2206/2206.01215.md

Title: A simple model for strange metallic behavior

URL Source: https://arxiv.org/html/2206.01215

Markdown Content: Sutapa Samanta samants2@wwu.eduDepartment of Physics and Astronomy, Western Washington University, 516 High Street, Bellingham, Washington 98225,USA Hareram Swain dhareram1993@physics.iitm.ac.inDepartment of Physics, Indian Institute of Technology Madras, Chennai 600036, India Benoît Douçot doucot@lpthe.jussieu.frLaboratoire de Physique Théorique et Hautes Energies,Sorbonne Université and CNRS UMR 7589, 4 place Jussieu, 75252 Paris Cedex 05, France Giuseppe Policastro giuseppe.policastro@ens.frLaboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris Cité, 24 rue Lhomond, F-75005 Paris, France Ayan Mukhopadhyay ayan@physics.iitm.ac.inDepartment of Physics, Indian Institute of Technology Madras, Chennai 600036, India

(September 20, 2024)

Abstract

A refined semi-holographic non-Fermi liquid model, in which carrier electrons hybridize with operators of a holographic critical sector, has been proposed recently for strange metallic behavior. The model, consistently with effective theory approach, has two couplings whose ratio is related to the doping. We explain the origin of the linear-in-T resistivity and strange metallic behavior as a consequence of the emergence of a universal form of the spectral function which is independent of the model parameters when the ratio of the two couplings take optimal values determined only by the critical exponent. This universal form fits well with photoemission data of copper oxide samples for under/optimal/over-doping with a fixed exponent over a wide range of temperatures. We further obtain a refined Planckian dissipation scenario in which the scattering time τ=f⋅ℏ/(k B⁢T)𝜏⋅𝑓 Planck-constant-over-2-pi subscript 𝑘 𝐵 𝑇\tau=f\cdot\hbar/(k_{B}T)italic_τ = italic_f ⋅ roman_ℏ / ( italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T ), with f 𝑓 f italic_f being 𝒪⁢(1)𝒪 1\mathcal{O}(1)caligraphic_O ( 1 ) at strong coupling, but 𝒪⁢(10)𝒪 10\mathcal{O}(10)caligraphic_O ( 10 ) at weak coupling.

I Introduction

An elegant explanation for universal unconventional transport phenomena of strange metals is based on the existence of a quantum critical point (QCP) at optimal doping Sachdev and Keimer (2011); Tailleffer; Michon et al. (2019); Hayes et al. (2021). It has been further emphasized via many tractable models that the quantum critical degrees of freedom may also provide the key to the emergence of unconventional superconducting and insulating phases. The spectral function measured via angle-resolved photoemission spectroscopy (ARPES) indeed provides evidence for absence of quasi-particles at the Fermi surface Damascelli et al. (2003); Vishik et al. (2010); Reber et al. (2015); Senthil (2008); Lee (2018); Varma (2020). Of late, there has been remarkable progress in achieving direct access to the quantum critical point via the application of a strong magnetic field Butch et al. (2012) and also a critical current Jung et al. (2018). However, it has been argued that an effective theory for universal transport properties of strange metals should also incorporate the degrees of freedom at intermediate scales, and which interacts with the infrared critical sector. This hypothesized feature is called Mottness Phillips (2006); Choy et al. (2008); Sakai et al. (2009); Yamaji and Imada (2011a, b); Hong and Phillips (2012).

Another influential paradigm for understanding transport phenomena in strange metals is Planckian dissipation Zaanen (2004); Sachdev (2011); Bruin JA and AP (2013); Hartnoll et al. (2016); Chowdhury et al. (2021); Hartnoll and Mackenzie (2021). Here, the absence of quasi-particles is assumed to coincide with the Mott-Ioffe-Regel (MIR) limit in which the scattering length approaches the interatomic spacing. It is also assumed that all intrinsic scales disappear as the system is close to a QCP. Therefore, a universal strong interaction limit emerges where the thermal energy k B⁢T subscript 𝑘 𝐵 𝑇 k_{B}T italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T is the only energy scale, and therefore the dissipation time-scale should be 𝒪⁢(ℏ/(k B⁢T))𝒪 Planck-constant-over-2-pi subscript 𝑘 𝐵 𝑇\mathcal{O}(\hbar/(k_{B}T))caligraphic_O ( roman_ℏ / ( italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T ) ). This readily accounts for the linear-in-T 𝑇 T italic_T dc resistivity at low temperatures. However, as discussed later, the abundant experimental data for heavy fermion compounds indicate Taupin and Paschen (2022) that this naive version of the paradigm may fail.

In this work, we analyze a simple effective model of strange metallic behavior proposed in Doucot et al. (2021) that captures features of Mottness. It incorporates lattice band electrons hybridizing with fermionic operators of a quantum critical sector, and can be viewed as an effective theory of the conducting electrons of the lower Hubbard band with only two couplings. It was shown that there always exists an optimal ratio of these two couplings at which the model exhibits universal transport properties, especially linear-in-T resistivity, over a very wide range of temperatures, irrespective of all parameters provided the critical exponent lies within a certain range. By optimal ratio, we simply imply that the ratio of couplings needs to be tuned such that we obtain linear-in-T resistivity. Here we will show that the fundamental origin of this feature in this model is the emergence of a universal form of the spectral function as a function of the frequency near the Fermi surface over a large range of temperatures. As discussed below, this universal behavior emerges at finite (non-vanishing) temperatures due to a competition between two different types of interactions, and this cannot be obtained from the self-energy contributions due to interactions with the critical sector alone. We also emphasize that the universal form is needed even at large frequencies to reproduce strange-metallic transport properties (although it is valid only near the Fermi surface) due to the underlying non-Fermi liquid nature of the system.

The critical sector of this model can be best viewed as a homogeneous bath of Sachdev-Ye-Kitaev (SYK) quantum dots Sachdev and Ye (1993); Kitaev and Suh (2018), each of which has a scaling symmetry in the large N 𝑁 N italic_N and infrared limit, and admits a dual holographic description in the form of a black hole in two-dimensional anti-de Sitter (A⁢d⁢S 2 𝐴 𝑑 subscript 𝑆 2 AdS_{2}italic_A italic_d italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT) space. The fermionic operators of this emergent holographic infrared conformal field theory (IR-CFT) are dual to free Dirac electrons living in the A⁢d⁢S 2 𝐴 𝑑 subscript 𝑆 2 AdS_{2}italic_A italic_d italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT black hole spacetimes whose masses determined by the dual scaling dimensions Lee (2009); Liu et al. (2011); Cubrovic et al. (2009); Faulkner et al. (2011, 2010); Iqbal et al. (2012).

The model is a special case of the more general semi-holographic semi-local non-Fermi liquids constructed in Mukhopadhyay and Policastro (2013) as extension of the simple Polchinski-Faulkner setup Faulkner and Polchinski (2011) in which only linear hybridization of the lattice electrons with fermionic operators of the critical sector was considered. In Mukhopadhyay and Policastro (2013); Doucot et al. (2017) it was shown that, in a suitable large N 𝑁 N italic_N limit, the low frequency behavior of the spectral function on the Fermi surface is given only by the linear hybridization considered in the Polchinski-Faulkner scenario even after incorporating arbitrary interactions between lattice electrons, and also between lattice electrons and the holographic IR CFT sector; thus implying the existence of generalized quasi-particles on the Fermi surface.

In Doucot et al. (2021), it was proposed that as a special case of Mukhopadhyay and Policastro (2013); Doucot et al. (2017), one can consider two irrelevant couplings on the Fermi surface which give rise to two distinct local self-energy terms, one of which is Fermi-liquid like and another given by the finite temperature fermionic holographic Green’s function. Since the latter term originates from the bath of SYK-type quantum dots, and we assume a smeared homogeneous distribution of these dots, the corresponding coupling is related to the density of these dots, and thus the doping strength.

Here we will explain the origin of the strange metallic behavior seen in Doucot et al. (2021) from the emergence of a universal form of the spectral function as a function of the frequency near the Fermi surface, which is independent of the temperature, the critical exponent and other model parameters, and which also fits nodal ARPES data well. Furthermore, we will obtain a refined Planckian dissipation paradigm from our effective approach in consistency with experimental data.

We will use units where ℏ=k B=1 Planck-constant-over-2-pi subscript 𝑘 𝐵 1\hbar=k_{B}=1 roman_ℏ = italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = 1.

II The refined semi-holographic model

In the model proposed in Doucot et al. (2021), the conducting electron (c^^𝑐\hat{c}over^ start_ARG italic_c end_ARG) interacts with a fermionic operator χ^CFT subscript^𝜒 CFT\hat{\chi}_{\rm CFT}over^ start_ARG italic_χ end_ARG start_POSTSUBSCRIPT roman_CFT end_POSTSUBSCRIPT of the holographic IR-CFT, and also with other filled lattice band electrons (f^^𝑓\hat{f}over^ start_ARG italic_f end_ARG). Crucially, all interactions scale with N 𝑁 N italic_N in a specific way. The Hamiltonian has the following generic form in the large N 𝑁 N italic_N limit:

