Abstract
In this paper, a fermionic hierarchical model is defined, inspired by the Kondo model, which describes a 1dimensional lattice gas of spin1/2 electrons interacting with a spin1/2 impurity. This model is proved to be exactly solvable, and is shown to exhibit a Kondo effect, i.e. that, if the interaction between the impurity and the electrons is antiferromagnetic, then the magnetic susceptibility of the impurity is finite in the 0temperature limit, whereas it diverges if the interaction is ferromagnetic. Such an effect is therefore inherently nonperturbative. This difficulty is overcome by using the exact solvability of the model, which follows both from its fermionic and hierarchical nature.
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Notes
The obstacle to a complete understanding of the model (with \(\lambda _0<0\)) being what would later be called the growth of a relevant coupling.
This means that all integrals will be defined and evaluated via the “Wick rule”.
References
Abraham, D., Baruch, E., Gallavotti, G., MartinLöf, A.: Dynamics of a local perturbation in the \(XY\) model (I). Stud. Appl. Math. 50, 121–131 (1971)
Anderson, P.: Local magnetized states in metals. Phys. Rev. 124, 41–53 (1961)
Anderson, P.: A poor man’s derivation of scaling laws for the Kondo problem. J. Phys. C 3, 2436–2441 (1970)
Anderson, P., Yuval, G.: Exact results in the Kondo problem: equivalence to a classical onedimensional Coulomb gas. Phys. Rev. Lett. 23, 89–92 (1969)
Anderson, P., Yuval, G., Hamann, D.: Exact results in the Kondo problem: equivalence to a classical onedimensional Coulomb gas. Phys. Rev. B 1, 4464–4473 (1970)
Andrei, N.: Diagonalization of the Kondo Hamiltonian. Phys. Rev. Lett. 45, 379–382 (1980)
Andrei, N., Furuya, K., Lowenstein, J.: Solution of the Kondo problem. Rev. Mod. Phys. 55, 331–402 (1983)
Benfatto, G., Gallavotti, G.: Perturbation theory of the Fermi surface in a quantum liquid. A general quasi particle formalism and one dimensional systems. J. Stat. Phys. 59, 541–664 (1990)
Benfatto, G., Gallavotti, G., Procacci, A., Scoppola, B.: Beta function and Schwinger functions for a many body system in one dimension. Anomaly of the Fermi surface. Commun. Math. Phys. 160, 93–172 (1994)
Dorlas, T.: Renormalization group analysis of a simple hierarchical fermion model. Commun. Math. Phys. 136, 169–194 (1991)
Dyson, F.: Existence of a phase transition in a onedimensional Ising ferromagnet. Commun. Math. Phys. 12, 91–107 (1969)
Kittel, C.: Introduction to Solid State Physics. Wiley, Hoboken (1976)
Kondo, J.: Resistance minimum in dilute magnetic alloys. Prog. Theoret. Phys. 32, 37–49 (1964)
Kondo, J.: Sticking to my bush. J. Phys. Soc. Jpn. 74, 1–3 (2005)
Nozières, P.: A Fermiliquid description of the Kondo problem at low temperatures. J. Low Temp. Phys. 17, 31–42 (1974)
Ruelle, D.: Statistical Mechanics. Benjamin, New York (1969, 1974)
Shankar, R.: Renormalizationgroup approach to interacting fermions. Rev. Mod. Phys. 66, 129–192 (1994)
Wilson, K.: Model Hamiltonians for local quantum field theory. Phys. Rev. 140, B445–B457 (1965)
Wilson, K.: Model of coupling constant renormalization. Phys. Rev. D 2, 1438–1472 (1970)
Wilson, K.: The renormalization group. Rev. Mod. Phys. 47, 773–840 (1975)
Acknowledgments
We are grateful to V. Mastropietro for suggesting the problem and to A. Giuliani, V. Mastropietro and R. Greenblatt for continued discussions and suggestions, as well as to J. Lebowitz for hospitality and support.
