correlated systems Reproduce long-range physics RG flow Aim! Remove short-range entanglement / correlations Generate proper RG flow & correct fixed points Recover scale invariance at criticality A C B
6 Real space renormalization group D quantum: real space NRG entanglement DMRG S. White (99) Real space analogue of NRG Wilson (975) breakdown for particle in a box problem D classical: block spin entanglement TRG Levin & Nave (7) L.P. Kadanoff (966)
7 Tensor renormalization group Levin & Nave (7) LN-TNR Three steps. Deform tensors, make a truncation. Coarse graining. Renormalize tensors (multiply tensor by a constant factor)
8 Tensor renormalization group SVD singular value decomposition (SVD) keep only keep the largest keep SVD original keep = S S T U singular values cost function keep keep p = p V S keep = S keep = /
9 Drawbacks of LN-TNR Off criticality cannot remove conner-double-line (CDL) tensors cannot give the correct structures of fixed points D D D D = D D D... D C A A C B
10 Drawbacks of LN-TNR At criticality cannot explicitly recover scale invariance cannot completely remove short-range entanglement
11 Drawbacks of LN-TNR At criticality cannot explicitly recover scale invariance cannot completely remove short-range entanglement Example classical Ising model partition function Z = X { } exp( X hiji i j)
12 Drawbacks of LN-TNR At criticality cannot explicitly recover scale invariance cannot completely remove short-range entanglement Example classical Ising model partition function Z = X { } exp( X i j) hiji local tensor T = T Ising u,l,d,r T Ising,,, = e, T Ising,,, = e, T Ising,,, = e, T Ising,,, = e, others =.
13 Drawbacks of LN-TNR Ising CFT central charge c =/ scaling dimensions I
14 Drawbacks of LN-TNR Ising CFT central charge c =/ scaling dimensions I Calculate scaling dimensions transfer matrix eigenvalues of the transfer matrix c, Zheng-Cheng Gu and Xiao-Gang Wen, Phys. Rev. B 8, 55 (9).
15 Drawbacks of LN-TNR Ising CFT central charge c =/ scaling dimensions I Calculate scaling dimensions transfer matrix eigenvalues of the transfer matrix c, Zheng-Cheng Gu and Xiao-Gang Wen, Phys. Rev. B 8, 55 (9). LN-TNR c, even, odd Loop-I, D= scaling dimensions change with RG step cannot explicitly recover scale invariance 5 5 RG step
16 Drawbacks of LN-TNR Ising CFT central charge c =/ scaling dimensions I Calculate scaling dimensions transfer matrix eigenvalues of the transfer matrix c, Zheng-Cheng Gu and Xiao-Gang Wen, Phys. Rev. B 8, 55 (9). LN-TNR c, even, odd Loop-I, D= scaling dimensions change with RG step cannot explicitly recover scale invariance high-index parts will destroy low-index parts accuracy is fine, stability is bad 5 5 RG step
17 Improvements of TRG off-critical critical TRG (Levin & Nave, 6) SRG (Xie, Jiang, Weng, Xiang, 8) TEFR (Gu & Wen, 9) TNR (Evenbly & Vidal, ) Loop-TRG! (current)
18 Improvements of TRG off-critical critical TRG (Levin & Nave, 6) SRG (Xie, Jiang, Weng, Xiang, 8) TEFR (Gu & Wen, 9) TNR (Evenbly & Vidal, ) Loop-TRG (current)
19 Renormalization Group Momentum space Tree level approximation no loop Real space Tree level approximation no loop Beyond tree level one loop Beyond tree level one loop
20 Renormalization Group Momentum space Tree level approximation no loop Real space Tree level approximation no loop momentum b ignored fast slow simply ignore the fast modes Beyond tree level one loop Beyond tree level one loop
21 Renormalization Group Momentum space Tree level approximation no loop Real space Tree level approximation no loop momentum b ignored fast slow simply ignore the fast modes Beyond tree level one loop Beyond tree level one loop momentum b ignored fast slow integrate over the fast modes
22 Renormalization Group Momentum space Tree level approximation no loop Real space Tree level approximation no loop momentum b ignored fast slow simply ignore the fast modes LN-TNR remove short-range entanglement by a local SVD Beyond tree level one loop Beyond tree level one loop momentum b ignored fast slow integrate over the fast modes
23 Renormalization Group Momentum space Tree level approximation no loop Real space Tree level approximation no loop momentum b ignored fast slow simply ignore the fast modes LN-TNR remove short-range entanglement by a local SVD Beyond tree level one loop Beyond tree level one loop momentum b ignored fast slow integrate over the fast modes LN-TNR +loop optimization further remove short-range entanglement inside a loop! This work!
