# ABCD of Kondo effect

Taro Kimura
木村太郎  Institut de Mathématiques de Bourgogne,
Université Bourgogne Franche-Comté

###### Abstract

We explore the Kondo effect incorporating the localized impurity transforming
under generic symmetry group $G$, that we call the $G$-Kondo effect. We derive
the one-dimensional effective model coupled with the impurity, and studied the
thermodynamic properties based on the boundary conformal field theory
approach. We in particular study the impurity entropy at the conformal fixed
point, and the low temperature behavior of the specific heat and the
susceptibility. We also consider the Wilson ratio based on these results, and
mention the case with exceptional group symmetry.

###### Contents

  1. 1 Introduction
  2. 2 Formulation of the impurity problem
    1. 2.1 Non-Abelian bosonization
    2. 2.2 Real fermion
  3. 3 Impurity entropy
    1. 3.1 SU($n$) theory
    2. 3.2 SO($n$) theory
      1. 3.2.1 Vector representation
      2. 3.2.2 Spinor representation
    3. 3.3 Sp($n$) theory
  4. 4 Low temperature thermodynamics
    1. 4.1 Bulk contribution
    2. 4.2 Impurity contribution
      1. 4.2.1 Specific heat
      2. 4.2.2 Susceptibility
      3. 4.2.3 Wilson ratio
  5. 5 EFG of Kondo effect
  6. 6 Discussion
  7. A Lie algebras
  8. B Character formula

##  1 Introduction

The Kondo effect (近藤効果) has been playing a central role in the study of
strongly correlated electron systems, providing a lot of innovative ideas for
our understanding of the quantum nature of many-body physics [1, 2]. It has
been originally observed and studied in the magnetic impurity system, where
the SU(2) spin degrees of freedom of the localized impurity spin interacting
with the conduction electron plays a crucial role in the non-trivial behavior
of the renormalization flow. In fact, the Kondo effect seems the first
experimental observation of the asymptotic freedom.

One of the interesting directions of the study of the Kondo effect is to
incorporate higher-rank SU$(n)$ symmetry, discussed in the context of the
quantum dot [3, 4], ultracold atomic system [5, 6], heavy-flavor QCD [7, 8],
etc. In addition, there has been a recent proposal to implement the Majorana
impurity, which realizes the SO($n$) Kondo system [9, 10, 11, 12, 13]. See
also [14, 15, 16, 17] for further discussions on unconventional symmetries,
including Sp$(n)$ symmetry. Such developments in the Kondo effect motivates us
to explore the quantum impurity system with generic symmetry group $G$, that
we call the $G$-Kondo system. In this paper, we mainly consider the classical
groups, namely, _ABCD of Kondo effect_ :

$\displaystyle A_{n}\ :\ \mathrm{SU}(n)\,,\qquad B_{n}\ :\
\mathrm{SO}(2n+1)\,,\qquad C_{n}\ :\ \mathrm{Sp}(n)\,,\qquad D_{n}\ :\
\mathrm{SO}(2n)\,.$ (1.1)

Here we apply the convention, s.t., $\mathrm{Sp}(1)=\mathrm{SU}(2)$,
$\mathrm{Sp}(2)=\mathrm{SO}(5)$. The key property of the Kondo effect is the
non-perturbative behavior below the characteristic temperature, called the
Kondo temperature $T_{\text{K}}$. Such a behavior is universal, and observed
in the system with arbitrary non-Abelian group $G$ in general: For the Kondo
coupling in the form of

$\displaystyle\mathsf{H}_{\text{K}}=\lambda_{\text{K}}\sum_{a=1}^{\dim
G}\sigma^{a}S^{a}\,,$ (1.2)

with the Kondo coupling $\lambda_{\text{K}}$, and the “spin” operators of the
electron and the impurity, $\sigma^{a}$ and $S^{a}$ (the generators of the Lie
algebra $\mathfrak{g}=\operatorname{Lie}G$), we obtain the scaling equation
[18] (See also [19]),

$\displaystyle\derivative{\lambda_{\text{K}}}{\log\Lambda}=-h^{\vee}\rho_{0}\,\lambda_{\text{K}}^{2}\,,$
(1.3)

where we define the energy scale $\Lambda$, the density of states $\rho_{0}$,
and the dual Coxeter number $h^{\vee}$ (See Tab. 1). The derivation of the
scaling equation (1.3) just uses the relation for the structure constant of
the Lie algebra $\mathfrak{g}=\operatorname{Lie}G$ shown in (A.1).

Since the electron-impurity coupling may be relatively small in the high
temperature regime compared to $T_{\text{K}}$, we can apply the perturbative
analysis there, whereas we should non-perturbatively explore the low
temperature regime due to the asymptotic freedom property. Conformal Field
Theory (CFT) is one of the non-perturbative approaches to the low temperature
regime of the Kondo system as demonstrated by Affleck–Ludwig [20, 21, 22, 23,
24, 25]. See also review articles [26, 27]. In this framework, the impurity
problem is replaced with the study of the boundary condition, which can be
analyzed based on Boundary CFT (BCFT).

In this paper, we formulate the Kondo effect with generic symmetry group
$G=ABCD$, and apply the BCFT approach in order to study the non-perturbative
aspects of the $G$-Kondo system. In Sec. 2, we first introduce the localized
impurity which transforms under the symmetry group $G=ABCD$, and then
formulate the quantum impurity problem based on the effective one-dimensional
theory. In particular, the impurity effect is described as the conformal
boundary condition of the effective theory, which allows us to apply the BCFT
techniques to the current problem. Based on such a description, we analyze the
thermodynamic properties of the $G$-Kondo system. In Sec. 3, we study the
impurity contribution to the thermodynamic entropy (the impurity entropy),
which would be a criterion whether the conformal fixed point is described as
the Fermi liquid (FL) or the non-Fermi liquid (NFL). In Sec. 4, we then study
the low temperature behaviors of the specific heat and the susceptibility
using the conformal perturbation theory with the leading irrelevant operator.
We clarify the parameter condition in order to obtain the FL/NFL behavior. We
also mention the case with the exceptional group symmetry, $G=EFG$, in Sec. 5.

##  2 Formulation of the impurity problem

The Kondo problem is originally formulated in three dimensions with a
conduction electron and a localized impurity. Assuming the electron-impurity
interaction takes a form of the $\delta$-function, we can focus on the
$s$-wave scattering, and the system is effectively reduced to the one-
dimensional model defined on a semi-infinite domain $x\in\mathbb{R}_{+}$.
Furthermore, since we are interested in the excitation in the vicinity of the
Fermi energy, we approximate the dispersion relation by $\epsilon(k)\approx
v_{\text{F}}(k-k_{\text{F}})$, which leads to the effective Hamiltonian,

$\displaystyle\mathsf{H}$
$\displaystyle=\frac{v_{\text{F}}}{2\pi}\int_{0}^{\infty}\differential{x}\quantity[\psi_{\text{L},i\alpha}^{\dagger}(x)\mathrm{i}\partial_{x}\psi_{\text{L},i\alpha}(x)-\psi_{\text{R},i\alpha}^{\dagger}(x)\mathrm{i}\partial_{x}\psi_{\text{R},i\alpha}(x)]+\mathsf{H}_{\text{K}}\,,$
(2.1)

where $v_{\text{F}}$ is the Fermi velocity, and
$\psi_{\text{L}/\text{R},i\alpha}$ is the left/right moving fermion with the
spin and channel (flavor) indices, $\alpha=1,\ldots,n$, $i=1,\ldots,k$. The
electron-impurity interaction is described by $\mathsf{H}_{\text{K}}$ defined
in (1.2).

