Renormalization of Tensor Network States II. RG of Tensor Network - PowerPoint PPT Presentation

Renormalization of Tensor Network States II. RG of Tensor Network States Tao Xiang Institute of Physics Chinese Academy of Sciences txiang@iphy.ac.cn

Tensor-Network Ansatz of the ground state wavefunction

1D: Matrix Product State (MPS) : m L-2 m 3 m 1 m 2 m 3 … … m L-1 m L … m 2 m L-1 m 1   d m L  A  [ m 2 ] D Virtual basis state      |  =  (𝑛 1 , … 𝑛 𝑀 )|𝑛 1 , … 𝑛 𝑀  Tr A m [ ]... [ A m ] m ... m 1 L 1 L 𝑛 1 ,…𝑛 𝑀 m m 1 L 𝑒𝐸 2 𝑀 parameters 𝑒 𝑀 parameters Parameter number grows Parameter number grows linearly with L exponentially with L MPS is the wave function generated by the DMRG

Example ： S=1 AKLT valence bond solid state 1 = 1 2  1   1 1 2        2  H S S S S     2  i i 1 i i 1  2 3 3 i m m     = A  [ m ] A  [ m ] virtual S=1/2 spin      Tr A m [ ]... [ A m ] m ... m 1 L 1 L A  [ m ] : m m 1 L To project two virtual S=1/2 states,  and  ,        0 0 1 0 0 2     onto a S=1 state m     [ 1] [0]   [1] A A A   0 1     2 0 0 0 Affleck, Kennedy, Lieb, Tasaki, PRL 59 , 799 (1987)

Gauge Invariance MPS wavefunction is unchanged if one replaces A[m] by 𝐵 𝑛 → 𝐵′ 𝑛 = 𝑄𝐵 𝑛 𝑄 -1 Matrix product state (MPS)      m Tr A m [ ]... [ A m ] m ... m 1 L 1 L   m m 1 L A  [ m ] m 1 m 2 m 3 … … m L-1 m L

MPS as a Projection of 2D Tensor-Network Model A i

Matrix Product Operator 𝑛 1 𝑛 2 𝑛 3 … … 𝑛 𝑀−1 𝑛 𝑀   M  ′ … … 𝑛 𝑀−1 ′ ′ ′ ′ 𝑛 1 𝑛 2 𝑛 3 𝑛 𝑀 ′ ⋯ 𝑁 𝑛 𝑀 , 𝑛 𝑀 ′ | ′ | ′ 𝑃 = 𝑈𝑠 𝑁 𝑛 1 , 𝑛 1 𝑛 1 ⋯ 𝑛 𝑀 𝑛 1 ⋯ 𝑛 𝑀 𝑛 Ground state eigen-operator:   ｜ 𝑃 ＝｜ =

2D: Projected Entangled Pair State (PEPS) Virtual spins at each bond form a maximally entangled state Physical Local Virtual basis basis tensor 𝑛 𝑧 𝑈 𝑦𝑦 ′ 𝑧𝑧 ′ [𝑛 ] = 𝑦 𝑦′ D y' Key parameter: virtual basis dimension D

Tensor-Network State as a Variational Ansatz  Tensor product states H. Niggemann and J. Zittarz, Z. Phys. B 101, 289 (1996) G. Sierra and M. Martin-Delgado, 1998  Variational approach: Nishino, Okunishi, Maeshima, Hieida, Akutsu, Gendiar (since 1998)  Projected entangled pair states (PEPS) : Area law obeys F. Verstraete and J. Cirac, cond-mat/0407066

PEPS: exact representation of Valence Bond Solid S = 2 Physical state Virtual basis state 𝑑 𝑏 𝑐 𝑈 𝑏𝑐𝑑𝑒 [𝑛 𝑗 ] = 𝑛 𝑗 d 1/2 𝐼 = 𝑄 4 (𝑗, 𝑘) 𝑗𝑘 To project two S=2 spins on sites i and j onto a total spin S=4 state

Bond Dimension Dependence of Physical Quantities Bond dimension dependence of the ground state energy and magnetization  Gapless: Power law dependent on D  Gapped: Exponential law dependent on D

Two Problems Need to Be Solved 1. Determine the local tensors 2. Evaluate the physical observables using the TRG or other tensor network RG methods

Evaluation of Expectation Values 𝑒 = 𝐸 2 𝐸  | 𝐸 2  |  | 

How Large Is the Virtual Bond Dimension Needed? 𝑒 = 𝐸 2  In the DMRG or all tensor-related approached, small physical dimension d 𝐸 2 is much earlier to study than a larger 𝑒 (accuracy drops quickly with 𝑒 )  The virtual bond dimension needed to obtain a converged result is roughly of order 𝑒 , i.e. 𝐸 2 , of course, the larger the  |  better