24 Algorithms of Loop-TNR
25 Algorithms of Loop-TNR Part One: Entanglement filtering
26 Algorithms of Loop-TNR Part One: Entanglement filtering Part Two: Optimizing tensors on a loop
27 Algorithms of Loop-TNR Part One: Entanglement filtering Together: Complete remove short-range entanglement Part Two: Optimizing tensors on a loop
28 Part One Entanglement filtering How. Find & insert projectors. Define new tensors P L P L P L P L T T T T P R P R P R P R
29 Part One Entanglement filtering How. Find & insert projectors. Define new tensors P L P L P L P L T T T T P R P R P R P R Aim Remove conner double line (CDL) tensors Generate local canonical gauge Zheng-Cheng Gu and Xiao-Gang Wen, Phys. Rev. B 8, 55 (9).
30 Part Two Optimizing tensors on a loop
31 Part Two Optimizing tensors on a loop cost function LN-TNR Loop-TNR
32 Part Two Optimizing tensors on a loop solve the linear equation
33 Part Two Optimizing tensors on a loop
34 Part Two Optimizing tensors on a loop Loop-TNR Results L =
35 Part Two Optimizing tensors on a loop Loop-TNR Results scaling dimensions does not change with scale ~ scale invariance a clear gap between high-level parts and low-level parts L =
36 Stability c, even, odd (a) LN-TNR =6, L= (b) Loop-TNR, =6, L= 5 5 c, even, odd (c) LN-TNR =, L= (d) Loop-TNR, =, L= (e) LN-TNR, =6, L= 6 (f) Loop-TNR, =6, L= c, even, odd Iteration step Iteration step
37 Stability c, even, odd (a) LN-TNR =6, L= (b) Loop-TNR, =6, L= = 6 remain accurate up to iteration steps 5 5 c, even, odd (c) LN-TNR =, L= 5 5 (d) Loop-TNR, =, L= even longer for = the proper RG flow last longer for larger 6 (e) LN-TNR, =6, L= 6 (f) Loop-TNR, =6, L= c, even, odd Iteration step Iteration step
38 Stability c, even, odd (a) LN-TNR =6, L= (b) Loop-TNR, =6, L= 5 5 c, even, odd (c) LN-TNR =, L= (d) Loop-TNR, =, L= increasing, more scaling dimensions can be resolved (e) LN-TNR, =6, L= 6 (f) Loop-TNR, =6, L= c, even, odd Iteration step Iteration step
39 Stability c, even, odd (a) LN-TNR =6, L= (b) Loop-TNR, =6, L= effectively, = c, even, odd (c) LN-TNR =, L= (d) Loop-TNR, =, L= c, even, odd (e) LN-TNR, =6, L= 6 5 (f) Loop-TNR, =6, L= effectively, = 6 = 56 much more scaling dimensions can be read off accuracy is higher 5 5 Iteration step Iteration step
40 Stability c, even, odd (a) LN-TNR =6, L= (b) Loop-TNR, =6, L= effectively, = c, even, odd (c) LN-TNR =, L= (d) Loop-TNR, =, L= c, even, odd (e) LN-TNR, =6, L= 5 5 Iteration step 6 5 (f) Loop-TNR, =6, L= Iteration step effectively, = 6 = 56 much more scaling dimensions can be read off accuracy is higher infinite ~ infinite dimensional fixed point tensor described by Ising CFT
41 Accuracy
42 Relative error of the free energy per site f (Tc) Loop-TNR (a) Critical LN-TNR 5 6 f (T) (b) Off-critical, =8 LN-TNR Loop-TNR T/T c
43 Relative error of the free energy per site At critical point, the error of Loop-TNR decays much faster than the error of LN-TNR f (Tc) Loop-TNR (a) Critical LN-TNR 5 6 f (T) (b) Off-critical, =8 LN-TNR Loop-TNR T/T c almost decays exponentially with bond dimension
44 Relative error of the free energy per site At critical point, the error of Loop-TNR decays much faster than the error of LN-TNR f (Tc) Loop-TNR (a) Critical LN-TNR 5 6 f (T) (b) Off-critical, =8 LN-TNR Loop-TNR T/T c almost decays exponentially with bond dimension At off-critical points, the errors almost remains a constant for all temperatures near the critical point
45 Loop TRG with reflection symmetry i i l j j k k l k j j k i i l l (a) (b) axis of symmetry global C symmetry (f) (e) (d) (c) fix the gauge (g) (h) j i k l T (i) S ideal case: explicitly invariant fixed point tensor S
46 Fixed point tensors LN-TNR absolute value LN-TNR absolute difference low-index parts are approximately invariant Loop-TNR absolute value Loop-TNR absolute difference all tensor elements are nearly invariant ~ scale invariance
47 Summary
48 Summary Real space TNR! Tree level: LN-TNR (Levin & Nave, 7) One loop: GW-TNR, EV-TNR, Loop-TNR
49 Summary Real space TNR! Tree level: LN-TNR (Levin & Nave, 7) One loop: GW-TNR, EV-TNR, Loop-TNR D algorithm LN-TNR + D algorithm itebd GW-TNR (Gu & Wen, 9) Loop-TNR Part One DMRG Varational MPS Loop-TNR Part Two MERA EV-TNR (Evenbly & Vidal, )