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Appendices
Appendix 1: Comparison with the Original Kondo Model
If the partition function for the original Kondo model in presence of a magnetic field h acting only on the impurity site and at finite L is denoted by \(Z^0_K(\beta ,\lambda _0,h)\) and the partition function for the model Eq. (2.1) with the same field h is denoted by \(Z_K(\beta ,\lambda _0,h)\), then
so that by defining
we get
In addition \(1\le \kappa \le 2\): indeed the first inequality is trivial and the second follows from the variational principle (see [16, Theorem 7.4.1, p. 188]):
where \(s(\mu )\) is the entropy of the state \(\mu \), and in which we used
Therefore, for \(\beta h^2\ll 1\) (which implies that if there is a Kondo effect then \(\beta m_K^2\ll 1\)), the model Eq. (2.1) exhibits a Kondo effect if and only if the original Kondo model does, therefore, for the purposes of this paper, both models are equivalent.
Appendix 2: Some Identities
In this appendix, we state three relations used to compute the flow equation Eq. (5.13), which follow from a patient algebraic meditation:
where the lower case \(\mathbf{a}\) denote \({\langle \,\mathbf{A}_1\,\rangle }\equiv {\langle \,\mathbf{A}_2\,\rangle }\) and \(a^{j_1,j_2}={\langle \,\psi ^+_1\sigma ^{j_1}\sigma ^{j_2}\psi ^_1\,\rangle } ={\langle \,\psi ^+_2\sigma ^{j_1}\sigma ^{j_2}\psi ^_2\,\rangle }\).
Appendix 3: Complete Beta Function
The beta function for the flow described in Sect. (6) is
in which we dropped the \(^{[m]}\) exponent on the right side. By considering the linearized flow equation (around \(\ell _j = 0\)), we find that \(\ell _0,\ell _4,\ell _6,\ell _8\) are marginal, \(\ell _2,\ell _5\) relevant and \(\ell _1,\ell _3,\ell _7\) irrelevant. The consequent linear flow is very different from the full flow discussed in Sect. 6.
The vector \({\varvec{\ell }}\) is related to \({\varvec{\alpha }}\) via the following map:
Appendix 4: The Algebra of the Operators \(O_{n,\pm }\)
Lemma 1
Given \(\eta \in \{,+\}\), \(m\le 0\) and \(\Delta \in \mathcal Q_m\), the span of the operators \(\{O_{n,\eta }^{[\le m]}(\Delta )\}_{n\in \{0,1,2,3\}}\) defined in Eq. (5.6) is an algebra, that is all linear combinations of products of \(O_{n,\eta }^{[\le m]}(\Delta )\)’s is itself a linear combination of \(O_{n,\eta }^{[\le m]}(\Delta )\)’s. The same result holds for the span of the operators \(\{O_{n,\eta }^{[\le m]}(\Delta )\}_{n\in \{0,\ldots ,8\}}\) defined in Eq. (6.5).
Proof
The only nontrivial part of this proof is to show that the product of two \(O_{n,\eta }\)’s is a linear combination of \(O_{n,\eta }\)’s. \(\square \)
Due to the anticommutation of Grassmann variables, any linear combination of \(\psi _{\alpha }^{[\le m]\pm }\) and \(\varphi _{\alpha }^{[\le m]\pm }\) squares to 0. Therefore, a straightforward computation shows that \(\forall (i,j)\in \{1,2,3\}^2\),
where the labels \(^{[\le m]}\) and \((\Delta )\) are dropped to alleviate the notation. In particular, this implies that any product of three \(A^{i}_\eta \) for \(i\in \{1,2,3\}\) vanishes (because the product of the right side of the first of Eq. (11.1) and any Grassmann field \(\psi _{\alpha }^{\pm }\) vanishes) and similarly for the product of three \(B^{i}_\eta \).
