Poor man's scaling: anisotropic Kondo and Coqblin–Schrieffer models

The Hamiltonian we start from is [4]

$\begin{eqnarray}H & = & {H}_{0}+V\\ & = & \displaystyle \sum _{{\bf{k}}\alpha }{\epsilon }_{{\bf{k}}}{c}_{{\bf{k}}\alpha }^{\dagger }{c}_{{\bf{k}}\alpha }+\displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{{\bf{kk}}^{\prime} }{\alpha \beta ,{ab}}}{V}_{\beta \alpha ,{ba}}{X}_{{ba}}{c}_{{\bf{k}}^{\prime} \beta }^{\dagger }{c}_{{\bf{k}}\alpha },\end{eqnarray} \tag{ 1 }$

where ${c}_{{\bf{k}}\alpha }^{\dagger }$ and c_kα are electron creation and annihilation operators of itinerant electron with wave vector k and internal quantum number α, _k is the energy of the electron; ${X}_{{ba}}=| b\rangle \langle a|$, where $| a\rangle ,\,| b\rangle$ are the internal states of the scattering system, is the Hubbard X-operator.

In this paper we use the old-fashioned on-the-energy-shell perturbation theory [29] following the paper by Kondo; similar approach was applied to the Anderson model by Haldane [30]. For the Hamiltonian (1) the transition probability per unit time from the initial state of the whole system m to the final state n is given to the second Born approximation by [1]

$\begin{eqnarray}&&{W}_{n\leftarrow m}=2\pi \delta ({E}_{n}-{E}_{m})| {T}_{{nm}}{| }^{2},\end{eqnarray} \tag{ 2 }$

where

$\begin{eqnarray}&&{T}_{{nm}}={V}_{{nm}}+\displaystyle \sum _{{\ell }}\displaystyle \frac{{V}_{n{\ell }}{V}_{{\ell }m}}{{E}_{m}-{E}_{{\ell }}}+...\end{eqnarray} \tag{ 3 }$

is the scattering matrix. Explicitly equation (3) takes the form [1]

$\begin{eqnarray}{T}_{\beta \alpha ,{ba}} & = & {V}_{\beta \alpha ,{ba}}+\displaystyle \sum _{\gamma ,c}{V}_{\beta \gamma ,{bc}}{V}_{\gamma \alpha ,{ca}}\displaystyle \sum _{{\epsilon }_{{\bf{q}}}\gt 0}\displaystyle \frac{1}{\epsilon -{\epsilon }_{{\bf{q}}}}\\ & & -\displaystyle \sum _{\gamma ,c}{V}_{\gamma \alpha ,{bc}}{V}_{\beta \gamma ,{ca}}\displaystyle \sum _{{\epsilon }_{{\bf{q}}}\lt 0}\displaystyle \frac{1}{{\epsilon }_{{\bf{q}}}-\epsilon }.\end{eqnarray} \tag{ 4 }$

Writing equation (4) we assumed that in the state m all the electron states below the Fermi surface (corresponding to _k = 0) are occupied, and there is an additional electron with the wave vector k and internal quantum number α; the scattering system is in the state $| a \rangle$. In the state n all the electron states below the Fermi surface are again occupied, and there is an additional electron with the wave vector ${\bf{k}}^{\prime}$ (${\epsilon }_{{\bf{k}}^{\prime} }={\epsilon }_{{\bf{k}}}=\epsilon$) and internal quantum number β; the scattering system is in the state $| b \rangle$.

In the R.H.S. of equation (4) the second term describes the processes when the electron with kα is first scattered to the unoccupied state qγ and then to ${\bf{k}}^{\prime} \beta$, and the third term describes the processes when an electron from an occupied state qγ is first scattered to ${\bf{k}}^{\prime} \beta$ and then the electron with kα fills up the state qγ which is now empty [1].

