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MPS & PEPS as a Laboratory for Condensed Matter

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

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MPS & PEPS as a Laboratory for Condensed Matter

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

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

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

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

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

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

  9. Part II Applications to Condensed Matter Theory

  10. Thm. TI SU(2) invariance Uniqueness injectivity EASY PROOF! State 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 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)

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

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

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

  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…

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