Applications of Matrix-Product States in Condensed Matter and AMO

129 Pages·2016·11.79 MB·English

Applications of Matrix-Product States
in Condensed Matter and AMO
Fabian Heidrich-Meisner
Ludwig-Maximilians-University Munich
Tensor16, MPIPKS Dresden, Nov. 4, 2016
Outline
1) Brief recap: MPS methods
and entanglement
H = E
| i | i
2) Overview over applications
Schmidt decomposition
m r
3) Entanglement scaling in
⌧
= s ↵ ↵
time-dependent simulations
↵ A B
! | i | i | i

↵
X
4) Nonequilibrium dynamics in optical
Entanglement
lattices
S = tr[⇢ ln⇢ ]
vN A A
5) Transport in nano-structures �
Hilbert-space
6) Finite temperatures
7) Electron-phonon problems
8) Outlook
sub-space of
m-dimensional MPS
Recap: Basics of MPS methods
“Structure” of wave-functions
Local degrees of freedom: � = ; site i
i
| i | "i | #i
= c � , � , . . . , �
� ,� ,� ,...,� 1 2 L
| i 1 2 3 L | i
� ,...,�
1 L
X
c
Non-local object, � ,� ,� ,...,�
1 2 3 L 2L coefficients
tensor of rank L
�
�
1
L
� �
2 . . . L 1
�
Use the methodology of SVD to obtain a local structure!
Key idea of density matrix renormalization group
Part A: “System”, length L Part B: “System”, length L
A B
N ,N m r
A B
⌧
= i j = s ↵ ↵
ij A B m ↵ A B
| i | i ⌦ | i ! | i ⇡ | i | i | i
i,j ↵
X X
Keep only m ~ 103 states with largest singular values
2
⇢ = s ↵ ↵
→ L ~ 102 ... 103 A ↵| iAAh |
↵
X
→ Ground states
& time evolution
Hilbert-space
→ T>0 in 1D
Sub-space of
→ Highly accurate
matrix-product states:
Small entanglement
White, Phys. Rev. Lett. 69, 2863 (1992)
Schollwöck, Annals of Physics 326, 96 (2011)
Matrix product states
Matrix dimensions (d=2) DMRG: Approximate wave-functions
with finite-dimensional MPS
Minimize energy by variation
over single A-matrices
Hilbert-space
� i 1
A L i 1 : (m d )
� � �
⇥
sub-space of
m-dimensional MPS
Connection to entanglement
Reduced DM:
2
⇢ = s ↵ ↵
A ↵ AA
| i h |
↵
X
Von-Neumann entropy:
2 2
S = tr[⇢ log ⇢ ] = s log s
vN A 2 A ↵ 2 ↵
� �
↵
X
Product state, no entanglement
= i j r = 1 : S = 0
A B vN
| i | i ⌦ | i
Worst case: Flat spectrum Estimate of “costs:
1
2
s = const =
S (L)
↵ m e vN
m
! /
Area laws
Estimate of “costs”:
2 2
S = tr[⇢ log ⇢ ] = s log s
vN A 2 A ↵ 2 ↵
� �
↵
S (L)
m e vN X
! /
Entanglement scaling for most states
Desired: costs should
not scale exponentially in L!
D
S L volume
vN
⇠ ⇠
B
A
L
Area laws
Estimate of “costs”:
2 2
S = tr[⇢ log ⇢ ] = s log s
vN A 2 A ↵ 2 ↵
� �
↵
S (L)
m e vN X
! /
Area law: Ground-states of short-ranged,
gapped Hamiltonians in D dimensions scales
Desired: costs should
with surface area not volume
not scale exponentially in L!
D 1
S L
B vN �
⇠
1D : m const. L > ⇠
⇠
A
L
z z
S S exp( i j /⇠)
i j
h i / �| � |
L
2D : m e
⇠
Review: Eisert, Cramer, Plenio Rev. Mod. Phys. 82, 277 (2010)
How to beat the area law in 2D: See P. Corboz’ lecture
Overview over applications