Equation (4) clearly explains the connection between the dynamics of the scattering system and the Kondo effect. For static impurity, equation (3) takes the form

$\begin{eqnarray}&&{T}_{\beta \alpha }={V}_{\beta \alpha }+\displaystyle \sum _{\gamma ,{\bf{q}}}\displaystyle \frac{{V}_{\beta \gamma }{V}_{\gamma \alpha }}{\epsilon -{\epsilon }_{{\bf{q}}}}+....\end{eqnarray} \tag{ 5 }$

Because all the integrals with respect to energy in perturbation series terms are understood in the Principal Value sense, the denominator going to zero is by itself not a problem, and the second order terms just gives a correction to the matrix element of the order of the ratio of the scattering energy V to the band width (we consider electron band  ∈ [−D₀, D₀] and assume that the ratio is V/D is small). In addition, this correction is only weakly -dependent. On the other hand, each of the second order terms in equation (4) contains large logarithmic multiplier $\mathrm{ln}(\epsilon /{D}_{0})$, because of strongly asymmetric range of integration [2]; due to the existence of the impurity quantum numbers, these terms do not add up to equation (5).

Equation (4), as it is written down, allows to calculate scattering at high temperatures. However, it allows more—to obtain a scaling equation for the Kondo model in the framework of the approach pioneered by Anderson [2, 3], which allows to selectively sum up the infinite perturbation series, using explicitly only the first two terms of such an expansion, as presented in equation (3).

We are interested only in the matrix elements between the electron states at a distance from the Fermi energy much less than the band width. The brilliant idea of Anderson, applied to the present situation, consists in reducing the band width of the itinerant electrons from [−D₀, D₀] to $[-{D}_{0}-{dD},{D}_{0}+{dD}]$ (dD < 0) and taking into account the terms which corresponded to summation in equation (4) with respect to the electron states in energy intervals $[-{D}_{0},-{D}_{0}-{dD}]$ and $[{D}_{0}+{dD},{D}_{0}]$ by renormalizing ${V}_{\beta \alpha ,{ba}}$. Notice that such renormalization can be performed provided that $| \epsilon | \ll {D}_{0}$ and hence can be discarded in the denominators of the second order terms in equation (4). In our particular case, like in general, renormalization is the reduction of the Hilbert space ${ \mathcal H }$ and changing the Hamiltonian H so as to keep the physical observable T constant. Thus we obtain

$\begin{eqnarray}&&{{dV}}_{\beta \alpha ,{ba}}=\rho \displaystyle \sum _{\gamma ,c}[{V}_{\beta \gamma ,{bc}}{V}_{\gamma \alpha ,{ca}}-{V}_{\gamma \alpha ,{bc}}{V}_{\beta \gamma ,{ca}}]\displaystyle \frac{{dD}}{{D}_{0}},\end{eqnarray} \tag{ 6 }$

where ρ is the density of states of itinerant electrons (assumed to be constant).

Poor man's scaling consists in changing equation (6) to

$\begin{eqnarray}&&{{dV}}_{\beta \alpha ,{ba}}(D)\\ &&\quad =\,\rho \displaystyle \sum _{\gamma ,c}[{V}_{\beta \gamma ,{bc}}(D){V}_{\gamma \alpha ,{ca}}(D)-{V}_{\gamma \alpha ,{bc}}(D){V}_{\beta \gamma ,{ca}}(D)]\displaystyle \frac{{dD}}{D},\end{eqnarray} \tag{ 7 }$

where D is now a running parameter. From equation (7) we obtain the scaling equation

$\begin{eqnarray}&&\displaystyle \frac{{{dV}}_{\beta \alpha ,{ba}}}{d\mathrm{ln}\,{\rm{\Lambda }}}=\rho \displaystyle \sum _{\gamma ,c}[{V}_{\beta \gamma ,{bc}}{V}_{\gamma \alpha ,{ca}}-{V}_{\gamma \alpha ,{bc}}{V}_{\beta \gamma ,{ca}}],\end{eqnarray} \tag{ 8 }$

where Λ = D/D₀ (actually, the change of (implied) argument of each matrix element D → Λ in equation (8) was done for no reason), and, of course, equation (1) should be now understood as

$\begin{eqnarray}&&H=\displaystyle \sum _{{\bf{k}}\alpha }{\epsilon }_{{\bf{k}}}{c}_{{\bf{k}}\alpha }^{\dagger }{c}_{{\bf{k}}\alpha }+\displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{{\bf{kk}}^{\prime} }{\alpha \beta ,{ab}}}{V}_{\beta \alpha ,{ba}}({\rm{\Lambda }}){X}_{{ba}}{c}_{{\bf{k}}^{\prime} \beta }^{\dagger }{c}_{{\bf{k}}\alpha }.\end{eqnarray} \tag{ 9 }$

Let the matrix V_{βα, ba} is presented as a sum of direct products of matrices, acting in ab and αβ spaces respectively

$\begin{eqnarray}&&V=\displaystyle \sum _{p\pi }{G}_{p}\otimes {{\rm{\Gamma }}}_{\pi }{c}_{p\pi },\end{eqnarray} \tag{ 10 }$

where the set of matrices {G_p} is closed with respect to commutation and hence generates some Lie algebra g; so is the set of matrices {Γ_p} (Lie algebra γ).