Using Eq. (11.1), we prove that \(\mathrm {span}\{O_{n,\eta }^{[\le m]}(\Delta )\}_{n\in \{0,1,2,3\}}\) is an algebra. For all \(n\in \{0,1,2,3\}\), \(p\in \{1,2,3\}\), \(l\in \{1,2\}\),
(here the \(^{[\le m]}\), \((\Delta )\) and \(_\eta \) are dropped). This concludes the proof of the first claim.
Next we prove that \(\mathrm {span}\{O_{n,\eta }^{[\le m]}(\Delta )\}_{n\in \{0,\ldots ,8\}}\) is an algebra. In addition to Eq. (11.2), we have, for all \(p\in \{0,\ldots ,8\}\),
This concludes the proof of the lemma.
Appendix 5: Fixed Points at \(h=0\)
We first compute the fixed points of Eq. (5.13) for \(\ell _2\ge 0\). It follows from Eq. (5.13) that if \({\varvec{\ell }}\) is a fixed point, then \(\ell _1=6\ell _3\), which implies
If \(\ell _2\ge 0\), Eq. (12.1) implies that either \(\ell _2=\ell _1=\ell _0=0\) or \(\ell _2=\frac{1}{3}\). In the latter case, either \(\ell _0=\ell _1=0\) or \(\ell _0\not =0\) and Eq. (5.13) becomes
In particular, \(\ell _1(112\ell _1)>0\), so that
which we inject into Eq. (12.2) to find that \(\ell _0<0\) and
Finally, we notice that \(\frac{1}{12}\) is a solution of Eq. (12.4), which implies that
which has a unique real solution. Finally, we find that if \(\ell _1\) satisfies Eq. (12.5), then
We have therefore shown that, if \(\ell _2\ge 0\), then Eq. (5.13) has three fixed points:
In addition, it follows from Eqs. (5.13) and (5.11) that, if \(\lambda _0<0\), then (recall that \(\alpha _0^{[0]}=\lambda _0\) and \(\alpha _i^{[0]}=0\), \(i=1,2,3\))
for all \(m\le 0\), which implies that the set \(\{{\varvec{\ell }}\ \ \ell _0<0,\ \ell _2\ge 0,\ \ell _1\ge 0,\ \ell _3\ge 0\}\) is stable under the flow. In addition, if \(\ell _0^{[m]}>\frac{2}{3}\), then \(\ell _0^{[m1]}<\ell _0^{[m]}\), so that the flow cannot converge to \({\varvec{\ell }}_0^*\) or \({\varvec{\ell }}_+^*\). Therefore if the flow converges, then it converges to \({\varvec{\ell }}^*\).
We now study the reduced flow Eq. (5.17), and prove that starting from \(2/3<\ell ^{[0]}_0<0\), \(\ell ^{[0]}_2=0\), the flow converges to \(f^*\). It follows from Eq. (5.17) that \(\ell ^{[m]}_0<0\), \(\ell ^{[m]}_2>0\) for all \(m<0\), so that if Eq. (5.17) converges to a fixed point, then it must converge to \(f^*\). In addition, by a straightforward induction, one finds that \(\ell _2^{[m1]}>\ell _2^{[m]}\) if \(\ell _2^{[m]}<\frac{1}{3}\). Furthermore, \((2\ell _2^{[m]}+(\ell _0^{[m]})^2)\le \frac{1}{3}C^{[m]}\), which implies that \(\ell _2^{[m]}\le \frac{1}{3}\). Therefore \(\ell _2^{[m]}\) converges as \(m\rightarrow \infty \). In addition, \(\ell _0^{[m1]}<\ell _0^{[m]}\) if \(\ell _0^{[m]}>\frac{2}{3}\), and \(\ell _0^{[m]}>\frac{1}{3}\ell _2^{[m]}\ge \frac{2}{3}\), so that \(\ell _0^{[m]}\) converges as well as \(m\rightarrow \infty \). The flow therefore tends to \(f^*\).
