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Physical review. B, Condensed matter

Boundary defects in integrable quantum spin ladders

Jianhui Dai, P. Schlottmann, Yupeng Wang

Published Mar 25, 2002• 3 citations
Publisher

Abstract

We study boundary rung defects in n-leg quantum spin ladders with $\mathrm{SU}{(2}^{n})$ leg symmetry. The rung interactions are parametrized by two coupling parameters J and ${J}^{\ensuremath{'}}$ interpolating between the uniform spin tube ${(J}^{\ensuremath{'}}=J)$ and the open spin ladder ${(J}^{\ensuremath{'}}=0).$ The rung defect at the edge of the ladder has rung coupling parameters ${J}_{d}$ and ${J}_{d}^{\ensuremath{'}}.$ The conditions for the integrability of the model are investigated as a function of n, J, ${J}^{\ensuremath{'}},$ ${J}_{d},$ and ${J}_{d}^{\ensuremath{'}},$ and the Bethe Ansatz equations for the exact solution are derived. The boundary quantum critical behavior is discussed. The conditions for the formation of a local moment (boundary bound state due to the defect) are studied. In the absence of a magnetic field such a bound state is only possible for even-legged spin tubes, while for odd-legged ladders a bound state can only occur in a large magnetic field. Depending on the parameters of the host this local moment may be screened in analogy to the Kondo effect.

DOI
10.1103/physrevb.65.144411

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We study boundary rung defects in n-leg quantum spin ladders with SU(2n)\mathrm{SU}{(2}^{n})SU(2n) leg symmetry. The rung interactions are parametrized by two coupling parameters J and J\ensuremath′{J}^{\ensuremath{'}}J\ensuremath′ interpolating between the uniform spin tube (J\ensuremath′=J){(J}^{\ensuremath{'}}=J)(J\ensuremath′=J) and the open spin ladder (J\ensuremath′=0).{(J}^{\ensuremath{'}}=0).(J\ensuremath′=0). The rung defect at the edge of the ladder has rung coupling parameters Jd{J}_{d}Jd​ and Jd\ensuremath′.{J}_{d}^{\ensuremath{'}}.Jd\ensuremath′​. The conditions for the integrability of the model are investigated as a function of n, J, J\ensuremath′,{J}^{\ensuremath{'}},J\ensuremath′, Jd,{J}_{d},Jd​, and Jd\ensuremath′,{J}_{d}^{\ensuremath{'}},Jd\ensuremath′​, and the Bethe Ansatz equations for the exact solution are derived. The boundary quantum critical behavior is discussed. The conditions for the formation of a local moment (boundary bound state due to the defect) are studied. In the absence of a magnetic field such a bound state is only possible for even-legged spin tubes, while for odd-legged ladders a bound state can only occur in a large magnetic field. Depending on the parameters of the host this local moment may be screened in analogy to the Kondo effect.
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.
A model for a spin-1/2 ladder system with two legs is introduced. It is demonstrated that this model is solvable via the Bethe ansatz method for arbitrary values of the rung coupling J. This is achieved by a suitable mapping from the Hubbard model with appropriate twisted boundary conditions. We determine that a phase transition between gapped and gapless spin excitations occurs at the critical value Jc=1/2{J}_{c}=1/2Jc​=1/2 of the rung coupling.
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.
The physical properties of the edge states of the cut two-leg spin ladder are investigated by means of the bosonization approach. By carefully treating boundary conditions, we derive the existence of spin-1/2 edge states in the spin ladder with a ferromagnetic rung exchange and for the open spin-1 Heisenberg chain. In contrast, such states are absent in the antiferromagnetic rung coupling case. The approach, based on a mapping onto decoupled semi-infinite off-critical Ising models, allows us to compute several physical quantities of interest. In particular, we determine the magnetization and dimerization profiles of the cut two-leg spin ladder and of the open biquadratic spin-1 chain in the vicinity of the SU(2)2\mathrm{SU}{(2)}_{2}SU(2)2​ Wess-Zumino-Novikov-Witten critical point.
The effects of multispin interactions upon quantum spin ladders are investigated theoretically. A two-leg S=12S=\frac{1}{2}S=21​ quantum spin ladder is taken to have intrarung (inter-rung) quadratic (biquadratic) Heisenberg exchange-type interactions and is placed in a uniform magnetic field. Due to the commutability properties of the Hamiltonian, the magnetic canonical partition function of the model is exactly calculable by a method of rung transfer matrix. Exact solutions are then derived for thermodynamic quantities like free and internal energies, magnetization, and magnetic susceptibilities (initial and nonlinear). The zero-temperature results comprise exact solutions in the complete interaction parameter space for the ground-state energy and degeneracy, zero-field spin gap, loci of quantum phase transitions, and the critical magnetic field.