The Lifshitz transition, where the Fermi surface (FS) topology changes,^[1] is beyond the paradigm of Landau’s symmetry breaking theory. This unconventional transition has been observed experimentally in cuprate superconductors,^[2–4] iron-based superconductors,^[5–13] topological insulator,^[14] graphene,^[15] and heavy-fermion compounds.^[16–21] Particularly, for some quantum critical heavy-fermion materials, such as YbRh₂Si₂, its magnetic field dependent thermopower, thermal conductivity, resistivity, and Hall effect show three transitions at high fields and the Lifshitz transitions are argued to be their origin.^[20] For CeRu₂Si₂, the high resolution Hall effect and magnetoresistance measurements across the metamagnetic transition are explained as an abrupt f-electron localization, where one of the spin-split sheets of the heaviest Fermi surface shrinks to a point.^[16] The Lifshitz transition leads to a way to understand the relation of the FS topology and the quantum critical point in heavy-fermion systems.^[22]

Theoretically, the Lifshitz transitions in heavy fermion systems have been carefully explored with mean-field theory and dynamical mean-field theory.^[23–34] At the mean-field level, the Lifshitz transition is triggered with the introduction of Heisenberg coupling into the usual Kondo lattice model, i.e., Kondo–Heisenberg model (KHM), and a case studying on square lattice suggests both first and second-order Lifshitz transitions.^[23–25] Interestingly, the appearance of Lifshitz transition with enhanced antiferromagnetic Heisenberg interaction preempts the disentanglement of Kondo singlet, thus the resulting Kondo breakdown mechanism predicted in literature should be reexamined.^[26,35,36]

Recently, non-Fermi liquid behaviors have been observed in triangular lattice heavy-fermion compounds like YbAgGe and YbAl₃C₃.^[37–42] Due to the frustration effect introduced by local f-electron spin located on the triangular lattice, the observed non-Fermi liquid phenomena could be linked to the idea of Kondo breakdown, where critical Kondo boson and deconfined gauge field induce singularity in thermodynamics and transport.^[37,39,42] However, as exemplified by the study on the square lattice, the topology of FS may change radically before any noticeable breakdown of the Kondo effect, therefore the possibility of Lifshitz transition on triangular lattice should be investigated first.

In the present work, we employ the large-N mean-field approach to study the KHM on the triangular lattice. As expected, we find that the Heisenberg antiferromagnetic interaction (J_H) induces twice FS topology changes at the case of the nearest-neighbour (NN) electron hopping but with next-nearest-neighbour (NNN) hole hopping and the case of the NN hole hopping but with NNN electron hopping. Both Lifshitz transitions are continuous. The density of state (DOS) of the conduction electron is changed by J_H. To meet with experiments, we give the STM line-shape of the differential conductance dI/V for different Kondo coupling (J_K) and the ratio of the tunneling amplitude of f-electron t_f versus conduction electron t_c. The calculated spectra are qualitatively consistent with data of CeCoIn₅.^[43]

Compared with the square lattice KHM, the triangular lattice KHM also has the Lifshitz transition, which is driven by the Heisenberg coupling J_H. The energy spectrum local maximum values of the triangular lattice and the square lattice KHM both move towards M →Γ at the case of the NN electron hopping but with NNN hole hopping. However, the triangular lattice has a continuous quantum phase transition around the Lifshitz transition points, while the square lattice has the first-order and second-order phase transitions.^[23] As changes of FS topology can be reflected in the effective mass, the effective mass enhancement m^*/m of the triangular lattice is smooth, leading to the continuous quantum phase transition around the Lifshitz transition, which is different from the square lattice in which m^*/m has the singularity at the Lifshitz transition points.^[23]

The paper is organized as follows. In Section 2, we describe the Kondo–Heisenberg model under the large-N mean-field theory. In Section 3, we present the Lifshitz transition and the DOS of conduction electron. In Section 4, we give the line-shape of differential conductance at different Heisenberg antiferromagnetic interaction, the Kondo coupling, and the ratio of the tunneling amplitude of f-electron to conduction electron. Finally, Section 5 is devoted to a brief conclusion and perspective.

The basic structure and the first Brillouin zone are shown in Fig. 1. The model Hamiltonian of the KHM is given by

Fig. 1.

