(Almost) 25 Years of DMRG - What Is It About? Ulrich Schollwck - PowerPoint PPT Presentation

(Almost) 25 Years of DMRG - 
 What Is It About? Ulrich Schollwöck University of Munich

Outline Concerning your talk (65min + 10 min for questions) we would prefer if you could provide a broad introduction into DMRG. This may include in particular: 1) Brief historic discussion 2) Why does one use DMRG? 3) Concept of entanglement and correlations and area laws 5) Details on how to implement DMRG/How does it work? 6) benchmark calculations and comparison to results obtained by resorting to different methods 7) comments on scaling behavior (w.r.t. number of electrons, dimension of truncated 1-particle Hilbert space and "bond length \chi") 8) Open problems While working out your presentation please be aware of related talks (for instance the one by Markus Reiher, "DMRG in Quantum Chemistry", building up on your presentation).


 
 fundamental problem of solid state what do we need DMRG for? problem class: 
 fundamental Hamiltonian (without lattice vibrations…!): e − e − p 2 q 2 + 1 1 j X X e H = | r i − r j | + V e ff ( r j ) 2 m e 2 4 ⇡✏ 0 j =1 j kinetic 
 electron-electron 
 lattice 
 energy interaction potential we don’t know how to solve the Schrödinger equation! 
 problem: electron-electron interactions


 many-body problem of solid state I scenario I 
 valence electrons well delocalized 
 interactions well screened electron cloud energy conductor half-filled lattice potential DOS many metals, semiconductors: single-electron picture OK 
 density functional theory (DFT)


 
 many-body problem of solid state II scenario II 
 valence electrons tightly bound 
 strong local interactions energy insulator 
 half-filled eg. high-Tc parent compounds lattice potential DOS many particle picture: strongly correlated materials 
 model Hamiltonian methods - here DMRG comes in!

why strong correlations? 1 dimension 0 dimensions spin chains & ladders magnetic impurity physics Luttinger liquid quantum dots 3 dimensions 2 dimensions T realistic modelling: frustrated magnets strange metal Fermi liquid Non-Fermi liquid transition metal, high- T c superconductors Néel order rare earth compounds pseudo gap superconductivity doping

transition metal oxides and rare earths belated filling of the d - and f- shells 
 valence electrons quite tightly bound


 
 
 cold atoms in optical lattices ultra-cold bosonic atoms form Bose-Einstein condensate 
 standing laser waves: optical lattice Greiner et al (Munich group), Nature ’02 
 interaction tunable U via lattice depth bosonic Hubbard t model extension to fermions (spin = hyperfine levels) e.g. M. Köhl et al (Esslinger group), PRL ‘05

tunability controlled tuning of interaction U/t in time via lattice depth adiabatic change of U/t : quantum phase transition momentum distribution function superfluid adiabatic increase of interaction U Mott insulator sudden change of U/t to Mott insulator: collapse and revival time „state engineering“ for generic quantum many-body systems

compression of information compression of information necessary and desirable diverging number of degrees of freedom emergent macroscopic quantities: temperature, pressure, ... classical spins thermodynamic limit: degrees of freedom (linear) 2 N N → ∞ quantum spins superposition of states N → ∞ 2 N thermodynamic limit: degrees of freedom (exponential)

classical simulation of quantum systems compression of exponentially diverging Hilbert spaces what can we do with classical computers? exact diagonalizations limited to small lattice sizes: 40 (spins), 20 (electrons) stochastic sampling of state space quantum Monte Carlo techniques negative sign problem for fermionic systems physically driven selection of subspace: decimation variational methods renormalization group methods how do we find the good selection? DMRG!

DMRG: a young adult 09.11.1992 S.R. White: Density Matrix Formulation for Quantum Renormalization Groups (PRL 69, 2863 (1992)) „This new formulation appears extremely powerful and versatile, and we believe it will become the leading numerical method for 1D systems; and eventually will become useful for higher dimensions as well.“ ~2004 old insight „DMRG is linked to MPS (Matrix Product States)“ goes viral Östlund, Rommer, PRL 75, 3537 (1995), Dukelsky, Martin-Delgado, Nishino, Sierra, EPL43, 457 (1998) Vidal, PRL 93, 040502 (2004), Daley, Kollath, Schollwöck, Vidal, J. Stat. Mech. P04005 (2004), 
 White, Feiguin, PRL 93, 076401 (2004), Verstraete, Porras, Cirac, PRL 93, 227205 (2004), Verstraete, Garcia-Ripoll, Cirac, PRL 93, 207204 (2004), Verstraete, Cirac, cond-mat/0407066 (2004) (some) reviews: U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005) - „old“ statistical physics perspective, applications U. Schollwöck, Ann. Phys. 326, 96 (2011) - „new“ MPS perspective, technical F. Verstraete, V. Murg, J. I. Cirac, Adv. Phys. 57, 143 (2008) - as seen from quantum information

matrix product states: definitions quantum system living on L lattice sites d local states per site { σ i } i ∈ { 1 , 2 , . . . , L } example: spin 1/2: d =2 | "i , | #i Hilbert space: H = ⌦ L i =1 H i H i = {| 1 i i , . . . , | d i i } most general state (not necessarily 1D): X c σ 1 ... σ L | σ 1 . . . σ L i | ψ i = σ 1 ,..., σ L c { σ } abbreviations: { σ } = σ 1 . . . σ L