50 Thank you!
Tensor network renormalization
Coogee'15 Sydney Quantum Information Theory Workshop Tensor network renormalization Guifre Vidal In collaboration with GLEN EVENBLY IQIM Caltech UC Irvine Quantum Mechanics 1920-1930 Niels Bohr Albert
More informationIntroduction to tensor network state -- concept and algorithm. Z. Y. Xie ( 谢志远 ) ITP, Beijing
Introduction to tensor network state -- concept and algorithm Z. Y. Xie ( 谢志远 ) 2018.10.29 ITP, Beijing Outline Illusion of complexity of Hilbert space Matrix product state (MPS) as lowly-entangled state
More informationRenormalization of Tensor- Network States Tao Xiang
Renormalization of Tensor- Network States Tao Xiang Institute of Physics/Institute of Theoretical Physics Chinese Academy of Sciences txiang@iphy.ac.cn Physical Background: characteristic energy scales
More informationScale invariance on the lattice
Coogee'16 Sydney Quantum Information Theory Workshop Feb 2 nd - 5 th, 2016 Scale invariance on the lattice Guifre Vidal Coogee'16 Sydney Quantum Information Theory Workshop Feb 2 nd - 5 th, 2016 Scale
More informationRenormalization of Tensor Network States
Renormalization of Tensor Network States I. Coarse Graining Tensor Renormalization Tao Xiang Institute of Physics Chinese Academy of Sciences txiang@iphy.ac.cn Numerical Renormalization Group brief introduction
More informationTensor network renormalization
Walter Burke Institute for Theoretical Physics INAUGURAL CELEBRATION AND SYMPOSIUM Caltech, Feb 23-24, 2015 Tensor network renormalization Guifre Vidal Sherman Fairchild Prize Postdoctoral Fellow (2003-2005)
More informationTensor network methods in condensed matter physics. ISSP, University of Tokyo, Tsuyoshi Okubo
Tensor network methods in condensed matter physics ISSP, University of Tokyo, Tsuyoshi Okubo Contents Possible target of tensor network methods! Tensor network methods! Tensor network states as ground
More informationc 2017 Society for Industrial and Applied Mathematics
MULTISCALE MODEL. SIMUL. Vol. 15, No. 4, pp. 1423 1447 c 2017 Society for Industrial and Applied Mathematics TENSOR NETWORK SKELETONIZATION LEXING YING Abstract. We introduce a new coarse-graining algorithm,
More informationRenormalization of Tensor Network States. Partial order and finite temperature phase transition in the Potts model on irregular lattice
Renormalization of Tensor Network States Partial order and finite temperature phase transition in the Potts model on irregular lattice Tao Xiang Institute of Physics Chinese Academy of Sciences Background:
More informationarxiv: v3 [cond-mat.str-el] 15 Sep 2015
Tensor Network Renormalization G. Evenbly 1 and G. Vidal 2 1 Institute for Quantum Information and Matter, California Institute of Technology, Pasadena CA 91125, USA 2 Perimeter Institute for Theoretical
More informationEfficient time evolution of one-dimensional quantum systems
Efficient time evolution of one-dimensional quantum systems Frank Pollmann Max-Planck-Institut für komplexer Systeme, Dresden, Germany Sep. 5, 2012 Hsinchu Problems we will address... Finding ground states
More informationTensor Networks, Renormalization and Holography (overview)
KITP Conference Closing the entanglement gap: Quantum information, quantum matter, and quantum fields June 1 st -5 th 2015 Tensor Networks, Renormalization and Holography (overview) Guifre Vidal KITP Conference
More informationarxiv: v1 [quant-ph] 18 Jul 2017
Implicitly disentangled renormalization arxiv:1707.05770v1 [quant-ph] 18 Jul 017 Glen Evenbly 1 1 Département de Physique and Institut Quantique, Université de Sherbrooke, Québec, Canada (Dated: July 19,
More informationarxiv: v1 [cond-mat.str-el] 24 Sep 2015
Algorithms for tensor network renormalization G. Evenbly 1 1 Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575 USA (Dated: September 25, 2015) arxiv:1509.07484v1 [cond-mat.str-el]
More informationFrom Path Integral to Tensor Networks for AdS/CFT
Seminar @ Osaka U 2016/11/15 From Path Integral to Tensor Networks for AdS/CFT Kento Watanabe (Center for Gravitational Physics, YITP, Kyoto U) 1609.04645v2 [hep-th] w/ Tadashi Takayanagi (YITP + Kavli
More informationAn introduction to tensornetwork
An introduction to tensornetwork states and MERA Sissa Journal Club Andrea De Luca 29/01/2010 A typical problem We are given: A lattice with N sites On each site a C d hilbert space A quantum hamiltonian
