A classification of gapped Hamiltonians in d = 1
|
|
- Jared Heath
- 6 years ago
- Views:
Transcription
1 A classification of gapped Hamiltonians in d = 1 Sven Bachmann Mathematisches Institut Ludwig-Maximilians-Universität München Joint work with Yoshiko Ogata NSF-CBMS school on quantum spin systems Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
2 Quantum spin systems A lattice Γ of finite dimensional quantum systems (spins), with Hilbert space H Λ = x Λ H x, Λ Γ, finite Observables on Λ Γ: A Λ = L(H Λ ) Local Hamiltonian: a sum of short range interactions Φ(X) = Φ(X) A X H Λ = X Λ Φ(X) The Heisenberg dynamics: τ t Λ(A) = exp(ith Λ )A exp( ith Λ ) Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
3 States The quasi-local algebra A Γ : A Γ = Λ Γ A Λ State ω: a positive, normalized, linear form on A Γ Finite volume Λ: A Λ = B(H Λ ) and ω(a) = Tr(ρ ω ΛA) where ρ ω Λ is a density matrix Infinite systems Γ: No density matrix in general But: Nets of states ω Λ on A Λ have weak-* accumulation points ω Γ as Λ Γ: states in the thermodynamic limit Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
4 The Ising model Now: Γ = Z, H x = C 2, spin 1/2, translation invariant Φ: sσx 3 if X = {x} for some x Z, h 0 Φ(X, s) = σx 1 1 σ1 x if X = {x 1, x} for some x Z 0 otherwise N N 1 H [0,N 1] (s) = σx 1σ 1 x 1 s (anisotropic XY chain) x=1 x=0 σ 3 x s 1: Unique ground state, gapped s 1: Two ground states, gapped Decay of correlations: exponential everywhere, polynomial at s c In between: the spectral gap closes: order - disorder QPT Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
5 Local vs topological order: physics Ordered phases non-unique ground state The usual picture: Local order parameter, e.g. ω(σ 3 0 ) Topological order : Local disorder, for A A X, X Λ, P Λ AP Λ C A 1 Cd(X, Λ) α, C A C, P Λ : The spectral projection associated to the ground state energy Cannot be smoothly deformed to a normal state Positive characterization of topological order (?) Topological degeneracy: if Λ g has genus g, then dimp Λg = f(g) Anyonic vacuum sectors Topological entanglement entropy Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
6 What is a (quantum) phase transition? A simple answer: A phase transition without temperature but under a continuous change of a parameter: Qualitative change in the set of ground states, parametrized by a coupling Localization-delocalization, parametrized by the disorder Percolation, parametrized by the probability Bifurcations in PDEs Higgs mechanism, parametrized by the coupling... Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
7 Ground state phases A slightly more precise answer: Consider: A smooth family of interactions Φ(s), s [0, 1] The associated Hamiltonians H Λ (s) = Φ(X, s) X Λ Spectral gap above the ground state energy γ Λ (s) such that { > 0 (s s c ) γ Λ (s) γ(s) C s s c µ (s s c ) QPT Associated singularity of the ground state projection P Λ (s) Basic question: What is a ground state phase? Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
8 Stability H Λ (s) = X Λ ( Φ(X) + sψ(x) ) If Ψ(X) is local, i.e. Ψ(X) = 0 whenever X Λ c 0, then usually Dynamics τ t Γ,s as a perturbation of τ t Γ,0 Continuity of the spectral gap at s = 0 Local perturbation of ground states Equilibrium states: ω β,s ω β,0 κs as s 0 Return to equilibrium: ω β,s τ t Γ,0 ω β,0 as t For translation invariant perturbations: No general stability results, but Perturbations of classical Hamiltonians Perturbations of frustration-free Hamiltonians Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
9 Automorphic equivalence Definition. Two gapped H, H are in the same phase if there is s Φ(s), C 0 and piecewise C 1, with Φ(0) = Φ, Φ(1) = Φ the Hamiltonians H(s) are uniformly gapped inf γ Λ(s) γ > 0 Λ Γ,s [0,1] The set of ground states on Γ: S Γ (s). Then there exists a continuous family of automorphism α s 1,s 2 Γ of A Γ α s 1,s 2 Γ S Γ (s 2 ) = S Γ (s 1 ) α s 1,s 2 Γ is local: satisfies a Lieb-Robinson bound Now: Invariants of the equivalence classes? Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
