Stable Flatbands, Topology, and Superconductivity of Magic Honeycomb Networks

TL;DR

This paper introduces a new mechanism for creating robust flatbands in materials using a network superstructure, explaining enhanced superconductivity and revealing higher-order topological corner states.

Contribution

It proposes a novel principle for flatband realization in real materials through network superstructures, linking it to superconductivity and topological corner states.

Findings

Flatbands are achievable via network superstructures in materials.

Enhanced superconductivity is explained by the flatband mechanism.

Higher-order topological corner states are demonstrated in the network.

Abstract

We propose a new principle to realize flatbands which are robust in real materials, based on a network superstructure of one-dimensional segments. This mechanism is naturally realized in the nearly commensurate charge-density wave of 1T-TaS${}_2$ with the honeycomb network of conducting domain walls, and the resulting flatband can naturally explain the enhanced superconductivity. We also show that corner states, which are a hallmark of the higher-order topological insulators, appear in the network superstructure.

Equations

$$H_0=-t_0\sum _{⟨\mathbf{r},{\mathbf{r}}^{\prime }⟩}{c}_{\mathbf{r},\sigma }^{†}{c}_{{\mathbf{r}}^{\prime },\sigma }-t_J\sum _{\{\mathbf{r},{\mathbf{r}}^{\prime }\}\in J}{c}_{\mathbf{r},\sigma }^{†}{c}_{{\mathbf{r}}^{\prime },\sigma }+H.c.,$$

$$H_{coup}=-t_d\sum _{⟨\mathbf{r},{\mathbf{r}}^{\prime }⟩}{c}_{\mathbf{r},\sigma }^{†}{d}_{{\mathbf{r}}^{\prime },\sigma }+H.c.,$$

$$H_{int}=U\sum _{\mathbf{r}}{n}_{\mathbf{r}}^2+V\sum _{⟨\mathbf{r},{\mathbf{r}}^{\prime }⟩}{n}_{\mathbf{r}}{n}_{{\mathbf{r}}^{\prime }}.$$

$$H_{int}\to H_{\text{BCS}}=\sum _{l\in 2\mathbb{Z}}{g}_l\sum _{\mathbf{p},\mathbf{k}}{\widehat{△}}_{l;\mathbf{p}}^{†}\cdot {\widehat{△}}_{l;\mathbf{k}},$$

$$\frac{1}{g_l}\sim {\int }_{\text{BZ}}\frac{d^{2}k|F_l(\mathbf{k}){|}^2}{\sqrt{({ϵ}_k-\mu {)}^2+|F_l(\mathbf{k}){△}_l{|}^2}}\sim \frac{1}{|{△}_l|},$$

$$\displaystyle H=\sum_{a,\sigma=c/s}\int dl~{}\frac{v_{\sigma}}{2}\Big{[}\frac{1}{K_{\sigma}}(\partial_{l}\theta_{a,\sigma})^{2}+K_{\sigma}(\partial_{l}\phi_{a,\sigma})^{2}\Big{]},$$

$$\displaystyle H_{eff}=J\sum_{\langle i,j\rangle}\Big{[}e^{\sqrt{2\pi}i(\theta_{i,c}-\theta_{j,c})}+h.c.\Big{]}$$

Full text

Band theory of electrons has gone through a renaissance in recent years, especially concerning its topological nature Hasan and Kane (2010); Qi and Zhang (2011). Band structures are also an important starting point to understand strongly correlated systems. In particular, when the Fermi level lies in a flatband, the correlation effects become dominant, leading to many interesting physics including superconductivity and magnetism. A few theoretical principles to realize flatbands are known, most notably the chiral (sublattice) symmetry and an imbalance between the number of sublattice sites Shima and Aoki (1993); Ramachandran et al. (2017). However, once further nearest-neighbor hoppings, which generally exist in real materials, are included, the chiral symmetry is lost and the flatbands in the simplified model acquire a substantial curvature. This is a major reason why it has been difficult to realize flatbands in real materials and observe resulting strong correlation effects experimentally, with few exceptions Cao et al. (2018a, b); Bistritzer and MacDonald (2011).

In this Letter, we propose a novel general principle to realize flatbands based on a network of one-dimensional segments. The resulting flatbands are protected by the combination of crystal and time-reversal symmetries, and are robust even in the presence of the further neighbor hoppings, as long as the effective hopping range is shorter than the segment length. We argue that this mechanism is naturally realized in the nearly commensurate charge-density wave (NC-CDW) phase of 1T-TaS2, in which the domain walls play the role of the one-dimensional metallic segments Park et al. (2019). The robust flatbands ensured by the new principle give a natural explanation of the observed superconductivity, which is strongly reminiscent of the moiré physics Cao et al. (2018a, b); Bistritzer and MacDonald (2011) of twisted bilayer graphene. Moreover, the network superstructure can also lead to corner states, i.e., a higher-order topological insulator.