###  2.1 Non-Abelian bosonization

For the complex (charged) fermion, the Hamiltonian (2.1) is invariant under
the following transformations,

$\displaystyle G=\mathrm{SU}(n):\ \psi_{\alpha}\mapsto(g\psi)_{\alpha}\,,\quad
G^{\prime}=\mathrm{SU}(k):\
\psi_{i}\mapsto(g^{\prime}\psi)_{i}\,,\quad\mathrm{U}(1):\
\psi\mapsto\mathrm{e}^{\mathrm{i}\phi}\psi\,.$ (2.2)

Namely, the fermion operator transforms in the bifundamental representation of
$G\times G^{\prime}$. Assuming the interaction $\mathsf{H}_{\text{K}}$ is
insensitive (singlet) to the channel symmetry $G^{\prime}$, we may consider
the right mover $\psi_{\text{R}}$ as the analytic continuation of the left
mover to the negative region,
$\psi_{\text{L},\alpha}(x)=(g\psi_{\text{R}})_{\alpha}(-x)=:\psi_{\alpha}(x)$
with $g\in G$. Then, we obtain the chiral expression of the Hamiltonian (2.1)
as follows:

$\displaystyle\mathsf{H}$
$\displaystyle=\frac{v_{\text{F}}}{2\pi}\int_{-\infty}^{\infty}\differential{x}\psi_{i\alpha}^{\dagger}(x)\mathrm{i}\partial_{x}\psi_{i\alpha}(x)+\mathsf{H}_{\text{K}}\,.$
(2.3)

The interaction term, originally introduced in (1.2), is rewritten as

$\displaystyle\mathsf{H}_{\text{K}}=\lambda_{\text{K}}\sum_{a=1}^{\dim
G}\psi_{i\alpha}^{\dagger}(x)(t^{a})_{\alpha\beta}\psi_{i\beta}(x)\,S^{a}\,\delta(x)=\lambda_{\text{K}}\,J^{a}(x)\,S^{a}\,\delta(x)\,,$
(2.4)

where the localized impurity spin operator is denoted by $S^{a}\,\delta(x)$
for $a=1,\ldots,\dim G$. We define the current operators with respect to
$(G,G^{\prime},\mathrm{U}(1))$-symmetry (non-Abelian bosonization; See, for
example, [28] for details):

$\displaystyle
J^{a}(x)={:\psi^{\dagger}_{i\alpha}(t^{a})_{\alpha\beta}\psi_{i\beta}:}(x)\,,\quad
J^{A}(x)={:\psi^{\dagger}_{i\alpha}(T^{A})_{ij}\psi_{j\alpha}:}(x)\,,\quad
J(x)={:\psi^{\dagger}_{i\alpha}\psi_{i\alpha}:}(x)$ (2.5)

where $(t^{a})_{a=1,\ldots,\dim G}$, $(T^{A})_{A=1,\ldots,\dim G^{\prime}}$
are the generators of the corresponding Lie algebras,
$\mathfrak{g}=\operatorname{Lie}G$ and
$\mathfrak{g}^{\prime}=\operatorname{Lie}G^{\prime}$, and the normal ordering
symbol denoted by $:\mathcal{O}:$. The levels of these currents are
respectively given by $k$, $n$, and $kn$. The level-$k$ $G$-symmetry current
($G_{k}$-current for short) obeys the operator product expansion (OPE),

$\displaystyle
J^{a}(z)J^{b}(w)=\frac{\frac{k}{2}\,\delta^{ab}}{(z-w)^{2}}+\frac{f^{ab}_{~{}~{}c}\,J^{c}(w)}{z-w}+\cdots$
(2.6)

where $(f^{ab}_{~{}~{}c})$ is the structure constant of the Lie algebra
$\mathfrak{g}=\operatorname{Lie}G$. Applying the mode expansion,

$\displaystyle
J_{n}^{a}=\oint\frac{\differential{z}}{2\pi\mathrm{i}}\,z^{n}\,J^{a}(z)\ \iff\
J^{a}(z)=\sum_{n\in\mathbb{Z}}J_{n}^{a}\,z^{-n-1}\,,$ (2.7)

the OPE is equivalent to the algebraic relation of the affine Lie algebra
$\widehat{\mathfrak{g}}_{k}$, which is an infinite dimensional extension of
$\mathfrak{g}$,

$\displaystyle\commutator{J_{n}^{a}}{J_{m}^{b}}=f^{ab}_{~{}~{}c}J_{n+m}^{c}+\frac{k}{2}n\,\delta^{ab}\delta_{n+m,0}\,.$
(2.8)

The same argument is applied to the $G^{\prime}$-current.

We put $v_{\text{F}}=1$ for simplicity. Then, the Hamiltonian (2.3) is
rewritten in the quadratic form of the current operators (Sugawara form),

$\displaystyle\mathsf{H}=\frac{1}{2\pi}\quantity[{\frac{1}{n+k}:J^{a}J^{a}:}(x)+{\frac{1}{k+n}:J^{A}J^{A}:}(x)+{\frac{1}{2kn}:JJ:}(x)]+\lambda_{\text{K}}J^{a}S^{a}\delta(x)\,.$
(2.9)

Such a decomposition of the $kn$-component complex fermion system is a
consequence of the conformal embedding

$\displaystyle\widehat{\mathfrak{su}(n)}_{k}\oplus\widehat{\mathfrak{su}(k)}_{n}\oplus\widehat{\mathfrak{u}(1)}_{kn}\subset\widehat{\mathfrak{u}(kn)}_{1}\,,$
(2.10)

which corresponds to the spin-charge(-channel) separation of the one-
dimensional fermion. We see the agreement of the central charges

$\displaystyle
c\quantity[\widehat{\mathfrak{su}(n)}_{k}]+c\quantity[\widehat{\mathfrak{su}(k)}_{n}]+c\quantity[\widehat{\mathfrak{u}(1)}_{kn}]$
$\displaystyle=\frac{k(n^{2}-1)}{k+n}+\frac{n(k^{2}-1)}{n+k}+1$
$\displaystyle=kn$
$\displaystyle=c\quantity[\widehat{\mathfrak{u}(kn)}_{1}]\,,$ (2.11)

where the central charge of the affine Lie algebra
$\widehat{\mathfrak{g}}_{k}$ is given by

$\displaystyle
c\quantity[\widehat{\mathfrak{g}}_{k}]=\frac{k\dim\mathfrak{g}}{k+h^{\vee}}$
(2.12)

with $h^{\vee}$ the dual Coxeter number of $\mathfrak{g}$, e.g., $h^{\vee}=n$
for $\mathfrak{su}(n)$. See Appendix A for details.

In fact, the current operator obeys the same OPE (2.6) under the redefinition,

$\displaystyle J^{a}(x)\ \longrightarrow\ J^{a}(x)+2\pi\delta(x)S^{a}\,.$
(2.13)

Therefore, if the coupling constant takes the critical value
$\lambda_{\text{K}}^{*}=2/(n+k)$, the interaction term in the Hamiltonian
(2.9) is absorbed into the quadratic term,

$\displaystyle\mathsf{H}=\frac{1}{2\pi}\quantity[{\frac{1}{n+k}:J^{a}J^{a}:}(x)+{\frac{1}{k+n}:J^{A}J^{A}:}(x)+{\frac{1}{2kn}:JJ:}(x)]\,,$
(2.14)

where we omit the non-dynamical term $S^{a}S^{a}$. In fact, the critical value
$\lambda_{\text{K}}$ is interpreted as the value of the coupling constant at
the conformal fixed point, and the impurity effect would be incorporated as
the conformal boundary condition labeled by the representation
$R_{\text{imp}}$ under the symmetry group $G$.

