Cluster Mean-field approach using DMRG

IFW Dresden, TU Dresden

Satoshi Nishimoto

Contents• Why cluster mean-field?

• Traditional Mean-field theory

• Cluster mean-field approach

• Application to quantum systems

- 1D charge-ordered systems

- 2D frustrated spin system

- 1D and 2D frustrated Hubbard system

- 2D Hubbard systems

- 3D Hubbard system

• Summary and outlook

Why cluster mean-field (CMF) ?

• System size is limited in numerical methods like exact-diagonalization, density-matrix renormalization group methods.

• In general, finite-size scaling analysis is necessary for quantitative discussion on physical quantities and quantum critical point. However, it is quite hard for 2D and 3D quantum systems.

• Can a quantitative discussion be allowed without finite-size scaling?

• High potential of mean-field theory + numerical simulations?

Our aim: to study 2D or 3D quantum systems with CMF + DMRG.

In this presentation, I report the performance of the CMF analysis using exact diagonalization and DMRG (with small clusters).

Mean-field approach a critical point can be estimated with a fixed system size.

MF approximation for the Hubbard model

Four-fermion terms of the interaction is split into two-fermion terms by replacing a pair of a creation and annihilation operator by its expectation value.

D.R. Pann, Phys. Rev. 142, 350 (1966).

Problems of the MF approximation

• The tendencies to symmetry breaking is overestimated since correlations between electrons are totally ignored.

• The importance of the magnetic degrees of freedom is overestimated.

In the Hubbard model,

The MF theory cannot provide a deeper understanding of magnetic moments and the metal-insulator transition, which is actually dominated by electron-electron correlations.

e.g.,- The gap opens for all U>0 on the bipartite-lattice Hubbard model.- It is difficult to apply to non-bipartite (frustrated) systems.

Cluster mean-field approach

• Only a part of cluster (site and/or bond) is replaced by mean field.

• Electron-electron correlations and charge fluctuations can be taken into account within the cluster size.

• As in the classical mean-field theory the local mean field is obtained self-consistently.

mean field

Exactly solved using ED, DMRG, QMC, etc.

cf. dynamical mean-field theory

DMFT can give accurate results even with small cluster

15-site cluster (1 solver + 14 bath sites) can reproduce the 1/U-expansion results very well.

U=5

U=6

Line: 2nd-order of 1/USquares: DMFT + ED

M.P. Eastwood, F. Gebhard, E. Kalinowski, SN, and R.M. Noack, Eur. Phys. J. B 35, 155 (2003).

Infinite-dimensional Hubbard model

Application to the 1D fermion systems- how to implement the CMF -

Spinless-fermion t-V model

The mean-field approximation is applied only on the edge sites 1 and L.

The mean field is replaced by the ``mean-field bond’’ connecting between the sites 1 and L:

The mean fields and are obtained self-consistently (iteratively).

0 1 2 3

-0.4

-0.2

0

0.2

0.4

Jz/J<

S z

i> L=8

L=12

Spinless t-V model vs. XXZ spin model1D XXZ Heisenberg model

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

V/t

<n

i> L=8

L=12

1D spinless t-V model

Entanglement spectra around the transition

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

V/t

<n

i> L=8

L=12

2.0 2.4 2.8

V t/

Entanglement entropy

: Eigenvalues of the reduced density matrix

1D t-U-V model at quarter filling

1D quarter-filled t-U-V model

S. Ejima, F. Gebhard, and SN, Europhys. Lett. 70, 492 (2005). H. Seo and H. Fukuyama, J. Phys. Soc. Jpn. 66, 1249 (1997).