With the help of equation (10) we can write down equation (8) in a more transparent form

$\begin{eqnarray}&&\displaystyle \sum _{p\pi }{G}_{p}\otimes {{\rm{\Gamma }}}_{\pi }\displaystyle \frac{{{dc}}_{p\pi }}{d\mathrm{ln}\,{\rm{\Lambda }}}\\ &&\quad =\,\displaystyle \frac{1}{2}\rho \displaystyle \sum _{s\sigma t\tau }[{G}_{s},{G}_{t}]\otimes [{{\rm{\Gamma }}}_{\sigma },{{\rm{\Gamma }}}_{\tau }]{c}_{s\sigma }{c}_{t\tau }.\end{eqnarray} \tag{ 11 }$

Introducing structure constants of the Lie algebras g and γ as f^p_st and ${\varphi }_{\sigma \tau }^{\pi }$

$\begin{eqnarray}&&[{G}_{s},{G}_{t}]=i\displaystyle \sum _{p}{f}_{{st}}^{p}{G}_{p},\ \ \ [{{\rm{\Gamma }}}_{\sigma },{{\rm{\Gamma }}}_{\tau }]=i\displaystyle \sum _{\pi }{\varphi }_{\sigma \tau }^{\pi }{{\rm{\Gamma }}}_{\pi },\end{eqnarray} \tag{ 12 }$

we can write down equation (8) in an even more transparent form

$\begin{eqnarray}&&\displaystyle \frac{{{dc}}_{p\pi }}{d\mathrm{ln}\,{\rm{\Lambda }}}=-\displaystyle \frac{1}{2}\rho \displaystyle \sum _{{st}\sigma \tau }{f}_{{st}}^{p}{\varphi }_{\sigma \tau }^{\pi }{c}_{s\sigma }{c}_{t\tau }.\end{eqnarray} \tag{ 13 }$

Typically, the {G_p} ({Γ_π}) appear as infinitesimal operators of some Lie group G (Γ), and hence are Hermitian. Actually, this can be said the other way round. Assuming that the matrices {G_p} ({Γ_π}) are Hermitian, we see that the algebra g (γ) is real, and, hence, by Lie's third theorem is the Lie algebra of some simply connected Lie group [31]. Anyway, if {G_p} ({Γ_π}) are Hermitian, the matrix c_pπ is real.

Consider an important particular case when γ ≡ g. (Matrices {G_p} and {Γ_π} not necessarily realize the same representation of the algebra, but have the same commutation relations; to emphasize that we will designate {Γ_π} as {Γ_p}.) If we assume that the matrix c_pπ, in addition to being real, is symmetric, it can be diagonalized by a unitary transformation of the generators (such transformation does not change the commutation relations), and the matrix keeps its diagonal form in the process of renormalization. Thus equation (10) can be 'reduced to the principal axes', that is to the form

$\begin{eqnarray}&&V=\displaystyle \sum _{p}{G}_{p}\otimes {{\rm{\Gamma }}}_{p}{c}_{p},\end{eqnarray} \tag{ 14 }$

and equation (13) takes the form

$\begin{eqnarray}&&\displaystyle \frac{{{dc}}_{p}}{d\mathrm{ln}\,{\rm{\Lambda }}}=-\displaystyle \frac{1}{2}\rho \displaystyle \sum _{{st}}{({f}_{{st}}^{p})}^{2}{c}_{s}{c}_{t}.\end{eqnarray} \tag{ 15 }$

Equation (15) will be solved in section 3. General analysis of the equation for possible three-dimensional Lie algebras will be presented in appendix A.

