Introduction to thedensity-matrix renormalization group
Eric Jeckelmann
Institute for Theoretical Physics
University of HanoverGermanyOutline
1. Introduction
2. Density-matrix renormalization group (DMRG):
Basic principle
3. DMRG code
4. DMRG truncation errors
5. Extension to two-dimensional systems
6. Overview of other extensions(TMRG, dynamics, bosons, non-local systems)
7. Conclusion

Extended one-dimensional Hubbard modelVUt
Electronic density0<ρ<2

HamiltonianĤ =XX †−tĉiσ ĉi+1σ + ĉ†i+1σ ĉiσ −t0ĉ†iσ ĉi+2σ + ĉ†i+2σ ĉiσiσiσXXρρ+Un̂i↑ − 2 n̂i↓ − 2 + V (n̂i − ρ)(n̂i+1 − ρ)iiQuantum many-body problem• Physical properties for a N-site lattice ?
∂ψih̄= Hψ ,∂tHψ = Eψ ,Z = T r exp(−βH)• Exact analytical results for special cases only(for instance, 1D Hubbard model, Tomonaga-Luttinger model).
• Hilbert space dimension = 4N
⇒ Exact diagonalizations up to N ≈ 16 only.
• “Exact” numerical methods for N  1 ?

Numerical renormalization group (NRG)[K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975)]

H2H3H4yy y y y y y y y

NRG for the particle-in-the-box problem[S.R. White and R.M. Noack, PRL 68, 3487 (1992)]
L−1XLXd2
H=−(|iihi + 1| + |i + 1ihi|) + 2|iihi| ≈ − 2dx
i=1
i=1
0.5

φ(x)

0.4

Lowest eigenstates (L=10,20)

0.3|2Li 6= |Li ⊗ |Li

0.2
0.1
0246810x1214161820

Low-energy states of system 6= Low-energy states of subsystems

Reduced density matrix
System + Environment|ψi =Pρi,j =PP

H = HS + HE + HSEi,α ψi,α |iiS |αiE∗ψψ
α i,α j,αj ρi,j φµ,j = λµ φµ,i0 ≤ λµ ≤ 1

Pµ λµ = 1

The most important states for the system are|φµ iS =Xi

φµ,i |iiS

for the largest density-matrix eigenvalues λµDensity-matrix eigenvalues
Gapped system of interacting harmonic oscillators[M. C. Chung and I. Peschel, Phys. Rev. B 62, 4191 (2000)]2wn ∼ exp −(const./N ) ln (n) , N = number of chains[Generalization of K. Okunishi, Y. Hieida, and Y. Akutsu, Phys. Rev. E 59, R6227 (1999)]
01010x1010x710x510x410x310x210x1−510

wn 10−10−15

10−20100250

500

n7501000

Density-matrix renormalization group (DMRG)[S.R. White, PRL 69, 2863 (1992); PRB 48, 10345 (1993)]

Infinite system

Finite system  y y y y y y y y y y
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DMRG code

From De Chiara et al.,e-print arXiv:cond-mat/0603842

Open source code (FORTRAN90)
available at
http://qti.sns.it/dmrg/phome.html

Optimization of DMRG calculations
1. Efficient iterative algorithms for superblock calculations (Lanczos or Davidson
algorithms, conjugate gradient, . . . )2. Use symmetries and quantum number conservation (block matrix) and/or
sparse matrix techniques[S. Ramasesha et al., Phys. Rev. B 54, 7598 (1996);
I. McCulloch and M. Gulácsi, Europhys. Lett. 57, 852 (2002)]
3. Use the wavefunction transformation technique[S.R. White, Phys. Rev. Lett. 77, 3633 (1996)]
4. Efficient dynamical memory management (cache, main memory, hard disk)
5. Parallelization[G. Hager, E. Jeckelmann, H.Fehske, and G. Wellein, J. Comp. Phys. 194, 795(2004)]

