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

Renormalization of Tensor Network States

Tao Xiang

Institute of Physics Chinese Academy of Sciences txiang@iphy.ac.cn

• II. RG of Tensor Network States

Tensor-Network Ansatz

• f the ground state wavefunction

:

1D: Matrix Product State (MPS)

| =

𝑛1,…𝑛𝑀

(𝑛1, … 𝑛𝑀)|𝑛1, … 𝑛𝑀



m1 m2 m3 mL-2 mL-1 mL

…

Parameter number grows exponentially with L

𝑒𝑀 parameters

 

1

1 1

[ ]... [ ] ...

L

L L m m

Tr A m A m m m   

𝑒𝐸2𝑀 parameters m1 m2 m3 … … mL-1 mL A[m2 ]

 

D d

Parameter number grows linearly with L

Virtual basis state

MPS is the wave function generated by the DMRG

 

1

1 1

[ ]... [ ] ... 1 2 [ 1] [0] [1] 1 2

L

L L m m

Tr A m A m m m A A A                         



 

2 1 1

1 1 2 2 3 3

i i i i i

H S S S S

 

          



Affleck, Kennedy, Lieb, Tasaki, PRL 59, 799 (1987)

Example：S=1 AKLT valence bond solid state

A[m]

  m

virtual S=1/2 spin A[m] : To project two virtual S=1/2 states,  and ,

• nto a S=1 state m

A[m]

  m

=

1 = 1 2  1 2

m1 m2 m3 … … mL-1 mL

 

1

1 1

[ ]... [ ] ...

L

L L m m

Tr A m A m m m   

A[m]

  m

Matrix product state (MPS)

Gauge Invariance

MPS wavefunction is unchanged if one replaces A[m] by

𝐵 𝑛 → 𝐵′ 𝑛 = 𝑄𝐵 𝑛 𝑄-1

MPS as a Projection of 2D Tensor-Network Model Ai

Matrix Product Operator

𝑛1

′

𝑛2

′

𝑛3

′ … … 𝑛𝑀−1 ′

𝑛𝑀

′

M

 

𝑛1 𝑛2 𝑛3 … … 𝑛𝑀−1 𝑛𝑀

𝑃 =

𝑛

𝑈𝑠 𝑁 𝑛1, 𝑛1

′ ⋯ 𝑁 𝑛𝑀, 𝑛𝑀 ′ |

𝑛1 ⋯ 𝑛𝑀 𝑛1

′

⋯𝑛𝑀

′ |

=

Ground state eigen-operator: 𝑃＝｜  ｜

𝑦 𝑦′

2D: Projected Entangled Pair State (PEPS)

𝑈𝑦𝑦′𝑧𝑧′ [𝑛] =

𝑧 y' 𝑛 Physical basis Local tensor Virtual basis D

Key parameter: virtual basis dimension D Virtual spins at each bond form a maximally entangled state

 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

Tensor-Network State as a Variational Ansatz

PEPS: exact representation of Valence Bond Solid

𝑈𝑏𝑐𝑑𝑒 [𝑛𝑗] =

𝑏 𝑐 𝑑 d 𝑛𝑗 Physical state Virtual basis state

S = 2

𝐼 =

𝑗𝑘

𝑄

4(𝑗, 𝑘) To project two S=2 spins on sites i and j onto a total spin S=4 state

1/2

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

Bond Dimension Dependence of Physical Quantities

• 1. Determine the local tensors
• 2. Evaluate the physical observables using the TRG
• r other tensor network RG methods

Two Problems Need to Be Solved

Evaluation of Expectation Values

| |

𝐸 𝑒 = 𝐸2

|

𝐸2

How Large Is the Virtual Bond Dimension Needed?

|

𝐸2 𝑒 = 𝐸2

 In the DMRG or all tensor-related approached, small physical dimension d is much earlier to study than a larger 𝑒 (accuracy drops quickly with 𝑒)  The virtual bond dimension needed to