More informationHolographic Branching and Entanglement Renormalization
KITP, December 7 th 2010 Holographic Branching and Entanglement Renormalization Glen Evenbly Guifre Vidal Tensor Network Methods (DMRG, PEPS, TERG, MERA) Potentially offer general formalism to efficiently
More informationSimulating Quantum Systems through Matrix Product States. Laura Foini SISSA Journal Club
Simulating Quantum Systems through Matrix Product States Laura Foini SISSA Journal Club 15-04-2010 Motivations Theoretical interest in Matrix Product States Wide spectrum of their numerical applications
More informationFermionic tensor networks
Fermionic tensor networks Philippe Corboz, Institute for Theoretical Physics, ETH Zurich Bosons vs Fermions P. Corboz and G. Vidal, Phys. Rev. B 80, 165129 (2009) : fermionic 2D MERA P. Corboz, R. Orus,
More informationThe density matrix renormalization group and tensor network methods
The density matrix renormalization group and tensor network methods Outline Steve White Exploiting the low entanglement of ground states Matrix product states and DMRG 1D 2D Tensor network states Some
More informationMachine Learning with Quantum-Inspired Tensor Networks
Machine Learning with Quantum-Inspired Tensor Networks E.M. Stoudenmire and David J. Schwab Advances in Neural Information Processing 29 arxiv:1605.05775 RIKEN AICS - Mar 2017 Collaboration with David
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 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 informationIt from Qubit Summer School
It from Qubit Summer School July 27 th, 2016 Tensor Networks Guifre Vidal NSERC Wednesday 27 th 9AM ecture Tensor Networks 2:30PM Problem session 5PM Focus ecture MARKUS HAURU MERA: a tensor network for
More informationQuantum Hamiltonian Complexity. Itai Arad
1 18 / Quantum Hamiltonian Complexity Itai Arad Centre of Quantum Technologies National University of Singapore QIP 2015 2 18 / Quantum Hamiltonian Complexity condensed matter physics QHC complexity theory
More informationThe uses of Instantons for classifying Topological Phases
The uses of Instantons for classifying Topological Phases - anomaly-free and chiral fermions Juven Wang, Xiao-Gang Wen (arxiv:1307.7480, arxiv:140?.????) MIT/Perimeter Inst. 2014 @ APS March A Lattice
More informationLecture 3: Tensor Product Ansatz
Lecture 3: Tensor Product nsatz Graduate Lectures Dr Gunnar Möller Cavendish Laboratory, University of Cambridge slide credits: Philippe Corboz (ETH / msterdam) January 2014 Cavendish Laboratory Part I:
More informationFrustration-free Ground States of Quantum Spin Systems 1
1 Davis, January 19, 2011 Frustration-free Ground States of Quantum Spin Systems 1 Bruno Nachtergaele (UC Davis) based on joint work with Sven Bachmann, Spyridon Michalakis, Robert Sims, and Reinhard Werner
More informationarxiv: v1 [quant-ph] 6 Jun 2011
Tensor network states and geometry G. Evenbly 1, G. Vidal 1,2 1 School of Mathematics and Physics, the University of Queensland, Brisbane 4072, Australia 2 Perimeter Institute for Theoretical Physics,
More informationTime Evolving Block Decimation Algorithm
Time Evolving Block Decimation Algorithm Application to bosons on a lattice Jakub Zakrzewski Marian Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Center, Jagiellonian University,
More informationQuantum Fields, Gravity, and Complexity. Brian Swingle UMD WIP w/ Isaac Kim, IBM
Quantum Fields, Gravity, and Complexity Brian Swingle UMD WIP w/ Isaac Kim, IBM Influence of quantum information Easy Hard (at present) Easy Hard (at present)!! QI-inspired classical, e.g. tensor networks
More informationarxiv: v2 [quant-ph] 31 Oct 2013
Quantum Criticality with the Multi-scale Entanglement Renormalization Ansatz arxiv:1109.5334v2 [quant-ph] 31 Oct 2013 G. Evenbly 1 and G. Vidal 2 1 The University of Queensland, Brisbane, Queensland 4072,
More informationManifestly diffeomorphism invariant classical Exact Renormalization Group
Manifestly diffeomorphism invariant classical Exact Renormalization Group Anthony W. H. Preston University of Southampton Supervised by Prof. Tim R. Morris Talk prepared for Asymptotic Safety seminar,
More information4 Matrix product states
Physics 3b Lecture 5 Caltech, 05//7 4 Matrix product states Matrix product state (MPS) is a highly useful tool in the study of interacting quantum systems in one dimension, both analytically and numerically.