###  2.2 Real fermion

We can similarly formulate the impurity problem with the real (Majorana)
fermions, by replacing the complex conjugate $\psi^{\dagger}$ with $\psi$. We
have seen that, the conformal embedding (2.10) plays a crucial role to specify
the symmetry of the system. In the case with the real fermions, we instead
have the following embeddings [29, 30]:

$\displaystyle\widehat{\mathfrak{so}(n)}_{k}\oplus\widehat{\mathfrak{so}(k)}_{n}$
$\displaystyle\subset\widehat{\mathfrak{so}(kn)}_{1}\quad\text{for}\quad kn:\
\text{even}\,,$ (2.15a)
$\displaystyle\widehat{\mathfrak{sp}(n)}_{k}\oplus\widehat{\mathfrak{sp}(k)}_{n}$
$\displaystyle\subset\widehat{\mathfrak{so}(4kn)}_{1}\,.$ (2.15b)

The central charges are given by

$\displaystyle
c\quantity[\widehat{\mathfrak{so}(n)}_{k}]+c\quantity[\widehat{\mathfrak{so}(k)}_{n}]$
$\displaystyle=\frac{k}{2}\frac{n^{2}-n}{n+k-2}+\frac{n}{2}\frac{k^{2}-k}{k+n-2}=\frac{kn}{2}=c\quantity[\widehat{\mathfrak{so}(kn)}_{1}]\,,$
(2.16a) $\displaystyle
c\quantity[\widehat{\mathfrak{sp}(n)}_{k}]+c\quantity[\widehat{\mathfrak{sp}(k)}_{n}]$
$\displaystyle=\frac{kn(2n+1)}{k+n+1}+\frac{nk(2k+1)}{n+k+1}=2kn=c\quantity[\widehat{\mathfrak{so}(4kn)}_{1}]\,.$
(2.16b)

The latter is also interpreted as the embedding for the $kn$-component
quaternionic fermion. Based on these embeddings, we have two options for the
symmetry groups for the real fermion system:

$\displaystyle(G,G^{\prime})=\begin{cases}(\mathrm{SO}(n),\mathrm{SO}(k))\\\
(\mathrm{Sp}(n),\mathrm{Sp}(k))\\\ \end{cases}$ (2.17)

We then consider the Sugawara Hamiltonian for the real fermion system:

$\displaystyle\mathsf{H}=\frac{1}{2\pi}\quantity[{\frac{1}{k+h^{\vee}(\mathfrak{g})}:J^{a}J^{a}:}(x)+{\frac{1}{n+h^{\vee}(\mathfrak{g}^{\prime})}:J^{A}J^{A}:}(x)]+\lambda_{\text{K}}J^{a}S^{a}\delta(x)\,.$
(2.18)

where $S^{a}$ is the impurity “spin” which transforms under $G=\mathrm{SO}(n)$
and Sp$(n)$, respectively. See Tab. 1 for the list of the dual Coxeter number.
We remark that, compared to the complex fermion system (2.9), there is no
$\mathrm{U}(1)$-current (the charge current) in this case. This is because the
analog of $\mathrm{U}(1)$ in this case is the discrete group
$\mathrm{O}(1)=\mathbb{Z}_{2}$. Precisely speaking, we should consider the
spin group (the universal cover of SO group) in the conformal embedding,
depending on the parity of the rank. See, for example, [31] for details.

If the coupling constant takes the critical value
$\lambda_{\text{K}}^{*}=2/(k+h^{\vee}(\mathfrak{g}))$, the interaction term is
combined into the quadratic form by the $G_{k}$-current shift as in (2.13),

$\displaystyle\mathsf{H}=\frac{1}{2\pi}\quantity[{\frac{1}{k+h^{\vee}(\mathfrak{g})}:J^{a}J^{a}:}(x)+{\frac{1}{n+h^{\vee}(\mathfrak{g}^{\prime})}:J^{A}J^{A}:}(x)]\,.$
(2.19)

Again, we consider the conformal boundary condition characterized by the
representation $R_{\text{imp}}$ of the impurity under the symmetry group $G$.

##  3 Impurity entropy

The impurity contribution to the entropy (impurity entropy; also called the
boundary entropy) provides a useful criterion for the FL/NFL behavior at the
conformal fixed point. In this Section, we compute the impurity entropy of
$G$-Kondo system based on the CFT formalism discussed in Sec. 2.

The impurity entropy of the Kondo system is obtained through the so-called
$g$-factor, which counts the boundary degeneracy [21]:

$\displaystyle S_{\text{imp}}=\log g(R_{\text{imp}})$ (3.1)

where we denote the representation of the impurity under the symmetry group
$G$ by $R_{\text{imp}}$. If the $g$-factor takes an integral value,
$g(R)\in\mathbb{N}$, the fixed point is supposed to be the FL, since one can
“count” the degeneracy of the impurity states, and the original picture which
consists of the conduction electron and the localized impurity still holds at
the fixed point. On the other hand, it would be described as the NFL if the
$g$-factor becomes an irrational number, $g(R)\not\in\mathbb{N}$, where the
original separated picture of the electron and the impurity is not available
any longer. We remark that there are several exceptional situations for the
NFL where the corresponding $g$-factor takes an integral value due to an
accidental symmetry.

The $g$-factor is given by the modular $\mathsf{S}$-matrix of the affine Lie
algebra $\widehat{\mathfrak{g}}_{k}$ denoted by $\mathsf{S}_{RR^{\prime}}$
[32]:

$\displaystyle
g(R)=\frac{\mathsf{S}_{0R}}{\mathsf{S}_{00}}=\chi_{R}\quantity(q^{\rho})\,,$
(3.2)

where $\chi_{R}(X)$ is the character of the representation $R$ of the Lie
algebra $\mathfrak{g}$ evaluated with the Cartan torus
$X\in(\mathbb{C}^{\times})^{\operatorname{rk}\mathfrak{g}}$, and $\rho$ is the
Weyl vector of $\mathfrak{g}$ given by (A.5). The character formula is
summarized in Appendix B for $G=ABCD$. The parameter
$q\in\mathbb{C}^{\times}$, $\absolutevalue{q}=1$, is defined with the level
and the dual Coxeter number $(k,h^{\vee})$ as follows:

$\displaystyle q=\exp\left(\frac{2\pi\mathrm{i}}{k+h^{\vee}}\right)\,.$ (3.3)

We remark that the expression of the $g$-factor (3.2) coincides with the
unknot Wilson loop expectation value of $G_{k}$ Chern–Simons theory on the
three-sphere $\mathbb{S}^{3}$ [33]. See also [34, 35, 36] for further
discussion on the relation to the line defect.

In fact, the $g$-factor is given as the principal specialization of the
character of the representation $R$, a.k.a., the quantum dimension of the
representation $R$,

$\displaystyle g(R)=\dim_{q}R\geq 1\,.$ (3.4)

The quantum dimension is reduced to the ordinary dimension in the limit $q\to
1$, $\lim_{q\to 1}\dim_{q}R=\dim R\in\mathbb{N}$, which is interpreted as the
$g$-factor in the weak coupling regime. In addition, since it is a polynomial
of $(q,q^{-1})$ with non-negative integer coefficients, it is smaller than the
classical value, $g(R)\leq\dim R$. This implies that the $g$-factor decreases
(possibly, monotonically) along the renormalization flow to the low-energy
regime. This is called the $g$-theorem, a boundary analog of the $c$-theorem
in CFT.