10 Frustration-free Hamiltonians in d = 1 Now Γ = Z, and H x H = C n Consider spaces {G N } N N such that G N H N and G N = N m x=0 for some m N; intersection property H x G m H (N m x) Natural positive translation invariant interaction: G m projection onto G m { τ x (1 G m ) X = [x, x + m 1] Φ(X) = 0 otherwise By the intersection property: KerH [1,N] = G N, parent Hamiltonian Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
11 Matrix product states Consider B = (B 1,..., B n ), B i M k and two projections p, q M k A CP map M k M k : B B n,k (p, q) if Ê B (a) = n B µ abµ 1. Spectral radius of ÊB is 1 and a non-degenerate eigenvalue 2. No peripheral spectrum: other eigenvalues have λ < 1 3. e B and ρ B : right and left eigenvectors of ÊB : pe B p and qρ B q invertible A map Γ k,b N,p,q : pm kq H N : Γ k,b N,p,q (a) = n µ 1,...µ N =1 µ=1 Tr(paqB µ N B µ 1 )ψ µ1 ψ µn Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
12 Gapped parent Hamiltonian Notation: and parent Hamiltonian H k,b N,p,q. ( ) G k,b N,p,q = Ran Γ k,b N,p,q H N Proposition. Assume that G k,b N,p,q satisfies the intersection property. Then i. H k,b N,p,q is gapped ii. S Z (H,p,q) k,b = { ω B } iii. Let d L = dim(p), d R = dim(q). There are affine bijections: E (M dl ) S (, 1] (H k,b,p,q), E (M dr ) S [0, ) (H k,b,p,q) i.e. Unique ground state on Z, edge states determined by p, q Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
13 Bulk state Given B, for A A {x}, n E B A(b) := ψ µ, Aψ ν B µ bbν µ,ν=1 Note: ÊB = E B 1 (b). ( ) ω (A B x A y ) = ρ B E B A x E B A y (e B ) ω B (Φ k,b m,p,q(x)) = 0: Ground state Exponential decay of correlations if σ(êb ) \ {1} {z C : z < 1} ( ) ω (A B x 1 y x 1 A y ) = ρ B E B A (ÊB ) y x 1 E B B(e) Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
14 Edge states Note: ω (A B x A y ) extends to Z: ( ) ( ) ρ B E B A x E B A y (e B ) = ρ B E B 1 E B A x E B A y (E B 1 (e B )) For the same E B, ω B ϕ(a 0 A x ) := ϕ ( ( ) (pe B p) 1/2 p E B A 0 E B A x (e B ) p(pe B p) 1/2) for any state ϕ on pm k p, and ω B ϕ(φ k,b m,p,q(x)) = 0 These extend to the right, but not to the left: S [0, ) (H k,b,p,q) E(M dl ) Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
15 A complete classification Theorem. Let H := H,p,q k,b and H := H k,b,p,q as in the proposition, with associated (d L, d R ), resp. (d L, d R ). Then, H H (d L, d R ) = (d L, d R) Remark: No symmetry requirement Proof by explicit construction of a gapped path of interactions Φ(s): on the fixed chain with A {x} = B(C n ) constant finite range translation invariant (no blocking) Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
16 Bulk product states Very simple representatives of each phase: Proposition. Let n 3, and (d L, d R ) N 2. Let k := d L d R. There exists B and projections p, q in Mat k (C) such that dimp = d L, dimq = d R B B n,k (p, q) G k,b m,p,q satisfy the intersection property the unique ground state ω B of the Hamiltonian H k,b,p,q on Z is the pure product state ω B (A x A y ) = y ψ 1, A i ψ 1 i=x Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
17 Example: the AKLT model SU(2)-invariant, antiferromagnetic spin-1 chain Nearest-neighbor interaction b 1 H[a,b] AKLT = x=a [ 1 2 (S x S x+1 ) (S x S x+1 ) ] b 1 = 3 x=a where P (2) x,x+1 is the projection on the spin-2 space of D 1 D 1 Uniform spectral gap γ of H [a,b], γ > H AKLT = H 2,B,1,1 with B B 3,2(1, 1) ( 1/3 0 B 1 = 0 1/3 ), B 2 = the AKLT model belongs to the phase (2, 2) P (2) x,x+1 ( ) ( ) 0 2/3, B 3 = /3 0 Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
18 About the proof B B n,k (p, q) {ÊB ω B Γ k,b N,p,q G k,b N,p,q H,p,q, k,b and by the proposition Gap(ÊB ) Gap(H k,b,p,q) Given B B n,k (p, q), B B n,k (p, q ), construct a path of gapped parent Hamiltonians H k,b(s),p(s),q(s) by embedding M k M k and interpolating interpolating p(s), q(s): dimensions interpolating B(s), keeping spectral properties of ÊB(s) Need pathwise connectedness of a certain subspace of (M k ) n Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
