DMFT reveals the non-Hermitian topology in heavy-fermion systems

A standard theory for metals describes Fermi surfaces that are manifolds with codimension-one [closed loops in two dimensions (2D) and closed surfaces in three dimensions (3D)] in momentum space. In semimetals, there are codimension-two Fermi nodes and band crossings: nodal points in 2D and nodal lines in 3D, respectively. However, the pseudogap phase of hole-doped cuprate high-$T_{c}$ superconductors exhibits Fermi arcs, which are opened Fermi surfaces with endpoints.Keimer .

In recent worksHuitao ; Vlad ; Papaj , it was revealed that nodal points in 2D Dirac semimetals can become Fermi arcs, the locations of which are determined by a topological theory of finite-lifetime quasiparticles. In quantum many-body systems, quasiparticles generically have finite lifetimes, resulting from scattering. Hence, they are described by non-Hermitian Hamiltonians, whose imaginary part reflects the finite lifetimes. These non-Hermitian matrices can have topological features called the “exceptional points” (EPs), where the Hamiltonian becomes non-diagonalizableKato . It was then shownHuitao ; Vlad ; Papaj that the existence of EPs lead to Fermi arcs, which end on the EPs. In previous studies, it was shown that two-lifetime modelsVlad , where two electron bands with finite hybridization have different lifetimes originating from the electron-phonon scattering, often have EPs and thus also Fermi arcs. Elastic electron-impurity scattering also induces EPs and Fermi arcsPapaj .

In this work, we demonstrate how two distinct lifetimes due to electron-electron interaction naturally appear in heavy fermion systems. Heavy-fermion systems can be described by the periodic Anderson modelTsunetsugu . This model naturally features two lifetimes, since there are two kinds of electrons, itinerant $c$-electrons and localized $f$-electrons. The $f$-electrons feel much stronger electron-electron interaction than the $c$-electrons, which leads to different lifetimes of the two bands.

We show that a Kondo semimetal, a Kondo lattice system with a small energy gap such as CeNiSnNakamoto , naturally has Fermi arc. The Kondo semimetal is understood as a Kondo insulator with a momentum-dependent hybridization gap. A Kondo insulator exhibits an insulating behavior at low temperatures, since the volume of the Fermi surface counts both $c$- and $f$-electrons and the hybridization of the Fermi surfaces makes this system gapped. However, it exhibits a metallic behavior with a $c$-electron Fermi surface at high temperatures, since the contribution from the $f$-electrons is blurred due to its finite lifetime, which decreases as the temperature raisesPColeman . The characteristic crossover temperature between the two behaviors is determined by comparing the magnitude of the hybridization gap and the temperature-dependent inverse lifetime of the $f$-electron. As we will show, in the Kondo semimetal with a momentum-dependent gap, the crossover happens at different temperatures for different points on the Fermi surface. Thus, a Kondo semimetal has a Fermi arc, which is a partial Fermi surface ending on EPs.

In this paper, we study a periodic Anderson model with a momentum-dependent hybridization gap in two dimensions. Using a perturbation theory, we argue that the difference between electron correlations in $c$- and $f$-bands naturally lead to two lifetimes (in fact, under certain assumptions, the self-energy of $c$-electrons vanishes to all orders of perturbations), which results in EPs and Fermi arcs at certain temperatures. To confirm our statement numerically, we adopt the dynamical mean field theory (DMFT) with the numerically exact continuous-time quantum Monte Carlo solver. The spectral function calculated by the DMFT clearly shows that there is the Fermi arc defined by the region where the spectral function has a peak at zero energy. These Fermi arcs due to the EPs can be observed by angle-resolved photoemission spectroscopy (ARPES).

We introduce the EPs and the finite-lifetime quasiparticle Hamiltonian for strongly correlated electron systemsHuitao ; Vlad .