• btain a converged result is roughly of
• rder 𝑒, i.e. 𝐸2, of course, the larger the

better

Computational Cost

Meth thod

• d

CP CPU U Time Minimum mum Memory

• ry

TMRG/CTMRG 𝐸12𝑀 𝐸10 TEBD 𝐸12𝑀 𝐸10 TRG 𝐸12ln𝑀 𝐸8 SRG 𝐸12ln𝑀 𝐸8 HOTRG 𝐸14ln𝑀 𝐸8 HOSRG 𝐸16ln𝑀 𝐸12 TNR 𝐸14ln𝑀 𝐸10 Loop-TNR 𝐸12ln𝑀 𝐸8

𝐸: bond dimension of PEPS bond dimension kept ~ 𝐸2 𝑀: lattice size

Tensor Network States in the Frustrated Lattices

• Z. Y. Xie et al, PRX 4, 011025 (2014)

Geometric frustration

1 1 N i i i

H J S S 



 



J J J Quantum frustration

e.g. S=1 bilinear-biquadratic Heisenberg model

 

2

cos sin

i j i j ij

H S S S S            



Two Kinds of Frustrations

More than two-body correlations/entanglements are important

PEPS on Kagome or Other Frustrated Lattices

 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

Max ( ) ~ 1 Max ( ) < 10-6

Cancellation in the PEPS

1 1 N i i i

H J S S 



 



Projected Entangled Simplex States (PESS)

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

• Z. Y. Xie et al, PRX 4, 011025 (2014)

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

𝐵𝑏𝑐[𝜏] = 1 1 2 𝑏 𝑐 𝜏 antisymmetric tensor C-G coefficients Local tensors Pn : projection operator Parent Hamiltonians

• r

Projection tensor Simplex tensor

To enlarge each simplex so that it contains more physical spins 5-PESS: a decorated square lattice 9-PESS: a honeycomb lattice

Larger Simplex PESS

• P. Corboz et al, PRB 86,

041106 (2012).

PESS on Triangular Lattice

PEPS PESS Order of local tensors: dD6 Simplex tensor: D3 Projection tensor: dD3

PESS on Square Lattice

J1 only J1- J2 model 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

𝐹 =  𝐼  |

 Accurate  Cost is high  D is generally less than 13 without using symmetries

[ ] [ ]

i i i i i i j j j

x y z x y z i x y z j i j i black j white

Tr A m B m m m   

 

 



Bond vectors: measure approximately the “entanglement”

• n the corresponding bonds

Simple Update: Entanglement Mean-Field Approach

Simple Update: Entanglement Mean-Field Approach state ground lim  

   H

e 



The local tensors are determined by projection

 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)

ij x y z ij ij i j

H H H H H H JS S      



Heisenberg model

Simple Update: Imaginary Time Evolution

 

state ground lim state ground lim    

      M H M H

e e

  

• 1. One iteration
• 2. Repeat the above iteration until converged

2 1 2 1

~         

  

z y x

H H H

e e e

  

Trotter-Suzuki decomposition

) , , ( ) (

, 2

z y x H H

• e

e e e

black i i i H H H H

x y z

   



     

 

     

Imaginary Time Evolution

,

ˆ

[ ] [ ]

i j x i i i i i i j j j

H H i j i j x y z x y y i x y y j i j i black j i x

e Tr m m e m m A m B m m m

 

  

    

     



Step I Step II Step III

SVD: singular value decomposition

Step I Step II Step III

Truncate basis space

One Step of Evolution

 To use bond vector  as effective fields to take into account the environment contribution  The projection is done locally. This keeps the locality of wavefunction, making the calculation efficient  Truncation error not accumulated

,

ˆ

[ ] [ ]

i j x i i i i i i j j j

H H i j i j x y z x y y i x y y j i j i black j i x

e Tr m m e m m A m B m m m

 

  

    

     



One Step of Evolution

Step I Step II Step III

SVD: singular value decomposition

1 1 N i i i

H J S S 



 



Application: Heisenberg Model on the Husimi Lattice

 Locally similar to the Kagome lattice, but less frustrated  Helpful to understand the Kagome Heisenberg model

Kagome lattice Husimi lattice

Heisenberg Model on the Husimi Lattice

Husimi lattice

1 1 N i i i

H J S S 



 



 PESS is defined on a unfrustrated lattice and can be easily studied  Simple update is rigorous

Is the Ground State a Spin Liquid?