19 Primitive maps Ê B = i.e. {B µ } are the Kraus operators n B µ Bµ µ=1 The spectral gap condition: Perron-Frobenius Irreducible positive map = 1. Spectral radius r is a non-degenerate eigenvalue 2. Corresponding eigenvector e > 0 3. Eigenvalues λ with λ = r are re 2πiα/β, α Z/βZ A primitive map is an irreducible CP map with β = 1 Lemma. ÊB is primitive iff there exists m N such that span {B µ1 B µm : µ i {1,..., n}} = M k Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
20 Primitive maps How to construct paths of primitive maps? Consider Y n,k := with the choice { B : B 1 = k α=1 } λ α e α e α, and B 2 e α, e β 0 (λ 1,..., λ k ) Ω := {λ i 0, λ i λ j, λ i /λ j λ k /λ l } Then, e α e β span {B µ1 B µm : µ i {1, 2}} for m 2k(k 1) + 3. Problem reduced to the pathwise connectedness of Ω C k Use transversality theorem Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
21 Consequences What we obtain: B(s) B n,k (1, 1) B n,k (p(s), q(s)) i.e. a good ÊB(s) For those: Γ k,b m,p(s),q(s) is injective dimgk,b m,p(s),q(s) = d Rd L G k,b m,p(s),q(s) satisfy the intersection property i.e. a good path H k,b,p(s),q(s) Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
22 Remarks More work at s = 0, 1, where the given B, B / Y n,k : Why is it hard? Because dimh = n is fixed Simpler problem for n k 2, i.e. by allowing periodic interactions Interaction range: m min = max{m, m, k 2 + 1, (k ) 2 + 1} all in all: (d L, d R ) = (d L, d R ) is sufficient (d L, d R ) = (d L, d R ) necessary: H H implies S [0, ) = S [0, ) α [0, ), S (, 1] = S (, 1] α (, 1] and α is bijective Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
23 Concrete representatives 1 S 0 (λ, d) = λ , S 0 1 +(d) =... 1, λ d 1 0 Let B 1 = S 0 (λ R, d R ) S 0 (λ L, d L ) B 2 = S + (d R ) S 0 (λ L, d L ) B 3 = S 0 (λ R, d R ) S + (d L ) B i = 0 if i 3. Properties: B d R 2 = 0, B d L 3 = 0, and B 1B 2 = λ R B 2B 1, B 1B 3 = λ L B 3B 1, B 2B 3 = ( λl λ R ) B 3B 2. Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
24 Concrete spectrum Simple consequence: Ê B = D + N R + N L with D = B 1 B1 diagonal, N R = B 2 B2, N L = B 3 B3, nilpotent, and DN R = λ 2 R N RD, DN L = λ 2 L N LD, N R N L = (λ R /λ L ) 2 N L N R. Then, Spectral gap if λ L, λ R 1 σ(êb ) = σ(d) Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
25 Product vacuum in the bulk Vectors? Recall Γ k,b N,p,q (a) = n µ 1,...µ N =1 Tr(paqB µ N B µ 1 )ψ µ1 ψ µn The product B µ1 B µn can have at most d R 1 B 2 s, and d L 1 B 1 s, so { G k,b N,p,q = span Γ k,b N,p,q (pbα 2 B β 3 }α=0,...,d q) R 1,β=0,...,d L 1 for α = β = 0, product vacuum: Γ k,b N,p,q (1) = Tr(p1q(B 1) m )ψ 1 ψ 1 Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
26 Conclusions Construction of gapped Hamiltonians from frustration-free states Unique ground state on Z Tunable number of edge states Edge index: Complete classification by the number of edge states (no symmetry) No bulk index: each phase has a representative with a pure product state on Z Many-body localization? Sven Bachmann (LMU) Invariants of ground state phases UAB / 26
Automorphic Equivalence Within Gapped Phases
1 Harvard University May 18, 2011 Automorphic Equivalence Within Gapped Phases Robert Sims University of Arizona based on joint work with Sven Bachmann, Spyridon Michalakis, and Bruno Nachtergaele 2 Outline:
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 informationFrustration-free Ground States of Quantum Spin Systems 1
1 FRG2011, Harvard, May 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
More informationIntroduction to Quantum Spin Systems
1 Introduction to Quantum Spin Systems Lecture 2 Sven Bachmann (standing in for Bruno Nachtergaele) Mathematics, UC Davis MAT290-25, CRN 30216, Winter 2011, 01/10/11 2 Basic Setup For concreteness, consider
More informationQuantum Phase Transitions
1 Davis, September 19, 2011 Quantum Phase Transitions A VIGRE 1 Research Focus Group, Fall 2011 Spring 2012 Bruno Nachtergaele See the RFG webpage for more information: http://wwwmathucdavisedu/~bxn/rfg_quantum_
More informationFRG Workshop in Cambridge MA, May
FRG Workshop in Cambridge MA, May 18-19 2011 Programme Wednesday May 18 09:00 09:10 (welcoming) 09:10 09:50 Bachmann 09:55 10:35 Sims 10:55 11:35 Borovyk 11:40 12:20 Bravyi 14:10 14:50 Datta 14:55 15:35
