MPS & PEPS as a Laboratory for Condensed Matter

Mikel SanzMPQ, Germany

Ignacio CiracMPQ, Germany

Michael WolfNiels Bohr Ins., Denmark

David Pérez-GarcíaUni. Complutense, Spain

II Workshop on Quantum Information, Paraty (2009)

Outline

I. Background1. Review about MPS/PEPS

• What, why, how,…

2. “Injectivity”• Definition, theorems and conjectures.

3. Symmetries• Definition and theorems

II. Applications to Condensed Matter1. Lieb-Schultz-Mattis (LSM) Theorem

• Theorem & proof, advantages.

2. Oshikawa-Tamanaya-Affleck (GLSM) Theorem• Theorem, fractional quantization of the magn., existence of plateaux.

3. Magnetization vs Area LawI. Theorem, discussion about generality

I. Others1. String order

Booooring

Review of MPS

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rank ρ n( ) ≈ dn

General

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rank ρ n( ) ≤ D2

MPS

Non-critical short range interacting ham.Hamiltonians with a unique gapped GS

Frustration-free hamiltonians

Review of MPS

Kraus Operators

BondDimension

PhysicalDimension

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d

€

D

€

D

€

Aαβi ∈ Cd ⊗CD ⊗CD

Translational Invariant (TI) MPS

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Ψ = tr Ai1L AiN[ ] i1L iN

i1L iN

d

∑

“Injectivity”

€

N €

Ai{ }i=1

d

€

Ai1L AiN

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Ψαβ = α Ai1L AiN

β i1L iN

i1L iN =1

d

∑

€

α

€

β

€

if ∃N : dim Ψαβ{ }α ,β =1

D= D2

€

if ∃N : rank ρ N( ) = D2

Injectivity!

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Γ X( ) = tr Ai1L AiN

X[ ] i1L iN

i1L iN =1

d

∑

Γ : MD → Cd( )

⊗N

Are they general?

Definition

SetMPSINJECTIVE!

Random MPS

Lemma

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if ∃N : rank ρ N( ) = D2 ⇒ rank ρ N +1( ) = D2

Injectivity reached never lost!

Definition (Parent Hamiltonian)

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ker ρ k( ) = v i{ }i=1

qAssume &

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h = ai v i v i , ai > 0i=1

q

∑

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ρ is a ground state (GS) of

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H = Ti h( )i

∑Translation

Operator

the

Thm. If injectivity is reached by blocking spins &

€

N

€

k > N

& gap

& exp. clustering

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h ≥ 0

“Injectivity”

Symmetries

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ug⊗N Ψ = e iNθ g Ψ

Definition

Thm.

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Ai uijg( )

i=1

d

∑ = e iθ g U g( )A jUg( )+

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e iθ g

€

U g( )

€

G

€

u g( )a group & two representations of dimensions d & D

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u g( )

Systematic Method to ComputeSU(2) Two-Body Hamiltonians

Density Matrix

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ρ L( ) = tr Ai1L AiL

ΛA jL

+ L A j1

+[ ] i1L iL j1L jL

i1L iLj1L jL

∑

Hamiltonian

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h = aijα( )

r S i o

r S j( )

α

i< j

∑α =1

2s

∑

Eigenvectors

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tr ρ n( )h2[ ] − tr ρ n( )h[ ]

2= 0 Quadratic Form!!

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L

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.

Proof1D Lieb, Schulz & Mattis (1963) 52 pages2D Hasting (2004), Nachtergaele (2005)

for semi-integer spins

Thm. TI

SU(2) invariance

Uniqueness injectivity

State EASY PROOF!

Nothing about the gap

Disadvantages Advantages

Thm enunciated for states instead HamiltoniansStraightforwardly generalizable to 2DDetailed control over the conditions

Oshikawa-Yamanaka-Affleck (GLSM) Theorem

U(1)

p - periodic

SU(2)

TImagnetization

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p s − m( )∈ Z

Fractional quantization of the

magnetization COOL!

Thm. (1D General)

Thm. (MPS)

U(1)

p - periodicMPS has magnetization

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p s − m( )∈ Z

Again Hamiltonians to statesGeneralizable to 2DWe can actually construct the examples

Advantages

Oshikawa-Yamanaka-Affleck (GLSM) Theorem

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m

€

h€

gap

10 particles

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H = HMG − hSz

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HMG = 2v σ i ⋅

v σ i+1 +

v σ i ⋅

v σ i+2( )

i

∑

Example

Ground State

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Δ > 0Gapped system: €

p = 2

m = 0

General Scheme

U(1)-invariant MPSWith given p and m

Parent Hamiltonian

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H0

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H = H0 − hSz

Magnetization vs Area Law

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A

€

B

Def. (Block Entropy)

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ρA = trB ρ

S = tr ρ A logρ A[ ]

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S Ψ Ψ( ) ≥ log p ≥ log 1

2m€

m ≠ 0

Thm. (MPS)

U(1)

p - periodic

magnetization m

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Ψ such that T Ψ = Ψ ⇒ S Ψ Ψ( ) ≥ log p

Thermodynamiclimit

Magnetization vs Area Law

How general is this theorem?

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6 particles 7 particles

8 particlesTheoretical

Minimal

Random StatesU(1)

TI

Spin 1/2

Block entropyL/2 - L/2

Thanks for your attention!!

Finally…