Exact solutions of Kondo problems in higher-order fermions

The Kondo effect Kondo (1964), which treats a quantum magnetic impurity in normal metals, invoked fundamental developments in condensed matter physics. Among these progresses, the conformal field theory (CFT) approach developed by Affleck and Ludwig (AL) unveiled the physical nature of the Kondo problem, and demonstrated the elegant connections between conformal symmetry and the Kondo fixed point Affleck and Ludwig (1991a, b, c); Affleck et al. (1992); Affleck and Ludwig (1993); Ludwig (1994); Affleck (1995); Delft (1995). In their original approach, AL mapped the multi-channel Kondo problem to an exactly-solvable CFT in the complex plane. In order to establish the mapping, they linearized the low-lying excitations in a narrow energy window $-\Lambda v_{F}<\epsilon<+\Lambda v_{F}$ about the Fermi surface, with $v_{F}$ being the Fermi velocity and $\Lambda$ being an artificially introduced cut-off, and ignored the higher-energy excitations, as a necessary approximation (Fig. 1(a)). Although this treatment works well for normal metals with well-defined Fermi surfaces, it will encounter difficulties in more exotic thermals baths, including the pseudogapped systems Yu et al. (2020a) and those with Fermi points, such as topological semimetals.

On the other hand, past years have witnessed increasing interests in the Kondo problem in topological materials with emergent particles, including graphene, Weyl and Dirac semimetals. In particular, most recently, even more exotic Weyl and Dirac semimetals which exhibit effective fermions with higher-order dispersion relations have been proposed. These higher-order dispersion relations take a form that is linear in one direction, but quadratic or cubic in the orthogonal plane Fang et al. (2012); Yang and Nagaosa (2014); Yang et al. (2015); Huang et al. (2016); Gao et al. (2016); Bradlyn et al. (2016); Liu and Zunger (2017); Ezawa (2017); Yu et al. (2018, 2019); Isobe and Fu (2019); Song et al. (2020); Wu et al. (2020); Wang et al. (2020); Wu et al. (2021). Such materials are named quadratic and cubic Weyl/Dirac semimetals respectively Yang and Nagaosa (2014); Liu and Zunger (2017); Song et al. (2020), with their corresponding emergent fermions referred to as quadratic and cubic Weyl/Dirac fermions respectively Fang et al. (2012); Yang and Nagaosa (2014); Gao et al. (2016), and they are classified as Weyl/Dirac according to their 2-fold/4-fold band degeneracies at their band crossings Yang and Nagaosa (2014); Yang et al. (2015); Gao et al. (2016). For example, in three dimensions, the cubic Weyl/Dirac semimetals display the dispersion relation, in its simplest form,

where $\mu$ is the band index, $a_{1\mu}$ and $a_{2\mu}$ are dispersion coefficients, $p\equiv|\vec{p}|=(p_{x}^{2}+p_{y}^{2}+p_{z}^{2})^{\frac{1}{2}}$. A number of materials for the quadratic and cubic Weyl/Dirac semimetals have been recently proposed and extensively studied. Examples of quadratic Weyl/Dirac semimetals include HgCr₂Se₄, SiSr₂, band-inverted $\alpha-$Sn and PdSb₂ Fang et al. (2012); Huang et al. (2016); Bradlyn et al. (2016), and examples of cubic Weyl/Dirac semimetals include LiO${}_{\text{S}}$O₃ and quasi-one-dimensional molybdenum monochalcogenide compounds A${}^{\text{I}}$(MoX${}^{\text{VI}}$)₃, where A${}^{\text{I}}$ = Na, K, Rb, In or Tl, X${}^{\text{VI}}$ = S, Se or Te Liu and Zunger (2017); Yu et al. (2018); Song et al. (2020); Wu et al. (2020). Moreover, various novel quantum phenomena are predicted to take place in these semimetals, such as charge density wave, non-Fermi liquid, and topological superconductivity Yang and Nagaosa (2014); Ezawa (2017); Yu et al. (2018); Song et al. (2020); Wu et al. (2020); Wang et al. (2020); Potel et al. (1980); Armici et al. (1980); Huang et al. (1983); Tarascon et al. (1984); Hor et al. (1985a, b); Brusetti et al. (1988); Tessema et al. (1991); Brusetti et al. (1994); Petrović et al. (2010). It is therefore timely and of theoretical importance to study Kondo problems in these higher-order fermion (KHOF) systems, in particular, using analytically exact methods.

