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 methodhψ|H|ψi
System energy E(ψ) =hψ|ψiMinimization w.r.t. |ψi ⇒ ground state |ψ0 i + O() and E0 + O(2 )DMRG wave function: Matrix-Product State|ψi =X{s1 ,s2 ,...,sN }(M1 [s1 ] · M2 [s2 ] · . . . · MN [sN ]) |s1 i|s2 i . . . |sN iwith Mi [si ] = m × m matrix and |si i, si = 1, . . . , ni = basis states of site i√√Exact wave function for m → D = n1 n2 . . . nN
[S. Östlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995); PRB 55, 2164 (1997)]

Photoemission

ARPES spectrum = spectral functions
Aσ (k, ω ≤ 0) = lim Im Gσ (k, h̄ω − iη)
η→0=X|hn|ĉk,σ |ψ0 i|2 δ(En − E0 + h̄ω)n11†ĉk,σ |ψ0 ihψ0 |ĉk,σGσ (k, h̄ω − iη) =π
Ĥ − E0 + h̄ω − iηDynamical correlation functions with DMRG
11†χÂ (ω + iη) = − hψ0 |ÂÂ|ψ0 iπ
E0 + ω + iη − Ĥ
Lanczos vector method

Correction vector method[K. Hallberg, PRB 52, 9827 (1995)][S. Ramasesha et al., Synth. Met. 85, 1019 (1997),T.D. Kühner and S.R. White, PRB 60, 335 (1999)]|φ0 i = Â|ψ0 i|φ1 i = Ĥ|φ0 i − a0 |φ0 i|φn+1 i = Ĥ|φn i − an |φn i − b2n |φn−1 iρL = ψ0 ψ0∗ + Lanczos vectors|CV i =

1E0 + ω + iη − ĤχÂ (ω + iη) =Â|ψ0 i−1hψ0 |Â† |CV iπρCV = ψ0 ψ0∗ + correction vectors

Variational principle for dynamical correlation functions[ E. Jeckelmann, PRB 66, 045114 (2002) ]Functional
WA,η
ˆ (ω, ψ) = hψ| Ĥ − ω − E0

2+ η 2 |ψi + ηhψ|Â|ψ0 i + ηhψ0 |Â† |ψi

Minimization with respect to ψ−ηÂ|ψ0 i +O()|ψmin i =22(E0 + ω − Ĥ) + ηWÂ,η (ω, ψmin ) = −πη Im χÂ (ω + iη) +O(2 )−1hψ0 |Â† (Ĥ − E0 − ω)|ψmin i
Re χÂ (ω + iη) =πηNote|ψmin i|CV i→−π hψn |Â|ψ0 i δ(ω − En + E0 ) |ψn i(η → 0)=Ĥ − E0 − ω + iη|ψmin iηDynamical DMRG (DDMRG)[E. Jeckelmann, PRB 66, 045114 (2002)]
For the spectral function (Â = ĉk,σ )2Wk,σ,η (ω, ψ) = hψ| Ĥ + ω − E0+ η 2 |ψi + ηhψ|ĉk,σ |ψ0 i + ηhψ0 |ĉ†k,σ |ψiMinimization of Wk,σ,η (ω, ψ) with a regular DMRG algorithm and∗ρCV = (ψ0 ψ0∗ + ψmin ψmin+ first Lanczos vector) yieldsAησ (k, ω) = Im Gσ (h̄ω − iη)−1=Wk,η (ω, ψmin )πη1=πηdω Aσ (ω )(ω − ω 0 )2 + η 2−∞

Z ∞

00

Spectrum in the thermodynamic limit[E. Jeckelmann, PRB 66, 045114 (2002)]

Lehmann spectral representation
I(ω) = ImχÂ (ω) = lim lim

η→0 N →∞X ηn|hψn |Â|ψ0 i|2π (ω − En + E0 )2 + η 2Using η(N ) > η0 (N ) with η(N → ∞) → 0I(ω) =limX η(N )N →∞ nπ|hψn |Â|ψ0 i|2(ω − En + E0 )2 + η(N )2Dense spectrum: η0 (N ) > En+1 (N ) − En (N )Discrete spectrum: η0 (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 vF ≈ W2a )NavF ∆t <2vF ~Na-Examples: Optical conductivity of one-dimensional insulators

