We present a numerical study of thermodynamical properties of dimerized frustrated Heisenberg chains down to extremely low temperatures with applications to ${\mathrm{CuGeO}}_{3}$. A variant of the finite temperature density matrix renormalization group (DMRG) allows the study of the dimerized phase previously unaccessible to ab initio calculations. We investigate static dimerized systems as well as the instability of the quantum chain towards lattice dimerization. The crossover from a quadratic response in the free energy to the distortion field at finite temperature to nonanalytic behavior at zero temperature is studied quantitatively. Various physical quantities are derived and compared with experimental data for ${\mathrm{CuGeO}}_{3}$ such as magnetic dimerization, critical temperature, susceptibility, and entropy.
We present a numerical study of thermodynamical properties of dimerized frustrated Heisenberg chains down to extremely low temperatures with applications to CuGeO3. A variant of the finite temperature density matrix renormalization group (DMRG) allows the study of the dimerized phase previously unaccessible to ab initio calculations. We investigate static dimerized systems as well as the instability of the quantum chain towards lattice dimerization. The crossover from a quadratic response in the free energy to the distortion field at finite temperature to nonanalytic behavior at zero temperature is studied quantitatively. Various physical quantities are derived and compared with experimental data for CuGeO3 such as magnetic dimerization, critical temperature, susceptibility, and entropy.
The density matrix renormalization group (DMRG) is a numerical method for studying low dimensional strongly correlated models. The study of these models is one of the most active areas of condensed matter physics today and has been spurred not only by the discovery of high temperature superconductivity but also by experiments on new compounds. In this thesis we study fundamentals of the density matrix renormalization group method as well as applications of the method to Heisenberg spin chains. The introductory text is meant as a brief overview of the DMRG as well as Heisenberg spin systems and provides an introduction to the papers. In papers I and II we investigate the density matrix renormalization group with the aim to better understand the underlying structure of the method. We find that if the renormalization converges to a fixed point in the thermodynamic limit, the resulting basis states can be written as a matrix product form. We also show an explicit relation of the DMRG to variational calculations using the fact that the matrix product states can be rederived through a variational ansatz without any reference to the DMRG calculation. It is then shown how the matrix product states can be used to define a set of Bloch waves for describing elementary excitations of the Heisenberg spin-1 chain. The low lying energy spectrum is calculated. In papers III and IV we investigate the thermodynamics of impurities in a spin-1/2 chain. For this purpose we have shown that a recently developed transfer matrix DMRG method can be used to calculate impurity properties. Both impurity corrections to thermodynamic properties and local properties at finite temperature can be obtained. The method gives quicker and more accurate results than quantum Monte Carlo. The numerical results show good agreement with field theory calculations for a number of impurity configurations and the cross-over function for the magnetic susceptibility from high- to low-temperature behavior shows the expected data collapse. Our results also give quantitative predictions for experiments on doped spin-1/2 chains.
Thermodynamical properties of the one-dimensional S=1/2 Heisenberg model with dimerized nearest and uniform next-nearest neighbor interactions are studied by the numerically exact quantum transfer-matrix method and the results are applied to CuGeO3. The Suzuki-Trotter formula is used to obtain a classical system with spin $\ensuremath{\sigma}=3/2$ and effective interactions between nearest neighbors only. The magnetic susceptibility curve is calculated and compared with experimental results in a wide temperature range, revealing the presence of frustration in the model proposed for CuGeO3. Temperature dependence of the dimerization parameter below the spin-Peierls transition point is also estimated.
We use the density matrix renormalization group (DMRG) method to study the ground and low-lying excited states of three kinds of uniform and dimerized alternating spin chain. The DMRG procedure is also employed to obtain low-temperature thermodynamic properties of these systems. We consider a 2N-site system with spins and alternating from site to site and interacting via a Heisenberg antiferromagnetic exchange. The three systems studied correspond to being equal to and ; all of them have very similar properties. The ground state is found to be ferrimagnetic with total spin . We find that there is a gapless excitation to a state with spin , and a gapped excitation to a state with spin . The DMRG analysis shows that the chain is susceptible to a conditional spin - Peierls instability. Furthermore, our studies of the magnetization, magnetic susceptibility and specific heat show strong magnetic field dependences. The product shows a minimum as a function of temperature T at low magnetic fields; the minimum vanishes at high magnetic fields. This low-field behaviour is in agreement with earlier experimental observations. The specific heat shows a maximum as a function of temperature, and the height of the maximum increases sharply at high magnetic fields. Although all three systems show qualitatively similar behaviour, there are some notable quantitative differences between the systems in which the site spin difference, , is large and small respectively.
Ferromagnetic, in contrast to antiferromagnetic, Heisenberg chains can undergo a spin-Peierls dimerization only at finite temperatures. They show reentrant behavior as a function of temperature, which might play a role for systems with small effective elastic constants as, for example, monatomic chains on surfaces. We investigate the physical properties of the dimerized ferromagnetic Heisenberg chain using a modified spin-wave theory. We calculate the exponentially decaying spin- and dimer-correlation functions, analyze the temperature dependence of the corresponding coherence lengths, the susceptibility, as well as the static and dynamic spin-structure factors. By comparing with numerical data obtained by the density-matrix renormalization group applied to transfer matrices, we find that the modified spin-wave theory yields excellent results for all these quantities for a wide range of dimerizations and temperatures.
The magnetic and thermodynamic properties of spin-1∕2 Heisenberg diamond chains are investigated in three different cases: (a) J1, J2, J3>0 (frustrated), (b) J1, J3<0, J2>0 (frustrated), and (c) J1, J2>0, J3<0 (nonfrustrated), where the bond coupling Ji(i=1,2,3)>0 stands for an antiferromagnetic (AF) interaction, and <0 for a ferromagnetic (F) interaction. The density-matrix renormalization-group (DMRG) technique is invoked to study the properties of the system in the ground state, while the transfer-matrix renormalization-group (TMRG) technique is applied to explore the thermodynamic properties. The local magnetic moments, spin-correlation functions, and static structure factors are discussed in the ground state for the three cases. It is shown that the static structure factor S(q) shows peaks at wave vectors $q=a\ensuremath{\pi}∕3$ (a=0,1,2,3,4,5) for different couplings in a zero magnetic field, which, however, in the magnetic fields where the magnetization plateau with m=1∕6 pertains, exhibits the peaks only at q=0, $2\ensuremath{\pi}∕3$, and $4\ensuremath{\pi}∕3$, which are found to be couplings independent. The DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes, where two fitting equations are proposed. It is observed that the six modes are closely related to the low-lying excitations of the system. At finite temperatures, the magnetization, susceptibility, and specific heat show various behaviors for different couplings. The double-peak structures of the susceptibility, and specific heat against temperature are obtained, where the peak positions and heights are found to depend on the competition of the couplings. It is also uncovered that the XXZ anisotropy of F and AF couplings leads the system of case (c) to display quite different behaviors. In addition, the experimental data of the susceptibility, specific heat, and magnetization for the compound Cu3(CO3)2(OH)2 are fairly compared with our TMRG results.
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). The physical picture of this behavior in this model is also discussed.