Tensor Renormalization Group and its Application

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Ground state energy of the 2D Heisenberg modelGround State Energy-0.34-0.36-0.38Square LatticeGround State Energy-0.18-0.2-0.22-0.24Triangle Lattice-0.40 0.1 0.2 0.3 0.41/LSquare TriangleDMRG -0.3346 -0.1814MC -0.334719 -0.1819SW -0.33475 -0.1822-0.260 0.1 0.2 0.31/LFree boundary conditionsΔE(L) ~1/LPeriodic boundary conditions:ΔE(L) ~ (1/L) 3
Why is the performance of DMRG poor in 2D?S 1S 2S 3S 4S 5S 61D4 5 12 133 6 11 142 7 10 151 8 9 162D1 2 3 4 5 6 72D 1D mapping• Introduce long-range coupling• Break the square lattice symmetry• Dem<strong>and</strong> grows exponentially withlattice size in order to satisfy theArea Law
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: Extend DMRG to 2D: Map 2D lattice t
Page 25 and 26: Tensor-Network WavefunctionΨ= Tr
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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