Philippe Corboz, Institute for Theoretical Physics, University of Amsterdam

76 Pages·2016·12.93 MB·English

Introduction to iPEPS
Philippe Corboz, Institute for Theoretical Physics, University of Amsterdam
Overview: tensor networks in 1D and 2D
1D
MPS 1D MERA
Matrix-product state Multi-scale entanglement renormalization ansatz
and more
1 2 3 4 5 6 7 8 ‣ 1D tree tensor
network
Underlying ansatz of the
density-matrix renormalization ‣ correlator
group (DMRG) method product states
i i i i i i i i i i i i i i i i i i
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
‣ ...
2D
PEPS (TPS) 2D MERA
and more
projected entangled-pair state
(tensor product state)
‣ Entangled-
plaquette states
‣ 2D tree tensor
network
‣ String-bond states
‣ ...
Outline of the 2 lectures
‣ Part I: iPEPS ansatz
✦ Repetition: area law of the entanglement entropy
‣ Part II: Contraction of PEPS / iPEPS
✦ MPS-MPO approach, corner-transfer-matrix (CTM) method,
Tensor Renormalization Group (TRG), Tensor network renormalization (TNR)
✦ Simple examples to get started:
➡ solving the 2D classical Ising model with the CTM method
➡ simple 2D quantum case (D=2, rotational symmetric)
‣ Interlude: Example application: the Shastry-Sutherland model
‣ Part III: Optimization
‣ Part IV: Computational cost & benchmarks
‣ Part V: iPEPS applications
‣ Outlook & summary
PART I:
iPEPS ansatz
“Corner” of the Hilbert space
Ground states (local H)
★ GS of local H’s are less entangled than a
Hilbert random state in the Hilbert space
★ Area law of the entanglement entropy
space
Area law of the entanglement entropy
.
. . E . .
.
1D 2D
A
E A E
. . . . . . L
L
. . . . . .
# relevant states
Entanglement entropy S(A) = tr[⇥ log ⇥ ] = � log �
A A i i
� exp(S)
� �
⇠
i
�
General (random) state Ground state (local Hamiltonian)
d d 1
S(L) L (volume) S(L) L (area law)
�
⇠ ⇠
Critical ground states: 1D S(L) = const � = const
(all in 1D but not all in 2D)
⇥ exp(�L)
2D S(L) �L
�
1D S(L) log(L) �
⇠
S(L) L log(L)
2D
⇠
MPS & PEPS
1D
MPS
Matrix-product state
Bond dimension D
1 2 3 4 5 6 7 8
Physical indices (lattices sites)
S. R. White, PRL 69, 2863 (1992)
Fannes et al., CMP 144, 443 (1992)
Östlund, Rommer, PRL 75, 3537 (1995)
✓ Reproduces area-law in 1D
S(L) = const
MPS & PEPS
1D
MPS
Matrix-product state
Bond dimension D
➡
One bond can contribute
1 2 3 4 5 6 7 8
at most log(D) to the
A E
entanglement entropy
L
S(A) log(D) = const
rank(⇢ ) D
A
 
✓ Reproduces area-law in 1D
S(L) = const
MPS & PEPS
1D 2D
MPS can we use
an MPS?
Matrix-product state
Bond dimension D
1 2 3 4 5 6 7 8
L
Physical indices (lattices sites)
S. R. White, PRL 69, 2863 (1992)
Fannes et al., CMP 144, 443 (1992)
Östlund, Rommer, PRL 75, 3537 (1995)
!!! Area-law in 2D !!!
✓ Reproduces area-law in 1D S(L) L
�
S(L) = const D exp(L)
⇠
MPS & PEPS
1D 2D
PEPS (TPS)
MPS
projected entangled-pair state
Matrix-product state
(tensor product state)
D
Bond dimension D Bond dimension
1 2 3 4 5 6 7 8
Physical indices (lattices sites)
S. R. White, PRL 69, 2863 (1992)
Fannes et al., CMP 144, 443 (1992)
Östlund, Rommer, PRL 75, 3537 (1995)
F. Verstraete, J. I. Cirac, cond-mat/0407066
Nishino, Hieida, et al., Prog. Theor. Phys. 105 (2001).
Nishio, Maeshima, Gendiar, Nishino, cond-mat/0401115
✓ Reproduces area-law in 1D ✓ Reproduces area-law in 2D
S(L) = const
S(L) L
