Z. Y. Xie ( 谢志远 ) Institute of Physics, Chinese Academy of Sciences Tensor Renormalization in classical statistical models and quantum lattice models

 Tensor network representation of classical statistical models.  Tensor renormalization in a tensor network model Part I  Projected entanglement simplex (PESS) rep. in frustrated system  Tensor renormalization in a quantum lattice model Part II

Basic notations in tensor graph: dot, free line, link

  Partition function as a network scalar: no free outer-line.   RJ. Baxter: vertex model,   All classical statistical models with only local interactions can be effectively written as a tensor-network model. Ising model on square lattice: already a tensor network: Periodic boundary condition is assumed. Ising model as tensor-network model

 Convert to a normal form  Introduce auxiliary DOF on each bond  This method is universal for all n. n. interaction. Ising model as tensor network model

How to evaluate the physical model How to contract the infinite tensor-network to get the partition function: It is a NP problem to contract it exactly! Tensor renormalization enters the evaluation of the partition function and expectation value.  Renormalization: Compression of DOF (information, Hilbert space) by discarding the irrelevant

How to evaluate the tensor-network model  There are at least 4 classes of methods: Transfer Matrix Renormalization Group(TMRG) Nishino(1995), XQWang, Txiang(1997) Time Evolving Block Decimation(TEBD) /boundary MPS vidal, PRL(2003) Corner Transfer Matrix Renormalization Group(CTMRG) Baxter(1968), Nishino(1996), Orus Vidal(2009)

Coarse-graining Tensor Renormalization Group(TRG). Kadanoff block spin decimation. Levin-Nave-TRG: PRL(2007) How to evaluate the tensor-network model

In 2D, they all work very well; In 3D or higher, they do not work so well! Tensor renormalization based on the higher-order singular value decomposition(HOSVD), i.e., HOTRG. our group(2012) scale1 scale2 scale3 Coarse-graining tensor renormalization

Block spin: how to decimate the local DOF. Bond dimension: DOF introduced on each bond Dimension scales super-exponentially! Block local spins in HOTRG

Decimation: shape down More than 1 cutoff simultaneously! Low rank approximation itself is an open problem! HOSVD: Nearly optimal, in most cases at least very good (1). Different blocks with the same shape are orthogonal (2). Blocks are sorted decreasingly according to norm Decimation of DOF Ref: L. de Latheauwer, B. de Moor, and J. Vandewalle, SIAM, J. Matrix Anal. Appl, 21, 1253 (2000).

How to calculate the HOSVD of a given tensor:  Successive SVD:  Independent SVD: HOSVD is of no mystery! Ref: L. de Latheauwer, B. de Moor, and J. Vandewalle, SIAM, J. Matrix Anal. Appl, 21, 1253 (2000).

How to use it in our case?

Coarse-grain in Y-direction. Repeat the same procedure in X-direction. A single renormalization step in 2D x’

Free energy Calculation of expectation value

Local physical quantity: definition required Calculation of expectation value

Coarse-grain in Z-direction. A renormalization step in 3D: cubic lattice  HOSVD of M, order-6, insert 4 isometry.  x, y, z axis direction

Magnetization Performance: cubic lattice Monte Carlo: Series Expansion: HOTRG(D=14):

Critical property Performance: cubic lattice former NRG

Renormalize the tensor network site by site independently, local update without considering the environment. Main Problem: local update system environment What we need: a decimation scheme which optimizes the global Z instead of the local system M itself! NRG -> DMRG!

Central idea: Note:  Do forward iteration (e.g., HOTRG) to construct tensor work at different RG scales.  Find the relation between the environment of two neighboring scales.  Using the relation to do backward iteration to get the environment of the targeted system (which need to be renormalized).  Use the environment to do global optimization of the system.

 SRG VERSION OF HOTRG? HOSRG!!!

Use the environment to do global optimization  There are several methods to modify the local decomposition by M env  Splitting Env to form an open system:  Cut at

Use the environment to do global optimization  There are several methods to modify the local decomposition by M env

Sweep scheme: Sweep scheme and performance Ising model on square: (D=20) NRG -> iDMRG -> fDMRG TRG -> SRG -> fSRG with sweep D = 24 HOSRG

 Tensor network representation of classical statistical models.  Tensor renormalization in a tensor network model Part I  Projected entangled simplex (PESS) rep. in frustrated system  Tensor renormalization in a quantum lattice model Part II

