Nematicity-enhanced superconductivity in systems with a non-Fermi liquid behavior

Strongly correlated systems have become an epitome in the condensed-matter physics, as exemplified by the high-temperature superconductivity in the cuprate [1], and iron-based [2] families. These compounds exhibit rich phase diagrams as hallmarked by the emergence of unconventional superconductivity, and a plethora of symmetry-broken phases such as spin and charge nematicity and stripe orders.

Quest for finding novel high-temperature superconductors spurs interests in exploring many-body systems with short-range repulsions but with (nearly) flat subregions in the band dispersion arising from hopping beyond nearest neighbors or from lattice structures [3, 4]. These systems with dispersionless band portions permit numerous scattering channels for the electrons and can give rise to various exotic quantum phases such as spin and charge density waves [5, 6], Mott insulating [7], and bad-metallic phases [8], as well as the formation of spatially extended Cooper pairs [9, 10].

Interaction and the flatness of the band structure can be intimately related to geometric and quantum frustration in producing strong correlation effects. The spin liquid behavior in hexagonal lattices, such as organic compounds [11, 12] and inorganic Herbertsmithites [13, 14], are typical examples. In these exotic liquids, the classical picture is no longer valid, and their quantum phase transitions cannot be described within Landau's phase transition theory.

Aside from these many-body phenomena, the electron nematicity, i.e. spontaneous breaking of spatial rotational symmetry triggered by many-body interactions, is another manifestation of the correlation effects [15, 16]. It is an intriguing direction to pursue the physical origins of these symmetry-broken phases [17–19], and to grasp the interplay between nematicity and other phases such as superconductivity [10, 20–24] and non-Fermi liquid [25]. Various studies report different roles of nematic fluctuations on the superconductivity including the competition between these two phases, e.g. in doped ${\textrm{BaFe}}_{2}{\textrm{As}}_{2}$ [26], the assistance of nematicity to enhance the superconductivity transition temperature, e.g. in twisted bilayer graphene [27], and the negligible effect of nematicity on the superconducting phase, e.g. in ${\textrm{FeSe}}$ [28]. One crucial aspect is figuring out which of these possibilities occur in systems that have a flat or partially flat band (PFB) in their dispersions [5, 6, 29, 30].

In this paper, we bring these features together to explore the interplay between the nematicity and superconductivity in PFB models on the triangular lattice, effective model for Moire-produced basis states [31]. Here the lattice structure frustrates magnetic orders, thereby giving opportunities for nematic instabilities to arise. As a key finding, we shall demonstrate that nematicity can significantly enhance transition temperatures (T_c) in the superconducting phase, with a $s_{x^2+y^2} -d_{x^2-y^2}-d_{xy}$-wave pairing symmetry. This occurs for an intermediate Hubbard repulsion and in a non-Fermi liquid regime.

The Hubbard Hamiltonian on the isotropic triangular lattice reads

$$\begin{equation} H = \sum_{\boldsymbol k, \sigma} \varepsilon_{\boldsymbol k} c^{\dagger}_{\boldsymbol k \sigma} c_{\boldsymbol k \sigma} +U \sum\limits_{i} n_{i \uparrow} n_{i \downarrow} - \mu \sum\limits_{i \sigma} n_{i \sigma}, \end{equation} \tag{ 1 }$$

where $c^{\dagger}_{\boldsymbol k \sigma}(c_{\boldsymbol k \sigma})$ creates (annihilates) an electron with momentum $\boldsymbol k = (k_{x}, k_{y})$ and spin σ at site i, $n_{i \sigma} \equiv c^{\dagger}_{i \sigma} c_{i \sigma}$. The repulsive Hubbard interaction is denoted as $U (\gt0)$, and µ is the chemical potential. The non-interacting band dispersion for the triangular lattice is given as

