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Physical review. B./Physical review. B

Kondo effect in the isotropic Heisenberg spin chain

Pradip Kattel, Parameshwar R. Pasnoori, J. H. Pixley, P. Azaria, Natan Andrei

Published May 8, 2024• 17 citations
Publisher

Abstract

We investigate the boundary effects that arise when spin-$\frac{1}{2}$ impurities interact with the edges of the antiferromagnetic spin-$\frac{1}{2}$ 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 ${T}_{K}$, is generated. As the impurity coupling strength increases, ${T}_{K}$ 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 ${E}_{b}$ that is greater than ${T}_{0}$. 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 $\frac{4}{5}$, the impurity can be screened by adding a bound mode that costs energy greater than ${T}_{0}$. 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.

DOI
10.1103/physrevb.109.174416

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We investigate the boundary effects that arise when spin-$\frac{1}{2}$ impurities interact with the edges of the antiferromagnetic spin-$\frac{1}{2}$ 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 $T_K$. As the impurity coupling strength increases, $T_K$ increases until it reaches its maximum value $T_0=2πJ$ 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 $\frac{4}{5}$, the impurity can be screened by adding a bound mode that costs energy greater than $T_0$. 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.

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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 T0=2\ensuremathπJ{T}_{0}=2\ensuremath{\pi}JT0​=2\ensuremathπJ 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.
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.
Abstract We investigate the boundary phenomena that arise in a finite-size XX spin chain interacting through an XX interaction with a spin − 1 2 impurity located at its edge. Upon Jordan–Wigner transformation, the model is described by a quadratic Fermionic Hamiltonian. Our work displays, within this ostensibly simple model, the emergence of the Kondo effect, a quintessential hallmark of strongly correlated physics. We also show how the Kondo cloud shrinks and turns into a single particle bound state as the impurity coupling increases beyond a critical value. In more detail, using both Bethe Ansatz and exact diagonalization techniques, we show that the local moment of the impurity is screened by different mechanisms depending on the ratio of the boundary and bulk coupling J imp J . When the ratio falls below the critical value 2 , the impurity is screened via the Kondo effect. However, when the ratio between the coupling exceeds the critical value 2 an exponentially localized bound mode is formed at the impurity site which screens the spin of the impurity in the ground state. We show that the boundary phase transition is reflected in local ground state properties by calculating the spinon density of states, the magnetization at the impurity site in the presence of a global magnetic field, and the finite temperature susceptibility of the impurity. We find that the spinon density of states in the Kondo phase has the characteristic Lorentzian peak that moves from the Fermi level to the maximum energy of the spinon as the impurity coupling is increased and becomes a localized bound mode in the bound mode phase. Moreover, the impurity magnetization and the finite temperature impurity susceptibility behave differently in the two phases. When the boundary coupling J imp exceeds the critical value 2 J , the model is no longer boundary conformal invariant as a massive bound mode appears at the impurity site.
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.
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.
We study the problem of a boundary magnetic impurity coupled with the solvable $t\ensuremath{-}J$ chain. This model provides a good starting point to understand the Kondo problem in a Luttinger liquid as well as in a strongly correlated host. As the Kondo coupling constant Ji{J}_{i}Ji​ may take arbitrary values without breaking the integrability, we can study the ferromagnetic and antiferromagnetic Kondo problems simultaneously. It is shown that the boundary coupling generally splits the impurity spin into two effective spins and induces the interaction-dependent residual entropy. The absence of Kondo screening in an antiferromagnetic Kondo coupling regime is found, which indicates a genuine competing effect between the Kondo coupling Ji{J}_{i}Ji​ and the impurity potential Vi.{V}_{i}.Vi​. A local Landau-Luttinger liquid description is proposed to calculate the specific heat of the impurity.
The Kondo effect in a one-dimensional spin-12\frac{1}{2}21​ XXZ\mathrm{XXZ}XXZ model in the gapless XY\mathrm{XY}XY regime $(\ensuremath{-}1<\ensuremath{\Delta}<~1)$ is studied both analytically and numerically. In our model an impurity spin (S=1/2)(S=1/2)(S=1/2) is coupled to a single spin in the XXZ\mathrm{XXZ}XXZ spin chain. Perturbative renormalization-group (RG) analysis is performed for various limiting cases to deduce low-energy fixed points. It is shown that in the ground state the impurity spin is screened by forming a singlet with a spin in the host XXZ\mathrm{XXZ}XXZ chain. In the antiferromagnetic side $(0<\ensuremath{\Delta}<~1)$ the host chain is cut into two semi-infinite chains by the singlet. In the ferromagnetic side $(\ensuremath{-}1<\ensuremath{\Delta}<0),$ on the other hand, the host XXZ\mathrm{XXZ}XXZ chain remains as a single chain through ``healing'' of a weakened bond in the low-energy (long-distance) limit. The density-matrix renormalization-group method is used to study the size scaling of finite-size energy gaps and the power-law decay of correlation functions in the ground state. The numerical results are in good agreement with the predictions of the RG analysis. Low-temperature behaviors of specific heat and susceptibility are also discussed.