Overall speedup ∼ 103

DMRG code parallelization
Hager, Jeckelmann, Fehske, Wellein, J. Comp. Phys. 2004
• Single-CPU performance = 40-80% of system peak performance• Shared-memory parallelization of BLAS DGEMM (dense matrix multiplications)• OpenMP parallelization of sparse matrix-vector multiplications (MVM) ψ 0 = HψBenchmarks for m = 2000 (Hubbard model)
8

1.1
1

1 CPU performanceHP rx5670
Intel Xeon DPIBM p690
SGI Origin 3400
SunFire 3800

Efficiency

0.80.76Speedup0.970.60.55idealAmdahl, s=0.02Amdahl, s=0.16
MVM OpenMPDavidson OpenMP
Total OpenMP
Total DGEMM

SGI Origin 3400

40.4

30.30.2

20.101234CPUs

567

DGEMM parallelization

811234

CPUs

56

MVM parallelization

78

Numerical errors
Superblock calculations with iterative procedures(Lanczos or Davidson algorithms, conjugate gradient)
Convergence (self-consistence) of DMRG basis
R[ρ] = ρ for all density matrices ρTruncation errors

Pm

Discarded weight (of density matrix ρ): Pm = 1 − k=1 λk√Scaling∆hHi ∼ Pmand∆hOi ∼ Pm
(Exact diagonalization for Pm → 0)

Finite-size effects(Boundary conditions, discrete spectrum)DMRG truncation error

0,4Energy vs. discarded weight

Staggered bond order vs discarded weight

12x3 Hubbard ladder, U=t, 6 holes

Half-filled extended Hubbard model, U=3t, V=1.5t

0,05

m=200
DMRG
Fit

m=400

0,30,040,2

δ(EDMRG-ECPMC)/t300

900
0,1 1800
0,00600

0,03300

4008006001024 sites

1200

1300
2200

12345-4Discarded weight Pm (x10 )Em = Eexact + cPm

60,020,00,5
Discarded weight (Pm)

1/2-31,0(x10 )√δ = δexact + c Pm

Two-dimensional systems
Multi-chain approach[S. Liang and H. Pang, Phys. Rev. B. 49, 9214 (1994)]

Figure fromU. Schollwöck,RMP 2005

Computational cost ∼ (chain length)c∼ exp(c×number of chains)Two dimensions vs one dimension
DMRG calculation for the ground state energy of a 16-site Hubbard modelat half filling and U = 12t

open chain

periodic square clusterm1202400

Pm

10−1210−6∼ 10 e10−2
∼ m

∆E/t

≈ 10−7≈ 10−3CPU (1999)

20 seconds

15 hours

Memory5 MB

250 MB−7 −m/10

Ladder systems[R. M. Noack, S. R. White, and D. J. Scalapino, Phys. Rev. Lett. 73, 882 (1994)]

21 × 6 ladder with 12 holes and U = 12t

Among the largest ladder systems studied with DMRG:• One-band Hubbard model with (28 × 6) sites[G. Hager, G. Wellein, E. Jeckelmann, and H. Fehske, Phys. Rev. B, 2005]• Three-band Hubbard ladder with (32 × 2) Cu-sites (and 162 O-sites)[S. Nishimoto, E. Jeckelmann, and D.J. Scalapino, Phys. Rev. B, 2002]

Transfer-matrix renormalization group(TMRG) methods
Density-matrix renormalization of a transfer matrix
1. TMRG for classical statistical systems[T. Nishino, J. Phys. Soc. Jpn. 64, 3598 (1995)]
2. Corner transfer matrix renormalization group[T. Nishino and K. Okunishi, J. Phys. Soc. Jpn. 65 891 (1996)]
3. TMRG for quantum statistical systems[R.J. Bursill, T. Xiang, and G. A. Gehring, J. Phys.: Condens. Matter 8, L583(1996); X.Q. Wang and T. Xiang, Phys. Rev. B 56, 5061 (1997);
N. Shibata, J. Phys. Soc. Jpn. 66, 2221 (1997)]
4. Quantum TMRG ⇒ Dynamical properties at finite temperature[T. Mutou, N. Shibata, and K. Ueda, Phys. Rev. Lett. 81, 4939 (1998);(E) 82, 3727 (1999)]

