Nematic Fluctuation Superconductivity

• Nematic-fluctuation-mediated superconductivity is characterized by Cooper pairing induced by enhanced nematic fluctuations that break rotational symmetry while preserving translation symmetry.
• Theoretical models reveal that forward scattering from soft nematic modes near the quantum critical point yields highly anisotropic gap structures, as seen in FeSe₁₋ₓSₓ systems.
• Experimental evidence, including rotated gap minima and non-Fermi-liquid behavior, supports the mechanism where nematic criticality enhances superconductivity.

Nematic-fluctuation-mediated superconductivity refers to Cooper pairing in metals (predominantly multi-orbital systems) that is induced or substantially enhanced by the critical quantum fluctuations associated with electronic nematic order—broken lattice rotational symmetry with preserved translation symmetry. This unconventional pairing mechanism is now supported by both theoretical frameworks and decisive experimental evidence demonstrating a distinct superconducting gap structure and phase diagrams in several families of quantum materials, most notably FeSe$_{1-x}$S$_x$ near its nematic quantum critical point (QCP) (Nag et al., 2024).

1. Nematic Order and Susceptibility

Nematic order in an electronic system involves the spontaneous breaking of discrete rotational symmetry ($C_4 \rightarrow C_2$), often manifested as unequal electronic occupation or hopping along orthogonal crystalline axes. The nematic order parameter is typically defined (one-band case) as $$\phi(\mathbf{r}) = \langle n_x(\mathbf{r}) - n_y(\mathbf{r}) \rangle$$. The dynamics and critical behavior of nematic fluctuations are described by the nematic susceptibility

$$\chi_{\text{nem}}(\mathbf{q},\omega) = \int d^dr\,dt\, e^{i(\mathbf{q}\cdot\mathbf{r} - \omega t)} \langle \phi(\mathbf{r},t) \phi(0,0) \rangle.$$

Near a nematic QCP, $\chi_{\text{nem}}(\mathbf{q},0) \sim 1/(r + c q^2)$, with $r \rightarrow 0$ at the QCP. These fluctuations are strongly enhanced near $q = 0$ and low energy, and couple to the conduction electrons through a symmetry-allowed channel, typically yielding a retarded, forward-scattering effective interaction (Lederer et al., 2014, Kuo et al., 2015).

2. Theoretical Framework for Nematic-Mediated Pairing

The minimal framework starts with electrons coupled to a soft bosonic nematic mode $\phi$ via coupling constant $g$. Soft nematic fluctuations exchanged between fermions mediate an effective attractive interaction: $$V_{\text{nem}}(\mathbf{k}, \mathbf{k}') \sim g^2 \chi_{\text{nem}}(\mathbf{q} = \mathbf{k} - \mathbf{k}', \omega = 0) = \frac{g^2}{r + c |\mathbf{k} - \mathbf{k}'|^2}.$$ This interaction peaks at small momentum transfer and is thus strongly forward-scattering, affecting predominantly intrapocket processes. The linearized superconducting gap equation in this context is: $$\Delta(\mathbf{k}) = -\sum_{\mathbf{k}'} V_{\text{nem}}(\mathbf{k}, \mathbf{k}') \frac{\Delta(\mathbf{k}')}{2E(\mathbf{k}')} \tanh\left( \frac{E(\mathbf{k}')}{2T} \right),$$ with $$E(\mathbf{k}) = \sqrt{\varepsilon^2(\mathbf{k}) + \Delta^2(\mathbf{k})}$$. In multi-orbital models, the nematic order is frequently of Pomeranchuk ($d$-wave) type, and the relevant nematic susceptibility and form factors reflect the orbital degrees of freedom (Islam et al., 2024, Yamase et al., 2013). At the QCP, this yields a strongly anisotropic gap with angular structure determined by the nematic form factor.

3. Experimental Evidence and Gap Structure

The FeSe$_{1-x}$S$_x$ material system provides an isolated nematic QCP (at $x_c \approx 0.17$), unclouded by competing magnetic or charge density wave orders (Nag et al., 2024). Scanning tunneling spectroscopy (STS) and Bogoliubov quasiparticle interference (BQPI) mapping of FeSe$_{0.81}$S$_{0.19}$ reveal:

• An exceptionally anisotropic, near-nodal superconducting gap with minima located at $\theta = \pm 45^\circ$ relative to the Fe-Fe crystalline axes, in direct contrast to the nearly isotropic spin-fluctuation-mediated gaps of other Fe-based superconductors (where minima align with Fe-Fe axes at $0^\circ$, $90^\circ$).
• The gap function fits a nematic form-factor model, such as $$\Delta(\theta) = \Delta_s + \Delta_s' \cos^2(2\theta)$$, with $\Delta_s'/\Delta_s \approx 8$, indicating deep, nearly nodal minima.
• Upon sulfur doping across the nematic QCP, the pairing mechanism shifts fundamentally from spin-fluctuation-dominated (aligned minima) to nematic-fluctuation-dominated (rotated minima) (Nag et al., 2024, Islam et al., 2024).

Key gap features, including deep minima at the “cold-spot” angles where the nematic form factor $f(\theta) = \cos 2\theta$ vanishes, are quantitatively reproduced by Eliashberg-type calculations with moderate coupling constants ($\lambda \approx 0.1$) (Nag et al., 2024).