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 informationMany-body entanglement witness
Many-body entanglement witness Jeongwan Haah, MIT 21 January 2015 Coogee, Australia arxiv:1407.2926 Quiz Energy Entanglement Charge Invariant Momentum Total spin Rest Mass Complexity Class Many-body Entanglement
More informationIPAM/UCLA, Sat 24 th Jan Numerical Approaches to Quantum Many-Body Systems. QS2009 tutorials. lecture: Tensor Networks.
IPAM/UCLA, Sat 24 th Jan 2009 umerical Approaches to Quantum Many-Body Systems QS2009 tutorials lecture: Tensor etworks Guifre Vidal Outline Tensor etworks Computation of expected values Optimization of
More informationRigorous free fermion entanglement renormalization from wavelets
Computational Complexity and High Energy Physics August 1st, 2017 Rigorous free fermion entanglement renormalization from wavelets arxiv: 1707.06243 Jutho Haegeman Ghent University in collaboration with:
More informationMERA for Spin Chains with Critical Lines
MERA for Spin Chains with Critical Lines Jacob C. Bridgeman Aroon O Brien Stephen D. Bartlett Andrew C. Doherty ARC Centre for Engineered Quantum Systems, The University of Sydney, Australia January 16,
More informationMachine Learning with Tensor Networks
Machine Learning with Tensor Networks E.M. Stoudenmire and David J. Schwab Advances in Neural Information Processing 29 arxiv:1605.05775 Beijing Jun 2017 Machine learning has physics in its DNA # " # #
More information3 Symmetry Protected Topological Phase
Physics 3b Lecture 16 Caltech, 05/30/18 3 Symmetry Protected Topological Phase 3.1 Breakdown of noninteracting SPT phases with interaction Building on our previous discussion of the Majorana chain and
More informationNews on tensor network algorithms
News on tensor network algorithms Román Orús Donostia International Physics Center (DIPC) December 6th 2018 S. S. Jahromi, RO, M. Kargarian, A. Langari, PRB 97, 115162 (2018) S. S. Jahromi, RO, PRB 98,
More informationMatrix-Product states: Properties and Extensions
New Development of Numerical Simulations in Low-Dimensional Quantum Systems: From Density Matrix Renormalization Group to Tensor Network Formulations October 27-29, 2010, Yukawa Institute for Theoretical
More informationFermionic tensor networks
Fermionic tensor networks Philippe Corboz, ETH Zurich / EPF Lausanne, Switzerland Collaborators: University of Queensland: Guifre Vidal, Roman Orus, Glen Evenbly, Jacob Jordan University of Vienna: Frank
More informationQuantum simulation with string-bond states: Joining PEPS and Monte Carlo
Quantum simulation with string-bond states: Joining PEPS and Monte Carlo N. Schuch 1, A. Sfondrini 1,2, F. Mezzacapo 1, J. Cerrillo 1,3, M. Wolf 1,4, F. Verstraete 5, I. Cirac 1 1 Max-Planck-Institute
More informationEntanglement signatures of QED3 in the kagome spin liquid. William Witczak-Krempa
Entanglement signatures of QED3 in the kagome spin liquid William Witczak-Krempa Aspen, March 2018 Chronologically: X. Chen, KITP Santa Barbara T. Faulkner, UIUC E. Fradkin, UIUC S. Whitsitt, Harvard S.
