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arXiv:cond-mat/9907057 · Journal of Physics Condensed Matter

New boundary bound states in an open quantum spin chain

Zhan-Ning Hu

Published Jan 1, 1999• 0 citations
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Abstract

New boundary bound states (BBS) are found of an integrable model with the magnetic impurities located at the edges of an open Heisenberg spin chain. These bound states carry the real energy and are formed by three or five imaginary modes of the rapidities. These imaginary modes of the rapidities give the non-zero antisymmetric wave functions and the moments of the centers of the bound states are zero. It means that these bound states are arisen by the magnetic impurities and localized at the edges of the correlated system. The Kondo screening occurs for the antiferromagnetic spin chain with the ferromagnetic impurities-electrons exchange interaction.

DOI
10.1088/0953-8984/11/29/102

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New boundary bound states (BBS) are found of an integrable model with the magnetic impurities located at the edges of an open Heisenberg spin chain. These bound states carry the real energy and are formed by three or five imaginary modes of the rapidities. These imaginary modes of the rapidities give the non-zero antisymmetric wave functions and the moments of the centers of the bound states are zero. It means that these bound states are arisen by the magnetic impurities and localized at the edges of the correlated system. The Kondo screening occurs for the antiferromagnetic spin chain with the ferromagnetic impurities-electrons exchange interaction.
We investigate the boundary effects that arise when spin-12\frac{1}{2}21​ impurities interact with the edges of the antiferromagnetic spin-12\frac{1}{2}21​ Heisenberg chain through spin exchange interactions. We consider both cases when the couplings are ferromagnetic or anti-ferromagnetic. We find that in the case of antiferromagnetic interaction, when the impurity coupling strength is much weaker than that in the bulk, the impurity is screened in the ground state via the Kondo effect. The Kondo phase is characterized by the Lorentzian density of states and dynamically generated Kondo temperature TKT_KTK​. As the impurity coupling strength increases, TKT_KTK​ increases until it reaches its maximum value T0=2πJT_0=2πJT0​=2πJ which is the maximum energy carried by a single spinon. When the impurity coupling strength is increased further, we enter another phase, the bound mode phase, where the impurity is screened in the ground state by a single particle bound mode exponentially localized at the edge to which the impurity is coupled. We find that the impurity can be unscreened by removing the bound mode. There exists a boundary eigenstate phase transition between the Kondo and the bound-mode phases, a transition which is characterized by the change in the number of towers of the Hilbert space. The transition also manifests itself in ground state quantities like local impurity density of states and the local impurity magnetization. When the impurity coupling is ferromagnetic, the impurity is unscreened in the ground state; however, when the absolute value of the ratio of the impurity and bulk coupling strengths is greater than 45\frac{4}{5}54​, the impurity can be screened by adding a bound mode that costs energy greater than T0T_0T0​. When two impurities are considered, the phases exhibited by each impurity remain unchanged in the thermodynamic limit, but nevertheless the system exhibits a rich phase diagram.
We investigate the boundary effects that arise when spin-12\frac{1}{2}21​ impurities interact with the edges of the antiferromagnetic spin-12\frac{1}{2}21​ Heisenberg chain through spin exchange interactions. We consider both cases when the couplings are ferromagnetic or antiferromagnetic. We find that in the case of antiferromagnetic interaction, when the impurity coupling strength is much weaker than that in the bulk, the impurity is screened in the ground state via the Kondo effect. The Kondo phase is characterized by the Lorentzian density of states and a dynamical scale, the Kondo temperature TK{T}_{K}TK​, is generated. As the impurity coupling strength increases, TK{T}_{K}TK​ increases until it reaches its maximum value ${T}_{0}=2\ensuremath{\pi}J$ which is the maximum energy carried by a single spinon. When the impurity coupling strength is increased