Recent developments in DMRG Eric Jeckelmann Institute for Theoretical Physics University of Hanover Germany
Outline 1. Introduction 2. Dynamical DMRG 3. DMRG and quantum information theory 4. Time-evolution 5. Finite-temperature dynamics 6. Conclusion
DMRG is a variational method System energy E(ψ) = ψ H ψ ψ ψ Minimization w.r.t. ψ ground state ψ 0 + O(ɛ) and E 0 + O(ɛ 2 ) DMRG wave function: Matrix-Product State ψ = {s 1,s 2,...,s N } (M 1 [s 1 ] M 2 [s 2 ]... M N [s N ]) s 1 s 2... s N with M i [s i ] = m m matrix and s i, s i = 1,..., n i = basis states of site i Exact wave function for m D = n 1 n 2... n N [S. Östlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995); PRB 55, 2164 (1997)]
Photoemission ARPES spectrum = spectral functions A σ (k, ω 0) = lim η 0 Im G σ (k, hω iη) = n n ĉ k,σ ψ 0 2 δ(e n E 0 + hω) G σ (k, hω iη) = 1 π ψ 0 ĉ k,σ 1 Ĥ E 0 + hω iη ĉk,σ ψ 0
Dynamical correlation functions with DMRG χâ(ω + iη) = 1 π ψ 1 0 Â E 0 + ω + iη Ĥ Â ψ 0 Lanczos vector method [K. Hallberg, PRB 52, 9827 (1995)] Correction vector method [S. Ramasesha et al., Synth. Met. 85, 1019 (1997), T.D. Kühner and S.R. White, PRB 60, 335 (1999)] φ 0 = Â ψ 0 φ 1 = Ĥ φ 0 a 0 φ 0 φ n+1 = Ĥ φ n a n φ n b 2 n φ n 1 CV = 1 E 0 + ω + iη Ĥ Â ψ 0 χâ(ω + iη) = 1 π ψ 0 Â CV ρ L = ψ 0 ψ 0+ Lanczos vectors ρ CV = ψ 0 ψ 0+ correction vectors
Variational principle for dynamical correlation functions [ E. Jeckelmann, PRB 66, 045114 (2002) ] Functional W ˆ A,η (ω, ψ) = ψ ( Ĥ ω E 0 ) 2 + η 2 ψ + η ψ Â ψ 0 + η ψ 0 Â ψ Minimization with respect to ψ ψ min = η (E 0 + ω Ĥ)2 + η 2 Â ψ 0 +O(ɛ) WÂ,η (ω, ψ min ) = πη Im χâ(ω + iη) +O(ɛ 2 ) Re χâ(ω + iη) = 1 πη ψ 0 Â (Ĥ E 0 ω) ψ min
Note ψ min π ψ n  ψ 0 δ(ω E n + E 0 ) ψ n (η 0) CV = Ĥ E 0 ω + iη η ψ min
Dynamical DMRG (DDMRG) [E. Jeckelmann, PRB 66, 045114 (2002)] For the spectral function (Â = ĉ k,σ ) W k,σ,η (ω, ψ) = ψ ( Ĥ + ω E 0 ) 2 + η 2 ψ + η ψ ĉ k,σ ψ 0 + η ψ 0 ĉ k,σ ψ Minimization of W k,σ,η (ω, ψ) with a regular DMRG algorithm and ρ CV = (ψ 0 ψ0 + ψ min ψmin + first Lanczos vector) yields A η σ(k, ω) = Im G σ ( hω iη) = 1 πη W k,η(ω, ψ min ) = 1 π dω A σ (ω ) η (ω ω ) 2 + η 2
Spectrum in the thermodynamic limit [E. Jeckelmann, PRB 66, 045114 (2002)] Lehmann spectral representation I(ω) = ImχÂ(ω) = lim η 0 lim N n η π ψ n  ψ 0 2 (ω E n + E 0 ) 2 + η 2 Using η(n) > η 0 (N) with η(n ) 0 I(ω) = lim N n η(n) π ψ n  ψ 0 2 (ω E n + E 0 ) 2 + η(n) 2 Dense spectrum: η 0 (N) > E n+1 (N) E n (N) Discrete spectrum: η 0 (N) = 0
Methods convolution of an infinite-system spectrum for comparison extrapolation or scaling analysis for N, η(n) 0 deconvolution of finite-system spectrum (fit to a smooth infinite-system spectrum) One-dimensional systems η(n) = cn 1 with c > effective band width W η 1 measurement time t One-dimensional electron gas (Fermi velocity v F W a 2 ) v F t < Na 2 7 v F Na
