MPS & PEPS as a Laboratory for Condensed Matter

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Presentation transcript:

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)

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

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

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

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

“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

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

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

Applications to Condensed Matter Theory 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 Lieb, Schulz & Mattis (1963) 52 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

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

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

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

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

Thanks for your attention!! Finally…