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arXiv:cond-mat/0305633 · New Journal of Physics

Note on the thermodynamic Bethe ansatz approach to the quantum phase diagram of the strong coupling ladder compounds

Murray T. Batchelor, Xi‐Wen Guan, Angela Foerster, Huan-Qiang Zhou

Published Aug 11, 2003• 12 citations
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Abstract

We investigate the low-temperature phase diagram of the exactly solved su (4) two-leg spin ladder as a function of the rung coupling and magnetic field H by means of the thermodynamic Bethe ansatz (TBA). In the absence of a magnetic field the model exhibits three quantum phases, while in the presence of a strong magnetic field there is no singlet ground state for ferromagnetic rung coupling. For antiferromagnetic rung coupling, there is a gapped phase in the regime H < H c1 , a fully polarized gapped phase for H > H c2 and a Luttinger liquid magnetic phase in the regime H c1 < H < H c2 . The critical behaviour derived using the TBA is consistent with the existing experimental, numerical and perturbative results for the strong coupling ladder compounds. This includes the spin excitation gap and the critical fields H c1 and H c2 , which are in excellent agreement with the experimental values for the known strong coupling ladder compounds (5IAP) 2 CuBr 4 ·2H 2 O, Cu 2 (C 5 H 12 N 2 ) 2 Cl 4 and (C 5 H 12 N) 2 CuBr 4 . In addition we predict the spin gap for the weak coupling compounds with , such as (VO) 2 P 2 O 7 , and also show that the gap opens for arbitrary .

