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.: Structure: Weakly coupled chains forming a triangular lattice Haldane’s conjecture (1983) for Heisenberg chains 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. 33 c Experiment at T=1.5 K: ~ % Classical model at T=0 : ~ % The quantitative inconsistency of the classical T=0 theory is certainly 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 study of the temperature dependence would be welcome. Above the reorientation transition all the chains respond the same way to the magnetic field, so the inter-chain coupling J’ 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 exponentially with the system size 1D: dimension Lanczos diagonalization S=1/2 Heisenberg: L~30 S=1 Heisenberg: L~20 Hubbard: L~14 Approximative methods: (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 dim = Mdim = d 1 2 L L+1 dim = M d 1 2 L L+1 dim = M truncation 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 mixes up 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 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 33 Note: Symmetry of the model depens on the field direction H // z: U(1) symmetry H = 6.85 T RbMnBr 3

Open boundary condition: strong boundary effect H = 6.85 T L = 100 RbMnBr 3 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-RVB x 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