Dynamical correlation functions with DMRG
11†
χÂ (ω + iη) = − hψ0 |ÂÂ|ψ0 iπE0 + 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

Improved methods: dynamical DMRG and t-DMRG → my second talk on ThursdayDMRG for bosonic systemsProblem: Hilbert space dimension D = ∞ for each boson site
Solution: Reduction of the boson Hilbert space with a density-matrix renormalization
1) Pseudo-site method[E. Jeckelmann and S.R. White, Phys. Rev. B 57, 6376 (1998)]
2) Optimal basis method[C. Zhang, E. Jeckelmann, and S.R. White, Phys. Rev. Lett. 80, 2661 (1998);
A. Weiße, H. Fehske, G. Wellein, and A. R. Bishop Phys. Rev. B 62, R747 (2000)]3) Four-block method [R. J. Bursill, Phys. Rev. B 60, 1643 (1999)]
Recent works:M. Tezuka, R. Arita, and H. Aoki, Phys. Rev. Lett. 95, 226401 (2005)
H. Fehske, G. Wellein, G. Hager, A. Weiße, and A. R. Bishop, Phys. Rev. B 69,165115 (2004)
Lecture notes: E. Jeckelmann and H. Fehske, cond-mat/0510637

Non-local systems
1. Momentum space (site = Bloch state)[S. Nishimoto, E. Jeckelmann, F. Gebhard, and R.M. Noack, Phys. Rev. B 65,165114 (2002); T. Xiang, Phys. Rev. B 53, R10445 (1996)]
2. Quantum chemistry (site = HF or DFT molecular orbital)[S. R. White and R. L. Martin, J. Chem. Phys. 110, 4127 (1999);
J. Hachmann, W. Cardoen, G.K.-L. Chan, e-print arXiv:cond-mat/0606115]
3. Nuclear physics (site = nuclear shell)[S. Pittel and N. Sandulescu, Phys. Rev. C 73 014301 (2006)]
4. Quantum Hall systems (site = orbital in a Landau level)[N. Shibata and D. Yoshioka, Phys. Rev. Lett. 86, 5755 (2001);
J. Phys. Soc. Jpn. 72 664 (2003) ]Possible but computationally demandingDMRG publications from 1994 to 2005
Number of papers, eprints or citations

100806040citations
paperseprints

2001994

1996

1998

2000
Year

2002

2004

2006

Annual number of• published papers on the topic ”density[-]matrix renormalization (OR renormalisation)”• citations to Steve White’s original paper [PRL 69, 2863-2866 (1992)], and• eprints with the string ”density matrix renormalization” in their title or abstract.
(a) and (b) from the ISI Web of Science database at http://www.isinet.com/, (c) from thecond-mat archive of the arXiv e-Print server at http://arxiv.org/.

More about DMRG
• Book: I. Peschel, X. Wang, M. Kaulke, and K. Hallberg (Eds.),
Density-Matrix Renormalization, Springer, Berlin, 1999.
• Review article: U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005).
E-print: cond-mat/0409292.
• Review article: R.M. Noack and S.R. Manmana, Diagonalization- andNumerical Renormalization-Group-Based Methods for Interacting Quantum
Systems in AIP Conf. Proc. 789, 93-163 (2005). E-print: cond-mat/0510321.
• DMRG homepage at http://dmrg.info• Tomotoshi Nishino’s DMRG homepage at
http://quattro.phys.sci.kobe-u.ac.jp/dmrg.html• A DMRG program (C++) for a particle in a box is available on Steve White’s
homepage at http://hedrock.ps.uci.edu/• A non-interacting DMRG package (C++) is included in the latest releaseof the ALPS project (http://alps.comp-phys.org/).

Summary• Density-matrix renormalization group (DMRG) method• Numerical method for correlated systems of spins and fermions• Highly accurate for static properties of one-dimensional local systems• Analysis of DMRG truncation errors• Extensions to finite temperature and 2D classical systems[transfer matrix DMRG (TMRG)] and to bosonic systems• Extensions to higher dimensions and non-local systems arecomputationally expensive• Dynamical properties and time evolution are now possible⇒ more in my second talk on Thursday