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	Fig. 1. (a) The real space triangular lattice, a1 = (0,a) and are the basic vectors. (b) The first Brillouin zone of the triangular lattice, and are the reciprocal vectors of the triangular lattice, the red dots are high symmetry points of the first Brillouin zone, where Γ = (0,0), , , , .

To proceed, we use the fermionic large-N mean-field method,^[48] which is believed to capture qualitative features in heavy Fermi liquid states. Introducing valence-bond order parameter and Kondo hybridization parameter ,^[23] and further considering the uniform resonance-valence-bond ansatz χ_ij = χ and the uniform Kondo hybridization V_i = V, as done in Refs. [23,49], one can obtain . Based on these mean-field formulations, equation (1) can be rewritten in the k-space as follows:

We consider the case of J_H≪ J_K, where the paramagnetic heavy Fermi liquid state is stable compared to other symmetry-breaking and exotic fractionalized states. When the Heisenberg interaction J_H increases, the band structure of quasiparticle evolves and a Lifshitz transition is expected to occur.

The FS is a normal circle when J_H is small as shown in Figs. 2(a) and 3(a), which means the influence of the short-range antiferromagntic correlation is negligible. Our system has twice Lifshitz transitions in two cases as shown in Figs. 2(d), 2(e), 3(b), and 3(e). In Figs. 2(d) and 3(b), there emerges a small circle below FS at the center, the quasiparticles begin to fill the area between the two loops. In Figs. 2(e) and 3(e), the FS happens to split into many Fermi pockets after this critical point, each pocket is the FS of hole quasiparticle. The Lifshitz quantum critical point for NN electron hopping with NNN hole hopping has the larger Heisenberg coupling J_H than the case of the NN hole hopping with NNN electron hopping.

Fig. 2.

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	Fig. 2. The FS evolution of the KHM versus the Heisenberg interaction JH for NN electron hopping but with NNN hole hopping, the first Brillouin zone of the triangular lattice is marked by the black line, where the parameters are t = 1, t1/t = 0.3, nc = 0.9, JK = 2.5.

Fig. 3.

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	Fig. 3. The FS evolution of the KHM versus the Heisenberg interaction JH for NN hole hopping but with NNN electron hopping, the first Brillouin zone of the triangular lattice is marked by the black line, where the parameters are t = –1, t1/t = 0.3, nc = 0.9, JK = 2.5.

We also give the quasiparticle energy spectra of these two cases to interpret the FS topology change as shown in Figs. 4 and 5. When J_H is increasing, the short-range antiferromagnetic correlation starts to change the electronic structure, and the FS begins to deform. In Figs. 4(a) and 4(b), the energy spectrum shows one local maximum across the Fermi energy around the K point, which gives the hole quasiparticle FS as shown in Figs. 2(a) and 4(b). When J_H > 0.38, there are two local maxima as shown in Figs. 4(c)–4(f) and 2(c)–2(f). In Fig. 5(a), the energy spectrum has one local maximum across the Fermi energy around the Γ point, which has the FS as shown in Fig. 3(a). When 0.055 < J_H < 0.18, there are two local maxima across the Fermi energy, which induces the hole quasiparticle occupation among two FSs as shown in Figs. 5(b)–5(e) and 3(b)–2(e). When J_H > 0.18, the maximum just crosses the Fermi energy as shown in Figs. 5(f) and 3(f). The FS moves towards M→Γ versus the Heisenberg coupling J_H at the case of NN electron hopping but with NNN hole hopping as shown in Figs. 2 and 4, in contrast to Γ→ M in the case of NN hole hopping but with NNN electron hopping as shown in Figs. 3 and 5.

Fig. 4.

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	Fig. 4. The quasiparticle energy spectra of the KHM from the direction for NN electron hopping but with NNN hole hopping, the blue dashed line denotes the Fermi energy level, where the parameters are t = 1, t1/t = 0.3, nc = 0.9, JK = 2.5.

Fig. 5.

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	Fig. 5. The quasiparticle energy spectra of the KHM from the direction for NN hole hopping but with NNN electron hopping, the blue dashed line denotes the Fermi energy level, where the parameters are t = –1, t1/t = 0.3, nc = 0.9, JK = 2.5.

According to many experiments on heavy-fermion quantum critical compounds YbRh₂Si₂ and CeRu₂Si₂,^[16,20] the FS change is related to the quantum phase transition. Thus, to identify the quantum phase transition around the Lifshitz transition, the ground-state energy E_g and its first-order derivative dE_g/dJ_H versus J_H are shown in Figs. 6 and 7, where dE_g/dJ_H is given by

Fig. 6.