Finally, we prove that starting from \(\ell ^{[0]}_0>0,\ell ^{[0]}_2=0\), the flow converges to \(f_+\). Similarly to the antiferromagnetic case, \(\ell ^{[m]}_2>0\) for all \(m<0\), \(\ell _2^{[m]}\le \frac{1}{3}\) and \(\ell _2^{[m1]}>\ell _2^{[m]}\). In addition, by a simple induction, if \(\lambda _0<1\), then \(\ell _0^{[m]}>0\) and \(\ell _0^{[m]}+\frac{1}{3}\ell _2^{[m]}\) is strictly decreasing and positive. In conclusion, \(\ell _0^{[m]}\) and \(\ell _2^{[m]}\) converge to \(f_+\).
Appendix 6: Asymptotic Behavior of \(n_j(\lambda _0)\) and \(r_j(h)\)
In this appendix, we show plots to support the claims on the asymptotic behavior of \(n_j(\lambda _0)\) (see Eq. 6.10, Fig. 7 and Eq. 6.11, Fig. 8) and \(r_j(h)\) (see Eq. 6.12, Fig. 9). The plots below have error bars which are due to the fact that \(n_j(\lambda _0)\) and \(r_j(h)\) are integers, so their value could be off by \({\pm }1\).
Appendix 7: Kondo Effect, XYModel, Free Fermions
In [1], given \(\nu \in [1,\ldots ,L]\), the Hamiltonian \(H_h=H_0 {h} \,\sigma _\nu ^z,\) with
has been considered with suitable boundary conditions, under which \(H_0\) and \({\sigma ^z_0} +1\) are unitarily equivalent to \(\sum _{q}{(\cos q)} \, a^+_qa^_q\) and, respectively, to \(\frac{2}{L} \sum _{q,q'} a^+_q a^_{q'} e^{i\nu (qq')}\) in which \(a^\pm _q\) are fermionic creation and annihilation operators and the sums run over q’s that are such that \(e^{iq L}=1\). It has been shown, [1]^{Footnote 3}, that, by defining
the partition function is equal to \(Z_L^0\zeta _L\) in which \(Z_L^0\) is the partition function at \(h=0\) and is extensive (i.e. of \(O(e^{const L})\)) and (see Appendix 8, Eq. 14.12)
where the contour C is a closed curve in the complex plane which contains the zeros of \(F_L(\zeta )\) (e.g. , for \(L\rightarrow \infty \), a curve around the real interval \([1,\sqrt{1+4h^2}]\) if \(h<0\) and \([\sqrt{1+4h^2},1]\) if \(h>0\)) but not those of \(1+e^{\beta z}\) (which are on the imaginary axis and away from 0 by at least \(\frac{\pi }{\beta }\)). In addition, it follows from a straightforward computation that \((F(z)1)/h\) is equal to the analytical continuation of \(2 (z^21)^{\frac{1}{2}}\) from \((1,\infty )\) to \(C\setminus [1,1]\).
At fixed \(\beta <\infty \) the partition function \(\zeta _L(\beta ,h)\) has a non extensive limit \(\zeta (\beta ,h)\) as \(L\rightarrow \infty \); \(\zeta (\beta ,h)\) and the susceptibility and magnetization values \(m(\beta ,h)\) and \(\chi (\beta ,h)\), are given in the thermodynamic limit by
so that \(\chi (\beta ,0)=\frac{2\sinh (\beta )}{(1+\cosh (\beta ))}\) and, in the \(\beta \rightarrow \infty \) limit,
both of which are finite. Adding an impurity at 0, with spin operators \({\varvec{\tau }}_0\), the Hamiltonian
is obtained. Does it exhibit a Kondo effect?
Since \({\varvec{\tau }}_0\) commutes with the \({\varvec{\sigma }}_n\) and, hence, with \(H_0\), the average magnetization and susceptibility, \(m^{int}(\beta ,h,\lambda )\) and \(\chi ^{int}(\beta ,h,\lambda )\), responding to a field h acting only on the site 0, can be expressed in terms of the functions \(\zeta (\beta ,h)\) and its derivatives \(\zeta '(\beta ,h)\) and \(\zeta ''(\beta ,h)\). By using the fact that \(\zeta (\beta ,h)\) and \(\zeta ''(\beta ,h)\) are even in h, while \(\zeta '(\beta ,h)\) is odd, we get:
Since \(\chi ^{int}(\beta ,0)\) is even in \(\lambda \), it diverges for \(\beta \rightarrow \infty \) independently of the sign of \(\lambda \), while \(\chi (\beta ,0)\) is finite. Hence, the model yields Pauli’s paramagnetism, without a Kondo effect.
