MPS & PEPS as a Laboratory
for Condensed Matter
Mikel Sanz
MPQ, Germany
David Pérez-García
Uni. Complutense, Spain
Michael Wolf
Ignacio Cirac
Niels Bohr Ins., Denmark
MPQ, Germany
II Workshop on Quantum Information, Paraty (2009)
Outline
I.
Background
1.
Review about MPS/PEPS
•
“Injectivity”
2.
•
3.
Definition, theorems and conjectures.
Symmetries
•
II.
What, why, how,…
Definition and theorems
Applications to Condensed Matter
1.
Lieb-Schultz-Mattis (LSM) Theorem
•
2.
Theorem & proof, advantages.
Oshikawa-Tamanaya-Affleck (GLSM) Theorem
•
3.
Theorem, fractional quantization of the magn., existence of plateaux.
Magnetization vs Area Law
•
4.
Theorem, discussion about generality
Others
•
String order
Review of MPS
General
rank n  d n
MPS

rank n  D2

Non-critical short range interacting ham.
Hamiltonians with a unique gapped GS
Frustration-free hamiltonians
Review of MPS
Kraus Operators
d
Physical
Dimension
Bond
Dimension
D 

i
A
 Cd  CD  CD
D
Translational
 Invariant (TI) MPS

 
d
tr A
i1
i1

iN
AiN  i1
iN
“Injectivity”
Definition


d
 Aii1 A
 
i1


N
if N : dim
 
D

 , 1

tr A
d
X  
D
2
i1
i1
iN 1
AiN X i1
 : M D  C d 
N
Injectivity!


Are they general?

INJECTIVE!
iN 1
d
i1
iNN
 i1

iN
if N : rankN   D2
Random
MPS
Set
MPS
iN
“Injectivity”
if N : rankN   D2
Lemma
rankN 1  D2

never lost!
Injectivity reached
h 0
Definition(Parent Hamiltonian)
Assume
ker 
k 


v i i1
q
&
is a ground state (GS) of
i
Translation
Operator



If injectivity is reached by blocking
N
spins
&
gap
&
exp. clustering

, ai  0
H  Ti h
the

Thm.
h   ai v i v i

i1
q

&
kN
Symmetries
Definition
ugN   e
iN g


Thm.

G
a group
u g 
&
u  g  U g 
two representations of dimensions d & D
 
e
d
 A u   e 
i g
i
i1



g
ij
i
g
U g  A jU g 
Systematic Method to Compute
SU(2) Two-Body Hamiltonians
Density Matrix
L  
tr A
i1
Aj1 i1
AiL AjL
i1 iL
j1 j L

L
Hamiltonian
2s
h

 1 i j
Eigenvectors

n 
 



n 

2
tr  h  tr  h  0
2

 aij Si S j
Quadratic Form!!
iL j1
jL
Part II
Applications to Condensed
Matter Theory
Lieb-Schulz-Mattis (LSM) Theorem
Thm.
The gap over the GS of an SU(2) TI
Hamiltonian of a semi-integer spin vanishes
in the thermodynamic limit as 1/N.
Proof
1D
2D
Thm.
Lieb, Schulz & Mattis (1963)
52 pages
Hasting (2004), Nachtergaele (2005)
TI
SU(2) invariance
Uniqueness
injectivity
State
EASY PROOF!
for semi-integer
spins
Disadvantages
Advantages
Nothing about the gap
Thm enunciated for states instead Hamiltonians
Straightforwardly generalizable to 2D
Detailed control over the conditions
Oshikawa-Yamanaka-Affleck (GLSM)
Theorem
Thm. (1D General)
SU(2)
TI
U(1)
p - periodic
ps  m  Z
magnetization

Fractional
quantization of the
magnetization
Thm. (MPS)
U(1)
p - periodic
MPS has magnetization

ps  m  Z
Advantages
Again Hamiltonians to states
Generalizable to 2D
We can actually construct the examples
Oshikawa-Yamanaka-Affleck (GLSM)
Theorem
m
Example

10 particles
HMG  2i  i1  i  i2 
gap
i
Ground State

h
H  H MG  hS z

p2
m0



Gapped system:

General Scheme
U(1)-invariant MPS
With given p and m
 0
H  H0  hSz
Parent Hamiltonian H 0


Magnetization vs Area Law
Def. (Block Entropy)
A  trB 
S  tr A log A 


Thm. (MPS)
U(1)
p - periodic
magnetization m

B
A

Thermodynamic
limit

such that T   
 S    logp
m0
 p  log
S     log
1
2m
Magnetization vs Area Law
How general is this theorem?
6 particles
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8 particles
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7 particles
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Theoretical
Minimal
Random States
U(1)
TI
Spin 1/2
Block entropy
L/2 - L/2
Thanks for
your attention!!
Finally…