# Incoherent transport in a model for the strange metal phase: Memory-matrix
formalism

Emile Pangburn  Institut de Physique Théorique, Université Paris-Saclay, CEA,
CNRS, F-91191 Gif-sur-Yvette, France  Anurag Banerjee  Department of Physics,
Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel  Herman Freire
Instituto de Física, Universidade Federal de Goiás, 74.001-970, Goiânia-GO,
Brazil  Catherine Pépin  Institut de Physique Théorique, Université Paris-
Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, France

###### Abstract

We revisit a phenomenological model of fermions coupled to fluctuating bosons
that emerges from finite-momentum particle-particle pairs for describing the
strange metal phase in the cuprates. The incoherent bosons dominate the
transport properties for the resistivity and optical conductivity in the non-
Fermi liquid phase. Within the Kubo formalism, the resistivity is
approximately linear in temperature with a Drude form for the optical
conductivity, such that the Drude lifetime is inversely proportional to the
temperature. Additionally, when the bosons emerge from electron pairs with
spin-triplet symmetry, the magnetoresistance has a linear magnetic field
dependence. Here, we compute the transport properties of such bosons within
the memory-matrix approach that successfully captures the hydrodynamic regime.
This technique emerges as the appropriate framework for describing the
transport coefficients of the strange metal phase. Our analysis confirms the
$T$-linear resistivity due to the Umklapp scattering that we obtained for this
effective model. Finally, we provide new predictions regarding the variation
of the thermal conductivity with temperature and examine the validity of the
Wiedemann-Franz law.

##  I Introduction

One of the most enduring mysteries of quantum condensed matter physics is
arguably the strange metal phase of the cuprate superconductors Keimer  _et
al._ (2015); Zaanen (2019); Norman and Pépin (2003); Lee  _et al._ (2006). The
conventional metal obeys universal laws for the variation of transport
coefficients with temperature. At low temperatures, the inverse of the
transport time typically goes like $\tau^{-1}\sim T^{2}$; hence, the
longitudinal conductivity displays $\sigma_{xx}\sim T^{-2}$, whereas the Hall
conductivity is given by $\sigma_{xy}\sim T^{-4}$ . Within the Fermi liquid
theory, which describes the behavior of conventional metals, electronic
quasiparticles are the sole type of charge carriers and, therefore, the Hall
angle becomes $\cot\theta_{H}=\sigma_{xx}/\sigma_{xy}\sim T^{2}$.

By contrast, the experimental data in the strange metal phase of the cuprates
display striking discrepancies with the standard Fermi liquid picture Gurvitch
and Fiory (1987); Emery and Kivelson (1995); Hussey  _et al._ (2004). Firstly,
the experimental observations demonstrate that $\sigma_{xx}\sim T^{-1}$ Legros
_et al._ (2018) and, at the same time, $\cot\theta_{H}\sim T^{2}$ Clayhold
_et al._ (1989); Ando and Murayama (1999); Barišić  _et al._ (2019).
Therefore, the transport time induced from the longitudinal conductivity
scales as $\tau\sim T^{-1}$, while the “Hall lifetime” varies as $\tau_{H}\sim
T^{-2}$, which is commonly referred to as the “separation of lifetimes” in the
literature Coleman  _et al._ (1996); Varma  _et al._ (1989); Abrahams and
Varma (2003). Furthermore, the Wiedemann-Franz law is satisfied in the strange
metal phase almost down to $T=0$, which apparently indicates the existence of
quasiparticles Proust  _et al._ (2002a); Michon  _et al._ (2018a);
Grissonnanche  _et al._ (2016a). The notable disagreement between the
different experimental data with the Fermi-liquid paradigm makes the strange
metal phase of the cuprates one of the biggest enigmas of correlated quantum
matter Keimer  _et al._ (2015); Varma  _et al._ (1989); Patel and Sachdev
(2014); Patel  _et al._ (2018); Zaanen (2019).

Interestingly, recent study Yang  _et al._ (2022) reveals a bosonic strange
metal phase in nanopatterned YBCO samples. In that work, the longitudinal
conductivity shows a linear temperature and field dependence along with a
vanishing Hall coefficient, as soon as the bosonic transport sets in Yang  _et
al._ (2022). In the present manuscript, we address the longstanding issue by
proposing a new type of bosonic excitation that can potentially describe the
strange metal phase in the cuprates Putzke  _et al._ (2021); Barišić  _et al._
(2019); Bozovic  _et al._ (2004); Caprara  _et al._ (2017); Efetov  _et al._
(2013); Kontani (2008); Merino and McKenzie (2000). The physical picture
underlying our phenomenological model Banerjee  _et al._ (2021); Pépin and
Freire (2022) can be summarized as follows: At high energies, the microscopic
lattice model generates fluctuating finite momenta bosons created from
particle-particle pairs (remnants of a pair-density-wave) and the constituent
fermions. These fluctuating bosons carry an electric charge of $2e$ and,
hence, also contribute to the charge transport. Moreover, due to the
scattering with fermions, such bosons become incoherent as the bosonic
propagator becomes
$D^{-1}(\mathbf{q},\omega)=-i\omega+\mathbf{q}^{2}+m_{b}(T)$ (where $m_{b}(T)$
is the temperature-dependent bosonic mass). Such bosons in two dimensions
indeed lead to a $T$-linear contribution to the longitudinal conductivity and
also the optical conductivity attains a Drude form $\sigma_{xx}^{\rm
boson}(\omega)\sim(\tau_{\rm boson}^{-1}-i\omega)^{-1}$, as shown in Ref.
Banerjee  _et al._ (2021).

Furthermore, since the bosons are incoherent, they do not contribute to the
transverse Hall conductivity $\sigma_{xy}^{\rm boson}=0$. Such a general
picture could explain the experimental data if the bosons are light enough to
short-circuit the fermions for the longitudinal conductivity. Since the
fermions are also present in this model, there must be scattering between
these two excitations. At low temperatures, scattering via incoherent bosons
with finite momentum produces a finite lifetime for the fermions, at least on
parts of the Fermi surface, with the generation of “cold spots” where the
lifetime is typical of the Fermi liquid theory with scattering rate
$\tau_{cold}^{-1}\sim T^{2}$, and “hot spots” where the dominant scattering is
through the incoherent bosons, leading to a scattering rate
$\tau_{hot}^{-1}\sim T^{\alpha}$, with $1<\alpha<1.5$ Hlubina and Rice (1995);
Rosch (1999). The modeling of the Hall conductivity on a Fermi surface with an
anisotropic lifetime has been treated in another study to describe the strange
metal phase of the cuprates Kokalj  _et al._ (2012). The angular average on
the Fermi surface favors Kokalj  _et al._ (2012) the “hot regions” for the
Hall conductivity, leading to an average Hall inverse lifetime
$\tau_{H}^{-1}\sim T^{3/2}$. Our study thus combines the two types of
excitations (bosons and fermions) to give a new perspective to the old
paradox, such that the Hall angle becomes $\cot\theta_{H}\sim T^{3}/T\sim
T^{2}$, consistent with the experiments.

In the present paper, we revisit this problem and confirm our key results that
such charge-two incoherent bosons in two dimensions contribute to the
longitudinal conductivity as $\sigma_{xx}\sim T^{-1}$. To this end, we
investigate the transport properties of such bosons using the memory-matrix
technique (that successfully captures the hydrodynamic regime). Consequently,
the $T$-linear resistivity regime of the incoherent charged bosons with finite
momentum due to Umklapp interactions stands on firm ground. Finally, we
provide new predictions regarding thermal conductivity as a function of
temperature in the model and also discuss the validity of the Wiedemann-Franz
law for this system.

