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Vertex corrections to the optical conductivity in the square lattice Hubbard model and their role for the high-temperature transport

J. Vučičević, J. Kokalj, Rok Žitko, Nils Wentzell, D. Tanasković, Jernej Mravlje

Published Nov 20, 2018• 0 citations
OpenAlex

Abstract

Calculating transport properties presents a great challenge in the study of strongly correlated electrons, particularly in two dimensions where the non-local processes and the vertex corrections are important, but remain poorly understood. We study the Hubbard model on the square lattice in the strongly correlated regime. We use several complementary state-of-the-art numerical methods to disentangle the different effects and resolve the problem conclusively at high temperatures. In this regime, non-local correlations and long-distance quantum fluctuations have little impact, yet the vertex corrections survive and substantially suppress resistivity. Our study becomes particularly relevant in the light of recent experiments on optical lattices that provide realizations of the two-dimensional Hubbard model in a similar parameter regime.

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Recent experiments on cold atoms in optical lattices allow for a quantitative comparison of the measurements to the conductivity calculations in the square lattice Hubbard model. However, the available calculations do not give consistent results, and the question of the exact solution for the conductivity in the Hubbard model remained open. In this Letter, we employ several complementary state-of-the-art numerical methods to disentangle various contributions to conductivity and identify the best available result to be compared to experiment. We find that, at relevant (high) temperatures, the self-energy is practically local, yet the vertex corrections remain rather important, contrary to expectations. The finite-size effects are small even at the lattice size 4×4, and the corresponding Lanczos diagonalization result is, therefore, close to the exact result in the thermodynamic limit.

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Publisher's Note: Cluster dynamical mean field theory study of antiferromagnetic transition in the square-lattice Hubbard model: Optical conductivity and electronic structure [Phys. Rev. B94, 085110 (2016)]

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We numerically study optical conductivity $σ(ω)$ 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 $ω$-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 $ω$-dependence, and sharpen the Drude peak and a high-$ω$ incoherent peak in the paramagnetic phase. They also enhance the dip in $σ(ω)$ at $ω$=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 $σ(ω)$ 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.

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Calculating transport properties presents a great challenge in the study of strongly correlated electrons, particularly in two dimensions where the non-local processes and the vertex corrections are important, but remain poorly understood. We study the Hubbard model on the square lattice in the strongly correlated regime. We use several complementary state-of-the-art numerical methods to disentangle the different effects and resolve the problem conclusively at high temperatures. In this regime, non-local correlations and long-distance quantum fluctuations have little impact, yet the vertex corrections survive and substantially suppress resistivity. Our study becomes particularly relevant in the light of recent experiments on optical lattices that provide realizations of the two-dimensional Hubbard model in a similar parameter regime.
Recent experiments on cold atoms in optical lattices allow for a quantitative comparison of the measurements to the conductivity calculations in the square lattice Hubbard model. However, the available calculations do not give consistent results, and the question of the exact solution for the conductivity in the Hubbard model remained open. In this Letter, we employ several complementary state-of-the-art numerical methods to disentangle various contributions to conductivity and identify the best available result to be compared to experiment. We find that, at relevant (high) temperatures, the self-energy is practically local, yet the vertex corrections remain rather important, contrary to expectations. The finite-size effects are small even at the lattice size 4×4, and the corresponding Lanczos diagonalization result is, therefore, close to the exact result in the thermodynamic limit.
High-temperature bad-metal transport has been recently studied both theoretically and in experiments as one of the key signatures of strong electronic correlations. Here we use the dynamical mean field theory and its cluster extensions, as well as the finite-temperature Lanczos method to explore the influence of lattice frustration on the thermodynamic and transport properties of the Hubbard model at high temperatures. We consider the triangular and the square lattices at half-filling and at 15% hole doping. We find that for $T\ensuremath{\gtrsim}1.5t$ the self-energy becomes practically local, while the finite-size effects become small at lattice size $4\ifmmode\times\else\texttimes\fi{}4$ for both lattice types and doping levels. The vertex corrections to optical conductivity, which are significant on the square lattice even at high temperatures, contribute less on the triangular lattice. We find approximately linear temperature dependence of dc resistivity in doped Mott insulator for both types of lattices.
The conductivity of the two-dimensional Hubbard model is particularly relevant for high-temperature superconductors. Vertex corrections are expected to be important because of strongly momentum-dependent self-energies. To attack this problem, one must also take into account the Mermin--Wagner theorem, the Pauli principle, and crucial sum rules in order to reach nonperturbative regimes. Here, we use the two-particle self-consistent approach that satisfies these constraints. This approach is reliable from weak to intermediate coupling. A functional derivative approach ensures that vertex corrections are included in a way that satisfies the fff-sum rule. The two types of vertex corrections that we find are the antiferromagnetic analogs of the Maki--Thompson and Aslamasov--Larkin contributions of superconducting fluctuations to the conductivity but, contrary to the latter, they include nonperturbative effects. The resulting analytical expressions must be evaluated numerically. The calculations are impossible unless a number of advanced numerical algorithms are used. These algorithms make extensive use of fast Fourier transforms, cubic splines, and asymptotic forms. A maximum entropy approach is specially developed for analytical continuation of our results. These algorithms are explained in detail in the appendices. The numerical results are for nearest-neighbor hoppings. In the pseudogap regime induced by two-dimensional antiferromagnetic fluctuations, the effect of vertex corrections is dramatic. Without vertex corrections the resistivity increases as we enter the pseudogap regime. Adding vertex corrections leads to a drop in resistivity, as observed in some high-temperature superconductors. At high temperatures, the resistivity saturates at the Ioffe--Regel limit. At the quantum critical point and beyond, the resistivity displays both linear and quadratic temperature dependence and there is a correlation between the linear term and the superconducting transition temperature. A hump is observed in the mid-infrared range of the optical conductivity in the presence of antiferromagnetic fluctuations.
We numerically study optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ 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 $\ensuremath{\omega}$ 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 $\ensuremath{\omega}$ dependence, and sharpen the Drude peak and a high-$\ensuremath{\omega}$ incoherent peak in the paramagnetic phase. They also enhance the dip in $\ensuremath{\sigma}(\ensuremath{\omega})$ at $\ensuremath{\omega}=0$ 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 $\ensuremath{\sigma}(\ensuremath{\omega})$ 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.
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.
Dynamical properties are notoriously difficult to compute in numerical treatments of the Fermi-Hubbard model, especially in two spatial dimensions. However, they are essential in providing us with insight into some of the most important and less well-understood phases of the model, such as the pseudogap and strange metal phases at relatively high temperatures, or unconventional superconductivity at lower temperatures, away from the commensurate filling. Here, we use the numerical linked-cluster expansions to compute spin and charge optical conductivities of the model at different temperatures and strong interaction strengths via the exact real-time-dependent correlation functions of the current operators. We mitigate systematic errors associated with having a limited access to the long-time behavior of the correlators by introducing fits and allowing for non-zero Drude weights when appropriate. We compare our results to available data from optical lattice experiments and find that the Drude contributions can account for the theory-experiment gap in the DC spin conductivity of the model at half filling in the strong-coupling region. Our method helps paint a more complete picture of the conductivity in the two-dimensional Hubbard model and opens the door to studying dynamical properties of quantum lattice models in the thermodynamic limit.