Examples: Optical conductivity of one-dimensional insulators Peierls insulator Half-filled 1D extended Hubbard model σ 1 (ω) 10 5 N=32 N=64 N=128 N=256 Exact =0.6t η(n)=7.68t/n σ 1 (ω) 1.0 0.8 0.6 0.4 U=8t, V 1 =2t η/t=0.4 (N=32) η/t=0.2 (N=64) η/t=0.1 (N=128) 0.2 0 0.4 0.8 1.2 1.6 2.0 ω/t 0.0 0 2 4 6 8 10 12 14 ω/t
Optical conductivity of the 1D Hubbard model (Half filling, U = 3t) Broadening η = 0.1t No broadening/deconvolved 1.0 1.5 σ 1 (ω) 0.8 0.6 0.4 Field Theory DDMRG σ 1 (ω) 1.0 0.5 DDMRG Field theory 0.2 0.0 0 1 2 3 4 5 6 ω/e g 0.0 0 1 2 3 4 5 6 ω/e g Jeckelmann, Gebhard and Essler, PRL 85, 3910 (2000) Jeckelmann and Fehske, cond-mat/0510637
Quasi-momentum for open boundary conditions [H. Benthien, F. Gebhard, and E. Jeckelmann, Phys. Rev. Lett. 92, 256401 (2004)] Periodic boundary conditions Bloch wavefunctions for momenta (wavevectors) ĉ kσ = 1 N n e ikn ĉ nσ k = 2π N z with z = 0, ±1, ±2,..., N 2 Open boundary conditions particle-in-the-box wavefunctions for quasi-momenta ĉ kσ = 2 N+1 n sin(kn) ĉ nσ, k = π N + 1 j with j = 1, 2,..., N
ARPES spectrum of TTF-TCNQ [R. Claessen et al., PRL 88, 096402 (2002); M. Sing et al., PRB 68, 125111 (2003) ] [Benthien, Gebhard, and Jeckelmann, PRL 92, 256401 (2004)] Experiment Hubbard model 0 ω -0.5-1 -1.5-2 -2.5 0-0.5-1 -1.5-2 log(a(k,w)) -3-3.5-4 0 0.5 1 1.5 κ/π
DDMRG vs Bethe Ansatz (ρ = 0.6, U = 4.9t) ω/t 0-0.5-1 -1.5-2 -2.5-3 -3.5-4 -4.5-5 -1-3k F -2k F -k F 0 k F 2k F 3k F 1 k/π
Spectral properties of SrCuO 2 [H. Benthien and E. Jeckelmann, e-print cond-mat/0606748] Quantitative description of optical absorption, angle-resolved photoemission spectrum, resonant inelastic X-ray scattering (RIXS), and neutron scattering with the one-dimensional half-filled extended Hubbard model RIXS spectrum [Y.-J. Kim et al., Phys. Rev. Lett. 92, 137402 (2004)] 6 20 4 Energy Loss (ev) 5.5 5 4.5 4 3.5 3 2.5 18 16 14 12 10 8 Intensity (per sec) ω (ev) 3.5 3 2.5 2 1.5 2 1.5 1 0.5 1 0.8 0.6 0.4 0.2 0 0.2 q/2π 6 4 2 1-1 -0.8-0.6-0.4-0.2 0 0.2 q/2π ɛ c (k) 2t cos(q) ɛ s (k) πj sin(q) 2
DMRG and quantum information theory: A new perspective The Hilbert space truncation in DMRG is optimal because ψ exact ψ DMRG is maximal [Noack and White in I. Peschel et al., Density Matrix Renormalization, Springer, 1999] the maximal entanglement between system and environment is preserved [G. Vidal et al., PRL 90, 227902 (2003)] Examples Measure of entanglement: Von Neumann entropy S = T rρ ln 2 (ρ) = µ λ µ ln 2 (λ µ ) λ 1 = 1, λ µ>1 = 0 S = 0, i.e. no entanglement λ µ = 1/m S = ln 2 (m), i.e. maximal entanglement
Entanglement entropy S L of a L-site cluster in an infinite chain S L diverges for L at criticality S L saturates for L in the non-critical regime Isotropic XX chain in a magnetic field Ising chain in a transverse field S L 3.5 3 2.5 2 1.5 1 0.5 0 0 50 100 150 200 L S L 1.8 1.6 1.4 1.2 1 0.8 0 50 100 150 200 L S L vanishes for H = H c S L = 0 for H = 0 S L diverges at criticality H = H c [Latorre, Rico, and Vidal, arxiv:quant-ph/0304098] Geometric arguments S L L d 1 for a hypercubic cluster in dimension d > 1