We first introduce quasiparticle Hamiltonian $H$:

which can be extracted from the retarded electron Green’s function $G^{R}(\bm{k},\omega)=(\omega-H(\bm{k},\omega))^{-1}$, where $H_{0}(\bm{k})$ is the single-particle Hamiltonian and $\Sigma(\bm{k},\omega)$ is the electron’s self-energy that includes the effect of electron-electron interaction. Note that the quasiparticle Hamiltonian $H(\bm{k},\omega)$ can be non-Hermitian and that its spectrum $\mathcal{E}_{n}(\bm{k},\omega)$ can be complex: In general, $\Sigma$ is a sum of the Hermitian part $\Sigma^{\prime}$ and the non-Hermitian part $\Gamma$: $\Sigma=\Sigma^{\prime}-i\Gamma$. $\Sigma^{\prime}$ renormalizes the bare band structure, while $\Gamma$ leads to finite quasiparticle lifetimes, resulting in the non-Hermitian self-energy and HamiltonianHuitao ; Vlad . The spectrum $\mathcal{E}_{n}(\bm{k},\omega)$ determines the complex poles $\omega=E_{n}(\bm{k})$ of the Green’s function according to $E_{n}=\mathcal{E}_{n}(\bm{k},E_{n})$. In the vicinity of a first-order pole, the Green’s function takes the form $G^{R}(\bm{k},\omega)\sim\frac{1}{\omega-E_{n}(\bm{k})}$. The real part of $E_{n}(\bm{k})$ determines the quasiparticle’s dispersion, while its imaginary part defines the quasiparticle’s inverse lifetime.

The important difference between Hermitian and non-Hermitian Hamiltonians is that the latter can be non-diagonalizable at certain momenta. Those points are called “exceptional points” in the mathematical physics literatureKato . At the EPs, linearly independent eigenstates do not span the full Hilbert space. One of the authors showed that these EPs in two and higher dimensions are topologically stableHuitao .

Topological exceptional points can appear in the quasiparticle spectrum of zero- and small-gap materials such as Dirac semimetals in two dimensionsVlad . We show that this scenario is also applicable to heavy Fermion systems, where the two orbitals correspond to itinerant and localized electrons and the hybridization gap can have a momentum-dependent form factorIkedaMiyake ; Dzero ; Zhong .

Let us now demonstrate how two different lifetimes naturally occur in heavy fermion systems. We discuss how the self-energy leads to EPs and Fermi arcs, when combined with a momentum-dependent hybridization gap. We consider a periodic Anderson model (PAM) defined as

where

and ${\cal H}_{int}=U\sum_{i}f_{i\uparrow}^{\dagger}f_{i\uparrow}f_{i\downarrow}^{\dagger}f_{i\downarrow}$. Here, $c_{\bm{k}\sigma}$ ($f_{\bm{k}\sigma}$) is an annihilation operator for itinerant $c$- (localized $f$-) electron with momentum $\bm{k}$ and spin $\sigma$, with dispersions $\epsilon_{c{\bm{k}}}=-2t(\cos k_{x}+\cos k_{y})-\mu$ and $\epsilon_{f\bm{k}}=2t_{f}(\cos k_{x}+\cos k_{y})+\epsilon_{f0}-\mu$, respectively. We use $t$ as the unit of the energy scale hereafter, and $\mu$ here is the chemical potential. We consider the case of half filling, which corresponds to $\mu=0$ and $\epsilon_{f0}=-U/2$, and is characterized by the particle-hole symmetry. At half filling, the term $\epsilon_{f0}=-U/2$ is exactly cancelled by the Hartree part of the electron’s self-energy, so we omit it for the rest of the paper for brevity. The hybridization gap is expressed as $v_{\bm{k}}=v$ for $s$-wave case which describes the Kondo insulators and $v_{\bm{k}}=v(\cos k_{x}-\cos k_{y})$ for $d$-wave case which describes Kondo semimetals, respectively. In this work, we focus on the $d$-wave gap, which has point nodes on the Fermi surface at $(k_{x},k_{y})=(\pm\pi/2,\pm\pi/2)$ sup .

