Version bump

Gallavotti_Jauslin_2015.tex

@@ -32,14 +32,14 @@ The $s-d$ model describes a chain of spin-1/2 electrons interacting magnetically
 
 \hugeskip
 
-\indent The $s-d$ model was introduced by Anderson [\cite{andSO}] and used by Kondo [\cite{konSF}] to study what would subsequently be called the {\it Kondo effect}. It describes a chain of electrons interacting with a fixed spin-1/2 magnetic impurity. One of the manifestations of the effect is that when the coupling is anti-ferrmoagnetic, the magnetic susceptibility of the impurity remains finite in the 0-temperature limit, whereas it diverges for ferromagnetic and for vanishing interactions.\par
+\indent The $s-d$ model was introduced by Anderson [\cite{An61}] and used by Kondo [\cite{Ko64}] to study what would subsequently be called the {\it Kondo effect}. It describes a chain of electrons interacting with a fixed spin-1/2 magnetic impurity. One of the manifestations of the effect is that when the coupling is anti-ferrmoagnetic, the magnetic susceptibility of the impurity remains finite in the 0-temperature limit, whereas it diverges for ferromagnetic and for vanishing interactions.\par
 
-\indent A modified version of the $s-d$ model was introduced by Andrei [\cite{andEZ}], which was shown to be exactly solvable by Bethe Ansatz. In [\cite{bgjOFi}], a hierarchical version of Andrei's model was introduced and shown to exhibit a Kondo effect. In the present letter, we show how the argument can be adapted to the $s-d$ model.\par
+\indent A modified version of the $s-d$ model was introduced by Andrei [\cite{An80}], which was shown to be exactly solvable by Bethe Ansatz. In [\cite{BGJ15}], a hierarchical version of Andrei's model was introduced and shown to exhibit a Kondo effect. In the present letter, we show how the argument can be adapted to the $s-d$ model.\par
 
-\indent We will show that in the hierarchical $s-d$ model, the computation of the susceptibility reduces to iterating an {\it explicit} map relating 6 {\it running coupling constants} (rccs), and that this map can be obtained by restricting the flow equation for the hierarchical Andrei model [\cite{bgjOFi}] to one of its invariant manifolds. The physics of both models are therefore very closely related, as had already been argued in [\cite{bgjOFi}]. This is particularly noteworthy since, at 0-field, the flow in the hierarchical Andrei model is relevant, whereas it is marginal in the hierarchical $s-d$ model, which shows that the relevant direction carries little to no physical significance.\par
+\indent We will show that in the hierarchical $s-d$ model, the computation of the susceptibility reduces to iterating an {\it explicit} map relating 6 {\it running coupling constants} (rccs), and that this map can be obtained by restricting the flow equation for the hierarchical Andrei model [\cite{BGJ15}] to one of its invariant manifolds. The physics of both models are therefore very closely related, as had already been argued in [\cite{BGJ15}]. This is particularly noteworthy since, at 0-field, the flow in the hierarchical Andrei model is relevant, whereas it is marginal in the hierarchical $s-d$ model, which shows that the relevant direction carries little to no physical significance.\par
 \bigskip
 