More informationThe AKLT Model. Lecture 5. Amanda Young. Mathematics, UC Davis. MAT290-25, CRN 30216, Winter 2011, 01/31/11
1 The AKLT Model Lecture 5 Amanda Young Mathematics, UC Davis MAT290-25, CRN 30216, Winter 2011, 01/31/11 This talk will follow pg. 26-29 of Lieb-Robinson Bounds in Quantum Many-Body Physics by B. Nachtergaele
More informationIntroduction to the Mathematics of the XY -Spin Chain
Introduction to the Mathematics of the XY -Spin Chain Günter Stolz June 9, 2014 Abstract In the following we present an introduction to the mathematical theory of the XY spin chain. The importance of this
More informationEntanglement spectrum and Matrix Product States
Entanglement spectrum and Matrix Product States Frank Verstraete J. Haegeman, D. Draxler, B. Pirvu, V. Stojevic, V. Zauner, I. Pizorn I. Cirac (MPQ), T. Osborne (Hannover), N. Schuch (Aachen) Outline Valence
More informationFerromagnetic Ordering of Energy Levels for XXZ Spin Chains
Ferromagnetic Ordering of Energy Levels for XXZ Spin Chains Bruno Nachtergaele and Wolfgang L. Spitzer Department of Mathematics University of California, Davis Davis, CA 95616-8633, USA bxn@math.ucdavis.edu
More informationStability of local observables
Stability of local observables in closed and open quantum systems Angelo Lucia anlucia@ucm.es Universidad Complutense de Madrid NSF/CBMS Conference Quantum Spin Systems UAB, June 20, 2014 A. Lucia (UCM)
More informationPropagation bounds for quantum dynamics and applications 1
1 Quantum Systems, Chennai, August 14-18, 2010 Propagation bounds for quantum dynamics and applications 1 Bruno Nachtergaele (UC Davis) based on joint work with Eman Hamza, Spyridon Michalakis, Yoshiko
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 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 informationQuantum Information and Quantum Many-body Systems
Quantum Information and Quantum Many-body Systems Lecture 1 Norbert Schuch California Institute of Technology Institute for Quantum Information Quantum Information and Quantum Many-Body Systems Aim: Understand
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 informationPhysics 239/139 Spring 2018 Assignment 2 Solutions
University of California at San Diego Department of Physics Prof. John McGreevy Physics 39/139 Spring 018 Assignment Solutions Due 1:30pm Monday, April 16, 018 1. Classical circuits brain-warmer. (a) Show
More informationSpin Superfluidity and Graphene in a Strong Magnetic Field
Spin Superfluidity and Graphene in a Strong Magnetic Field by B. I. Halperin Nano-QT 2016 Kyiv October 11, 2016 Based on work with So Takei (CUNY), Yaroslav Tserkovnyak (UCLA), and Amir Yacoby (Harvard)
More informationFeshbach-Schur RG for the Anderson Model
Feshbach-Schur RG for the Anderson Model John Z. Imbrie University of Virginia Isaac Newton Institute October 26, 2018 Overview Consider the localization problem for the Anderson model of a quantum particle
More informationMATRIX LIE GROUPS AND LIE GROUPS
MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either
More informationQuantum Chaos and Nonunitary Dynamics
Quantum Chaos and Nonunitary Dynamics Karol Życzkowski in collaboration with W. Bruzda, V. Cappellini, H.-J. Sommers, M. Smaczyński Phys. Lett. A 373, 320 (2009) Institute of Physics, Jagiellonian University,
More informationComplex Algebraic Geometry: Smooth Curves Aaron Bertram, First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool
Complex Algebraic Geometry: Smooth Curves Aaron Bertram, 2010 12. First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool for classifying smooth projective curves, i.e. giving
More informationMET Workshop: Exercises
MET Workshop: Exercises Alex Blumenthal and Anthony Quas May 7, 206 Notation. R d is endowed with the standard inner product (, ) and Euclidean norm. M d d (R) denotes the space of n n real matrices. When
More informationQuantum Many Body Systems and Tensor Networks
Quantum Many Body Systems and Tensor Networks Aditya Jain International Institute of Information Technology, Hyderabad aditya.jain@research.iiit.ac.in July 30, 2015 Aditya Jain (IIIT-H) Quantum Hamiltonian
More informationALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes!