Computational Cost Meth thod od CP CPU U Time Minimum mum Memory ory TMRG/CTMRG 𝐸 12 𝑀 𝐸 10 TEBD 𝐸 12 𝑀 𝐸 10 TRG 𝐸 12 ln𝑀 𝐸 8 SRG 𝐸 12 ln𝑀 𝐸 8 HOTRG 𝐸 14 ln𝑀 𝐸 8 HOSRG 𝐸 16 ln𝑀 𝐸 12 TNR 𝐸 14 ln𝑀 𝐸 10 Loop-TNR 𝐸 12 ln𝑀 𝐸 8 𝐸 : bond dimension of PEPS bond dimension kept ~ 𝐸 2 𝑀 : lattice size

Tensor Network States in the Frustrated Lattices Z. Y. Xie et al, PRX 4, 011025 (2014)

Two Kinds of Frustrations More than two-body correlations/entanglements are important Geometric frustration J J N    H J S S  i i 1  i 1 J Quantum frustration e.g. S=1 bilinear-biquadratic Heisenberg model      2       H cos S S sin S S     i j i j ij

PEPS on Kagome or Other Frustrated Lattices Max ( ) ~ 1 Max ( ) < 10 -6  There is a serious cancellation in the tensor elements if three tensors on a simplex (triangle here) are contracted  3-body (or more-body) entanglement is important

Cancellation in the PEPS N    H J S S  1 i i  1 i

Projected Entangled Simplex States (PESS) Z. Y. Xie et al, PRX 4, 011025 (2014) Projection tensor Simplex tensor  Virtual spins at each simplex form a maximally entangled state  Remove the geometry frustration: The PESS is defined on the decorated honeycomb lattice  Only 3 virtual bonds, low cost

PESS as an exact representation of Simplex Solid States D. P. Arovas, Phys. Rev. B 77 , 104404 (2008) Example: S = 2 spin model on the Kagome lattice A S = 2 spin is a symmetric superposition of two virtual S = 1 spins Three virtual spins at each triangle form a spin singlet Projection tensor Simplex tensor

S=2 Simplex Solid State on the Kagome Lattice Local tensors Parent Hamiltonians or antisymmetric tensor 𝐵 𝑏𝑐 [𝜏] = 1 1 2 C-G coefficients 𝑏 𝑐 𝜏 P n : projection operator Projection tensor Simplex tensor

Larger Simplex PESS To enlarge each simplex so that it contains more physical spins 5-PESS: a decorated square lattice P. Corboz et al, PRB 86 , 041106 (2012). 9-PESS: a honeycomb lattice

PESS on Triangular Lattice PEPS PESS dD 6 Simplex tensor: D 3 Order of local tensors: Projection tensor: dD 3

PESS on Square Lattice J 1 - J 2 model J 1 only Vertex-sharing Edge-sharing

How to Determine the Tensor-Network Wave Function

Determination of Tensor Network Wavefunction 1. Imaginary time evolution Simple update (entanglement mean-field approach)  Jiang, Weng, Xiang, PRL 101 , 090603 (2008) the solution can be used as the initial input of local tensors in I or in the full update calculation Full update  Murg, Verstraete, Cirac, PRA 75 , 033605 (2007) 2. Minimize the ground state energy Nishino et al, Nuclear Physics B 575 [FS] 504 (2000) F. Verstraete and J. Cirac, cond-mat/0407066

Variational Minimization of Ground State Energy Determine local tensors by minimizing the ground state energy  𝐼  𝐹 =  |   Accurate  Cost is high  D is generally less than 13 without using symmetries

Simple Update: Entanglement Mean-Field Approach       Tr A [ m B ] [ m ] m m x y z x y z i x y z j i j i i i i i i j j j  i black  j white Bond vectors: measure approximately the “entanglement” on the corresponding bonds

Simple Update: Entanglement Mean-Field Approach The local tensors are determined by projection e     H lim ground state     Converge fast  D as large as 100 can be calculated (more if symmetry is considered)  Exact on the Bethe lattice Li, von Delft, Xiang, PRB 86 , 195137 (2012)

Canonical Form on the Bethe Lattice Li, von Delft, Xiang, PRB 86 , 195137 (2012)

Simple Update: Imaginary Time Evolution      H H H H H Heisenberg model ij x y z ij   H JS S ij i j     H lim e ground state      M     H lim e ground state   M

Imaginary Time Evolution            H H H H 2 ( ) e e e y e o z x Trotter-Suzuki     H H ( x , y , z ) decomposition    i , i  i black 1. One iteration      H e x 1 0      H e y 2 1 ~      H e z 0 2 2. Repeat the above iteration until converged

One Step of Evolution               H H e Tr m m e i j , m m A [ m B ] [ m ] m m x i j i j x y z x y y i x y y j i j i i i i i i j j j  i black   ˆ j i x Step I Step I Step II Step II Step III Truncate basis space Step III SVD: singular value decomposition

One Step of Evolution               H H e Tr m m e i j , m m A [ m B ] [ m ] m m x i j i j x y z x y y i x y y j i j i i i i i i j j j  i black   ˆ j i x  To use bond vector  as effective fields to take into account the Step I environment contribution  The projection is done locally. This Step II keeps the locality of wavefunction, making the calculation efficient Step III  Truncation error not accumulated SVD: singular value decomposition