$$\begin{align} \varepsilon_{\boldsymbol k}(t,t^{^{\prime}}) & = -t \big[-2 \cos(k_{x}) -4 \cos(k_{x}/2) \cos(\sqrt{3} k_{y}/2) \big] \nonumber \\ & \quad +t^{^{\prime}} \big[-2 \cos(\sqrt{3} k_{y}) - 4\cos(3k_{x}/2) \cos(\sqrt{3} k_{y}/2) \big], \nonumber \\ \end{align} \tag{ 2 }$$

where t is the nearest-neighbor hopping (taken as a unit of energy) and tʹ is the second-neighbor hopping. Here, we consider $(t,t^{^{\prime}}) = (1.0,0.15)$, which possesses a nearly flat region along $K-K^{^{\prime}}-K^{^{\prime\prime}}$, see dashed line in figure 1. For the interaction, we set an intermediate $U = 4.5t$, with the inverse temperature set to be $\beta\equiv1/(k_{B}T) = 30/t$ except in figure 3(c).

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To study paramagnetic phases with no spin imbalance, we employ the dynamical mean-field theory (DMFT) combined with the fluctuation exchange approximation (FLEX), known as the FLEX+DMFT [32]. This method comprises DMFT and FLEX double loops, solved self-consistently at each FLEX+DMFT iteration. In this work, we solve the DMFT impurity problem by the modified iterative perturbation theory [33, 34]. The momentum-dependent FLEX self-energy is constructed from the bubble and ladder diagrams. After removing the doubly-counted diagrams in the local FLEX self-energy, the FLEX+DMFT self-energy is updated. The momentum-dependent self-energy in the FLEX+DMFT incorporates vertex corrections generated from the DMFT iterations into the local part of the FLEX self-energy. Even though our FLEX+DMFT method does not deal with spatial vertex corrections, larger coordination number and frustrated magnetic fluctuations in the triangular lattice give rise to more local self-energies and less dominant spatial vertex corrections than in the square lattice [35]. As a result, the FLEX+DMFT is considered to be a reliable approach that incorporates local and nonlocal correlations.

When we start from the non-interacting tight-binding Hamiltonian, equation (1), that has the sixfold rotational (C₆) lattice symmetry, the solution of the many-body problem may exhibit a lower symmetry. To study the phases with/without C₆ symmetry, we solve the FLEX+DMFT loops with/without imposing the C₆ constraints. To explore the Pomeranchuk instability with the broken C₆ symmetry, we take an initial self-energy as $\Sigma_{\textrm{in}} = 0.05 [\cos(k_{x}) - \cos(\sqrt{3}k_{y}/2) \cos(k_{x}/2) ]$ which acts as a seed for distorting the Fermi surface for the FLEX+DMFT iterations. The FLEX+DMFT calculations are here performed on a $64 \times 64$ momentum grid and an energy mesh with 2048 points.

We start with presenting the momentum distribution function plotted in panels (a–c) in figure 2 (top rows). For a system with a well-defined Fermi surface, $\langle n_{k} \rangle$ should take the value of unity (zero) inside (outside) the Fermi surface for $T\rightarrow0$. For all band fillings in our results, the maxima of the momentum-dependent distribution function are below unity. The system exhibits a filling-dependent degrading of C₆ down to a twofold C₂ symmetry in $\langle n_{k} \rangle$. Namely, we have here an emergence of nematicity, or a Pomeranchuk instability. The breaking of C₆ is seen to occur even right at half-filling, while the electron-doped case shows a preserved C₆.

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To quantify the broken C₆ symmetry, we introduce point-group resolved Pomeranchuk order parameters defined as $\xi_{d_{x^2-y^2}} = \sum_{\boldsymbol k} d_{x^2-y^2} (\boldsymbol k) n_{\boldsymbol k}$ and $\xi_{d_{xy}} = \sum_{\boldsymbol k} d_{xy}(\boldsymbol k) n_{\boldsymbol k},$ with $\sum_{\boldsymbol k} = 1$ [36]. The form factors, $d_{x^2-y^2} = \cos(k_x)-\cos(\sqrt{3}k_y/2)\cos(k_x/2)$ and $d_{xy} = \sqrt{3}\sin(\sqrt{3}k_y/2)\sin(k_x/2)$, describe the distortion of the Fermi surface in the point group C₆, and ξ is a real number with values between zero (when C₆ is preserved) and unity.