The results of the previous section can be easily generalized to the case when the electron dispersion law determines the power law dependence of the DOS upon the energy

$\begin{eqnarray}&&\rho (\epsilon )=C| \epsilon {| }^{r},\ \ \ \mathrm{if}\ \ | \epsilon | \lt {D}_{0},\end{eqnarray} \tag{ 16 }$

where r can be either positive or negative [12]. (We consider in this paper only particle-hole symmetric DOS; the influence of high particle-hole asymmetry on Kondo effect was studied in [32].) Notice that r = 0 corresponds to the previously considered case of flat DOS. In this case instead of equation (8) we obtain

$\begin{eqnarray}&&\displaystyle \frac{{{dV}}_{\beta \alpha ,{ba}}}{d\mathrm{ln}\,{\rm{\Lambda }}}={{rV}}_{\beta \alpha ,{ba}}+G\displaystyle \sum _{\gamma ,c}[{V}_{\beta \gamma ,{bc}}{V}_{\gamma \alpha ,{ca}}-{V}_{\gamma \alpha ,{bc}}{V}_{\beta \gamma ,{ca}}],\end{eqnarray} \tag{ 17 }$

where G = CD₀^r is the DOS at the original band edges. Equation (17) is invariant with respect to simultaneous change of sign of r, of all matrix elements of V and of the direction of flow. Hence further on we consider explicitly only the case of positive r.

The appearance of the linear term in the R.H.S. of equation (17) demands explanation [12, 33]. To get scaling equation, in addition to integrating out the states at the band edges, another procedure is needed, both for the case of flat DOS, and for the case of power law DOS. We did not mention it in the previous section, because it does not change the equation, but in the case of power law DOS it does. After integrating out the states at the band edges we have to restore the original band width, which demands decrease of the unit of energy by a factor of ${D}_{0}/D$. To keep the DOS constant we should increase the unit of volume by the factor ${({D}_{0}/D)}^{r+1}$. One should understand that in equation (1), V has units of energy multiplied by volume, so after all the rescalings the perturbation is multiplied by the factor of ${(D/{D}_{0})}^{r}$, which explains the appearance of the linear term.

To make equation (17) look similar to equation (8) we introduce new variables $\lambda ={(D/{D}_{0})}^{r}$ and $\widetilde{V}=V/r\lambda$, after which we can write down equation (17) as

$\begin{eqnarray}&&\displaystyle \frac{d{\widetilde{V}}_{\beta \alpha ,{ba}}}{d\lambda }=G\displaystyle \sum _{\gamma ,c}[{\widetilde{V}}_{\beta \gamma ,{bc}}{\widetilde{V}}_{\gamma \alpha ,{ca}}-{\widetilde{V}}_{\gamma \alpha ,{bc}}{\widetilde{V}}_{\beta \gamma ,{ca}}].\end{eqnarray} \tag{ 18 }$

Previously in this section and in section 2 we studied how the effective perturbation at a given energy changes, when the cut-off changes? Alternatively, we can ask ourselves: how does the effective perturbation at a given cut-off change, when the energy changes? That is we are interested in

$\begin{eqnarray}&&{{dT}}_{\beta \alpha ,{ba}}(\epsilon )\equiv {T}_{\beta \alpha ,{ba}}(\epsilon +d\epsilon )-{T}_{\beta \alpha ,{ba}}(\epsilon )\end{eqnarray} \tag{ 19 }$

(d can be of any sign). Looking at equation (4) we understand that

$\begin{eqnarray}{{dT}}_{\beta \alpha ,{ba}}(\epsilon ) & = & \displaystyle \sum _{\gamma ,c}{V}_{\beta \gamma ,{bc}}{V}_{\gamma \alpha ,{ca}}\displaystyle \sum _{-d\epsilon \lt {\epsilon }_{{\bf{q}}}\lt 0}\displaystyle \frac{1}{\epsilon -{\epsilon }_{{\bf{q}}}}\\ & & -\displaystyle \sum _{\gamma ,c}{V}_{\gamma \alpha ,{bc}}{V}_{\beta \gamma ,{ca}}\displaystyle \sum _{0\lt {\epsilon }_{{\bf{q}}}\lt d\epsilon }\displaystyle \frac{1}{{\epsilon }_{{\bf{q}}}-\epsilon }\\ & & =\rho \displaystyle \sum _{\gamma ,c}[{V}_{\beta \gamma ,{bc}}{V}_{\gamma \alpha ,{ca}}-{V}_{\gamma \alpha ,{bc}}{V}_{\beta \gamma ,{ca}}]\displaystyle \frac{d\epsilon }{\epsilon }.\end{eqnarray} \tag{ 20 }$