In ideal KHOF systems, the Fermi energy is located at the Dirac point (DP), with higher-order dispersions along certain directions in momentum space. In these cases, the aforementioned linearization approximation in AL’s original CFT approach becomes inapplicable. On one hand, it generates an artificial flat bands along certain directions, as shown by Fig. 1(b). Clearly, the flat band is not sufficient to capture the realistic low-energy excitations of the higher-order fermions. On the other hand, in contrast to the Kondo problems in normal metals with a single band, anisotropic multi-bands with touching nodes must be taken into account. These unique features of the KHOF systems pose severe challenges for the conventional CFT approach. Does there exist any alternative CFT scheme that overcome these difficulties?

In this work, we propose a full-energy mapping CFT (FEMCFT) method. This method solves all the above problems, and produces exact solutions to Kondo problems in a large class of exotic semimetals, in particular, the KHOF models. In contrast with the conventional CFT scheme that only establishes conformal invariance for the linearized low-lying excitations within some artificially-introduced energy cut-off $\Lambda$ near the Fermi surface, our approach is able to map the full energy spectrum of the KHOF model into a form that observes conformal invariance. Consequently, instead of artificially introducing a cut-off $\Lambda$, and only mapping the low energy degrees of freedom of the bath bounded by $\Lambda$ to the complex plane, we can map the full energy spectrum into the complex plane without introducing any artificial cut-offs in the spectrum, as shown by Fig. 1(c) and 1(d). As a result, our approach is free from the flat band issue shown in Fig. 1(b). An additional advantage of our method is that, by rigorously taking into account the entire energy spectrum of the bath, we can make more accurate predictions about the low-energy fixed point and the thermodynamic quantities at finite temperatures, compared to conventional CFT techniques.

We emphasize that our FEMCFT approach is analytically exact for ideal isotropic KHOF systems. It also works well for highly realistic anisotropic KHOF materials, after systematically treating the anisotropic effects as corrections to the isotropic parts. Our work therefore rigorously solves the KHOF and related models for higher-order topological semimetals, and significantly advances the scope of CFT approaches to many-body resonances, bringing about new applications in previously inaccessible systems.

The remaining part of the manuscript is organized as follows. In Sec.II, we define the general Hamiltonian for the KHOF systems, which describes a multichannel Kondo impurity in higher-order Weyl/Dirac systems. In Sec.III, we present our FEMCFT approach in the isotropic limit, which establishes an exact mapping of an ideal isotropic KHOF system to a 2D CFT in the complex plane. The calculations of thermodynamic quantities are discussed in Sec.IV. In Sec.V, we tackle realistic anisotropic KHOF systems, and demonstrate our method with an explicit example, namely, the Kondo problem in the anisotropic cubic Weyl/Dirac fermion system in three dimensions, whose dispersion is given by (1), which has attracted great interests recently.

For the sake of generality, we consider a KHOF system in $n$ spatial dimensions, whose Hamiltonian is given by

where $H_{0}$ is the bath Hamiltonian, and $H_{I}$ is the interaction Hamiltonian. $H_{0}$ is given by

where $\vec{p}=(p_{1},...,p_{n})$, $c^{\dagger}_{\alpha,i,\mu}(\vec{p})$ and $c_{\alpha,i,\mu}(\vec{p})$ are conduction electron creation and annihilation operators respectively, with $\alpha\in\{\uparrow,\downarrow\}$ labelling the up/down spin index, $i\in\{1,2,...,k\}$ labelling the channel index, and $\mu\in\{+,-\}$ labelling the band index. Summation over repeated indices is implied.

