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Computational approaches for quantum many-body systems

Learn about computational methods for quantum many-body systems in this course. Topics include quantum Ising model, tensor structure, Monte Carlo methods, entanglement, tensor network states, and more.

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Computational approaches for quantum many-body systems

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  1. Computational approaches for quantum many-body systems HGSFP Graduate Days SS2019 Martin Gärttner

  2. Course overview Lecture 1: Introduction to many-body spin systems Quantum Isingmodel,Bloch sphere, tensor structure, exact diagonalization Lecture 2: Collective spin models LMG model, symmetry, semi-classical methods,Monte Carlo Lecture 3: Entanglement Mixed states, partial trace, Schmidt decomposition Lecture 4: Tensor network states Area laws, matrix product states,tensor contraction, AKLT model Lecture 5: DMRG and other variational approaches Energy minimization, PEPS and MERA, neural quantum states

  3. Learning goals After today you will be able to … • … explain the matrixproduct state ansatz. • … translate between tensors and their diagrammatic representation. • … derive that the bond dimension bounds the Schmidt rank. • … evaluate observables for MPS, especially with periodic boundaries.

  4. Unreasonably large Hilbert space Hilbert space: Physical corner Martin Gärttner - BDC 2018

  5. Unreasonably large Hilbert space Physical states: Can be reached through local Hamiltonians in polynomial time. Ground states of local gapped Hamiltonians. Locally entangled. Area law states Martin Gärttner - BDC 2018

  6. Area law vs. volume law volume law area law Ground states of local gapped Hamiltonians Only k-body terms, k~1 Excitation gap does not go to zero as Area law → half-chain entropy doesn’t grow with system size 1D

  7. Representing quantum states: MPS • Consider N spins in 1D parameters

  8. Representing quantum states: MPS . . . • Consider N spins in 1D . . .

  9. Representing quantum states: MPS A B • Consider N spins in 1D parameters parameters • Finite entanglement capacity:

  10. References Roman Orus: A practical introduction to tensor networks Very nice tutorial! AKLT model: English Wikipedia Original AKLT paper: Phys. Rev. Lett. 59, 799 (1987)

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