Peierls insulator∆=0.6tη(N)=7.68t/N

N=32
N=64
N=128
N=256
Exact

η/t=0.4 (N=32)
η/t=0.2 (N=64)
η/t=0.1 (N=128)

0.8

σ1(ω)

σ1(ω)

U=8t, V1=2t

1.0

105

Half-filled 1D extended Hubbard model

0.6
0.4
0.2

00.4

0.8

1.2

ω/t

1.62.00.00246ω/t

8101214Optical conductivity of the 1D Hubbard model(Half filling, U = 3t)

Broadening η = 0.1t

No broadening/deconvolved

1.0

1.5

Field Theory
DDMRG

0.6
0.4
0.2
0.00

DDMRG
Field theory

1.0

σ1(ω)

σ1(ω)

0.8

0.5

0.0
1

23

ω/Eg

4560123ω/Eg

4Jeckelmann, Gebhard and Essler, PRL 85, 3910 (2000)
Jeckelmann and Fehske, cond-mat/0510637

56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)ĉkσ =

s1 X ikne ĉnσN n2πN
k=z with z = 0, ±1, ±2, . . . ,N2

Open boundary conditions⇒ particle-in-the-box wavefunctions
for quasi-momentaĉkσ =s2 X
sin(kn) ĉnσ ,N +1 nπk=j with j = 1, 2, . . . , N
N +1

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

ω

-0.5

0-1-0.5-1.5-1-1.5-2-2-2.5-3-3.5-4
00.5

κ/π

1

1.5log(A(k,w))

0

DDMRG vs Bethe Ansatz (ρ = 0.6, U = 4.9t)
0-0.5-1-1.5

ω/t-2-2.5-3-3.5-4-4.5-5-1 -3kF-2kF-kF

0k/πkF

2kF

3kF 1

Spectral properties of SrCuO2[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 scatteringwith the one-dimensional half-filled extended Hubbard modelRIXS spectrum[Y.-J. Kim et al., Phys. Rev. Lett. 92, 137402 (2004)]
65.5
5

20

4

18

3.5

16

3

12

3.5
3

10

2.5

8

2

6

1.5

ω (eV)

144

Intensity (per sec)

Energy Loss (eV)

4.5

2.5
2
1.5
1-1-0.8

-0.6

-0.4

-0.2

00.2

q/2π4

12
0.5
−1−0.8−0.6−0.4 −0.2q/2π00.2

c (k) ∼ 2t cos(q)s (k) ∼ πJ| sin(q)|2

DMRG and quantum information theory: A new perspective
The Hilbert space truncation in DMRG is optimal because• ||ψexact − ψDMRG || 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)]

Measure of entanglement: Von Neumann entropyP

S = −T rρ ln2 (ρ) = −µ λµ ln2 (λµ )Examples• λ1 = 1, λµ>1 = 0 ⇒ S = 0, i.e. no entanglement• λµ = 1/m ⇒ S = ln2 (m), i.e. maximal entanglement

Entanglement entropy SL of a L-site cluster in an infinite chain• SL diverges for L → ∞ at criticality• SL saturates for L → ∞ in the non-critical regime
Isotropic XX chain in a magnetic field

Ising chain in a transverse field

3.5

SL

3

1.8

2.5

1.6

2SL

1.5

1.4
1.2

11

0.5

0.8

0050100
L150

SL vanishes for H = Hc200

050100
L150200

SL = 0 for H = 0SL diverges at criticality H = Hc

[Latorre, Rico, and Vidal, arXiv:quant-ph/0304098]
Geometric arguments ⇒ SL ∼ Ld−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, limL→∞ m(L) = ∞• For d > 1, m(L) ∼ nLd−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: Pm ≤ 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 SL 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)]

4080
200
400

0.1

1.9
1.85

0.05
01

1.2

1.4

1.6

1.8

U/∆2

2.2

2.4

2.6

Site entropy

1.3
1.25

4080
200400

1.2
1.15
1.1

11.2

1.4

1.6

1.8U/∆

Two−site entropy

Dimer order

Ionic Hubbard model (∆ = t)