4. Phase Diagram and Competing Orders

The domelike enhancement of $T_c$ centering on the nematic QCP, with $T_c$ maximized where the Weiss temperature $T^* \to 0$ and nematic susceptibility diverges, is a pervasive feature across several families:

• FeSe$_{1-x}$S$_x$: Nematic fluctuations peak at $x \gtrsim x_c$; superconductivity exhibits maximal gap anisotropy and $T_c$ enhancement in this regime (Nag et al., 2024, Islam et al., 2024).
• Ba$_{1-x}$Sr$_x$Ni$_2$As$_2$: Sixfold enhancement in $T_c$ is observed coincident with suppression of nematic order to $T=0$ (Eckberg et al., 2019).
• Cu$_x$TiSe$_2$: Maximal $T_c$ occurs where the elastoresistive nematic Weiss temperature crosses zero (Lv et al., 2 Jan 2026).
• Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$, Cs(V$_{1-x}$Ti$_x$)$_3$Sb$_5$: Curie–Weiss divergent nematic susceptibility and maximal $T_c$ at the QCP are observed (Kuo et al., 2015, Lv et al., 2 Jan 2026, Philippe et al., 2022, Wuttke et al., 2022).

Some systems host intertwined nematic, charge density wave (CDW), or magnetic orders, but nematic fluctuations can dominate pairing when isolated or when competing orders are suppressed (Eckberg et al., 2019, Lv et al., 2 Jan 2026).

5. Symmetry and Anisotropy of the Gap

Nematic-fluctuation-mediated interactions generally favor sign-preserving (“$s_{++}$”) or extended $s$-wave states, with the gap strongly modulated by the nematic form factor. Depending on the detailed momentum-space structure of the electron–nematic vertex (e.g., $f(\mathbf{k})$), the pairing symmetry can be highly anisotropic:

• For simple momentum-independent couplings, the $s$-wave channel is favored, often with an anisotropic angular profile, possibly with nodes or deep minima.
• For orbital-dependent or higher-rank coupling, $d$-wave or nematic $p$-wave states can be stabilized, especially when competing density wave fluctuations feedback into the quartic terms of the free energy (Kozii et al., 2018, Kang et al., 2016).
• In particular cases (e.g., FeSe thin films), absence of competing magnetism allows degenerate $s$/$d$ states, which are split by spin–orbit or interface symmetry-breaking effects, locking the pairing into either a nearly isotropic $s$-wave or a nodal nematic form depending on microscopic parameters (Kang et al., 2016).

Microscopically, the dominance of forward scattering renders the gap nodeless yet deeply anisotropic unless the form factor vanishes on portions of the Fermi surface—the so-called “cold spots” (Islam et al., 2024, Nag et al., 2024).

6. Non-Fermi-Liquid Behavior and Thermodynamic Anomalies

Nematic quantum criticality not only enhances pairing but also generates non-Fermi-liquid normal state phenomena:

• The self-energy near the nematic QCP shows strong momentum and frequency dependence, with robust non-Fermi-liquid scaling away from “protected” points on the Fermi surface (Lederer et al., 2016).
• Thermodynamic and spectroscopic observables deviate strongly from BCS expectations: the specific heat displays a pronounced maximum well below $T_c$, residual density of states at low energy, enhanced power-law behavior in quantities such as the penetration depth ($\delta\lambda(T) \sim T^{1.5}$), and a lack of sharp mean-field jumps at $T_c$ (Islam et al., 2024, Nag et al., 2024).
• Transport anomalies include bad-metallic conductivity and breakdown of coherent quasiparticles outside “cold” Fermi surface regions, confirmed by both DQMC simulations and experiment (Lederer et al., 2016).

These signatures differentiate nematic-fluctuation-mediated superconductors from systems dominated by either phonon or spin fluctuation pairing.

7. Broader Implications and Directions

Theoretically, nematic fluctuations universally enhance $T_c$ in any pairing channel selected by non-critical interactions. In 2D systems, the enhancement is particularly large, diverging logarithmically with the correlation length, while in 3D it is more modest (Lederer et al., 2014). Experimental confirmation of the one-to-one scaling between the nematic susceptibility (via, e.g., elastoresistivity or Nernst effect) and $T_c$ underlies the identification of nematicity as a primary pairing “glue” in these materials (Wuttke et al., 2022, Lv et al., 2 Jan 2026, Eckberg et al., 2019).

This mechanism is now central to the understanding of superconductivity in pure nematic systems (FeSe$_{1-x}$S$_x$, Ba$_{1-x}$Sr$_x$Ni$_2$As$_2$, Cu$_x$TiSe$_2$), intertwined orders (twisted bilayer graphene (Kozii et al., 2018)), and heavy fermion systems, and posits that tuning toward a pure nematic QCP is a viable route to high-$T_c$ superconductivity not accessible in spin- or charge-ordered quantum critical systems (Islam et al., 2024, Nag et al., 2024).

Alternative scenarios include the local stabilization of vestigial charge-$4e$ superconductivity at nematic domain walls, in which competing nematic order suppresses quartet order in the bulk but locally enables its condensation at structural boundaries; this has distinct spectroscopic fingerprints and is relevant for a broader class of multi-component superconductors (Hecker et al., 2023). The crucial experimental signatures for nematic-fluctuation-mediated pairing remain the angle-resolved gap structure, strain and disorder dependencies of $T_c$, and thermodynamic anomalies traceable to the underlying nematic criticality.