Entanglement Chern numbers for random systems

POSTECH, Korea, July 31 (2015) Ψ = 1 D D Entanglement Chern numbers for random systems j Ψ j Ψj Yasuhiro Hatsugai Institute of Physics, Univ. of Tsukuba Ref: T. Fukui & Y. Hatsugai, J. Phys. Soc. Jpn. 84, 043703 (2015), ibid. 83, 113705 (2014) open select

Plan Quantum Hall plateau transition & topology Finite Chern number by the extended states Sum rule of the Chern number Technical developments in the decade (2005-2015) Topological phases: symmetry & quantization with disorder Von Neumann-Wigner theorem diabatic principle with symmetry Entanglement Chern numbers for random systems Entanglement entropy & E. hamiltonian Extensive partition with symmetry breaking Entanglement Chern numbers

Plan Quantum Hall plateau transition Finite Chern number by the extended states Sum rule of the Chern number Numerical developments in the decade (2005-2015) Topological phases: symmetry & quantization Von Neumann-Wigner theorem diabatic principle with symmetry Entanglement Chern numbers for random systems Entanglement entropy & E. hamiltonian Extensive partition with symmetry breaking Entanglement Chern numbers

Quantum Hall plateau transition Long history Quantum phase transition driven by disorder among states with different topological numbers Focus: Delocalized states within Landau Levels Khmelnitskii 84 Laughlin 84 Kivelson-Lee-Zhang 92 Ludwig-Fisher-Shankar-Grinstein 94 Yang-hatt 96 Sheng-Wen 97 Hatsugai-Ishibashi-Morita 99 Hall plateau transition of graphene Many... & effects of randomness in graphene Suzuura-ndo 02 McCann et al. 06 leiner-efetov 06 ltland 06 Sheng-Sheng-Wen 06 Schweitzer-Markos 08 Nomura-Ryu-Koshino-Mudry-Furusaki 08

R. Willett et al. 80, K.v.Klitzing et al. V topologically cylinder

Quantum Hall plateau transition potential shape (periodic in x and y) localised extended Most of the states are localised except band enters x =(e i x,e i y ) wave functions ex. Song-Maruyama-YH PR 07

Energy spectrum of loch Electrons in a magnetic field Hofstadter 76 H = ij c i ei ij c j + h.c. c i : electron annihilation operator Electrons in a 2D periodic potential + magnetic field It might be realized by cold atoms? Landau levels The spectrum is fractal as a function of magnetic flux per plaquette

Hofstadter s utterfly Self-similar Fractal in condensed matter

Graphene in a magnetic field Tight-binding model on a honeycomb lattice H = t ij e i ij ij 2 P = S t t ij ij P P = S S t = S 0 = p q (p, q) = 1 Rammal 1985 φ E=0 Landau level : outside Onsager s semiclassical quantisation scheme 0 : flux quantum

Observation of nomalous QHE in Graphene nomalous QHE of gapless Dirac Fermions xy = e2 h (2n + 1), n = 0, ±1, ±2, = 2 e2 h (n + 1 2 ) Zhang et al. Nature 2005 Novoselov et al. Nature 2005

Conventional QHE Landau Level and Integer QHE k E E( = 0) = µ F 2 2m k2 D(E) D(E) = n (E n ) n = (n + 1 2 ) xy [e 2 /h] 1 2 3 1/2 3/2 5/2 µ F E [ ] µ F E [ ] xy (µ F ) = e2 n, n = 1, 2, 3, n 1 < µ F < n 1/2 3/2 5/2

QHE of Graphene (Gapless Semiconductor) Landau Level of Doubled Dirac Fermions k E( = 0) = ± c k µ F E (E n) n : No zero point energy shift n = ±C n D(E) = n 1 2 3 2 5 C = c (2)e McClure, 1956 xy (µ F ) = e2 (2n + 1) xy [e 2 /h] 1 3 5 7 9 xy : e2 h odd integer n = 0, 1, 2, 3, n 1 < µ F < n Zheng-ndo 2002 Gusynin-Sharapov, 2005 Peres-Guinea-Neto 2006...

