Holographic encoding of universality in corner spectra National - PowerPoint PPT Presentation

Holographic encoding of universality in corner spectra National Center for Theoretical Sciences (NCTS),Taiwan Ching-Yu Huang ( 黃靜瑜 ) work with 
 Román Orús (Donostia International Physics Center) 
 Tzu-Chieh Wei (Stony Brook university) 
 (TNSAA) 2018-2019 2018/12/03 Reference: PRB 95, 195170 (2017)

Outline Introduction 
 • - Phase transition 
 - Tensor network state 
 - corner tensor The fingerprints of universal physics are encoded holographically in numerical • CTMs and CTs. 
 - 1d quantum Ising universality class 
 - classical and quantum Ising (quantum-classical correspondence) 
 - deformed symmetry protected topological order phase 
 - 2d quantum XXZ model Quantum state renormalization in 2D using corner tensors 
 • - chiral topological PEPS Summary •

Motivation Different phases - rich world • Why the different phases exist 
 • - Symmetry breaking theory [Landau ] Magnets: rotation symmetry breaking • Crystals: translation symmetry breaking… • Local order parameters distinguish different phases 
 •

Motivation Phase transition: 
 • - Type of transition 
 - Characteristic properties: 
 Symmetry, Order parameters, Critical points, 
 Critical exponents and Universality classes,… [Sac hd ev,’11] Example : 
 Example : 
 • • 2D quantum Ising model with Transverse 2D classical Ising model 
 field h X ( σ z i σ z i +1 + h σ x H = − i ) X ( σ z i σ z i +1 + h σ x H = − i ) h=0 i i - 2nd order transition 
 - critical exponents 
 - CFT (central charge c=1/2) ordered disordered ordered disordered [Gu, Levin,& Wen’ 08]

Motivation To study quantum many-body system • The Hilbert space grows exponentially with system size H ∼ d N • To e ffi cient simulation (polynomial in memory and time) • To study various Hamiltonians (e.g. Bosons and Fermions) and measure • physical properties and observables Find the ground state (approximation) 
 Measure physical observables (approximation) 
 exact diagonalization (ED), 
 Density matrix renormalization group (DMRG) 
 Tensor network state (simple update, full update)+ 
 Tensor network algorithm (PEPS,TRG,SRG,HOTRG,TNR,…..) 
 …………

Matrix/Tensor Product States The numerical implementation for finding the ground states of spin systems are • based on the matrix/tensor product states (MPS/TNS) . These states can be understood from a series of Schmidt (bi-partite) • decomposition . It is QIS inspired. The ground state is approximated by the relevance of entanglement. • TNS MPS … tTr ( A s 1 A s 2 ...A s N ) | s 1 , s 2 , ..., s N i X | Ψ i = s 1 ,s 2 ,...,s N

Infinite system finite TNS infinite TNS … … … … [F. Verstraete, I. Cirac 06’] Unit cell of tensors is repeated 
 periodically over the whole PEPS: 
 translational invariance [J. Jordan, R. Orus, G. Vidal, F. Verstraete, I. Cirac, 08’]

Tensor network algorithm Structure (Anstaz) 1d: MPS 2d: TNS X | Ψ i = tTr ( A s 1 A s 2 ...A s n ) | s 1 , s 2 , ...., s n i s 1 ,s 2 ,....,s n Find the best ground Compute observable stats imaginary time iterative optimization Contraction of the evolution of individual tensors 
 tensor network 
 (energy minimization) exact / approximate

Determine the observables 
 - Contracting the infinite 2d lattice Observable … h Ψ | ˆ O | Ψ i h Ψ | Ψ i Norm … … … h Ψ | h Ψ | … … … ˆ O … … … … … | Ψ i | Ψ i … … … … … … … … … To determine observables … … (double indices) Contraction of this infinite lattice … … … … …

Contracting the infinite 2d lattice from infinite TNS corner transfer matrix … … … … … C 1 C 2 T 1 T 4 T 2 … … … half-row 
 transfer matrix C 4 C 3 T 3 … … … … half-column 
 transfer matrix Renormalized Corner Transfer Matrices CTM method [R. Orús, G. Vidal,09’, R. Orús,12’] • The corner tensors C → one quadrant of the 2D TNS 
 The half-row/half-column tensors T → half an infinite row/column of TNS

CTM method • Follow this procedure by absorbing rows and columns towards the left, up, right, and down directions until convergence is reached • At every step we need to renormalize with isometries • In the end, the corner tensors C represent one quadrant of the 2D TNS, and the half-row/ half-column tensors T to the renormalization of half an infinite row/column of TNS [R. Orús, G. Vidal,09’]

Outline Introduction 
 • - Tensor network state 
 - corner tensor 
 - Symmetry protected topological order phases The fingerprints of universal physics are encoded • holographically in numerical CTMs and CTs. 
 - 1d quantum Ising universality class 
 - classical and quantum Ising (quantum-classical correspondence) 
 - deformed symmetry protected topological order phase 
 - 2d quantum XXZ model Quantum state renormalization in 2D using corner tensors 
 • - chiral topological PEPS Summary •