We consider a general higher-order dispersion $\epsilon_{\mu}(\vec{p})$ of the form

where $p\equiv|\vec{p}|$. The constants $a_{1\mu}$ and $a_{2\mu}$ satisfy $a_{1+}=-a_{1-}$, $a_{2+}=-a_{2-}$. The dispersion (4) is of particular interest, because for $n=3$, it reduces to a cubic fermion system in three-dimensions (1). For $n=2$, it produces a quadratic band crossing point, which also has attracted great interests in the past decade Sun et al. (2009); Uebelacker and Honerkamp (2011); Wang et al. (2015). Moreover, for $n\geq 3$, it describes even higher order emergent fermions which can be realized in artificial electric circuits with auxiliary dimensions Yu et al. (2020).

$H_{I}$ describes the exchange interaction between an impurity spin and the bath electrons:

where $\vec{S}$ is the spin of the magnetic impurity, $\lambda_{i}$ is the Kondo coupling constant for the $i$-th channel.

Eq. (2) - (5) constitute the most general KHOF model in $n$-dimensions. Our focus is to look for exact CFT solutions to Eq. (2) - (5).

In this section, we outline our FEMCFT approach for an ideal isotropic KHOF system, i.e. the case where $a_{2\mu}=0$ in (4), so that

More details of the derivations in this section can be found in Appendix B. We shall treat the anisotropic case $a_{2\mu}\neq 0$ in Sec.V.

We expand $c_{\alpha,i,\mu}(\vec{p})$ as a linear combination of the spherical harmonics, which form a complete set of orthonormal functions on the $(n-1)-$ sphere $S^{n-1}$:

Here $Y_{l_{1},...,l_{n-1}}(\theta_{1},...,\theta_{n-1})$ are the spherical harmonics in $n$ dimensions, see Appendix A and Higuchi (1987) for detailed derivations of their important properties relevant for this work. $\theta_{1},...,\theta_{n-1}$ are the $n-1$ angular coordinates of $S^{n-1}$, with $0\leq\theta_{1}<2\pi$, $0\leq\theta_{j}\leq\pi$ for $j=2,...,n-1$. The integers $|l_{1}|\leq l_{2}\leq...\leq l_{n-1}$ denote different partial waves, and are the analogues of the quantum numbers $m,l$ in the three-dimensional spherical harmonics $Y_{l}^{m}(\theta,\phi)$: in 3 dimensions, $n=3$, $l_{1}\equiv m,l_{2}\equiv l$, $\theta_{1}\equiv\phi$ is the azimuthal angle, and $\theta_{2}\equiv\theta$ is the polar angle. $H_{0}$ and $H_{I}$ can be written in terms of $c_{l_{1},...,l_{n-1},\alpha,i,\mu}(p)$ as

Next we perform a change of variables on the fields $c_{l_{1},...,l_{n-1},\alpha,i,\mu}(p)$ from $p$ to $\epsilon$, by defining

In this way, the $c_{l_{1},...,l_{n-1},\alpha,i,\mu}(\epsilon_{\mu})$ fields also satisfy the proper fermionic anti-commutation relations. With the definition

we combine the fields from the + and - bands into one single composite fermionic field $\Phi_{0,...,0,\alpha,i}(\epsilon)$. In terms of $\Phi_{0,...,0,\alpha,i}(\epsilon)$, $H_{0}$ and $H_{I}$ become

where $a\equiv|a_{1\mu}|$. Notice that we have combined the $+$ and $-$ bands into a single band, and eliminated the band index $\mu\in\{+,-\}$ from our model.

