# Tensor Renormalization Group and its Application

Tensor Renormalization Group and its Application

Tensor Renormalization Group and its Application

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<strong>Tensor</strong> <strong>Renormalization</strong> <strong>Group</strong> <strong>and</strong> <strong>its</strong><strong>Application</strong>Tao Xiang ( 向 濤 )Institute of Physics / Institute of TheoreticalPhysicsChinese Academy of Sciencestxiang@aphy.iphy.ac.cn

Strong Coupling Approach2 64 → a particularly selected set of basis statesQuantum Monte CarloNumerical <strong>Renormalization</strong> <strong>Group</strong>K. 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 <strong>Renormalization</strong> <strong>Group</strong> 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

Concept of renormalization groupEnergyH 0H 1H 2••• ••• H effRG iteration• 1943 Ernst Stueckelberg initialized a renormalizationprogram to attack the problems of infinities in QEDbut his paper was rejected by Physical Review.• 1953 Ernst Stueckelberg <strong>and</strong> Andre Petermannopened the field of renormalization groupErnst Stueckelberg

Numerical <strong>Renormalization</strong> <strong>Group</strong> Method• 1974 Wilson opened the field of numericalRG by combining the idea of renormalizationgroup with numerical calculations.Kenneth Wilson• He solved the famous single-impurity Kondomodel (Kondo effect).

Basic Idea of Numerical <strong>Renormalization</strong> <strong>Group</strong>To represent a targeted stateψ0∞= ∑l=1f nllby an approximate wavefunction using alimited number of basis statesψ%0M≈ ∑ f % nl=1llsuch that their overlap is maximizedψ%ψ0 0M=∑ %l=1ffllRefine a basis set by performing a series of basis transformations

Wilson block-spin NRGH 2keep the lowest D states of H 2H = H + H + HL R LR4 2 2 4keep the lowest D states of H 4H = H + H + HL R LR2n n n 2nDiagonalize H,keep the states according to their energy

Reasons for the Failure of Wilson block-spin NRG1. Interface effect is too big2. Big truncation error:total D 2 states, but only Dof them retained3. Criterion of truncation:energy is not a goodindicator of basis statesi( +)i+1 i.H = ∑ a a + hcExample: Particle on a lattice

Density Matrix <strong>Renormalization</strong> <strong>Group</strong> (DMRG)System Environmentm 1m 2n 2m 3S. White, PRL (1992)Use Environment as apump to probe thebasis states in Systemn 3m 4|i〉 sys|j〉 envψ = ∑ f i ji,jij sys env

NRG versus DMRGWilson NRGSystemSysDMRGSystemEnvEnergy is the only quantity thatcan be used to measure theweight of a basis stateρ=e− β HUse a sub-system as a pump toprobe the other part of the systemρsys=Trenve− β HThe weight is measured by theentanglement between Sys <strong>and</strong> Env

What is a “Density Matrix” measurement?System Environment|i〉 sys|j〉 env,ψ = ∑ f i ji jij sys envQuantum Information:Schmidt decomposition∑ψ = Λnn sys envΛ n2is the eigenvalue of reduceddensity matrixnnMathematician:Singular value decompositionij in , n j,nn=1Dn=1Nf = U Λ V≈∑∑UΛVin , n j,n

DMRG Wavefunction: Matrix Product StateS Ostlund, S Rommer, PRL 1995 The wavefunction generated by the DMRG iteration is a matrix-productstate It can be also regarded as a variational ansatz of many-body wavefunction( )Ψ =∑ Tr Am [ ]... Am [ ] m...mm Lm1L1 L 1LAxx[ m ]1 2 1D×D matrixx ix i+1 bond variablesm ilocal basis

Valence bond solid stateS=11 ⎡H = ∑ S 1 ( ) 2 2iSi+ 1Si Si+1i 2 ⎢⋅ + ⋅ +⎣ 3 3⎤⎥⎦Ψ =∑m Lm1L( )Tr Am [ ]... Am [ ] m...m1 L 1L⎛ 0 0⎞ ⎛−1 0⎞⎛0 2⎞A[ − 1] = ⎜ ⎟ A[0] = ⎜ A[1]=2 0 0 1⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝0 0 ⎠S=1/2S=1/2Affleck, Kennedy, Lieb, Tasaki, PRL 59, 799 (1988)