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 informationIntroduction to Tensor Networks: PEPS, Fermions, and More
Introduction to Tensor Networks: PEPS, Fermions, and More Román Orús Institut für Physik, Johannes Gutenberg-Universität, Mainz (Germany)! School on computational methods in quantum materials Jouvence,
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 informationQuantum quenches in 2D with chain array matrix product states
Quantum quenches in 2D with chain array matrix product states Andrew J. A. James University College London Robert M. Konik Brookhaven National Laboratory arxiv:1504.00237 Outline MPS for many body systems
More informationHolographic Geometries from Tensor Network States
Holographic Geometries from Tensor Network States J. Molina-Vilaplana 1 1 Universidad Politécnica de Cartagena Perspectives on Quantum Many-Body Entanglement, Mainz, Sep 2013 1 Introduction & Motivation
More informationZhong-Zhi Xianyu (CMSA Harvard) Tsinghua June 30, 2016
Zhong-Zhi Xianyu (CMSA Harvard) Tsinghua June 30, 2016 We are directly observing the history of the universe as we look deeply into the sky. JUN 30, 2016 ZZXianyu (CMSA) 2 At ~10 4 yrs the universe becomes
More informationPositive Tensor Network approach for simulating open quantum many-body systems
Positive Tensor Network approach for simulating open quantum many-body systems 19 / 9 / 2016 A. Werner, D. Jaschke, P. Silvi, M. Kliesch, T. Calarco, J. Eisert and S. Montangero PRL 116, 237201 (2016)
More informationTowards a manifestly diffeomorphism invariant Exact Renormalization Group
Towards a manifestly diffeomorphism invariant Exact Renormalization Group Anthony W. H. Preston University of Southampton Supervised by Prof. Tim R. Morris Talk prepared for UK QFT-V, University of Nottingham,
More informationNUMERICAL METHODS FOR QUANTUM IMPURITY MODELS
NUMERICAL METODS FOR QUANTUM IMPURITY MODELS http://www.staff.science.uu.nl/~mitch003/nrg.html March 2015 Andrew Mitchell, Utrecht University Quantum impurity problems Part 1: Quantum impurity problems
More informationLoop corrections in Yukawa theory based on S-51
Loop corrections in Yukawa theory based on S-51 Similarly, the exact Dirac propagator can be written as: Let s consider the theory of a pseudoscalar field and a Dirac field: the only couplings allowed
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 informationThe Density Matrix Renormalization Group: Introduction and Overview
The Density Matrix Renormalization Group: Introduction and Overview Introduction to DMRG as a low entanglement approximation Entanglement Matrix Product States Minimizing the energy and DMRG sweeping The
More informationVI.D Self Duality in the Two Dimensional Ising Model
VI.D Self Duality in the Two Dimensional Ising Model Kramers and Wannier discovered a hidden symmetry that relates the properties of the Ising model on the square lattice at low and high temperatures.
More informationarxiv: v4 [cond-mat.stat-mech] 13 Mar 2009
The itebd algorithm beyond unitary evolution R. Orús and G. Vidal School of Physical Sciences, The University of Queensland, QLD 4072, Australia arxiv:0711.3960v4 [cond-mat.stat-mech] 13 Mar 2009 The infinite
More informationRenormalization Group for the Two-Dimensional Ising Model
Chapter 8 Renormalization Group for the Two-Dimensional Ising Model The two-dimensional (2D) Ising model is arguably the most important in statistical physics. This special status is due to Lars Onsager
More informationMatrix Product Operators: Algebras and Applications
Matrix Product Operators: Algebras and Applications Frank Verstraete Ghent University and University of Vienna Nick Bultinck, Jutho Haegeman, Michael Marien Burak Sahinoglu, Dominic Williamson Ignacio
More informationGravity Actions from Tensor Networks
Gravity Actions from Tensor Networks [hep-th/1609.04645] with T. Takayanagi and K. Watanabe and ongoing work with P. Caputa, N. Kundu, T. Takayanagi, K. Watanabe Masamichi Miyaji Yukawa Institute for Theoretical
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 informationVI.D Self Duality in the Two Dimensional Ising Model
VI.D Self Duality in the Two Dimensional Ising Model Kramers and Wannier discovered a hidden symmetry that relates the properties of the Ising model on the square lattice at low and high temperatures.