We then define the left and right moving fields, respectively, as

Also, by introducing the imaginary time $\tau\equiv it$, we define the complex plane $\mathbb{C}$, to which we map our KHOF model, by

i.e. The horizontal axis of $\mathbb{C}$ is the imaginary time $\tau$, and the vertical axis of $\mathbb{C}$ is $r$ introduced in (III). We then view $\Psi_{\leftarrow/\rightarrow\alpha i}(r)$ in the Heisenberg picture, so that they now have time-dependence, and live on $\mathbb{C}$. $\Psi_{\leftarrow\alpha i}(\tau,r)$ and $\Psi_{\rightarrow\alpha i}(\tau,r)$ are related to each other by

so $\Psi_{\rightarrow\alpha i}(\tau,r)$ can be eliminated in terms of $\Psi_{\leftarrow\alpha i}(\tau,r)$. $H_{0}$ and $H_{I}$ can be written in terms as $\Psi_{\leftarrow\alpha i}(\tau,r)$ as

The $n$-dimensional KHOF model is thus mapped into the complex plane $\mathbb{C}$ defined in (15), with $H_{0}$ and $H_{I}$ taking the forms of (17) and (18) respectively. These are in similar forms as the Hamiltonians mapped from single band normal metals. In particular, we see from (18) that the Kondo exchange coupling remains short-ranged in the new fermionic degrees of freedom; the impurity spin $\vec{S}$ only couples to the new fermionic field $\Psi_{\leftarrow\alpha i}(\tau,r)$ at $r=0$. This means that $H_{I}$ is only confined to the boundary $r=0$, with $H=H_{0}$ in the bulk $r\neq 0$, and the problem is suitable for further analysis by techniques in 2D boundary CFT.

We note that our FEMCFT approach is analytically exact; we have not introduced any artificial cut-off $\Lambda$ in the momentum or energy. Our integrals $\int_{0}^{\infty}dp$ and $\int_{-\infty}^{\infty}d\epsilon$ are over the entire spectrum. Also, we have transformed the 2-band KHOF system into an effective one-band system. This further facilitates the analysis of KHOF systems using CFT techniques.

After mapping KHOF systems into the form (17) + (18) on the complex plane via our FEMCFT approach, we can proceed to determine the thermodynamic quantities at $T>0$. This is carried out via AL’s standard CFT techniques, which we briefly summarize here. More details on their approach can be found in Affleck and Ludwig (1991a, b, c); Affleck et al. (1992); Affleck and Ludwig (1993); Ludwig (1994); Affleck (1995); Delft (1995). The underlying geometry of the $T>0$ physics is the infinite cylinder with circumference $\beta=1/T$. The system is further mapped from the complex plane onto the infinite cylinder via a conformal map. The complete list of boundary operators for the system with a particular boundary condition can be then obtained by applying “double fusion” to the free system (i.e. the system with trivial boundary condition). From the list of boundary operators, we can determine the leading irrelevant operator with coupling constant $\tilde{\lambda}$, whose Green’s functions allow us to compute the thermodynamic quantities of interest. The circumference of the cylinder $\beta=1/T$ enters the calculation of the thermodynamic quantities as a finite size of the system, giving rise to their $T$-dependences. For example, the resistivity $\rho$ is found as $\rho=\rho_{U}\left\{1-\alpha\tilde{\lambda}^{2}T^{2}+...\right\}$ in the Fermi liquid case and $\rho=\rho_{U}\frac{1-S^{(1)}}{2}\left\{1+\alpha\tilde{\lambda}T^{\frac{2}{2+k}}+...\right\}$ in the non-Fermi liquid case. Here $k$ is the number of channels, $S^{(1)}=\frac{\cos\{\pi(2s+1)/(2+k)\}}{\cos\{\pi/(2+k)\}}$ with $s$ being the impurity spin, $\rho_{U}$ is the unitary limit resistivity, i.e. the greatest resistivity possibly achievable, and $\alpha$ is a dimensionless constant, with $\alpha=4\sqrt{\pi}$ in the special case $k=2$. These results are in excellent agreements with results obtained from numerical renormalization group (NRG) analysis.