##  II The model

We consider here a two-dimensional phenomenological model Banerjee  _et al._
(2021) of fluctuating charge-two bosons interacting with each other and among
themselves. The bosons are in a “soup” of fermions, and the corresponding
fermion-boson scattering affects the boson lifetime significantly (to be
explained below). The bosonic part of the Hamiltonian of the model with $N$
“flavors” is given by

$\displaystyle\hat{\mathcal{H}}$
$\displaystyle=\sum_{\mathbf{q}}b_{\mathbf{q}}^{\dagger}\left[\frac{\text{$\left|\mathbf{q}\right|$}^{2}}{2m_{b}}+\mu_{0}\right]b_{\mathbf{q}}+\frac{\lambda^{2}}{2N}\sum_{\mathbf{q}}Q_{\text{{q
}}}Q_{-\mathbf{q}},$ (1)

where $\mu_{0}$ and $\lambda$ denote, respectively, the bare bosonic mass term
and the boson-boson interaction, and
$Q_{\mathbf{q}}=\sum_{\mathbf{k}}b_{\mathbf{k}+\mathbf{q}}^{\dagger}b_{\mathbf{k}}$,
where $b_{\mathbf{k}}^{\dagger}$ ($b_{\mathbf{k}}$) is the creation
(annihilation) operator for a boson with momentum $\mathbf{k}$. Although the
spin index is not shown for simplicity, we allow for the possibility that the
bosons have either spin-zero or spin-one. As mentioned above, the model also
possesses a “background” of fermions (not shown in the Hamiltonian of Eq. (1))
and the corresponding fermion-boson scattering processes lead to retardation
effects, which are taken into account via the one-loop bosonic self-energy
$\Sigma\left(\omega\right)=-i\gamma\omega$, where the Landau-damping constant
$\gamma$ is given by
${\gamma=g_{I}^{2}\mathcal{N}(\epsilon_{F})/(2\pi\sqrt{(2k_{F}Q_{0})^{2}-Q_{0}^{4}})}$,
with $\mathcal{N}(\epsilon_{F})$ being the density of states at the Fermi
energy, $k_{F}$ is the corresponding Fermi momentum, $Q_{0}$ is the finite
momentum of the bosons and $g_{I}$ is the fermion-boson interaction. In a
previous work Banerjee  _et al._ (2021), we have demonstrated using the Kubo
formula that this effective model indeed displays a quantum critical phase
with approximately $T$-linear resistivity and shows a Drude form for the
optical conductivity.

We now proceed to calculate the transport properties of this effective model
within the memory-matrix (MM) formalism Forster (1975); Götze and Wölfle
(1972); Rosch and Andrei (2000) (for more information about the technicalities
of this method, see, e.g., Refs. Mahajan  _et al._ (2013); Patel and Sachdev
(2014); Lucas and Sachdev (2015); Hartnoll  _et al._ (2014); Vieira  _et al._
(2020); Freire (2017a, b, 2018, 2014); Zaanen  _et al._ (2015); Hartnoll  _et
al._ (2018); Mandal and Freire (2021); Freire and Mandal (2021); Mandal and
Freire (2022); Wang and Berg (2019, 2022)). The MM approach emerges as a more
suitable framework to describe the non-Fermi liquid phase exhibited, since
$(i)$ it does not rely on the existence of well-defined quasiparticles at low
energies, and $(ii)$ it successfully captures the hydrodynamic regime that is
expected to describe the non-equilibrium dynamics of this strongly correlated
metallic phase.

Here, we will follow an approach similar to that used in the recent work by
Wang and Berg Wang and Berg (2019) to calculate transport properties in the
context of an Ising-nematic quantum critical theory. In this way, we will
project the non-equilibrium dynamics of the present model in terms of slowly-
varying operators that are nearly conserved. Naturally, the boson operators
$n_{\mathbf{k}}=b_{\mathbf{k}}^{\dagger}b_{\mathbf{k}}$ turn out to be nearly
conserved in the limit of either small $\lambda$ or large $N$, since the
equation of motion for these operators is given by

$\displaystyle\dot{n}_{\mathbf{k}}$ $\displaystyle=i\left[\hat{{\cal
H}},n_{\mathbf{k}}\right]$
$\displaystyle=\frac{2\lambda^{2}i}{N}\sum_{\mathbf{q}}Q_{-\mathbf{q}}\left(b_{\mathbf{k}+\mathbf{q}}^{\dagger}b_{\mathbf{k}}-b_{\mathbf{k}}^{\dagger}b_{\mathbf{k}-\mathbf{q}}\right)-h.c..$
(2)

In the MM formalism, to leading order in $1/N$, the memory matrix writes

$\displaystyle M_{\mathbf{k}\mathbf{k}^{\prime}}\left(\Omega\right)$
$\displaystyle=\frac{1}{i\Omega}\left[G_{\dot{n}_{\mathbf{k}}\dot{n}_{\mathbf{k^{\prime}}}}^{R}\left(\Omega\right)-G_{\dot{n}_{\mathbf{k}}\dot{n}_{\mathbf{k^{\prime}}}}^{R}\left(0\right)\right],$
(3)

where $G_{AB}^{R}\left(\Omega\right)$ is the retarded Green’s function for
nearly-conserved operators $A$ and $B$, which is calculated to zeroth order in
$1/N$. The MM turns out to be a generalization of the scattering rate in
Boltzmann theory (but applicable to strongly correlated systems in which this
quantity may not be well-defined) and enters as a retardation process in the
calculation of the optical conductivity $\sigma(\Omega)$ and the thermal
conductivity at zero electric field $\bar{\kappa}(\Omega)$ in the following
way

$\displaystyle\sigma\left(\Omega\right)$
$\displaystyle=\sum_{\mathbf{k}\mathbf{k^{\prime}}}\chi_{J_{\mathbf{k}}n_{\mathbf{k}}}\left(\frac{1}{M_{\mathbf{k}\mathbf{k^{\prime}}}\left(\Omega\right)-i\Omega\chi_{\mathbf{k}\mathbf{k^{\prime}}}}\right)\chi_{J_{\mathbf{k^{\prime}}}n_{\mathbf{k^{\prime}}}},$
(4) $\displaystyle\bar{\kappa}\left(\Omega\right)$
$\displaystyle=\frac{1}{T}\sum_{\mathbf{k}\mathbf{k^{\prime}}}\chi_{J_{\mathbf{k}}^{\mathcal{Q}}n_{\mathbf{k}}}\left(\frac{1}{M_{\mathbf{k}\mathbf{k^{\prime}}}\left(\Omega\right)-i\Omega\chi_{\mathbf{k}\mathbf{k^{\prime}}}}\right)\chi_{J_{\mathbf{k^{\prime}}}^{\mathcal{Q}}n_{\mathbf{k^{\prime}}}},$
(5)

where $J_{\mathbf{k}}$ and $J_{\mathbf{k}}^{\mathcal{Q}}$ are, respectively,
the electric current and the thermal current operators of the model, with the
corresponding susceptibilities given by
$\chi_{J_{\mathbf{k}}n_{\mathbf{k}}}=\int_{0}^{\beta}d\tau\left\langle
J_{\mathbf{k}}\left(\tau\right)n_{\mathbf{k}}\left(0\right)\right\rangle$,
$\chi_{\mathbf{k}\mathbf{k^{\prime}}}=\int_{0}^{\beta}d\tau\left[\left\langle
n_{\mathbf{k}}\left(\tau\right)n_{\mathbf{k^{\prime}}}\text{$\left(0\right)$}\right\rangle-\left\langle
n_{\mathbf{k}}\right\rangle\left\langle
n_{\mathbf{k^{\prime}}}\right\rangle\right]$, and
$\chi_{J_{\mathbf{k}}^{\mathcal{Q}}n_{\mathbf{k}}}=\int_{0}^{\beta}d\tau\left\langle
J_{\mathbf{k}}^{\mathcal{Q}}\left(\tau\right)n_{\mathbf{k}}\left(0\right)\right\rangle$.

