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arXiv:cond-mat/9610164 · Physical Review Letters

Thermodynamic Density Matrix Renormalization Group Study of the Magnetic Susceptibility of Half-Integer Quantum Spin Chains

S. Moukouri, Laurent G. Caron

Published Nov 25, 1996• 39 citations
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Abstract

It is shown that White's density matrix renormalization group technique can be adapted to obtain thermodynamic quantities. As an illustration, the magnetic susceptibility of Heisenberg $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ and $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3/2$ spin chains are computed. A careful finite size analysis is made to determine the range of temperatures where the results are reliable. For the $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ chain, the comparison with the exact Bethe ansatz curve shows an agreement within $1%$ down to $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.05J$.

DOI
10.1103/physrevlett.77.4640

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It is shown that White's density matrix renormalization group technique can be adapted to obtain thermodynamic quantities. As an illustration, the magnetic susceptibility of Heisenberg S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 and S=3/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3/2S=3/2 spin chains are computed. A careful finite size analysis is made to determine the range of temperatures where the results are reliable. For the S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 chain, the comparison with the exact Bethe ansatz curve shows an agreement within 11%1 down to T=0.05JT\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.05JT=0.05J.
Thermodynamic properties of the quantum Heisenberg spin chains with S=1/2,S=1/2,S=1/2, 1, and 3/23/23/2 are investigated using the transfer-matrix renormalization-group method. The temperature dependence of the magnetization, susceptibility, specific heat, spin-spin correlation length, and several other physical quantities in a zero or finite applied field are calculated and compared. Our data agree well with the Bethe ansatz, exact diagonalization, and quantum Monte Carlo results and provide further insight into the quantum effects in the antiferromagnetic Heisenberg spin chains.
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Exact diagonalization of small model systems gives the thermodynamics of spin chains or quantum cell models at high temperatures TTT. Density matrix renormalization group calculations of progressively larger systems are used to obtain excitations up to a cutoff WC{W}_{C}WC​ and the low-TTT thermodynamics. The hybrid approach is applied to the magnetic susceptibility $\ensuremath{\chi}(T)$ and specific heat C(T)C(T)C(T) of spin-1/21/21/2 chains with isotropic exchange such as the linear Heisenberg antiferromagnet and the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ model with ferromagnetic J1<0{J}_{1}<0J1​<0 and antiferromagnetic J2>0{J}_{2}>0J2​>0. The hybrid approach is fully validated by comparison with Heisenberg antiferromagnet results. It extends ${J}_{1}\ensuremath{-}{J}_{2}$ thermodynamics down to $T\ensuremath{\sim}0.01|{J}_{1}|$ for ${J}_{2}/|{J}_{1}|\ensuremath{\ge}{\ensuremath{\alpha}}_{c}=1/4$ and is consistent with other methods. The criterion for the cutoff WC(N){W}_{C}(N)WC​(N) in systems of NNN spins is discussed. The cutoff leads to bounds for the thermodynamic limit that are best satisfied at a specific T(N)T(N)T(N) at system size NNN.
We calculate the impurity susceptibility for a spin-1/2 magnetic impurity symmetrically coupled to its two nearest-neighbor spins in an antiferromagnetic Heisenberg spin-1/2 chain. This is done by extending White's density-matrix renormalization group (DMRG) technique to finite temperatures. It is found that the impurity susceptibility satisfies Curie law at low temperatures, and that the Curie constant decreases rapidly with increasing coupling constant between the impurity spin and the spins in the chain. For strong couplings, a logarithmic dependence of the susceptibility on temperature emerges. The Curie law behavior agrees with the DMRG analysis at zero temperature but deviates from the prediction of conformal field theory.
By making use of the Lanczos and the density-matrix renormalization-group methods, we study the zero-temperature properties of a spin-1/2 antiferromagnetic Heisenberg chain system with a spin-1/2 magnetic impurity symmetrically coupled to its two nearest-neighbor spins. A comparison is made with the results for a magnetic impurity coupled only to the nearest-neighbor spin in the chain. The binding energy and the spin correlations are calculated. The matrix elements of the impurity spin operators are also calculated between several low-energy states. By carefully analyzing finite-size effects, we show that the dynamical structure factor of the impurity contains a \ensuremath{\delta}-function contribution at zero frequency which does not vanish in the thermodynamic limit for the case of symmetrical coupling. This indicates a partial decoupling of the impurity spin from the spins in the chain, and therefore the impurity susceptibility is predicted to show a Curie behavior. This contradicts the two-channel Kondo behavior suggested from analytic considerations.