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Physical Review B

Signature of the Luttinger liquid phase in alternating Heisenberg spin- 12 chains: Analytical and numerical approaches

Jahanfar Abouie, S. Mahdavifar

Published Nov 25, 2008• 19 citations
Publisher

Abstract

We have studied the zero- and low-temperature behaviors of anisotropic alternating antiferromagnetic-ferromagnetic Heisenberg spin-$\frac{1}{2}$ chains in a transverse magnetic field. Using the analytical spinless fermion approach, the thermodynamic behavior of the model has been studied. We have introduced a different order parameter to distinguish the gapless Luttinger liquid phase from the other gapped phases. The exact diagonalization Lanczos results are also used to compare with the spinless fermion ones. We have found a double peak structure in the specific-heat curves for the region between the two quantum critical points. Using the numerical full diagonalization results, the existence of the double peak structure is confirmed.

DOI
10.1103/physrevb.78.184437

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We have studied the zero- and low-temperature behaviors of anisotropic alternating antiferromagnetic-ferromagnetic Heisenberg spin-12\frac{1}{2}21​ chains in a transverse magnetic field. Using the analytical spinless fermion approach, the thermodynamic behavior of the model has been studied. We have introduced a different order parameter to distinguish the gapless Luttinger liquid phase from the other gapped phases. The exact diagonalization Lanczos results are also used to compare with the spinless fermion ones. We have found a double peak structure in the specific-heat curves for the region between the two quantum critical points. Using the numerical full diagonalization results, the existence of the double peak structure is confirmed.
We have studied the zero and low-temperature behavior of anisotropic alternating antiferromagnetic-ferromagnetic Heisenberg spin-1/2 chains in a transverse magnetic field. Using the analytical spinless fermion approach, the thermodynamic behavior of the model has been studied. We have introduced a new order parameter to distinguish the gapless Luttinger liquid phase from the other gapped phases. The exact diagonalization Lanczos results are also used to compare with the spinless fermion ones. We have found a double peak structure in the specific heat curves for the region between the two quantum critical points. Using the numerical full diagonalization results, the existence of the double peak structure is confirmed.
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.
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.
The S =1/2 bond alternating Heisenberg spin chain in a magnetic field is studied by the numerical diagonalization and the finite size scaling based on the conformal field theory. It is found that the magnetic state (0< m <1) is gapless (the central charge c =1) and described by the Luttinger liquid, which corresponds to a spinless Fermion system with repulsive interaction for antiferromagnetic-antiferromagnetic alternating chains, but attractive one for ferromagnetic-antiferromagnetic chains. The critical exponents δ and η are determined at the critical fields. The results suggest that the phase transitions belong to the same universality class as the S =1 antiferromagnetic chain for both alternating cases.
The ground state phase diagrams and thermodynamic properties of a spin-1 bond-alternating Heisenberg antiferromagnetic chain with a single-ion anisotropy in longitudinal and transverse magnetic fields are investigated jointly by means of the infinite time evolving block decimation, the linearized tensor renormalization group, and the density matrix renormalization group methods. It is found that in the magnetic field--bond alternating ratio plane six phases such as singlet dimer, Haldane, two Tomonaga-Luttinger liquid, 1/2 magnetization plateau, and spin-polarized phases are identified in a longitudinal field, while in a transverse field there are five phases including singlet dimer, Haldane, Z2{Z}_{2}Z2​ symmetry breaking, quasi-1/2 magnetization plateau, and quasi-spin-polarized phases. A reentrant behavior of the staggered magnetization in a transverse field is observed. The quantum critical behaviors in longitudinal and transverse magnetic fields are disclosed to fall into different universality classes with corresponding conformal field central charge c=1c=1c=1 and 1/21/21/2, respectively. The experimental data of the compound NTENP under both longitudinal and transverse magnetic fields are nicely fitted, and the Luttinger liquid behavior of low-temperature specific heat experimentally observed is also confirmed.
The thermodynamics and quantum phase transitions of two typically alternating double-chain systems are investigated by Green’s function theory.(i) For the completely antiferromagnetic(AFM) alternating double-chain, the low-temperature antiferromagnetism with gapped behavior is observed, which is in accordance with the experimental result. In a magnetic field, we unveil the ground state phase diagram with zero plateau, 1/2 plateau, and polarized ferromagnetic(FM) phases,as a result of the intra-cluster spin-singlet competition. Furthermore, the Gr ¨uneisen ratio is an excellent tool to identify the quantum criticality and testify various quantum phases.(ii) For the antiferromagnetically coupled FM alternating chains,the 1/2 magnetization plateau and double-peak structure of specific heat appear, which are also observed experimentally.Nevertheless, the M–h curve shows an anomalous behavior in an ultra-low field, which is ascribed to the effectively weak Haldane-like state, demonstrated by the two-site entanglement entropy explicitly.