further, we enter another phase, the bound mode phase, where the impurity is screened in the ground state by a single particle bound mode exponentially localized at the edge to which the impurity is coupled. We find that, in contrast to the Kondo phase, the impurity can be unscreened by unoccupying the single particle bound mode. This costs an energy Eb{E}_{b}Eb​ that is greater than T0{T}_{0}T0​. There exists a boundary eigenstate phase transition between the Kondo and the bound-mode phases, a transition which is characterized by the change in the number of towers of the Hilbert space. The transition also manifests itself in local thermodynamic quantities---local impurity density of states and the local impurity magnetization in the ground state. When the impurity coupling is ferromagnetic, the impurity is unscreened in the ground state; however, when the absolute value of the ratio of the impurity and bulk coupling strengths is greater than 45\frac{4}{5}54​, the impurity can be screened by adding a bound mode that costs energy greater than T0{T}_{0}T0​. When two impurities are considered, the phases exhibited by each impurity remain unchanged in the thermodynamic limit, but nevertheless the system exhibits a rich phase diagram.
The authors study here boundary phenomena in a gapped anisotropic Heisenberg spin chain, revealing a rich phase diagram from the interplay of fractionalized spin-\textonequarter{} edge modes and a spin-\textonehalf{} impurity. The study uncovers distinct Kondo and bound mode phases, and a novel phase with midgap excitations for nonintegrable antiferromagnetic coupling, providing new insights into impurity screening and Kondo effects. Surprisingly, impurity screening can occur with nonintegrable ferromagnetic coupling, while the impurity remains unscreened for integrable ferromagnetic coupling.
Boundary phases and thermodynamics of the Kondo spin-sss chain: from overscreened Kondo to boundary-bound states
We study a spin-12\frac1221​ impurity coupled to the boundary of a strongly correlated spin-sss Takhtajan--Babujian chain, an integrable model whose low-energy physics is described by a perturbed SU(2)2sSU(2)_{2s}SU(2)2s​ Wess--Zumino--Witten conformal field theory. While boundary conformal field theory determines the low-energy universality class of the weak-coupling regime, exact Bethe Ansatz methods reveal a sequence of boundary quantum phase transitions in which impurity-bound states emerge and reorganize the Hilbert space into multiple excitation towers built on distinct boundary configurations. This tower restructuring provides the organizing principle for a rich boundary phase diagram extending beyond the conventional Kondo regime. Weak antiferromagnetic coupling realizes the overscreened 2s2s2s-channel Kondo universality class, whereas stronger couplings generate localized boundary modes and qualitatively new screening mechanisms. To describe the resulting thermodynamics, we develop a generalized thermodynamic Bethe Ansatz framework that captures the multi-tower structure across all regimes. The impurity entropy reproduces the boundary conformal field theory prediction in the overscreened Kondo regime but develops pronounced nonmonotonic temperature dependence once boundary-bound states appear, in quantitative agreement with large-scale finite-temperature matrix-product-operator simulations. Complementary dynamical calculations reveal sharp threshold features in the impurity spectral function that directly track the underlying tower structure. Together, boundary conformal field theory, exact Bethe Ansatz, generalized thermodynamic Bethe Ansatz, and tensor-network simulations provide a unified description of impurity screening, boundary-bound-state formation, and excitation-tower reconstruction in a correlated spin-sss chain.
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We study the Heisenberg ferromagnetic spin chain coupled with a boundary impurity. Via Bethe ansatz solution, it is found that (i) for J>0J>0J>0, the impurity spin behaves as a diamagnetic center and is completely screened by 2S2S2S bulk spins in the ground state, no matter how large the impurity spin is; (ii) the specific heat of the local composite (impurity plus 2S2S2S bulk spins which form bound state with it) shows a simple power law ${C}_{\mathrm{loc}}\ensuremath{\sim}{T}^{3/2}$; (iii) for J<0J<0J<0, the impurity is locked into the critical behavior of the bulk. Possible phenomena in higher dimensions are discussed.