H^=^𝐻 absent\displaystyle\hat{H}=over^ start_ARG italic_H end_ARG =∑𝐤 ϵ(𝐤)c^†(𝐤)c^(𝐤)+N∑𝐤(g c^†(𝐤)χ^CFT(𝐤)+c.c.+⋯)\displaystyle\sum_{\mathbf{k}}\epsilon(\mathbf{k})\hat{c}^{\dagger}(\mathbf{k}% )\hat{c}(\mathbf{k})+N\sum_{\mathbf{k}}\left(g\hat{c}^{\dagger}(\mathbf{k})% \hat{\chi}{\text{CFT}}(\mathbf{k})+c.c.+\cdots\right)∑ start_POSTSUBSCRIPT bold_k end_POSTSUBSCRIPT italic_ϵ ( bold_k ) over^ start_ARG italic_c end_ARG start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( bold_k ) over^ start_ARG italic_c end_ARG ( bold_k ) + italic_N ∑ start_POSTSUBSCRIPT bold_k end_POSTSUBSCRIPT ( italic_g over^ start_ARG italic_c end_ARG start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( bold_k ) over^ start_ARG italic_χ end_ARG start_POSTSUBSCRIPT CFT end_POSTSUBSCRIPT ( bold_k ) + italic_c . italic_c . + ⋯ ) +N 2 H^IR CFT+∑i,j,k(λ i⁢j⁢k,𝐤 1,𝐤 2,𝐤 3\displaystyle+N^{2}\hat{H}{\text{IR CFT}}+\sum_{i,j,k}\left(\lambda_{ijk,% \mathbf{k}{1},\mathbf{k}{2},\mathbf{k}{3}}\right.+ italic_N start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT over^ start_ARG italic_H end_ARG start_POSTSUBSCRIPT IR CFT end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_i , italic_j , italic_k end_POSTSUBSCRIPT ( italic_λ start_POSTSUBSCRIPT italic_i italic_j italic_k , bold_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , bold_k start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT(1) c^†(𝐤 1)f^i(𝐤 2)f^j†(𝐤 3)f^k(𝐤 1−𝐤 2+𝐤 3)+c.c)+⋯,\displaystyle\left.\hat{c}^{\dagger}(\mathbf{k}{1})\hat{f}{i}(\mathbf{k}{2}% )\hat{f}{j}^{\dagger}(\mathbf{k}{3})\hat{f}{k}(\mathbf{k}{1}-\mathbf{k}{2% }+\mathbf{k}{3})+c.c\right)+\cdots,over^ start_ARG italic_c end_ARG start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( bold_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) over^ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) over^ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( bold_k start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) over^ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - bold_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + bold_k start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) + italic_c . italic_c ) + ⋯ ,

in which interactions involving c 𝑐 c italic_c that are multi-linear in χ CFT subscript 𝜒 CFT\chi_{\rm CFT}italic_χ start_POSTSUBSCRIPT roman_CFT end_POSTSUBSCRIPT are suppressed, those which are linear in χ CFT subscript 𝜒 CFT\chi_{\rm CFT}italic_χ start_POSTSUBSCRIPT roman_CFT end_POSTSUBSCRIPT are 𝒪⁢(N)𝒪 𝑁\mathcal{O}(N)caligraphic_O ( italic_N ), and those which do not involve χ CFT subscript 𝜒 CFT\chi_{\rm CFT}italic_χ start_POSTSUBSCRIPT roman_CFT end_POSTSUBSCRIPT are 𝒪⁢(N 0)𝒪 superscript 𝑁 0\mathcal{O}(N^{0})caligraphic_O ( italic_N start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ). We have removed c†⁢f superscript 𝑐†𝑓 c^{\dagger}f italic_c start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_f type interactions by diagonalizing the 𝒪⁢(N 0)𝒪 superscript 𝑁 0\mathcal{O}(N^{0})caligraphic_O ( italic_N start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ) quadratic terms. The above is a special case of the semi-holographic models analyzed in Mukhopadhyay and Policastro (2013); Doucot et al. (2017) implying that we can infer its low energy behavior via Wilsonian RG approach in the large N 𝑁 N italic_N limit.

The propagating degree of freedom on the Fermi surface is essentially c 𝑐 c italic_c plus a 𝒪⁢(1/N)𝒪 1 𝑁\mathcal{O}(1/N)caligraphic_O ( 1 / italic_N ) contribution from χ 𝜒\chi italic_χ Mukhopadhyay and Policastro (2013). The resulting retarded Green’s function on the Fermi surface takes the following form in the large N 𝑁 N italic_N limit (see the supplemental material of Doucot et al. (2021) for a derivation by considering perturbative corrections in λ 𝜆\lambda italic_λ and other couplings):

G R⁢(ω,𝐤)=subscript 𝐺 𝑅 𝜔 𝐤 absent\displaystyle G_{R}(\omega,\mathbf{k})=italic_G start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_ω , bold_k ) =(ω+i γ(ω 2+π 2 T 2)+α𝒢(ω)\displaystyle\Bigg{(}\omega+i\tilde{\gamma}(\omega^{2}+\pi^{2}T^{2})+\tilde{% \alpha}\mathcal{G}(\omega)( italic_ω + italic_i over~ start_ARG italic_γ end_ARG ( italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_T start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) + over~ start_ARG italic_α end_ARG caligraphic_G ( italic_ω ) −1 2⁢m(𝐤 2−k F 2)+𝒪(1 N))−1.\displaystyle-\tfrac{1}{2m}(\mathbf{k}^{2}-k_{F}^{2})+\mathcal{O}\left(\frac{1% }{N}\right)\Bigg{)}^{-1}.- divide start_ARG 1 end_ARG start_ARG 2 italic_m end_ARG ( bold_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_k start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) + caligraphic_O ( divide start_ARG 1 end_ARG start_ARG italic_N end_ARG ) ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT .(2)

Here γ∼𝒪⁢(λ 2)similar-to𝛾 𝒪 superscript 𝜆 2\tilde{\gamma}\sim\mathcal{O}(\lambda^{2})over~ start_ARG italic_γ end_ARG ∼ caligraphic_O ( italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) and α∼𝒪⁢(g 2)similar-to𝛼 𝒪 superscript 𝑔 2\tilde{\alpha}\sim\mathcal{O}(g^{2})over~ start_ARG italic_α end_ARG ∼ caligraphic_O ( italic_g start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) are the two leading (irrelevant) couplings determining the Fermi-liquid and holographic components of the self-energy respectively with the latter given by Iqbal and Liu (2009)

𝒢⁢(ω)=e i⁢(ϕ+π⁢ν/2)⁢(2⁢π⁢T)ν⁢Γ⁢(1 2+ν 2−i⁢ω 2⁢π⁢T)Γ⁢(1 2−ν 2−i⁢ω 2⁢π⁢T).𝒢 𝜔 superscript 𝑒 𝑖 italic-ϕ 𝜋 𝜈 2 superscript 2 𝜋 𝑇 𝜈 Γ 1 2 𝜈 2 𝑖 𝜔 2 𝜋 𝑇 Γ 1 2 𝜈 2 𝑖 𝜔 2 𝜋 𝑇\displaystyle\mathcal{G}(\omega)=e^{i(\phi+\pi\nu/2)}(2\pi T)^{\nu}\frac{% \Gamma\left(\tfrac{1}{2}+\tfrac{\nu}{2}-i\tfrac{\omega}{2\pi T}\right)}{\Gamma% \left(\tfrac{1}{2}-\tfrac{\nu}{2}-i\tfrac{\omega}{2\pi T}\right)},.caligraphic_G ( italic_ω ) = italic_e start_POSTSUPERSCRIPT italic_i ( italic_ϕ + italic_π italic_ν / 2 ) end_POSTSUPERSCRIPT ( 2 italic_π italic_T ) start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT divide start_ARG roman_Γ ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG + divide start_ARG italic_ν end_ARG start_ARG 2 end_ARG - italic_i divide start_ARG italic_ω end_ARG start_ARG 2 italic_π italic_T end_ARG ) end_ARG start_ARG roman_Γ ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG - divide start_ARG italic_ν end_ARG start_ARG 2 end_ARG - italic_i divide start_ARG italic_ω end_ARG start_ARG 2 italic_π italic_T end_ARG ) end_ARG .(3)

In order that the spectral function ρ=−2⁢Im⁢G R 𝜌 2 Im subscript 𝐺 𝑅\rho=-2\text{Im }G_{R}italic_ρ = - 2 Im italic_G start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT is non-negative, we require 0<ϕ<π⁢(1−ν)0 italic-ϕ 𝜋 1 𝜈 0<\phi<\pi(1-\nu)0 < italic_ϕ < italic_π ( 1 - italic_ν ). We choose ϕ≈π⁢(1−ν)/2 italic-ϕ 𝜋 1 𝜈 2\phi\approx\pi(1-\nu)/2 italic_ϕ ≈ italic_π ( 1 - italic_ν ) / 2 as discussed in Doucot et al. (2021) so that (only) at the Fermi momentum, the spectral function has particle-hole symmetry. However, we do not need significant fine-tuning of ϕ italic-ϕ\phi italic_ϕ for our conclusions to hold.

It is convenient to re-write the propagator(II) with variables re-scaled with respect to the Fermi energy scale (E F=k F 2/2⁢m subscript 𝐸 𝐹 superscript subscript 𝑘 𝐹 2 2 𝑚 E_{F}=k_{F}^{2}/2m italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT = italic_k start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_m) as follows:

G R⁢(x,𝐲)=subscript 𝐺 𝑅 𝑥 𝐲 absent\displaystyle G_{R}(x,\mathbf{y})=italic_G start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_x , bold_y ) =E F−1(x+i γ(x 2+(π x T)2)+α e i⁢(ϕ+π⁢ν/2)(2 π x T)ν\displaystyle E_{F}^{-1}\left(x+i\gamma(x^{2}+(\pi x_{T})^{2})+\alpha e^{i(% \phi+\pi\nu/2)}(2\pi x_{T})^{\nu}\right.italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_x + italic_i italic_γ ( italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_π italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) + italic_α italic_e start_POSTSUPERSCRIPT italic_i ( italic_ϕ + italic_π italic_ν / 2 ) end_POSTSUPERSCRIPT ( 2 italic_π italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT Γ⁢(1 2+ν 2−i⁢x 2⁢π⁢x T)Γ⁢(1 2−ν 2−i⁢x 2⁢π⁢x T)−(|𝐲|2−1))−1.\displaystyle\left.\frac{\Gamma\left(\tfrac{1}{2}+\tfrac{\nu}{2}-i\tfrac{x}{2% \pi x_{T}}\right)}{\Gamma\left(\tfrac{1}{2}-\tfrac{\nu}{2}-i\tfrac{x}{2\pi x_{% T}}\right)}-(|\mathbf{y}|^{2}-1)\right)^{-1},.divide start_ARG roman_Γ ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG + divide start_ARG italic_ν end_ARG start_ARG 2 end_ARG - italic_i divide start_ARG italic_x end_ARG start_ARG 2 italic_π italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT end_ARG ) end_ARG start_ARG roman_Γ ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG - divide start_ARG italic_ν end_ARG start_ARG 2 end_ARG - italic_i divide start_ARG italic_x end_ARG start_ARG 2 italic_π italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT end_ARG ) end_ARG - ( | bold_y | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ) ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT .(4)