Figure 3(a) displays ξ against the band filling. We can see that, as the band filling is reduced, ξ starts to grow, and at a critical band filling $\langle n_{{{\textrm{c}}_{1}}} \rangle = 1.02$ (vertical blue line in figure 3(a)), $\xi_{d_{xy}}$ undergoes a first-order phase transition [37, 38]. At this filling, the onset of nematicity is accompanied by a Lifshitz transition, where the Fermi surface delineated by the ridges in $|G_{\boldsymbol k}|^2$(not shown) is not only distorted but undergoes a topological change from closed to open structures.

We further notice that the filling dependence of the nematicity differs between $\xi_{d_{x^{2}-y^{2}}}$ and $\xi_{d_{xy}}$ in the PFB model; compare purple and magenta lines in figure 3(a). For $0.986\lt\langle n \rangle \lt\langle n_{{\textrm{c}}_{1}} \rangle$, $\xi_{d_{x^{2}-y^{2}}}$ is dominant, while $\xi_{d_{xy}}$ takes over below $\langle n \rangle = 0.986$, which we call the second characteristic band filling, $\langle n_{{\textrm{c}}_{2}} \rangle$ (vertical dashed sky-blue line in figure 3(a)). While $\xi_{d_{x^{2}-y^{2}}}$ displays a first-order transition at $\langle n_{{{\textrm{c}}_{1}}} \rangle$, $\xi_{d_{xy}}$ exhibits a crossover at $\langle n_{{\textrm{c}}_{2}} \rangle$. This suggests that thermodynamic parameters such as temperature at which $\xi_{d_{xy}}$ and $\xi_{d_{x^{2}-y^{2}}}$ experience the first-order transitions are different from each other.

To trace back the origin of the nematic phases, let us next present the momentum-dependent spin susceptibility $\chi_{s}(\boldsymbol k)$ for the PFB model in figures 2(d)–(f). In the electron-doped regime where the Pomeranchuk instability is absent, χ_s respects the sixfold rotational symmetry of the lattice, with peaks at $\boldsymbol k = (\sqrt{3}\pi/2,0)$ and its equivalent positions under C₆. As band filling is decreased below the half-filling, the spin susceptibility develops spikes around $\boldsymbol k = (\sqrt{3}\pi/3,2\pi/3)$ and the equivalent places under a C₂ subgroup of the original C₆ rotational symmetry. The appearance of spikes at mid-momenta in the spin susceptibilities indicates the presence of long-range spin fluctuations in our systems [39].

In general, an electronic nematicity without breaking the translational symmetry can be driven by structural transitions, charge [40] or spin [41] fluctuations. Our Hamiltonian does not deal with the distortion of the lattice or phonons and thus precludes structural transitions. We have checked that the charge susceptibility is at least an order of magnitude smaller than the spin susceptibility. Thus the spin-mediated correlations should be responsible for the emergence of the Pomeranchuk instability [24].

