† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11674139, 11704166, and 11834005), the Fundamental Research Funds for the Central Universities, China, and PCSIRT, China (Grant No. IRT-16R35).
Motivated by recent experimental progress on triangular lattice heavy-fermion compounds, we investigate possible Lifshitz transitions and the scanning tunnel microscope (STM) spectra of the Kondo–Heisenberg model on the triangular lattice. In the heavy Fermi liquid state, the introduced Heisenberg antiferromagnetic interaction (JH) results in the twice Lifshitz transition at the case of the nearest-neighbour electron hopping but with next-nearest-neighbour hole hopping and the case of the nearest-neighbour hole hopping but with next-nearest-neighbour electron hopping, respectively. Driven by JH, the Lifshitz transitions on triangular lattice are all continuous in contrast to the case on square lattice. Furthermore, the STM spectra shows rich line-shape which is influenced by the Kondo coupling JK, the Heisenberg antiferromagnetic interaction JH, and the ratio of the tunneling amplitude of f-electron tf versus conduction electron tc. Our work provides a possible scenario to understand the Fermi surface topology and the quantum critical point in heavy-fermion compounds.
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 (JH) 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 JH. To meet with experiments, we give the STM line-shape of the differential conductance dI/V for different Kondo coupling (JK) and the ratio of the tunneling amplitude of f-electron tf versus conduction electron tc. 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 JH. 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
The basic structure and the first Brillouin zone are shown in Fig.
![](cpb_29_7_077102/cpb_29_7_077102_ieqn1.gif)
![](cpb_29_7_077102/cpb_29_7_077102_ieqn2.gif)
![](cpb_29_7_077102/cpb_29_7_077102_ieqn3.gif)
![](cpb_29_7_077102/cpb_29_7_077102_ieqn4.gif)
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
![](cpb_29_7_077102/cpb_29_7_077102_ieqn15.gif)
![](cpb_29_7_077102/cpb_29_7_077102_ieqn16.gif)
![](cpb_29_7_077102/cpb_29_7_077102_ieqn17.gif)
![](cpb_29_7_077102/cpb_29_7_077102_ieqn18.gif)
![](cpb_29_7_077102/cpb_29_7_077102_ieqn19.gif)
We consider the case of JH≪ JK, where the paramagnetic heavy Fermi liquid state is stable compared to other symmetry-breaking and exotic fractionalized states. When the Heisenberg interaction JH increases, the band structure of quasiparticle evolves and a Lifshitz transition is expected to occur.
The FS is a normal circle when JH is small as shown in Figs.
We also give the quasiparticle energy spectra of these two cases to interpret the FS topology change as shown in Figs.
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 Eg and its first-order derivative dEg/dJH versus JH are shown in Figs.
The DOS of the conduction electron is shown in Figs.
Therefore, under the MF method,[23,49] when Heisenberg superexchange JH 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
![](cpb_29_7_077102/cpb_29_7_077102_ieqn22.gif)
![](cpb_29_7_077102/cpb_29_7_077102_ieqn23.gif)
![](cpb_29_7_077102/cpb_29_7_077102_ieqn24.gif)
![](cpb_29_7_077102/cpb_29_7_077102_ieqn25.gif)
![](cpb_29_7_077102/cpb_29_7_077102_ieqn26.gif)
![](cpb_29_7_077102/cpb_29_7_077102_ieqn27.gif)
To calculate DOS, it is helpful to introduce fermionic quasiparticles Akσ and Bkσ with the following transformation:
![](cpb_29_7_077102/cpb_29_7_077102_ieqn28.gif)
![](cpb_29_7_077102/cpb_29_7_077102_ieqn29.gif)
![](cpb_29_7_077102/cpb_29_7_077102_ieqn30.gif)
When increasing tf/tc, the line-shape changes quickly. Figures
We also give the STM spectra of the different Kondo coupling JK as shown in Figs.
Among these figures, the gap is increasing against the Heisenberg coupling JH and the Kondo coupling JK. The STM line-shape of the differential conductance dI/dV is mainly influenced by the ratio tf/tc 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 (JH) 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 JK and the Heisenberg interaction JH. 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 JH, the ratio of the amplitudes of the f-electron to the amplitude of the the conduction electron tf/tc, and the Kondo correlation JK 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.
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