Above x=ω/E F 𝑥 𝜔 subscript 𝐸 𝐹 x=\omega/E_{F}italic_x = italic_ω / italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT, x T=T/E F subscript 𝑥 𝑇 𝑇 subscript 𝐸 𝐹 x_{T}=T/E_{F}italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = italic_T / italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT, 𝐲=𝐤/k F 𝐲 𝐤 subscript 𝑘 𝐹\mathbf{y}=\mathbf{k}/k_{F}bold_y = bold_k / italic_k start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT, γ=γ⁢E F 𝛾𝛾 subscript 𝐸 𝐹\gamma=\tilde{\gamma}E_{F}italic_γ = over~ start_ARG italic_γ end_ARG italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT, and α=α⁢E F−(1−ν)𝛼𝛼 superscript subscript 𝐸 𝐹 1 𝜈\alpha=\tilde{\alpha}E_{F}^{-(1-\nu)}italic_α = over~ start_ARG italic_α end_ARG italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - ( 1 - italic_ν ) end_POSTSUPERSCRIPT. Therefore, the dimensionless effective couplings of the model are α 𝛼\alpha italic_α and γ 𝛾\gamma italic_γ. For this effective description to be valid at temperatures smaller than the Fermi energy, we require both of these couplings to be small.

The ratio α/γ 𝛼 𝛾\alpha/\gamma italic_α / italic_γ is naturally related to the doping strength following our previous discussion. The Fermi energy is the only scale of the model, however the scale at which this model should be matched to the microscopic description must be lower.

III Universality and fits to ARPES data

In Doucot et al. (2021) it was shown that for any value of ν 𝜈\nu italic_ν in the range 0.66⪅ν⪅0.95 less-than-or-approximately-equals 0.66 𝜈 less-than-or-approximately-equals 0.95 0.66\lessapprox\nu\lessapprox 0.95 0.66 ⪅ italic_ν ⪅ 0.95, there exists an optimal ratio α/γ 𝛼 𝛾\alpha/\gamma italic_α / italic_γ at which the model II exhibits linear-in-T d.c. resistivity over a very wide range of temperatures. At this optimal ratio (for its dependence on ν 𝜈\nu italic_ν see plots in Doucot et al. (2021)), the Fermi liquid and holographic self-energy contributions finite temperature self-energy contributions shown in II (and II) balance out to produce new scaling behavior. Here we show that the key to such strange metallic behavior is the emergence of a universal form of the spectral function.

Let us define new variables x=ω/T=x/x T𝑥 𝜔 𝑇 𝑥 subscript 𝑥 𝑇\tilde{x}=\omega/T=x/x_{T}over~ start_ARG italic_x end_ARG = italic_ω / italic_T = italic_x / italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT and 𝐲~~𝐲\tilde{\mathbf{y}}over~ start_ARG bold_y end_ARG via |𝐲|2−1=x T−1⁢(|𝐲|2−1)superscript𝐲 2 1 superscript subscript 𝑥 𝑇 1 superscript 𝐲 2 1|\tilde{\mathbf{y}}|^{2}-1=x_{T}^{-1}\left(|\mathbf{y}|^{2}-1\right)| over~ start_ARG bold_y end_ARG | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 = italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( | bold_y | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ). Note |𝐲|=1𝐲 1|\tilde{\mathbf{y}}|=1| over~ start_ARG bold_y end_ARG | = 1 corresponds to the Fermi surface. Furthermore, we define the function F 𝐹 F italic_F from the retarded propagator (II) as follows:

F⁢(x,𝐲):=T⁢G R⁢(ω,𝐤)=T⁢G R⁢(x,𝐲)assign 𝐹𝑥𝐲 𝑇 subscript 𝐺 𝑅 𝜔 𝐤 𝑇 subscript 𝐺 𝑅𝑥𝐲\displaystyle F(\tilde{x},\tilde{\mathbf{y}}):=TG_{R}(\omega,\mathbf{k})=TG_{R% }(\tilde{x},\tilde{\mathbf{y}})italic_F ( over~ start_ARG italic_x end_ARG , over~ start_ARG bold_y end_ARG ) := italic_T italic_G start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_ω , bold_k ) = italic_T italic_G start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( over~ start_ARG italic_x end_ARG , over~ start_ARG bold_y end_ARG ) =(x+i γ x T(x2+π 2)+α x T−1(2 π x T)ν\displaystyle=\Big{(}\tilde{x}+i\gamma x_{T}(\tilde{x}^{2}+\pi^{2})+\alpha x_{% T}^{-1}(2\pi x_{T})^{\nu}= ( over~ start_ARG italic_x end_ARG + italic_i italic_γ italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( over~ start_ARG italic_x end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) + italic_α italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 2 italic_π italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT e i⁢(ϕ+π⁢ν/2)Γ⁢(1 2+ν 2−i⁢x2⁢π)Γ⁢(1 2−ν 2−i⁢x2⁢π)−(|𝐲|2−1))−1.\displaystyle\hskip 14.22636pte^{i(\phi+\pi\nu/2)}\frac{\Gamma\left(\tfrac{1}{% 2}+\tfrac{\nu}{2}-i\tfrac{\tilde{x}}{2\pi}\right)}{\Gamma\left(\tfrac{1}{2}-% \tfrac{\nu}{2}-i\tfrac{\tilde{x}}{2\pi}\right)}-(|\tilde{\mathbf{y}}|^{2}-1)% \Big{)}^{-1}.italic_e start_POSTSUPERSCRIPT italic_i ( italic_ϕ + italic_π italic_ν / 2 ) end_POSTSUPERSCRIPT divide start_ARG roman_Γ ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG + divide start_ARG italic_ν end_ARG start_ARG 2 end_ARG - italic_i divide start_ARG over start_ARG italic_x end_ARG end_ARG start_ARG 2 italic_π end_ARG ) end_ARG start_ARG roman_Γ ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG - divide start_ARG italic_ν end_ARG start_ARG 2 end_ARG - italic_i divide start_ARG over~ start_ARG italic_x end_ARG end_ARG start_ARG 2 italic_π end_ARG ) end_ARG - ( | over~ start_ARG bold_y end_ARG | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ) ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT .(5)

By universality, we mean that −2⁢T⁢Im⁢G R⁢(ω,𝐤)=T⁢ρ⁢(ω,𝐤)=−2⁢I⁢m⁢F⁢(x,𝐲)2 𝑇 Im subscript 𝐺 𝑅 𝜔 𝐤 𝑇 𝜌 𝜔 𝐤 2 I m 𝐹𝑥𝐲-2T{\rm Im}G_{R}(\omega,\mathbf{k})=T\rho(\omega,\mathbf{k})=-2{\rm Im}F(% \tilde{x},\tilde{\mathbf{y}})- 2 italic_T roman_Im italic_G start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_ω , bold_k ) = italic_T italic_ρ ( italic_ω , bold_k ) = - 2 roman_I roman_m italic_F ( over~ start_ARG italic_x end_ARG , over~ start_ARG bold_y end_ARG ) is approximately the same function of x~~𝑥\tilde{x}over~ start_ARG italic_x end_ARG (i.e. ω/T 𝜔 𝑇\omega/T italic_ω / italic_T) when (i) |𝐲|∼1 similar-to𝐲 1|\tilde{\mathbf{y}}|\sim 1| over~ start_ARG bold_y end_ARG | ∼ 1 and fixed (i.e. the momentum is near the Fermi surface) and (ii) the ratio of the couplings α/γ 𝛼 𝛾\alpha/\gamma italic_α / italic_γ is in a narrow optimal range of values corresponding to the choice of the scaling exponent ν 𝜈\nu italic_ν. The explicit dependence on x T=T/E F subscript 𝑥 𝑇 𝑇 subscript 𝐸 𝐹 x_{T}=T/E_{F}italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = italic_T / italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT thus disappears in the universal regime to a good approximation, and so does the explicit dependence on the scaling exponent ν 𝜈\nu italic_ν, and also the overall scale of the couplings (as only their ratio matters). We find that the universality emerges only at finite temperatures for t c<x T(=T/E F)<10 subscript 𝑡 𝑐 annotated subscript 𝑥 𝑇 absent 𝑇 subscript 𝐸 𝐹 10 t_{c}<x_{T}(=T/E_{F})<10 italic_t start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT < italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( = italic_T / italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ) < 10, and also when 0.66⪅ν⪅0.95 less-than-or-approximately-equals 0.66 𝜈 less-than-or-approximately-equals 0.95 0.66\lessapprox\nu\lessapprox 0.95 0.66 ⪅ italic_ν ⪅ 0.95 and when both couplings are small (γ⪅0.01 𝛾 0.01\gamma\lessapprox 0.01 italic_γ ⪅ 0.01). The lower end of x T subscript 𝑥 𝑇 x_{T}italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT, i.e. t c subscript 𝑡 𝑐 t_{c}italic_t start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT depends on the choice of ν 𝜈\nu italic_ν. For ν≈0.66 𝜈 0.66\nu\approx 0.66 italic_ν ≈ 0.66, t c∼0.01 similar-to subscript 𝑡 𝑐 0.01 t_{c}\sim 0.01 italic_t start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ∼ 0.01. However, for higher values of ν≈0.95 𝜈 0.95\nu\approx 0.95 italic_ν ≈ 0.95, universality emerges at much lower temperatures x T(=T/E F)∼0.01 similar-to annotated subscript 𝑥 𝑇 absent 𝑇 subscript 𝐸 𝐹 0.01 x_{T}(=T/E_{F})\sim 0.01 italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( = italic_T / italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ) ∼ 0.01. We emphasize that both the optimal ratio α/γ 𝛼 𝛾\alpha/\gamma italic_α / italic_γ and t c subscript 𝑡 𝑐 t_{c}italic_t start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT depend only on ν 𝜈\nu italic_ν, and not on the overall scale of the couplings (i.e. γ 𝛾\gamma italic_γ) in quantitative agreement with Doucot et al. (2021).