Now let us turn to a non-Fermi liquid character of the present electronic systems, since the flat portions of the band may well exert peculiar effects. We can quantify this in terms of the impurity self-energy in DMFT by fitting the imaginary part of the self-energy on Matsubara axis to $|{\textrm{Im}} \Sigma_{\textrm{DMFT}} ({\textrm{i}} \omega_{n})| \propto \omega_{n}^{\alpha}$, and present the result for the exponent α in figure 3(b). In general, α = 1 at small ω_n (c.f., α = 2 on real frequency axis as $|{\textrm{Im}} \Sigma_{\textrm{DMFT}} ( \omega)| \approx \max(\omega^2, T^2)$ at small T) characterizes the Fermi liquid, while α < 0.5 will signify a non-Fermi liquid (bad metal) behavior [42–45]. Above the first order Pomeranchuk transition for $\langle n \rangle\gt\langle n_{{{\textrm{c}}_{1}}} \rangle$, α's computed for systems with (dashed lines) and without (solid lines) the enforced C₆ constraint trivially coincide with each other. We can see that both systems display strong non-Fermi liquid behavior with α well below 1. If we turn to $\langle n \rangle\lt\langle n_{{{\textrm{c}}_{1}}} \rangle$ for which we have revealed the nematicity, figure 3(b) shows notable differences in α between the cases where C₆ is enforced or not. After a sharp drop at $\langle n_{{{\textrm{c}}_{1}}} \rangle$ as the band filling is reduced, α gradually increases (decreases) in the presence (absence) of the imposed sixfold constraint. Eventually α starts to decrease with decreasing $\langle n \rangle$ at $\langle n \rangle \approx 0.85$ in the PFB system. The persistent α < 0.5 for $\langle n \rangle\lt\langle n_{{{\textrm{c}}_{1}}} \rangle$ implies that the nematic phase resides in the non-Fermi liquid regime.

Now let us come to our key interest in pairing instabilities, for which we solve the linearized Eliashberg equation, $\lambda \Delta (k) = -\frac{1}{\beta} \sum_{k^{^{\prime}}} V_{\textrm{eff}}(k-k^{^{\prime}}) G_{k^{^{\prime}}} G_{-k^{^{\prime}}} \Delta(k^{^{\prime}}),$ to find the largest eigenvalue λ for the spin-singlet, even-frequency superconducting gap function Δ. Here, $k\equiv(\boldsymbol k, {\textrm{i}} \omega_{n})$ with ω_n the fermionic Matsubara frequency, and the effectiv interaction for singlets given as $V_{\textrm{eff}}(k) = U+3 U^2 \chi_{s} (k) /2 - U^2 \chi_{c} (k)/2$. The pairing is identified when λ exceeds unity [46]. Figure 3(c) depicts λ in the presence (dashed green lines) or absence (solid blue) of imposed C₆ symmetry in PFB.

When the sixfold rotational symmetry is enforced, we get λ < 0.8 in PFB model, indicating that the singlet superconductivity does not arise for the temperature ($k_BT = t/30$) considered here. We can still notice that λ displays a double-peak structure with a minimum at $\langle n \rangle_{\textrm{min}} = 0.95$. The dip is shown to occur at the band filling at which the $d_{x^2-y^2}$ gap function with two-nodal lines for $\langle n \rangle\gt\langle n \rangle_{\textrm{min}}$ changes into a more complicated multi-nodal-line gap functions for $\langle n \rangle\lt\langle n \rangle_{\textrm{min}}$, see [47] supplemental material for details. This behavior of the gap function reflects a crossover from the antiferromagnetic spin structure with a single nesting vector for $\langle n \rangle\gt\langle n \rangle_{\textrm{min}}$, to a more complex spin structure for $\langle n \rangle\lt\langle n \rangle_{\textrm{min}}$ where single peaks in the spin susceptibility evolve into extended structures (see figures 2(d)–(f)). Thus the system for $\langle n \rangle\lt\langle n \rangle_{\textrm{min}}$ goes beyond the conventional nesting physics. Similar structure in λ and associated gap function have also been reported for PFB systems on the square lattice [9], again in the absence of nematicity.

In a dramatic contrast, if we allow the C₆ symmetry to be broken spontaneously, λ soars from those with C₆ restriction, as seen for $\langle n_{\textrm{c}} \rangle \lt \langle n \rangle \lt1.15$. This occurs concomitantly with the Pomeranchuk order parameters (ξ's), which grow precisely in this filling region. Just below $\langle n_{\textrm{c}} \rangle$, λ in the systems with broken C₆ (solid blue lines in figure 3(c)) exhibits a rapid growth and exceeds unity. This is inherited in the superconducting transition temperatures $(T_{\textrm{c}})$, presented in figure 4(a).