Kagome lattice

If yes, then the g.s of the Kagome Heisenberg should also be a spin liquid since the Kagome is more frustrated than the Husimi lattice

1 1 N i i i

H J S S 



 



Husimi lattice

E0 converges algebraically with D  the excitation is gapless

Ground State Energy

 1200 Neel ordered at any finite 𝐸, vanishes in the limit 𝐸 → ∞  𝑁 ~ 𝐸𝛽 with 𝛽 = −0.588(2))

Magnetization: Ground State is a Gapless Spin Liquid

Kagome Lattice

• Z. Y. Xie et al, PRX 4, 011025 (2014)

1 1 N i i i

H J S S 



 



Issue under debate: Is the ground state a spin liquid?

S=1/2 Kagome Heisenberg Model

Possible Ground States

Valence bond crystal

• R. R. P. Singh and D. A. Huse, PRB 77, 144415 (2008) series expansion
• G. Evenbly and G. Vidal, PRL 104, 187203 (2010) MERA
• Y. Iqbal, F. Becca, and D. Poilblanc, PRB 83, 100404 (2011) VMC

Z2 Gapped spin liquid

• S. Yan, D. A. Huse, and S. R. White, Science 332, 1173 (2011) DMRG

Depenbrock,McCulloch,Schollwock, PRL 109, 067201 (2012) DMRG ………….

U(1) Gapless spin liquid

• Y. Ran, M. Hermele, P. A. Lee, and X.-G. Wen, PRL 98, 117205 (2007) Gutzwiller
• Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, PRB 87, 060405 (2013) VMC+Lanczos

………….

Difficulty: Lack of Good Numerical Methods

 Mean field or variational approach:

need accurate guess of trial wavefunction

 Quantum Monte Carlo:

suffers from the minus sign problem on frustrated systems

 Density Matrix Renormalization Group (DMRG):

limited to small lattice systems (area law), the number of states need to be retained grows exponentially with the circumference

 Upper bound: iDMRG (cylinder Ly=12), D=5000, E=-0.4332  Yan et al: -0.4379(3)  Depenbrock et al: -0.4386(5), D=16000, Ly=17

DMRG results

Valence bond solid Gapped spin liquid Gapless spin liquid

Ground state energy of the Kagome Heisenberg model

DMRG

Valence bond solid Gapped spin liquid Gapless spin liquid

3-PESS

Ground state energy of the Kagome Heisenberg model

Ground state energy of the Kagome Heisenberg model

The ground state energy converges algebraically with D, indicating that the system is gapless.

Comparison between Kagome and Husimi Lattices

Bond Dimension D

 No long-range magnetic order in the infinite D limit  The ground state is more likely a gapless spin liquid

120 Degree Neel Magnetization

Deconfined Quantum Point

 

2

cos sin

i j i j ij

H S S S S            



• Honeycomb lattice: each site has only 3 neighbors,

quantum fluctuation is strong

• Quantum Monte Carlo has the minus sign problem

when sin > 0

S = 1 Bilinear-biquadratic Heisenberg model on Honeycomb Lattice

 

2

cos sin

i j i j ij

H S S S S            





FQ

IV Ferro-magnetic Antiferro-magnetic Ferro-quadrupolar Antiferro-quadrupolar

( )i

i

Q 

i

Q

AFQ Classical phase diagram Quantum fluctuation may lead to a deconfined quantum critical point

Ground State Phase Diagram in the Classical Limit

2nd order

There are four phases, but the AFQ phase is killed by quantum fluctuation d  0.19

Ground State Energy and Orders

 d

FQ

Magnetic Orders

Continuous phase transition point: deconfined critical point?

red bon black o d b nd

1 2

i j i j

S S P S S    

 

d

Plaquette Valence Bond Crystal

 Proper tensor-network wave function in treating a frustrated quantum lattice system  The simple update an approximate and efficient algorithm for determining the local tensors  Kagome Heisenberg model is likely to be a gapless spin liquid, but more study needed

Summary