The origin of universality is the competition of the two finite temperature effects in the local self-energy contributions, one which is Fermi-liquid like and another which is holographic. It is also important to note that although unievrsality emerges only near the Fermi surface, it should be valid for the full range of frequencies, as the tail of the spectral function also contributes to transport properties, as we will see soon. It is important that we are at finite temperature, as otherwise ω 2 superscript 𝜔 2\omega^{2}italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT (the Fermi-liquid contribution) and ω 2⁢ν superscript 𝜔 2 𝜈\omega^{2\nu}italic_ω start_POSTSUPERSCRIPT 2 italic_ν end_POSTSUPERSCRIPT (the holographic contribution) in the self-energy, cannot compete with each other at arbitrary values of the frequency.

We can readily see from Fig1 that as a function of the frequency, the scaled spectral function T⁢(ρ/2)=−T⁢Im⁢G R=−Im⁢F 𝑇 𝜌 2 𝑇 Im subscript 𝐺 𝑅 Im 𝐹 T(\rho/2)=-T{\rm Im}G_{R}=-{\rm Im}F italic_T ( italic_ρ / 2 ) = - italic_T roman_Im italic_G start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT = - roman_Im italic_F is independent of the temperature to an excellent approximation, for T>0.01⁢E F 𝑇 0.01 subscript 𝐸 𝐹 T>0.01E_{F}italic_T > 0.01 italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT when plotted as a function of x=ω/T𝑥 𝜔 𝑇\tilde{x}=\omega/T over~ start_ARG italic_x end_ARG = italic_ω / italic_T on the Fermi surface (|𝐲|=1𝐲 1|\tilde{\mathbf{y}}|=1| over~ start_ARG bold_y end_ARG | = 1) and also near it (|𝐲|=0.95𝐲 0.95|\tilde{\mathbf{y}}|=0.95| over~ start_ARG bold_y end_ARG | = 0.95 and |𝐲|=1.05𝐲 1.05|\tilde{\mathbf{y}}|=1.05| over~ start_ARG bold_y end_ARG | = 1.05), at the optimal ratio α/γ=100 𝛼 𝛾 100\alpha/\gamma=100 italic_α / italic_γ = 100 corresponding to ν=0.95 𝜈 0.95\nu=0.95 italic_ν = 0.95, as claimed above. Furthermore, from plots in Fig. 2, we find that T⁢(ρ/2)𝑇 𝜌 2 T(\rho/2)italic_T ( italic_ρ / 2 ) is also independent of ν 𝜈\nu italic_ν at the corresponding optimal ratios α/γ 𝛼 𝛾\alpha/\gamma italic_α / italic_γ (at a representative temperature T=0.16⁢E F 𝑇 0.16 subscript 𝐸 𝐹 T=0.16E_{F}italic_T = 0.16 italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT) both at the Fermi surface and near it. Together these plots validate that F 𝐹 F italic_F is indeed independent of x T subscript 𝑥 𝑇 x_{T}italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT and also the model parameters like ν 𝜈\nu italic_ν in the universal regime.

Figure 1: Top: Plot of (1/2)⁢T⁢ρ 1 2 𝑇 𝜌(1/2)T\rho( 1 / 2 ) italic_T italic_ρ as a function of ω/T 𝜔 𝑇\omega/T italic_ω / italic_T for various T≥0.01⁢E F 𝑇 0.01 subscript 𝐸 𝐹 T\geq 0.01E_{F}italic_T ≥ 0.01 italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT at the optimal α/γ=100 𝛼 𝛾 100\alpha/\gamma=100 italic_α / italic_γ = 100 for ν=0.95 𝜈 0.95\nu=0.95 italic_ν = 0.95 on the Fermi surface (left), as well as near it (right). In all plots, γ=0.001 𝛾 0.001\gamma=0.001 italic_γ = 0.001.

Figure 2: Universality for different values of ν 𝜈\nu italic_ν on the Fermi surface (top) as well as near it (bottom) for corresponding optimal α/γ 𝛼 𝛾\alpha/\gamma italic_α / italic_γ, and γ=0.001 𝛾 0.001\gamma=0.001 italic_γ = 0.001. We have chosen T=0.16⁢E F 𝑇 0.16 subscript 𝐸 𝐹 T=0.16E_{F}italic_T = 0.16 italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT.

Table 1: Left: δ 𝛿\delta italic_δ for “optimal set of values” of α 𝛼\alpha italic_α for given choices of ν 𝜈\nu italic_ν and γ=0.001 𝛾 0.001\gamma=0.001 italic_γ = 0.001. We see that δ<10 𝛿 10\delta<10 italic_δ < 10. Right: The same for some set of values which are “not optimal” for which δ 𝛿\delta italic_δ is significantly larger than 10 10 10 10.

Although the universality is well demonstrated in Figs. 1 and 2 by how well the spectral functions multiplied by the temperature collapse on to a single curve, a more quantitative study is also illuminating. To develop a quantitative measure, we first define a standard curve,

f s:=(−2⁢T⁢Im⁢G R⁢(ω,𝐤))⁢[ν=0.8,α=0.023,γ=0.001,T/E F=0.16],assign subscript 𝑓 𝑠 2 𝑇 Im subscript 𝐺 𝑅 𝜔 𝐤 delimited-[]formulae-sequence 𝜈 0.8 formulae-sequence 𝛼 0.023 formulae-sequence 𝛾 0.001 𝑇 subscript 𝐸 𝐹 0.16 f_{s}:=(-2T{\rm Im}G_{R}(\omega,\mathbf{k}))[\nu=0.8,\alpha=0.023,\gamma=0.001% ,T/E_{F}=0.16],italic_f start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT := ( - 2 italic_T roman_Im italic_G start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_ω , bold_k ) ) [ italic_ν = 0.8 , italic_α = 0.023 , italic_γ = 0.001 , italic_T / italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT = 0.16 ] ,(6)

which is in the universal regime, Furthermore, let

δ 100=∫−∞∞|(−2⁢I⁢m⁢F−f s)|⁢d x∫−∞∞F s⁢d x,𝛿 100 superscript subscript 2 𝐼 𝑚 𝐹 subscript 𝑓 𝑠 differential-d𝑥 superscript subscript subscript 𝐹 𝑠 differential-d𝑥\frac{\delta}{100}=\frac{\int_{-\infty}^{\infty}|(-2ImF-f_{s})|{\rm d}\tilde{x% }}{\int_{-\infty}^{\infty}F_{s}{\rm d}\tilde{x}},divide start_ARG italic_δ end_ARG start_ARG 100 end_ARG = divide start_ARG ∫ start_POSTSUBSCRIPT - ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT | ( - 2 italic_I italic_m italic_F - italic_f start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) | roman_d over~ start_ARG italic_x end_ARG end_ARG start_ARG ∫ start_POSTSUBSCRIPT - ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_F start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT roman_d over~ start_ARG italic_x end_ARG end_ARG ,(7)

where F 𝐹 F italic_F is defined in (III). A simple quantitative test of universality of F 𝐹 F italic_F is then to check if δ<10 𝛿 10\delta<10 italic_δ < 10.

Table 2: δ 𝛿\delta italic_δ for optimal and non-optimal values of γ 𝛾\gamma italic_γ for corresponding values of ν 𝜈\nu italic_ν with α=0.023 𝛼 0.023\alpha=0.023 italic_α = 0.023 and T/E F=0.16 𝑇 subscript 𝐸 𝐹 0.16 T/E_{F}=0.16 italic_T / italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT = 0.16 fixed. We see that δ<10 𝛿 10\delta<10 italic_δ < 10 for optimal cases, while for non-optimal cases δ 𝛿\delta italic_δ is much larger than 10 10 10 10. This also provides evidence that the universality is determined only by the ratio of α/γ 𝛼 𝛾\alpha/\gamma italic_α / italic_γ.

In Table 1, we have studied the values of δ 𝛿\delta italic_δ with variation of α 𝛼\alpha italic_α for various fixed values of ν 𝜈\nu italic_ν at x T=T/E F=0.16 subscript 𝑥 𝑇 𝑇 subscript 𝐸 𝐹 0.16 x_{T}=T/E_{F}=0.16 italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = italic_T / italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT = 0.16 both in and away from the optimal regime. In Table 2, we have studied the values of δ 𝛿\delta italic_δ when γ 𝛾\gamma italic_γ is varied from the optimal value at various fixed values of ν 𝜈\nu italic_ν while keeping α=0.023 𝛼 0.023\alpha=0.023 italic_α = 0.023 and x T=T/E F=0.16 subscript 𝑥 𝑇 𝑇 subscript 𝐸 𝐹 0.16 x_{T}=T/E_{F}=0.16 italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = italic_T / italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT = 0.16 fixed as in the standard curve. We readily note that in the optimal regime, δ 𝛿\delta italic_δ is typically much smaller than 10 10 10 10 while in the non-optimal regime it is significantly larger. As we keep α 𝛼\alpha italic_α instead of γ 𝛾\gamma italic_γ fixed in Table 2 unlike in our plots and Table 1, the two tables together provide evidence that universality is determined only by the ratio α/γ 𝛼 𝛾\alpha/\gamma italic_α / italic_γ (at the corresponding ν 𝜈\nu italic_ν) and not by their absolute values (although we need γ<0.01 𝛾 0.01\gamma<0.01 italic_γ < 0.01 for the universality to emerge).

