Efficient Representation of Ground States of Many-body Quantum Systems: Matrix-Product Projected States Ansatz

Transcription

1 Efficient Representation of Ground States of Many-body Quantum Systems: Matrix-Product Projected States Ansatz Systematic! Fermionic! D>1?! Chung-Pin Chou 1, Frank Pollmann 2, Ting-Kuo Lee 1 1 Institute of Physics, Academia Sinica, Taipei, Taiwan 2 Max-Planck-Institut fur Physik komplexer Systeme, Dresden, Germany arxiv: , submitted to PRL Thanks: Peter Fulde NSYSU 3/8/2012

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3 Outline Introduction Why tensor-product states (TPS)? How to improve matrix-product states (MPS)? Matrix-product projected states (MPPS) Variational Monte Carlo technique Example: 1D spinless fermion t-v model Perspective

4 Solid State Condensed Matter Single-particle approximation Correlated electronic state 2 ˆ pˆ P i j 1 e 1 ZiZje 1 Zje 2m 2M 8 r r 8 R R 4 r R i j j 0 ij i j 0 ij i j 0 i, j i j Paradigm from Landau and Anderson Symmetry Breaking, Rigidity BEC, Magnetic orders in spin systems, Conventional SC, New Paradigm? Topological Order X.G. Wen No broken symmetry, No local order parameter, Ground-state degeneracy

5 Quantum Chemistry 1. Many electrons in an atom: Pauli principle + Hund s rule Periodic Table! 2. Diatomic molecule: Molecular orbital method (linear combination of atomic orbitals, LCAO) Bonding, anti-bonding, non-bonding, Heitler-London method (Heisenberg) 3. Polyatomic molecule: Valence-bond orbital theory (Hybridization of polyatomic orbitals) 4. Crystal field: Ligand field theory, Jahn-Teller effect

6 Solid-State Theory 1. Nearly-free (NF) and tight-binding (TB) electrons in the periodic potential: Energy band! 2. Energy-band calculation: Orthogonalized plane wave (OPW) Pseudo-potential NF Augmented plane wave (APW) Muffin-Tin potential NF + TB k p method k=0 perturbation KKR, MTO, LAPW, LMTO, 3. Self-consistent calculation: Valence electron approximation Adiabatic approximation (Born-Oppenheimer) Hartree-(Fock) approximation 4. Density functional theory (DFT): Y(R) vs n(r) Hohenberg-Kohn theorem, Kohn-Sham equation Local Density Approximation (LDA), Molecular dynamics + DFT,

7 Strong Electronic Correlation 1. Metal or insulator? CoO = (3d 7 4s 2 )+(2s 2 2p 4 ) = 15/cell 2. Mott-Insulator transition: ZSA classification (Zaanen, Sawatsky, Allen) of Mott insulator Orbital degeneracy Orbital ordering Spin ordering 3. Doped Mott insulator: H c c U n n Hubbard ij i j i i i,, j i H c c J S S tj ij i j ij i j i,, j i j Cuprates: High-temperature superconductivity Manganites: Colossal magnetoresistance (CMR) Charge ordering, stripe, phase separation 4. Kondo effect (magnetic impurity): H c c J s S KLM ij i j i i i,, j i Anderson model, Kondo Lattice model, Kondo-Heisenberg model, RKKY, Heavy fermion: anomalous metallic behavior; Kondo singlet, Kondo insulator,

8 Exotic phases and quantum phase transitions? Mott transition at finite U? H c c Un n 2 D Hubbard ij i j i i i,, j i low energy sectors Miyagawa and Yokoyama, JPSJ 2011 How many phases left? H2 D Heisenberg Jij Si Sj... i j Neel phase, spiral antiferromagnet, valence bond solid, quantum spin liquid, Sachdev, arxiv 2009

9 Bottleneck in numerical algorithms For N spin-1/2 systems, Y a Wa There are 2 N coefficients W(a)! (The number of operations grows exponentially with N!) a a n, n,..., n n i, 1 2 N Up to now, solutions: 1. Mean-field Approximation: 2N coefficients now! QMC Y 1 2 N 2. Renormalization Group Method: Grouping and Re-defining! NRG, DMRG

10 How to write the correlated electronic state? Strategy: Hamiltonian Mean-field decoupling Correlated factor Hubbard, Anderson, Order parameters, Mean-field states Jastrow, Gutzwiller, Usually, it is just a good trial state If more correlations are considered, Weak and intermediate correlation ---! Strong correlation ---? P. W. Anderson and J. R. Schrieffer, Physics Today, 1991 Based on quantum information theory TPS

11 Pictorial language for tensor network TPS Node: Each tensor (site) Edge: The length of each index (bond) bond physical Contraction: MPS with periodic boundary condition (PBC)

12 Matrix Product State (MPS, TPS with rank 2) Y ˆ ˆ ˆ L MPS 1 2 a a n n n W Tr A[1] A[2] A[ L] a a a U. Schollwock, Ann. Phys Aˆ n i [] i ab, a n 1~ d, e. g., d 2 i a, b 1~ d The upper bound scales exponentially with L unless truncates to the bond length. (limit the entanglement) exponentially decaying eigenvalue spectra of reduced density operators! The number of parameters scales polynomially with L. (exact when ) Memory requirement: Ld 2 ; Time cost: Ld 2 5 a L /2 n1, n2,..., nl

13 History of the TPS CGTN DMRG MPS MPS Marti, 2010 White, 1992 Ostlund, 1995 Takahashi, 1999 TTS SBS MPS Vertex-MPS FOC Schuch, 2008 Verstraete, 2004 MPS GMPS Sierra, 1998 Fannes, 1992 FCS 2D MPS Lange, 1994 Barthel, 2009 Liang, 1994 Vertex-MPS Schuch, 2007 CTMRG MERA Fannes, D MPS VDMA Niggemann, 1997 RTM Hieida, 1999 TPA Nishino, 2001 PEPS Martin-Delgado, 2001 Verstraete, 2004 Vidal, 2007 Xiang, 2001 TPS Maeshima, 2001 Generalize 1D MPS to 2D quantum or 3D classical systems