-\indent The $s-d$ model [\cite{konSF}] represents a chain of non-interacting spin-1/2 fermions, called {\it electrons}, which interact with an isolated spin-1/2 {\it impurity} located at site 0. The Hilbert space of the system is $\mathcal F_L\otimes\mathbb C^2$ in which $\mathcal F_L$ is the Fock space of a length-$L$ chain of spin-1/2 fermions (the electrons) and $\mathbb C^2$ is the state space for the two-level impurity. The Hamiltonian, in the presence of a magnetic field of amplitude $h$ in the direction $\bm\omega\equiv(\bm\omega_1,\bm\omega_2,\bm\omega_3)$, is
+\indent The $s-d$ model [\cite{Ko64}] represents a chain of non-interacting spin-1/2 fermions, called {\it electrons}, which interact with an isolated spin-1/2 {\it impurity} located at site 0. The Hilbert space of the system is $\mathcal F_L\otimes\mathbb C^2$ in which $\mathcal F_L$ is the Fock space of a length-$L$ chain of spin-1/2 fermions (the electrons) and $\mathbb C^2$ is the state space for the two-level impurity. The Hamiltonian, in the presence of a magnetic field of amplitude $h$ in the direction $\bm\omega\equiv(\bm\omega_1,\bm\omega_2,\bm\omega_3)$, is
 \begin{equation}\begin{array}{r@{\ }>{\displaystyle}l}
 H_K=&H_0+V_0+V_h=:H_0+V\\[0.3cm]
 H_0=&\sum_{\alpha\in\{\uparrow,\downarrow\}}\sum_{x=-{L}/2}^{{L}/2-1} c^+_\alpha(x)\,\left(-\frac{\Delta}2-1\right)\,c^-_\alpha(x)\\[0.5cm]
@@ -48,7 +48,7 @@ V_h=&-h \,\sum_{j=1,2,3}\bm\omega_j \tau^j
 \end{array}\label{eqhamdef}\end{equation} 
 where $\lambda_0$ is the interaction strength, $\Delta$ is the discrete Laplacian $c_\alpha^\pm(x),\,\alpha=\uparrow,\downarrow$ are creation and annihilation operators acting on {\it electrons}, and $\sigma^j=\tau^j,\,j=1,2,3$, are Pauli matrices. The operators $\tau^j$ act on the {\it impurity}. The boundary conditions are taken to be periodic.\par
 
-\indent In the {\it Andrei model} [\cite{andEZ}], the impurity is represented by a fermion instead of a two-level system, that is the Hilbert space is replaced by $\mathcal F_L\otimes\mathcal F_1$, and the Hamiltonian is defined by replacing $\tau^j$ in~(\ref{eqhamdef}) by $d^+\tau^jd^-$ in which $d_\alpha^\pm(x),\,\alpha=\uparrow,\downarrow$ are creation and annihilation operators acting on the impurity.\par
+\indent In the {\it Andrei model} [\cite{An80}], the impurity is represented by a fermion instead of a two-level system, that is the Hilbert space is replaced by $\mathcal F_L\otimes\mathcal F_1$, and the Hamiltonian is defined by replacing $\tau^j$ in~(\ref{eqhamdef}) by $d^+\tau^jd^-$ in which $d_\alpha^\pm(x),\,\alpha=\uparrow,\downarrow$ are creation and annihilation operators acting on the impurity.\par
 \bigskip
 
 \indent The partition function $Z={\rm Tr}\, e^{-\beta H_K}$ can be expressed formally as a functional integral:
@@ -62,7 +62,7 @@ $$
 and the trace is over the state-space of the spin-1/2 impurity, that is a trace over $\mathbb C^2$.\par
 \bigskip
 
-\indent We will consider a {\it hierarchical} version of the $s-d$ model. The hierarchical model defined below is {\it inspired} by the $s-d$ model in the same way as the hierarchical model defined in [\cite{bgjOFi}] was inspired by the Andrei model. We will not give any details on the justification of the definition, as such considerations are entirely analogous to the discussion in [\cite{bgjOFi}].\par
+\indent We will consider a {\it hierarchical} version of the $s-d$ model. The hierarchical model defined below is {\it inspired} by the $s-d$ model in the same way as the hierarchical model defined in [\cite{BGJ15}] was inspired by the Andrei model. We will not give any details on the justification of the definition, as such considerations are entirely analogous to the discussion in [\cite{BGJ15}].\par
 
 \indent The model is defined by introducing a family of {\it hierarchical fields} and specifying a {\it propagator} for each pair of fields. The average of any monomial of fields is then computed using the Wick rule.\par
 