We remark that, in the large $k$ limit, the asymptotic behavior of the
impurity entropy is given by

$\displaystyle S_{\text{imp}}\ \xrightarrow{k\to\infty}\ \log\dim R\,.$ (3.5)

Hence, the $g$-factor takes an integral value, and thus the the fixed point is
described as the FL in the limit $k\to\infty$. This implies that the quantum
correction to the $g$-factor is largely suppressed in in this case.

###  3.1 SU($n$) theory

We consider the $g$-factor for $G=\mathrm{SU}(n)$ theory. Let us consider the
impurity, which transforms in the $n$-dimensional (natural) representation of
$\mathrm{SU}(n)$ denoted by $\mathbf{n}=R(1,0,\ldots,0)$. Then, applying the
formula shown in Appendix B, we obtain the $g$-factor as follows:

$\displaystyle
g(\mathbf{n})=\frac{q^{\frac{1}{2}(n-1)}-q^{-\frac{1}{2}(n-1)}}{q^{\frac{1}{2}}-q^{-\frac{1}{2}}}=q^{\frac{1}{2}(n-1)}+q^{\frac{1}{2}(n-3)}+\cdots+q^{-\frac{1}{2}(n-1)}\
\xrightarrow{q\to 1}\ n\,.$ (3.6)

We remark that its conjugate representation gives rise to the same $g$-factor,
$g(\bar{\mathbf{n}})=g(\mathbf{n})$. The numerical values of the $g$-factor
with several $(n,k)$ are given as follows [37]:

(3.13)

We see that the $g$-factor takes irrational values for generic $(n,k)$, except
at $k=1,\infty$, $n\to\infty$. This implies that it behaves as the FL at large
$(n,k)$ and at $k=1$, while it would be described as the NFL in other cases.
(The case with $(n,k)=(3,3)$ is exceptional.) The symmetry $n\leftrightarrow
k$ is a consequence of the level-rank duality of the affine Lie algebras,
$\widehat{\mathfrak{su}(n)}_{k}$ and $\widehat{\mathfrak{su}(k)}_{n}$, which
is also related to the conformal embedding (2.10) [38, 39, 40].

###  3.2 SO($n$) theory

We consider the $n$-dimensional vector representation and the spinor
representation of $\mathrm{SO}(n)$ theory.

####  3.2.1 Vector representation

The $g$-factor for the vector representation of $\mathrm{SO}(n)$ denoted by
$\mathbf{n}=R(1,\ldots,0)$ is given by

$\displaystyle g(\mathbf{n})$ $\displaystyle=\begin{cases}\displaystyle
q^{m-1/2}+\cdots+q^{1/2}+1+q^{-1/2}+\cdots+q^{-m+1/2}&(n=2m+1)\\\
q^{m-1}+\cdots+q+2+q^{-1}+\cdots+q^{-m+1}&(n=2m)\end{cases}$
$\displaystyle\xrightarrow{q\to 1}\ n\,,$ (3.14)

and the numerical values are given as follows:

(3.21)

In this case, the $g$-factor becomes $g=1,2$ for $k=1,2$, and it is expected
to be described as the FL. The remaining cases would be the NFL. We can see
the level-rank duality between $\widehat{\mathfrak{so}(n)}_{k}$ and
$\widehat{\mathfrak{so}(k)}_{n}$ similarly to SU($n$) theory. We remark that
$\mathrm{SO}(4)_{4}$ exceptionally gives an integral value due to the
isomorphism, $\mathrm{SO}(4)=\mathrm{SU}(2)\times\mathrm{SU}(2)$.

####  3.2.2 Spinor representation

We then study the spinor representation,
$\mathscr{S}=R\quantity(\frac{1}{2},\ldots,\frac{1}{2})$, whose dimension is
$2^{h/2}$ with the Coxeter number $h$ (Tab. 1): It is given by $2^{m}$ and
$2^{m-1}$ for SO($2m+1$) and SO($2m$). Precisely speaking, SO$(2m)$ theory has
two spin representations, $\mathscr{S}^{+}$ and $\mathscr{S}^{-}$, labeled by
$\mathscr{S}^{\pm}=R\quantity(\frac{1}{2},\ldots,\pm\frac{1}{2})$, which give
rise to the same $g$-factor.

The numerical values of the $g$-factor are given by

(3.27)

We remark that the spinor representation of SO(6) is equivalent to the
$\mathbf{4}$-representation of SU(4) as in (3.13) as a consequence of the
isomorphism between SO(6) and SU(4).

In particular, we see the peculiar behavior, $g(\mathscr{S})=1$ for
SO$(2m)_{k=1}$ and $g(\mathscr{S})=\sqrt{2}=1.4142...$ for SO$(2m+1)_{k=1}$.
This behavior is consistent with the analysis in [12] studying the
$\mathrm{SO}(n)_{k=1}$ impurity system: The Majorana impurity system would be
in general described as the NFL except for $k=1$ of SO($2m$) theory. In fact,
the central charges of SO($n$) theories are given by

$\displaystyle c\left[\widehat{\mathfrak{so}(2m)}_{k}\right]$
$\displaystyle=\frac{km(2m-1)}{k+2m-2}\ \xrightarrow{k\to 1}\ m\,,$ (3.28a)
$\displaystyle c\left[\widehat{\mathfrak{so}(2m+1)}_{k}\right]$
$\displaystyle=\frac{km(2m+1)}{k+2m-1}\ \xrightarrow{k\to 1}\
m+\frac{1}{2}\,.$ (3.28b)

This is interpreted as follows: For the SO$(2m)$ theory, there are $2m$
Majorana fermions, and they make pairs to form $m$ complex fermions to be
described as the FL. In the case of SO$(2m+1)$ theory, on the other hand,
$2m+1$ Majorana fermions similarly gives rise to $m$ pairs ($m$ complex
fermions) and a single Majorana fermion. The factor $\frac{1}{2}$ is actually
interpreted as the contribution of the Majorana fermion, which leads to the
NFL behavior.

###  3.3 Sp($n$) theory

Let us turn to the $\mathrm{Sp}(n)$ theory. In this case, we consider the
$2n$-dimensional (natural) representation of $\mathrm{Sp}(2n)$,
$\mathbf{2n}=R(1,0,\ldots,0)$. The $g$-factor is given as follows:

$\displaystyle g(\mathbf{2n})=q^{n}+\cdots+q+q^{-1}+\cdots q^{-n}\
\xrightarrow{q\to 1}\ 2n\,.$ (3.29)

The numerical values of the $g$-factor are given as follows:

(3.36)

In order to obtain consistent results for Sp$(n)$ theory, we have to rescale
the $q$-parameter as $q\to q^{1/2}$, verified with the isomorphisms,
$\mathrm{Sp}(1)=\mathrm{SU}(2)$ and $\mathrm{Sp}(2)=\mathrm{SO}(5)$. We can
see the level-rank duality between $\widehat{\mathfrak{sp}(n)}_{k}$ and
$\widehat{\mathfrak{sp}(k)}_{n}$ as well. The limit values are
$\displaystyle\lim_{n\to\infty}g=2k$ and $\displaystyle\lim_{k\to\infty}g=2n$.
We remark that the cases with $n=1$ and $n=2$ are equivalent to SU(2) and (the
spin representation of) SO(5) shown in (3.13) and (3.27), respectively, due to
the isomorphisms.

