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Analytical expression for $\pi$-ton vertex contributions to the optical conductivity
by Juraj Krsnik, Anna Kauch, Karsten Held
Submission summary
| Authors (as registered SciPost users): | Anna Kauch · Juraj Krsnik |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202409_00019v1 (pdf) |
| Date submitted: | 2024-09-17 14:46 |
| Submitted by: | Krsnik, Juraj |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Vertex corrections from the transversal particle-hole channel, so-called $\pi$-tons, are generic in models for strongly correlated electron systems and can lead to a displaced Drude peak (DDP). Here, we derive the analytical expression for these $\pi$-tons, and how they affect the optical conductivity as a function of correlation length $\xi$, fermion lifetime $\tau$, temperature $T$, and coupling strength to spin or charge fluctuations $g$. In particular, for $T\rightarrow T_c$, the critical temperature for antiferromagnetic or charge ordering, the dc vertex correction is algebraic $\sigma_{VERT}^{dc}\propto \xi \sim (T-T_c)^{-\nu}$ in one dimension and logarithmic $\sigma_{VERT}^{dc}\propto \ln\xi \sim \nu \ln (T-T_c)$ in two dimensions. Here, $\nu$ is the critical exponent for the correlation length. If we have the exponential scaling $\xi \sim e^{1/T}$ of an ideal two-dimensional system, the DDP becomes more pronounced with increasing $T$ 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: $\omega_{DDP} \sim 1/\tau$. These characteristic dependencies can guide experiments to evidence $\pi$-tons in actual materials.
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Report
The paper analytically evaluates the pi-ton vertex corrections to optical conductivity and discusses the implications of those corrections for the displaced Drude peak phenomenology. Earlier work by some of the authors has established the existence of pi-ton vertex corrections, the novel idea of this work is to introduce additional approximations that allow for analytical evaluation.
Overall, the topic of the investigation is interesting and progress obtained from a simplified evaluation of momentum integrals significant, but I feel there are parts where the discussion could be improved.
Concerning the derivation, I do not understand the separation into hole and electron parts well. Paper says that taking w_n > 0 implies a restriction concerning which fermionic frequencies contribute. Is the statement that contributions from other frequencies vanishes strictly or is this an approximation? If the latter, is this approximation necessary, is it better at lower temperatures? What does this imply for the vanishing bosonic Matsubra frequency?
To me it is insufficiently clear what does "adaptive integration" result really mean. Is Ornstein Zernike form for the susceptibility assumed also there and the analytical approximation refers mainly to simplification of momentum integrals? A reader could benefit by more explicit discussion of where approximations are.
Given the simple forms assumed for the Green's function and vertex in analytics, could one write the expressions for them on the real frequency axis and perform the integrals there?
Concerning the physics, the paper gives a criterion for the occurrence of the Drude peak in terms of that the vertex correction needs to exceed a given threshold. On the other hand, in the low T limit it also gives unphysical negative conductivity. What breaks down? Do not all approximations introduced by the authors become better at low temperature?
Concerning the Ward identity sum-rule, perhaps one could explicitly write that the vanishing of zero-Matsubara frequency value is consistent with the absence of momentum dependence of self-energy. Which aspect of the vertex function in the approximation guarantees that sigma_vert(w) indeed behaves in this way?
Authors consider 1d and 2d. What happens in 3d? Some of the materials that display displaced Drude peak phenomenology actually exist in 3d.
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Report #1 by Simone Fratini (Referee 1) on 2024-11-15 (Invited Report)
Report
This paper addresses theoretically one of the possible microscopic scenarios leading to displaced Drude peaks (DDP), a widespread experimental phenomenon occurring in a variety of quantum materials. The focus here is on a specific class of vertex corrections arising in the particle-hole channel, that have previously been shown by the authors to affect the optical conductivity beyond the common "bubble" approximation.
Numerous examples have been analyzed in recent times by the authors as well as by other groups, illustrating the realization of this general mechanism in different microscopic models and dimensionalities. The present long paper rationalizes these observations, setting up an approximate theory that: (i) provides a unified framework to interpret the existing data; (ii) serves as a useful parametrization of the numerical results, including a formula (Eq.21) that could be applied directly to the interpretation of experiments; (iii) provides indications on how to tweak the microscopic parameters in order to match the experimental observations.
The paper is written in a very pedagogical format, that is well suited for this journal. The topic itself also nicely matches the recent trend set by related DDP studies also published in SciPost.
Although I strongly recommend publication of this work, I have some general and few more specific questions:
1. Are the vertex corrections addressed by the authors in any way related to Bang and Kotliar's corrections to the optical conductivity found via the slave boson technique [PRB 48 9898 (1993)], originating there from the incoherent motion of the charge carriers and their coupling to low-energy spin excitations?
2. Eqs. 1-2 bear some similarity with the starting point of the phenomenological theory of strange metals given by S. Caprara and collaborators [Communications Physics 5 (1), 10 (2022)]. Can the authors show whether and how the pi-tons would affect the temperature dependence of the resistivity (i.e. illustrate the impact of Eq. 22 on an otherwise T^2-dependence)? Or does the Fermi liquid assumption underlying their approach automatically lead to a Fermi liquid-like behavior in transport?
3. l.145, even though it has appeared elsewhere, a figure depicting the corresponding Feynman diagrams could be useful here
4. Based on Eq. 21, is it possible to obtain an anomalous frequency decay of the Drude peak in the optical conductivity? In particular, could the authors comment if this can be made to match the shape that is often seen in experiments when DDPs are absent, and that was phenomenologically analyzed for example in [Nature Communications 14 (1), 3033 (2023)]?
5. In Eq. 22, the pi-ton contribution always suppresses the d.c. conductivity, while in other references cited in this mannuscript it was found that pi-tons can either suppress or enhance the conductivity depending on the temperature range. Could the authors comment on this?
6. l.290 is it obvious that the same parametric form of the lifetime assumed here should hold both in 1D and 2D?
7. l.362 this should probably read "at the beginning of Section 3.2"
8. l.369 what is the extra microscopic ingredient used in Ref. 32 that is missed here, and why should it be more correct than the original assumption Eq. 25, therefore motivating a modification of the temperature behavior of the correlation length?
9. From the comparison of the analytical Eqs. 7 and 21, is it possible to derive precise conditions for the emergence of a DDP? And for a possible coexistence of a Drude peak together with a finite-frequency maximum?
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