Tensor Renormalization Group and its Application

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Area Law of Entanglement EntropyFor a gapped systemd −1Sentanglement~ L ~lnDminenvironmentsystem( )Ψ =∑ Tr Am [ ]... Am [ ] m...mm Lm1L1 L 1‣ 1D (d=1) D min ~ L 0LAxx[ m ]1 2 1D×D matrix‣ 2D (d=2)D min ~ e LD min grows exponentially with the system size
<strong>Tensor</strong>-Network WavefunctionΨ= Tr∏λi ∈j∈blackwhitexiλyiλziAxiyiyi[ m ] Bixjyjyj[ mj]mimj keep the locality of localinteractions satisfy the area law:The number of dangling bondsis proportional to the crosssection<strong>Tensor</strong> network state = tensor product state: vertex state model Niggemann, Zittartz, 96: projected entangled-pair state (PEPS) Cirac, Verstraete, 04
Page 1: Tensor Renormalization Group and it
Page 5 and 6: Characteristic Energy ScalesNumeric
Page 8 and 9: Numerical Renormalization Group Met
Page 10 and 11: Wilson block-spin NRGH 2keep the lo
Page 13 and 14: Density Matrix Renormalization Grou
Page 15 and 16: What is a “Density Matrix” meas
Page 17 and 18: Valence bond solid stateS=11 ⎡H =
Page 19 and 20: Hidden OrderHidden “antiferromagn
Page 21 and 22: Extend DMRG to 2D: Map 2D lattice t
Page 23: Why is the performance of DMRG poor
Page 27 and 28: How to determine the local tensor?[
Page 29 and 30: Projection Approachlimβ →∞e−
Page 31 and 32: Projection: x-bond∏−τH xi,je
Page 33 and 34: Expectation ValueΨ = Tr T [ m ] m
Page 35 and 36: Step I: RewiringM=∑kj,il mji mlkm
Page 37 and 38: Accuracy of TRGD = 24Ising model on
Page 39 and 40: I. Poor Man’s SRG: entanglement m
Page 41 and 42: II. More accurate treatment of SRGE
Page 43 and 44: Accuracy of SRGD = 24Ising model on
Page 45 and 46: Specific Heat of the Ising model on
Page 47 and 48: Heisenberg Model on Honeycomb Latti
Page 49 and 50: Staggered MagnetizationThe tensor-n
Page 51 and 52: Square Lattice
Page 53 and 54: Summary DMRG is an accurate numeric
Page 55 and 56: S 1S 2 S 3σ 1Tensor-network repres

Tensor Renormalization Group and its Application

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