Experimentally, 1T-TaS2 has a rich phase diagram of CDW orders Wilson et al. (1975) and superconductivity (SC) Sipos et al. (2008); Yu et al. (2015); Liu et al. (2016), which are accessible by tuning temperature, pressure, and doping. There are three distinct regimes: commensurate CDW (C-CDW), NC-CDW, and incommensurate CDW. The C-CDW phase has a long-ranged charge ordering and appears at the lowest temperature with the ambient pressure. This state is the correlation-driven Mott insulator Fazekas and Tosatti (1980); Law and Lee (2017); He et al. (2018). When the C-CDW ordering is slightly suppressed by pressure or doping, then the domain walls appear in between the locally charge-ordered domains, i.e., it enters into the NC-CDW state. If the pressure or doping increase further, the SC emerges from this NC-CDW state. Note that a similar phase diagram is obtained in another CDW material, namely 1T-TiSe2 Li et al. (2016). This suggests that the NC-CDW Sipos et al. (2008); Yu et al. (2015); Liu et al. (2016); Li et al. (2016); Chen et al. (2019) is somehow essential for realizing SC in these CDW materials though how this actually happens has been unclear. We will point out that a honeycomb network of metallic domain walls in the NC-CDW phase Spijkerman et al. (1997); Park et al. (2019); Wu and Lieber (1989) hosts a series of robust flatbands ensured by the new principle, giving a natural explanation for the observed SC.

1. Model and Flat Bands: A recent STM study Park et al. (2019) combined with DFT calculation provided an unprecedented detail of the electronic structure of the network, where the metallic nature of the domain walls and trijunctions is clearly exposed. Similar structure emerges in, e.g., helium mixture absorbed to graphite Morishita (2019). Here we consider a minimal tight-binding model, which consists of the low-energy modes inside the domain walls; see Fig. 1(A).

[TABLE]

where the second sum over $\{\bm{r},\bm{r}^{\prime}\}\in J$ runs over the sites around the junctions $J$ and $t_{J}\leq t_{0}$. Here $\sigma$ represents the spin and the sum over it is assumed. Here we assumed the spin rotational symmetry for simplicity. The model also has time-reversal symmetry $\mathcal{T}$, and crystalline $D_{6}$ symmetry, which are the symmetries observed in the STM experiment of 1T-TaS2 Park et al. (2019). Diagonalizing Eq.(1), we find a cascade of flatbands, Dirac and quadratic band crossings; see Fig. 1(C). On top of this, a three-component spin-$1$ Dirac fermion Dóra et al. (2011) can appear when $t_{J}/t_{0}\to 0$. The number of the flatbands is proportional to the number of the sites between the junctions. The topological band crossings are protected by symmetries. On the other hand, the flat dispersion cannot be generally protected because it requires infinitely many parameters to be tuned. Hence, they are generically fragile, e.g., against the second neighbor hoppings SI . Despite of the fragile nature, flatbands are an ideal stage to realize correlation-dominated physics, such as ferromagnetism, and thus have been studied vigorously Leykam et al. (2018). A well-known mechanism which gives rise to flatbands is an imbalance between sublattice sites in bipartite lattices. This was applied  Shima and Aoki (1993) to hyperhoneycomb systems which have some similarities with the systems we study in this paper. However, such flatbands can be generically removed by inclusion of short-ranged further neighbor hoppings, which exist in real materials.

Remarkably, our flatbands from the network defy this standard phenomenology and are stable against $D_{6}$-symmetric local perturbations. For example, addition of the third-neighbor hoppings $t_{3}$ near the nodes do not disperse the flatbands. We can even include the insulating electronic states from the domain area (described by $H_{\text{dom}}$); see Fig. 1(A). The full Hamiltonian is now $H=H_{0}+H_{\text{dom}}+H_{\text{coup}}$ with

[TABLE]

where $c_{\bm{r},\sigma}$ ($d_{\bm{r},\sigma}$) electrons belong to the network (to the domains). Here $H_{\text{dom}}$ is described by the band insulator which has the two energy-split states per site and the different sites are connected by the small hopping $t_{d}$ [Fig. 1(A)]. From the perturbative reasoning, we expect that this model includes all the possible symmetric local perturbations. Notably, the flatbands and overall band shapes remain almost intact inside the gap [Fig.1(D)]. This result implies that in NC-CDW 1T-TaS2, even if the bulk bands from the domain region are included, flatbands inside the gap are almost intact. Finally, the flatbands survive SI against the Rashba spin-orbit coupling. Such stability is absent for other networks SI . In passing, we note that this is consistent with our previous phenomenological approach; see SI .