It is known that the PAM exhibits a number of strongly correlated phenomena showing the non-Fermi-liquid behavior Amaricci2008 and undergoing the Mott transition Sordi2007 ; Amaricci2017 ; Amaricci2012 near some particular fillings. In our work, we stay away from these special fillings, i.e., in the regime where the Fermi liquid description remains adequate.

Due to the fact that the $c$ electrons in model (5) are not interacting, only $f$ electrons have a finite lifetime: This implies the following form of

This point is explained in Sec. S2. in the supplemental material. Thus, the two-lifetime description naturally arises in the periodic Anderson model.

The $f$-electron self-energy is computed using a second-order perturbation theory in Sec. S2 of the supplemental material (see also, Ref. Yamada1986 ). At temperatures comparable or higher than the hybridization scale $v$, we can ignore the momentum dependence of $\Sigma(\omega)$. This is because the finite-temperature broadening renders the $f$-band approximately flat. Hence, the onsite interaction and a flat band together result in a local self-energy. Since we are interested in the spectrum of low-energy electron excitations, we focus on low-frequency self energies. In particular, in the zero-frequency limit, $\Sigma^{f}$ can be treated as a purely imaginary constant, $\Sigma^{f}(\bm{k},0)=-i\Gamma$, where the real part vanishes because the particle-hole symmetry pins the energy of the $f$-electron at zero. Furthermore, at small frequencies, one can expand $\Sigma^{f}$ as the following:

Our DMFT calculations presented later show that coefficients $a_{1}$ and $a_{2}$ are generally positive. The inclusion of the positive higher-order terms $a_{1}$ and $a_{2}$ will not change the qualitative features of the electron spectrum function discussed next. In particular, the effective of including $a_{1}$, which is the leading term of the real part of $\Sigma^{f}$, is a reduction of the widths of some bands: it renormalizes $t_{f}$ to $Zt_{f}$, $\Gamma$ to ${Z}\Gamma$, and $v_{k}$ to $Z^{1/2}v_{k}$, respectively, while keeping $t_{c}$ unaffected. (This is analyzed in details in Sec. S2 of the SM.) Here, $Z\equiv(1+a_{1})^{-1}$.

We now discuss the electron spectral function, and show the Fermi arcs in the $d$-wave PAM. We plot the electron spectral function from (1). The result is shown in Fig. 1.

In general, the inverse lifetime $\Gamma$ is small at low temperatures and large at high temperatures, reflecting a long and a short quasiparticle lifetime, respectively. Hence, we demonstrate low-temperature and high-temperature phenomena using $\Gamma=1$, $\Gamma=2$ and $\Gamma=6$, respectively. In the $s$-wave PAM, there is a gap with $\Gamma=1$ in the whole momentum space as shown in Figs.1(a) and (g). With increasing $\Gamma$, the gap is closed on a Fermi surface. This is the standard crossover behavior between a Kondo insulator at a low temperature and a small $c$-electron Fermi surface at a high temperature. In the $d$-wave PAM, on the other hand, we can clearly see the bulk Fermi arcs at the low temperature, as shown in Figs.1(d) and (j). The Fermi arcs grow into a full $c$-electron Fermi surface at the high temperature.

The Fermi arcs observed in the $d$-wave PAM can be explained as follows. With $\Sigma=-i\Gamma-a_{1}\omega$, the complex poles of the Green’s function are given by

with $M_{\bm{k}}\equiv\epsilon_{f\bm{k}}-Z^{-1}\epsilon_{c\bm{k}}$. This complex spectrum is similar to that for the Dirac semimetals in two dimensions if we replace $M_{\bm{k}}$ and $|v_{\bm{k}}|$ with $v_{x}k_{x}$ and $v_{y}k_{y}$, respectively (and assuming $Z=1$)Vlad .

The spectral function $\rho(\omega,\bm{k})=-\frac{1}{\pi}\text{Im Tr }(G^{R}-G^{A})$ with $G^{A}\equiv\left[G^{R}\right]^{\dagger}$ is given by

with

The form-factors $A_{\bm{k}}$ and $B_{\bm{k}}$ originate from the non-zero real part of self-energy, and both equal to 1 in case when $a_{1}=0$.