DOI
10.1088/1367-2630/5/1/107

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We investigate the low-temperature phase diagram of the exactly solved su (4) two-leg spin ladder as a function of the rung coupling and magnetic field H by means of the thermodynamic Bethe ansatz (TBA). In the absence of a magnetic field the model exhibits three quantum phases, while in the presence of a strong magnetic field there is no singlet ground state for ferromagnetic rung coupling. For antiferromagnetic rung coupling, there is a gapped phase in the regime H < H c1 , a fully polarized gapped phase for H > H c2 and a Luttinger liquid magnetic phase in the regime H c1 < H < H c2 . The critical behaviour derived using the TBA is consistent with the existing experimental, numerical and perturbative results for the strong coupling ladder compounds. This includes the spin excitation gap and the critical fields H c1 and H c2 , which are in excellent agreement with the experimental values for the known strong coupling ladder compounds (5IAP) 2 CuBr 4 ·2H 2 O, Cu 2 (C 5 H 12 N 2 ) 2 Cl 4 and (C 5 H 12 N) 2 CuBr 4 . In addition we predict the spin gap for the weak coupling compounds with , such as (VO) 2 P 2 O 7 , and also show that the gap opens for arbitrary .
This article considers recent advances in the investigation of the thermal and magnetic properties of integrable spin ladder models and their applicability to the physics of strong coupling ladder compounds. For this class of compounds the rung coupling J ⊥ is much stronger than the coupling J ∥ along the ladder legs. The ground state properties of the integrable two-leg spin- and the mixed spin-() ladder models at zero temperature are analysed by means of the Thermodynamic Bethe Ansatz (TBA). Solving the TBA equations yields exact results for the critical fields and critical behaviour. The thermal and magnetic properties of the models are discussed in terms of the recently introduced High Temperature Expansion (HTE) method, which is reviewed in detail. In the strong coupling region the integrable spin- ladder model exhibits three quantum phases: (i) a gapped phase in the regime , (ii) a fully polarized phase for , and (iii) a Luttinger liquid magnetic phase in the regime H c1<H<H c2. The critical behaviour in the vicinity of the critical points H c1 and H c2 is of Pokrovsky-Talapov type. The temperature-dependent thermal and magnetic properties are directly evaluated from the exact free energy expression and compared to known experimental results for the strong coupling ladder compounds (5IAP)2CuBr4· 2H2O, Cu2(C5H12N2)2Cl4, (C5H12N)2CuBr4, BIP-BNO and [Cu2(C2\O 2)(C10H8N2)2)](NO3)2. Similar analysis of the mixed spin-() ladder model reveals a rich phase diagram, with a and a full saturation magnetization plateau within the strong antiferromagnetic rung coupling regime. For weak rung coupling, the fractional magnetization plateau is diminished and a new quantum phase transition occurs. The phase diagram can be directly deduced from the magnetization curve obtained from the exact result derived from the TBA and HTE. The results are applied to the mixed ferrimagnetic ladder compound PNNBNO. The thermodynamics of the spin-orbital model with different single-ion anisotropies is also discussed. For this model single-ion anisotropy can trigger different quantum phase transitions within the spin and orbital degrees of freedom, with magnetization plateaux arising from different spin and orbit Landé g-factors.
We investigate the thermodynamics of an integrable spin ladder model which possesses a free parameter besides rung and leg couplings. The model is exactly solvable by means of the Bethe ansatz and exhibits a phase transition between a gapped and a gapless spin excitation spectrum. The magnetic susceptibility is obtained numerically and its dependence on the anisotropy parameter is determined. The spin gap obtained from the susceptibility curve and the one obtained from the Bethe ansatz equations are in very good agreement. Our results for the magnetic susceptibility fit well the experimental data for the organometallic compounds (5IAP) 2 CuBr 4 ·2H 2 O (Landee C. P. et al. , Phys. Rev. B , 63 (2001) 100402(R)) Cu 2 (C 5 H 12 N 2 ) 2 Cl 4 (Hayward C. A., Poilblanc D. and Lèvy L. P., Phys. Rev. B , 54 (1996) R12649, Chaboussant G. et al. , Phys. Rev. Lett. , 19 (1997) 925; Phys. Rev. B , 55 (1997) 3046.) and (C 5 H 12 N) 2 CuBr 4 (Watson B. C. et al. , Phys. Rev. Lett. , 86 (2001) 5168) in the strong-coupling regime.
We examine the properties of the Bethe ansatz solvable two- and three-leg spin-SSS ladders. These models include Heisenberg rung interactions of arbitrary strength and thus capture the physics of the spin-SSS Heisenberg ladders for strong rung coupling. The discrete values derived for the magnetization plateaux are seen to fit with the general prediction based on the Lieb-Schultz-Mattis theorem. We examine the magnetic phase diagram of the spin-1 ladder in detail and find an extended magnetization plateau at the fractional value 〈M〉=12〈M〉=\frac{1}{2}〈M〉=21​ in agreement with the experimental observation for the organic polyradical spin-1 ladder compound BIP-TENO.
We present an integrable spin ladder model which possesses a free parameter besides the rung and leg couplings.The model is exactly solvable by means of the Bethe ansatz method.We determine the dependence on the anisotropy parameter of the phase transition between gapped and gapless spin excitations and present the phase diagram.Finally, we show that the model is a special case of a more general Hamiltonian with six free parameters.We also investigate the thermodynamics of the model.Specifically, the magnetic susceptibility as a function of the temperature is obtained numerically.The influence of this anisotropy parameter on these physical curves is determined explicitly.A comparsion between the spin gap obtained from the susceptibility curve and that one obtained from the Bethe ansatz equations is performed and a good agreement is found.
The integrable su(1|3)-invariant spin-ladder model with boundary defect is studied using the Bethe ansatz method. The exact phase diagram for the ground state is obtained and the boundary quantum critical behavior is discussed. It consists of a gapped phase in which the rungs of the ladder form singlet states and a gapless Luttinger liquid phase. Depending on the boundary potential of the first rung a boundary bound state may occur. If populated at low temperatures this bound state represents a localized magnetic moment of spin one. In the Luttinger liquid phase the local moment is screened at low temperatures in analogy to the Kondo effect.
The integrable su(4)-invariant spin-ladder model with a boundary defect is studied using the Bethe ansatz method. The exact phase diagram for the ground state is given and the boundary quantum critical behavior is discussed. It consists of a gapped phase in which the rungs of the ladder form singlet states and a gapless Luttinger liquid phase. It is found that in the gapped phase the boundary bound state corresponds to an unscreened local moment, while in the Luttinger liquid phase the local moment is screened at low temperatures in analogy to the Kondo effect.