The application of tensor network method in three dimensional - PowerPoint PPT Presentation

The application of tensor network method in three dimensional quantum system Ching-Yu Huang 黃靜瑜 Department of Applied Physics, Tunghai University, Taiwan 2019/12/06 TNSAA 2019-2020

Motivation The classification of 3d bosonic topological order (TO)& symmetry protected • topological order(SPT) is well known ( fixed point wave function ) We would like to study quantum system (with topological order) in 3D • But How to detect those topological order phase numerically? Numerical tool: 3D HOTRG , 3D CTM,… • To simplify our problem, we will consider fixed point wave function with deformation (not from Hamiltonian)

Outline Introduction : 
 topological order 
 2D and 3D toric code ℤ N Numerical method: 
 Tensor-Network scheme for modular S and T matrices (tnST) 
 3D high order tensor renormalization group Numerical results: 
 Case study: , , topological order in 3D 
 ℤ 2 ℤ 3 ℤ 4 Dimensional Reduction to 2D 
 3D AKLT (symmetry ) state and deformation Summary

Introduction: Topological order Beyond Landau (symmetry-breaking) paradigm 
 [Tsui,Stomer,Gossard ‘82,Laughlin ‘83, eg. Fractional Quantum Hall, Spin Liquid, ... Anderson ‘73,...] Topological order characterized by: N g Topology-dependent ground-state degeneracy ( ) Nontrivial excitations and statistics (usually in 2d) Long-range entanglement [Wen ’90] Potential application in fault-tolerant quantum computation [Wen and Niu ‘90 ] g=0 g=2 g=1 • |Ψ� ¡→ � �� |Ψ� anyon |Ψ� ¡→ |Ψ� boson self statistics |Ψ� ¡→ �|Ψ� mutual statistics fermion • •

ℤ N Topological order: Toric code 2D and 3D: spins reside on edges 
 N -state degrees of freedom located on the link | q ⟩ i The Hamiltonian of the toric code 
 ℤ N H = − J e s ) − J m 2 ∑ 2 ∑ ( A s + A † ( B p + B † p ) s s The operators and as 
 Z i X i Z i | q ⟩ i = ω q | q ⟩ i ; ω = 2 e 2 π i / N X i | q ⟩ i = | q − 1 ⟩ i ; Ground state satisfy 
 A s | G . S . ⟩ = B p | G . S . ⟩ = | G . S . ⟩

ℤ N Topological order: Toric code Degeneracy on 2,3-torus 
 2D: # deg = N 2 3 D: # deg = N 3 Representative ground states can be written as a tensor network X O O G s i ) | s 1 , s 2 , ... i , | ψ i = tTr ( P s i v l @ each site:p @ each link ( 3 direction ) G s α , β = δ s , α δ s , β P xx ′ zz ′ = 1 yy ′ s only if α β x − x ′ + y − y ′ + z − z ′ = 0 ( mod n ) ➔ Deform toric G s α , β = f s δ s , α δ s , β Ground state: 
 → use the string operater to get other ground state | ψ α , β i = ( Z 1 ) α ( Z 2 ) β | ψ 0 , 0 i e.g. 2d TC

Order parameter: from wave function overlap Topological order characterized by its quasiparticle excitations- anyons (with nontrivial braiding statistics) i e θ Mathematically, the braiding statistics is encoded in the modular matrices. The modular matrices, or S and T matrices, are generated respectively by the 90º rotation and Dehn twist on torus. h ψ a | ˆ S | ψ b i = e � α S V + o (1 /V ) S ab h ψ a | ˆ T | ψ b i = e � α T V + o (1 /V ) T ab , need to first st] es {| ψ a i } N :degenerate ground state a =1 be given by [Hung & Wen ’14; Moradi & Wen ‘14]

Previous work: 2D topological order with deformation h ψ a | ˆ S | ψ b i = e � α S V + o (1 /V ) S ab Start from a wave function in 2D with deformation 
 h ψ a | ˆ T | ψ b i = e � α T V + o (1 /V ) T ab , ⇒ By tuning a parameter to study the phase transition (a) (b) (c) We propose a way -tnST “Tensor network scheme for modular S and T matrices” to detect quantum phase transition numerically. (d) [ Huang and Wei 2016] (e) (f) How to describe a quantum state? 
 (g) Tensor product states [F. Verstraete, Murg, & Cirac 2008] What is the “order parameter”? 
 Modular matrices [Zhang,Grover, Turner, Oshikawa, & Vishwanath 2012] How to calculate the observable? 
 Higher order tensor renormalization group [Xie, Chen, Qin, Zhu, Yang, & Xiang, 2012 ]

ℤ N 2D Topological order phase S & T from wave function overlaps (string/membranes as “symmetry twists”): 
 ➔ use real space renormalization to obtain fixed-point values 
 (as number of RG steps ); 
 n RG → ∞ (note: symmetry twists are also coarse-grained) Ground-state degeneracy & modular matrices/invariants believed to be su ffi cient to characterize topological order [ Huang and Wei 2016] (a) 4.5 topological order phase: ℤ 2 RG=2 RG=4 4.0 | Ψ⟩ = ∑ RG=6 Wave function RG=10 | ψ c ⟩ 3.5 trivial phase Topological order c 3.0 1 0 0 0 0 1 Deformed wave function 0 0 1 0 S = 1 1 1 1 B C 0 1 0 1 0 0 2.5 @ A 1 1 1 1 0 0 0 1 S = T = A , B C 1 1 1 1 @ 2.0 1 1 1 1 Q = | 0 ih 0 | + g | 1 ih 1 | 1.5 0.70 0.75 0.80 0.85 0.90 (b)