(for the sake of definiteness we have chosen d > 0). Scaling equation we obtain by changing V in the last line of equation (20) to T. Thus we recover equation (8), only this time ${\rm{\Lambda }}=| \epsilon | /{D}_{0}$. Similarly, for the power law DOS, we recover equation (17), with $\lambda ={(| \epsilon | /{D}_{0})}^{r}$.

The CS model [15] is represented by the Hamiltonian

$\begin{eqnarray} & & H=\displaystyle \sum _{m}\epsilon {c}_{m}^{\dagger }{c}_{m}+J\displaystyle \sum _{{mm}^{\prime} }{X}_{{mm}^{\prime} }{c}_{m^{\prime} }^{\dagger }{c}_{m}\\ & & -{(J/N)\displaystyle \sum _{{mm}^{\prime} }{X}_{{mm}}{c}_{m^{\prime} }^{\dagger }{c}_{m^{\prime} }},\end{eqnarray} \tag{ 29 }$

where quantum number m changes from 1 to N. (To avoid cluttering, we omit in the equations in this section the wave vector indices.) We changed sign of the constant J with respect to the original paper [15], so that the Hamiltonian (29) would look like the Hamiltonian (21). The last term in the R.H.S. of equation (29) (and in similar equations further on) is crossed-out to show that it does not give any contribution to the scaling equation, which for the Hamiltonian (29) (and the constant DOS) has the form [2]

$\begin{eqnarray}&&\displaystyle \frac{{dJ}}{d\mathrm{ln}\,{\rm{\Lambda }}}=-N\rho {J}^{2}.\end{eqnarray} \tag{ 30 }$

For N = 2 the model coincides with the spin-isotropic Kondo model.

If we demand that interaction (24) has SU(2) symmetry, it takes the form

$\begin{eqnarray}&&V=J\overrightarrow{S}\cdot \overrightarrow{\sigma }.\end{eqnarray} \tag{ 31 }$

If we reduce the symmetry to U(1), interaction (24) takes the form

$\begin{eqnarray}&&V={J}_{x}({S}^{x}{\sigma }^{x}+{S}^{y}{\sigma }^{y})+{J}_{z}{S}^{z}{\sigma }^{z}.\end{eqnarray} \tag{ 32 }$

we will call such exchange interaction the XXZ Kondo model.

Equation (32) can be written down using Hubbard operators

$\begin{eqnarray}V & = & {J}_{x}({X}_{+-}{c}_{-}^{\dagger }{c}_{+}+{X}_{-+}{c}_{+}^{\dagger }{c}_{-})\\ & & +{J}_{z}({X}_{++}{c}_{+}^{\dagger }{c}_{+}+{X}_{--}{c}_{-}^{\dagger }{c}_{-})\\ & & -{\displaystyle \frac{1}{2}{J}_{z}({X}_{++}+{X}_{--})({c}_{+}^{\dagger }{c}_{+}+{c}_{-}^{\dagger }{c}_{-}).}\end{eqnarray} \tag{ 33 }$

Motivated by equation (33) we suggest the following Hamiltonian for arbitrary N, which we for obvious reasons will call the XXZ CS model,

$\begin{eqnarray}H & = & \displaystyle \sum _{m}\epsilon {c}_{m}^{\dagger }{c}_{m}+{J}_{x}\displaystyle \sum _{m\ne m^{\prime} }{X}_{{mm}^{\prime} }{c}_{m^{\prime} }^{\dagger }{c}_{m}\\ & & +{J}_{z}\displaystyle \sum _{m}{X}_{{mm}}{c}_{m}^{\dagger }{c}_{m}-{\displaystyle \frac{{J}_{z}}{N}\displaystyle \sum _{{mm}^{\prime} }{X}_{{mm}}{c}_{m^{\prime} }^{\dagger }{c}_{m^{\prime} }}.\end{eqnarray} \tag{ 34 }$

(An alternative motivation for introducing the model can be found in appendix C.) Further on in this section we'll present the calculations only for the case of constant DOS, and only the final results will be written down for the case of the power law DOS.