Haldane ConjectureInteger Heisenberg spin chain has a finite excitation gap= ∑ ⋅H SiSi + 1iHaldaneNi 2+ :S = 1Energy gapY 2BaNiO 5

Hidden OrderHidden “antiferromagnetic” order↑ 0 0 0 ↓ 0 ↑ 0 0 ↓ 0 ↑ 0 0 0 ↓ 0 ↑ 0 0 0 0 ↓Is there any order in theHaldane spin chain?↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓S = -1 (↑) , 0, 1 (↓)Nd 2BaNiO 5Antiferromagneti Neel orderNeelNobel 1970YBa 2Cu 3O 6

Accuracy of the DMRG in 1DDMRG: Can calculate all static, thermodynamic, dynamic quantities1D S=1 Heisenberg modelGround state energyQuantum Monte Carlo -1.4015(5)DMRG -1.401484038971(4)(keep D = 100 states)HeisenbergNH = J∑S ri⋅S ri + 1i=1

Extend DMRG to 2D: Map 2D lattice to 1DDMRG is essentially a 1D methodMulti-chain mapping:The width of the lattice is fixedDiagonal-path:Lattice grows in both directionsT Xiang, J Z Lou, Z B Su, PRB 64, 104414 (2001)

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

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

Two Problems Need To Be SolvedΨ= Tr∏λi ∈j∈blackwhitexiλyiλziAxiyiyi[ m ] Bixjyjyj[ mj]mimj1. How to determine the local tensor?2. How to evaluate the expectation values, given atensor-product wavefunction?

How to determine the local tensor?[ ]Ψ = Tr∏T m mix yz i iii i1. Variational approachto minimize• Converging slowly• The bond dimension that can be treated is smallΨ H ΨΨ | ΨD ≤ 5Gu, Levin, Wen, PRB (2008)

How to determine the local tensor?Ψ= Tr∏λi ∈j∈blackwhitexiλyiλziAxiyiyi[ m ] Bixjyjyj[ mj]mimj2. Projection approach−lim e βH Ψ = ground stateβ →∞• Very accurate• Fast converging• Large bond dimension can be treated(more if symmetry is considered)Projection OperatorD ~ 70 (honeycomb lattice)D ~ 20 (square or Kagome lattice)Jiang, Weng, Xiang, PRL 101, 090603 (2008)1D: Vidal, PRL 98, 070201 (2007)

Projection Approachlimβ →∞e−βHΨ=ground statelimM →∞(−τH)Me Ψ = ground stateHeisenberg model∑H = H = H + H + HijH = JS ⋅Sij i jij x y z

Projection IterationeH−τHα=≈e−τH∑i∈blackzeH−τHi,i+αye−τHx2+ o(τ )( α = x,y,z)Trotter-Suzuki decomposition1. One iterationΨ1Ψ~Ψ20= e= e= e−τH−τH−τHxyzΨΨ0Ψ122. Repeat the above iteration until converged

Projection: x-bond∏−τH xi,je Ψ = Tr mm e mm λλλA [ m ] B [ m ] mmi i i i i i j j ji∈blackj=+i xˆ′ ′ ′ ′−τHi j i j x y z x yy i x y y j i jStep IStep IStep IIStep IIStep IIIStep IIITruncate basis spaceSVD: singular value decomposition

How accurate is this projection approachλ MFWithout performing thecanonical transformationfor the matrix productstateUsing the canonicalrepresentation of thematrix product state:λ is the eigenvalue of thedensity matrix1D Heisenberg model

Expectation ValueΨ = Tr T [ m ] mΨΨ =∏iTr∑xyy i i∏i i ixx′ , yy′ , zz′i i i i i iA = T [ m] T [ m]xx′ , yy′ , zz′ xyz x′ y′ z′mBond dimension D 2iATo evaluate〈Ψ |Ψ 〉 <strong>and</strong> 〈Ψ |O|Ψ 〉is equivalent toevaluating the partitionfunctions of classicalstatistical models,such as the S=1/2 Isingmodel

Coarse Grain <strong>Tensor</strong> <strong>Renormalization</strong> <strong>Group</strong>Z= Tr∏Tixyzii iRewiring DecimationLevin & Nave, PRL (2007)

Step I: RewiringM=∑kj,il mji mlkmN∑n=1T T= U ΛVkj, n n il,nBond field: measures theentanglement between U <strong>and</strong> V