A specific feature of Sp($n$) theory is that the $g$-factor becomes irrational
even for $k=1$ (except for $n=1$, which is isomorphic to SU(2)). Therefore,
the Sp($n$) fixed point would be described as the NFL in general. The central
charge of the Sp($n$) theory is given by

$\displaystyle
c\quantity[\widehat{\mathfrak{sp}(n)}_{k}]=\frac{kn(2n+1)}{k+n+1}\
\xrightarrow{k\to 1}\ \frac{n(2n+1)}{n+2}\,.$ (3.37)

In fact, it is a non-integer value even at $k=1$, which also suggests that the
$\mathrm{Sp}(n)_{k=1}$ theory would behave as the NFL.

##  4 Low temperature thermodynamics

In addition to the conformal fixed point corresponding to zero temperature
$T=0$, we can also study the low temperature behavior of the thermodynamic
quantities based on the conformal perturbation theory.

###  4.1 Bulk contribution

We first consider the bulk contribution to the thermodynamic quantities, the
specific heat and the susceptibility [41, 42, 43]:

$\displaystyle
C_{\text{bulk}}=\frac{\pi}{3}c\,T\,,\qquad\chi_{\text{bulk}}=\frac{k}{2\pi}\,,$
(4.1)

where $c$ is the total central charge, corresponding to the conformal
embeddings, (2.11) and (2.16),

$\displaystyle c=bkn=\begin{cases}kn&(\mathrm{SU}(n)_{k})\\\\[5.0pt]
\displaystyle\frac{kn}{2}&(\mathrm{SO}(n)_{k})\\\
2kn&(\mathrm{Sp}(n)_{k})\end{cases}$ (4.2)

with the symmetry parameter

$\displaystyle\mathrm{SU}(n):\ b=1\,,\qquad\mathrm{SO}(n):\
b=\frac{1}{2}\,,\qquad\mathrm{Sp}(n):\ b=2\,.$ (4.3)

We remark that the parameter $\beta=2b$ is called the Dyson index in the
context of random matrix theory. See [44] for details.

The susceptibility considered here is with respect to the external $G$-field
(magnetic field), which is coupled to the $G$-current in the form of
$h^{a}J^{a}$. The $G$-symmetry allows us to take the specific direction for
the coupling term $h^{1}J^{1}=:hJ^{1}$ without loss of generality. See [9, 10,
12, 13] for the proposal to realize the magnetic field in $\mathrm{SO}(n)$
Kondo system.

###  4.2 Impurity contribution

In order to compute the finite temperature behavior, we apply the perturbative
analysis with the leading irrelevant operators as follows [20, 22]:

$\displaystyle\mathcal{O}_{1}$ $\displaystyle={:J^{a}J^{a}:(x)}\,\delta(x)\,,$
(4.4a) $\displaystyle\mathcal{O}_{2}$
$\displaystyle={:J_{-1}^{a}\phi^{a}:(x)}\,\delta(x)\,,$ (4.4b)

with

$\displaystyle\delta\mathsf{H}=-\lambda_{\sigma}\,\mathcal{O}_{\sigma}\qquad(\sigma=1,2)\,.$
(4.5)

The first operator $\mathcal{O}_{1}$ is the quadratic operator with the
dimension $\Delta_{1}=2$. Therefore, the dimensional analysis shows that the
corresponding coupling constant $\lambda_{1}$ should be proportional to
$T_{\text{K}}^{-1}$.

The second operator $\mathcal{O}_{2}$ is the first descendent of the adjoint
primary operator $\phi^{a}(x)$ with the (conformal) dimension [45],

$\displaystyle\Delta_{\phi}=\frac{h^{\vee}}{k+h^{\vee}}\,,$ (4.6)

for $\widehat{\mathfrak{g}}_{k}$ theory, where we apply the conformal
dimension formula of the primary operator associated with the representation
$R$ of the Lie algebra $\mathfrak{g}$, $\Delta_{R}=C_{2}(R)/(k+h^{\vee})$.
$C_{2}(R)$ is the quadratic Casimir of the representation $R$. For the adjoint
representation, it is given by the dual Coxeter number (A.1).

The dimension of the second operator becomes $\Delta_{2}=1+\Delta_{\phi}$, and
the corresponding coupling constant $\lambda_{2}$ should be proportional to
$T_{\text{K}}^{-\Delta_{\phi}}$. We remark that this operator
$\mathcal{O}_{2}$ may be identified with the so-called $\hat{h}$-scalar in the
context of the coset CFT [46]. Since $\Delta_{\phi}<1$ for $k\geq 1$, and thus
$\Delta_{2}<\Delta_{1}$, the second operator $\mathcal{O}_{2}$ is the leading
irrelevant operator if it exists. In fact, the adjoint representation is not a
fundamental representation for $G=\mathrm{SU}(n)$ nor $\mathrm{Sp}(n)$: The
adjoint representation is given by the tensor product of n and
$\overline{\textbf{n}}$ representations for $\mathrm{SU}(n)$, and the degree
two symmetric tensor product of the natural representation for
$\mathrm{Sp}(n)$. Therefore, the operator $\mathcal{O}_{2}$ is not generated
by the fusion process if $k=1$. Hence, we should apply the operator
$\mathcal{O}_{1}$ for the perturbative expansion to discuss the finite
temperature behavior for $\mathrm{SU}(n)_{1}$ and $\mathrm{Sp}(n)_{1}$, while
we should use the other operator $\mathcal{O}_{2}$ for the remaining cases.

For the complex fermion system, the total partition function in the presence
of the perturbative term $\delta\mathsf{H}$ and the external $G$-field
coupling is given by the path integral over the fermion field,

$\displaystyle Z$ $\displaystyle=\exp\quantity(-\beta F(\lambda_{\sigma},h))$
$\displaystyle=\int\mathcal{D}\psi\mathcal{D}\psi^{\dagger}\,\exp\quantity[-\int_{-\beta/2}^{\beta/2}\differential{\tau}\int_{-L/2}^{L/2}\differential{x}\quantity(\psi^{\dagger}(\partial_{\tau}+\mathrm{i}\partial_{x})\psi-\lambda_{\sigma}\mathcal{O}_{\sigma}+\frac{h}{2\pi}J^{1}(\tau,x))]$
(4.7)

where $L$ is the system size, and $\beta=T^{-1}$ is the inverse temperature.
Splitting the free energy into the bulk and the impurity contributions,
$F=Lf_{\text{bulk}}+f_{\text{imp}}$, we obtain

$\displaystyle\mathrm{e}^{-\beta f_{\text{imp}}}$
$\displaystyle=\left<\exp\quantity[-\int_{-\beta/2}^{\beta/2}\differential{\tau}\int_{-L/2}^{L/2}\differential{x}\quantity(-\lambda_{\sigma}\mathcal{O}_{\sigma}+\frac{h}{2\pi}J^{1}(\tau,x))]\right>\,,$
(4.8)

where the average is taken with respect to the path integral with
$(\lambda_{\sigma},h)=(0,0)$. We obtain the same correlator expression in the
real fermion system. We will perturbatively evaluate this path integral to
compute the low temperature behavior of the thermodynamic quantities.