We point out that the thermal conductivity at zero electric current of the
model (which will be denoted here by $\kappa$) is given by
$\kappa=\bar{\kappa}-T\alpha^{2}/\sigma$, where $\alpha$ is the thermoelectric
coefficient. Since we have demonstrated in a previous work Banerjee  _et al._
(2021) that the present model has particle-hole symmetry, the critical
contribution to the thermoelectric response is expected to vanish. Therefore,
in this case, the thermal conductivity at zero electric current will be equal
to the thermal conductivity at zero electric field (i.e.,
$\kappa=\bar{\kappa}$).

Furthermore, for clean systems, if no coupling to the lattice is present, the
memory matrix of the model also vanishes identically (see Appendix A).
However, if Umklapp terms are taken into account, we get after contracting the
vertices the following result:

$\displaystyle
M_{\mathbf{k}\mathbf{k^{\prime}}}\left(\Omega\right)=8\,\delta_{\mathbf{k}\mathbf{k^{\prime}}}\sum_{\mathbf{p},\mathbf{q}}{\cal
M}_{\mathbf{k},\mathbf{p},\mathbf{q}},$ (6)

with the corresponding Feynman diagram given by Fig. 1.

Figure 1: The Feynman diagram for the calculation of the memory matrix in the
present model.

##  III Results

###  III.1 Evaluations

We now compute the various terms in Eqs. (4) and (5) within our MM formalism.
For convenience, we use units such that $e=k_{B}=\hbar=1$ from now on. We
start from the renormalized propagator for the incoherent bosons given by

$\displaystyle D_{\mathbf{k}}^{-1}$
$\displaystyle=\left|\omega_{n}\right|+\mu_{k}(T),$ (7)

with $\mu_{k}(T)=k^{2}+\frac{\lambda^{2}}{2N}T$. The damping term
$\left|\omega_{n}\right|$ (we have set $\gamma$=1) comes from the scattering
via fermionic carriers, and it is responsible for the incoherent character of
the bosons. The potential $\mu_{k}$ has a term $k^{2}$, which refers to the
bosonic dispersion. Note that the bosons have a mass scaling with temperature
(the $T$ term in $\mu_{k}$). It comes mainly from the Hartree diagram
generated by the four-boson interaction.

We begin with the evaluation of

$\displaystyle\chi_{J_{\mathbf{k}}n_{\mathbf{k}}}$
$\displaystyle={v}_{\mathbf{k}}\int_{0}^{\beta}d\tau\left\langle
n_{\mathbf{k}}\left(\tau\right)\right\rangle\left\langle
n_{\mathbf{k}}\left(0\right)\right\rangle+{v}_{\mathbf{k}}\chi_{\mathbf{k}\mathbf{k}}$
(8)
$\displaystyle=\chi_{J_{\mathbf{k}}n_{\mathbf{k}}}^{(a)}+{v}_{\mathbf{k}}\chi_{\mathbf{k}\mathbf{k}}.$

with $v_{\mathbf{k}}=\mathbf{k}/m_{b}$. The first term in Eq. (8) can be found
to be equal to (see Appendix B)

$\displaystyle\chi_{J_{\mathbf{k}}n_{\mathbf{k}}}^{(a)}$
$\displaystyle=\frac{{v}_{\mathbf{k}}n_{B}\left(\mu_{k}\right)}{\pi}\log\left(\frac{D}{\mu_{k}}\right),$
(9)

where $D$ is the bandwidth of the boson dispersion and $n_{B}$ is the Bose-
Einstein distribution. For the next generalized susceptibility, we have
$\chi_{\mathbf{k}\mathbf{k^{\prime}}}=\delta_{\mathbf{k}\mathbf{k^{\prime}}}T\sum_{\omega_{n}}D_{\mathbf{k}}^{2}$,
which gives (see Appendix B)

$\displaystyle\chi_{\mathbf{k}\mathbf{k^{\prime}}}$
$\displaystyle=\delta_{\mathbf{k}\mathbf{k^{\prime}}}\frac{T}{\mu_{k}^{2}}\left(\tanh^{-1}\frac{T}{\mu_{k}}\right)^{2}.$
(10)

###  III.2 Conductivity in the critical regime

In order to complete the evaluation of the optical conductivity, we first
notice that the summations over $\mathbf{k}$ and $\mathbf{k}^{\prime}$ in Eqs.
(4) and (5) vanish identically if the coupling to the lattice is not taken
into account. Umklapp terms with $\mathbf{k}^{\prime}=\mathbf{k}\pm
n\mathbf{U}$, with $\mathbf{U}$ being a reciprocal lattice wave vector,
generate a finite result for the optical conductivity. This result is obtained
by the scaling displayed in Eqs. (8)-(12) in the critical regime. In such a
regime, we assume that the amplitude of the wave vector $\mathbf{k}$ is of the
order of the bandwidth, whereas the temperature is smaller than the bandwidth
$T\ll D$. In this situation, the second term
$\chi_{J_{\mathbf{k}}n_{\mathbf{k}}}^{(b)}$ in Eq. (8) dominates over the
first term. Moreover, the MM evaluates to

$\displaystyle
M_{\mathbf{k}\mathbf{k^{\prime}}}\left(\Omega_{n}\right)=\delta_{\mathbf{k}\mathbf{k^{\prime}}}\frac{\lambda^{4}T^{2}}{N^{2}\Omega_{n}}\text{$\sum_{\omega_{n},p_{n}}\sum_{\mathbf{p},\mathbf{q}}D_{\mathbf{p}}D_{\mathbf{k}}D_{\mathbf{p}+\mathbf{q}}D_{\mathbf{k+}\mathbf{q}}$},$
(11)

which finally yields

$\displaystyle M_{\mathbf{k}\mathbf{k^{\prime}}}\left(\Omega\right)$
$\displaystyle=\frac{\delta_{\mathbf{k}\mathbf{k}^{\prime}}\lambda^{4}}{N^{2}}\frac{T^{3}}{192\pi^{3}\mu_{k}^{3}}\left(\frac{\mu_{k}T}{\mu_{k}^{2}+T^{2}}+\tanh^{-1}\frac{T}{\mu_{k}}\right).$
(12)

Lastly, the optical conductivity can be rewritten as

$\displaystyle\sigma\left(\Omega\right)$
$\displaystyle=\sum_{\mathbf{k}}\frac{\chi_{\mathbf{k}\mathbf{k}}}{\chi_{\mathbf{k}\mathbf{k}}^{-1}M_{\mathbf{k}\mathbf{k}}\left(\Omega\right)-i\Omega}.$
(13)

Noticing that the typical scaling relation $\mu_{k}\sim T$ holds in the
critical regime (because $T$ is the only energy scale in the problem and thus
$k^{2}\sim T$), the summation over $\mathbf{k}$ in Eq. (13) can be finally
performed. Scalings arguments lead to $\sum_{\mathbf{k}}\sim T$,
$M_{\mathbf{k}\mathbf{k}}\sim T^{0}$, $\chi_{\mathbf{k}\mathbf{k}}\sim
T^{-1}$, which result in a typical form for the optical conductivity given by

$\displaystyle\sigma\left(\Omega\right)$
$\displaystyle\sim\frac{1}{T-i\Omega}.$ (14)

The above result has also been confirmed in the dc limit by numerical means.

