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arXiv:cond-mat/0512489 · Advances In Physics

Integrable models and quantum spin ladders: comparison between theory and experiment for the strong coupling ladder compounds

Murray T. Batchelor, X. W. Guan, N. Oelkers, Z. Tsuboi

Published May 1, 2007• 63 citations
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Abstract

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.

DOI
10.1080/00018730701265383

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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)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 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 .
We study the field-induced quantum phase transitions (QPT) and the relevant magnetic properties of a spin-1/2 two-leg integrable spin ladder (ISL), of which the system parameters in bridging to the real compounds are determined by setting the extra interactions in the Hamiltonian of the ISL relative to the Heisenberg spin ladder to vanish in the expectation in the ground state (GS). Such an ISL analytically has the correct leading terms of both the critical fields of the two QPT's as in the real strongly-coupled compounds: gμ_BH_{c1}=J_{\perp}-J_{\parallel} and gμ_BH_{c2}=J_{\perp}+2J_{\parallel} in terms of the experimental leg (J_{\parallel}) and rung (J_{\perp}) interactions. The symmetric magnetization inflection point is located at gμ_BH_{IP}=J_{\perp}+J_{\parallel}/2. The magnetizations for the GS and at finite temperatures, as well as the susceptibility, show good agreements in various comparisons with the finite-site exact diagonalization, the transfer-matrix renormalization group numerical result, the perturbation theory, and the compounds (5IAP)_2CuBr_4\cdot 2H_2O, Cu_2(C_5H_{12}N_2)_2Cl_4 and (C_5H_{12}N)_2CuBr_4.
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 investigate the thermal and magnetic properties of the integrable su(4)su(4)su(4) ladder model by means of the quantum transfer matrix method. The magnetic susceptibility, specific heat, magnetic entropy, and high field magnetization are evaluated from the free energy derived via the recently proposed method of high temperature expansion for exactly solved models. We show that the integrable model can be used to describe the physics of the strong coupling ladder compounds. Excellent agreement is seen between the theoretical results and the experimental data for the known ladder compounds $(5\mathrm{I}\mathrm{A}\mathrm{P}{)}_{2}{\mathrm{C}\mathrm{u}\mathrm{B}\mathrm{r}}_{4}\ifmmode\cdot\else\textperiodcentered\fi{}2{\mathrm{H}}_{2}\mathrm{O}$, Cu2(C5H12N2)2Cl4{\mathrm{C}\mathrm{u}}_{2}({\mathrm{C}}_{5}{\mathrm{H}}_{12}{\mathrm{N}}_{2}{)}_{2}{\mathrm{C}\mathrm{l}}_{4}Cu2​(C5​H12​N2​)2​Cl4​, etc.
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.
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.