To explain the unusual stability, we look into the structure of the wave functions inside the flatbands. We find that those wave functions vanish at the junctions SI . Hence, when the low-energy modes are entirely from the network and only the nearest neighbor hoppings are included, i.e., $t_{J}=0$ in Eq.(1), the wave function is a standing wave $\psi(l)$ [Fig. 1(B)]. From such standing waves, we can construct a set of localized states Bergman et al. (2008) which consist of the flatbands: we consider a linear combination around the honeycomb plaquette with a sign oscillation, i.e., $\Psi\sim\sum_{a}(-1)^{a}\psi(l)$ for $a\in\{1,2,\cdots 6\}$ labeling the six links around the plaquette; see Fig. 1(B). Then we see that this state cannot disperse into the neighboring plaquettes because of the destructive interference. Such destructive interference persists as far as the hopping distance is shorter than the length of the wire. Finally, the intrawire further neighbor terms do not alter this conclusion because they affect only the energy and intrawire structure of the standing waves SI . This illustrates the importance of the symmetry and the locality on the stability of the flatbands. We can also understand that the cascade of the flatbands must appear because there are many standing wave solutions per wire Katsura and Maruyama .

We expect the flatbands to be removed either by long-ranged direct hopping across the domains or by breaking symmetries. Indeed, we can show SI that the flatbands are lifted when the symmetries are broken, or when such long-ranged hoppings [e.g., $t_{f}$ in Fig. 1(B)] are included. This means that, for the NC-CDW state of 1T-TaS2, we need a sizable hopping between the two sites apart by $\sim O(80)\r{A}$ Park et al. (2019) to remove the flatbands, which is not realistic. This explains why the dispersion of the “flatbands” are very small but still finite when the domain sites are included. When the time-reversal symmetry is broken, the band touchings are gapped out and it results in dispersive Chern bands SI , which is a natural platform for fractionalization Tang et al. (2011).

For 1T-TaS2, we constructed a tight-binding model in the Supplemental Material SI , which fits reasonably well with DFT+U calculation on the domain wall. The tight-binding parameters scale as $1/d^{5}$ Skolimowski et al. (2019) where $d$ is the distance between the atomic sites. From this, we find that the cascade of the flatbands emerges SI . Next, we comment on the effect Ritschel et al. (2015, 2018); Le Guyader et al. (2017); Lee et al. (2019) of the interlayer coupling. The interlayer interaction has been suggested to be important in 1T-TaS2. However, we remark that there are plenty of experimental data and theory Park et al. (2019); He et al. (2018) suggesting that the main physics is essentially 2D. For example, the resistivity along the $c$ axis is much larger, e.g., by 500 times Hambourger and Di Salvo (1980), than the intralayer resistivity Hambourger and Di Salvo (1980); Svetin et al. (2017) and anisotropic 2D charge transfer is observed for the NC-CDW state Kühn et al. (2019). Further, the SC $T_{c}$ is almost insensitive to the pressure Sipos et al. (2008) and not much affected under the dimensional reduction Yu et al. (2015). Based on these, we focus on the 2D physics here.

2. Superconducting states: Having established the stability of the flatbands, we now discuss the many-body physics when the Fermi level is near one of the flatbands. Such a system is unstable toward various particle-hole and particle-particle channels. However, guided by experiments, we mainly focus on superconductivity in this paper.

First, we perform the simplest BCS mean-field theory with the phenomenological interaction

[TABLE]

Projecting to the BCS channel, we obtain SI

[TABLE]

where $g_{l}$ is the interaction strength along the pairing channel of the angular momentum $l$, which we compute numerically. $\hat{\triangle}_{l;\bm{p}}$ is the corresponding pairing order parameter. Note that, within the mean-field decomposition of Eq.(2), only the spin-singlet sector appears. Below we consider only the $s$-wave and $(d+id)$-wave pairing channels, i.e., $g_{0}$ and $g_{2}=g_{-2}$ of Eq.(3). Higher angular momentum pairing channels ($|l|>2$) will belong to the same representations of $s$- or $(d\pm id)$-wave pairing channels in the honeycomb symmetry. The magic of the flatbands appears when the gap equation is solved.