More informationMomentum-space and Hybrid Real- Momentum Space DMRG applied to the Hubbard Model
Momentum-space and Hybrid Real- Momentum Space DMRG applied to the Hubbard Model Örs Legeza Reinhard M. Noack Collaborators Georg Ehlers Jeno Sólyom Gergely Barcza Steven R. White Collaborators Georg Ehlers
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 informationRichard Williams C. S. Fischer, W. Heupel, H. Sanchis-Alepuz
Richard Williams C. S. Fischer, W. Heupel, H. Sanchis-Alepuz Overview 2 1.Motivation and Introduction 4. 3PI DSE results 2. DSEs and BSEs 3. npi effective action 6. Outlook and conclusion 5. 3PI meson
More informationQuantum many-body systems and tensor networks: simulation methods and applications
Quantum many-body systems and tensor networks: simulation methods and applications Román Orús School of Physical Sciences, University of Queensland, Brisbane (Australia) Department of Physics and Astronomy,
More informationManuscript code: BH11160 RECVD: Mon Feb 8 16:14: Resubmission to: Physical Review B Resubmission type: resubmit
1 of 6 05/17/2012 09:32 AM Date: Mon, 8 Feb 2010 21:14:55 UT From: esub-adm@aps.org To: prbtex@ridge.aps.org CC: wen@dao.mit.edu Subject: [Web] resub BH11160 Gu Subject: BH11160 Manuscript code: BH11160
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 informationPlaquette Renormalized Tensor Network States: Application to Frustrated Systems
Workshop on QIS and QMP, Dec 20, 2009 Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao and Center for Quantum Science and Engineering Hsin-Chih Hsiao, Ji-Feng
More informationMatrix product approximations to multipoint functions in two-dimensional conformal field theory
Matrix product approximations to multipoint functions in two-dimensional conformal field theory Robert Koenig (TUM) and Volkher B. Scholz (Ghent University) based on arxiv:1509.07414 and 1601.00470 (published
More informationSpectral action, scale anomaly. and the Higgs-Dilaton potential
Spectral action, scale anomaly and the Higgs-Dilaton potential Fedele Lizzi Università di Napoli Federico II Work in collaboration with A.A. Andrianov (St. Petersburg) and M.A. Kurkov (Napoli) JHEP 1005:057,2010
More informationRG Limit Cycles (Part I)
RG Limit Cycles (Part I) Andy Stergiou UC San Diego based on work with Jean-François Fortin and Benjamín Grinstein Outline The physics: Background and motivation New improved SE tensor and scale invariance
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 informationUniversal Dynamics from the Conformal Bootstrap
Universal Dynamics from the Conformal Bootstrap Liam Fitzpatrick Stanford University! in collaboration with Kaplan, Poland, Simmons-Duffin, and Walters Conformal Symmetry Conformal = coordinate transformations
More informationExact Functional Renormalization Group for Wetting Transitions in Dimensions.
EUROPHYSICS LETTERS Europhys. Lett., 11 (7), pp. 657-662 (1990) 1 April 1990 Exact Functional Renormalization Group for Wetting Transitions in 1 + 1 Dimensions. F. JULICHER, R. LIPOWSKY (*) and H. MULLER-KRUMBHAAR
More informationarxiv: v2 [cond-mat.str-el] 6 Nov 2013
Symmetry Protected Quantum State Renormalization Ching-Yu Huang, Xie Chen, and Feng-Li Lin 3 Max-Planck-Institut für Physik komplexer Systeme, 087 Dresden, Germany Department of Physics, University of
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 information(r) 2.0 E N 1.0
The Numerical Renormalization Group Ralf Bulla Institut für Theoretische Physik Universität zu Köln 4.0 3.0 Q=0, S=1/2 Q=1, S=0 Q=1, S=1 E N 2.0 1.0 Contents 1. introduction to basic rg concepts 2. introduction
More informationLocal Algorithms for 1D Quantum Systems
Local Algorithms for 1D Quantum Systems I. Arad, Z. Landau, U. Vazirani, T. Vidick I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 1 / 14 Many-body physics Are physical
More informationarxiv: v1 [cond-mat.str-el] 7 Aug 2011
Topological Geometric Entanglement of Blocks Román Orús 1, 2 and Tzu-Chieh Wei 3, 4 1 School of Mathematics and Physics, The University of Queensland, QLD 4072, Australia 2 Max-Planck-Institut für Quantenoptik,
More informationGapped spin liquid with Z[subscript 2] topological order for the kagome Heisenberg model
Gapped spin liquid with Z[subscript 2] topological order for the kagome Heisenberg model The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters.