We remark that, although our $T>0$ thermodynamic quantities take the same form as AL’s, they are valid over greater ranges of temperatures compared to AL’s. This is because AL’s original approach to Kondo problems in normal metals requires the introduction of a narrow cutoff $\Lambda$ about the Fermi surface. Thus in their calculated thermodynamic quantities, AL can only consider temperatures $T$ satisfying $T\ll\Lambda v_{F}$, and ignore terms of order $T/\Lambda v_{F}$. In comparison, we did not introduce any artificial cut-offs in the spectrum. As a result, our calculated thermodynamic quantities are more accurate and are valid over greater ranges of temperatures, without the restriction of $T\ll\Lambda v_{F}$. Moreover, conventional CFT approaches are not applicable at all in KHOF systems with coinciding Fermi energies and DPs. Our theory fills this research gap, and enables an exact CFT analysis for these novel phases with emergent higher-order fermions.

Realistic topological semimetals with higher order fermions are always anisotropic. In this section, we therefore present a general framework to treat this anisotropy, followed by an explicit application of our framework to a concrete example, namely the anisotropic cubic fermion model in three dimensions, realized by setting $n=3$ and $a_{2\mu}\neq 0$ in (4). More details of the derivations can be found in Appendix C.

For anisotropic KHOF systems, $H_{I}$ remains the same as before, since the dispersion relation does not enter the expression of $H_{I}$. However, $H_{0}$ can no longer be reduced to the simple form (9), due to the lack of spherical symmetry. In general, we can express $H_{0}$ into a matrix form in the partial wave basis, i.e.,

where $C_{\alpha,i,\mu}(p)$ is a column vector whose $L-$th element is given by $c_{L,\alpha,i,\mu}(p)$, where for brevity we have used the multi-index notation $L\equiv(l_{1},...,l_{n-1})$. $C^{\dagger}_{\alpha,i,\mu}(p)$ is the Hermitian conjugate of $C_{\alpha,i,\mu}(p)$ and $A_{\mu}(p)$ is a matrix whose $(L,L^{\prime})-$th element, $A_{L,L^{\prime},\mu}(p)$, describes the coupling between the $L$ and $L^{\prime}-$th partial waves. We remark that for all Hermitian systems, the only partial waves that couple to the impurity are those with $l_{1}=0$. Thus the multi-index $L$ only includes $(l_{1}=0,l_{2},...,l_{n-1})$. In particular, in three dimensions, $L$ only includes $(m=0,l)\rightarrow l$, i.e. the multi-index $L$ reduces to the single index $l$: $L=l$.

We separate $H_{0}$ in (19) into two terms

where the term $H_{00}$ consists of only the $L=(0,...,0)$ partial waves, thus describing the “isotropic part” of $H_{0}$. $\Sigma_{0}$ consists of all the remaining partial waves with $L\neq(0,...,0)$, and captures the “anisotropic part” of $H_{0}$, as well as its couplings to $H_{00}$, as shown by left picture of Fig.2.

Due to anisotropy, partial waves with different quantum numbers are coupled to each other, forming a hierarchical structure depicted in the left picture of Fig. 2 (in general, different models can give rise to different coupling hierarchical structures. The left picture of Fig. 2 shows the case for the cubic fermion model in three dimensions (22), which suffices to illustrate our framework). As shown in Fig. 2, the impurity firstly couples to the isotropic sector, which further couples to the higher components. Owing to the hierarchical nature of the couplings, we can integrate out the anisotropic part by down-folding the Hamiltonian matrix in Eq.(19) to the isotropic subspace. It can be shown that in the static limit, the anisotropic effects of $\Sigma_{0}$ can be casted into an additional renormalization term $\epsilon_{\Sigma\mu}(p)$, which serves as a modification to the isotropic part of the dispersion relation $\epsilon_{\mu}(p)$.

