Entanglement Behavior of 2D Quantum Models Shu Tanaka (YITP, Kyoto - PowerPoint PPT Presentation

Entanglement Behavior of 2D Quantum Models Shu Tanaka (YITP, Kyoto University) Collaborators : Hosho Katsura (Univ. of Tokyo, Japan) Anatol N. Kirillov (RIMS, Kyoto Univ., Japan) Vladimir E. Korepin (YITP, Stony Brook, USA) Naoki Kawashima (ISSP, Univ. of Tokyo, Japan) Lou Jie (Fudan Univ., China) Ryo Tamura (NIMS, Japan) VBS on symmetric graphs, J. Phys. A, 43 , 255303 (2010) “VBS/CFT correspondence”, Phys. Rev. B, 84 , 245128 (2011) Quantum hard-square model, Phys. Rev. A, 86 , 032326 (2012) Nested entanglement entropy, Interdisciplinary Information Sciences, 19 , 101 (2013) 論文での使用姓 ミドルネーム 所属部署 職 名 見込み 受入希望研究室 審査結果通知先

Digest Entanglement properties of Physical properties of 2D quantum systems 1D quantum systems VBS on square lattice VBS on hexagonal lattice Quantum lattice gas on ladder Volume exclusion e ff ect VBS state on 2D lattice Quantum lattice gas on ladder Entanglement Entanglement Total system Total system Hamiltonian Hamiltonian 1D AF Heisenberg 2D Ising Square lattice Square ladder Hexagonal lattice 1D F Heisenberg 2D 3-state Potts Triangle ladder

Introduction - Entanglement - Motivation - Preliminaries

Introduction EE is a measure to quantify entanglement. Total Divide | Ψ � system Subsystem Subsystem A B Schmidt decomposition Reduced density matrix � λ α | φ [A] α � � | φ [B] | Ψ � = α � � λ 2 α | φ [A] α �� φ [A] ρ A = Tr B | Ψ �� Ψ | = α | α α Normalized GS φ [A] ∈ H A , φ [B] ∈ H B α α {| φ [A] α � } , {| φ [B] : Orthonormal basis α � } von Neumann entanglement entropy � λ 2 α ln λ 2 S = Tr ρ A ln ρ A = − α α

Introduction Entanglement properties in 1D quantum systems!! 1D gapped systems: EE converges to some value. 1D critical systems: EE diverges logarithmically with L. coe ffi cient is related to the central charge. XY( a = ∞ , γ = 0) XXZ model under magnetic fi eld 2.5 ENTROPY − S − � i +1 + σ y i σ y ( σ x i σ x i +1 + ∆ σ z i σ z i +1 − λσ z H XXZ = i ) ) 0 = λ 1 = ∆ , ( i Z X 2 X ) 0 XY model under magnetic fi eld = λ 5 2 XY( a = 1 , γ = 1) = , ∆ . ( Z X X 1.5 XY( a = 1 . 1 , γ = 1) 1 A B L 10 20 30 40 NUMBER OF SITES − L − G. Vidal et al. PRL 90 , 227902 (2003) Entanglement properties in 2D quantum systems??

Preliminaries: re fl ection symmetric case Pre-Schmidt decomposition Re fl ection symmetry {| φ [A] α � } , {| φ [B] α � } � | φ [A] α � � | φ [B] | Ψ � = α � Linearly independent α (but not orthonormal) Subsystem Subsystem A B Overlap matrix α | φ [A] α | φ [B] ( M [A] ) αβ := � φ [A] β � , ( M [B] ) αβ := � φ [B] β � Re fl ection symmetry M [A] = M [B] = M Useful fact M [A] = M [B] = M If and is real symmetric matrix, M d 2 � α S = − p α = p α ln p α , � α d 2 α α where are the eigenvalues of . d α M J. Phys. A, 43 , 255303 (2010)

Digest Entanglement properties of Physical properties of 2D quantum systems 1D quantum systems VBS on square lattice VBS on hexagonal lattice Quantum lattice gas on ladder Volume exclusion e ff ect VBS state on 2D lattice Quantum lattice gas on ladder Entanglement Entanglement Total system Total system Hamiltonian Hamiltonian 1D AF Heisenberg 2D Ising Square lattice Square ladder Hexagonal lattice 1D F Heisenberg 2D 3-state Potts Triangle ladder

VBS (Valence-Bond-Solid) state Valence bond = Singlet pair AKLT (A ffl eck-Kennedy-Lieb-Tasaki) model I. A ffl eck, T. Kennedy, E. Lieb, and H. Tasaki, PRL 59 , 799 (1987). � � 2 � S i +1 + 1 � � S i · � � S i · � � ( S = 1) S i +1 H = 3 i Valence bond Ground state: VBS state (projection) S = 1 - Exact unique ground state; S=1 VBS state - Rigorous proof of the “Haldane gap” - AFM correlation decays fast exponentially

VBS (Valence-Bond-Solid) state VBS state = Singlet-covering state 2D square lattice 2D hexagonal lattice MBQC using VBS state T-C. Wei, I. A ffl eck, and R. Raussendorf, Phys. Rev. Lett. 106 , 070501 (2011). A. Miyake, Ann. Phys. 326 , 1656 (2011).