###  III.3 Lorenz ratio in the critical regime

The Wiedemann-Franz law Chester and Thellung (1961) for the Lorenz ratio
$\text{L}=\frac{\kappa}{\sigma T}$ is one of the most fundamental properties
of a Fermi liquid. It states that at low temperatures

$\displaystyle\lim_{T\rightarrow
0}\text{L}=\frac{\pi^{2}}{3}\equiv\text{L}_{0},$ (15)

in units of $k_{B}=e=1$. It reflects the fact that energy and charge are
carried by the same degrees of freedom. To compute this ratio, we compute the
thermal conductivity $\kappa$ using the same method with the following
substitution of the susceptibility
$\chi_{J_{\mathbf{k}}^{\mathcal{Q}}n_{{\mathbf{k}}}}=\epsilon_{{\mathbf{k}}}\chi_{J_{\mathbf{k}}n_{{\mathbf{k}}}}$.
At the critical regime, we get within the MM approach

$\displaystyle\kappa\left(\Omega\right)$
$\displaystyle=\frac{1}{T}\sum_{\mathbf{k}}\epsilon_{\mathbf{k}}^{2}\frac{\chi_{\mathbf{k}\mathbf{k}}}{\chi_{\mathbf{k}\mathbf{k}}^{-1}M_{\mathbf{k}\mathbf{k}}\left(\Omega\right)-i\Omega}.$
(16)

Scaling arguments lead to $\epsilon_{\mathbf{k}}\sim T$, which finally gives
for the thermal conductivity

$\displaystyle\kappa\left(\Omega\right)$
$\displaystyle\sim\frac{T}{T-i\Omega}.$ (17)

In the critical regime, the incoherent boson system obeys the correct scaling,
$\frac{\kappa}{\sigma T}\sim C$, with $C$ being a constant as $T\rightarrow
0$. However, it strongly violates the WF law because this constant is strongly
dependent on the boson-boson coupling $\dfrac{\lambda^{2}}{2N}$ (as shown in
Fig. 2) and so is model-dependent. This result is also confirmed using Kubo
linear response in Appendix C. Similar violation has been seen in the $t-J$
model Houghton  _et al._ (2002) and near heavy fermion quantum critical point
Kim and Pépin (2009).

In the cuprates, the WF law is satisfied in the overdoped Proust  _et al._
(2002b); Nakamae  _et al._ (2003); Sun  _et al._ (2009); Michon  _et al._
(2018b) and underdoped Grissonnanche  _et al._ (2016b) regime. This
experimental fact constrains the bosonic coupling $\lambda$ of our model. We
point out that this result can also be traced to the fact that we considered
Umklapp scattering as the sole mechanism for momentum relaxation in the
present calculation. Presumably, adding disorder via spatially random
interactions Chowdhury  _et al._ (2022) in our present model might lead to a
universal behavior in the Lorenz ratio, which would then be consistent with
experimental data. However, we leave the latter investigation for a future
work.

Figure 2: The Lorenz ratio $\text{L}=\kappa/(\sigma T)$ as a function of the
bosonic coupling $g=\dfrac{\lambda^{2}}{2N}$, computed within the Kubo
formalism. The same results are obtained with the memory matrix formalism. $T$
is fixed such that it satisfies $T>\dfrac{\mu_{0}}{g}$, in order to be in the
critical regime for all coupling values used.

##  IV Conclusions

In this paper, we have computed the transport properties of an effective model
when charged fluctuating bosons (charge-two particle-particle bosons in this
special case) are present at high energies in the phase diagram of the cuprate
superconductors. The main results are as follows:

  * •

A regime with approximately $T$-linear resistivity is obtained from the
transport properties of the boson-fermion “soup”. The charged bosons scatter
with the fermions and become overdamped via the Landau damping
$-i\gamma\omega$ (where $\omega$ is the real frequency). Optical conductivity
was also evaluated Banerjee  _et al._ (2021) and yields a Drude-like
conductivity for $\omega<T$ with a lifetime given by $\tau_{xx}=\tau_{b}\sim
T^{-1}$. This agrees with experimental observation Michon  _et al._ (2022).

  * •

At low temperatures, the boson transport in the conductivity is “short-
circuited” by the fermions that possess a transport lifetime $\tau_{f}\sim
T^{-2}$. Hence, the regime where the bosons dominate the conductivity has a
finite temperature range, which needs to be compared with the experimental
data (this will be performed below).

  * •

Due to Landau damping, the charged bosons do not contribute to the Hall
conductivity. This quantity is thus dominated by the fermions via hot-spot and
cold-spot physics. It has been argued by other authors Kokalj  _et al._ (2012)
that a phenomenological estimate of the Hall conductivity from the fermions
gives $\tau_{xy}\sim T^{-3/2}$, since the Hall (weighted) average around the
Fermi surface scans at the same time both the hot and cold region. Altogether,
in this regime, the cotangent Hall goes as
$\cot\theta_{H}\sim\tau_{xx}/\tau_{xy}^{2}\sim T^{3}/T\sim T^{2}$, which
corresponds to the experimental observation.

  * •

The thermal transport has also been calculated and it has been verified that
in the regime analyze here (i.e., the critical case) the Wiedemann-Franz law
is strongly violated at low temperatures with the dependence on the bosonic
coupling. By constraining this coupling, the WF law can be recovered to agree
with recent experimental data Michon  _et al._ (2018c), but as mentioned in
the main text, there is still room for improvement here. It would be
interesting to investigate also the effects of adding disorder via spatially
random interactions in our model.

Finally, we point out that although our present study might not yet be the
final solution for the strange metal phase of the cuprates, this perspective
opens a new viewpoint on the physics of those compounds. The main idea is that
at high energy scales, due to the strong superexchange interaction which
brings the system to the regime of strong coupling, bosons of charge-zero
(particle-hole) and of charge-two (particle-particle) with a spectrum of wave
vectors are generated. When the temperature is lowered, some of those bosons
condense at a specific wave vector, which give rise to various kind of orders,
like, e.g., charge modulations or stripe physics depending on the compounds.
Some bosons are unstable, like the charge-two finite vector ones related,
e.g., to pair-density wave (PDW) order. In a recent line of ideas Chakraborty
_et al._ (2019); Grandadam  _et al._ (2020), the instability of the finite
momentum charge-two bosons could be described by a theory of
“fractionalization” in which these excitations become entangled, opening a gap
in the antinodal region of the Brillouin zone. In the strange metal phase,
they also give rise to the regime of $T$-linear resistivity described in the
present paper.

As mentioned above, it is worth plugging numbers to determine the possible
boundaries of the obtained strange metal regime, which is valid between
$T_{min}<T<T_{max}$. If we assume that $T_{max}$ can be as high as 1000 K, the
constraint on $T_{min}$ is that the fermions should short-circuit the bosons
for $T<T_{min}$. So we may impose that, for $T\lesssim T_{min}$, we have that
$\sigma_{f}\gtrsim\sigma_{b}$. For a rough estimate, we can then use for the
fermions that $\sigma_{f}=ne^{2}\tau_{f}/m_{f}^{*}$, where $m_{f}^{*}$ is the
quasiparticle effective mass. For the bosons, since we showed that they also
obey the Drude form for the conductivity, we have that
$\sigma_{b}=n_{b}\left(2e\right)^{2}\tau_{b}/m_{b}$, where $m_{b}$ is the mass
introduced in Eq. (1). From Ref. Kokalj  _et al._ (2012), one obtains that
$\sigma_{f}^{-1}=A\,T^{2}$, with $A\simeq 5\cdot
10^{-3}\mu\Omega\cdot\text{cm}$. Thus, the reasonable range allowed
experimentally for $T_{min}$ is $1\text{ K}<T_{min}<10\text{ K}$. If
$T_{min}=1\text{ K}$, one would get $m_{b}/m_{f}^{*}\sim 0.02$, whereas if
$T_{min}=10\text{ K}$ the ratio of masses should be given by
$m_{b}/m_{f}^{*}\sim 0.2$. Given the strong mass renormalization of the
fermions inside the strange metal phase of the present model, this range of
the ratio of masses is conceivable in order to allow for a wide fluctuation
regime where the incoherent bosons dominate the transport properties with
respect to the fermions in the context of the resistivity of this non-Fermi
liquid phase.

##  V Acknowledgments

We are grateful for discussion with G. Grissonnanche, N. Hussey, B. Ramshaw,
L. Taillefer on various experimental issues. H.F. acknowledges funding from
CNPq under Grant No. 311428/2021-5. A.B. acknowledges support from the
Kreitman School of Advanced Graduate Studies and European Research Council
(ERC) Grant Agreement No. 951541, ARO (W911NF-20-1-0013).

