Long-range (LR) interactions are no longer just a theoretical idea: they can be engineered in several low-dimensional platforms and offer a new way to control heat transport. At the same time, they push beyond the standard picture developed mainly for short-range, momentum-conserving lattices. In this review, we consider one-dimensional Fermi-Pasta-Ulam-Tsingou (FPUT)-type lattices where LR effects enter mainly through a quartic anharmonic coupling that decays as a power law, characterized by an exponent \sigma. Our goal is to explain, in a unified way, how LR anharmonicity changes microscopic energy exchange, collective dynamics, and the macroscopic scaling laws of heat conduction--while also clarifying what is well established and what is still debated.
We first introduce the main models and the quantities used to discribe transport in LR-FPUT systems. These include the nonequilibrium steady-state thermal conductivity \kappa(N), the equilibrium Green–Kubo (GK) approach based on the heat-current autocorrelation C_JJ(t), and spatiotemporal correlations of heat/energy fluctuations \rho_Q(m,t), together with the scaling of the mean-squared displacement (MSD). A key technical point is that, in LR lattices, the microscopic heat current contains explicit nonlocal contributions. As a result, different (formally related) implementations of the current may lead to noticeable differences at finite system sizes and finite times. To extract the intrinsic transport behavior, we therefore stress practical checks: (i) the stability of scaling exponents against boundary and driving protocols, (ii) validation among all the three methods, i.e., \kappa(N), C_JJ(t), and \rho_Q(m,t), and (iii) careful control of finite-size and finite-time crossovers.
Based on these methods, we review representative \sigma-dependent transport regimes. (i) The special point \sigma=2 is central and remains controversial. Early work reported ballistic-like behavior under certain boundary conditions. More recent studies using more controlled numerical method (e.g., periodic driving together with correlation scaling) tend to support strongly enhanced superdiffusion, with an effective divergence exponent around \alpha \simeq 0.7, rather than ballistic transport. The most consistent physical picture links this enhancement to weak nonintegrability and the presence of long-lived coherent carriers, such as mobile breathers or related nonlinear excitations, while recognizing that finite-size results are unusually sensitive to boundary implementation and to how the heat current is defined. We summarize the current consensus and the main remaining disagreements, and we argue that ballistic vs strong superdiffusion should be decided by consistency across multiple observables, not by temperature profiles alone. (ii) In the weak-LR regime (1 \leq \sigma \leq 3), a striking result is a near-diffusive window around \sigma \simeq 1.25. This is conceptually important because it suggests a possible route toward quasi-normal transport within Hamiltonian, momentum-conserving one-dimensional lattices, which usually show anomalous conduction in the short-range case. We discuss how this challenges the common intuition that momentum conservation implies anomalous transport, but we also emphasize that the current evidence should be viewed as a strong candidate that still needs careful confirmation against finite-size crossovers and method dependence. (iii) In the strong-LR regime (0 \leq \sigma \leq 1), simulations show clear antipersistent (negative) structures in C_JJ(t) and a trend toward slower transport, sometimes described as subdiffusive-like transport. (iv) We also discuss an inverse LR coupling design LR FPUT model that can invoke ballistic transport, suggesting that LR nonlocality combined with structural modulation may open new transport channels and may be useful for thermal rectification.
Finally, we outline key open directions: establishing reliable finite-size scaling to distinguish genuine \sigma-driven dynamical transitions from broad crossovers; mapping transport regimes in a (\sigma,T,\textnonlinearity) phase diagram; and identifying the microscopic carriers (phonons, stationary/mobile breathers) and their scattering dynamics that underpin the observed superdiffusive, near-diffusive, and antipersistent slow-transport behaviors. These needs also connect to experiments, since LR couplings can be realized in platforms such as Coulomb crystals and frustrated magnets, and LR interaction engineering has already enabled efficient thermal rectification--suggesting practical routes to tune heat conduction and build thermal diodes/switches based on nonlocality and nonlinearity.