We see that the spectral function is given by the sum of two Lorentzians, whose locations and widths are given by the real and imaginary parts of $E_{\pm}$, respectively. In particular, we focus on the line $\epsilon_{c\bm{k}}=Z\epsilon_{f\bm{k}}$, which is the location of the high-temperature $c$-electron Fermi surface. Along this line, Eq. (11) is simplified to the following form,

We first focus at a fixed momentum $k$, and discuss how a temperature-dependent $\Gamma$ affects the electron spectrum function. For simplicity, we consider $Z=1$($a_{1}=0$). In general, we expect $\Gamma$ to increase with temperature. In particular, it should approach zero in the $T=0$ limit. The competition between the temperature-dependent $\Gamma$ and the hybridization $v_{k}$ can explain the Kondo crossover. At low temperatures, when $4|v_{\bm{k}}|^{2}>\Gamma^{2}$, $E_{\pm}$ have different real parts, and the spectrum is a superposition of two peaks. The distance between the two peaks, $\sqrt{4|v_{\bm{k}}|^{2}-\Gamma^{2}}$, represents a hybridization gap. This indicates that the $f$ and $c$ electrons are hybridized, and the $f$ electrons join the $c$ electrons to contribute to the volume of the Fermi surface. At high temperature, when $4|v_{\bm{k}}|^{2}<\Gamma^{2}$, $E_{\pm}$ have the same real part, and the spectrum exhibits a single peak located at $\omega=0$ if $\epsilon_{c\bm{k}}=\epsilon_{f\bm{k}}$. This indicates that there is no gap, and $k$ is on a Fermi surface. This also means that the $f$-electrons form local moments, which are detached from the Fermi surface formed solely by the $c$ electron.

For the $s$-wave PAM, the crossover between the two behaviors happens at the same temperature where $\Gamma=2v$ for all $\bm{k}$, and a complete $c$-electron Fermi surface emerges from a Kondo insulator as the temperature raises. For the $d$-wave PAM, however, at different momenta on the Fermi surface, the crossover happens at different temperatures determined by $v(\bm{k})$. In particular, for $\Gamma<2v$, the Fermi surface is separated into two regions by EPs located at $2|v_{\bm{k}}|=\Gamma$, where $E_{\pm}$ become degenerate. The hybridization gap only appears on part of the crossing lines, where $4|v_{\bm{k}}|^{2}>\Gamma^{2}$. The rest of the crossing lines, where $4|v_{\bm{k}}|^{2}<\Gamma^{2}$, then forms Fermi arcs. Hence, the arcs are longer with bigger $\Gamma$. At zero temperature, the model is a Kondo semimetal with point nodes. As temperature raises and $\Gamma$ increases, the Fermi arcs grow longer, until the full $c$-electron Fermi surface emerges at $\Gamma=2v$.

We notice that, phenomenologically, when two peaks in the spectrum function are closer than their peak widths, they become inseparable experimentally. Hence, the exact locations of the ending points of the Fermi arcs may differ from the EPs, and they depend on the details of data analysis. However, the existence of the Fermi arcs is tied to EPs in the quasiparticle Hamiltonian.

To treat the heavy fermion systems more accurately, we use the dynamical mean field theory (DMFT). In the DMFT, we assume that the self-energy does not depend on the momentum:

This assumption is appropriate when the temperature is not too low comparison with the maximum of hybridization energysup . The momentum dependence does not affect the appearance of the bulk Fermi arcs, qualitatively. We set $v=0.75$ and $t_{f}=0$. To calculate the local and momentum-dependent spectral functions $\rho(\omega)$ and $\rho(\omega,\bm{k})$, we utilize the DMFT with the numerically exact segment-based hybridization-expansion continuous-time quantum Monte Carlo impurity solver (ct-HYB) Werner . We point out that there is no fermion sign problem because of the off-diagonal elements of the self-energy in the spin space are zero in our two-orbital systemNagaiDMFT . After confirming that the self-energy for $c$ electrons is zero in the two-orbital DMFT calculation, we use one-orbital DMFT calculation with integrating out the $c$-electron degree of freedom. Here, the effective impurity problem is solved by an open-source program package, $i$QISTiqist . We set $U=8$.