Number of density matrix eigenstates m(l) for a fixed accuracy in the thermodynamic limit L In d = 1 away from criticality, m(l) finite value In d = 1 at criticality, lim L m(l) = For d > 1, m(l) n Ld 1
Optimization using entanglement entropy (for non-local systems) DMRG procedure = successive density-matrix truncations Fixed number m of density-matrix eigenstates is not optimal Dynamical block state selection is better: P m const. [Legeza et al., PRB 67, 125114 (2003)] Even better: quantum information loss χ(s) const. [Legeza and Sólyom, PRB 70, 205118 (2004)] Site ordering in non-local systems Optimal ordering using quantum information entropy [Legeza and Sólyom, PRB 68, 195116 (2003)]
Entropy and quantum phase transitions Quantum phase transition singularity in the ground state entanglement [Wu, Sarandy, and Lidar, PRL 93, 250404 (2004)] Problem: measure of entanglement? Cluster entropy S L for L [G. Vidal et al., PRL 90, 227902 (2003)] Concurrence (only for spin systems) [A. Osterloh, Nature 416, 608 (2005)] One-site entropy (often insensitive) [Gu et al., PRL 93, 086402 (2004)] Two-site entropy [Legeza and Sólyom, PRL 96, 116401 (2006)] Ionic Hubbard model ( = t) Dimer order Site entropy 0.1 0.05 1.3 1.25 1.2 1.15 40 80 200 400 0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 U/ 40 80 200 400 1.1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 U/ Two site entropy 1.9 1.85 1.8 1.75 1.7 1.65 1.6 1.55 40 1.5 80 200 1.45 400 1.4 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 U/
Variational matrix-product states Reformulation of DMRG as a variational optimization of matrix-product states [F. Verstraete, D. Porras, and J.I. Cirac, PRL 93, 227205 (2004)] As good as DMRG for open chains In theory, improvement for periodic systems In theory, extension to higher dimensions with projected entangled pair states (PEPS) [F. Verstraete and J. I. Cirac, cond-mat/0407066]
Time-dependent DMRG i h ψ(t) t = H(t)ψ(t) First approach: integration in a static superblock (Runge-Kutta) ρ = ψ 0 ψ 0 (incorrect) [Cazalilla and Marston, PRL 88, 256403 (2002)] ρ = i ψ ( t i )ψ (t i ) (not practical) [Luo, Xiang, and Wang, PRL 91, 049701 (2003)] Second approach: Taylor-expansion of exp( iht) ρ = n ψ n ψ n with ψ n = H n ψ(t = 0) (not practical) [Schmitteckert, PRB 70, 121302 (2004)]
Vidal s algorithm [G. Vidal, PRL 93, 040502 (2004)] Only one-dimensional nearest-neighbor Hamiltonians H = N 1 k=1 H k,k+1 exp( ihτ) = N/2 k=1 Trotter-Suzuki decomposition exp( iτh 2k 1,2k ) N/2 1 k=1 exp( iτh 2k,2k+1 ) + O(τ 2 ) (More accurate for higher order decomposition) Matrix-product representation exp( iτh k,k+1 )ψ(t) M k [s k ], M k+1 [s k+1 ] M k[s k ], M k+1[s k+1 ] Time step = sweep through the lattice k = 1,..., N 1 ψ(t + τ) = exp( iτh)ψ(t)