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	Fig. 6. (a) The ground-state energy Eg and (b) the first derivative dEg/dJH versus the Heisenberg interaction JH for NN electron hopping but with NNN hole hopping, where the parameters are t = 1, t1/t = 0.3, nc = 0.9, JK = 2.5.

Fig. 7.

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	Fig. 7. (a) The ground-state energy Eg and (b) the first derivative dEg/dJH versus the Heisenberg interaction JH for NN hole hopping but with NNN electron hopping, where the parameters are t = –1, t1/t = 0.3, nc = 0.9, JK = 2.5.

Fig. 8.

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	Fig. 8. The effective mass enhancement m*/m versus the Heisenberg coupling JH. Panel (a) is for NN electron hopping but with NNN hole hopping, where the parameters are t = 1, t1/t = 0.3, nc = 0.9, JK = 2.5. Panel (b) is for NN hole hopping but with NNN electron hopping, where the parameters are t = –1, t1/t = 0.3, nc = 0.9, JK = 2.5.

The DOS of the conduction electron is shown in Figs. 9 and 10. The Heisenberg interaction J_H has an effect on the DOS of the conduction electron ρ_c, the larger J_H induces the larger gap. Thus, the DOS is changed after the Lifshitz transition. In Figs. 9(a) and 9(b), the DOS has a gap and two peaks. At J_H = 0.4255, it develops a new small peak as shown in Fig. 9(c). The conduction electron DOS has one gap and three peaks when J_H > 0.4255 as shown in Figs. 9(c)–9(f). In Fig. 10, the DOS always has a gap and two peaks, but the left peak is lower than the right peak in Figs. 10(a)–10(d), while becomes higher than the right peak in Figs. 10(e) and 10(f). Both peaks are increasing versus the Heisenberg coupling J_H, and the peak at ω < 0 arises from the van Hove singularity of the large (hybridized) FS.^[50]

Fig. 9.

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	Fig. 9. The conduction electron DOS πc for the KHM versus the Heisenberg interaction JH for NN electron hopping but with NNN hole hopping, where the parameters are t = 1, t1/t = 0.3, nc = 0.9, JK = 2.5. It is shown that the conduction electron DOS is changed after the Lifshitz transition.

Fig. 10.

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	Fig. 10. The DOS of the conduction electron πc for the KHM versus the Heisenberg interaction JH for NN hole hopping but with NNN electron hopping, where the parameters are t = –1, t1/t = 0.3, nc = 0.9, JK = 2.5. It is shown that the conduction electron DOS is changed after the Lifshitz transition.

Therefore, under the MF method,^[23,49] when Heisenberg superexchange J_H increases, the presence of the short-range antiferromagnetic correlation gradually changes the electronic structure, and leads to the mentioned two kinds of Lifshitz transitions, which is similar to Ref. [23]. However, our work finds the continuous transition around the Lifshitz transition, which is different from that in the square lattice, i.e., the first-order and second-order phase transitions.^[23]

With the FS topology of the quasiparticles changed, the area of FS varies at some critical values. To get more insight into the Lifshitz transition, it is helpful to use an effective low-energy theory to grasp the basic physical feature. Since the Lifshitz transition is mainly a single particle problem, one may use the following simple action:

We should note that the change of the FS topology, i.e., the Lifshitz transition, has a direct experimental implication. The Hall coefficient will change its sign when the electronic FS transforms into the hole-type one or some parts of FS disappear. Besides this, one can use the quantum oscillation to measure the effective mass of the quasi-particle as the signal of the Lifshitz transitions discussed here.

The STM spectrum is one of the indispensable tools in the study of correlated quantum matter, especially for several quantum critical heavy-electron compounds, which is a real-space probe that measures a local conductance.^[51–53] In the linear-response regime, the current–voltage characteristics is related to the local DOS of the material.^[54] There are also many STM experiments on the heavy-fermion compounds like YbRh₂Si₂ and CeCoIn₅.^{[22,43,55–58]} Those results coincide with angle-resolved photoemission spectroscopy to understand the physics of quantum critical point in heavy-fermion compounds.^[53]

Here, we follow Ref. [50] to get the differential conductance dI(V)/dV on the triangular lattice by

To calculate DOS, it is helpful to introduce fermionic quasiparticles A_kσ and B_kσ with the following transformation:

Fig. 11.