Topological meaning of the Hall Conductance TKNN formula: Kubo formula xy = e2 C = 1 h : (k)<e F C 2 i T 2 :Z as a topological invariant F F = d = d d = d xy H(k) (k) = (k) (k) Thouless-Kohmoto-Nightingale-den Nijs 82 Sum over the bands below E F :First Chern number of the -th and intrinsically integer unless the energy gap collapses k, (k) = ±1 (k) regularity of the erry connection k T 2 = {k =(k x,k y ) 0 k x,k y 2 } Z d = dk µ k µ

erry Connection parameter space Eigenvectors ( space ) with Parameters x y H(x) (x) = (x) (x) Information between nearby states erry connection : = d = d dx Gauge Transformation i C ( ) = H(x) and H(y) are independent Fiber undle gauge potential Geometrical quantities C : erry phase (x) = (x) e i (x) dx. = + id = + i d dx dx C( )= 1 Parameters: twisted boundary conditions: 2 i x =(e i x,e i y ) Z S (belian) d : Chern # S

One-body to Many-body (Sum rule) erry phases,... one particle problem H( ) j ( )i = j ( ) j ( )i xy = e2 h 2 i Niu-Thouless-Wu formula ZT 1 d = e2 1 2 Gi = h 2 i MY (c `) 0i `=1 Twisted boundary conditions =( x, y) Z Tr M d FS T 2 TKNN formula M particle state: filled Dirac sea = hg dgi =Tr M FS Many-body from one-body =( 1,, M ) Collect M states below the Fermi level FS d = 1 d 1 1 d M......... M d 1 M d M Matrix vector potential of the Fermi ( Dirac ) Sea Non belian extension for the Chern numbers YH 2004 Wilczek-Zee 84

Hall conductance as a topological invariant y Chern numbers of loch electrons j xy = e2 h j =1 C, C = 1 (k) < µ F, = 1,, j 2 i d, Counting vortices in the band = d Z Thouless-Kohmoto-Nightingale-den Nijs 1982 Sum over the filled bands Need to sum many bands until E=0 E=0 Numerical difficulty for the weak field (experimental situation) { graphene with randomness oki-ndo 1986 Need to fill negative energy Dirac sea Rammal 1985 Need to sum over them van Hove singularity E=0 {

ulk xy of the Filled Fermi sea & Dirac Sea Integration of the Nonbelian erry Connection of the Fermi Sea & Dirac Sea H j (k) j (k) = j (k) j (k) Technology 1 Physical gap (many body) Numerical advantage for graphene =( 1,, M ) Collect M states below the Fermi level Non belian extension (Manybody) FS d = Technology 1 1 d 1 1 d M......... M d 1 M d M Very small gaps due to Matrix disorder vector potential do not modify of the Fermi ( Dirac ) Sea Non belian extension for the Chern numbers xy = e2 h 1 2 i T 2 Tr M d FS Hatsugai 2004

Fukui-Hatsugai-Suzuki, JPSJ, 74, 1674 (2005) Chern Numbers on a Discrete rillouin Zone U µ (k l ) : Link variable & Field Strength on Plaquette U µ (k l ) n(k l ) n(k l +ˆµ) /N µ (k l ) F 12 (k l ) N µ (k l )= n(k l ) n(k l +ˆµ) : F 12 (k l ) ln U 1 (k l )U 2 (k l + ˆ1)U 1 (k l + ˆ2) 1 U 2 (k l ) 1 k l 2 1 N π < F 12 (k l )/i π (principal value) Chern Number on Lattice c 1 2πi l F 12 (k l ) lso Topological: Intrinsically Integer Do not Use ny Gauge Dependent Quantities, Gauge Invariant, No Use of erry s Connection

Fukui-Hatsugai-Suzuki, JPSJ, 74, 1674 (2005) Chern Numbers on a Discrete rillouin Zone U µ (k l ) : F 12 (k l ) Link variable & Field Strength on Plaquette : U µ (k l ) n(k l ) n(k l +ˆµ) /N µ (k l ) Chern Number on Lattice N µ (k l )= n(k l ) n(k l +ˆµ) Convergence of integers 1,2,1,-1,1,2,1,1,1,1,1... F 12 (k l ) ln U 1 (k l )U 2 (k l + ˆ1)U 1 (k l + ˆ2) 1 by the mesh sizes U 2 (k 1 l ) 1 IT SHOULD π < F E 12 (k 1 l )/i! π k l 2 (principal value) N c 1 2πi l F 12 (k l ) lso Topological: Intrinsically Integer Do not Use ny Gauge Dependent Quantities, Gauge Invariant, No Use of erry s Connection