(Adaptive) time-dependent DMRG (t-dmrg) [A.J. Daley, C. Kollath, U. Schollwöck, and G. Vidal, JSTAT (2004); S.R. White and A.E. Feiguin, PRL 93, 076401 (2004)] Efficient implementation of Vidal s algorithm within DMRG The DMRG superblock basis is adapted at every step ρ(t, k) = ψ(t, k)ψ (t, k) DMRG step: Adding one site to the left block or to the right block ψ(t, k+1) = exp( iτh k,k+1 )ψ(t, k) or ψ(t, k) = exp( iτh k,k+1 )ψ(t, k+1) Time step = one sweep from (L) left to right or (R) right to left (L) ψ(t + τ) = ψ(t, k = N) = exp( iτh)ψ(t, k = 1) = exp( iτh)ψ(t) (R) ψ(t + τ) = ψ(t, k = 1) = exp( iτh)ψ(t, k = N) = exp( iτh)ψ(t) For longer-range interactions or higher dimensions: Adaptive Runge-Kutta approach [Feiguin and White, PRB 72, 020404 (2005)]
t-dmrg code From De Chiara et al., e-print arxiv:cond-mat/0603842
Spin chain dynamics t=14 <S z (x)> 0.5 0 t=0 t=2 50 100 150 x Time evolution of the local magnetization S z (x) of a 200 site spin-1 Heisenberg chain after S + (100) is applied (τ = 0.1). From S.R. White and A.E. Feiguin, PRL 93, 076401 (2004)
Problem: runaway time 1 L=100, dt = 0.05 0.1 1e 02 S z deviation 1e 03 1e 04 1e 05 1e 06 1e 07 m=10 m=20 m=30 m=40 m=50 m=60 m=70 m=80 1e 08 0 10 20 30 time runaway time 30 20 10 B A m 20 30 40 50 Error in the local magnetization as a function of time in the XX model (τ = 0.05). From Gobert, Kollath, Schollwöck, Schütz, PRE 71, 036102 (2005)
Transport through nanostructures Interacting spinless fermion system coupled to non-interacting leads Left Lead Nano System Right Lead M L /2 sites M S sites M L /2 sites t C t C 1 2 n 1 n m 1 m M Finite-size effects are important no steady state Differential conductance vs bias voltage (M S = 7) (M = 144, m = 600 for weak coupling, M = 192, m = 800 for strong coupling) 1 0.8 V/ t S = 3 V/ t S = 1 g [ e 2 / h ] 0.6 0.4 0.2 0-0.4-0.2 0 0.2 0.4 µ SD [ t ] From Schneider and Schmitteckert, arxiv:cond-mat/0601389
ρ = Finite-temperature dynamics Density operator ρ in Hilbert space of dimension D wavefunction in space of dimension D 2 {s 1,...,s N } {t 1,...,t N } matrix-product representation of ρ (M 1 [s 1, t 1 ]... M N [s N, t N ]) s 1... s N t 1... t N with M i [s i, t i ] = m m matrix and s i, s i = 1,..., n i = basis states of site i dρ dt dρ dβ Imaginary and real time evolution = Hρ and ρ(β = 0) = Id ρ = exp( βh) = i[h, ρ] ρ(t + τ) = exp( ihτ)ρ(t) exp(ihτ) [Verstraete et al., PRL 93, 207204 (2004); Zwolak and Vidal, PRL 93, 207205 (2004); Feiguin and White, PRB 72, 220401 (2005)]
Summary DMRG is a variational method Connection to matrix-product states Relation to quantum information theory (entanglement) Optimization using entanglement entropy Criticality and entanglement Time-dependent DMRG Finite-temperature dynamics Bright prospects for the future
Thanks Holger Benthien (Marburg) Florian Gebhard (Marburg) Satoshi Nishimoto (Dresden) Fabian Eßler (Oxford)