Particularly, Table 2 is also remarkable because we see that, although γ 𝛾\gamma italic_γ is small, a small variation in γ 𝛾\gamma italic_γ can produce a significant departure from universality. Therefore, the effect of γ 𝛾\gamma italic_γ is significant not only at infinitesimal temperatures (recall that in this table x T=T/E F=0.16 subscript 𝑥 𝑇 𝑇 subscript 𝐸 𝐹 0.16 x_{T}=T/E_{F}=0.16 italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = italic_T / italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT = 0.16). At this point, it is appropriate to emphasize again that δ<10 𝛿 10\delta<10 italic_δ < 10 provides a good physical test of universality because the contribution to transport comes also from higher frequencies, which constitute the tail of the spectral function, as will be discussed below.

Furthermore, one can similarly verify quantitatively by computing δ 𝛿\delta italic_δ explicitly that the universality emerges only when t c<x T(=T/E F)<10 subscript 𝑡 𝑐 annotated subscript 𝑥 𝑇 absent 𝑇 subscript 𝐸 𝐹 10 t_{c}<x_{T}(=T/E_{F})<10 italic_t start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT < italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( = italic_T / italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ) < 10, and that t c subscript 𝑡 𝑐 t_{c}italic_t start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT is determined by ν 𝜈\nu italic_ν alone.

III.1 Comparing with ARPES data

As shown in Fig.3, our spectral function fits well with the nodal ARPES data for the imaginary part of the self-energy for different underdoped, overdoped and optimally doped samples of B⁢i 2⁢S⁢r 2⁢C⁢a⁢C⁢u 2⁢O 8+δ 𝐵 subscript 𝑖 2 𝑆 subscript 𝑟 2 𝐶 𝑎 𝐶 subscript 𝑢 2 subscript 𝑂 8 𝛿 Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta}italic_B italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_S italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_C italic_a italic_C italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_O start_POSTSUBSCRIPT 8 + italic_δ end_POSTSUBSCRIPT at various temperatures, reported in Reber et al. (2019). The fits work well keeping the Fermi energy and the value of α 𝛼\alpha italic_α fixed for each doping (for all temperatures), and with γ=0.001 𝛾 0.001\gamma=0.001 italic_γ = 0.001 chosen in all cases 1 1 1 The data has been extracted form the paper using WebPlotDigitizer tool. https://automeris.io/WebPlotDigitizer/.. The exponent ν=0.95 𝜈 0.95\nu=0.95 italic_ν = 0.95 is remarkably stable across doping, and the fitted value of α 𝛼\alpha italic_α indeed corresponds to the optimal ratio at which we obtain universality. However, α 𝛼\alpha italic_α does decrease very mildly in the overdoped region implying more Fermi-liquid like behavior. In Reber et al. (2019), the fits were obtained with varying exponents and different phenomenological ansatz which was improved in Smit et al. (2021) from theoretical considerations. However, our fits work well with more constraints.

We have not done a quantitative study of goodness of fit due to the significant noisiness of the data, but our fitting is physically motivated. For the universality to emerge at the temperatures reported in the plot (assuming E F∼10 4⁢K similar-to subscript 𝐸 𝐹 superscript 10 4 𝐾 E_{F}\sim 10^{4}K italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_K ), we need ν 𝜈\nu italic_ν to be closer to 0.95 0.95 0.95 0.95 as discussed above. The choice ν=0.95 𝜈 0.95\nu=0.95 italic_ν = 0.95 is then at the edge of universality. Furthermore, we expect α 𝛼\alpha italic_α to depend on the doping strength, as discussed before, and not γ 𝛾\gamma italic_γ. Therefore, the fits with varying α 𝛼\alpha italic_α and fixed γ 𝛾\gamma italic_γ suggest the validity of the underlying physical intuition for the model.

Although our model needs to include order parameters for description of various phases, the very mild variation of α 𝛼\alpha italic_α and hence the ratio of couplings with the doping in our fits could suggest the that strange metallic behavior can be restored to a good approximation even away from optimal doping if order parameters are suppressed. Indeed restoration of strange metallic behavior by suppression of charged density waves has been reported for Y⁢B⁢a 2⁢C⁢u 3⁢O 7−δ 𝑌 𝐵 subscript 𝑎 2 𝐶 subscript 𝑢 3 subscript 𝑂 7 𝛿 YBa_{2}Cu_{3}O_{7-\delta}italic_Y italic_B italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_C italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_O start_POSTSUBSCRIPT 7 - italic_δ end_POSTSUBSCRIPT in underdoped samples below the pseudogap temperature Wahlberg et al. (2021). See Wú et al. (2022); Husain et al. (2020) for the overdoped regime.

Figure 3: Fit of the imaginary part of self energy with the ARPES data from Reber et al. (2019). The labels OPT, OD and UD denote optimal/over/under-doping and the following numbers denote T c subscript 𝑇 𝑐 T_{c}italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT in K.

IV Transport and Planckian dissipation

Ignoring vertex corrections and assuming that charge transport is primarily due to the propagating degrees of freedom on the Fermi surface, the dc conductivity can be readily obtained from the spectral function via (n F subscript 𝑛 𝐹 n_{F}italic_n start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT is the Fermi-Dirac distribution):

σ dc=e 2 m 2⁢∫d⁢ω 2⁢π⁢∫d 2⁢k 4⁢π 2⁢k 2⁢ρ⁢(ω,𝐤,T)2⁢(−∂n F⁢(ω,T)∂ω).subscript 𝜎 dc superscript 𝑒 2 superscript 𝑚 2 𝑑 𝜔 2 𝜋 superscript 𝑑 2 𝑘 4 superscript 𝜋 2 superscript 𝑘 2 𝜌 superscript 𝜔 𝐤 𝑇 2 subscript 𝑛 𝐹 𝜔 𝑇 𝜔\displaystyle\sigma_{\text{dc}}=\frac{e^{2}}{m^{2}}\int\frac{d\omega}{2\pi}% \int\frac{d^{2}k}{4\pi^{2}}k^{2}\rho(\omega,\mathbf{k},T)^{2}\left(-\dfrac{% \partial n_{F}(\omega,T)}{\partial\omega}\right).italic_σ start_POSTSUBSCRIPT dc end_POSTSUBSCRIPT = divide start_ARG italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∫ divide start_ARG italic_d italic_ω end_ARG start_ARG 2 italic_π end_ARG ∫ divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_k end_ARG start_ARG 4 italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ ( italic_ω , bold_k , italic_T ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( - divide start_ARG ∂ italic_n start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ω , italic_T ) end_ARG start_ARG ∂ italic_ω end_ARG ) .(8)

The linear-in-T behavior of the dc resistivity over a wide range of temperatures at the optimal ratio of the couplings simply follows from the universal form of the spectral function (III) given that the above integral gets its major contribution from momenta near the Fermi surface. The latter implies that k 2 superscript 𝑘 2 k^{2}italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT in the integrand can be replaced by k F 2 superscript subscript 𝑘 𝐹 2 k_{F}^{2}italic_k start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. To see this, we can replace ρ 𝜌\rho italic_ρ by (−1/2⁢T)⁢Im⁢F⁢(x,𝐲)1 2 𝑇 Im 𝐹𝑥𝐲(-1/2T){\rm Im}F(\tilde{x},\tilde{\mathbf{y}})( - 1 / 2 italic_T ) roman_Im italic_F ( over~ start_ARG italic_x end_ARG , over~ start_ARG bold_y end_ARG ) following (III) in the above integral. A factor of T 2 superscript 𝑇 2 T^{2}italic_T start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT comes from changing the loop frequency and momenta variables to x=ω/T𝑥 𝜔 𝑇\tilde{x}=\omega/T over~ start_ARG italic_x end_ARG = italic_ω / italic_T and 𝐲≈n^⁢k F⁢(1+(E F/T)⁢((k/k F)−1))𝐲^𝑛 subscript 𝑘 𝐹 1 subscript 𝐸 𝐹 𝑇 𝑘 subscript 𝑘 𝐹 1\tilde{\mathbf{y}}\approx\hat{n}k_{F}(1+(E_{F}/T)((k/k_{F})-1))over~ start_ARG bold_y end_ARG ≈ over^ start_ARG italic_n end_ARG italic_k start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( 1 + ( italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT / italic_T ) ( ( italic_k / italic_k start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ) - 1 ) ) respectively (with n^=𝐤/k^𝑛 𝐤 𝑘\hat{n}=\mathbf{k}/k over^ start_ARG italic_n end_ARG = bold_k / italic_k). An additional factor of T−1 superscript 𝑇 1 T^{-1}italic_T start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT arises from the ω 𝜔\omega italic_ω-derivative of n F subscript 𝑛 𝐹 n_{F}italic_n start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT. The integral thus behaves as 1/T 1 𝑇 1/T 1 / italic_T for the very wide range of temperatures over which the universal form of the spectral function (III) is valid (e.g. for T>0.01⁢E F 𝑇 0.01 subscript 𝐸 𝐹 T>0.01E_{F}italic_T > 0.01 italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT, α/γ≈100 𝛼 𝛾 100\alpha/\gamma\approx 100 italic_α / italic_γ ≈ 100 and ν≈0.95 𝜈 0.95\nu\approx 0.95 italic_ν ≈ 0.95).

However, explicit computations shown in Fig.4 imply that the linear-in-T 𝑇 T italic_T behavior of the resistivity holds only up to an upper value of temperature because the integral over loop momenta gets contributions eventually from momenta far away from the Fermi surface where universality breaks down. In reality, the integral over loop momenta has cutoff due to the bandwidth, and therefore the linear-in-T 𝑇 T italic_T scaling should hold to a higher approximation. We note that since the linear-in-T resistivity is obtained only when the integral gets major contribution from the Fermi surface, our assumption, that only the propagating degrees of freedom on the Fermi surface contribute to the charge transport, is consistent.