##  Appendix A Diagram contractions

We define the vertices as follows

and their conjugate counterpart. The vertices write

$\displaystyle
V_{\mathbf{k}}=\sum_{\mathbf{p},\mathbf{q}}V_{\mathbf{k},\mathbf{p},\mathbf{q}},$
(18) $\displaystyle\mbox{with
}V_{\mathbf{k,}\mathbf{p},\mathbf{q}}=V_{\mathbf{k},\mathbf{p},\mathbf{q}}^{a}-V_{\mathbf{k},\mathbf{p},\mathbf{q}}^{b}-h.c..$

We construct the function ${\cal M}_{\mathbf{k},\mathbf{p},\mathbf{q}}$ by
contracting

$\displaystyle{\cal M}_{\mathbf{k},\mathbf{p},\mathbf{q}}$
$\displaystyle=\left\langle
V_{\mathbf{k},\mathbf{p},\mathbf{q}}\cdot\bar{V}_{\mathbf{k^{\prime},}\mathbf{p^{\prime}},\mathbf{q^{\prime}}}\right\rangle,$
(19) $\displaystyle=\left<-V_{\mathbf{k},\mathbf{p},\mathbf{q}}^{a}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{a}-V_{\mathbf{k},\mathbf{p},\mathbf{q}}^{b}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{b}\right.$
$\displaystyle+\bar{V}_{\mathbf{k},\mathbf{p},\mathbf{q}}^{a}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{a}+\bar{V}_{\mathbf{k},\mathbf{p},\mathbf{q}}^{b}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{b}$
$\displaystyle+V_{\mathbf{k},\mathbf{p},\mathbf{q}}^{a}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{b}+V_{\mathbf{k},\mathbf{p},\mathbf{q}}^{b}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{a}$
$\displaystyle-\left.\bar{V}_{\mathbf{k},\mathbf{p},\mathbf{q}}^{a}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{b}-\bar{V}_{\mathbf{k},\mathbf{p},\mathbf{q}}^{b}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{a}+h.c.\right>.$

There are eight types of contractions

$\displaystyle 1)$ $\displaystyle\left\langle
V_{\mathbf{k},\mathbf{p},\mathbf{q}}^{a}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{a}\right\rangle=\delta_{\mathbf{k},\mathbf{k^{\prime}}-\mathbf{q}}\delta_{\mathbf{p^{\prime},}\mathbf{p}+\mathbf{q}}\delta_{\mathbf{q^{\prime},-}\mathbf{q}}{\cal
M}_{\mathbf{k},\mathbf{p},\mathbf{q}},$ (20) $\displaystyle 2)$
$\displaystyle\left\langle V_{\mathbf{k},\mathbf{p},\mathbf{q}}^{b}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{b}\right\rangle=\delta_{\mathbf{k},\mathbf{k^{\prime}}+\mathbf{q}}\delta_{\mathbf{p^{\prime},}\mathbf{p}+\mathbf{q}}\delta_{\mathbf{q^{\prime},-}\mathbf{q}}{\cal
M}_{\mathbf{k},\mathbf{p},\mathbf{q}},$ $\displaystyle 3)$
$\displaystyle\left\langle V_{\mathbf{k},\mathbf{p},\mathbf{q}}^{a}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{b}\right\rangle=\delta_{\mathbf{k},\mathbf{k^{\prime}}}\delta_{\mathbf{p^{\prime},}\mathbf{p}+\mathbf{q}}\delta_{\mathbf{q^{\prime},-}\mathbf{q}}{\cal
M}_{\mathbf{k},\mathbf{p},\mathbf{q}},$ $\displaystyle 4)$
$\displaystyle\left\langle V_{\mathbf{k},\mathbf{p},\mathbf{q}}^{b}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{a}\right\rangle=\left\langle
V_{\mathbf{k},\mathbf{p},\mathbf{q}}^{a}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{b}\right\rangle,$
$\displaystyle 5)$
$\displaystyle\left\langle\bar{V}_{\mathbf{k},\mathbf{p},\mathbf{q}}^{a}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{a}\right\rangle=\delta_{\mathbf{k},\mathbf{k^{\prime}}}\delta_{\mathbf{p^{\prime},}\mathbf{p}}\delta_{\mathbf{q^{\prime},q}}{\cal
M}_{\mathbf{k},\mathbf{p},\mathbf{q}},$ $\displaystyle 6)$
$\displaystyle\left\langle\bar{V}_{\mathbf{k},\mathbf{p},\mathbf{q}}^{b}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{b}\right\rangle=\left\langle\bar{V}_{\mathbf{k},\mathbf{p},\mathbf{q}}^{a}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{a}\right\rangle,$
$\displaystyle 7)$
$\displaystyle\left\langle\bar{V}_{\mathbf{k},\mathbf{p},\mathbf{q}}^{a}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{b}\right\rangle=\delta_{\mathbf{k},\mathbf{k^{\prime}}-\mathbf{q}}\delta_{\mathbf{p^{\prime},}\mathbf{p}+\mathbf{q}}\delta_{\mathbf{q^{\prime},q}}{\cal
M}_{\mathbf{k},\mathbf{p},\mathbf{q}},$ $\displaystyle 8)$
$\displaystyle\left\langle\bar{V}_{\mathbf{k},\mathbf{p},\mathbf{q}}^{b}\cdot
V_{\mathbf{k^{\prime}},\mathbf{p^{\prime}},\mathbf{q^{\prime}}}^{a}\right\rangle=\delta_{\mathbf{k},\mathbf{k^{\prime}}+\mathbf{q}}\delta_{\mathbf{p^{\prime},}\mathbf{p}+\mathbf{q}}\delta_{\mathbf{q^{\prime},q}}{\cal
M}_{\mathbf{k},\mathbf{p},\mathbf{q}}.$

Altogether, we finally have

$\displaystyle M_{\mathbf{k}\mathbf{k^{\prime}}}\left(\Omega\right)$
$\displaystyle=4\sum_{\mathbf{p},\mathbf{q}}\left(\delta_{\mathbf{k},\mathbf{k^{\prime}}}-\delta_{\mathbf{k},\mathbf{k^{\prime}}\pm\mathbf{q}}\right){\cal
M}_{\mathbf{k},\mathbf{p},\mathbf{q}},$ (21)

which vanishes if the coupling to the lattice is absent. Umklapp terms provide
a nonzero value to $M_{\mathbf{k}\mathbf{k^{\prime}}}\left(\Omega\right)$,
leading to Eq. (12) presented in the main part of the manuscript.

##  Appendix B Evaluations

###  B.1 Susceptibilities

After performing a spectral decomposition of the boson Green’s function Eq.
(7), the first part of the susceptibility
$\chi_{J_{\mathbf{k}}n_{\mathbf{k}}}$ writes

$\displaystyle\chi_{J_{\mathbf{k}}n_{\mathbf{k}}}^{(a)}$
$\displaystyle={v}_{\mathbf{k}}\int_{-D}^{D}\frac{dE}{2\pi}n_{B}\left(E\right)\frac{E}{E^{2}+\mu_{k}^{2}}$
(22)
$\displaystyle={v}_{\mathbf{k}}\frac{n_{B}\left(\mu_{k}\right)}{\pi}\log\left(\frac{D}{\mu_{k}}\right),$

with the convention $n_{B}\left(E\right)=\coth\frac{\beta E}{2}$.

As for the susceptibility $\chi_{\mathbf{k}\mathbf{k}}$, we proceed in the
same way with a spectral decomposition of each boson propagator given by

$\displaystyle\chi_{\mathbf{k}\mathbf{k}}$
$\displaystyle=\int\frac{dE_{1}dE_{2}}{\left(2\pi\right)^{2}}\frac{n_{B}\left(E_{1}\right)-n_{B}\left(E_{2}\right)}{-E_{1}+E_{2}}\frac{E_{1}}{E_{1}^{2}+\mu_{k}^{2}}\frac{E_{2}}{E_{2}^{2}+\mu_{k}^{2}}$
(23)
$\displaystyle\simeq\int_{-T}^{T}\int_{-T}^{T}\frac{dE_{1}dE_{2}}{\left(2\pi\right)^{2}}\frac{T}{\left(E_{1}^{2}+\mu_{k}^{2}\right)\left(E_{2}^{2}+\mu_{k}^{2}\right)}$
$\displaystyle=\frac{T}{\mu_{k}^{2}}\left(\tanh^{-1}\frac{T}{\mu_{k}}\right)^{2},$

where in the second line we have used the approximation
$n_{B}\left(x\right)\simeq T/x$ for $\left|x\right|<T$.