Numerical Technique from the Lattice gauge theory Topological Invariant on Discretized Lattice Lattice in k space ( discretization for the integral ) xy = e2 h 1 2 i F 1234 F 1234 = Im log U 12 U 23 U 34 U 41 U mn = det j Technical dvantage for large Chern Numbers k l gauge invariant Technology 2 Chern number extension of the KSV formula for polarization m n, n = ( 1 (k n ),, j (k n )) 2 1 N rillouin Zone Fermi Sea of j filled bands U µ (k l ) F 12 (k l ) U µ (k l ) n(k l ) n(k l +ˆµ) /N µ (k l ) N µ (k l )= n(k l ) n(k l +ˆµ) F 12 (k l ) ln U 1 (k l )U 2 (k l + ˆ1)U 1 (k l + ˆ2) 1 U 2 (k l ) 1 π < F 12 (k l )/i π Fukui-Hatsugai-Suzuki 2005 (principal value)

Hall Conductace vs chemical potential ccurate Hall conductance over whole spectrum(graphene) D(E) single band model Electron Like in this region -2 30 2 20 10-3 -2-1 1 2 3-10 xy [e 2 /h] Hole Like in this region µ/t, t 1[eV] for graphene = 1/31 Dirac Like -20 in this region Hatsugai-Fukui-oki 06

Chern numbers ( xy) based on Realistic and Calc. Fermi surface Fermi surface xy xy xy E F E F quantized everywhere M.rai and Y.Hatsugai, Phys.Rev. 79, 075429 (200

Correlated Random Hopping (distribution of gauge field ) Landscape of hopping amplitude T. Kawarabayashi,Y.H.,H. oki, Phys.Rev. Lett. 103, 156804,(2009) W δt /t = 2.0 for calculation of density of states

W δt /t = 0.4 φ uni /φ 0 =1/50 System size 20 20 c.f. TrMdFS Chern Number with spatial correlation without spatial correlation n=0 n=1 n=2 without chiral symmetry Nomura-Ryu-Koshino-Mudry-Furusaki, Phys. Rev. Lett. 100 246806 (2008) xy = e2 h 1 2 i T 2 n= ---1 Spatial correlation makes n=0 LL special n= ---2 No critical region only for n=0 100 samples E F /t T. Kawarabayashi,Y.H.,H. oki, Phys.Rev. Lett. 103, 156804,(2009)

Plan Quantum Hall plateau transition Finite Chern number by the extended states Sum rule of the Chern number Numerical developments in the decade (2005-2015) Topological phases: symmetry & quantization with disorder Von Neumann-Wigner theorem diabatic principle with symmetry Entanglement Chern numbers for random systems Entanglement entropy & E. hamiltonian Extensive partition with symmetry breaking Entanglement Chern numbers

Quantum Liquids without Symmetry reaking Quantum Liquids in Low Dimensional Quantum Systems Low Dimensionality, Quantum Fluctuations No Symmetry reaking No Local Order Parameter Various Phases & Quantum Phase Transitions Gapped Quantum Liquids in Condensed Matter Integer & Fractional Quantum Hall States RV & Dimer Models of Fermions and Spins Lots of topological phases Integer spin chains Correlated Electrons & Spins with Frustrations Half filled Kondo Lattice Quantum Spin Hall phase Topological Order X.G.Wen List at 03 Kane & Mele 05, ernevig, Hughes & Zhang 06 How to characterize Topological phases? Edge states for characterization : bulk-edge correspondence Quantized quantities for topological order parameters

Example from history diabatic principle for quantum Hall states flux attachment (Jain) diabatic heuristic argument (Greiter-Wen-Wilczek) phases (filling & statistical parameter) 2 3 FQHE 3/7 FQHE 2/5 FQHE 1/3 IQHE (3rd LL) IQHE (LLL) Semion IQHE (2nd LL) 1/2 state Free boson LLL boson Free fermion 1 1/ 2 3