For the interaction (34) scaling equation (11) becomes

$\begin{eqnarray}\displaystyle \frac{{{dJ}}_{x}}{d\mathrm{ln}\,{\rm{\Lambda }}} & = & -(N-2)\rho {J}_{x}^{2}-2\rho {J}_{x}{J}_{z}\\ \displaystyle \frac{{{dJ}}_{z}}{d\mathrm{ln}\,{\rm{\Lambda }}} & = & -N\rho {J}_{x}^{2}.\end{eqnarray} \tag{ 35 }$

For the case of isotropic CS model and for the case N = 2, equation (35) is reduced to the well established results [2].

Dividing two equations in (35) by each other we obtain homogeneous differential equations of the first degree, which can be easily integrated

$\begin{eqnarray}&&{J}_{z}=\displaystyle \frac{C}{\sqrt[N+2]{| w-1{| }^{N}{\left|w+\tfrac{2}{N}\right|}^{2}}}\end{eqnarray} \tag{ 36 }$

$\begin{eqnarray}&&{J}_{x}={{wJ}}_{z},\end{eqnarray} \tag{ 37 }$

where C is an arbitrary constant. Substituting the solution (36), (37) into equation (35) we obtain

$\begin{eqnarray}&&\displaystyle \frac{{dw}}{d\mathrm{ln}\,{\rm{\Lambda }}}=\displaystyle \frac{{CN}\rho w(w-1)\left(w+\tfrac{2}{N}\right)}{\sqrt[N+2]{| w-1{| }^{N}{\left|w+\tfrac{2}{N}\right|}^{2}}}.\end{eqnarray} \tag{ 38 }$

(For N = 2 equation (38) can be easily integrated in terms of elementary transcendental functions, otherwise in quadratures.) equation (38) has three fixed points: $w=1,w=-2/N$, and w = 0. The first two are stable for C > 0, the last one is stable for C < 0. (One should keep in mind that Λ decreases in the process of evolution.)

At the phase plane with the coordinates J_x, J_z the fixed point w = 1 turns into the flow line J_x = J_z, and the fixed point w = −2/N turns into the flow line J_x = −(2/N)J_z. Both lines are attractors for J_z > 0, and repellers (serving as the phase boundaries) for J_z < 0. The w = 0 fixed point turns into the line of fixed points J_x = 0, stable for J_z < 0, and unstable for J_z > 0. (The half-line J_x = 0, J_z > 0 serves as the phase boundary.) Notice that J_x = J_z = 0 is a degenerate fixed point [37].

Stable fixed point correspond to phases of equation (35), characterized by the asymptotic behaviour of the flow lines. The part of the phase plane J_x > −(2/N) J_z, 0 is characterized by the attractor J_x = J_z > 0, and will be called the Kondo phase I. Notice that in this phase, the SU(N) symmetry, which is absent for the microscopic Hamiltonian, is being recovered in the process of scaling. The part J_x < −J_z, 0 is characterized by the attractor J_x = −(2/N)J_z < 0, and will be called the Kondo phase II. In the part −(2/N)J_z ≥ J_x ≥ J_z the flow lines are attracted to the fixed points at the axis J_x = 0.

A flow line can reach the fixed point w = 0 only in the end of infinitely long evolution, which is a common situation for a fixed point. Hence Ising model is obtained only in the infrared limit. However, in a Kondo phase a flow line reaches fixed point w = 1 or w = −2/N after finite evolution (the R.H.S. of equation (38) being non-analytic function of w at these points). Singularity of J_x, J_z at finite value of Λ is another indication of the limited applicability (in the Kondo phases) of the truncated to second order scaling equation.

To the general solution (36) we should add singular solutions

$\begin{eqnarray}{J}_{x} & = & {J}_{z}=\displaystyle \frac{1}{N\rho \mathrm{ln}\,{\rm{\Lambda }}+{C}_{1}}\\ {J}_{x} & = & -\displaystyle \frac{2{J}_{z}}{N}=\displaystyle \frac{-1}{2\rho \mathrm{ln}\,{\rm{\Lambda }}+{C}_{2}}.\end{eqnarray} \tag{ 39 }$

Once again we see that evolution starting in one of the Kondo phases hits the singular point at finite value of lnΛ.