ALGEBRAIC GROUPS Disclaimer: There are millions of errors in these notes! 1. Some algebraic geometry The subject of algebraic groups depends on the interaction between algebraic geometry and group theory.
More informationvon Neumann algebras, II 1 factors, and their subfactors V.S. Sunder (IMSc, Chennai)
von Neumann algebras, II 1 factors, and their subfactors V.S. Sunder (IMSc, Chennai) Lecture 3 at IIT Mumbai, April 24th, 2007 Finite-dimensional C -algebras: Recall: Definition: A linear functional tr
More informationNon-magnetic states. The Néel states are product states; φ N a. , E ij = 3J ij /4 2 The Néel states have higher energy (expectations; not eigenstates)
Non-magnetic states Two spins, i and j, in isolation, H ij = J ijsi S j = J ij [Si z Sj z + 1 2 (S+ i S j + S i S+ j )] For Jij>0 the ground state is the singlet; φ s ij = i j i j, E ij = 3J ij /4 2 The
More informationClusters and Percolation
Chapter 6 Clusters and Percolation c 2012 by W. Klein, Harvey Gould, and Jan Tobochnik 5 November 2012 6.1 Introduction In this chapter we continue our investigation of nucleation near the spinodal. We
More informationAditi Mitra New York University
Entanglement dynamics following quantum quenches: pplications to d Floquet chern Insulator and 3d critical system diti Mitra New York University Supported by DOE-BES and NSF- DMR Daniel Yates, PhD student
More informationSpring 2018 CIS 610. Advanced Geometric Methods in Computer Science Jean Gallier Homework 3
Spring 2018 CIS 610 Advanced Geometric Methods in Computer Science Jean Gallier Homework 3 March 20; Due April 5, 2018 Problem B1 (80). This problem is from Knapp, Lie Groups Beyond an Introduction, Introduction,
More informationHeat full statistics: heavy tails and fluctuations control
Heat full statistics: heavy tails and fluctuations control Annalisa Panati, CPT, Université de Toulon joint work with T.Benoist, R. Raquépas 1 2 3 Full statistics -confined systems Full (counting) Statistics
More informationCOMPLEX MULTIPLICATION: LECTURE 13
COMPLEX MULTIPLICATION: LECTURE 13 Example 0.1. If we let C = P 1, then k(c) = k(t) = k(c (q) ) and the φ (t) = t q, thus the extension k(c)/φ (k(c)) is of the form k(t 1/q )/k(t) which as you may recall
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 informationDetermining a local Hamiltonian from a single eigenstate
Determining a local Hamiltonian from a single eigenstate Xiao-Liang Qi 1,2 and Daniel Ranard 1 arxiv:1712.01850v1 [quant-ph] 5 Dec 2017 1 Stanford Institute for Theoretical Physics, Stanford University,
More informationarxiv:quant-ph/ v2 24 Dec 2003
Quantum Entanglement in Heisenberg Antiferromagnets V. Subrahmanyam Department of Physics, Indian Institute of Technology, Kanpur, India. arxiv:quant-ph/0309004 v2 24 Dec 2003 Entanglement sharing among
More informationquasi-particle pictures from continuous unitary transformations
quasi-particle pictures from continuous unitary transformations Kai Phillip Schmidt 24.02.2016 quasi-particle pictures from continuous unitary transformations overview Entanglement in Strongly Correlated
More informationEntanglement Dynamics for the Quantum Disordered XY Chain
Entanglement Dynamics for the Quantum Disordered XY Chain Houssam Abdul-Rahman Joint with: B. Nachtergaele, R. Sims, G. Stolz AMS Southeastern Sectional Meeting University of Georgia March 6, 2016 Houssam
More informationLecture 12: Detailed balance and Eigenfunction methods
Miranda Holmes-Cerfon Applied Stochastic Analysis, Spring 2015 Lecture 12: Detailed balance and Eigenfunction methods Readings Recommended: Pavliotis [2014] 4.5-4.7 (eigenfunction methods and reversibility),
More informationClassical and quantum simulation of dissipative quantum many-body systems
0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 Classical and quantum simulation of dissipative quantum many-body systems
More informationA Brief Introduction to Functional Analysis