1.8
1.75
1.7
1.65
1.6
1.55

40
80
200400
∞

1.5
1.45

2

2.2

2.4

2.6

1.4

1

1.2

1.4

1.6

1.8

U/∆

2

2.2

2.4

2.6

Variational matrix-product states
Reformulation of DMRG as a variational optimizationof 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 entangledpair states (PEPS)[F. Verstraete and J. I. Cirac, cond-mat/0407066]

Time-dependent DMRG
∂ψ(t)
ih̄
= H(t)ψ(t)∂tFirst approach: integration in a static superblock (Runge-Kutta)• ρ = ψ0 ψ0∗ (incorrect)[Cazalilla and Marston, PRL 88, 256403 (2002)]

• ρ=P

∗
i ψ( ti )ψ (ti ) (not practical)[Luo, Xiang, and Wang, PRL 91, 049701 (2003)]

Second approach: Taylor-expansion of exp(−iHt)• ρ=P∗nψψwithψ=H
ψ(t = 0) (not practical)
nn n n[Schmitteckert, PRB 70, 121302 (2004)]

Vidal’s algorithm [G. Vidal, PRL 93, 040502 (2004)]
Only one-dimensional nearest-neighbor Hamiltonians

H=N−1X

Hk,k+1k=1

Trotter-Suzuki decomposition
N/2exp(−iHτ ) =Y

k=1

N/2−1exp(−iτ H2k−1,2k )Yexp(−iτ H2k,2k+1 ) + O(τ 2 )k=1(More accurate for higher order decomposition)
Matrix-product representation
0exp(−iτ Hk,k+1 )ψ(t) ⇒ Mk [sk ], Mk+1 [sk+1 ] → Mk0 [sk ], Mk+1
[sk+1 ]

Time step = sweep through the lattice k = 1, . . . , N − 1

ψ(t + τ ) = exp(−iτ H)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 stepρ(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(−iτ Hk,k+1 )ψ(t, k) or ψ(t, k) = exp(−iτ Hk,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(−iτ H)ψ(t, k = 1) = exp(−iτ H)ψ(t)(R) ψ(t + τ ) = ψ(t, k = 1) = exp(−iτ H)ψ(t, k = N ) = exp(−iτ H)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

z<S (x)>t=14

t=20.5t=0050100x150

Time evolution of the local magnetization hS z (x)i 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 time1

L=100, dt = 0.05

0.1

1e−03
1e−04
1e−05

m=10
m=20
m=30
m=40
m=50
m=60
m=70
m=80

1e−06
1e−07
1e−08

runaway time

S z deviation

1e−02

030

B20A
102010time

2030m

405030Error 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
ML /2 sites

Nano System
MS sites
tC

1n−1

2

Right Lead
ML /2 sitestCnm−1m

M

Finite-size effects are important ⇒ no steady state
Differential conductance vs bias voltage (MS = 7)
(M = 144, m = 600 for weak coupling, M = 192, m = 800 for strong coupling)
1V/ tS = 3
V/ tS = 12g [e /h]

0.8
0.6
0.4
0.2
0-0.4

-0.2

0µSD [ t ]

0.2

0.4

From Schneider and Schmitteckert, arXiv:cond-mat/0601389

Finite-temperature dynamics
Density operator ρ in Hilbert space of dimension D≡ wavefunction in space of dimension D 2

ρ=X⇒ matrix-product representation of ρX(M1 [s1 , t1 ] · . . . · MN [sN , tN ]) |s1 i . . . |sN iht1 | . . . htN |{s1 ,...,sN } {t1 ,...,tN }with Mi [si , ti ] = m × m matrix and |si i, si = 1, . . . , ni = basis states of site i
Imaginary and real time evolutiondρ= −Hρ and ρ(β = 0) = Id ⇔ ρ = exp(−βH)
dβdρ= −i[H, ρ] ⇔ ρ(t + τ ) = exp(−iHτ )ρ(t) exp(iHτ )dt[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)