The problem of the effect of inhomogeneous impurities in a lattice structure on the critical behaviour of systems which undergo a second-order phase transition is studied on a model system, the 2D Ising Model with impurity bonds randomly distributed over the lattice. It is known that in the critical region the 2D Ising Model is equivalent to the model of free fermions. We show that the effect of impurities is to add a four-fermion interaction with the corresponding charge proportional to the concentration of impurities. The resulting fermion model is simple enough and can be studied exactly by renormalization group methods. We show that for any small concentration of impurity bonds a new critical regime is established if we go sufficiently close to the phase-transition point. In particular we find that the specific heat singularity changes from C ˜ ln 1/|τ|(τ=(Τ − Τc/Τc to C ˜ ln ln 1/|τ| for τ≪τi ˜ exp(−const/c i), c i being the concentration of impurities, while the spin-spin correlation function in the critical point changes more seriously: from <σ R σΟ> ˜ 1/R1/4 to <σ R σΟ> ˜ exp { − (const/c i)(ln ln R2 )} for R≫R i˜ exp (const/c i), which corresponds to a change of the critical exponent η from 1/4 to 0.The problem of the effect of inhomogeneous impurities in a lattice structure on the critical behaviour of systems which undergo a second-order phase transition is studied on a model system, the 2D Ising Model with impurity bonds randomly distributed over the lattice. It is known that in the critical region the 2D Ising Model is equivalent to the model of free fermions. We show that the effect of impurities is to add a four-fermion interaction with the corresponding charge proportional to the concentration of impurities. The resulting fermion model is simple enough and can be studied exactly by renormalization group methods. We show that for any small concentration of impurity bonds a new critical regime is established if we go sufficiently close to the phase-transition point. In particular we find that the specific heat singularity changes from C ˜ ln 1/|τ|(τ=(Τ − Τc/Τc to C ˜ ln ln 1/|τ| for τ≪τi ˜ exp(−const/c i), c i being the concentration of impurities, while the spin-spin correlation function in the critical point changes more seriously: from <σ R σΟ> ˜ 1/R1/4 to <σ R σΟ> ˜ exp { − (const/c i)(ln ln R2 )} for R≫R i˜ exp (const/c i), which corresponds to a change of the critical exponent η from 1/4 to 0.

Advances In PhysicsJan 1, 1983285 citations

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Magnetic Impurities in the Honeycomb Kitaev Model

Kusum Dhochak, Ravi Shankar, Vikram Tripathi

We study the effect of coupling magnetic impurities to the honeycomb lattice spin- $1/2$ Kitaev model in its spin-liquid phase. We show that a spin- $S$ impurity coupled to the Kitaev model is associated with an unusual Kondo effect with an intermediate coupling unstable fixed point ${K}_{c}\ensuremath{\sim}J/S$ separating topologically distinct sectors of the Kitaev model. We also show that the massless spinons in the spin-liquid mediate an interaction of the form ${S}_{i\ensuremath{\alpha}}^{2}{S}_{j\ensuremath{\beta}}^{2}/{R}_{ij}^{3}$ between distant impurities unlike the usual dipolar RKKY interaction ${S}_{i\ensuremath{\alpha}}{S}_{j\ensuremath{\alpha}}/{R}_{ij}^{3}$ noted in various 2D impurity problems with a pseudogapped density of states of the spin bath. Furthermore, this long-range interaction is possible only if the impurities (a) couple to more than one neighboring spin on the host lattice and (b) the impurity spin $S\ensuremath{\ne}1/2$.We study the effect of coupling magnetic impurities to the honeycomb lattice spin-$1/2$ Kitaev model in its spin-liquid phase. We show that a spin-$S$ impurity coupled to the Kitaev model is associated with an unusual Kondo effect with an intermediate coupling unstable fixed point ${K}_{c}\ensuremath{\sim}J/S$ separating topologically distinct sectors of the Kitaev model. We also show that the massless spinons in the spin-liquid mediate an interaction of the form ${S}_{i\ensuremath{\alpha}}^{2}{S}_{j\ensuremath{\beta}}^{2}/{R}_{ij}^{3}$ between distant impurities unlike the usual dipolar RKKY interaction ${S}_{i\ensuremath{\alpha}}{S}_{j\ensuremath{\alpha}}/{R}_{ij}^{3}$ noted in various 2D impurity problems with a pseudogapped density of states of the spin bath. Furthermore, this long-range interaction is possible only if the impurities (a) couple to more than one neighboring spin on the host lattice and (b) the impurity spin $S\ensuremath{\ne}1/2$.

arXiv:0912.3881Physical Review LettersSep 7, 201050 citations

We study the effect of coupling magnetic impurities to the honeycomb lattice spin-

1/2

Kitaev model in its spin-liquid phase. We show that a spin-

S

impurity coupled to the Kitaev model is associated with an unusual Kondo effect with an intermediate coupling unstable fixed point ${K}_{c}\ensuremath{\sim}J/S$ separating topologically distinct sectors of the Kitaev model. We also show that the massless spinons in the spin-liquid mediate an interaction of the form ${S}_{i\ensuremath{\alpha}}^{2}{S}_{j\ensuremath{\beta}}^{2}/{R}_{ij}^{3}$ between distant impurities unlike the usual dipolar RKKY interaction ${S}_{i\ensuremath{\alpha}}{S}_{j\ensuremath{\alpha}}/{R}_{ij}^{3}$ noted in various 2D impurity problems with a pseudogapped density of states of the spin bath. Furthermore, this long-range interaction is possible only if the impurities (a) couple to more than one neighboring spin on the host lattice and (b) the impurity spin $S\ensuremath{\ne}1/2$.