A Brief Introduction to Functional Analysis Sungwook Lee Department of Mathematics University of Southern Mississippi sunglee@usm.edu July 5, 2007 Definition 1. An algebra A is a vector space over C with
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 informationCONSTRAINED PERCOLATION ON Z 2
CONSTRAINED PERCOLATION ON Z 2 ZHONGYANG LI Abstract. We study a constrained percolation process on Z 2, and prove the almost sure nonexistence of infinite clusters and contours for a large class of probability
More informationQuantum numbers for relative ground states of antiferromagnetic Heisenberg spin rings
Quantum numbers for relative ground states of antiferromagnetic Heisenberg spin rings Klaus Bärwinkel, Peter Hage, Heinz-Jürgen Schmidt, and Jürgen Schnack Universität Osnabrück, Fachbereich Physik, D-49069
More informationScaling analysis of snapshot spectra in the world-line quantum Monte Carlo for the transverse-field Ising chain
TNSAA 2018-2019 Dec. 3-6, 2018, Kobe, Japan Scaling analysis of snapshot spectra in the world-line quantum Monte Carlo for the transverse-field Ising chain Kouichi Seki, Kouichi Okunishi Niigata University,
More informationDecoherence and Thermalization of Quantum Spin Systems
Copyright 2011 American Scientific Publishers All rights reserved Printed in the United States of America Journal of Computational and Theoretical Nanoscience Vol. 8, 1 23, 2011 Decoherence and Thermalization
More informationLinear Algebra M1 - FIB. Contents: 5. Matrices, systems of linear equations and determinants 6. Vector space 7. Linear maps 8.
Linear Algebra M1 - FIB Contents: 5 Matrices, systems of linear equations and determinants 6 Vector space 7 Linear maps 8 Diagonalization Anna de Mier Montserrat Maureso Dept Matemàtica Aplicada II Translation:
More information12. Hilbert Polynomials and Bézout s Theorem
12. Hilbert Polynomials and Bézout s Theorem 95 12. Hilbert Polynomials and Bézout s Theorem After our study of smooth cubic surfaces in the last chapter, let us now come back to the general theory of
More informationLECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C)
LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C) IVAN LOSEV Introduction We proceed to studying the representation theory of algebraic groups and Lie algebras. Algebraic groups are the groups
More informationRings and groups. Ya. Sysak
Rings and groups. Ya. Sysak 1 Noetherian rings Let R be a ring. A (right) R -module M is called noetherian if it satisfies the maximum condition for its submodules. In other words, if M 1... M i M i+1...
More informationQuantum Statistical Mechanical Systems
Sfcs Quantum Statistical Mechanical Systems Mentor: Prof. Matilde Marcolli October 20, 2012 Sfcs 1 2 Quantum Statistical Mechanical Systems 3 4 and Uniformization 5 6 of the System 7 of 8 and Further Study
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 informationCommutative Banach algebras 79
8. Commutative Banach algebras In this chapter, we analyze commutative Banach algebras in greater detail. So we always assume that xy = yx for all x, y A here. Definition 8.1. Let A be a (commutative)
More informationLarge automorphism groups of 16-dimensional planes are Lie groups
Journal of Lie Theory Volume 8 (1998) 83 93 C 1998 Heldermann Verlag Large automorphism groups of 16-dimensional planes are Lie groups Barbara Priwitzer, Helmut Salzmann Communicated by Karl H. Hofmann
More informationSSH Model. Alessandro David. November 3, 2016
SSH Model Alessandro David November 3, 2016 Adapted from Lecture Notes at: https://arxiv.org/abs/1509.02295 and from article: Nature Physics 9, 795 (2013) Motivations SSH = Su-Schrieffer-Heeger Polyacetylene
More informationarxiv: v2 [math-ph] 26 Feb 2017
February 28, 207 Localization Properties of the Disordered XY Spin Chain A review of mathematical results with an eye toward Many-Body Localization Houssam Abdul-Rahman, Bruno Nachtergaele 2, Robert Sims,
More informationOn the Random XY Spin Chain
