MPS & PEPS as a Laboratory for Condensed Matter
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Transcript of MPS & PEPS as a Laboratory for Condensed Matter
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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)
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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
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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
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Review of MPS
Kraus Operators
BondDimension
PhysicalDimension
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d
€
D
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D
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Aαβi ∈ Cd ⊗CD ⊗CD
Translational Invariant (TI) MPS
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Ψ = tr Ai1L AiN[ ] i1L iN
i1L iN
d
∑
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“Injectivity”
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N €
Ai{ }i=1
d
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Ai1L AiN
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Ψαβ = α Ai1L AiN
β i1L iN
i1L iN =1
d
∑
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α
€
β
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if ∃N : dim Ψαβ{ }α ,β =1
D= D2
€
if ∃N : rank ρ N( ) = D2
Injectivity!
€
Γ X( ) = tr Ai1L AiN
X[ ] i1L iN
i1L iN =1
d
∑
Γ : MD → Cd( )
⊗N
Are they general?
Definition
SetMPSINJECTIVE!
Random MPS
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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
∑
€
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 &
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N
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k > N
& gap
& exp. clustering
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h ≥ 0
“Injectivity”
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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
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U g( )
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G
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u g( )a group & two representations of dimensions d & D
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u g( )
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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
€
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!!
€
L
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Part II
Applications to Condensed Matter Theory
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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
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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
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Oshikawa-Yamanaka-Affleck (GLSM) Theorem
€
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
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Magnetization vs Area Law
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A
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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
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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
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Thanks for your attention!!
Finally…