(matrix) product states exponentially many coefficients! standard approximation: mean-field approximation d L → dL coefficients c σ 1 ... σ L = c σ 1 · c σ 2 · . . . · c σ L often useful, but misses essential quantum feature: entanglement consider 2 spin 1/2: H i = {| " i i , | # i i } H = H 1 ⊗ H 2 | ψ i = c ↑↑ | ""i + c ↑↓ | "#i + c ↓↑ | #"i + c ↓↓ | ##i 1 | "#i � 1 c ↑↓ 6 = c ↑ c ↓ singlet state: p p | ψ i = | #"i 2 2 generalize product state to matrix product state: c σ 1 · c σ 2 · . . . · c σ L → M σ 1 · M σ 2 · . . . · M σ L

matrix product states useful generalization even for matrices of dimension 2: AKLT (Affleck-Kennedy-Lieb-Tasaki) model 
 general matrix product state (MPS): 
 M σ 1 M σ 2 . . . M σ L | σ 1 σ 2 . . . σ L i X | ψ i = σ 1 ,..., σ L matrix dimensions: 
 (1 × D 1 ) , ( D 1 × D 2 ) , . . . , ( D L − 2 × D L − 1 ) , ( D L − 1 × 1) non-unique: gauge degree of freedom 
 XX − 1 = 1 M σ i +1 → X − 1 M σ i +1 M σ i → M σ i X

matrix product states Why are matrix product states interesting? 
 any state can be represented as an MPS 
 (even if numerically inefficiently) MPS are hierarchical: matrix size related to degree of entanglement MPS emerge naturally in renormalization groups MPS can be manipulated easily and efficiently MPS can be searched efficiently: 
 which MPS has lowest energy for a given Hamiltonian?

singular value decomposition (SVD) key workhorse of MPS manipulation and generally very useful! general matrix A of dimension ( m × n ) k = min( m, n ) A = USV † then UU † = I U † U = I with dim. (ON col); if : ( m × k ) U m = k dim. diagonal: non-neg.: ( k × k ) S s 1 ≥ s 2 ≥ s 3 ≥ . . . s i ≥ 0 singular values, non-vanishing = rank r ≤ k V † V † V = I V V † = I dim. (ON row); if : 
 ( k × n ) k = n popular notation: (left) singular vectors 
 | u i i U = [ | u 1 i | u 2 i . . . ]

SVD and EVD (eigenvalue decomp.) singular value decomposition (always possible): 
 A = USV † s 1 ≥ s 2 ≥ s 3 ≥ . . . s i ≥ 0 eigenvalue decomposition (for special square matrices): 
 eigenvectors 
 λ i U = [ | u 1 i | u 2 i . . . ] AU = U Λ A † A AA † connection by „squaring“ A : 
 AA † = USV † V SU † = US 2 U † ⇒ ( AA † ) U = US 2 A † A = V SU † USV † = V S 2 V † ⇒ ( A † A ) V = V S 2 eigenvalues = singular values squared eigenvectors = left, right singular vectors 


any state can be decomposed as MPS reshape coefficient vector into matrix of dimension and SVD: 
 ( d × d L − 1 ) X U σ 1 ,a 1 S a 1 ,a 1 V † c σ 1 σ 2 ... σ L → Ψ σ 1 , σ 2 ... σ L = a 1 , σ 2 ... σ L slice U into d row vectors: 
 a 1 U σ 1 ,a 1 → { A σ 1 } A σ 1 1 ,a 1 = U σ 1 ,a 1 with rearrange SVD result: 
 X c a 1 σ 2 σ 3 ... σ L = S a 1 ,a 1 V † A σ 1 c σ 1 σ 2 ... σ L = 1 ,a 1 c a 1 σ 2 σ 3 ... σ L a 1 , σ 2 ... σ L a 1 reshape coefficient vector into matrix of dim. and SVD: 
 ( d 2 × d L − 2 ) X U a 1 σ 2 ,a 2 S a 2 ,a 2 V † c a 1 σ 2 σ 3 ... σ L → Ψ a 1 σ 2 , σ 3 ... σ L = a 2 , σ 3 ... σ L slice U into d matrices: 
 a 2 A σ 2 a 1 ,a 2 = U a 1 σ 2 ,a 2 rearrange SVD result: and so on! 
 X A σ 1 c σ 1 σ 2 ... σ L = 1 ,a 1 A σ 2 a 1 ,a 2 c a 2 σ 3 σ 3 ... σ L a 1 ,a 2

Schmidt decomposition bipartition of „universe“ AB into subsystems A and B: {| i 〉 A } {| j 〉 B } L 1 ℓ +1 ℓ dim H A dim H B X X | ψ i = ψ ij | i i A | j i B i =1 j =1 read coefficients as matrix entries, carry out SVD: 
 r X Schmidt decomposition 
 | ψ i = s α | α i A | α i B α =1 dim H A dim H B orthonormal X X | α i A = U i α | i i A | α i B = V ∗ j α | j i B sets! 
 i =1 j =1