Some partial history: AKLT authors: PRL(1987), Commun. Math. Phys.(1988) prototype of matrix product state and honeycomb tensor-network Niggemann: Z. Phys. B: CMP (1996,1997) special tensor-network wavefunction for honeycomb Heisenberg model, equivalence between expectation value and classical partition function Sierra and Martin-Delgado: general ansatz Proceedings on the ERG(1998) Nishino: variational ansatz to study 3D classical lattice Prog.Theor.Phys(2001) PEPS, MERA, and so on. F. Verstraete, arXiv: ； Vidal, PRL(2007) Note: Wavefunction itself does not provide any intuition about the entanglement structure between its constituents, the only constrain comes from the area law: It doesn't matter whether the cat is black or white, as long as it catches mice! Only a cat that can catch rats is a good cat! Tensor network states and quantum lattice models

Reinterpret by the concept of space Projector and local Entangled Pair Some critical properties:  Satisfies the area law  Formally has no sign problem  Local pair entanglement.  Can have power-law decay correlation function  Ground state of any local Hamiltonian and PEPS, as long as a very large D.  ‘Can be represent’ does not mean equal, might does not mean effective! Projected Entangled Pair State construction on square lattice F. Verstraete, J. I. Cirac, arXiv:

Choose a wavefunction ansatz/form of the targeted state. Determine the unknown parameters in the wavefunction form. Good review: R. Orus, Annals of physics, 349, 117 (2014) Tensor renormalization in tensor network states  1. global variational extremum problem find a PEPS which minimize the energy : Ax = E*Bx  2. imaginary time evolution In practice the central problem is reduced to how to update/renormalize the wavefunction after a small evolution step (1). Global variational extremum problem find a PEPS which minimize the difference: Ax=Y (2). Reduced to a standard local SVD/HOSVD problem by mean field entanglement approximation: bond vector projection/simple update (3). Regard the surrounding cluster as the whole environment.

Calculate the expectation value: figure Again 2D network scalar: identical to classical partition function! Tensor renormalization in tensor network states

Calculate the expectation value:  Biggest obstacle for application in QLM: D->D^2, MERA Tensor renormalization in tensor network states

PEPS ansatz: Degeneracy: entanglement spectra at each bond has full double degeneracy. Information cancelation: Similar as sign/frustration as in MC! Local pair entanglement; PEPS! PEPS on Kagome lattice: hidden frustration A, B, C has dominant element 1

Structure Property all the advantage of PEPS: area law, no fermion sign, power-law decay correlator… our group, PRX 4, (2014) Projected Entangled Simplex State(PESS)  Introduce a simplex tensor S: the triangle/simplex entanglement, instead of pair  Ansatz is defined on unfrustrated lattice: honeycomb, no hidden frustration here! Simplex ~ possible building block, such as triangle for Kagome

Local update scheme with mean-field entanglement approximation: Determination of PESS ground state wavefunction

Local update scheme with mean-field entanglement approximation: Determination of PESS ground state wavefunction

Local update scheme with mean-field entanglement approximation: figure Determination of PESS ground state wavefunction

Spin-2 Valence Bond Solid (VBS): Spin-2 Simplex Solid (SS): 2:4*1/2, ½+1/2:pair singlet 2:2*1, 1+1+1: simplex singlet two exact examples Kagome lattice

Other possible PESS by choosing larger simplex Simplest PESS: 3PESS, smallest simplex: triangle Multi-site interaction and longer-range can be encoded easily if needed (a) 3-PESS

RVB trial wf. (2013) MERA (2010) Series expansion (2008) iDMRG m=5k (2011) VMC+Lanczos (2013) Extra. HOCC (2011) Extra. fDMRG m=1.6w (2012) 3PESS, 5PESS, and 9PESS all effective 5PESS, 9PESS < 3PESS 13-state PESS: promising and competitive

If you like extrapolation, as in DMRG… Note here: 1. D=19, better 2. Extrapolation gives lower energy Convergence? regime? DMRG: Extrap. finite SU(2) 16,000-state

PESS on other lattices: e.g.,

Difference between PEPS and PESS: PESS is more flexible than PEPS.

Many other things not included here: MERA and branch MERA Variation in PEPS Exact PEPS representation of some special states TEBD, CTM(RG) fPEPS and grassmann TPS Topological measure and entanglement Continuous-space matrix product state Thermal and excited states, non-equilibrium process, spin glass Tree tensor network, and Correlator product states Combination of tensor renormalization with Monte Carlo Symmetry Canonical form Real time evolution RG flow Filtering scheme in TRG... Young, need more effort and deeper mathematical foundation!