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	Fig. 11. The STM spectra of the KHM versus the ratio of amplitudes tf/tc for NN electron hopping but with NNN hole hopping, where the parameters are t = 1, t1/t = 0.3, nc = 0.9, JK = 2.5. The ratio of amplitudes tf/tc influences the line-shape of the dI/dV.

Fig. 12.

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	Fig. 12. The STM spectra of the KHM versus the ratio of amplitudes tf/tc for NN hole hopping but with NNN electron hopping, where the parameters are t = –1, t1/t = 0.3, nc = 0.9, JK = 2.5. The ratio of amplitudes tf/tc influences the line-shape of the dI/dV.

When increasing t_f/t_c, the line-shape changes quickly. Figures 11(a)–11(c) have three peaks and figures 11(h)–11(i) have two peaks. There emerges a peak when ω > 0, and the peak is increasing versus t_f/t_c, which is the precursor of the emerging f-electron band.^[50] Figures 12(a)–12(d) have two peaks and figures 12(h)–12(i) show three peaks. Figure 11(e) becomes two peaks while figure 12(e) begins to have three peaks. In Figs. 11(f) and 12(f), the left resonance peak nearly vanishes, which means the suppression of the differential conductance around the Fermi energy.^[50]

We also give the STM spectra of the different Kondo coupling J_K as shown in Figs. 13 and 14. Compared with Figs. 11 and 12, the line-shape varies versus t_f/t_c. There also exists the suppression of the differential conductance around the Fermi energy as shown in Figs. 13(e) and 14(e). Figures 11(b), 12(b), 13(b), and 14(b) are the DOS of the conduction electron.

Fig. 13.

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	Fig. 13. The STM spectra of the KHM versus the ratio of amplitudes tf/tc for NN electron hopping but with NNN hole hopping, where the parameters are t = 1, t1/t = 0.3, nc = 0.9, JK = 2. The ratio of amplitudes tf/tc influences the shape-line of the dI/dV.

Fig. 14.

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	Fig. 14. The STM spectra of the KHM model versus the ratio of amplitudes tf/tc for NN hole hopping but with NNN electron hopping, where the parameters are t = –1, t1/t = 0.3, nc = 0.9, JK = 2. The ratio of amplitudes tf/tc influences the shape-line of the dI/dV.

Among these figures, the gap is increasing against the Heisenberg coupling J_H and the Kondo coupling J_K. The STM line-shape of the differential conductance dI/dV is mainly influenced by the ratio t_f/t_c in which the f-electron component becomes more and more significant. We also find that these spectra are qualitatively similar with those of CeCoIn₅.^[43] These results show the existence of two resonance peaks structure in differential conductance as Refs. [50,59], which gives the insight to the heavy-fermion compounds by STM to examine the correlated electrons with high energy and spatial resolutions.^[43]

In summary, we have investigated the KHM on triangular lattice with the fermionic large-N mean-field theory at the case of the NN electron hopping with NNN hole hopping and the case of the NN hole hopping and NNN electron hopping. At the heavy-fermion liquid state, the Heisenberg antiferromagnetic interaction (J_H) induces twice FS topology changes, i.e., the Lifshitz transition, where goes through the continuous quantum phase transition. In two cases, the conduction electron DOS is changed after the Lifshitz transition, the gap is influenced by the Kondo coupling J_K and the Heisenberg interaction J_H. The line-shape of the differential conductance dI/dV shows that the existence of two resonance peaks structure in differential conductance as in Refs. [50,59]. The short-range antiferromagnetic correlation coupling J_H, the ratio of the amplitudes of the f-electron to the amplitude of the the conduction electron t_f/t_c, and the Kondo correlation J_K influence the shape-line of the differential conductance dI/dV, which gives the insight to detect the heavy-fermion compounds STM spectra for examining the correlated electrons with high energy and spatial resolutions.^[43]

As some triangular heavy-fermion compounds like YbAgGe^[37–40] and YbAl₃C₃^[41,42] have been found, we expect that our results may be confirmed by many FS measurements (Hall coefficient, de Haas–van Alphen measurements, angle-resolved photoemission spectroscopy, quasiparticle interference, and STM spectrum experiments) in those compounds.