###  B.2 Memory Matrix

The MM, after spectral decomposition, writes

$\displaystyle M_{\mathbf{k}\mathbf{k}}\left(\Omega_{n}\right)$
$\displaystyle=\frac{\lambda^{4}}{N^{2}}\frac{1}{\Omega_{n}}T\sum_{\omega_{q}}\sum_{\mathbf{p},\mathbf{q}}\int\prod_{i=1}^{4}\frac{dE_{i}}{2\pi}F\left(E_{i}\right)$
$\displaystyle\times\frac{n_{B}\left(E_{3}\right)-n_{B}\left(E_{4}\right)}{E_{3}-E_{4}+i\omega_{q}}\frac{n_{B}\left(E_{1}\right)-n_{B}\left(E_{2}\right)}{E_{1}-E_{2}+i\omega_{q}+i\Omega_{n}},$

with $F\left(E\right)=\frac{E}{E^{2}+\mu_{k}^{2}}$. Performing the sum over
$\omega_{q}$, we get

$\displaystyle M_{\mathbf{k}\mathbf{k}}\left(\Omega_{n}\right)$
$\displaystyle=\frac{\lambda^{4}}{N^{2}}\frac{1}{\Omega_{n}}\sum_{\mathbf{p},\mathbf{q}}\int\prod_{i=1}^{4}\frac{dE_{i}}{2\pi}F\left(E_{i}\right)$
$\displaystyle\times\left[n_{B}\left(E_{1}\right)-n_{B}\left(E_{2}\right)\right]\left[n_{B}\left(E_{3}\right)-n_{B}\left(E_{4}\right)\right]$
$\displaystyle\times\frac{n_{B}\left(E_{1}-E_{2}\right)-n_{B}\left(E_{3}-E_{4}\right)}{E_{1}-E_{2}-E_{3}+E_{4}+i\Omega_{n}}.$
(25)

We now perform analytic continuation $i\Omega_{n}\rightarrow\Omega+i$$\delta$.
We obtain, after taking the limit $\Omega\rightarrow 0$,

$\displaystyle M_{\mathbf{k}\mathbf{k}}\left(\Omega\right)$
$\displaystyle=\frac{\lambda^{4}}{N^{2}}\pi\sum_{\mathbf{p},\mathbf{q}}\int\prod_{i=1}^{4}\frac{dE_{i}}{2\pi}F\left(E_{i}\right)$
$\displaystyle\times\left[n_{B}\left(E_{1}\right)-n_{B}\left(E_{2}\right)\right]\left[n_{B}\left(E_{3}\right)-n_{B}\left(E_{4}\right)\right]$
$\displaystyle\times\delta\left(E_{1}-E_{2}-E_{3}+E_{4}\right)\frac{\partial
n_{B}}{\partial E}\bigg{|}_{E_{1}-E_{2}}.$

We now use $\partial n_{B}/\partial E\simeq-T/E^{2}$, if $\left|E\right|<T$
and zero elsewhere. We have a factor $-T/(E_{1}-E_{2})^{2}$, with the
condition that $\left|E_{1}-E_{2}\right|<T$. We thus have $E_{2}\simeq
E_{1}\pm T$ and form resolving the $\delta$-function, $E_{4}\simeq E_{3}\pm
T$. This in turn gives
$n_{B}\left(E_{1}\right)-n_{B}\left(E_{2}\right)\simeq\left(\partial
n_{B}/\partial E_{1}\right)T\simeq-T^{2}/E_{1}^{2}$ and likewise
$n_{B}\left(E_{3}\right)-n_{B}\left(E_{4}\right)\simeq-T^{2}/E_{3}^{2}$.
Eliminating two variables in the integral Eq. (LABEL:eq:B5), but remembering
that $\left|E_{1}-E_{2}\right|<T$ and $\left|E_{3}-E_{4}\right|<T$, leads to

$\displaystyle M_{\mathbf{k}\mathbf{k}}\left(\Omega\right)$
$\displaystyle=\frac{\lambda^{4}}{N^{2}}\frac{\pi}{\left(2\pi\right)^{4}}\sum_{\mathbf{p},\mathbf{q}}\int_{-T}^{T}\int_{-T}^{T}dE_{1}dE_{3}T^{4}$
$\displaystyle\times\frac{1}{\left(E_{1}^{2}+\mu_{p}^{2}\right)\left(E_{1}^{2}+\mu_{p+q}^{2}\right)\left(E_{3}^{2}+\mu_{k}^{2}\right)\left(E_{3}^{2}+\mu_{k+q}^{2}\right)}$
$\displaystyle=\frac{\lambda^{4}}{N^{2}}\frac{\pi}{\left(2\pi\right)^{4}}T^{4}\sum_{\mathbf{p},\mathbf{q}}I_{\mathbf{p},\mathbf{q}}I_{\mathbf{k},\mathbf{q}}.$
(27)

With $\left(a,b\right)=\left(\mu_{p},\mu_{p+q}\right)$ for the first integral
and $\left(a,b\right)=\left(\mu_{k},\mu_{k+q}\right)$ for the second integral,
we obtain the following result

$\displaystyle I_{a,b}$
$\displaystyle=2\int_{0}^{T}\frac{dx}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)}$
$\displaystyle=2\left[\frac{-b\tan^{-1}\frac{T}{a}+a\tan^{-1}\frac{T}{b}}{ab\left(a^{2}-b^{2}\right)}\right].$
(28)

In order to go further, we need to perform the integration over $\mathbf{p}$
and $\mathbf{q}$. For this reason, we assume (and this can be also checked at
the end of the calculation) that the order of magnitude of the various wave
vectors is such that they are all scaling with temperature as $k\sim q\sim
p\sim\sqrt{T}$. We can thus use $\mu_{k+q}\simeq\mu_{k}$ inside the integrals.
We then have

$\displaystyle M_{\mathbf{k}\mathbf{k}}\left(\Omega\right)$
$\displaystyle=\frac{\lambda^{4}}{N^{2}}\frac{\pi}{\left(2\pi\right)^{4}}T^{4}J_{\mathbf{p},\mathbf{q}}\frac{1}{\mu_{k}^{3}}\left[\frac{\mu_{k}T}{\mu_{k}^{2}+T^{2}}+\tan^{-1}\frac{T}{\mu_{k}}\right],$
(29)

with

$\displaystyle J_{\mathbf{p},\mathbf{q}}$
$\displaystyle=\int_{0}^{T}\int_{0}^{T}\frac{dp^{2}dq^{2}}{\left(\mu_{p}^{2}-\mu_{p+q}^{2}\right)}\left[-\frac{1}{\mu_{p+q}}\tan^{-1}\frac{T}{\mu_{p}}\right.$
$\displaystyle\left.+\frac{1}{\mu_{p}}\tan^{-1}\frac{T}{\mu_{p+q}}\right].$
(30)

Recalling that $\mu_{p}=p^{2}+T$ and $\mu_{p+q}=\left(p+q\right)^{2}+T$, and
expanding $\tan^{-1}\left(T/\mu_{p}\right)\sim
T/\mu_{p}-\left(T/\mu_{p}\right)^{3}$, we get

$\displaystyle J_{\mathbf{p},\mathbf{q}}$
$\displaystyle\simeq\frac{T^{3}}{3}\int_{0}^{T}\frac{dp^{2}d(p+q)^{2}}{\mu_{p}^{3}\mu_{p+q}^{3}}$
$\displaystyle=\frac{1}{12T}.$ (31)

Putting together Eqs. (29) and (31), it leads to

$\displaystyle M_{\mathbf{k}\mathbf{k}}\left(\Omega\right)$
$\displaystyle=\frac{\lambda^{4}}{N^{2}}\frac{\pi}{12\left(2\pi\right)^{4}}\frac{T^{3}}{\mu_{k}^{3}}\left(\frac{\mu_{k}T}{\mu_{k}^{2}+T^{2}}+\tanh^{-1}\frac{T}{\mu_{k}}\right),$
(32)

which is the result presented in the main part of the manuscript.