Remark

(1)
Finally an analysis essentially identical to the above can be performed to study the model in Eq. (2.1) without impurity (and with or without spin) to check that the magnetic susceptibility to a field h acting only at a single site is finite: the result is the same as that of the XY model above: the single site susceptibility is finite and, up to a factor 2, given by the same formula \(\chi (\beta ,0)=\frac{4\sinh \beta }{1+\cosh \beta }\).

(2)
The latter result makes clear both the essential roles for the Kondo effect of the spin and of the noncommutativity of the impurity spin components.
Appendix 8: Some Details on Appendix 7
The definition of \(H_h\) has to be supplemented by a boundary condition to give a meaning to \({\varvec{\sigma }}_{L+1}\). If \(\sigma ^\pm _n=(\sigma ^x\pm i\sigma ^y_n)/2\) define \(\mathcal{N}_{<n}\) as \(\sum _{i<n}\sigma ^+_i\sigma ^_i=\sum _{i<n}\mathcal{N}_i\) and \(\mathcal{N}=\mathcal{N}_{\le L}\). Then set as boundary condition
(parityantiperiodic b.c.) so that \(H_h\) becomes
Introducing the Pauli–Jordan transformation
In these variables
Assume \(L=\)even and let \(I\,\,{\buildrel def\over =}\,\,\{q q= \pm \frac{(2n+1)\pi }{L}, \, n=0,1,\ldots ,\frac{L}{2} 1\}\); then
In diagonal form let \(U_{jq}\) be a suitable unitary matrix such that
Then \(\lambda _j\) must satisfy
\(\forall q\in I\), where we used the fact that \(A^_p A^+_q {\vert 0\rangle } =\delta _{p,q}{\vert 0\rangle }\). We consider the two cases \(\lambda _j\ne \cos q\) for all \(q\in I\) or \(\lambda =\cos q_0\) for some \(q_0\in I\).
In the first case:
where \({N(\lambda _j)}\) is set in such a way that U is unitary, or, in the second case,
Since \(\cos q\) takes \(\frac{1}{2}L\) values and the equation \(F_L(\lambda )=0\) has \(\frac{L}{2}\) solutions, the spectrum of \(H_h\) is completely determined and given by the \(2^L\) eigenvalues
and the partition function is
On the other hand, since the function \(F'_L(z)/F_L(z)\) has L / 2 poles with residue \(+1\) (those corresponding to the zeros of \(F_L(z)\)) and L / 2 poles with residue \(1\) (those corresponding to the poles of \(F_L(z)\)), the contour integral in the r.h.s. of Eq. (13.3) is equal to
Appendix 9: meankondo: A Computer Program to Compute Flow Equations
The computation of the flow equation Eq. (10.1) is quite long, but elementary, which makes it ideally suited for a computer. We therefore attach a program, called meankondo and written by I.Jauslin, used to carry it out (the computation has been checked independently by the other authors). One interesting feature of meankondo is that it has been designed in a modelagnostic way, that is, unlike its name might indicate, it is not specific to the Kondo model and can be used to compute and manipulate flow equations for a wide variety of fermionic hierarchical models. It may therefore be useful to anyone studying such models, so we have thoroughly documented its features and released the source code under an Apache 2.0 license. See http://ian.jauslin.org/software/meankondo for details.
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Benfatto, G., Gallavotti, G. & Jauslin, I. Kondo Effect in a Fermionic Hierarchical Model. J Stat Phys 161, 1203–1230 (2015). https://doi.org/10.1007/s1095501513787
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DOI: https://doi.org/10.1007/s1095501513787