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arXiv:cond-mat/9712202 · Journal of Physics Condensed Matter

Mixed Heisenberg chains: II. Thermodynamics

H. Niggemann, G. Uimin, J. Zittartz

Published Jun 15, 1998• 29 citations
arXivHTMLPDF

Abstract

We consider thermodynamic properties, e.g. specific heat and magnetic susceptibility, of alternating Heisenberg spin chains. Due to a hidden Ising symmetry, these chains can be decomposed into a set of finite chain fragments. The problem of finding the thermodynamic quantities is effectively separated into two parts. First we deal with finite objects; secondly we can incorporate the fragments into a statistical ensemble. As functions of the coupling constants, the models exhibit special features in the thermodynamic quantities, e.g. the specific heat displays double peaks at low enough temperatures. These features stem from first-order quantum phase transitions at zero temperature, which have been investigated in the first part of this work.

DOI
10.1088/0953-8984/10/23/021

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We consider thermodynamic properties, e.g. specific heat and magnetic susceptibility, of alternating Heisenberg spin chains. Due to a hidden Ising symmetry, these chains can be decomposed into a set of finite chain fragments. The problem of finding the thermodynamic quantities is effectively separated into two parts. First we deal with finite objects; secondly we can incorporate the fragments into a statistical ensemble. As functions of the coupling constants, the models exhibit special features in the thermodynamic quantities, e.g. the specific heat displays double peaks at low enough temperatures. These features stem from first-order quantum phase transitions at zero temperature, which have been investigated in the first part of this work.
Thermodynamic properties of S=1 antiferromagnetic Heisenberg chains with free and periodic boundaries are investigated by a quantum Monte Carlo method. In particular, temperature dependences of the specific heat, the magnetic susceptibility, and the hidden order parameter, which are inherent to the Haldane phase, are investigated. The specific heat turns out to have a peak at a temperature Tpeak{\mathit{T}}_{\mathrm{peak}}Tpeak​\ensuremath{\sim}2\ensuremath{\Delta}, where \ensuremath{\Delta} is the energy gap of the present model, although the temperature dependence of the specific heat is very similar to the Schottky type. In open chains, due to the fourfold degeneracy of the ground state, the magnetic susceptibility shows a Curie-like divergence regardless of the number of spins in the chain. The amplitude of the Curie-like divergence is consistent with the edge moments (S=1/2). On the other hand, it is confirmed that the long-range hidden order exists only in the ground state. How the hidden order grows as the temperature goes to zero (T\ensuremath{\rightarrow}0) is investigated by introducing an intrinsic correlation length ${\ensuremath{\xi}}_{\mathit{D}}$ of the ordering, which diverges as T\ensuremath{\rightarrow}0. A qualitative difference in fluctuations between the ground state and the finite-temperature states is discussed making use of snapshots of the transformed two-dimensional Ising system on which a Monte Carlo simulation is being performed.
The magnetic and thermodynamic properties of spin-1∕21∕21∕2 Heisenberg diamond chains are investigated in three different cases: (a) J1{J}_{1}J1​, J2{J}_{2}J2​, J3>0{J}_{3}>0J3​>0 (frustrated), (b) J1{J}_{1}J1​, J3<0{J}_{3}<0J3​<0, J2>0{J}_{2}>0J2​>0 (frustrated), and (c) J1{J}_{1}J1​, J2>0{J}_{2}>0J2​>0, J3<0{J}_{3}<0J3​<0 (nonfrustrated), where the bond coupling Ji{J}_{i}Ji​ (i=1,2,3)>0(i=1,2,3)>0(i=1,2,3)>0 stands for an antiferromagnetic (AF) interaction, and <0<0<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)S(q)S(q) shows peaks at wave vectors $q=a\ensuremath{\pi}∕3$ (a=0,1,2,3,4,5)(a=0,1,2,3,4,5)(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∕6m=1∕6m=1∕6 pertains, exhibits the peaks only at q=0q=0q=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 XXZXXZXXZ 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{\mathrm{Cu}}_{3}{(\mathrm{C}{\mathrm{O}}_{3})}_{2}{(\mathrm{O}\mathrm{H})}_{2}Cu3​(CO3​)2​(OH)2​ are fairly compared with our TMRG results.
Thermodynamic properties of a tetrameric bond-alternating Heisenberg spin chain with ferromagnetic-ferromagnetic-antiferromagnetic-antiferromagnetic exchange interactions are studied using the transfer-matrix renormalization group and compared to experimental measurements. The temperature dependence of the uniform susceptibility exhibits typical ferrimagnetic features. Both the uniform and staggered magnetic susceptibilities diverge in the limit $T\ensuremath{\rightarrow}0$, indicating that the ground state has both ferromagnetic and antiferromagnetic long-range orders. A double-peak structure appears in the temperature dependence of the specific heat. Our numerical calculation gives a good account for the temperature and field dependence of the susceptibility, the magnetization, and the specific heat for $\mathrm{Cu}{(3\text{\ensuremath{-}}\mathrm{Clpy})}_{2}{({\mathrm{N}}_{3})}_{2}$ $(3\text{\ensuremath{-}}\mathrm{Clpy}=3\text{\ensuremath{-}}\text{Chloroyridine})$.
The thermodynamic properties of a spin S=1/2S=1/2S=1/2 tetrameric Heisenberg antiferromagnetic chain with alternating interactions AF1-AF2-AF1-F{\text{AF}}_{1}{\text{-AF}}_{2}{\text{-AF}}_{1}\text{-F}AF1​-AF2​-AF1​-F (AF and F denote the antiferromagnetic and ferromagnetic couplings, respectively) are studied by means of the transfer-matrix renormalization-group method and Jordan-Wigner transformation. It is found that in the absence of magnetic field, the thermodynamic behaviors are closely related to the gapped low-lying excitations, and a novel structure with three peaks in the temperature dependence of specific heat is unveiled. In a magnetic field, a phase diagram in the temperature-field plane for the couplings satisfying JAF1=JAF2=JF{J}_{{\text{AF}}_{1}}={J}_{{\text{AF}}_{2}}={J}_{\text{F}}JAF1​​=JAF2​​=JF​ is obtained, in which various phases are identified. The temperature dependence of thermodynamic quantities including the magnetization, susceptibility, and specific heat are studied to characterize the corresponding phases. It is disclosed that the magnetization has a crossover behavior at low temperature in the Luttinger liquid phase, which is shown falling into the same class as that in the S=1S=1S=1 Haldane chain. In the plateau regime, the thermodynamic behaviors alter at a certain field, which results from the crossing of two excitation spectra. By means of the fermion mapping, it is uncovered that the system has four spectra from fermion and hole excitations that are responsible for the observed thermodynamic behaviors.
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.
The frustrated ferromagnetic spin-1∕21∕21∕2 Heisenberg chain is studied by means of a low-energy field theory as well as the density-matrix renormalization group and exact diagonalization methods. First, we study the ground-state phase diagram in a magnetic field and find an ``even-odd'' (EO) phase characterized by bound pairs of magnons in the region of two weakly coupled antiferromagnetic chains. A jump in the magnetization curves signals a first-order transition at the boundary of the EO phase, but otherwise the curves are smooth. Second, we discuss thermodynamic properties at zero field, where we confirm a double-peak structure in the specific heat for moderate frustrating next-nearest-neighbor interactions.