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arXiv:cond-mat/0703431 · Physical Review Letters

Effects of Two Energy Scales in Weakly Dimerized Antiferromagnetic Quantum Spin Chains

Andreas Brühl, B. Wolf, В. А. Пащенко, M. Anton, C. Gross, W. Aßmus, Roser Valentí, S. Glocke, Andreas Klümper, Tanusri Saha‐Dasgupta, Badiur Rahaman, Michael Lang

Published Jul 30, 2007• 9 citations
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Abstract

By means of thermal expansion and specific heat measurements on the high-pressure phase of (VO)(2)P(2)O(7), the effects of two energy scales of the weakly dimerized antiferromagnetic S=1/2 Heisenberg chain are explored. The low-energy scale, given by the spin gap Delta, is found to manifest itself in a pronounced thermal expansion anomaly. A quantitative analysis, employing the density-matrix renormalization-group approach for transfer matrices calculations, shows that this feature originates from changes in the magnetic entropy with respect to Delta, partial differentialS(m)/partial differentialDelta. This term, inaccessible by specific heat, is visible only in the weak-dimerization limit, where it reflects peculiarities of the excitation spectrum and its sensitivity to variations in Delta.

DOI
10.1103/physrevlett.99.057204

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By means of thermal expansion and specific heat measurements on the high-pressure phase of (VO)(2)P(2)O(7), the effects of two energy scales of the weakly dimerized antiferromagnetic S=1/2 Heisenberg chain are explored. The low-energy scale, given by the spin gap Delta, is found to manifest itself in a pronounced thermal expansion anomaly. A quantitative analysis, employing the density-matrix renormalization-group approach for transfer matrices calculations, shows that this feature originates from changes in the magnetic entropy with respect to Delta, partial differentialS(m)/partial differentialDelta. This term, inaccessible by specific heat, is visible only in the weak-dimerization limit, where it reflects peculiarities of the excitation spectrum and its sensitivity to variations in Delta.
Combining the thermal coherent states with the bosonization technique,we develop a self_consistent quantum approach to investigate the dimerized antiferromagnetic Heisenberg chain at finite temperature.In the case of weak dimerization,temperature dependences of the magnetic excited energy gap and the specific heat are calculated analytically.The energy gap monotonically decreases with increasing temperature until it vanishes at the Peierls transition temperature T_P.The specific heat is strongly enhanced near T_P.It is also found that there is a scaling behavior of T_P as a function of the zero_temperature dimerization δ_0 as T_P ∝δ~v_0 with v=(0.66) for small δ_0.Our results are qualitatively consistent with experimental data observed in CuGeO_3.
Thermodynamic properties of S=2 antiferromagnetic Heisenberg chains are studied not only under the periodic boundary condition but also under the open one employing a quantum Monte Carlo method. Temperature and size dependences of the energy, the specific heat, and the magnetic susceptibility are calculated and edge effects on them are investigated in detail. The specific heat shows a well-pronounced Schottky anomaly but the maximum is located at a temperature much larger than the Haldane gap of the system. As temperature goes to zero, the magnetic susceptibility vanishes for the periodic chains, while it diverges for the open chains. The edge contribution in the open-chain susceptibility is attributed to the two S=1 effective spins localized at the chain ends at low temperatures, while to an S=2 free spin at high temperatures. This is a visualization of a quantum-classical crossover and an evidence that the present model is a Haldane system. High-temperature behaviors of the thermodynamic quantities are also discussed with the help of a series-expansion method. \textcopyright{} 1996 The American Physical Society.
The effect of dimerization on the random antiferomagnetic Heisenberg chain with spin 1/2 is studied by the density matrix renormalization group method. The ground state energy, the energy gap distribution and the string order parameter are calculated. Using the finite size scaling analysis, the dimerization dependence of the these quantities are obtained. The ground state energy gain due to dimerization behaves as u a with a > 2 where u denotes the degree of dimerization, suggesting the absence of spin-Peierls instability. It is explicitly shown that the string long range order survives even in the presence of randomness. The string order behaves as u 2β with β∼0.37 in agreement with the recent prediction of real space renormalization group theory (β=(3−5)/2≃0.382\beta =(3-\sqrt{5})/2 \simeq 0.382β=(3−5​)/2≃0.382). The physical picture of this behavior in this model is also discussed.
The low energy properties of the spin-1/2 random Heisenberg chain with ferromagnetic and antiferromagnetic interactions are studied by means of the density matrix renormalization group (DMRG) and real space renormalization group (RSRG) method for finite chains. The results of the two methods are consistent with each other. The deviation of the gap distribution from that of the random singlet phase and the formation of the large-spin state is observed even for relatively small systems. For a small fraction of the ferromagnetic bond, the effect of the crossover to the random singlet phase on the low temperature susceptibility and specific heat is discussed. The crossover concentration of the ferromagnetic bond is estimated from the numerical data.
The magnetization process, the susceptibility, and the specific heat of the spin-1∕21∕21∕2 antiferromagnet (AF)-AF-ferromagnet (F) and F-F-AF trimerized quantum Heisenberg chains have been investigated by means of the transfer matrix renormalization group (TMRG) technique as well as the modified spin-wave (MSW) theory. A magnetization plateau at m=1∕6m=1∕6m=1∕6 for both trimerized chains is observed at low temperature. The susceptibility and the specific heat show various behaviors for different ferromagnetic and antiferromagnetic interactions and in different magnetic fields. The TMRG results of susceptibility and the specific heat can be nicely fitted by a linear superposition of double two-level systems, where two fitting equations are proposed. Three branch excitations, one gapless excitation and two gapful excitations, for both systems are found within the MSW theory. It is observed that the MSW theory captures the main characteristics of the thermodynamic behaviors at low temperatures. The TMRG results are also compared with the possible experimental data.
A zero-temperature renormalization group (RG) approach is applied to the one-dimensional, spin-1/2 antiferromagnetic Heisenberg dimerized (alternating) chain. Specifically, the ground state energy and lowest-lying spectral excitations are examined. The calculation indicates the existence of a gap in the spectrum of the dimerized chain which vanishes only in the limit of a uniform spin chain, in contrast to a recent Green’s function approach. The RG results are in reasonable agreement with numerical extrapolations on the exact eigenvalue spectrum of finite chains of up to 12 spins. Both methods are compared with several other approximate treatments of the Heisenberg system, and tested by comparison with exact results for the spin-1/2 XY dimerized chain.