VBS (Valence-Bond-Solid) state VBS state = Singlet-covering state Schwinger boson representation n ( b ) = b † k b k k | �� = a † | vac � , | �� = b † | vac � 4 a † k a k + b † k b k = 2 S k 3 2 Valence bond solid (VBS) state � � 1 � a † k b † l � b † k a † | VBS � = | vac � l n ( a ) = a † k a k � k,l � k 0 0 1 2 3 4 S=0 1/2 1 3/2 2

VBS (Valence-Bond-Solid) state Re fl ection symmetry Subsystem Subsystem A B 2D square lattice 2D hexagonal lattice Subsystem A Subsystem B Subsystem A Subsystem B

VBS (Valence-Bond-Solid) state � � a † k b † l � b † k a † � | VBS � = | vac � l - Local gauge transformation � k,l � - Re fl ection symmetry � | φ [A] α � � | φ [B] α � = { α } � � { α } = α 1 , · · · , α | Λ A | Auxiliary spin: α i = ± 1 / 2 #bonds on edge: | Λ A | Overlap matrix Subsystem A Subsystem B : 2 | Λ A | × 2 | Λ A | matrix M { α } , { β } Each element can be obtained by Monte Carlo calculation!! Phys. Rev. B, 84 , 245128 (2011) SU(N) case can be also calculated. cf. H. Katsura, arXiv:1407.4262

Entanglement properties - Entanglement entropy - Entanglement spectrum - Nested entanglement entropy

Entanglement properties of 2D VBS states VBS state = Singlet-covering state 2D square lattice 2D hexagonal lattice L y PBC Subsystem A Subsystem B Subsystem A Subsystem B L x OBC

Entanglement entropy of 2D VBS states cf. Entanglement entropy of 1D VBS states N � S � � a † i b † i +1 � b † i a † | VBS � = | vac � i +1 i =0 Subsystem A Subsystem B S=8 S=6 S=4 S=3 S=2 S=1 H. Katsura, T. Hirano, and Y. Hatsugai, PRB 76 , 012401 (2007). S = ln (# Edge states)

Entanglement entropy of 2D VBS states S | Λ A | = ln 2 − σ σ ≥ 0 σ 1D = 0 ξ square > ξ hexagonal σ square > σ hexagonal #bonds on edge: | Λ A | 2D square lattice 2D hexagonal lattice L y PBC or OBC Subsystem A Subsystem B Subsystem A Subsystem B L x OBC

Entanglement spectra of 2D VBS states H. Li and F. D. M. Haldane, Phys. Rev. Lett. 101 , 010504 (2008). � e − λ α | φ [A] α �� φ [A] α | Reduced density matrix ρ A = α ρ A = e − H E Entanglement Hamiltonian ( H E = − ln ρ A ) Square Hexagonal (L x =5, L y =16) (L x =5, L y =32) 1D antiferro 1D ferro Heisenberg Heisenberg des Cloizeaux- Spin wave Pearson mode cf. J. I. Cirac, D. Poilbranc, N. Schuch, and F. Verstraete, Phys. Rev. B 83 , 245134 (2011).

Nested entanglement entropy “Entanglement” ground state := g.s. of : H E | Ψ 0 � H E = − ln ρ A H E | Ψ 0 � = E gs | Ψ 0 � ρ A | Ψ 0 � = ρ 0 | Ψ 0 � Maximum eigenvalue Nested entanglement entropy Nested reduced density matrix S ( � , L ) = − Tr 1 , ··· , � [ � ( � ) ln � ( � )] � ( � ) := Tr � +1 , ··· ,L [ | Ψ 0 �� Ψ 0 | ] 1D quantum critical system (periodic boundary condition) A B � P . Calabrese and J. Cardy, J. Stat. Mech. (2004) P06002.

Nested entanglement entropy S ( � , L ) = − Tr 1 , ··· , � [ � ( � ) ln � ( � )] 1.4 (a) (b) 0.8 1.2 A B � 0.7 S(l,16) S(l,16) 1 0.6 Square lattice Square ladder 0.8 (PBC) (OBC) c=1.01(7) L x =5, L y =16 0.5 a=0.393(1) L x =5, L y =16 s 1 =0.77(4) c 1 /v=0.093(3) fitting fitting 0.6 0 2 4 6 8 10 12 14 16 0 2 4 6 8 1 l l 1D antiferromagnetic Heisenberg Central charge: c = 1 des Cloizeaux-Pearson mode in ES supports this result. VBS/CFT correspondence

Digest Entanglement properties of Physical properties of 2D quantum systems 1D quantum systems VBS on square lattice VBS on hexagonal lattice Quantum lattice gas on ladder Volume exclusion e ff ect VBS state on 2D lattice Quantum lattice gas on ladder Entanglement Entanglement Total system Total system Hamiltonian Hamiltonian 1D AF Heisenberg 2D Ising Square lattice Square ladder Hexagonal lattice 1D F Heisenberg 2D 3-state Potts Triangle ladder

Digest Entanglement properties of Physical properties of 2D quantum systems 1D quantum systems VBS on square lattice VBS on hexagonal lattice Quantum lattice gas on ladder Volume exclusion e ff ect VBS state on 2D lattice Quantum lattice gas on ladder Entanglement Entanglement Total system Total system Hamiltonian Hamiltonian 1D AF Heisenberg 2D Ising Square lattice Square ladder Hexagonal lattice 1D F Heisenberg 2D 3-state Potts Triangle ladder

Rydberg Atom Rydberg atom (excited state) Interaction Ground state Max Planck Institute n i n j � � � σ x H = Ω i + ∆ n i + V | r i − r j | γ i ∈ Λ i ∈ Λ i,j