[TABLE]

where $F_{l}(\bm{k})$ is the form factor SI , e.g., $F_{2}(\bm{k})\to(k_{x}+ik_{y})^{2}$ for $|\bm{k}|\ll 1$ and $\epsilon_{\bm{k}}=\mu$. Hence, the gap is linearly enhanced, i.e., $|\triangle_{l}|\sim|g_{l}|$ (if $g_{l}<0$), instead of the standard exponential suppression $\sim\exp(-1/|g_{l}|\nu_{l})$. Because there is no other scale, the mean-field energy of the SC is linear in $g_{l}$ and so is the $T_{c}$ SI . Hence, the honeycomb network strongly enhances the SC $T_{c}$. The phase diagram for one flatband is in Fig.2(A). Thus we conclude that, within the mean-field theory, the $s$-wave SC is strongly enhanced when $U$ is attractive Julku et al. (2016). On the repulsive interaction side, the system exhibits ferromagnetism. While the repulsive $U$ opens up a small window for the $(d\pm id)$-channel pairing, the dominance of the $s$-wave SC is a characteristic feature of the present flat-band system. SI . In passing, the BdG fermion spectrum is also computed SI .

Although the mean-field theory ignores the fluctuation, it is a good starting point for revealing possible phases and phase diagram. In fact, the mean-field theory qualitatively agrees with a rigorous result on flat-band ferromagnetism Tasaki (1992) with a repulsive $U$, and our findings in this Letter are consistent with recent numerical studies in other flat-band systems Julku et al. (2016); Mondaini et al. (2018); Huang et al. (2019); Tasaki (1994).

Let us comment on the interaction Eq.(2). The strong electron-phonon coupling in 1T-TaS2 is known as the driving force for the formation of the CDW states Liu (2009). However, after the formation of the CDW clusters through the electron-phonon coupling, the soft phonons are naturally hardened Lazar et al. (2015). In particular, the domain wall in the NC-CDW state is also structurally reconstructed by forming its own CDW clusters from electron-phonon coupling, Park et al. (2019) which will reduce the coupling more. Hence, we expect that the electron-phonon coupling becomes inactive for the physics within the NC-CDW phase. Nevertheless, the phonon-electron interaction was essential in forming the parent CDW domains and the domain wall network, and hence its effect is already encoded the model Eq.(1). Next, it is well known that the coupling of electrons and (optical or gapped) phonons effectively plays the same role as the attractive $U$, which favors the $s$-wave SC.SI This will effectively renormalize $U$ in Eq.(2) toward negative. Hence, explicitly including the effects of these phonons does not alter the dominance of the $s$-wave SC.SI On the other hand, the long-range Coulomb interaction will be efficiently screened due to the large density of states of flatbands and it will be rendered into a short-ranged interaction, which can be effectively encoded by Eq.(2).

It is instructive to consider the strong-coupling limit, Wu et al. (2019) where the interaction is bigger than the hopping integrals. For this, we start from the decoupled strongly correlated wires, each of which is described by a Tomonaga-Luttinger liquid (TLL)

[TABLE]

where the Luttinger parameters $\{K_{c},K_{s}\}$ capture the correlation nonperturbatively Fradkin (2013). This is the correlated version of the scattering problem Park et al. (2019), which faithfully reproduced the band structures.

The two-dimensional SC can be preempted by the spin gap Fradkin (2013); Kivelson et al. (1998) in each wire. Once the spin gap forms, the low-energy physics of each wire is described by a single-component TLL of $\theta_{a,c}$ describing the fluctuating SC pairs Fradkin (2013). Since the SC pair is bosonic, the junction of three TLLs at each vertex of the honeycomb network corresponds to the bosonic $Y$ junction  Tokuno et al. (2008), rather than the fermionic one  Chamon et al. (2003); Oshikawa et al. (2006) where the fermion statistics plays an important role. When each wire has sufficient attraction, i.e., $K_{c}\leq 1$ Fradkin (2013), then the interwire coupling between the SC fluctuations, namely, the Josephson coupling $J$, becomes relevant Fradkin (2013). Only interested in the pattern of the phases, we note that the problem is symmetrically equivalent to the $XY$ model on the kagome lattice, while leaving the full quasi-1D treatment Kivelson et al. (1998) to the future study

[TABLE]

Depending on the sign of $J$, either the 2D $s$-wave or $(d\pm id)$-wave SCs can emerge. When $J>0$, then the so-called $\sqrt{3}\times\sqrt{3}$ order appears Reimers and Berlinsky (1993), which translates as the $(d\pm id)$-SC. If $J<0$, the conventional $s$-wave SC emerges. When the repulsive-$U$ dominates the junction region and the region becomes Mott insulating Berg et al. (2009), $J>0$ can appear. From this strong-coupling limit, we can learn how the $2k_{F}$-density wave of 1d wires competing with SC is suppressed. For the generic filling of each wire, the momentum $k_{F}$ will not be commensurate with the wire length $L$, i.e., $\phi_{L}=k_{F}L$ is not a rational number. This frustrates the phases of the density waves and thus their true two-dimensional order is strongly suppressed to develop. Because the density waves are the main competitors of the SC in one dimension, this gives a natural favor on the SC.

