MPS PEPS as a Laboratory for Condensed Matter
MPS & PEPS as a Laboratory for Condensed Matter Mikel Sanz MPQ, Germany David Pérez-García Uni. Complutense, Spain Michael Wolf Ignacio Cirac Niels Bohr Ins. , Denmark MPQ, Germany II Workshop on Quantum Information, Paraty (2009)
Outline I. g n ri Background 1. Review about MPS/PEPS • 2. “Injectivity” • 3. II. Definition, theorems and conjectures. Symmetries • o o o Bo What, why, how, … Definition and theorems Applications to Condensed Matter 1. Lieb-Schultz-Mattis (LSM) Theorem • 2. Theorem & proof, advantages. Oshikawa-Tamanaya-Affleck (GLSM) Theorem • 3. Theorem, fractional quantization of the magn. , existence of plateaux. Magnetization vs Area Law I. Theorem, discussion about generality Others 1. String order
Review of MPS General MPS Non-critical short range interacting ham. Hamiltonians with a unique gapped GS Frustration-free hamiltonians
Review of MPS Kraus Operators Physical Dimension Bond Dimension Translational Invariant (TI) MPS
“Injectivity” Definition Injectivity! Are they general? INJECTIVE! Random MPS Set MPS
“Injectivity” Lemma never lost! Injectivity reached Definition (Parent Hamiltonian) Assume & is a ground state (GS) of the Thm. Translation Operator If injectivity is reached by blocking spins & gap & exp. clustering &
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!!
Part II Applications to Condensed Matter Theory
Lieb-Schulz-Mattis (LSM) Theorem Thm. Proof Thm. The gap over the GS of an SU(2) TI Hamiltonian of a semi-integer spin vanishes in thermodynamic limit as 1/N. 1 D 2 D Lieb, Schulz & Mattis (1963) 52 pages Hasting (2004), Nachtergaele (2005) TI SU(2) invariance Uniqueness injectivity State EASY PROOF! for semi-integer spins Disadvantages Advantages Nothing about the gap Thm enunciated for states instead Hamiltonians Straightforwardly generalizable to 2 D Detailed control over the conditions
Oshikawa-Yamanaka-Affleck (GLSM) Theorem Thm. (1 D General) SU(2) TI U(1) p - periodic magnetization Fractional quantization of the magnetization ! L OO C Thm. (MPS) U(1) p - periodic MPS has magnetization Advantages Again Hamiltonians to states Generalizable to 2 D We can actually construct the examples
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 theorem? 6 particles 8 particles 7 particles Theoretical Minimal Random States U(1) TI Spin 1/2 Block entropy L/2 - L/2
Thanks for your attention!! Finally…
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