We also emphasize that although the integral (8) receives contribution from momenta only near the Fermi surface, the large frequency tail of the spectral function contributes significantly. Therefore, for the analytic argument for the linear-in-T resistivity to be valid, the universal form of the spectral function should be a good approximation at all frequencies, as stated before.

The Hall conductivity σ H subscript 𝜎 𝐻\sigma_{H}italic_σ start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT for small magnetic fields similarly takes the form (ω c=e⁢B/m subscript 𝜔 𝑐 𝑒 𝐵 𝑚\omega_{c}=eB/m italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = italic_e italic_B / italic_m):

σ H=2⁢ω c⁢e 2 m⁢∫d⁢ω 2⁢π⁢∫d 2⁢k 4⁢π 2⁢k x⁢ρ⁢(ω,𝐤,T)⁢∂n F⁢(ω,T)∂ω subscript 𝜎 H 2 subscript 𝜔 𝑐 superscript 𝑒 2 𝑚 𝑑 𝜔 2 𝜋 superscript 𝑑 2 𝑘 4 superscript 𝜋 2 subscript 𝑘 𝑥 𝜌 𝜔 𝐤 𝑇 subscript 𝑛 𝐹 𝜔 𝑇 𝜔\displaystyle\sigma_{\rm H}={2\omega_{c}\frac{e^{2}}{m}}\int\frac{d\omega}{2% \pi}\int\frac{d^{2}k}{4\pi^{2}}k_{x}\rho(\omega,\mathbf{k},T)\dfrac{\partial n% {F}(\omega,T)}{\partial\omega}italic_σ start_POSTSUBSCRIPT roman_H end_POSTSUBSCRIPT = 2 italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT divide start_ARG italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_m end_ARG ∫ divide start_ARG italic_d italic_ω end_ARG start_ARG 2 italic_π end_ARG ∫ divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_k end_ARG start_ARG 4 italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_k start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_ρ ( italic_ω , bold_k , italic_T ) divide start_ARG ∂ italic_n start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ω , italic_T ) end_ARG start_ARG ∂ italic_ω end_ARG ×(∂∂k x⁢Re⁢G R⁢(ω,𝐤,T)).absent subscript 𝑘 𝑥 Re subscript 𝐺 𝑅 𝜔 𝐤 𝑇\displaystyle\hskip 85.35826pt\times\left(\dfrac{\partial}{\partial k{x}}% \text{Re }G_{R}(\omega,\mathbf{k},T)\right)\ .× ( divide start_ARG ∂ end_ARG start_ARG ∂ italic_k start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_ARG Re italic_G start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_ω , bold_k , italic_T ) ) .(9)

The above readily follows from considering the real time version of the corresponding expression obtained in Fukuyama et al. (1969); Itoh (1985) (see Appendix A for reproduction of the Fermi liquid result). Using the Kramers-Kronig relation for Re⁢G R Re subscript 𝐺 𝑅{\rm Re},G_{R}roman_Re italic_G start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT, one can argue similarly to the case of the dc-resistivity, that T−2 superscript 𝑇 2 T^{-2}italic_T start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT behavior of Hall conductivity should hold if the momentum integral receives major contribution from near the Fermi surface, and (III) is valid.

Figure 4: The dc conductivity (top) and Hall conductivity (bottom) show T−1 superscript 𝑇 1 T^{-1}italic_T start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT and T−2 superscript 𝑇 2 T^{-2}italic_T start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT behaviors respectively at the optimal ratios of α/γ 𝛼 𝛾\alpha/\gamma italic_α / italic_γ. Here γ=0.001.𝛾 0.001\gamma=0.001.italic_γ = 0.001 .

Here, we have assumed a spherical Fermi surface. However, when the Fermi surface has equal amounts of positive and negative curvatures, the integral (IV) can be vanishingly small, and the Hall conductivity could behave as T−3 superscript 𝑇 3 T^{-3}italic_T start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT instead of T−2 superscript 𝑇 2 T^{-2}italic_T start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT as observed in Abrahams and Varma (2003). We leave this for a future investigation.

Fig.4 shows the log-log plots of dc conductivity and Hall conductivity as a function of temperature at the optimal ratios of the couplings corresponding to various ν 𝜈\nu italic_ν, and with γ=0.001 𝛾 0.001\gamma=0.001 italic_γ = 0.001. At low temperatures, the dc and Hall conductivities behave as T−ν superscript 𝑇 𝜈 T^{-\nu}italic_T start_POSTSUPERSCRIPT - italic_ν end_POSTSUPERSCRIPT and T−2⁢ν superscript 𝑇 2 𝜈 T^{-2\nu}italic_T start_POSTSUPERSCRIPT - 2 italic_ν end_POSTSUPERSCRIPT respectively 2 2 2 However higher loops contribute significantly at very low T 𝑇 T italic_T giving residual resistivity, etc.. Crucially, there is a wide range of temperatures with the lower end coinciding with that of the universal regime (thus starting from 0.01⁢E F 0.01 subscript 𝐸 𝐹 0.01E_{F}0.01 italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT) for higher ν 𝜈\nu italic_ν) for which the d.c. and Hall conductivities behave as T−1 superscript 𝑇 1 T^{-1}italic_T start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT and T−2 superscript 𝑇 2 T^{-2}italic_T start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT, respectively. For higher γ 𝛾\gamma italic_γ, the temperature range over which the universal scaling behavior is obtained, decreases although the lower end of this range does not change (recall the discussion in the previous section that the latter is determined only by ν 𝜈\nu italic_ν and not the scale of couplings).

To make contact with Planckian dissipation, we recall the Fermi liquid results obtained from (8) and (IV) with Fermi liquid type spectral functions (see Fukuyama et al. (1969) and Appendix A for derivations):

σ dc=n⁢e 2 m⁢b⁢τ,σ H=n⁢e 2 m⁢ω c⁢τ 2,formulae-sequence subscript 𝜎 dc 𝑛 superscript 𝑒 2 𝑚 𝑏 𝜏 subscript 𝜎 H 𝑛 superscript 𝑒 2 𝑚 subscript 𝜔 𝑐 superscript 𝜏 2\sigma_{\rm dc}=\frac{ne^{2}}{m}b,\tau,\quad\sigma_{\rm H}=\frac{ne^{2}}{m}% \omega_{c}\tau^{2},italic_σ start_POSTSUBSCRIPT roman_dc end_POSTSUBSCRIPT = divide start_ARG italic_n italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_m end_ARG italic_b italic_τ , italic_σ start_POSTSUBSCRIPT roman_H end_POSTSUBSCRIPT = divide start_ARG italic_n italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_m end_ARG italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,(10)

where τ 𝜏\tau italic_τ is the scattering time, b 𝑏 b italic_b is the renormalization factor for the density of states at the Fermi surface and n 𝑛 n italic_n is the carrier density. For the Fermi liquid, n/m=E F/π 𝑛 𝑚 subscript 𝐸 𝐹 𝜋 n/m=E_{F}/\pi italic_n / italic_m = italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT / italic_π. Motivated by the observation that the integrals (8) and (IV) get their major contributions from the Fermi surface in the regime where they show T−1 superscript 𝑇 1 T^{-1}italic_T start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT and T−2 superscript 𝑇 2 T^{-2}italic_T start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT dependence, we can verify whether (11) can be consistent with our results although they are not derivable from a quasi-particle picture. Setting b=1 𝑏 1 b=1 italic_b = 1, (10) yields

τ=σ H ω c⁢σ dc,n m=σ dc 2⁢ω c e 2⁢σ H.formulae-sequence 𝜏 subscript 𝜎 H subscript 𝜔 𝑐 subscript 𝜎 dc 𝑛 𝑚 superscript subscript 𝜎 dc 2 subscript 𝜔 𝑐 superscript 𝑒 2 subscript 𝜎 H\tau=\frac{\sigma_{\rm H}}{\omega_{c}\sigma_{\rm dc}},,,,,\frac{n}{m}=% \frac{\sigma_{\rm dc}^{2}\omega_{c}}{e^{2}\sigma_{\rm H}}.italic_τ = divide start_ARG italic_σ start_POSTSUBSCRIPT roman_H end_POSTSUBSCRIPT end_ARG start_ARG italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT roman_dc end_POSTSUBSCRIPT end_ARG , divide start_ARG italic_n end_ARG start_ARG italic_m end_ARG = divide start_ARG italic_σ start_POSTSUBSCRIPT roman_dc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG start_ARG italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT roman_H end_POSTSUBSCRIPT end_ARG .(11)

These can be used to extract τ 𝜏\tau italic_τ and the effective n/m 𝑛 𝑚 n/m italic_n / italic_m from the explicit σ dc subscript 𝜎 dc\sigma_{\rm dc}italic_σ start_POSTSUBSCRIPT roman_dc end_POSTSUBSCRIPT and σ H subscript 𝜎 H\sigma_{\rm H}italic_σ start_POSTSUBSCRIPT roman_H end_POSTSUBSCRIPT obtained from (8) and (IV), respectively, with the spectral function obtained from (II).