The domain wall states of 1T-TaS2 presumably experience small correlation effect and the junction regions are quite metallic from the STM data Park et al. (2019). Furthermore, no magnetism is observed in experiments, i.e., interaction is dominantly attractive. Furthermore, the SC in 1T-TaS2 is experimentally observed for a broad range of the parameters Sipos et al. (2008); Yu et al. (2015). When compared with Fig.2(B), the SC is consistent with the $s$ wave. These suggest that the SC state of 1T-TaS2 is likely an $s$-wave SC.

3. Higher-order topology: When the filling per wire is commensurate, then the network can develop its own CDW order and become an insulator. We highlight here that the insulating network hosts an interesting possibility, namely emergent corner states which are akin to those of the 2d higher-order topological insulators Benalcazar et al. (2017); Schindler et al. (2018a, b). We illustrate this on the half-filled spinless fermion model, whose generalization to the spinful systems is straightforward. In such wires, the dimerization is expected, which manifests as the staggered nearest-neighbor hopping parameters; see Fig. 2(B). To clearly expose the corner states, we perform a finite-size calculation on a single unit cell of the network. In the spectrum Fig. 2(D), we see four in-gap modes, which are localized at the trijunction and the boundary Fig. 2(C). Those states are protected by crystal symmetries. We can also show that the above system similarly supports an in-gap mode at the corner of the edge. See Supplemental Material SI where the junction of two gapped domain walls supports a corner mode protected by a reflection symmetry. Such 0D states protected by crystalline symmetries are the hallmarks of higher-order topology.

The above higher-order topology of gapped domain wall networks may be relevant to various materials with similar structures such as C-CDW 1T-TaS2 Cho et al. (2017) and twin boundary networks of MoSe2 grown on MoS2 Ma et al. (2017). Spectacularly, it has been noted Cho et al. (2017); Skolimowski et al. (2019) that the insulating domain walls of C-CDW 1T-TaS2 have precisely the same structure as the dimerization, and they form junctions and networks. We believe that the confirmation of the higher-order topology in these systems will be an extremely interesting future problem, given that there is no concrete experimental demonstration of the 2D higher-order topology so far.

4. Conclusions: We considered the electronic structure of a conducting honeycomb network, where we uncovered the emergence of the cascade of the flatbands that are stable against various local perturbations. Compared to the previous studies Leykam et al. (2018); Barreteau et al. (2017); Sun et al. (2016); Zheng et al. (2014); Maruyama et al. (2016); Shima and Aoki (1993); Bergman et al. (2008), our work reveals that the flatbands can emerge in much broader sets of the models beyond the models with chiral symmetry or only nearest-neighbor hopping terms. We also demonstrated that the domain wall network is an ideal place to find diverse topological band structures: topological band crossings, and higher-order corner states. We find that the robust flatbands and network geometry can explain the coexistence of SC and CDW states in the network materials, which goes beyond the previous Landau-Ginzburg theory Chen et al. (2019), which is blind to the emergent electronics from the network.

The signature of the network and its role in SC can be detected in various experiments in 1T-TaS2. First, the combination of our STM data, DFT calculation, Park et al. (2019) and the result of our current paper already points strongly toward the existence of the flatbands. In particular, the scanning tunneling spectroscopy Park et al. (2019) showed that the band gap is filled, which again supports the emergence of cascade of band structures. In particular, one nearly flatband is observed right below the chemical potential in the currently available photoemission data Hu et al. (2018); Perfetti et al. (2006); Sohrt et al. (2014) though further investigation will be desirable. Second, magnetotransport and oscillations can provide the information about the primary conducting and SC channels. They have been applied in the small twist-angle bilayer graphene Rickhaus et al. (2018); Efimkin and MacDonald (2018) and textured superconducting states of 1T-TiSe2 Li et al. (2016); Chen et al. (2019). Also the flatbands show a few characteristic behaviors in thermodynamic quantities, which we summarize in the Supplemental Material SI . It would be also interesting to perform numerical calculation on our network model with interactions, to confirm our predictions.