@@ -115,7 +115,7 @@ O_{0,\eta}^{[\le 0]}(\Delta):=\frac12\mathbf A^{[\le 0]}_\eta(\Delta)\cdot\bm\ta
 O_{4,\eta}^{[\le 0]}(\Delta):=\frac12\mathbf A^{[\le 0]}_\eta(\Delta)\cdot\bm\omega,& O_{5,\eta}^{[\le 0]}(\Delta):=\frac12\mathbf \bm\tau\cdot\bm\omega,\\[0.3cm]
 O_{6,\eta}^{[\le 0]}(\Delta):=\frac12(\mathbf A^{[\le 0]}_\eta(\Delta)\cdot\bm\omega)(\bm\tau\cdot\bm\omega),& O_{7,\eta}^{[\le 0]}(\Delta):=\frac12(\mathbf A^{[\le 0]}_\eta(\Delta)^2)(\bm\tau\cdot\bm\omega)
 \end{array}\label{eqOdef}\end{equation} 
-(the numbering is meant to recall that in [\cite{bgjOFi}]) in which $\bm\tau=(\tau^1,\tau^2,\tau^3)$ and $\mathbf A_\eta^{[\le 0]}(\Delta)$ is a vector of polynomials in the fields whose $j$-th component for $j\in\{1,2,3\}$ is 
+(the numbering is meant to recall that in [\cite{BGJ15}]) in which $\bm\tau=(\tau^1,\tau^2,\tau^3)$ and $\mathbf A_\eta^{[\le 0]}(\Delta)$ is a vector of polynomials in the fields whose $j$-th component for $j\in\{1,2,3\}$ is 
 \begin{equation}
 A_\eta^{[\le 0]j}(\Delta):=\sum_{(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2} \psi_\alpha^{[\le 0]+}(\Delta_\eta)\sigma^j_{\alpha,\alpha'}\psi_{\alpha'}^{[\le 0]-}(\Delta_\eta)
 \label{eqAdef}\end{equation} 
@@ -141,7 +141,7 @@ in terms of which
 \begin{equation}
 Z=C^{2|\mathcal Q_0|}\prod_{m=-N(\beta)+1}^0(C^{[m]})^{|\mathcal Q_{m-1}|}
 \label{eqZind}\end{equation} 
-in which $|\mathcal Q_m|=2^{N(\beta)-|m|}$ is the cardinality of $\mathcal Q_m$. In addition, similarly to [\cite{bgjOFi}], the map relating $\ell_p^{[m]}$ to $\ell_p^{[m-1]}$ and $C^{[m]}$ can be computed explicitly from~(\ref{eqindW}):
+in which $|\mathcal Q_m|=2^{N(\beta)-|m|}$ is the cardinality of $\mathcal Q_m$. In addition, similarly to [\cite{BGJ15}], the map relating $\ell_p^{[m]}$ to $\ell_p^{[m-1]}$ and $C^{[m]}$ can be computed explicitly from~(\ref{eqindW}):
 \begin{equation}\begin{array}{r@{\ }>{\displaystyle}l}
 C^{[m]} =& 1 +\frac{3}{2}\ell_{0}^2 +\ell_{0}\ell_{6} +9\ell_{1}^2 +\frac{\ell_{4}^2}{2} +\frac{\ell_{5}^2}{4} +\frac{\ell_{6}^2}{2} +9\ell_{7}^2 \\[0.3cm]
 \ell^{[m-1]}_{0} =& \frac1C\left(\ell_{0} -\ell_{0}^2 +3\ell_{0}\ell_{1} -\ell_{0}\ell_{6} \right)\\[0.3cm]
@@ -155,9 +155,9 @@ in which the $^{[m]}$ have been dropped from the right hand side.\par
 \bigskip
 
 
-\indent The flow equation~(\ref{eqbetafun}) can be recovered from that of the hierarchical Andrei model studied in [\cite{bgjOFi}] (see in particular [\cite{bgjOFi}, (C1)] by restricting the flow to the invariant submanifold defined by \begin{equation} \ell_2^{[m]}=\frac13,\quad \ell_3^{[m]}=\frac16\ell_1^{[m]},\quad \ell_8^{[m]}=\frac16\ell_4^{[m]}. \label{e18}\end{equation} This is of particular interest since $\ell_2^{[m]}$ is a relevant coupling and the fact that it plays no role in the $s-d$ model indicates that it has little to no physical relevance.\par
+\indent The flow equation~(\ref{eqbetafun}) can be recovered from that of the hierarchical Andrei model studied in [\cite{BGJ15}] (see in particular [\cite{BGJ15}, (C1)] by restricting the flow to the invariant submanifold defined by \begin{equation} \ell_2^{[m]}=\frac13,\quad \ell_3^{[m]}=\frac16\ell_1^{[m]},\quad \ell_8^{[m]}=\frac16\ell_4^{[m]}. \label{e18}\end{equation} This is of particular interest since $\ell_2^{[m]}$ is a relevant coupling and the fact that it plays no role in the $s-d$ model indicates that it has little to no physical relevance.\par
 