##  Appendix C Comparison with Kubo formalism

Figure 3: The electrical conductivity and thermal conductivity as a function
of $T$ computed within Kubo formalism. For $T>\mu_{0}$, in the critical
regime, the constant behavior of $\kappa$ and the $\dfrac{1}{T}$ behavior of
$\sigma$ set in.

Here, we compare the results obtained within the memory matrix formalism with
the standard Kubo formalism Kubo (1957). Following our previous work Banerjee
_et al._ (2021), the optical conductivity and the thermal conductivity (as we
pointed out in the main text, the thermal conductivity at zero electric field
$\bar{\kappa}$ and the thermal conductivity at zero electric current $\kappa$
are equal in the present calculation) are given, respectively, by

$\displaystyle\sigma(\omega)$
$\displaystyle=-\dfrac{\mathcal{K}_{0}(\omega_{n})}{\omega_{n}}\bigg{|}_{i\omega_{n}\rightarrow\omega+i0^{+}},$
(33) $\displaystyle{\kappa}(\omega)$
$\displaystyle=-\dfrac{1}{T}\dfrac{\mathcal{K}_{2}(\omega_{n})}{\omega_{n}}\bigg{|}_{i\omega_{n}\rightarrow\omega+i0^{+}},$
(34)

with $\mathcal{K}_{\alpha}$ computed using the spectral decomposition of the
boson propagator
$D^{-1}(\mathbf{q},\omega_{n})=|\omega_{n}|+\mu_{\mathbf{q}}(T)$ used in Eq.
(22)

$\displaystyle\mathcal{K}_{\alpha}(\omega_{n})=$ (35)
$\displaystyle-\sum\limits_{\mathbf{q}}\epsilon_{\mathbf{q}}^{\alpha}\int\frac{dE_{1}dE_{2}}{\left(2\pi\right)^{2}}\frac{n_{B}\left(E_{1}\right)-n_{B}\left(E_{2}\right)}{-E_{1}+E_{2}+\omega_{n}}\frac{E_{1}}{E_{1}^{2}+\mu_{\mathbf{q}}^{2}}\frac{E_{2}}{E_{2}^{2}+\mu_{\mathbf{q}}^{2}}.$
(36)

Performing an analytical continuation $\omega_{n}\rightarrow\omega+i\delta$
and considering the limit $\omega\rightarrow 0$, we get:

$\displaystyle\mathcal{K}_{\alpha}(\omega)=\dfrac{T\omega}{4\pi}\sum\limits_{\mathbf{q}}\epsilon_{\mathbf{q}}^{\alpha}\int\frac{dE}{\left(2\pi\right)^{2}}\frac{T}{E^{2}}\left(\frac{E}{E^{2}+\mu_{\mathbf{q}}^{2}}\right)^{2}.$
(37)

$\mathcal{K}_{\alpha}$ can be integrated explicitly and we obtain the optical
and thermal conductivities $\sigma$ and $\kappa$ shown in Fig. 3.

The $T$-dependence of $\mu$ is given by

$\mu\equiv\mu(T)=\left\\{\begin{array}[]{ll}\mu_{0}+\dfrac{\lambda^{2}}{2N}T\,\text{log}\left(\dfrac{T}{\mu_{0}}\right)&\text{for
}T\gg\mu_{0},\\\ \mu_{0}&\text{for }T\ll\mu_{0}.\end{array}\right.$ (38)

We consider the critical regime to compare our results with those obtained
from the memory matrix formalism. $T$ satisfies $T>\mu_{0}$ and
$T>\dfrac{\lambda^{2}\mu_{0}}{2N}$; thus, we can approximate
$\dfrac{\mu}{T}\approx\dfrac{\lambda^{2}}{2N}$ by neglecting logarithmic
corrections. In this regime, we have
$\sigma=\dfrac{1}{T}\tilde{\sigma}(\mu/T)$ and $\kappa=\tilde{\kappa}(\mu/T)$,
in agreement with the scaling obtained using the memory matrix formalism in
the critical regime in Eqs. (14) and (17). If we consider logarithmic
corrections in $\mu(T)$, the results obtained remain qualitatively similar up
to a weak dependence on $T$.

In contrast to the Fermi liquid case, the Lorenz ratio
$L=\dfrac{\kappa}{\sigma T}=\dfrac{\tilde{\kappa}}{\tilde{\sigma}}$ is not
given by a universal constant but depends on the interaction strength
$g=\dfrac{\lambda^{2}}{2N}$. This is due to the infrared (IR) dependence on
the integration over $\mathbf{q}$ for the optical conductivity. Because the
integral is divergent, it explicitly depends on the IR cutoff which is
${\mu}/{T}$, and leads to the $g$-dependence of L. The Lorenz ratio increases
with the bosonic coupling in this regime, as shown in Fig. 2.

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# Higher Level Completeness for Permutation Polynomials

S Rajagopal and P Vanchinathan

###### Abstract

Generalising the concept of a complete permutation polynomial over a finite
field, we define completness to level $k$ for $k\geq 1$ in fields of odd
characteristic. We construct two families of polynomials that satisfy the
condition of high level completeness for all finite fields, and two more
families complete to the maximum level possible for large collection of finite
fields.

Under the binary operation of composition of functions one family of
polynomials is an abelian group isomorphic to the additive group, while the
other is isomorphic to the multiplicative group.

Division of Mathematics

VIT University

Vandalur–Kelambakkam Road

Chennai, 600 127 INDIA

<EMAIL_ADDRESS><EMAIL_ADDRESS>Keywords:
permutation polynomial; complete permutation polynomial

AMS Subject Classification: 11T06

##  1 Introduction

A polynomial over a finite field is said to be a permutation polynomial if it
gives rise to a bijective function on that field. The monograph [8] has a
whole chapter devoted to this. The handbook has a large collection of results
and bibliography [9]. The survey articles by Hou [6], [7], gives the status of
the theory of permutation in general and binomials and trinomials in
particular. Due to its applications to combinatorics, cryptology and coding
theory lot of effort is put in constructing them.

A permutation polynomial $f(x)\in\mathbf{F}_{q}[x]$ is called a complete
permutation polynomial, if $f(x)+x$ is also a permutation polynomial.

Complete permutation polynomials seem to have certain advantages. In
applications to cryptography they are used for constructing bent functions.
Results on complete permutation polynomials can be found, for example, in
[1],[2], [3], [4], [10].

In this paper our objective is to find large family of complete permutation
polynomials in each finite field and we achieve it and as a bonus, they are
higher level complete polynomials:

###### Definition 1

For a positive integer $k$, we say a polynomial $f(x)\in\mathbf{F}_{q}[x]$ is
complete to level $k$, or simply $k$-complete, if along with $f(x)$ we have
$f(x)+x,f(x)+2x,\ldots,f(x)+kx$ are also permutation polynomials.

Clearly complete polynomials of level upto $p-1$ where $p$ is the
characteristic of $\mathbf{F}_{q}$, only need to be studied.

Here are some easy-to-verify statements of higher level completeness property
of a permutation polynomial:

  * •

Higher completeness is same as usual completeness in characterstic $2$.

  * •

Usual complete polynomials are 1-complete.

  * •

If a permutation polynomial is $k$-complete, then it is also
$k^{\prime}$-complete for positive integers $k^{\prime}<k$.

###### Definition 2

A permutation polynomial that is complete to level $k$ for $k=p-1$ is said to
be a maximally complete permutation polynomial.