̂ ̂ ̂ ̂ ̂ ̂ Topological invariant (Modular Matrices) in three dimension SL (3, ) group : generated by a and s t ℤ t = ( s = ( 1 0 0 1 ) 0 1 0 1 0 0 ) 1 1 0 0 0 1 0 0 shear along y direction 
 cyclic shift of z,y,x axes ⊥ on surface x axis Modular matrices S and T are representations using degenerate ground states ➔ also give exchange/braiding statistics of anyonic excitations 
 T i , j = ⟨Ψ i | ̂ S i , j = ⟨Ψ i | ̂ s | Ψ j ⟩ t | Ψ j ⟩ s t Ground states: membrane { ̂ h x , ̂ h y , ̂ operators acting on h z } reference G.S. 
 | Ψ j ⟩ = ̂ h x ̂ h y ̂ h z | Ψ 0 ⟩ Use 3D HOTRG and 3D tnST scheme !!


 Numerical method: 3D renormalization group 3D high order tensor renormalization group ( HOTRG ) [ Xie,Chen, Qin, Zhu, Yang , Xiang,2012] D 11 ➜ In the 3D calculation, the computational time scales with D 6 and the memory scales with . 3D tnST scheme : 3d HOTRG

Numerical results: 
 ℤ 2 3D topological order with deformation on cubic lattice Use tr(S) and tr(T) as “order parameters” [He,Moradi &Wen, PRB 14’] in 2D Z 2 Deform the 3D toric-code ground state by local operator on each spin 
 Q ( g ) | Ψ ( g ) ⟩ = Q ( g ) ⊗ N | Ψ TC ⟩ Q ( g ) = | 0 ⟩⟨ 0 | + g 2 | 1 ⟩⟨ 1 | (g=1: undeformed; g=0: product state) Trivial phase S,T =identity Trivial Topological Trivial Topological order D cut = 8 2 3 n RG E ff ective lattice size: (fixed point as RG steps ) 
 n RG → ∞ ➔ transition at g ≈ 0.68 from topological (e.g. g=1) to trivial phase (e.g. g=0)

Numerical results: 
 ℤ 3 ℤ 4 Deforming and topological order Deform : ℤ 3 Deform : ℤ 4 Q ( g ) ℤ 3 = | 0 ⟩⟨ 0 | + g 2 | 1 ⟩⟨ 1 | + g 4 | 2 ⟩⟨ 2 | Q ( g ) ℤ 4 = | 0 ⟩⟨ 0 | + g 2 | 1 ⟩⟨ 1 | + g 4 | 2 ⟩⟨ 2 | + g 6 | 3 ⟩⟨ 3 | g c ≈ 0.66 g c ≈ 0.65 D cut = 8 D cut = 9

ℤ N 3D topological order with deformation Transitions agree with mapping to 3D Ising/Potts models N − 1 Under such deformation and ( ) ∑ q i = g 2 and q i | i ⟩⟨ i | q i ≥ 0 Q = q 0 = 1 i =0 Potts partition function ⟨Ψ GS ( g ) | Ψ GS ( g ) ⟩ ⟺ ℤ D cut = 8 D cut = 9 D cut = 8

̂ ̂ → Dimensional reduction: 3D 2D Compactify z-direction to small radius: 
 (i) 3D 2D (ii) SL(3, ) reduces to SL(2, ) → ℤ ℤ 2D braiding is associated with SL(2, ) group, which is generated by ℤ s yx = ( t yx = ( 0 0 1 ) 1 0 1 0 0 1 ) | G | ⨁ C 3 D C 2 D G = [Moradi & Wen 2015, − 1 0 0 1 1 0 ➜ Reduction G Wang & Wen 2015] 0 0 0 n =1 ℤ N ℤ N ➜ We verify that 3D topological order is decomposed into copies of 2D topological order via block structure of S & T (showing real parts) ℤ 2 ℤ 3 2

Other lattice structure Diamond lattice ➜ Combing two tensors to form a new tensor. The diamond lattice deforms into a cubic lattice. 3d HOTRG


 ℤ 2 Deforming topological order in diamond lattice g c ≈ 0.771 Deform : 
 ℤ 2 Q ( g ) ℤ 2 = | 0 ⟩⟨ 0 | + g 2 | 1 ⟩⟨ 1 |

Conclusion: part I Main result: 
 tensor-network scheme for modular matrices (tnST) to diagnose 3D topological order → ℤ N successfully applied to transitions in 3D toric code under string tension Future: 
 1. Twisted “quantum double” models 
 2. Fixed point wave function with deformation 
 -> exact MPO/ PEPO

Twisted topological models TC : | Ψ⟩ = ∑ DS : | Ψ⟩ = ∑ ( − 1) # loops | ψ c ⟩ | ψ c ⟩ 2d Twisted by 3-cocyle c c [oliver, 2016] 3d: Twisted by 4-cocyle • The tensor representation of = := the basis vector Need more e ffj cient 3 D tensor RG !! • The membrane operator ATRG, BTRG !! 2d twisted TO with the physical index g g g is called a 2d twisted TO Tensor on cubic lattice: large physical degree and bond dimension 3d twisted TO 3d twisted TO

3D Twisted Z 2 × Z 2 topological order From exact TO wave function • GSD = 4 3 =64 • 0 1 1 1 B C H 4 (Z 2 × Z 2 ,U(1)) = (Z 2 ) 2 , • B C 1 B C B C B C 1 The T matrix of w 00 , from fixed B C • B C 1 B C B C point wave function 1 B C B C B 1 C B C 4 B C 1 M B C T = B C 1 B C i =1 B C 1 B C B C B 1 C B C B C 1 B C B C 1 B C B C 1 B C B C B C 1 @ A 1 (40)