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T_c, with the broken C₆, exhibits a single-dome structure as a function of band filling. We can observe that the presence of a flat portion in PFB or a van Hove singularity for $t^{^{\prime}} = 0$ have similar effects on the largest values of T_c when $\xi_{d_{xy} } \geqslant \xi_{d_{x^2-y^2}}$. One should note that, while a van Hove singularity at E_F only occurs at a single point on the filling axis, a flat portion of the band can accommodate a range of band filling. This difference is reflected in the width of the T_c dome at a given temperature; see figure 4(a) and SM. The maximum of T_c in the PFB is seen to take place close to $\langle n_{{\textrm{c}}_{2}} \rangle$ at which $\xi_{d_{xy} }$ exceeds $\xi_{d_{x^2-y^2}}$. Note that the superconducting transition temperature becomes almost doubled as we pass through $\langle n_{{\textrm{c}}_{1}} \rangle$, see figure 4(a), which should come from the interplay between nematicity, spin fluctuations, and superconductivity.

Let us now delve into the gap function in momentum space in figures 4(b)–(d). In the electron-doped regime, the PFB model exhibits a conventional $d_{x^2-y^2}$ paring symmetry [48]. This behavior of the gap function persists for $\langle n \rangle \gt\langle n_{\textrm{c}} \rangle$. On the other hand, below $\langle n_{{\textrm{c}}_{1}} \rangle$ where the C₆ symmetry is broken down to its C₂ subgroup, the dominant channel of instability is a mixture of $s_{x^2+y^2}$, $d_{x^2-y^2}$ and d_xy -wave symmetries, see figure S16 in SM.

To better understand the role of nematicity in superconducting phases, we can look at $\Delta V_{\textrm{eff}} = V_{\textrm{eff}}- V_{\textrm{eff}}^{\,C_{6}}$, where $V_{\textrm{eff}}^{\,C_{6}}$ is the effective interaction with the imposed C₆ constraint. As shown in figure S19 in SM, V_eff is much intensified when C₆ is lifted. Since χ_c is much smaller than χ_s , the effective pairing interaction reflects the momentum-dependence of the spin-susceptibility under the Pomeranchuk distortions. This effective interaction assists electrons to nonlocally form Cooper pairs [15, 38, 49]. The deformation in $\Delta V_{\textrm{eff}}$ allows first-order perturbation corrections in the distortion, which should be responsible for the drastic changes in λ below $\langle n_{\textrm{c}} \rangle$. This contrasts with the previous study on the interplay between nematicity and superconductivity, where the enhancement of λ originates from the second-order perturbation corrections and thus results in much smaller changes [24].

We have studied whether and how an emergent nematicity affects superconductivity in PFBs on the regular triangular lattice. We have shown with the FLEX+DMFT that nematicity dramatically affects pairing symmetry, and the T_C is significantly enhanced by the lowered point-group symmetry in the electronic structure. This is shown to occur in a non-Fermi liquid regime, which is characterized by blurred Fermi surfaces, momentum-dependent fractional occupations of the band, and a fractional power-law in the self-energy. In the presence of nematic order, the superconducting symmetry changes from an (extended) $d_{x^2-y^2}$-wave to a $s_{x^2+y^2} -d_{x^2-y^2}-d_{xy}$-wave, where unlike the conventional nesting-driven case, the pairing interaction is governed by an intricate spin susceptibility structure.

Future works should include the elaboration of the way in which the non-Fermi liquid property affects the superconductivity, and exploration of the interplay between Pomeranchuk instability and superconductivity in multi-band/orbital systems with flat regions.

Sh S acknowledges financial support from the ANR under the Grant ANR-18-CE30-0001-01 (TOPODRIVE) and the European Union Horizon 2020 research and innovation program under Grant Agreement No. 829044 (SCHINES). M K was supported by JSPS KAKENHI Grand Numbers JP20K22342 and JP21K13887. H A thanks CREST (Core Research for Evolutional Science and Technology) 'Topology' project (Grant Number JPMJCR18T4) from Japan Science and Technology Agency, and JSPS KAKENHI Grant JP17H06138.

The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors. The data that support the findings of this study are available upon request from the authors.