More informationIsotropic Entanglement
Isotropic Entanglement Ramis Movassagh 1 and Alan Edelman 1 1 Department of Mathematics, M.I.T. QIP, January 2011 Acknowledgments Peter W. Shor Jerey Goldstone X-G Wen, Patrick Lee, Peter Young, Mehran
More information4 4 and perturbation theory
and perturbation theory We now consider interacting theories. A simple case is the interacting scalar field, with a interaction. This corresponds to a -body contact repulsive interaction between scalar
More informationCritical Region of the QCD Phase Transition
Critical Region of the QCD Phase Transition Mean field vs. Renormalization group B.-J. Schaefer 1 and J. Wambach 1,2 1 Institut für Kernphysik TU Darmstadt 2 GSI Darmstadt 18th August 25 Uni. Graz B.-J.
More informationImplications of Poincaré symmetry for thermal field theories
L. Giusti STRONGnet 23 Graz - September 23 p. /27 Implications of Poincaré symmetry for thermal field theories Leonardo Giusti University of Milano-Bicocca Based on: L. G. and H. B. Meyer JHEP 3 (23) 4,
More informationDephasing, relaxation and thermalization in one-dimensional quantum systems
Dephasing, relaxation and thermalization in one-dimensional quantum systems Fachbereich Physik, TU Kaiserslautern 26.7.2012 Outline 1 Introduction 2 Dephasing, relaxation and thermalization 3 Particle
More informationExcursion: MPS & DMRG
Excursion: MPS & DMRG Johannes.Schachenmayer@gmail.com Acronyms for: - Matrix product states - Density matrix renormalization group Numerical methods for simulations of time dynamics of large 1D quantum
More informationTensor network simulations of strongly correlated quantum systems
CENTRE FOR QUANTUM TECHNOLOGIES NATIONAL UNIVERSITY OF SINGAPORE AND CLARENDON LABORATORY UNIVERSITY OF OXFORD Tensor network simulations of strongly correlated quantum systems Stephen Clark LXXT[[[GSQPEFS\EGYOEGXMZMXMIWUYERXYQGSYVWI
More informationApproaching conformality in lattice models
Approaching conformality in lattice models Yannick Meurice The University of Iowa yannick-meurice@uiowa.edu Work done in part with Alexei Bazavov, Zech Gelzer, Shan-Wen Tsai, Judah Unmuth-Yockey, Li-Ping
More informationREVIEW REVIEW. Quantum Field Theory II
Quantum Field Theory II PHYS-P 622 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory Chapters: 13, 14, 16-21, 26-28, 51, 52, 61-68, 44, 53, 69-74, 30-32, 84-86, 75,
More informationQuantum Field Theory II
Quantum Field Theory II PHYS-P 622 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory Chapters: 13, 14, 16-21, 26-28, 51, 52, 61-68, 44, 53, 69-74, 30-32, 84-86, 75,
More informationTensor network simulation of QED on infinite lattices: learning from (1 + 1)d, and prospects for (2 + 1)d
Tensor network simulation of QED on infinite lattices: learning from (1 + 1)d, and prospects for (2 + 1)d Román Orús University of Mainz (Germany) K. Zapp, RO, Phys. Rev. D 95, 114508 (2017) Goal of this
More informationPhase Transitions and Renormalization:
Phase Transitions and Renormalization: Using quantum techniques to understand critical phenomena. Sean Pohorence Department of Applied Mathematics and Theoretical Physics University of Cambridge CAPS 2013
More informationInverse square potential, scale anomaly, and complex extension
Inverse square potential, scale anomaly, and complex extension Sergej Moroz Seattle, February 2010 Work in collaboration with Richard Schmidt ITP, Heidelberg Outline Introduction and motivation Functional
More informationValence Bonds in Random Quantum Magnets
Valence Bonds in Random Quantum Magnets theory and application to YbMgGaO 4 Yukawa Institute, Kyoto, November 2017 Itamar Kimchi I.K., Adam Nahum, T. Senthil, arxiv:1710.06860 Valence Bonds in Random Quantum
More informationThe Ising model Summary of L12
The Ising model Summary of L2 Aim: Study connections between macroscopic phenomena and the underlying microscopic world for a ferromagnet. How: Study the simplest possible model of a ferromagnet containing
More informationDynamical mean field approach to correlated lattice systems in and out of equilibrium
Dynamical mean field approach to correlated lattice systems in and out of equilibrium Philipp Werner University of Fribourg, Switzerland Kyoto, December 2013 Overview Dynamical mean field approximation
More information