1. MPS & PEPS as a Laboratory for Condensed Matter
Mikel Sanz
MPQ, Germany
David Pérez-García
Uni. Complutense, Spain
Michael Wolf
Niels Bohr Ins., Denmark
Ignacio Cirac
MPQ, Germany
II Workshop on Quantum Information, Paraty (2009)

2. Booooring Outline Background Applications to Condensed Matter
Review about MPS/PEPS
What, why, how,…
“Injectivity”
Definition, theorems and conjectures.
Symmetries
Definition and theorems
Applications to Condensed Matter
Lieb-Schultz-Mattis (LSM) Theorem
Theorem & proof, advantages.
Oshikawa-Tamanaya-Affleck (GLSM) Theorem
Theorem, fractional quantization of the magn., existence of plateaux.
Magnetization vs Area Law
Theorem, discussion about generality
Others
String order

3. Review of MPS Non-critical short range interacting ham.
General
MPS
Non-critical short range interacting ham.
Hamiltonians with a unique gapped GS
Frustration-free hamiltonians

4. Review of MPS Kraus Operators Translational Invariant (TI) MPS
Physical
Dimension
Bond
Dimension
Translational Invariant (TI) MPS

5. “Injectivity” Injectivity! Definition Are they general? INJECTIVE!
Set
MPS
Random MPS
Are they general?
INJECTIVE!

6. “Injectivity” Lemma Definition (Parent Hamiltonian) Thm.
Injectivity reached
never lost!
Definition (Parent Hamiltonian)
Assume
&
is a ground state (GS) of
the
Thm.
If injectivity is reached by blocking spins &
& gap
& exp. clustering
Translation
Operator

7. Symmetries Definition Thm. a group &
two representations of dimensions d & D

8. Systematic Method to Compute SU(2) Two-Body Hamiltonians
Density Matrix
Hamiltonian
Eigenvectors
Quadratic Form!!

9. Applications to Condensed Matter Theory
Part II
Applications to Condensed Matter Theory

10. 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 Lieb, Schulz & Mattis (1963) pages
2D Hasting (2004), Nachtergaele (2005)
Thm. TI
SU(2) invariance
Uniqueness injectivity
State
for semi-integer spins
EASY PROOF!
Nothing about the gap
Disadvantages
Advantages
Thm enunciated for states instead Hamiltonians
Straightforwardly generalizable to 2D
Detailed control over the conditions

11. Oshikawa-Yamanaka-Affleck (GLSM) Theorem
Thm. (1D General)
SU(2) TI
U(1)
p - periodic
magnetization
Fractional quantization of the magnetization
COOL!
Thm. (MPS)
U(1)
p - periodic
MPS has magnetization
Again Hamiltonians to states
Generalizable to 2D
We can actually construct the examples
Advantages

12. Oshikawa-Yamanaka-Affleck (GLSM) Theorem
Example
10 particles
Ground State
Gapped system:
General Scheme
U(1)-invariant MPS
With given p and m
Parent Hamiltonian

13. Magnetization vs Area Law
Def. (Block Entropy)
Thm. (MPS)
U(1)
p - periodic
magnetization m
Thermodynamic
limit

14. Magnetization vs Area Law
How general is this theorem?
6 particles
7 particles
Theoretical
8 particles
Minimal
U(1) TI
Spin 1/2
Random States
Block entropy
L/2 - L/2

15. Thanks for
your attention!!
Finally…