Tensor Renormalization Group and its Application

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Strong Coupling Approach2 64 → a particularly selected set of basis statesQuantum Monte CarloNumerical Renormalization GroupK. Wilson• Wilson NRG 1960s–1980s‣ Kondo problem• Density-matrix renormalization group (White 1992)‣ Most accurate numerical method for studying 1Dquatnum systems‣ Poor performance in 2D
Characteristic Energy ScalesNumerical Renormalization Group Density Functional TheoryQuantum Monte Carlo Dynamical Mean FieldEnergy10 -6 10 -4 10 -2 10 0 10 2 10 4 T(K)Cold atoms 3 He-superfluid 4 He-superfluid CMR simple metalheavy fermions QHE high-Tc semiconductorsuperconductorStrong coupling Weak couplinglow Dimensionality highstrongQuantum fluctuations weakweakCoulomb screening strong
Page 1: Tensor Renormalization Group and it
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 and 24: Why is the performance of DMRG poor
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
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Tensor Renormalization Group and its Application

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