Many-Body Fermion Density Matrix: Operator-Based Truncation Scheme

Transcription

1 Many-Body Fermion Density Matrix: Operator-Based Truncation Scheme SIEW-ANN CHEONG and C. L. HENLEY, LASSP, Cornell U March 25, 2004 Support: NSF grants DMR , DMR

2 The Big Picture GOAL Ground state of infinite-sized interacting system; LIMITATION Numerically, can only solve finite-sized interacting system; Pure state (infinite system) = mixed state (finite subsystem) = natural description using density matrices; Review the use of density matrices in quantum mechanics; DOWN & UP Truncation & Renormalization: smaller subsystem in small system (Q1) small subsystem in infinite system (Q2)? Learning from noninteracting systems. Already know answer to Q2; Statistical mechanics analogy: density matrix eigenstates manybody energy eigenstates of noninteracting system; Operator-Based Density Matrix Truncation Scheme & results for 1D free spinless fermions; Cluster density matrices on 2D square lattice: Compare noninteracting and strongly interacting systems. Attempt to answer Q1. 1

3 Density Matrix & Quantum Mechanics Quantum mechanics: state wave function Ψ density matrix (DM) ρ pure 4 4 mixed 8 4 If system S = subsystem A ({ a, a,... }) + subsystem B ( b, b,... }), state of system ρ state of subsystem ρ A : Partial trace over subsystem B, i.e. ρ A = Tr B ρ; Expectation of referencing operators, i.e. (ρ A ) aa = K a K a ρ [S.-A. Cheong and C. L. Henley, Phys. Rev. B 69, (2004)]. Applications of ρ A : DM-based renormalization group [S. R. White, PRL 69, 2863 (1992); R. J. Bursill, PRB 60, 1643 (1999)]; Diagnostic & extraction of important correlations [Vidal et al, PRL 90, (2003)]. 2

4 DM of Free Spinless Fermions Free spinless fermions Fermi sea ground state Ψ F. For block of B sites identified as subsystem, DM found to have the structure [M.-C. Chung and I. Peschel, PRB 64, (2001)] [ B ] ρ B exp ϕ l f l=1 l f l, { f l, f l } = 1. Relation to correlation function G(i, j) = Ψ F c i c j Ψ F, i, j = 1,..., B, found to be [S.-A. Cheong and C. L. Henley, Phys. Rev. B 69, (2004); I. Peschel, J. Phys. A: Math. Gen 36, L205 (2003)] ϕ l = ln [ λ l (1 λ l ) 1], B λ l = λ l ( f l 0 ), ( B) i j = G(i, j). A particular eigenstate of ρ B described by a set of numbers (n 1,..., n l,..., n B ), n l = 0, 1, w = f l 1 f l 2 f l P 0, n l = δ l,li, and its weight is w exp ( Φ), Φ = 3 B l=1 n lϕ l.

5 Statistical Mechanics Analogy [S.-A. Cheong and C. L. Henley, Phys. Rev. B 69, (2004)] free spinless fermion ρ B Hamiltonian H = k ɛ k c k c k H = l ϕ l f l f l pseudo-hamiltonian 1-particle energy ɛ k ϕ l 1-particle pseudo-energy 1-particle operator c k f l 1-particle pseudo-operator occupation number n k n l pseudo-occupation number total energy E = l n k ɛ k Φ = l n l ϕ l total pseudo-energy Fermi level ɛ F ϕ F pseudo-fermi level Based on analogy, average pseudo-occupation is n l = 1 exp ϕ l + 1. Most probable eigenstate of ρ B has structure of Fermi sea: ϕ l ϕ F occupied, ϕ l > ϕ F empty. Other eigenstates look like excitations about Fermi sea. 4

6 Operator-Based DM Truncation Scheme DM eigenstates with largest weights always have ϕ l ϕ l ϕ F empty. These differ in n l for ϕ l ϕ F ; Keep only f l with ϕ l ϕ F : ϕ F occupied and n l = 0 ϕ ϕ F n l truncate ϕ F } m = γb f l s retained n l = 1 Compare with weight-ranked truncation: eigenstates with largest weights all kept; some eigenstates with intermediate weights not kept, but replaced with eigenstates with slightly smaller weights; eigenstates with small weights not kept. 5

7 Results 1D Free Spinless Fermions ε(k) ε F E k E γ = 0.25 γ = 0.33 γ = 0.50 γ = 0.67 γ = 0.75 γ = B 6

8 Cluster DM on 2D Square Lattice Definition of system: R 2 R 1 5-site cluster, various system sizes N = R 1 R 2. Computation of cluster DM ρ C : obtain ground state Ψ (exact diagonalization or otherwise); ρ 0 = Ψ Ψ partial trace translational invariance; degeneracy and shape averaging. ρ C (care with fermion sign!); 7

9 2D Cluster DM 1-Particle Weights nearest neighbor hopping (noninteracting) and nearest neighbor hopping + infinite nearest neighbor repulsion (strongly interacting); 0-particle weight not interesting monotonic decreasing with filling n, very similar for noninteracting and strongly interacting systems; Look at 1-particle weights: 5 of these, characterized by angular momentum quantum numbers s 1, p x, p y, d, s 2. Infinite system limit for noninteracting system, 200 sites for a squarish finite system; Small finite systems (noninteracting & interacting) of 20 sites, strong influence from: finite size effect; shell effect (most severe for d state, least severe for s 1 state). 8

10 2D Cluster DM 1-Particle Weights 0.5 noninteracting strongly interacting w s (4, 1) (1,4) (8, 2) (2,8) (16, 4) (4,16) (32, 8) (8,32) free (32, 8) (8,32) (4, 1) (1,4) (3, 2) (2,3) (4,1) (1,4) n n

11 2D Cluster DM 1-Particle Weights 0.04 noninteracting strongly interacting (4, 1) (1,4) (8, 2) (2,8) (16, 4) (4,16) (32, 8) (8,32) 0.06 w d free (32, 8) (8,32) (4, 1) (1,4) (3, 2) (2,3) (4,1) (1,4) n n 10

12 Conclusions Much learnt from noninteracting spinless fermions (Q2): 1D 2D structure of ρ B ; Operator-Based DM Truncation Scheme; when infinite system limit reached; shell effect & its persistence; How much of this understanding can be applied to interacting fermions (Q1)? finite size effect & shell effect entangled; adaptation and extension of Operator-Based DM Truncation Scheme; Still far from eventual goal: ground state of infinite system of interacting fermions. 11