####  4.2.1 Specific heat

We consider the low temperature behavior of the specific heat. From the
operator $\mathcal{O}_{1}$, we obtain [20]

$\displaystyle
C_{\text{imp}}=\lambda_{1}\frac{k\dim\mathfrak{g}}{3}\pi^{2}T\,,$ (4.9)

which exhibits the linear dependence on the temperature, a.k.a., the FL
property. As discussed above, this result is applied only to the case with
$k=1$ for $G=\mathrm{SU}(n)$ and $\mathrm{Sp}(n)$:

$\displaystyle
C_{\text{imp}}=\begin{cases}\displaystyle\lambda_{1}\frac{n^{2}-1}{3}\pi^{2}T&(\mathrm{SU}(n)_{1})\\\\[5.0pt]
\displaystyle\lambda_{1}\frac{n(2n+1)}{3}\pi^{2}T&(\mathrm{Sp}(n)_{1})\end{cases}$
(4.10)

From the operator $\mathcal{O}_{2}$, we instead obtain the following
expression:

$\displaystyle
C_{\text{imp}}=\begin{cases}\displaystyle\lambda_{2}^{2}\pi^{2\Delta_{\phi}+1}\Delta_{\phi}^{2}\quantity(1-2\Delta_{\phi})\frac{\Gamma(1/2-\Delta_{\phi})\Gamma(1/2)}{\Gamma(1-\Delta_{\phi})}\dim\mathfrak{g}\quantity(h^{\vee}+\frac{k}{2})\,T^{2\Delta_{\phi}}&(\Delta_{\phi}<1/2)\\\\[10.00002pt]
\displaystyle\lambda_{2}^{2}\pi^{2}\dim\mathfrak{g}\quantity(h^{\vee}+\frac{k}{2})\,T\log\quantity(\frac{T_{\text{K}}}{T})&(\Delta_{\phi}=1/2)\\\\[10.00002pt]
\displaystyle\frac{4\lambda_{2}^{2}\pi^{2}\Delta_{\phi}\,T_{\text{K}}^{2\Delta_{\phi}-1}}{(2\Delta_{\phi}+1)(2\Delta_{\phi}-1)}\dim\mathfrak{g}\quantity(h^{\vee}+\frac{k}{2})\,T&(\Delta_{\phi}>1/2)\end{cases}$
(4.11)

In the derivation of this expression, we use the formula,

$\displaystyle\left<\,J_{-1}^{a}\phi^{a}(\tau_{1},0)J_{-1}^{b}\phi^{b}(\tau_{2},0)\,\right>=\frac{(h^{\vee}+k/2)\dim\mathfrak{g}}{\absolutevalue{[\tau_{1}-\tau_{2}]}^{2(\Delta_{\phi}+1)}}\qquad\text{with}\qquad[x]=\frac{\beta}{\pi}\sin\quantity(\frac{\pi}{\beta}x)\,.$
(4.12)

The remaining computation is parallel with the earlier analysis. See, for
example, [22, 37] for details. Compared to the dimension of the adjoint
primary operator (4.6), we have $\Delta_{\phi}\lessgtr 1/2$ $\iff$
$h^{\vee}\lessgtr k$. Then, focusing on the temperature dependence, it shows

$\displaystyle
C_{\text{imp}}\propto\begin{cases}T^{2\Delta_{\phi}}&(\Delta_{\phi}<1/2;\
h^{\vee}<k)\\\ T\log(T_{\text{K}}/T)&(\Delta_{\phi}=1/2;\ h^{\vee}=k)\\\
T&(\Delta_{\phi}>1/2;\ h^{\vee}>k)\end{cases}$ (4.13)

Therefore, for $h^{\vee}\leq k$, we have the fractional power (logarithmic)
dependence on the temperature, which is a specific feature of the NFL, while
it shows the standard FL behavior for $h^{\vee}>k$, which is compatible with
the contribution of the operator $\mathcal{O}_{1}$ (4.9).

####  4.2.2 Susceptibility

We can similarly compute the susceptibility. From the operator
$\mathcal{O}_{1}$, we obtain [20]

$\displaystyle\chi_{\text{imp}}=\lambda_{1}\frac{k(h^{\vee}+k)}{2}\,,$ (4.14)

which is indeed the FL behavior as expected. As well as the specific heat,
this result is applied only to the case with $k=1$ for $G=\mathrm{SU}(n)$ and
$\mathrm{Sp}(n)$:

$\displaystyle\chi_{\text{imp}}=\begin{cases}\displaystyle\lambda_{1}\frac{n+1}{2}&(\mathrm{SU}(n)_{1})\\\\[5.0pt]
\displaystyle\lambda_{1}\frac{n+2}{2}&(\mathrm{Sp}(n)_{1})\end{cases}$ (4.15)

From the operator $\mathcal{O}_{2}$, we obtain the following expression:

$\displaystyle\chi_{\text{imp}}=\begin{cases}\displaystyle\lambda_{2}^{2}\pi^{2\Delta_{\phi}-1}\quantity(\frac{1}{2}-\Delta_{\phi})\frac{\Gamma(1/2-\Delta_{\phi})\Gamma(1/2)}{\Gamma(1-\Delta_{\phi})}\quantity(h^{\vee}+\frac{k}{2})^{2}\,T^{2\Delta_{\phi}-1}&(\Delta_{\phi}<1/2)\\\\[10.00002pt]
\displaystyle
2\lambda_{2}^{2}\quantity(h^{\vee}+\frac{k}{2})^{2}\,\log\quantity(\frac{T_{\text{K}}}{T})&(\Delta_{\phi}=1/2)\\\\[10.00002pt]
\displaystyle\frac{2\lambda_{2}^{2}\,T_{\text{K}}^{2\Delta_{\phi}-1}}{2\Delta_{\phi}-1}\quantity(h^{\vee}+\frac{k}{2})^{2}&(\Delta_{\phi}>1/2)\end{cases}$
(4.16)

In this case, we use the formula

$\displaystyle\left<\,J^{1}(z_{1})J^{1}(z_{2})J_{-1}^{a}\phi^{a}(z_{3})J_{-1}^{b}\phi^{b}(z_{4})\,\right>_{\text{connected}}$
$\displaystyle=\quantity(\frac{1}{[z_{1}-z_{4}]^{2}[z_{2}-z_{3}]^{2}}+\frac{1}{[z_{1}-z_{3}]^{2}[z_{2}-z_{4}]^{2}})\frac{(h^{\vee}+k/2)^{2}}{[z_{3}-z_{4}]^{2\Delta_{\phi}}}$
(4.17)

where the function $[x]$ is defined in (4.12). See [22, 37] for details. The
temperature dependence is given by

$\displaystyle\chi_{\text{imp}}\propto\begin{cases}T^{2\Delta_{\phi}-1}&(\Delta_{\phi}<1/2;\
h^{\vee}<k)\\\ \log(T_{\text{K}}/T)&(\Delta_{\phi}=1/2;\ h^{\vee}=k)\\\
\text{const.}&(\Delta_{\phi}>1/2;\ h^{\vee}>k)\end{cases}$ (4.18)

Then, for the NFL regime $h^{\vee}\leq k$, it shows the fractional power
(logarithmic) dependence as well as the specific heat, while it shows no
temperature dependence for the FL regime $h^{\vee}>k$.