 Corner tensor Z = tr( C 1 C 2 C 3 C 4 ) , (a) … … … … • Corner transfer matrices (CTMs) Z C 1 C 2 method can be used to study physical system. 
 C 4 C 3 … … [R. J. Baxte 1968; T. Nishino and K. Okunishi 1996; R. Orús 2012] … … • CTMs can be defined for any 2d tensor | ψ (0) i network h Ψ | Ψ i U ( δτ ) U ( δτ ) ✓ The Partition function of classical lattice model U ( δτ ) U ( δτ ) ∗ ✓ The time-evolution of a 1d quantum system U ( δτ ) ∗ U ( δτ ) ∗ ✓ The norm of 2d PEPS h ψ (0) | [R. Orús 2012]

Corner tensor • Corner transfer matrices and corner tensor contain a great amount of holographic information about the bulk properties of the system • Bulk information is encoded at the “ boundary ” corners - similar to “entanglement spectrum” and “entanglement Hamiltonians [H. Li and F.D.M. Haldane 2008; J. I. Cirac, D. Poilblanc, N. Schuch, and F. Verstraete 2011] • For example, Peschel showed that the entanglement spectrum of a quantum spin chain is identical to the spectrum of some Corner transfer matrices in 2d [I. Peschel, M. Kaulke, and Ö. Legeza 1999; I. Peschel 2012] 1d quantum Ising model [ H q , H c ] = 0 2d classical Ising model Hc 
 with an isotropic coupling K L − 1 L − 1 � � σ [ i ] x − δσ [ L ] σ [ i ] z σ [ i + 1] H q = − , − λ x z where δ = cosh 2 K and λ = sinh 2 K , i = 1 i = 1 matrix technique. The transverse field labeled

Corner tensor • Start from a Hamiltonian 
 or a wave function 
 - form a tensor network 
 - use CTM method to 
 study the properties of 
 [R. Orús, G. Vidal,09’, R. Orús,12’] ground state A reduced density matrix ρ of a system can • We then use these corner be given, and we have 
 tensor (e.g. corner entropy) to corner spectra and corner entropy pinpoint quantum phase transitions 3d TN 2d TN C 1 C 2 ρ C 4 C 3

CTM from tensor network (TN) • For 1d quantum: from Hamiltonian ➜ (1+1)d TN exp (-Ht) Ψ | ψ (0) i 0 | Ψ 〉 = τ exp (-Ht) U ( δτ ) Ψ 0 U ( δτ ) C 1 C 2 T 1 U ( δτ ) CTM method U ( δτ ) ∗ T 4 T 2 U ( δτ ) ∗ C 4 C 3 T 3 U ( δτ ) ∗ h ψ (0) | [R. Orús 2012] corner spectrum | ψ f i C 1 C 2 = ρ entanglement spectrum h ψ f | C 4 C 3

CTM from tensor network (TN) corner spectrum • For 2d quantum: 
 = from Hamiltonian ➜ (2+1)d TN entanglement spectrum from wave function (a) use the 2d quantum state renormalization ➜ 3d TN (b) the norm of PEPS ➜ 2d TN • For 2d classical: 
 • For 3d classical: 
 partition function ➜ 2d TN partition function ➜ 3d TN A reduced density (a) … … … … matrix ρ of a system Z C 1 C 2 C 1 C 2 can be given, and ρ we have 
 C 4 C 3 corner spectra and C 4 C 3 corner entropy … … … …

1d quantum Ising universality class Entanglement spectra and the entanglement entropy obtained from iTEBD • 1d quantum Ising model: σ [ i ] x σ [ i + 1] σ [ i ] � � H q = − − h z , x i i Get the ground state via T c imaginary time evaluation t ( h ) = p arcsin 1 /h exp (-Ht) Ψ 0 | Ψ 〉 = τ exp (-Ht) Ψ 0 δ t ⌧ 1 Divide into small time-steps | Ψ 0 i ! | Ψ 1 i ! ... ! | Ψ m i ! | Ψ m +1 i ! ... | E m +1 E 0 ≥ E 1 ≥ ... ≥ E m ≥ ≥ ... iTEBD MPS … … …

1d quantum Ising universality class The corner spectrum obtained from the time-evolution of a 1d quantum system • 1d quantum Ising model: σ [ i ] x σ [ i + 1] σ [ i ] � � H q = − − h z , x i i Get the ground state via T c imaginary time evaluation t ( h ) = p arcsin 1 /h exp (-Ht) Ψ 0 | Ψ 〉 = τ exp (-Ht) Ψ 0 | ψ (0) i U ( δτ ) U ( δτ ) U ( δτ ) U ( δτ ) ∗ C 1 C 2 T 1 C 1 C 2 U ( δτ ) ∗ T 4 T 2 ρ U ( δτ ) ∗ C 4 C 3 C 4 C 3 h ψ (0) | T 3


 
 
 1d quantum Ising universality class • 2d classical model The partition function 
 X e − β H c ( { s } ) Z c = { s } with classical Hamiltonian 
 X s [ i ] s [ j ] H c { s } = − h i,j i The critical point is β c = 1 /T c = 1 ⇣ ⌘ √ 2 log 1 + 2 (a) t = T/T c … … … … (b) Z C 1 C 2 C 1 C 2 ρ C 4 C 3 C 4 C 3 … … … … Z = tr( C 1 C 2 C 3 C 4 )