Example from history diabatic principle for quantum Hall states flux attachment (Jain) diabatic heuristic argument (Greiter-Wen-Wilczek) Connect states by adiabatic process 2 3 FQHE 3/7 FQHE 2/5 FQHE 1/3 IQHE (3rd LL) IQHE (LLL) Semion IQHE (2nd LL) 1/2 state Free boson LLL boson Free fermion 1 1/ 2 3

Example from history diabatic principle for quantum Hall states flux attachment (Jain) diabatic heuristic argument (Greiter-Wen-Wilczek) Collect gapped phases by adiabatic continuation Label of the Class : diabatic invariant (topological number)

Example from history diabatic principle for quantum Hall states flux attachment (Jain) diabatic heuristic argument (Greiter-Wen-Wilczek) = =0 Collect gapped phases by adiabatic continuation Label of the Class : diabatic invariant (topological number)

Locality of gapped ground state Dimer chains KLT & VS solids (1,1) gapped integer spin chain Something complicated but gapped many-body gap small

Locality of gapped ground state diabatic deformation! gap remains open Something complicated but gapped many-body gap

Locality of gapped ground state diabatic deformation! Decoupled! gap remains open Something very simple & gapped big! many-body gap

Locality of gapped ground state diabatic process to be decoupled: gap remains open short range entangled state

Use of diabatic invariant & Quantization YH, JPSJ 73 2604 ( 04), 74 2374 ( 05), 75 123601 ( 06) Topological order : free fermion is still non trivial ex. IQHE physical Physical System free Trivial phase Fixed point diabatic continuation Quantized quantity : diabatic invariant Remains the same, unless the gap close.

TRULY GENERIC phase without any symmetry protection topologically single phase (too simple?) With some symmetry,, C Von Neumann-Wigner theorem 1.Discrete symmetry Time reversal Charge conjugation Space inversion Reflection 2.Gauge symmetry U(1) : QHE (TR ) Sp(1) : QSHE (TR ) YH, 06 Chen-Gu-Wen, 10 Pollmann-erg-Turner-Oshikawa., 10

Gapped Topological! diabatic invariants Parameter dependent hamiltonian erry connection Intrinsically quantized Chern numbers: 1st, 2nd, 3rd,... QHE... Z Symmetry protected quantization erry phases & generalization: Quantum spin chains, Spin-QHE, topological insulators... Z 2 1 C 1 = 2 i 1 = 1 2 i Z Z M 2 F...,-2,-1,0,1,2,... M 1 0 or 1/2

Up to this point symmetry helps a lot!

Quantum Spin Hall Effect = Quantum Hall Effect without magnetic field = Quantum Hall Effect with time-reversal invariance No external net magnetic field +1 QSHE spin up spin down -1 TR invariant 1-st Chern number is vanishing with TR NEED Z 2 invariants Fu-Kane 06

Symmetry restriction can be too strong & severe 1-st Chern number is vanishing with TR NEED more freedom! Z 2 invariants Kane-Mele 05 Fu-Kane 06 Entanglement Chern number T. Fukui & Y. Hatsugai, J. Phys. Soc. Jpn. 84, 043703 (2015), ibid. 113705 (2014)

Partition for entanglement entropy & hamiltonian =Tr e H /Z Entanglement hamiltonian Emergence of effective boundary degrees of freedom: edge states Something for classification Topological order Edge states erry connections

Entanglement Entropy Mixed State From Entanglement Direct Product State Ψ = Ψ Ψ Holzhey, Larsen and Wilczek, 94 System = State = Ψ Ψ Entangled State Partial Trace ρ = Ψ Ψ ρ =Tr ρ Ψ = 1 D D j Ψ j Ψj Pure State D =1 D = 1 D j Ψ j Ψj Mixed State ρ = 1 D D Ψ j Ψk Ψ j Ψk jk How much is the State Entangled? Entanglement Entropy : S = log ρ = log D