Plots for τ 𝜏\tau italic_τ and effective n/m 𝑛 𝑚 n/m italic_n / italic_m as functions of T/E F 𝑇 subscript 𝐸 𝐹 T/E_{F}italic_T / italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT are shown in Fig.5 for γ=0.001 𝛾 0.001\gamma=0.001 italic_γ = 0.001, and for the optimal ratios α/γ 𝛼 𝛾\alpha/\gamma italic_α / italic_γ corresponding to various values of ν 𝜈\nu italic_ν. Indeed, τ≈f⋅T−1 𝜏⋅𝑓 superscript 𝑇 1\tau\approx f\cdot T^{-1}italic_τ ≈ italic_f ⋅ italic_T start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT (in units ℏ=k B=1 Planck-constant-over-2-pi subscript 𝑘 𝐵 1\hbar=k_{B}=1 roman_ℏ = italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = 1) with f 𝑓 f italic_f in the range 2⁢π×0.9−2⁢π×1.3 2 𝜋 0.9 2 𝜋 1.3 2\pi\times 0.9-2\pi\times 1.3 2 italic_π × 0.9 - 2 italic_π × 1.3, and is thus approximately independent of ν 𝜈\nu italic_ν and other details of the critical sector. Planckian behavior with such higher value of f≈𝒪⁢(10)𝑓 𝒪 10 f\approx\mathcal{O}(10)italic_f ≈ caligraphic_O ( 10 ) can indeed be found in many heavy fermion compounds according to Taupin and Paschen (2022). Furthermore, the effective n/m 𝑛 𝑚 n/m italic_n / italic_m is approximately constant for a very wide range of temperatures and also lies in the range 2⁢π−2.5⁢π 2 𝜋 2.5 𝜋 2\pi-2.5\pi 2 italic_π - 2.5 italic_π times the corresponding Fermi liquid value (=E F/π absent subscript 𝐸 𝐹 𝜋=E_{F}/\pi= italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT / italic_π) as we vary ν 𝜈\nu italic_ν from 0.66 0.66 0.66 0.66 to 0.95 0.95 0.95 0.95, confirming a Drude-like phenomenology.

The value of f 𝑓 f italic_f decreases as we increase γ 𝛾\gamma italic_γ keeping the optimal ratio α/γ 𝛼 𝛾\alpha/\gamma italic_α / italic_γ fixed, while the effective n/m 𝑛 𝑚 n/m italic_n / italic_m remains almost invariant. When γ=0.01 𝛾 0.01\gamma=0.01 italic_γ = 0.01, f≈1 𝑓 1 f\approx 1 italic_f ≈ 1 as found in cuprates Taupin and Paschen (2022). However, f 𝑓 f italic_f cannot be made smaller than 𝒪⁢(1)𝒪 1\mathcal{O}(1)caligraphic_O ( 1 ), as increasing γ 𝛾\gamma italic_γ beyond 0.01 0.01 0.01 0.01 drastically shrinks the range of temperatures over which universality of the spectral function holds and strange metallic behavior is exhibited. So we naturally obtain the expected strong coupling Planckian limit f≈1 𝑓 1 f\approx 1 italic_f ≈ 1. At smaller γ 𝛾\gamma italic_γ and therefore for weaker coupling, f 𝑓 f italic_f increases as pointed above, in consistency with the analysis of Taupin and Paschen (2022). In contrast, we find that the effective n/m 𝑛 𝑚 n/m italic_n / italic_m is almost independent of the overall strength of the couplings at their optimal ratio.

Figure 5: The plots of effective 4⁢π 2⁢n/(m⁢E F)4 superscript 𝜋 2 𝑛 𝑚 subscript 𝐸 𝐹 4\pi^{2}n/(mE_{F})4 italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n / ( italic_m italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ) (left) and τ⁢E F/(2⁢π)𝜏 subscript 𝐸 𝐹 2 𝜋\tau E_{F}/(2\pi)italic_τ italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT / ( 2 italic_π ) (right) as functions of T/E F 𝑇 subscript 𝐸 𝐹 T/E_{F}italic_T / italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT are shown for various values of ν 𝜈\nu italic_ν (at γ=0.001 𝛾 0.001\gamma=0.001 italic_γ = 0.001) and the corresponding optimal ratios α/γ 𝛼 𝛾\alpha/\gamma italic_α / italic_γ.

V Discussion

For our simple model to be more realistic and to capture the various phases of cuprates and other materials, we should consider a lattice of A⁢d⁢S 2 𝐴 𝑑 subscript 𝑆 2 AdS_{2}italic_A italic_d italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT black holes (dual to the quantum dots) which couple to the itinerant electrons via a critical fermionic operator, include Yukawa type interactions with bulk scalars, and analyze the system with a novel dynamical mean field theory (DMFT) type formulation. This approach is similar to those in Song et al. (2017); Patel et al. (2018a); Chowdhury et al. (2018); Patel et al. (2018b); Cha et al. (2020a); Mousatov et al. (2019); Patel and Sachdev (2019); Cha et al. (2020b); Patel et al. (2022) and also to quantum black hole microstate models discussed in Kibe et al. (2020, 2022). In the latter case, the strange metallic regime can emerge naturally when the throats behave as decoupled units leading to a local self-energy for the itinerant electrons. The decoupling of the throats has been shown to be necessary for the emergence of black hole complementarity without encountering information paradoxes Kibe et al. (2020, 2022, 2023) in the microstate models 3 3 3 In such microstate models, after perturbations due to infalling matter, the throats decouple from each other absorbing energy from infalling matter, while the latter also decouples from the throats Kibe et al. (2023). Furthermore, the decoupled residue of the infalling matter (absorbed in the hair) is a time-dependent quantum state which non-isometrically encodes its initial state. The ringdown of the decoupled throats also encodes the same information transitorily and transfers the same to Hawking radiation. Thus information is replicated (but not cloned) as black hole complimentarity demands. The decoupling of all degrees of freedom provides the underlying mechanism for the black hole complimentarity to emerge and in our present context could be relevant for the robustness of the locality of the self-energy.

Our simple model with only two effective couplings, nevertheless, provides an effective theory to understand many features of strange metallic transport and a refined version of Planckian dissipation. A deeper understanding could arise from applications of quantum information theory to the lattice models.

Acknowledgements.

We thank Subhasis Ghosh for comments on the manuscript, and also for very stimulating and helpful discussions. AM acknowledges support from the Ramanujan Fellowship of the Science and Engineering Board (SERB) of the Department of Science and Technology of India. HS acknowledges support from the INSPIRE PhD Fellowship of Department of Science and Technology of India. AM and GP also acknowledge support from IFCPAR/CEFIPRA funded project no 6304-3. SS acknowledges support by the US National Science Foundation under Grant DMR-1945395.

Appendix A The conductivities of the Fermi liquid

In order to derive the Fermi liquid dc and Hall conductivities, we start from the Green’s function (as parametrized in Fukuyama et al. (1969))

G R⁢(ω,𝐤)=(a−1⁢ω−b−1⁢ϵ k+i⁢τ−1)−1,subscript 𝐺 𝑅 𝜔 𝐤 superscript superscript 𝑎 1 𝜔 superscript 𝑏 1 subscript italic-ϵ 𝑘 𝑖 superscript 𝜏 1 1\displaystyle G_{R}(\omega,\mathbf{k})=\left(a^{-1}\omega-b^{-1}\epsilon_{k}+i% \tau^{-1}\right)^{-1},italic_G start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_ω , bold_k ) = ( italic_a start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ω - italic_b start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ϵ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_i italic_τ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ,(12)

where τ 𝜏\tau italic_τ is the scattering time, a 𝑎 a italic_a and b 𝑏 b italic_b are renormalization factors. The specific heat particularly is renormalized by b/a 𝑏 𝑎 b/a italic_b / italic_a and b 𝑏 b italic_b is simply the factor renormalizing the density of states at the Fermi surface. To see the latter, we note that when τ 𝜏\tau italic_τ is large, we can approximate

ρ⁢(0,𝐤)=−2⁢I⁢m⁢G R⁢(0,𝐤)≈2⁢π⁢δ⁢(b−1⁢ϵ k)𝜌 0 𝐤 2 I m subscript 𝐺 𝑅 0 𝐤 2 𝜋 𝛿 superscript 𝑏 1 subscript italic-ϵ 𝑘\displaystyle\rho(0,\mathbf{k})=-2{\rm Im},G_{R}(0,\mathbf{k})\approx 2\pi% \delta(b^{-1}\epsilon_{k})italic_ρ ( 0 , bold_k ) = - 2 roman_I roman_m italic_G start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( 0 , bold_k ) ≈ 2 italic_π italic_δ ( italic_b start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ϵ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) =2⁢π⁢b⁢δ⁢(ϵ k).absent 2 𝜋 𝑏 𝛿 subscript italic-ϵ 𝑘\displaystyle=2\pi b\delta(\epsilon_{k}).= 2 italic_π italic_b italic_δ ( italic_ϵ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) .(13)

In order to do the integral Eq.(8), we proceed at follows. Firstly, at zero temperature

∂∂ω⁢n F⁢(ω)=−δ⁢(ω).𝜔 subscript 𝑛 𝐹 𝜔 𝛿 𝜔\displaystyle\frac{\partial}{\partial\omega}n_{F}(\omega)=-\delta({\omega}).divide start_ARG ∂ end_ARG start_ARG ∂ italic_ω end_ARG italic_n start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ω ) = - italic_δ ( italic_ω ) .(14)

Secondly, since the integral gets contribution Eq.(8) would get contribution only from ω=0 𝜔 0\omega=0 italic_ω = 0 and also from the Fermi surface when τ 𝜏\tau italic_τ is large, we can approximate

ρ⁢(ω,𝐤)2≈−2⁢π⁢b⁢δ⁢(ϵ k)×2⁢τ 𝜌 superscript 𝜔 𝐤 2 2 𝜋 𝑏 𝛿 subscript italic-ϵ 𝑘 2 𝜏\displaystyle\rho(\omega,\mathbf{k})^{2}\approx-2\pi b\delta(\epsilon_{k})% \times 2\tau italic_ρ ( italic_ω , bold_k ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≈ - 2 italic_π italic_b italic_δ ( italic_ϵ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) × 2 italic_τ(15)

where we have utilized (A) for the first factor, and then substituted ω=0 𝜔 0\omega=0 italic_ω = 0 and ϵ k=0 subscript italic-ϵ 𝑘 0\epsilon_{k}=0 italic_ϵ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 0 in the second factor. It is then easy to see that Eq.(8) reduces to