####  4.2.3 Wilson ratio

As seen in the analysis above, the thermodynamic quantities explicitly depend
on the coupling constants $\lambda_{1,2}$, which are non-universal parameters
of the model, although the temperature dependence seems universal. In order to
cancel such a coupling dependence, we consider the Wilson ration as a
combination of the specific heat and the susceptibility:

$\displaystyle
R_{\text{W}}=\quantity(\frac{\chi_{\text{imp}}}{C_{\text{imp}}})\Big{/}\quantity(\frac{\chi_{\text{bulk}}}{C_{\text{bulk}}})\,.$
(4.19)

From the perturbative analysis with the operator $\mathcal{O}_{1}$, we obtain
the Wilson ratio as follows:

$\displaystyle
R_{\text{W}}=\frac{h^{\vee}+1}{\dim\mathfrak{g}}\,c=\begin{cases}\displaystyle\frac{n}{n-1}&(\mathrm{SU}(n)_{1})\\\\[5.0pt]
\displaystyle\frac{2(n+2)}{2n+1}&(\mathrm{Sp}(n)_{1})\end{cases}$ (4.20)

where $c$ is the total central charge (4.2). Together with the operator
$\mathcal{O}_{2}$, we obtain the Wilson ratio for $h^{\vee}\leq k$ as follows:

$\displaystyle
R_{\text{W}}=\frac{bn}{3}\frac{(h^{\vee}+k/2)(h^{\vee}+k)^{2}}{h^{\vee
2}\dim\mathfrak{g}}\,c=\begin{cases}\displaystyle\frac{(n+k/2)(n+k)^{2}}{3n(n^{2}-1)}&(\mathrm{SU}(n)_{k})\\\\[10.00002pt]
\displaystyle\frac{(n-2+k/2)(n-2+k)^{2}}{3(n-1)(n-2)^{2}}&(\mathrm{SO}(n)_{k})\\\\[10.00002pt]
\displaystyle\frac{2}{3}\frac{(n+2+k/2)(n+1+k)^{2}}{(n+1)^{2}(2n+1)}&(\mathrm{Sp}(n)_{k})\\\\[10.00002pt]
\end{cases}$ (4.21)

where $b$ is the symmetry parameter defined in (4.3). For $h^{\vee}>k$, on the
other hand, the contributions from the operators $\mathcal{O}_{1}$ and
$\mathcal{O}_{2}$ are compatible, so that we should take into account both of
them. This makes the Wilson ratio non-universal: It depends on the ratio of
the coupling constants $\lambda_{1}$ and $\lambda_{2}$. This phenomenon,
called the Fermi/non-Fermi mixing, occurs in particular in this parameter
regime [37].

##  5 EFG of Kondo effect

We briefly mention the case with the exceptional group symmetry, $G=EFG$. In
this case, there is no systematic conformal embedding as in (2.10) and (2.15).
Hence, it may have a difficulty in the formulation of the Kondo problem in the
sense discussed there. Nevertheless, the analysis discussed in Sec. 3 and Sec.
4 is similarly applicable to the exceptional groups $G=EFG$.

One thing we should specifically clarify is the leading irrelevant operator
discussed in Sec. 4.2. In fact, since the adjoint representation is one of the
fundamental representations for $G=EFG$, we should take the operator
$\mathcal{O}_{2}$ as the leading irrelevant operator for $\forall k\geq 1$,
which implies the NFL behavior for arbitrary $k\geq h^{\vee}$.111This argument
is not applied to $E_{8}$ since the unique primary field of
$\widehat{\mathfrak{e}}_{8,k}$ is the identity at level one $k=1$.  In the
case with $k<h^{\vee}$, on the other hand, it would be described as the FL.

##  6 Discussion

In this paper, we have formulated the Kondo problem with generic symmetry
group $G=ABCD$, and studied its thermodynamic properties based on the BCFT
analysis. Although BCFT provides a powerful tool to study the non-perturbative
aspects of the Kondo effect, it would be instructive to verify the results
obtained in this paper with other methodologies, e.g., numerical
renormalization group, Bethe ansatz analysis. In particular, since the exact
method based on the Bethe ansatz is formulated for the integrable system with
generic $G$ [47], the $G$-Kondo system could be similarly explored with the
Bethe ansatz with inhomogeneity. See a review article [48] for details.

Even though we have theoretically studied the Kondo system with generic
symmetry group $G$, it would be important to examine how to realize such a
generalized situation. There are several realizations for the SU($n$) systems
as mentioned earlier (Sec. 1), where a similar analysis is also applicable
[49]. For the SO($n$) Kondo system, the Majorana fermion realizes the quantum
impurity in the spinor representation of SO($n$). From this point of view, it
would be also interesting to consider the impurity which transforms in other
representations of SO$(n)$, e.g., the vector representation. For the Sp($n$)
theory, it might be possible to use a similar setup to the SO$(n)$ case:
Actually both cases are based on the conformal embedding of the real fermion
as shown in Sec. 2.2. For example, one can use the isomorphism,
$\mathrm{Sp}(2)=\mathrm{SO}(5)$. In this case, the (four-dimensional) spinor
representation of $\mathrm{SO}(5)$ is directly identified with the natural
representation of Sp$(2)$ (See Sec. 3.3). This suggests the realization of the
Sp$(n)$ Kondo system with the Majorana impurities by imposing the symplectic
Majorana condition.

#### Acknowledgments

We would like to thank E. Kogan and A. Tsvelik for their kind correspondences.
We are also grateful to Sho Ozaki for the earlier collaboration, which leads
to a lot of results presented in this paper. In addition, we would appreciate
useful comments by the anonymous referee. This work has been supported in part
by “Investissements d’Avenir” program, Project ISITE-BFC (No.
ANR-15-IDEX-0003), and EIPHI Graduate School (No. ANR-17-EURE-0002).

##  Appendix A Lie algebras

We summarize the properties of Lie algebras used in this paper. See, for
example, [50] for detailed information about the Lie algebras.

In the CFT analysis, we often use the dual Coxeter number, which is given as
the quadratic Casimir element of the adjoint representation,

$\displaystyle f^{acd}f^{bcd}=h^{\vee}\,\delta^{ab}\,,$ (A.1)

where $f^{abc}$ is the structure constant of the Lie algebra $\mathfrak{g}$.
We remark that it agrees with the Coxeter number $h=h^{\vee}$ for $G=ADE$,
which is related to the rank, and the dimension of $\mathfrak{g}$,

$\displaystyle\dim\mathfrak{g}=n(h+1)\stackrel{{\scriptstyle
G=ADE}}{{=}}n(h^{\vee}+1)\,.$ (A.2)

Therefore, the central charge (2.12) of the level $k=1$ affine Lie algebra for
$\mathfrak{g}=\mathfrak{ade}(n)$ coincides with the rank,

$\displaystyle c\quantity[\widehat{\mathfrak{ade}(n)}_{1}]=n\,,$ (A.3)

which implies the realization with $n$ complex fermions (Frenkel–Kac
construction [51]).

We summarize these properties in Tab. 1. We notice that the dual Coxeter
number and the dimension of SO$(n)$ are in general given as

$\displaystyle
h^{\vee}=n-2\,,\qquad\operatorname{dim}\mathrm{SO}(n)=\frac{1}{2}(n^{2}-n)\,.$
(A.4)

The Weyl vector is defined as

$\displaystyle\rho=\frac{1}{2}\sum_{\alpha\in\Delta_{+}(\mathfrak{g})}\alpha\,,$
(A.5)

with the set of the positive roots of $\mathfrak{g}$ denoted by
$\Delta_{+}(\mathfrak{g})$. It is also possible to express the Weyl vector
$\rho$ as the sum of the fundamental weights. See [52] for details. The Weyl
vector is explicitly given as

$\displaystyle\mathrm{SU}(n)\ :\quad$
$\displaystyle\rho=\quantity(\frac{n-1}{2},\frac{n-3}{2},\ldots,-\frac{n-3}{2},-\frac{n-1}{2})\,,$
(A.6a) $\displaystyle\mathrm{SO}(2n+1)\ :\quad$
$\displaystyle\rho=\quantity(n-\frac{1}{2},n-\frac{3}{2},\ldots,\frac{1}{2})\,,$
(A.6b) $\displaystyle\mathrm{Sp}(n)\ :\quad$
$\displaystyle\rho=\quantity(n,n-1,\ldots,2,1)\,,$ (A.6c)
$\displaystyle\mathrm{SO}(2n)\ :\quad$
$\displaystyle\rho=\quantity(n-1,n-2,\ldots,1,0)\,.$ (A.6d)

Classification | Lie group $G$ | Coxeter $h$ | dual Coxeter $h^{\vee}$ | dimension
---|---|---|---|---
$A_{n}$ | SU($n+1$) | $n+1$ | $n+1$ | $n(n+2)$
$B_{n}$ | SO($2n+1$) | $2n$ | $2n-1$ | $n(2n+1)$
$C_{n}$ | Sp($n$) | $2n$ | $n+1$ | $n(2n+1)$
$D_{n}$ | SO($2n$) | $2n-2$ | $2n-2$ | $n(2n-1)$
$E_{6,7,8}$ | – | 12, 18, 30 | 12, 18, 30 | 78, 133, 248
$F_{4}$ | – | 12 | 9 | 52
$G_{2}$ | – | 6 | 4 | 14
Table 1: Properties of Lie algebras.