To further enable a CFT analysis, we approximate the value of $\epsilon_{\Sigma\mu}(p)$ by evaluating it at the Fermi wave number, $p_{F}$. This fully captures the low-energy Kondo physics, since only the excitations near the Fermi point are important at low temperatures. It should be noted that, although this approximation is in the same spirit as AL’s linearization approximation near the Fermi surface, it is made only to the anisotropic part of the dispersion. Our method still admits an exact treatment of the isotropic part by mapping the entire spectrum into a two-dimensional CFT. Therefore, it is free from the flat band issue illustrated in Fig. 1(b), and overcomes the major difficulty present in conventional CFT techniques.

The above procedures generate an renormalized bath coupled to the impurity, as indicated by the right picture of Fig. 2. The renormalized bath enjoys the effective dispersion relation $\tilde{\epsilon}_{\mu}(p)\equiv\epsilon_{\mu}(p)+\epsilon_{\Sigma\mu}(p_{F})$, which contains the isotropic part of the dispersion, as well as the corrections from the anisotropy. Correspondingly, the bath is now effectively described by

The above steps constitute a general framework that takes into account the anisotropy in the CFT analysis.

We now apply our approach to a concrete example, namely the cubic fermion system in three dimensions. This is realized by setting $n=3$ and turning on $a_{2\mu}$ in (4), namely

which is linear in the $p_{1}-$direction but cubic in the orthogonal plane, a typical low-energy dispersion of cubic fermion materials.

This model admits a simple coupling hierarchical structure as shown in Fig. 2. Here, the impurity is coupled only to the isotropic component, which is further connected to the higher-order partial waves via nearest-neighbour hoppings. Moreover, it can be shown that the hopping strength decays as $l$ increases. As a result, $A_{\mu}(p)$ in (19) can be casted into the simple tridiagonal form

We then integrate out the higher-order partial waves by down-folding the matrix $A_{\mu}(p)$ to the isotropic subspace. Interestingly, because each entry of $A_{\mu}(p)$ is proportional to $p$, the renormalization $\epsilon_{\Sigma_{\mu}}(p)$ is also found to be proportional to $p$, and thus vanishes at the DP with $p_{F}=0$. This essentially indicates that the contributions from different high-order partial waves display a “destructive interference”, in the sense that they completely cancel with each other at the DP. This is an interesting feature of cubic Dirac fermions, which greatly simplifies the treatment of anisotropy.

The effective dispersion relation is eventually obtained as

which is clearly of the form (6) in three dimensions. As a result, all subsequent derivations following Eq. (6) hold, and our FEMCFT approach for isotropic KHOF systems applies. $H_{0}$ and $H_{I}$ will be mapped to the forms of (17) and (18) respectively. Correspondingly, the impurity ground state and the thermodynamic quantities can be readily obtained using the previously discussed methods.

In conclusion, we have developed a full-energy mapping conformal field theory (FEMCFT) approach to tackle Kondo problems in higher order fermion and related systems. Our FEMCFT approach is capable of solving these models exactly, which are inaccessible by using conventional CFT approaches. Moreover, the FEMCFT approach is able to make more accurate predictions about the thermodynamic quantities over greater ranges of temperatures. We applied our FEMCFT approach to a specific KHOF system, namely the cubic Weyl/Dirac fermion system in three dimensions. The high efficiency of our approach to realistic KHOF systems is clearly justified.

The FEMCFT approach significantly broadens the scope of existing CFT methods in the study of Kondo problems. We anticipate developments of novel CFT techniques for treating Kondo problems in more complicated systems, such as those displaying pseudogaps, as possible future directions of research. Such advancements would undoubtedly further fortify the connections between the mathematically elegant CFTs and the physically intriguing fixed points in strong-correlated problems, such as novel quantum criticalities and many-body resonances in strong-coupling limits.