Equation (36) allows us to plot the flow diagram of the scaling equation, which is presented on figure 1. For the sake of definiteness we have chosen N = 4. Notice that the flow diagram is qualitatively similar to that of the XXZ Kondo model [2].

Zoom In Zoom Out Reset image size

For the case of the power law DOS equations (36) and (38) become

$\begin{eqnarray}&&{J}_{z}=\displaystyle \frac{C\lambda }{\sqrt[N+2]{| w-1{| }^{N}{\left|w+\tfrac{2}{N}\right|}^{2}}}\end{eqnarray} \tag{ 40 }$

and

$\begin{eqnarray}&&r\displaystyle \frac{{dw}}{d\lambda }=\displaystyle \frac{{CNG}\omega (w-1)\left(w+\tfrac{2}{N}\right)}{\sqrt[N+2]{| w-1{| }^{N}{\left|w+\tfrac{2}{N}\right|}^{2}}}\end{eqnarray} \tag{ 41 }$

(equation (39) is changed similarly).

Equations (40) and (41) is a rigorous but a bit formal result. In particular, it demands some effort to extract out of them the fixed points of the original scaling equation

$\begin{eqnarray}r\displaystyle \frac{{{dJ}}_{x}}{d\lambda } & = & {{rJ}}_{x}-(N-2){{GJ}}_{x}^{2}-2{{GJ}}_{x}{J}_{z}\\ r\displaystyle \frac{{{dJ}}_{z}}{d\lambda } & = & {{rJ}}_{z}-{{NGJ}}_{x}^{2},\end{eqnarray} \tag{ 42 }$

which can be easily found by inspection of equation (42). The equation has a trivial fixed point $({J}_{x}^{* },{J}_{z}^{* })=(0,0)$, which is stable for r > 0 and unstable for r < 0, and two semi-stable (critical) non-trivial fixed points

$\begin{eqnarray}{J}_{x}^{* } & = & \displaystyle \frac{2-N\pm \sqrt{{(N-2)}^{2}+8{{Nr}}^{2}}}{4{NG}}\\ {J}_{z}^{* } & = & \displaystyle \frac{{{{NGJ}}_{x}^{* }}^{2}}{r}.\end{eqnarray} \tag{ 43 }$

To formulate the general anisotropic CS model let us return to the spin-anisotropic Kondo model. The interaction can be written down using Hubbard operators

$\begin{eqnarray}V & = & \displaystyle \frac{{J}_{x}}{4}({X}_{+-}+{X}_{-+})({c}_{+}^{\dagger }{c}_{-}+{c}_{-}^{\dagger }{c}_{+})\\ & & -\displaystyle \frac{{J}_{y}}{4}({X}_{+-}-{X}_{-+})({c}_{+}^{\dagger }{c}_{-}-{c}_{-}^{\dagger }{c}_{+})\\ & & +{J}_{z}({X}_{++}{c}_{+}^{\dagger }{c}_{+}+{X}_{--}{c}_{-}^{\dagger }{c}_{-})\\ & & -{\displaystyle \frac{{J}_{z}}{2}({X}_{++}+{X}_{--})({c}_{+}^{\dagger }{c}_{+}+{c}_{-}^{\dagger }{c}_{-})}.\end{eqnarray} \tag{ 44 }$

Motivated by equation (44) we suggest the following interaction for arbitrary N

$\begin{eqnarray}V & = & \displaystyle \frac{{J}_{x}}{2}\displaystyle \sum _{m\ne m^{\prime} }{X}_{{mm}^{\prime} }({c}_{m^{\prime} }^{\dagger }{c}_{m}+{c}_{m}^{\dagger }{c}_{m^{\prime} })\\ & + & \displaystyle \frac{{J}_{y}}{2}\displaystyle \sum _{m\ne m^{\prime} }{X}_{{mm}^{\prime} }({c}_{m^{\prime} }^{\dagger }{c}_{m}-{c}_{m}^{\dagger }{c}_{m^{\prime} })\\ & + & {J}_{z}\displaystyle \sum _{m}{X}_{{mm}}{c}_{m}^{\dagger }{c}_{m}-{\displaystyle \frac{{J}_{z}}{N}\displaystyle \sum _{{mm}^{\prime} }{X}_{{mm}}{c}_{m^{\prime} }^{\dagger }{c}_{m^{\prime} }}.\end{eqnarray} \tag{ 45 }$