##  Appendix B Character formula

The character of the representation $R$ of the Lie algebra $\mathfrak{g}$ has
several equivalent expressions. We here present the determinantal formula for
the character of the classical groups, $G=ABCD$, a.k.a., the Schur, symplectic
Schur, and orthogonal Schur functions.

Let
$X=(x_{1},\ldots,x_{n})\in(\mathbb{C}^{\times})^{\operatorname{rk}\mathfrak{g}}$
be an element of the maximal Cartan torus of $G$. Then the character of the
representation $R$ parametrized by a (half-)integer sequence $\lambda$ is
given as follows:

$\displaystyle\chi_{R(\lambda)}(X)=\begin{cases}s_{\lambda}(X)&(\mathrm{SU}(n))\\\
o^{(\text{o})}_{\lambda}(X)&(\mathrm{SO}(2n+1))\\\
sp_{\lambda}(X)&(\mathrm{Sp}(n))\\\
o^{(\text{e})}_{\lambda}(X)&(\mathrm{SO}(2n))\end{cases}$ (B.1)

where $s_{\lambda}(X)$, $o^{(\text{o/e})}_{\lambda}(X)$, and $sp_{\lambda}(X)$
are the Schur, orthogonal Schur, and symplectic Schur functions, defined as
the ratio of the determinants [52] (See also [53, 54]):

$\displaystyle s_{\lambda}(X)$ $\displaystyle=\frac{\det_{1\leq i,j\leq
n}\quantity(x_{j}^{\lambda_{i}+n-i})}{\det_{1\leq i,j\leq
n}\quantity(x_{j}^{n-i})}\,,$ (B.2a) $\displaystyle
o^{(\text{o})}_{\lambda}(X)$ $\displaystyle=\frac{\det_{1\leq i,j\leq
n}\quantity(x_{j}^{\lambda_{i}+n-i+\frac{1}{2}}-x_{j}^{-(\lambda_{i}+n-i+\frac{1}{2})})}{\det_{1\leq
i,j\leq n}\quantity(x_{j}^{n-i+\frac{1}{2}}-x_{j}^{-(n-i+\frac{1}{2})})}\,,$
(B.2b) $\displaystyle sp_{\lambda}(X)$ $\displaystyle=\frac{\det_{1\leq
i,j\leq
n}\quantity(x_{j}^{\lambda_{i}+n-i+1}-x_{j}^{-(\lambda_{i}+n-i+1)})}{\det_{1\leq
i,j\leq n}\quantity(x_{j}^{n-i+1}-x_{j}^{-(n-i+1)})}\,,$ (B.2c) $\displaystyle
o^{(\text{e})}_{\lambda}(X)$ $\displaystyle=\frac{\det_{1\leq i,j\leq
n}\quantity(x_{j}^{\lambda_{i}+n-i}+x_{j}^{-(\lambda_{i}+n-i)})+\det_{1\leq
i,j\leq
n}\quantity(x_{j}^{\lambda_{i}+n-i}-x_{j}^{-(\lambda_{i}+n-i)})}{\det_{1\leq
i,j\leq n}\quantity(x_{j}^{n-i}+x_{j}^{-(n-i)})}\,.$ (B.2d)

The highest weight representation $R(\lambda)$ is parametrized by a sequence
of (half-)integers, $\lambda=(\lambda_{1},\ldots,\lambda_{n})$:

$\displaystyle\mathrm{SU}(n)\ :\quad$
$\displaystyle\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}\geq
0)\in\mathbb{Z}_{\geq 0}^{n}\,,$ (B.3a) $\displaystyle\mathrm{SO}(2n+1)\
:\quad$
$\displaystyle\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}\geq
0)\in\mathbb{Z}_{\geq 0}^{n}\ \text{or}\ \quantity(\mathbb{Z}_{\geq
0}+\frac{1}{2})^{n}\,,$ (B.3b) $\displaystyle\mathrm{Sp}(n)\ :\quad$
$\displaystyle\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}\geq
0)\in\mathbb{Z}_{\geq 0}^{n}\,,$ (B.3c) $\displaystyle\mathrm{SO}(2n)\ :\quad$
$\displaystyle\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq|\lambda_{n}|\geq
0)\in\mathbb{Z}_{\geq 0}^{n}\ \text{or}\ \quantity(\mathbb{Z}_{\geq
0}+\frac{1}{2})^{n}\,.$ (B.3d)

For $\mathrm{SU}(n)$ and $\mathrm{Sp}(n)$, $\lambda$ is a partition, a
sequence of non-negative non-increasing integers. The denominators of the
Schur functions have the following expressions:

$\displaystyle\det_{1\leq i,j\leq n}\quantity(x_{j}^{n-i})$
$\displaystyle=\prod_{i<j}^{n}(x_{i}-x_{j})\,,$ (B.4a)
$\displaystyle\det_{1\leq i,j\leq
n}\quantity(x_{j}^{n-i+\frac{1}{2}}-x_{j}^{-(n-i+\frac{1}{2})})$
$\displaystyle=\prod_{i<j}^{n}(x_{i}^{\frac{1}{2}}x_{j}^{\frac{1}{2}}-x_{i}^{-\frac{1}{2}}x_{j}^{-\frac{1}{2}})(x_{i}^{\frac{1}{2}}x_{j}^{-\frac{1}{2}}-x_{i}^{-\frac{1}{2}}x_{j}^{\frac{1}{2}})\prod_{i=1}^{n}(x_{i}^{\frac{1}{2}}-x_{i}^{-\frac{1}{2}})\,,$
(B.4b) $\displaystyle\det_{1\leq i,j\leq
n}\quantity(x_{j}^{n-i+1}-x_{j}^{-(n-i+1)})$
$\displaystyle=\prod_{i<j}^{n}(x_{i}^{\frac{1}{2}}x_{j}^{\frac{1}{2}}-x_{i}^{-\frac{1}{2}}x_{j}^{-\frac{1}{2}})(x_{i}^{\frac{1}{2}}x_{j}^{-\frac{1}{2}}-x_{i}^{-\frac{1}{2}}x_{j}^{\frac{1}{2}})\prod_{i=1}^{n}(x_{i}-x_{i}^{-1})\,,$
(B.4c) $\displaystyle\det_{1\leq i,j\leq
n}\quantity(x_{j}^{n-i}+x_{j}^{-(n-i)})$
$\displaystyle=2\prod_{i<j}^{n}(x_{i}^{\frac{1}{2}}x_{j}^{\frac{1}{2}}-x_{i}^{-\frac{1}{2}}x_{j}^{-\frac{1}{2}})(x_{i}^{\frac{1}{2}}x_{j}^{-\frac{1}{2}}-x_{i}^{-\frac{1}{2}}x_{j}^{\frac{1}{2}})\,,$
(B.4d)

which correspond to the Weyl denominator formula (also known as the
generalized Vandermonde determinant).

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