Partition for entanglement entropy & hamiltonian =Tr e H /Z Entanglement hamiltonian Emergence of effective boundary degrees of freedom: edge states Extensive partition Hsieh-Fu 13 Derived new system of the edge states? Li-Haldane 08

Partition for entanglement entropy & hamiltonian Extensive partition Hsieh-Fu 13 = ih H i = E i pure state

Novel uses of entanglement hamiltonian T. Fukui & Y. Hatsugai, J. Phys. Soc. Jpn. 84, 043703 (2015), ibid. 113705 (2014 Derived new system of the edge states? H i = E i = ih pure state =Tr e H /Z Entanglement hamiltonian mixed state : finite temperature

symmetric H Chern number is vanishing due to the TR invariance

H! H =Tr e H /Z Symmetry breaking by partial trace Chern number can be finite (integer)

H i = E i = ih Entanglement hamiltonian =Tr e H /Z pure state T =0 T>0 H! H mixed state : finite temperature Consider a ground state of H purification! T =0

Even though C =0 due to the symmetry, C can be finite with interaction in principle Construct entanglement hamiltonian! T. Fukui & Y. Hatsugai, J. Phys. Soc. Jpn. 83, 113705 (2014) Symmetry restriction can be too strong! Derived symmetry breaking in a symmetric phase H i = E i = ih extensive partition =Tr e H /Z Derived symmetry breaking Entanglement hamiltonian symmetric H! H Symmetry breaking by partial trace Calculate Chern number for H :entanglement Chern number

Chern number is calculable stably and easily Numerical Technique from the Lattice gauge theory Topological Invariant on Discretized Lattice Lattice in k space ( discretization for the integral ) xy = e2 h 1 2 i F 1234 F 1234 = Im log U 12 U 23 U 34 U 41 U mn = det j U µ (k l ) F 12 (k l ) Technical dvantage for large Chern Numbers k l gauge invariant m n, n = ( 1 (k n ),, j (k n )) U µ (k l ) n(k l ) n(k l +ˆµ) /N µ (k l ) N µ (k l )= n(k l ) n(k l +ˆµ) Fukui-Hatsugai-Suzuki J. Phys. Soc. Jpn. 74, 1674 (2005) Fermi Sea of j filled bands F 12 (k l ) ln U 1 (k l )U 2 (k l + ˆ1)U 1 (k l + ˆ2) 1 U 2 (k l ) 1 π < F 12 (k l )/i π 2 1 N rillouin Zone Ex. -0.321+0.22-0.1+0.0032+ +0.83 =3.0000000 (principal value)

T. Fukui & Y. Hatsugai, J. Phys. Soc. Jpn. 84, 113705 (2014) One can get phase diagrams for example 1: weak phases in layered 2D topological systems Fuku-Imura-Hatsugai. Phys. Soc. Jpn. 82, 073708 (2013) Yoshimura-Imura-Fukui-Hatsugai, Phys. Rev. 90, 155443 (2005) example 2: Kane-Mele 2D topological insulator Kane-Mele,Phys. Rev. Lett. 95 (2005)

example 1: weak phases in layered 2D topological systems 0 +1 - -1 + -1 + 0 +1 - T. Fukui & Y. Hatsugai, J. Phys. Soc. Jpn. 84, 113705 (2014)

example 1: weak phases in layered 2D topological systems 0(0,0) +1 - -1 + -1 + 0(+1,-1) +1 - T. Fukui & Y. Hatsugai, J. Phys. Soc. Jpn. 84, 113705 (2014)

example 2:Z 2 topological phases of the Kane-Mele model KM phase diagram without Z 2 invariants H.raki, T. Kariyado, T. Fukui,Y. Hatsugai 15

H! H! Pure state Mixed state T =0 T>0 purification! ground state Chern # T =0 Purification of entanglement hamiltonian is a new tool! Interaction can be included (Manybody effects) c.f Z 2 invariant is only for free cases

H! H E-Chern # should be useful for random systems? extended states for topological insulators?? Purification of entanglement hamiltonian is a new tool! Interaction can be includes (Manybody effects) c.f Z 2 invariant is only for free cases

Symmetry Topology H! H new tool (entanglement) Disorder Thank you