Quasi-1D antiferromagnets in a magnetic field a DMRG study

Institute of Theoretical Physics

University of Lausanne

Switzerland

G. Fath

Spin chains

Motivations:

Quasi-1D AF materials

e.g.: S

S

1

5 2

8 2 2 2 4: )

/ :

Ni(C H N NO ClO / NENP /

CsNiCl

CsMnBr

RbMnBr

2

3

3

3

Structure: Weakly coupled chains forming a triangular lattice

T T

T T

N

N

3D ordering

non - collinear magnetic order

quasi - independent 1D chains

no magnetic LRO due to Coleman' s theorem

(anisotropies may induce order)

:

:

Haldane’s conjecture (1983) for Heisenberg chains

S

S

half - integer

integer different low - energy properties

Colorful T=0 phase diagram in the space of couplings

Simplest toy-models for interacting many-body systems

The compounds of the family ABX are not only studied in relation with the Haldane gap, but because of the interesting phenomenon of spin reorientation in the presence of a magnetic field.

3

The experiments on CsMnBr and RbMnBr showed that the magnetization process is qualitatively well reproduced by a classical spin model at T=0. While the classical calculation overestimates the magnetization for a given field, it under-estimates seriously its directional anisotropy above H .

3 3

c

Experiment at T=1.5 K: ~ 7--10 %Classical model at T=0 : ~ 0--0.5 %

The quantitative inconsistency of the classical T=0 theory iscertainly due to thermal and/or quantum fluctuations.

The experimental temperature T=1.5 K is comparable to the characteristic energy of the anisotropy terms in the Hamiltonian, so thermal fluctuations may also have an important effect.

classical

renormalized classical

experimental points

How to estimate the role of quantum fluctuation:

-- Spin-wave theory is unreliable due to the quasi-1D character of the problem -- strong spin reduction

Effect of thermal fluctuation:

Conclusion:-- Magnetization remains overestimated-- The directional anisotropy is strongly enhanced at T=1.5K

-- The 1D Hamiltonian was studied numerically at T=0 by the DMRG method

/Santini et al, PRB 54, 6327 (1996)/

/Santini et al, to appear in PRB/

Conclusion:-- Magnetization is in accordance with the experimental values-- The directional anisotropy is strongly enhanced

Both the thermal and quantum fluctuations can be responsible for the anisotropy. Experimental studyof the temperature dependence would be welcome.

Above the reorientation transition all the chains respond thesame way to the magnetic field, so the inter-chain couplingJ’ has a negligable effect of O(J’/J ).2

Fluctuation effects above H can be studied using a strictly 1D model.c

Density Matrix Renormalization Group Method

Goal: Find the ground state and low-energy excitations of low-dimensional quantum lattice problems

Difficulty: The number of degrees of freedom increases exponentiallywith the system size

1D: dimension Lanczos diagonalization

S=1/2 Heisenberg: L~30

S=1 Heisenberg: L~20

Hubbard: L~14

2L

3L

4L

Approximative methods:

QMC: error

Wilson's RG: error

White's RG: error

L

L

L

10 10 10

10 10

10 10 10

2 2 4

1 2

2 4 12

(DMRG)

Numerical RG methods

Idea:

Build up the lattice systematically

B

B’

B’’

etc

Truncate the degrees of freedom in the block & Renormalize the block operators

~B

dim = M dim = d1 2 L L+1

1 2 L L+1

dim = M d

B'

1 2 L L+1

dim = Mtruncation

M “important”degrees of freedom

Keep

(d-1)M “unimportant”degrees of freedom

Discard

Note: The accuracy of the RG procedure depens on how we choose the “inportant” degrees of freedom to keep

White’s innovations:

Problem of the boundary condition

Old RG: Block + site is renormalized with open bc independently of the environment

DMRG: Block + site is embedded into a large environment (superblock) to avoid the restrictions coming from a fixed bc

Which states to keep

The question is the optimal unitary transformation which mixesup the degrees of freedom before the truncation process.

Old RG: Diagonalize the block + site Hamiltonian and keep the M lowest energy states

DMRG: Diagonalize the superblock Hamiltonian, form the reduced density matrix of the block + site, diagonalize it, and keep the M states with the highest probability

Approximate number of publications using the DMRG method

0

10

20

30

40

50

60

70

80

'92 '93 '94 '95 '96

Spin chains with different S, Dimerization and frustration, Coupled spin chains, Models with itinerant fermions, Kondo systems, Coupled fermion chains, Systems with single and randomly distributed impurities, Disordered systems, Dynamical correlation functions,Spin chains coupled to phonons, Anderson’s orthogonality chatastrophy problem, 2D classical critical phenomena, 2D quantum problems,

DMRG calculation on RbMnBr and CsMnBr 3 3

Total spin vs chain length x,ztotx,z S LS L/

Note: Symmetry of the model depens on the field direction

H // z: U(1) symmetry Stotz varies discontinuously

0.00 0.02 0.04 0.06 0.08 0.101/L

0.03

0.04

0.05

0.06

0.07

0.08

x,z

z

x

H = 6.85 T

RbMnBr3

Local magnetization as a function of position S nnx,z

Open boundary condition: strong boundary effect

0 20 40 60 80 100n

-1 .0

-0.5

0.0

0.5

1.0

1.5

S zn

H = 6.85 T L = 100RbMnBr3

The rapidly decaying oscillations around the ends are due to egde spins

Edge spins (in the isotropic case):

S=1: Short-range RVB

S=5/2:

effective spin-1/2

effective spin-1

2 SR-RVB + LR-RVBx

Bulk magnetization can be measured here

Conclusions

Problems for further study

• Localization length of edge spins as a function of the magnetic field, screening of edge spins

• Question of a possible quantum phase transition induced by the magnetic field

• Crossover phenomenon in the bulk and surface correlation functions

• Quasi-1D materials behave as 3D or 1D systems depending on the actual parameters (temperature, magnetic field)

• The effect of fluctuations (thermal, quantum) is usually very strong

• Approximations that work very well in 3D may fail for these materials

• The DMRG method proved to be very efficient in simulating low-dimensional quantum lattice problems