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arXiv:2409.11158 · SciPost Physics

Analytical expression for $\pi$-ton vertex contributions to the optical conductivity

Juraj Krsnik, Anna Kauch, Karsten Held

Published Apr 25, 2025• 0 citations
arXivHTMLPDF

Abstract

Vertex corrections from the transversal particle-hole channel, so-called \pi π -tons, are gene- ric in models for strongly correlated electron systems and can lead to a displaced Drude peak (DDP). Here, we derive the analytical expression for these \pi π -tons, and how they affect the optical conductivity as a function of correlation length \xi ξ , fermion lifetime \tau τ , temperature T T , and coupling strength to spin or charge fluctuations g g . In particular, for T→ T_c T → T c , the critical temperature for antiferromagnetic or charge ordering, the dc vertex correction is algebraic \sigma_{VERT}^{dc}\propto \xi \sim (T-T_c)^{-\nu} σ V E R T d c ∝ ξ ∼ ( T − T c ) − ν in one dimension and logarithmic \sigma_{VERT}^{dc}\propto \ln\xi \sim \nu \ln (T-T_c) σ V E R T d c ∝ ln ξ ∼ ν ln ( T − T c ) in two dimensions. Here, \nu ν is the critical exponent for the correlation length. If we have the exponential scaling \xi \sim e^{1/T} ξ ∼ e 1 / T of an ideal two-dimensional system, the DDP becomes more pronounced with increasing T T but fades away at low temperatures where only a broadening of the Drude peak remains, as it is observed experimentally, Further, we find the maximum of the DPP to be given by the inverse lifetime: \omega_{DDP

DOI
10.21468/scipostphys.18.4.138

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Analytical expression for π\piπ-ton vertex contributions to the optical conductivity
Vertex corrections from the transversal particle-hole channel, so-called \pi π -tons, are gene- ric in models for strongly correlated electron systems and can lead to a displaced Drude peak (DDP). Here, we derive the analytical expression for these \pi π -tons, and how they affect the optical conductivity as a function of correlation length \xi ξ , fermion lifetime \tau τ , temperature T T , and coupling strength to spin or charge fluctuations g g . In particular, for T→ T_c T → T c , the critical temperature for antiferromagnetic or charge ordering, the dc vertex correction is algebraic \sigma_{VERT}^{dc}\propto \xi \sim (T-T_c)^{-\nu} σ V E R T d c ∝ ξ ∼ ( T − T c ) − ν in one dimension and logarithmic \sigma_{VERT}^{dc}\propto \ln\xi \sim \nu \ln (T-T_c) σ V E R T d c ∝ ln ξ ∼ ν ln ( T − T c ) in two dimensions. Here, \nu ν is the critical exponent for the correlation length. If we have the exponential scaling \xi \sim e^{1/T} ξ ∼ e 1 / T of an ideal two-dimensional system, the DDP becomes more pronounced with increasing T T but fades away at low temperatures where only a broadening of the Drude peak remains, as it is observed experimentally, Further, we find the maximum of the DPP to be given by the inverse lifetime: \omega_{DDP
Correlated electron systems often show strong bosonic fluctuations, e.g., of antiferromagnetic nature, around a large wave vector such as $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi}...)$. These fluctuations can give rise to vertex corrections to the optical conductivity through the (transversal) particle-hole channel, coined $\ensuremath{\pi}$-ton contributions. Previous numerical results differed qualitatively on how such vertex corrections alter the optical conductivity. Here we clarify that $\ensuremath{\pi}$-ton vertex corrections lead to a displaced Drude peak, by studying the weakly correlated regime of a one-dimensional single-orbital Hubbard model. The proximity and enhancement of the effect when approaching a phase transition of, e.g., antiferromagnetic nature can be utilized for discriminating $\ensuremath{\pi}$-tons in experiments from other physics leading to a displaced Drude peak.
Antiferromagnetic or charge density wave fluctuations couple with light through the recently discovered $\ensuremath{\pi}$-ton contribution to the optical conductivity, and quite generically constitute the dominant vertex corrections in low-dimensional correlated electron systems. Here we study the arguably simplest version of these $\ensuremath{\pi}$-tons based on the semianalytical random phase approximation (RPA) ladder in the transversal particle-hole channel. The vertex corrections to the optical conductivity are calculated directly for real frequencies. We validate that the RPA qualitatively reproduces the $\ensuremath{\pi}$-ton vertex corrections to the Drude peak in the Hubbard model. Depending on the temperature we find vertex corrections to broaden or sharpen the Drude peak.
The optical conductivity contains information about energy absorption and the underlying physical processes. In finite-dimensional systems, vertex corrections to the bare bubble need to be considered, which is a computationally challenging task. Recent numerical studies showed that in the weak-coupling limit, near an ordering instability with wave-vector $\ensuremath{\pi}$, the vertical ladder describing particle-hole pairs interacting via the exchange of this wave vector becomes the dominant vertex correction. The corresponding Maki-Thompson-like diagram has been dubbed $\ensuremath{\pi}$-ton. Here we add the $\ensuremath{\pi}$-ton ladder vertex correction to dynamical mean-field theory estimates of the optical conductivity. By performing calculations on the Kadanoff-Baym contour, we reveal the characteristic spectral signatures of the $\ensuremath{\pi}$-tons and their evolution under nonequilibrium conditions. We consider interaction quenches of the weakly correlated Hubbard model near the antiferromagnetic phase boundary and analyze the evolution of the Drude and $\ensuremath{\pi}$-ton features. While the bubble contribution to the optical conductivity is found to thermalize rapidly, after some oscillations with frequencies related to the local spectral function, the $\ensuremath{\pi}$-ton contribution exhibits a slower evolution. We link this observation to the prethermalization phenomenon which has been previously studied in weakly interacting, quenched Hubbard models.
The optical conductivity contains relevant information on the properties of correlated electron systems. In infinite dimensions, where dynamical mean field theory becomes exact, vertex corrections can be neglected and the conductivity computed from particle-hole bubbles. An interesting question concerns the nature and effect of the most relevant vertex corrections in finite-dimensional systems. A recent numerical study showed that the dominant vertex correction near an ordering instability with wave vector $\ensuremath{\pi}$ comes from a vertical ladder, analogous to the Maki-Thompson diagram. Since the random phase approximation version of this ladder diagram, dubbed $\ensuremath{\pi}$-ton, can be easily evaluated, this suggests a simple procedure for incorporating antiferromagnetic or charge density wave fluctuations into dynamical mean field estimates of the optical conductivity and related susceptibilities. We implement this procedure for the half-filled Hubbard model, considering the $\ensuremath{\pi}$-ton, a double-ladder extension of the $\ensuremath{\pi}$-ton and Aslamazov-Larkin-type diagrams, and reveal the spectral signatures of these vertex corrections.
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.