1 CBMS: B ham, AL June 17, 2014 On the Random XY Spin Chain Robert Sims University of Arizona 2 The Isotropic XY-Spin Chain Fix a real-valued sequence {ν j } j 1 and for each integer n 1, set acting on
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 informationTime-dependent DMRG:
The time-dependent DMRG and its applications Adrian Feiguin Time-dependent DMRG: ^ ^ ih Ψ( t) = 0 t t [ H ( t) E ] Ψ( )... In a truncated basis: t=3 τ t=4 τ t=5τ t=2 τ t= τ t=0 Hilbert space S.R.White
More informationH ψ = E ψ. Introduction to Exact Diagonalization. Andreas Läuchli, New states of quantum matter MPI für Physik komplexer Systeme - Dresden
H ψ = E ψ Introduction to Exact Diagonalization Andreas Läuchli, New states of quantum matter MPI für Physik komplexer Systeme - Dresden http://www.pks.mpg.de/~aml laeuchli@comp-phys.org Simulations 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 informationMAT265 Mathematical Quantum Mechanics Brief Review of the Representations of SU(2)
MAT65 Mathematical Quantum Mechanics Brief Review of the Representations of SU() (Notes for MAT80 taken by Shannon Starr, October 000) There are many references for representation theory in general, and
More informationNotes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers
Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop Eric Sommers 17 July 2009 2 Contents 1 Background 5 1.1 Linear algebra......................................... 5 1.1.1
More informationarxiv:physics/ v1 [math-ph] 17 May 1997
arxiv:physics/975v1 [math-ph] 17 May 1997 Quasi-Exactly Solvable Time-Dependent Potentials Federico Finkel ) Departamento de Física Teórica II Universidad Complutense Madrid 84 SPAIN Abstract Niky Kamran
More informationRegularization and Inverse Problems
Regularization and Inverse Problems Caroline Sieger Host Institution: Universität Bremen Home Institution: Clemson University August 5, 2009 Caroline Sieger (Bremen and Clemson) Regularization and Inverse
More informationRepresentation theory & the Hubbard model
Representation theory & the Hubbard model Simon Mayer March 17, 2015 Outline 1. The Hubbard model 2. Representation theory of the symmetric group S n 3. Representation theory of the special unitary group
More informationElectrons in a periodic potential
Chapter 3 Electrons in a periodic potential 3.1 Bloch s theorem. We consider in this chapter electrons under the influence of a static, periodic potential V (x), i.e. such that it fulfills V (x) = V (x
More informationDecay of correlations in 2d quantum systems
Decay of correlations in 2d quantum systems Costanza Benassi University of Warwick Quantissima in the Serenissima II, 25th August 2017 Costanza Benassi (University of Warwick) Decay of correlations in
More information(x k ) sequence in F, lim x k = x x F. If F : R n R is a function, level sets and sublevel sets of F are any sets of the form (respectively);
STABILITY OF EQUILIBRIA AND LIAPUNOV FUNCTIONS. By topological properties in general we mean qualitative geometric properties (of subsets of R n or of functions in R n ), that is, those that don t depend
More informationNOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
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 informationThe Unitary Group In Its Strong Topology
The Unitary Group In Its Strong Topology Martin Schottenloher Mathematisches Institut LMU München Theresienstr. 39, 80333 München schotten@math.lmu.de, +49 89 21804435 Abstract. The unitary group U(H)
More informationLECTURE NOTE #11 PROF. ALAN YUILLE
LECTURE NOTE #11 PROF. ALAN YUILLE 1. NonLinear Dimension Reduction Spectral Methods. The basic idea is to assume that the data lies on a manifold/surface in D-dimensional space, see figure (1) Perform
More informationAnderson átmenet a kvark-gluon plazmában
Anderson átmenet a kvark-gluon plazmában Kovács Tamás György Atomki, Debrecen 25. április 22. Kovács Tamás György Anderson átmenet a kvark-gluon plazmában /9 Együttműködők: Falk Bruckmann (Regensburg U.)