14 News from the TPS: Stripes in 2D t-j Model? J/t=0.4 TPS: MFS: x=1/8 x=1/8 x=1/6 P. Corboz et al., PRB 2011 W.J. Hu et al., arxiv:

15 Idea from the TPS The number of the relevant states of short-range Hamiltonians is not too huge, The number to parameterize all possible Hamiltonians with m-body interactions: N Cm s m Can we find the corner of the Hilbert space using simple product state? NO! How to extend product states to cover the corner? Thanks to quantum information theory Area law!

16 Area Law SvN L D 1 Finite-range Hamiltonians with a gap at T=0 1D + gapless (or criticality): S ~ log(l) 2D + 1D Fermi Surface: S ~ L log(l) Why DMRG works in D=1 and fails in D>1? A B Two d-dim. state spaces max 2 S log d d 2 S Dimensions 2 S to encode entanglement! Fulfilling the area law, Matrix Product State (MPS) in 1D, Projected Entangled Pair States (PEPS) in 2D

17 A more efficient way to cover the corner?! Current Problems on TPS: 1. It is not easy to understand the complicated fermionic structure arising from fermionic PEPS! 2. The minimal distance, determined by bond dimensions, from the ground states is strongly limited by computer resources! Solution: Try other initial states from mean-field theory! 1D systems could be the 1 st test area!

18 Matrix-product Projected State (MPPS) Initial state: 1 1 ˆ n1ˆ n2 ˆ nl,, Y Tr A A A n n 1 MPS [1] [2] [ L] 1 L n,, n L n,, n L ˆ ˆ 0 1 L 1 n,, n ˆ n1 n2 nl Tr P P P [1] [2] [ L] 0 S n,, n n,, n L L L d ˆni ˆ ni P A n n [ i] [ i] i i n n,, n 1 L 0 i 1 i1 n 1 n,, n i Srandom S(a) : Slater determinant L L D 1 ˆ n1ˆ n2 ˆ nl,,,, Y Tr A A A Sn n n n MPPS [1] [2] [ L] 1 L 1 L n,, n L Short-range entanglement Long-range entanglement

19 Pictorial representation

20 Variational Monte Carlo Method a 1 2 Y Oˆ Y a Y Y Oˆ a Y Y a Y Y a Y 1 M Y Oˆ a a 0... M Y a Relatively large system size Large interaction Geometrical frustration Optimization: Stochastic reconfiguration method Truncation in the parameter space Stochastic annealing approach

21 1D Half-filled spinless fermion t-v model i i 1.. i i1 i i H t c c h c V nn t a a h. c. V n n i Hard-core bosonic model Fermi Liquid Exactly solved in 1D (Bethe Ansatz) Holstein-Primakoff b b i i1 i i1 i a a z 1 z 1 Jxy Si Si 1 Jz Si Si 1 i, a x, y i 2 2 Spin-1/2 XXZ model V/t 0 2 Gapless Luttinger Liquid Gapped CDW with 2a 0 Jordan-Wigner S z i Jxy Jz n i 1 2 2t V

22 Jastrow-type wave functions: Pˆ J h h Pˆ Pˆ S a a Y J 0 J a ijnn ˆi ˆj a ji, i j e ij ln rij v r ij ( xi xj) sin L In the following, exact solutions are obtained by ED or infinite time-evolving block decimation (itebd) algorithm. MPS with large bond dimensions G. Vidal, PRL 2007

23 MPPS show more efficient than MPS for small V/t! MPPS capture the phase transition to CDW at V/t=2! Jastrow-type state fails to describe the gapped phase! L=18

24 MPPS show much faster convergence for the gapless phases than MPS! At the critical point, the convergence for MPPS is still better than MPS! L=202

25 MPPS converge to the exact results with a 1/L 2 scaling in the gapless phase! MPPS converge exponentially once L is of the order of the correlation length!

26 L=202 1 C R nn n n L nn i i R i ir i C nn A R B R ( 1) R 2 2 R 2 cos 2V t 1 1 MPO do not need to transform the state into one with algebraic correlation but they only need to modify the exponent! MPPS obtain a better estimation for the CDW order parameter than MPS! R: odd sites L R sin L

27 How to extend to 2D? W a Tr Aˆ Aˆ Aˆ ttr Tˆ Tˆ Tˆ 1 2 L 1 2 n n n n n nn [1] [2] [ L] [1] [2] [ N] A n n [] i T[] i ˆ i ab, ˆ i a, b, c, d L. Wang, et al., PRB 2011 Tensorial trace is an exponentially long calculation! An approximate contraction is necessary, Tensor-Entanglement Renormalization Group (TERG)

28 Many references on TRG a) M. Levin and C. P. Nave, PRL 2007 b) H. C. Jiang et al., PRL 2008 c) Z.-C. Gu et al., PRB 2008 d) A. W. Sandvik, PRL 2008 e) L. Wang, et al., PRE 2011 f) L. Wang et al., PRB 2011

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30 Summaries 1) We introduce the MPPS based on matrix-product projection operators. 2) The MPPS allows to variationally improve the short range entanglement of a given trial state by optimizing the operators, while the long range entanglement is already contained in the initial trial state. 3) We demonstrate the usefulness of the MPPS by considering a model of 1D interacting spinless fermions and show that the convergence in terms of the bond dimension is much faster as compared to standard MPS. 4) This approach can be generalized to higher dimensions using tensorproduct projection operators.