-\indent The qualitative behavior of the flow is therefore the same as that described in [\cite{bgjOFi}] for the hierarchical Andrei model. In particular the susceptibility, which can be computed by deriving $-\beta^{-1}\log Z$ with respect to $h$, remains finite in the 0-temperature limit as long as $\lambda_0<0$, that is as long as the interaction is anti-ferromagnetic.\par
+\indent The qualitative behavior of the flow is therefore the same as that described in [\cite{BGJ15}] for the hierarchical Andrei model. In particular the susceptibility, which can be computed by deriving $-\beta^{-1}\log Z$ with respect to $h$, remains finite in the 0-temperature limit as long as $\lambda_0<0$, that is as long as the interaction is anti-ferromagnetic.\par
 \hugeskip
 
 {\bf Acknowledgements}: We are grateful to G.~Benfatto for many enlightening discussions on the $s-d$ and Andrei's models.

bibliography.BBlog.tex

@@ -1,20 +1,16 @@
 \hrefanchor
-\outdef{citeandSO}{And61}
-\hbox{\parbox[t]{\rw}{[\cite{andSO}]}\parbox[t]{\colw}{P.~Anderson - {\it Localized magnetic states in metals}, Physical Review, Vol.~124, n.~1, p.~41-53, 1961.}}\par
+\outdef{citeAn61}{An61}
+\hbox{\parbox[t]{\rw}{[\cite{An61}]}\parbox[t]{\colw}{P.W. Anderson - {\it Localized magnetic states in metals}, Physical Review, Vol.~124, n.~1, p.~41-53, 1961.}}\par
 \bigskip
-
 \hrefanchor
-\outdef{citeandEZ}{And80}
-\hbox{\parbox[t]{\rw}{[\cite{andEZ}]}\parbox[t]{\colw}{N.~Andrei - {\it Diagonalization of the Kondo Hamiltonian}, Physical Review Letters, Vol.~45, n.~5, 1980.}}\par
+\outdef{citeAn80}{An80}
+\hbox{\parbox[t]{\rw}{[\cite{An80}]}\parbox[t]{\colw}{N. Andrei - {\it Diagonalization of the Kondo Hamiltonian}, Physical Review Letters, Vol.~45, n.~5, 1980.}}\par
 \bigskip
-
 \hrefanchor
-\outdef{citebgjOFi}{BGJ15}
-\hbox{\parbox[t]{\rw}{[\cite{bgjOFi}]}\parbox[t]{\colw}{G.~Benfatto, G.~Gallavotti, I.~Jauslin - {\it Kondo effect in a Fermionic hierarchical model}, arXiv 1506.04381, 2015.}}\par
+\outdef{citeBGJ15}{BGJ15}
+\hbox{\parbox[t]{\rw}{[\cite{BGJ15}]}\parbox[t]{\colw}{G. Benfatto, G. Gallavotti, I. Jauslin - {\it Kondo effect in a Fermionic hierarchical model}, arXiv 1506.04381, 2015.}}\par
 \bigskip
-
 \hrefanchor
-\outdef{citekonSF}{Kon64}
-\hbox{\parbox[t]{\rw}{[\cite{konSF}]}\parbox[t]{\colw}{J.~Kondo - {\it Resistance minimum in dilute magnetic alloys}, Progress of Theoretical Physics, Vol.~32, n.~1, 1964.}}\par
+\outdef{citeKo64}{Ko64}
+\hbox{\parbox[t]{\rw}{[\cite{Ko64}]}\parbox[t]{\colw}{J. Kondo - {\it Resistance minimum in dilute magnetic alloys}, Progress of Theoretical Physics, Vol.~32, n.~1, 1964.}}\par
 \bigskip
-