Figure 3 shows the spectral functions and the Fermi arcs. To calculate the real-frequency self-energy $\Sigma(\omega)$ and the spectral function $\rho(\omega,\bm{k})$, we use the Padé approximation as the method for the numerical analytic continuation of the self-energy $\Sigma(i\omega_{n}\rightarrow\omega)$. After the Padé approximation, we fit the self-energy with the form $\Sigma_{f}(\omega)=-(a_{1}\omega+a_{3}\omega^{3})-i(\Gamma+a_{2}\omega^{2}+a_{4}\omega^{4})$ in the region $-1<\omega<1$. We find that the set of parameters $(\Gamma,a_{1},a_{2},a_{3},a_{4})$ equals (5.70895, 2.28419, 1.01368, 1.15838, 0.580893) at $T=1/4$, (2.70929, 9.54693, 5.49701, -2.19662, 5.38239) at $T=1/20$, and (2.06142, 11.8766, 16.49, -10.2929, -6.21199) at $T=1/60$, respectively. Along the Fermi line without hybridization defined by $M_{\bm{k}}=0$ close to $\bm{k}=(0,\pm\pi)$ and $(\pm\pi,0)$, the spectral function $\rho(\omega,\bm{k})$ begins to have a dip at $\omega=0$ by decreasing temperature, as shown in Fig. 3(d)-(f). We confirm that the spectral functions as a function of $\omega$ and $\bm{k}$ show that there is a gap at zero energy in the low temperature region. These results indicate the existence of the Fermi arc, since the line where the spectral function has a peak at the zero energy has boundaries.

In this work, we studied heavy-electron systems using PAMs. Although the microscopic many-body Hamiltonian is Hermitian, the one-body quasiparticle Hamiltonian is non-Hermitian because of finite quasiparticle lifetimes, which is due to electron interactions. In particular, we demonstrate that the difference in electron correlations between $c$- and $f$-electrons in PAMs naturally leads to two lifetimes in the non-Hermitian quasiparticle Hamiltonian. This, combined with a $d$-wave hybridization gap, leads to the existence of EPs in the Hamiltonian and “Fermi arcs” in electron spectral function, at suitable temperatures. Our DMFT calculation with a numerically exact continuous-time quantum Monte Carlo solver confirmed our statement. These unusual Fermi arcs due to EPs can be observed by ARPES, which measures electron spectral function. Our discussion is applicable to other systems, such as two- and three-dimensional Kondo insulators with $p$-wave hybridization. In a two-dimensional system with $p$-wave hybridization ($V_{\bm{k}}=\bm{v}_{\bm{k}}\cdot\bm{\sigma}$ with $\bm{v}_{\bm{k}}=(\sin k_{x},\sin k_{y})$), cross-shaped Fermi arcs appear around $\bm{k}=(\pm\pi,0)$ and $(0,\pm\pi)$. In a three-dimensional system, the Fermi arc becomes the three-dimensional “Fermi arc” defined by the surface with the boundaries in momentum space, which will be shown elsewhere.

Y.N. would like to acknowledge H. Shen for helpful discussions and comments. The calculations were performed by the supercomputing system SGI ICE X at the Japan Atomic Energy Agency. YN was partially supported by JSPS KAKENHI Grant Number 15K00178 and 18K03552, the ”Topological Materials Science” (No. JP16H00995 and No. 18H04228) KAKENHI on Innovative Areas from JSPS of Japan. V. K. was partially supported by the Quantum Materials program at LBNL, funded by the US Department of Energy under Contract No. DE-AC02-05CH11231.

Y.N. and Y.Q. contributed equally to this work.

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