More informationTopological Defects inside a Topological Band Insulator
Topological Defects inside a Topological Band Insulator Ashvin Vishwanath UC Berkeley Refs: Ran, Zhang A.V., Nature Physics 5, 289 (2009). Hosur, Ryu, AV arxiv: 0908.2691 Part 1: Outline A toy model of
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 informationPart III Symmetries, Fields and Particles
Part III Symmetries, Fields and Particles Theorems Based on lectures by N. Dorey Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often
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 informationChapter 4. Inverse Function Theorem. 4.1 The Inverse Function Theorem
Chapter 4 Inverse Function Theorem d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d dd d d d d This chapter
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 informationHIGHER SPIN CORRECTIONS TO ENTANGLEMENT ENTROPY
HIGHER SPIN CORRECTIONS TO ENTANGLEMENT ENTROPY JHEP 1406 (2014) 096, Phys.Rev. D90 (2014) 4, 041903 with Shouvik Datta ( IISc), Michael Ferlaino, S. Prem Kumar (Swansea U. ) JHEP 1504 (2015) 041 with
More informationAntiferromagnetic Potts models and random colorings
Antiferromagnetic Potts models and random colorings of planar graphs. joint with A.D. Sokal (New York) and R. Kotecký (Prague) October 9, 0 Gibbs measures Let G = (V, E) be a finite graph and let S be
More informationResolvent Algebras. An alternative approach to canonical quantum systems. Detlev Buchholz
Resolvent Algebras An alternative approach to canonical quantum systems Detlev Buchholz Analytical Aspects of Mathematical Physics ETH Zürich May 29, 2013 1 / 19 2 / 19 Motivation Kinematics of quantum
More informationThe Quantum Heisenberg Ferromagnet
The Quantum Heisenberg Ferromagnet Soon after Schrödinger discovered the wave equation of quantum mechanics, Heisenberg and Dirac developed the first successful quantum theory of ferromagnetism W. Heisenberg,
More informationNecessary and sufficient conditions for strong R-positivity
Necessary and sufficient conditions for strong R-positivity Wednesday, November 29th, 2017 The Perron-Frobenius theorem Let A = (A(x, y)) x,y S be a nonnegative matrix indexed by a countable set S. We
More informationThe Quantum Hall Conductance: A rigorous proof of quantization
Motivation The Quantum Hall Conductance: A rigorous proof of quantization Spyridon Michalakis Joint work with M. Hastings - Microsoft Research Station Q August 17th, 2010 Spyridon Michalakis (T-4/CNLS
More informationThe 1+1-dimensional Ising model
Chapter 4 The 1+1-dimensional Ising model The 1+1-dimensional Ising model is one of the most important models in statistical mechanics. It is an interacting system, and behaves accordingly. Yet for a variety
More informationRandom Lifts of Graphs
27th Brazilian Math Colloquium, July 09 Plan of this talk A brief introduction to the probabilistic method. A quick review of expander graphs and their spectrum. Lifts, random lifts and their properties.
More informationOverview of Markovian maps
Overview of Markovian maps Mike Boyle University of Maryland BIRS October 2007 *these slides are online, via Google: Mike Boyle Outline of the talk I. Subshift background 1. Subshifts 2. (Sliding) block
More informationQuantum spin systems - models and computational methods
Summer School on Computational Statistical Physics August 4-11, 2010, NCCU, Taipei, Taiwan Quantum spin systems - models and computational methods Anders W. Sandvik, Boston University Lecture outline Introduction
More informationSpectral theory for compact operators on Banach spaces
68 Chapter 9 Spectral theory for compact operators on Banach spaces Recall that a subset S of a metric space X is precompact if its closure is compact, or equivalently every sequence contains a Cauchy
More information5 Topological defects and textures in ordered media
5 Topological defects and textures in ordered media In this chapter we consider how to classify topological defects and textures in ordered media. We give here only a very short account of the method following
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 informationGLOBAL TOPOLOGICAL CONFIGURATIONS OF SINGULARITIES FOR THE WHOLE FAMILY OF QUADRATIC DIFFERENTIAL SYSTEMS
GLOBAL TOPOLOGICAL CONFIGURATIONS OF SINGULARITIES FOR THE WHOLE FAMILY OF QUADRATIC DIFFERENTIAL SYSTEMS JOAN C. ARTÉS1, JAUME LLIBRE 1, DANA SCHLOMIUK 2 AND NICOLAE VULPE 3 Abstract. In [1] the authors
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 informationLet us recall in a nutshell the definition of some important algebraic structure, increasingly more refined than that of group.
Chapter 1 SOME MATHEMATICAL TOOLS 1.1 Some definitions in algebra Let us recall in a nutshell the definition of some important algebraic structure, increasingly more refined than that of group. Ring A
More informationTopological phases of SU(N) spin chains and their realization in ultra-cold atom gases
Topological phases of SU(N) spin chains and their realization in ultra-cold atom gases Thomas Quella University of Cologne Workshop on Low-D Quantum Condensed Matter University of Amsterdam, 8.7.2013 Based
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 informationQuantum Mechanics Solutions
Quantum Mechanics Solutions (a (i f A and B are Hermitian, since (AB = B A = BA, operator AB is Hermitian if and only if A and B commute So, we know that [A,B] = 0, which means that the Hilbert space H
More informationNematicity and quantum paramagnetism in FeSe
Nematicity and quantum paramagnetism in FeSe Fa Wang 1,, Steven A. Kivelson 3 & Dung-Hai Lee 4,5, 1 International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China.
More informationP3317 HW from Lecture and Recitation 10
P3317 HW from Lecture 18+19 and Recitation 10 Due Nov 6, 2018 Problem 1. Equipartition Note: This is a problem from classical statistical mechanics. We will need the answer for the next few problems, and
More information