DMRG

3.75

5

4kF Chrage ordering for large U/t and V/t

Fast convergence with system size

2 3 4 5 60

0.2

0.4

0.6

0.8

1U/t = 5

V/t

L = 8

L = 12

DMRG(up to 256 sites)

Application to the 2D frustrated spin system- the CMF works for higher dimensional system? -

Neel order of the J1-J2 model on honeycomb lattice

24-site PBC cluster

3 4 6

87 9

5

10

22

11 12 13 14

15 16 17 18

19 20 21

23 24

1 2

N=24, 54, 96 with DMRG + finite-size scaling

R. Ganesh, J. van den Brink, and SN, PRL 110, 127203 (2013).

Suppression of the Neel order by the frustration

0 0.1 0.20

0.1

0.2

0.3

J2/J1

24-site CMF

DMRG(thermodynamic limit)

<m

1>

1. Staggered moment at the unfrustrated case (J2=0):

m=0.20 (24-site CMF)m=0.28 (DMRG)

Energy gain by the formation of global long-range Neel order is underestimated.

2. Disappearance point of the Neel order:

J2/J1=0.09 (24-site CMF)J2/J1=0.22 (DMRG)

The effect of the geometrical frustration may be overestimated.

Application to the 1D frustrated Hubbard system- Improvement of the CMF for frustrated systems -

CMF approach for frustrated systems

• Incommensurability of frustrated systems is inconvenient to periodic (small) systems.

• Open boundary conditions could (sometimes) resolve the incom-mensurability problem.

• Note that the open-edge effects do not pose other crucial problems.

Does the CMF work for the open system?

L/2 L/2+1

t

UMott transition of the 1D half-filled Hubbard model

1. Taking one solver site:

2. Taking two solver sites:

Eventually,

Wave function is even for open chain

10-4 10-3 10-2 10-1 10010-6

10-5

10-4

10-3

10-2

10-1

<m

L/2

>

U/t

2 solvers

1 solver

PBC

0 1 2 3 4 5 610-2

10-1

1

U/t

OBC

Neel state is incompatible with even-site open chain.

S.R. White and R.M. Noack, PRL 68, 3487 (1992)

CMF for a chain with odd number of sites

10-4 10-3 10-2 10-1 1 1010-6

10-5

10-4

10-3

10-2

10-1

1

U/t

OBC

<m

(L+

1)/

2>

9 sites, 1 solver

Taking the central site as one solver:

solver site

CMF calculation may be possible for open system with odd number of sites.

Mott transition of 1D frustrated t1-t2-U model

In the U=0+ limit, the Mott transition occurs at t2/t1=0.5.

For t2/t1>0.5, the Mott transition is expected to occur at finite U.

- Large quantum fluctuations- Non-systematic size-dependence

One of the most difficult 1D cases for DMRG.

Correlation strength at the Mott transition

mean-field solver

12 sites, PBC

9 sites, OBC

9 sites, OBC

12 sites, PBC

The 9-site OBC points look closer to the DMRG ones than the 12-site PBC results.

G.I. Japaridze, R.M. Noack, D. Baeriswyl, and L. Tincani, PRB 76, 115118 (2007)

1D half-filled t-U-V model

BOW

CDW-BOW transition

SDW-BOW transition

8-site results agree to large DMRG calculations.

charge disproportionation ?

‘Neel’ order ?

Application to the 2D and 3D Hubbard models- Mott transition for higher dimensional system -

Half-filled Hubbard model on (1) square lattice(2) triangular lattice(3) honeycomb lattice(4) cubic lattice

2D half-filled Hubbard model on square lattice

10-1 1 1010-2

10-1

1

<m

cen

ter>

3x3 5x5

mean-field solver

Uc = 0 can be reproduced.Open cluster with odd x odd sites

2D half-filled Hubbard model on triangular lattice

mean-field solver

N = 28

N = 25

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

U/t

N = 28 N = 25

2D half-filled Hubbard model on honeycomb lattice

Z.Y. Meng et al, Nature 464, 847 (2010)

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

U/t

N = 13 N = 37mean-field solver

3D half-filled Hubbard model on cubic lattice

1 1010-2

10-1

1

U/t<

mce

nte

r>mean-field solver

Summary and outlook

• (Semi-)quantitative estimation of phase boundary may be possible in some measure within relatively small clusters.

• CMF can take open clusters, which may be more convenient for frustrated systems.

• Extension to the multi-orbital systems.

• Extension to 3D systems.