For this interaction scaling equation (11) becomes

$\begin{eqnarray}\displaystyle \frac{{{dJ}}_{x}}{d\mathrm{ln}\,{\rm{\Lambda }}} & = & -(N-2)\rho {J}_{x}{J}_{y}-2\rho {J}_{y}{J}_{z}\\ \displaystyle \frac{{{dJ}}_{y}}{d\mathrm{ln}\,{\rm{\Lambda }}} & = & -(N-2)\rho {J}_{x}{J}_{y}-2\rho {J}_{x}{J}_{z}\\ \displaystyle \frac{{{dJ}}_{z}}{d\mathrm{ln}\,{\rm{\Lambda }}} & = & -N\rho {J}_{x}{J}_{y}.\end{eqnarray} \tag{ 46 }$

Like in the previous Subsection we obtain after integration

$\begin{eqnarray}&&{J}_{z}=\displaystyle \frac{{C}_{1}}{\sqrt[N+2]{| w-1{| }^{N}{\left|w+\tfrac{2}{N}\right|}^{2}}}=\displaystyle \frac{{C}_{2}}{\sqrt[N+2]{| v-1{| }^{N}{\left|v+\tfrac{2}{N}\right|}^{2}}}\end{eqnarray} \tag{ 47 }$

$\begin{eqnarray}&&{J}_{x}={{wJ}}_{z},\qquad \qquad \qquad {J}_{y}={{vJ}}_{z}.\end{eqnarray} \tag{ 48 }$

Analog of equation (38) can be presented as

$\begin{eqnarray}&&\displaystyle \frac{{dw}}{d\mathrm{ln}\,{\rm{\Lambda }}}=\displaystyle \frac{{C}_{1}N\rho {vw}(w-1)\left(w+\tfrac{2}{N}\right)}{\sqrt[N+2]{| w-1{| }^{N}{\left|w+\tfrac{2}{N}\right|}^{2}}}.\end{eqnarray} \tag{ 49 }$

Thus in the phase space with the coordinates J_x, J_y, J_z there are 4 Kondo phases, each defined by the attractor of all the flow lines, given by one of the vectors: $(1,1,1),\left(-\tfrac{2}{N},-\tfrac{2}{N},1\right),\left(-1,\tfrac{2}{N},-1\right),\left(\tfrac{2}{N},-1,-1\right)$. In addition there are 3 Ising phases, each defined by the fixed points of all the flow lines, lying on one of the lines: ${J}_{x}={J}_{y}=0,{J}_{x}={J}_{z}=0,{J}_{y}={J}_{z}=0$. Further analysis of the phase diagram we postpone until later.

For the case of the power law DOS equations (47) and (49) become

$\begin{eqnarray}&&{J}_{z}=\displaystyle \frac{{C}_{1}\lambda }{\sqrt[N+2]{| w-1{| }^{N}{\left|w+\tfrac{2}{N}\right|}^{2}}}=\displaystyle \frac{{C}_{2}\lambda }{\sqrt[N+2]{| v-1{| }^{N}{\left|v+\tfrac{2}{N}\right|}^{2}}}\end{eqnarray} \tag{ 50 }$

and

$\begin{eqnarray}&&r\displaystyle \frac{{dw}}{d\lambda }=\displaystyle \frac{{C}_{1}{NGv}(w-1)\left(w+\tfrac{2}{N}\right)}{\sqrt[N+2]{| w-1{| }^{N}{\left|w+\tfrac{2}{N}\right|}^{2}}}.\end{eqnarray} \tag{ 51 }$

In the end, notice that it would be interesting to apply the approach presented in this paper to the case, where the exchange is influenced by Rashba [38] or Dzyaloshinsky-Morya-Kondo interaction [36], to the case when the DOS has a logarithmic singularity [39], to the multi-channel Kondo model [40], and also to the problem of competition between the Kondo effect and RKKY interaction [41]. Another possibly interesting field for application of the presented approach is the models when orbital and spin degrees of freedom of the impurity co-exist [42].