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arXiv:2507.16673 · Physical Review Letters

Nonlocal Correlation Effects in dc and Optical Conductivity of the Hubbard Model

Nagamalleswararao Dasari, Hugo U. R. Strand, Martin Eckstein, Alexander I. Lichtenstein, E. A. Stepanov

Published Feb 17, 2026• 3 citations
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Abstract

Conductivity is one of the most direct probes of electronic systems, yet its theoretical description remains challenging in the presence of strong nonlocal correlations. In this Letter, we analyze the conductivity of the half-filled single-band Hubbard model and identify the role of spatial correlations across the Mott transition. We show that in the correlated metallic regime, an accurate description of the conductivity requires not only the correct spectral function but also the inclusion of complex multielectron processes encoded in vertex corrections. The crossover to the Mott insulating regime is marked by a vanishing contribution of vertex corrections to the dc conductivity. However, in the Mott insulating case vertex corrections remain significant for the optical conductivity.

DOI
10.1103/gzwp-zd6t

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Conductivity is one of the most direct probes of electronic systems, yet its theoretical description remains challenging in the presence of strong nonlocal correlations. In this Letter, we analyze the conductivity of the half-filled single-band Hubbard model and identify the role of spatial correlations across the Mott transition. We show that in the correlated metallic regime, an accurate description of the conductivity requires not only the correct spectral function but also the inclusion of complex multielectron processes encoded in vertex corrections. The crossover to the Mott insulating regime is marked by a vanishing contribution of vertex corrections to the dc conductivity. However, in the Mott insulating case vertex corrections remain significant for the optical conductivity.
In this paper, we investigate the impact of nonlocal correlations on charge fluctuations in the two-dimensional single-band Hubbard model close to the Mott metal-to-insulator transition, employing the ladder dynamical vertex approximation. At half filling and for interaction strengths and temperatures where the system is in the Mott insulating phase, charge fluctuations are strongly suppressed. Under these conditions, dynamical mean-field theory (DMFT) calculations predict a strong enhancement of the charge susceptibility at small (electron or hole) doping. However, these DMFT results include only the effects of purely local correlations despite the importance of nonlocal correlations in two-dimensional systems. We have, hence, carried out ladder dynamical vertex approximation $(\mathrm{lD}\mathrm{\ensuremath{\Gamma}}\mathrm{A})$ simulations which allow for the inclusion of such nonlocal correlation effects while retaining the local ones of DMFT. Our $\mathrm{lD}\mathrm{\ensuremath{\Gamma}}\mathrm{A}$ numerical data show that close to half filling the large uniform charge susceptibility of DMFT is strongly suppressed by nonlocal fluctuations but gradually increases with (electron) doping. At a certain doping value, charge fluctuations eventually become larger in $\mathrm{lD}\mathrm{\ensuremath{\Gamma}}\mathrm{A}$ with respect to DMFT, indicating that the absence of nonlocal correlations underestimates the mobility of the charge carriers in this parameter regime. This metallization effect is also reflected in an enhancement of the $\mathrm{lD}\mathrm{\ensuremath{\Gamma}}\mathrm{A}$ kinetic and potential energies and a corresponding reduction of the (absolute value of the) $\mathrm{lD}\mathrm{\ensuremath{\Gamma}}\mathrm{A}$ Matsubara self-energy with respect to DMFT.
The dc conductivity for the t-J model is exactly obtained at t=J and for the Hubbard model in one dimension. Our calculation is carried out for arbitrary band filling by use of the finite-size-scaling method based on the Bethe-ansatz solution under twisted boundary conditions. The effective transport mass defined as the inverse of the conductivity serves as a sensitive probe of the metal-insulator transition. The effective mass is extremely enhanced near half-filling both for the t-J model at t=J and the repulsive Hubbard model, which is a direct evidence of the metal-insulator transition due to the electron correlation. We observe that the large antiferromagnetic coupling J in the t-J model increases the effective mass. On the other hand, the conductivity in the attractive Hubbard model exhibits quite different properties from the repulsive models.
We analyze the role of spatial electronic correlations and, in particular, of the magnetic fluctuations in Mott insulators. A half-filled Hubbard model is solved at large strength of the repulsion UUU on a two-dimensional square lattice using an advanced diagrammatic approach capable of going beyond Hartree-Fock and single-site dynamical mean-field theories. We show that at high temperatures, when the magnetic fluctuations are weak, the electronic self-energy of the system is mainly local and is well reproduced by the atomic (Hubbard-I) approximation. Upon lowering the temperature toward the magnetically ordered phase, the nonlocality of the self-energy becomes crucial in determining the momentum dispersion of the Hubbard bands and the Green's function zeros. We therefore establish a precise link between Luttinger surface, nonlocal correlations and spectral properties of the Hubbard bands.
We numerically study optical conductivity σ(ω)σ(ω)σ(ω) near the "antiferromagnetic" phase transition in the square-lattice Hubbard model at half filling. We use a cluster dynamical mean field theory and calculate conductivity including vertex corrections, and to this end, we have reformulated the vertex corrections in the antiferromagnetic phase. We find that the vertex corrections change various important details in temperature- and ωωω-dependencies of conductivity in the square lattice, and this contrasts sharply the case of the Mott transition in the frustrated triangular lattice. Generally, the vertex corrections enhance variations in the ωωω-dependence, and sharpen the Drude peak and a high-ωωω incoherent peak in the paramagnetic phase. They also enhance the dip in σ(ω)σ(ω)σ(ω) at ωωω=0 in the antiferromagnetic phase. Therefore, the dc conductivity is enhanced in the paramagnetic phase and suppressed in the antiferromagnetic phase, but this change occurs slightly below the transition temperature. We also find a temperature region above the transition temperature in which the dc conductivity shows an insulating behavior but σ(ω)σ(ω)σ(ω) retains the Drude peak, and this region is stabilized by the vertex corrections. We also investigate which fluctuations are important in the vertex corrections and analyze momentum dependence of the vertex function in detail.
Elucidating the physics of the single-orbital Hubbard model in its intermediate-coupling regime is a key missing ingredient to our understanding of metal-insulator transitions in real materials. Using recent nonperturbative many-body techniques that are able to interpolate between the spin-fluctuation-dominated Slater regime at weak coupling and the Mott insulator at strong coupling, we obtain the momentum-resolved spectral function in the intermediate regime and disentangle the effects of antiferromagnetic fluctuations and local electronic correlations in the formation of an insulating state. This allows us to identify the Slater and Heisenberg regimes in the phase diagram, which are separated by a crossover region of competing spatial and local electronic correlations. We identify the crossover regime by investigating the behavior of the local magnetic moment, shedding light on the formation of the insulating state at intermediate couplings.
Results for the optical conductivity and resistivity of the Hubbard model in infinite spatial dimensions are presented. At half-filling we observe a gradual crossover from a metal with a Drude peak at ω = 0 in the optical conductivity to an insulator as a function of U for temperatures above the antiferromagnetic phase transition. When doped, the "insulator" becomes a Fermi liquid with a corresponding temperature dependence of the optical conductivity and resistivity. We find a T 2 -coefficient in the low-temperature resistivity which suggests that the carriers in the system acquire a considerable mass-enhancement due to the strong local correlations. At high temperatures, a crossover into a semi-metallic regime takes place.