Step II: Decimation

Accuracy of TRGD = 24Ising model on triangular lattice

Second renormalization of tensor-network state (SRG)‣ TRG:truncation error of M isminimized(envZ=Tr MM )What needs to be minimizedis the error of Z!‣ SRG:The renormalization effect ofM env to M is consideredsystemenvironmentXie et al, PRL 103, 160601 (2009)

I. Poor Man’s SRG: entanglement mean-field approach(env) ij, kl ,Z=Tr MM = ∑ M Mijklenvkl ijM = ∑ U Λ Vkj, il kj, n n il,n4n=1... DM ≈ Λ Λ Λ Λenv 1/ 2 1/ 2 1/ 2 1/2kl,ij k l i jMean field (or cavity) approximationBond field – mean field variableΛΛ = Λ 1/2 Λ 1/2From environmentFrom system

Accuracy of Poor Man’s SRGD = 24Ising model on triangular lattice

II. More accurate treatment of SRGEvaluate the environment contribution M env using TRGTRGM env

( n 1)M −ijkl1. Forward iterationM→M(0) (1)→L →M( N )( n)M i ' j ' kl ' '2. Backward iterationM→M( N) ( N−1)M(0)→L→ =MenvM M S S S S= ∑ ∑( n−1) ( n)ijkl i' j ' kl ' ' k' jp j ' pi i' lq l ' qki' j' k' l'pq

Accuracy of SRGD = 24Ising model on a triangular lattice

Error versus D

Specific Heat of the Ising model on Triangular LatticesD = 24

Quantum Heisenberg Model on Honeycomb LatticeH = J∑S r⋅ SrGround State Energy i jijSRG D = 14E = -0.54439SRG D = 30E = -0.54442Monte Carlo:E = -0.54454 ( ± 20 )Lattice size30N = 2×3U. Low, Condensed Matter Physics2009 Vol 12, 497

Heisenberg Model on Honeycomb LatticeQuantum Monte Carlo ResultE = -0.36303 (± 13) per bond= -0.54454 ( ± 20 ) per siteU. Low, Condensed MatterPhysics 2009 Vol 12, 497

Staggered MagnetizationSRG D = 15M = 0.31003Monte Carlo:M = 0.2681U. Low, Condensed MatterPhysics 2009 Vol 12, 497M = 0.22Reger, Riera, Young,JPC 1989Spin Wave:M = 0.24Series expansionM = 0.27

Staggered MagnetizationThe tensor-networkstate cuts the longrangecorrelationThe bond dimension isroughly of the order ofthe correlation length ofthe tensor-network stateThe logarithmiccorrection to the AreaLaw is important here

Staggered Magnetization4 th order polynomial fitM = 0.263Monte Carlo:M = 0.2681

Square Lattice

Kagome Lattice

Summary DMRG is an accurate numerical method for studying 1Dquantum systems In 2D, the tensor-network representation of quantum manybodystates is a good starting point The quantum tensor-network wavefunction can beaccurately <strong>and</strong> efficiently evaluated by the projectionmethod The partition function or expectation values of tensornetworkmodel/state can be accurately determined by theSRG methodAcknowledgement:Zhiyuan Xie, Qiaoni Chen, Huihai Zhao IOP / ITP, CASZhengyu Weng, Hongchen Jiang TsinghuaUniversity

<strong>Tensor</strong>-Network Representation of Classical Statistical ModelH= -J∑SiSijIsing modeljZ= Tr∏Tixyzii i<strong>Tensor</strong>-network model in dual latticeTriangular latticeDual lattice: honeycomb lattice

S 1S 2 S 3σ 1<strong>Tensor</strong>-network representationH= -J∑S Si jσ 3 σ ij2Z = Trexp− βH = Tr∏exp −βHΔ( ) ( )σσσ=SS1 2 3=SS2 3 1=SS3 1 2H ( Δ=− J σ1+ σ2 + σ3)/2σσσ = SSSSSS = 11 2 3 2 3 3 1 1 2Δ

<strong>Tensor</strong>-network representationZ= Tr T∏ixyz( )− Jβσ+ σ + σ / 2ii i1 2 3Tσσσ= e δ σ1 2 31σ2σ3−1( )S 1σ 3 σ 2S 2 S 3σ 1