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A new method for quantum calculation using the tree tensor network

A new method for quantum calculation using the tree tensor network. Masashi Orii Kusakabe laboratory. Contents. Magnetic order Spin Hamiltonian Computational cost tree tensor network (TTN). Introduction Research Summary. Sz projection method Result of the calculation.

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A new method for quantum calculation using the tree tensor network

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  1. A new method for quantum calculation using the tree tensor network Masashi OriiKusakabe laboratory

  2. Contents • Magnetic order • Spin Hamiltonian • Computational cost • tree tensor network (TTN) Introduction Research Summary • Sz projection method • Result of the calculation

  3. Magnetic Order • Microscopic order in a magnet • Ferro magnetic state • Antiferro magnetic state

  4. Classical and Quantum state For ２spin system (S=1/2), zero temperature

  5. Spin Hamiltonian • Schrödinger equation The basic equation for the microscopic particles • Heisenberg model • Effective Hamiltonian for the spin system

  6. Spin Hamiltonian We don’t consider the orbital of the electron but only the state of the spins. Ψ＝Φ⊗Θ (State=orbital⊗ spin) Constant J<０,antiferro

  7. Computational Cost • Exponential growth of the dimension １spin problem : System size Dimension of State vector １ ２ ３ ４ ３２ ２＾１＝２ ２＾２＝４ ２＾３＝８ ２＾４＝１６ ２＾３２ (２ dimension) １ or … ２spinproblem: … Reduce as finite , , or (２ dimension) ２ PC has 2^(32)=2G memory (limitation of PC) ３３ ２＾３３ Nspinproblem requires … … N ２dimension

  8. Motivationand conclusion • To limit the dimension, numerical methods are widely researched. [5] ΔE ΔE PEPS[2] DMRG[1] TTN VP （our work） MERA[3] TTN[4] Degree of freedom Degree of freedom [1] Steven R. White , Phys. Rev. Lett. 69, 2863-2866 (1992). [2] F. Verstraete et al. PRL 96, 220601 (2006) [3] G. Evenbly and G. Vidal, Phys.Lev. B 79, 144108 (2009). [4] L. Tagliacozzo, G. Evenbly, and G.Vidal , Phys. Rev. B 80, 235127 (2009). [5] F. Verstraete et al . PRL 93,227205(2004)

  9. Tree Tensor Network (TTN) • We consider the spin system using a network like a tree. TTN for 2×2lattice TTN for 4×4lattice A picture of a real tree

  10. Tree Tensor Network (TTN) １site of αdimension (α＜４) Spin Reduce! ２spin ;４dimension , , or αdimension ２ ４ ２ dimension

  11. Research • Sz projection method • Result of the calculation

  12. Our idea : Limiting the wave function • In case of N=４,Exact state has total Sz=０ • Using only the state of totalSz =０(projection), • calculation would get better.

  13. Result of calculation 4 spin system, periodic boundary condition α＝１dimension 2 3 1 4 Projection recovers the entropy and made better energy.

  14. Summary and future work • We have developed a new TTN method with Sz projection (TTN VP). • TTN VP shows a better result,because its entropy becomes larger. • Future work: to calculate more than 4 spins

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