σ dc=e 2 m 2⁢τ⁢b 4⁢π×2×m⁢∫0∞d y⁢y⁢δ⁢(y−2⁢m⁢E F)subscript 𝜎 dc superscript 𝑒 2 superscript 𝑚 2 𝜏 𝑏 4 𝜋 2 𝑚 superscript subscript 0 differential-d 𝑦 𝑦 𝛿 𝑦 2 𝑚 subscript 𝐸 𝐹\displaystyle\sigma_{\rm dc}=\frac{e^{2}}{m^{2}}\tau\frac{b}{4\pi}\times 2% \times m\int_{0}^{\infty}{\rm d}y,,y\delta(y-2mE_{F})italic_σ start_POSTSUBSCRIPT roman_dc end_POSTSUBSCRIPT = divide start_ARG italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_τ divide start_ARG italic_b end_ARG start_ARG 4 italic_π end_ARG × 2 × italic_m ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT roman_d italic_y italic_y italic_δ ( italic_y - 2 italic_m italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT )(16)

with y=k 2 𝑦 superscript 𝑘 2 y=k^{2}italic_y = italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT (note we do the ω 𝜔\omega italic_ω integral first). Therefore

σ dc=e 2⁢E F π⁢b⁢τ=n⁢e 2 m⁢b⁢τ,subscript 𝜎 dc superscript 𝑒 2 subscript 𝐸 𝐹 𝜋 𝑏 𝜏 𝑛 superscript 𝑒 2 𝑚 𝑏 𝜏\displaystyle\sigma_{\rm dc}=e^{2}\frac{E_{F}}{\pi}b\tau=\frac{ne^{2}}{m}b\tau,italic_σ start_POSTSUBSCRIPT roman_dc end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT end_ARG start_ARG italic_π end_ARG italic_b italic_τ = divide start_ARG italic_n italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_m end_ARG italic_b italic_τ ,(17)

since n/m=E F/π 𝑛 𝑚 subscript 𝐸 𝐹 𝜋 n/m=E_{F}/\pi italic_n / italic_m = italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT / italic_π for the Fermi liquid.

We can similarly proceed to evaluate Eq.(IV) and obtain the Hall conductivity.

We use the approximation (A) for ρ⁢(ω,𝐤)𝜌 𝜔 𝐤\rho(\omega,\mathbf{k})italic_ρ ( italic_ω , bold_k ). Since the integral then gets contribution from ω=0 𝜔 0\omega=0 italic_ω = 0 and the Fermi surface only, we can use

∂∂k x⁢Re⁢G R⁢(ω,𝐤)≈−b−1⁢τ 2⁢∂ϵ k∂k x=−b−1⁢τ 2⁢k x m subscript 𝑘 𝑥 Re subscript 𝐺 𝑅 𝜔 𝐤 superscript 𝑏 1 superscript 𝜏 2 subscript italic-ϵ 𝑘 subscript 𝑘 𝑥 superscript 𝑏 1 superscript 𝜏 2 subscript 𝑘 𝑥 𝑚\displaystyle\frac{\partial}{\partial k_{x}}{\rm Re}G_{R}(\omega,\mathbf{k})% \approx-,b^{-1}\tau^{2}\frac{\partial,\epsilon_{k}}{\partial k_{x}}=-,b^{-1% }\tau^{2}\frac{k_{x}}{m}divide start_ARG ∂ end_ARG start_ARG ∂ italic_k start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_ARG roman_Re italic_G start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_ω , bold_k ) ≈ - italic_b start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG ∂ italic_ϵ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_k start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_ARG = - italic_b start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_k start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_ARG start_ARG italic_m end_ARG(18)

Therefore,

σ H=ω c⁢e 2⁢τ 2 2⁢π 2⁢∫d 2⁢k⁢δ⁢(ϵ k)⁢(k x m)2 subscript 𝜎 H subscript 𝜔 𝑐 superscript 𝑒 2 superscript 𝜏 2 2 superscript 𝜋 2 superscript d 2 𝑘 𝛿 subscript italic-ϵ 𝑘 superscript subscript 𝑘 𝑥 𝑚 2\displaystyle\sigma_{\rm H}=\frac{\omega_{c}e^{2}\tau^{2}}{2\pi^{2}}\int{\rm d% }^{2}k,,\delta(\epsilon_{k})\left(\frac{k_{x}}{m}\right)^{2}italic_σ start_POSTSUBSCRIPT roman_H end_POSTSUBSCRIPT = divide start_ARG italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∫ roman_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_k italic_δ ( italic_ϵ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ( divide start_ARG italic_k start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_ARG start_ARG italic_m end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT

Now

∫d 2⁢k⁢δ⁢(ϵ k)⁢(k x m)2=2 m⁢∫d 2⁢k⁢δ⁢(k x 2+k y 2−2⁢m⁢E F)⁢k x 2 superscript d 2 𝑘 𝛿 subscript italic-ϵ 𝑘 superscript subscript 𝑘 𝑥 𝑚 2 2 𝑚 superscript d 2 𝑘 𝛿 superscript subscript 𝑘 𝑥 2 superscript subscript 𝑘 𝑦 2 2 𝑚 subscript 𝐸 𝐹 superscript subscript 𝑘 𝑥 2\displaystyle\int{\rm d}^{2}k,,\delta(\epsilon_{k})\left(\frac{k_{x}}{m}% \right)^{2}=\frac{2}{m}\int{\rm d}^{2}k,,\delta(k_{x}^{2}+k_{y}^{2}-2mE_{F})% ,{k_{x}}^{2}∫ roman_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_k italic_δ ( italic_ϵ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ( divide start_ARG italic_k start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_ARG start_ARG italic_m end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG 2 end_ARG start_ARG italic_m end_ARG ∫ roman_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_k italic_δ ( italic_k start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_k start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_m italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ) italic_k start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT =2 m⁢∫0∞d k⁢k 3⁢δ⁢(k 2−2⁢m⁢E F)absent 2 𝑚 superscript subscript 0 differential-d 𝑘 superscript 𝑘 3 𝛿 superscript 𝑘 2 2 𝑚 subscript 𝐸 𝐹\displaystyle\hskip 85.35826pt=\frac{2}{m}\int_{0}^{\infty}{\rm d}k,,k^{3},% \delta(k^{2}-2mE_{F})= divide start_ARG 2 end_ARG start_ARG italic_m end_ARG ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT roman_d italic_k italic_k start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_δ ( italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_m italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ) ×∫0 2⁢π d θ cos 2 θ\displaystyle\hskip 142.26378pt\times\int_{0}^{2\pi}{\rm d}\theta,{\rm cos}^{% 2}\theta× ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 italic_π end_POSTSUPERSCRIPT roman_d italic_θ roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ =π m⁢∫0∞d y⁢y⁢δ⁢(y−2⁢m⁢E F)absent 𝜋 𝑚 superscript subscript 0 differential-d 𝑦 𝑦 𝛿 𝑦 2 𝑚 subscript 𝐸 𝐹\displaystyle\hskip 85.35826pt=\frac{\pi}{m}\int_{0}^{\infty}{\rm d}y,,y,% \delta(y-2mE_{F})= divide start_ARG italic_π end_ARG start_ARG italic_m end_ARG ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT roman_d italic_y italic_y italic_δ ( italic_y - 2 italic_m italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ) =2⁢π⁢E F.absent 2 𝜋 subscript 𝐸 𝐹\displaystyle\hskip 85.35826pt=2,\pi E_{F}.= 2 italic_π italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT .

Therefore,

σ H=E F π⁢e 2⁢ω c⁢τ 2=n⁢e 2 m⁢ω c⁢τ 2.subscript 𝜎 H subscript 𝐸 𝐹 𝜋 superscript 𝑒 2 subscript 𝜔 𝑐 superscript 𝜏 2 𝑛 superscript 𝑒 2 𝑚 subscript 𝜔 𝑐 superscript 𝜏 2\displaystyle\sigma_{\rm H}=\frac{E_{F}}{\pi}e^{2}\omega_{c}\tau^{2}=\frac{ne^% {2}}{m}\omega_{c}\tau^{2}.italic_σ start_POSTSUBSCRIPT roman_H end_POSTSUBSCRIPT = divide start_ARG italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT end_ARG start_ARG italic_π end_ARG italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG italic_n italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_m end_ARG italic_ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .(19)

The Hall coefficient of the Fermi liquid is

R=σ H B⁢σ dc 2=1 n⁢e×1 b 2.𝑅 subscript 𝜎 H 𝐵 superscript subscript 𝜎 dc 2 1 𝑛 𝑒 1 superscript 𝑏 2 R=\frac{\sigma_{\rm H}}{B\sigma_{\rm dc}^{2}}=\frac{1}{ne}\times\frac{1}{b^{2}}.italic_R = divide start_ARG italic_σ start_POSTSUBSCRIPT roman_H end_POSTSUBSCRIPT end_ARG start_ARG italic_B italic_σ start_POSTSUBSCRIPT roman_dc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG = divide start_ARG 1 end_ARG start_ARG italic_n italic_e end_ARG × divide start_ARG 1 end_ARG start_ARG italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG .(20)

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• Note (3)In such microstate models, after perturbations due to infalling matter, the throats decouple from each other absorbing energy from infalling matter, while the latter also decouples from the throats Kibe et al. (2023). Furthermore, the decoupled residue of the infalling matter (absorbed in the hair) is a time-dependent quantum state which non-isometrically encodes its initial state. The ringdown of the decoupled throats also encodes the same information transitorily and transfers the same to Hawking radiation. Thus information is replicated (but not cloned) as black hole complimentarity demands. The decoupling of all degrees of freedom provides the underlying mechanism for the black hole complimentarity to emerge and in our present context could be relevant for the robustness of the locality of the self-energy.