Example: Any linear polynomial $ax+b\in\mathbf{F}_{q}[x]$ is maximally
complete if $a$ is not in the prime subfield; in case $a$ is in the prime
subfield, it will be complete to level $k=p-1-a^{\prime}$, with $a^{\prime}$
being the least positive integer representing $a\in\mathbf{F}_{p}$.

Question: Do complete permutation polynomials of level $k>1$ and degree${}>1$
exist?

Our main results of this paper are the following affirmative answers to the
above question:

Theorem A: For any prime $p\geq 3$ and a composite integer $n$, finite fields
of order $p^{n}$ admit maximally complete permutation polynomials which are
non-linear.

In case the finite field does not have a proper subfield bigger than the prime
subfield (the case where $n$ in Theorem A is a prime) we can find polynomials
that are almost maximally complete:

Theorem B: In all finite fields of characteristic $\geq 3$, there exist non-
linear complete permutation polynomials of level $p-2$.

For a more precise and constructive version of the above, see Theorems 1, 2
and 3 in the next section.

##  2 Main Results and Proofs

Notation: $\mathbf{F}_{q}$ will denote a finite field of prime power order
$q$, assumed to be of odd characteristic $p$.

For an extension field $\mathbf{F}_{q^{n}}$ of $\mathbf{F}_{q}$. we denote by
$m=q+q^{2}+q^{3}+\cdots+q^{n-1}$. Note that $m+1=q^{n}-1$ and that $m$ is a
multiple of the characteristic $p$.

###### Lemma 1

For a finite field $\mathbf{F}_{q}$ and an extension field
$\mathbf{F}_{q^{n}}$ and any $c\in\mathbf{F}_{q}$ the following polynomial

$f_{c+}(x)=x+c\sum\limits_{j=1}^{m}x^{j(q-1)}$

is a permutaion polynomial over $\mathbf{F}_{q^{n}}$.

Proof: It will follow once we show that the polynomial $f_{c+}(x)$ actually is
the function given below:

$f_{c+}(a)=\left\\{\begin{array}[]{lll}a-c&\mbox{for
}&a\in\mathbf{F}_{q^{n}}\setminus\mathbf{F}_{q}\\\ a&\mbox{for
}&a\in\mathbf{F}_{q}\end{array}\right.$

When $a$ is in the base field $\mathbf{F}_{q}$, we claim that all the terms in
the summation are 1. That is so because we are summing the $r$-th powers of
$a$ where $r$ is always a multiple of $q-1$. Thus $f_{c+}(a)=a+cm=a$.

Next we will move on to evaluating it on the remaining elements. So consider
$a\in\mathbf{F}_{q^{n}}\setminus\mathbf{F}_{q}$. The summation actually is a
sum of $m$ terms of a geometric progression. So

$\displaystyle f_{c+}(a)$
$\displaystyle=a+c\left[a^{(q-1)}+a^{2(q-1)}+\cdots+a^{m(q-1)}\right]$
$\displaystyle=a+ca^{q-1}\Biggl{[}\frac{\left(a^{q-1}\right)^{m}-1}{a^{q-1}-1}\Biggr{]}$
$\displaystyle=a+c\Biggl{[}\frac{a^{q^{n}-1}-a^{q-1}}{a^{q-1}-1}\Biggr{]}\mbox{
as }(m+1)(q-1)=q^{n}-1$
$\displaystyle=a+c\Biggl{[}\frac{1-a^{q-1}}{a^{q-1}-1}\Biggr{]}$
$\displaystyle=a-c\hskip 56.9055pt\square$

Summarising we see that $f_{c+}(x)$ when restricted to the base field is the
identity function, and, on the elements not in the base field, is a
translation by an element of the base field. Thus $f_{c+}(x)$ is a permutation
polynomial of $\mathbf{F}_{q^{n}}$ for all $c\in\mathbf{F}_{q}$. Though we
make no use of it further, it is now clear that the permutation given by this
polynomial for $c\neq 0$, has order $p$, with exactly $q$ fixed points.

Next we provide a multiplicative analogue of the polynomials constructed in
the previous theorem.

###### Lemma 2

Under the same notation as earlier, we define for $c\in\mathbf{F}_{q},c\neq 1$
a polynomial by

$f_{c*}(x)=x+cx\sum\limits_{j=1}^{m}x^{j(q-1)}.$

This is a permutaion polynomial over $\mathbf{F}_{q^{n}}$.

In fact this polynomial is the function given by the following description:

$f_{c*}(x)=\left\\{\begin{array}[]{lll}x\left(1-c\right)&\mbox{for
}&x\in\mathbf{F}_{q^{n}}\setminus\mathbf{F}_{q}\\\ x&\mbox{for
}&x\in\mathbf{F}_{q}\end{array}\right.$

Case 1 :
To prove $f_{c*}(a)=a\forall a\in\mathbf{F}_{q}$.
As $f_{c*}(0)=0$, we consider $a\in\mathbf{F}_{q}^{*}$.

$\displaystyle
f_{c*}(a)=a+c\left[a^{q}+a^{(2q-1)}+\cdots+a^{m.q-(m-1)}\right]$
$\displaystyle=a+ca\big{[}a^{q-1}+a^{2(q-1)}+a^{3(q-1)}+\cdots+a^{m(q-1)}\big{]}$
$\displaystyle=a+ca\left[\underbrace{a+a+\cdots+a}_{m\rm\ times}\right]$
$\displaystyle=a+cma=a$

Case 2 : $a\in\mathbf{F}_{q^{n}}\setminus\mathbf{F}_{q}$.

$\displaystyle f_{c*}(a)$
$\displaystyle=a+ca\left[a^{(q-1)}+a^{2(q-1)}+a^{3(q-1)}+\cdots+a^{m(q-1)}\right]$
$\displaystyle=a+ca\Biggl{[}\frac{a^{(m+1)(q-1)}-a^{q-1}}{a^{q-1}-1}\Biggr{]}$
$\displaystyle=a+ca\Biggl{[}\frac{1-a^{q-1}}{a^{q-1}-1}\Biggr{]}$
$\displaystyle=a\left(1-c\right)\hskip 56.9055pt\square$

Again we can specify the cycle type of this permutation $f_{c*}$ in terms of
the order of $(1-c)$ as an element of the multiplicative group
$\mathbf{F}_{q}^{*}$.

The permutation polynomials constructed above are actually complete to higher
level. Here is the precise result:

###### Theorem 1

The polynomial $f_{c+}(x)$ is $(p-2)$-complete, that is, $f_{c+}(x)$ along
with the set of $(p-2)$ polynomials
$x+f_{c+}(x),2x+f_{c+}(x),3x+f_{c+}(x),\ldots,(p-2)x+f_{c+}(x)$ are all
permutation polynomials.

Proof: For a positive integer $k$ from the definition we get

$\displaystyle kx+f_{c+}(x)$
$\displaystyle=(k+1)x+c\left[x^{q-1}+x^{2(q-1)}+\cdots+x^{m(q-1)}\right]$
$\displaystyle=(k+1)\Biggl{[}x+\frac{c}{k+1}\left(x^{q-1}+x^{2(q-1)}+\cdots+x^{m(q-1)}\right)\Biggr{]}$

But, visual inspection makes it clear that, the expression inside the square
brackets is actually $f_{c^{\prime}+}$ for $c^{\prime}=c/(k+1)$. Thus we
arrive at an identity

$\displaystyle kx+f_{c+}(x)$
$\displaystyle=\left(k+1\right)f_{c^{\prime}+}(x)$

By Lemma 1 RHS is a permutation polynomial and hence because the LHS is too.
Of course we need $k+1\neq 0$, and so this is true for $k=1,2,\ldots,p-2$. So
$f_{c+}(x)$ is $(p-2)$-complete. $\square$

###### Theorem 2

The polynomial $f_{c*}(x)$ are complete to level $p-2$, i.e., $f_{c*}(x)$
along with the set of $(p-2)$ polynomials
$x+f_{c*}(x),2x+f_{c*}(x),3x+f_{c*}(x),\ldots,(p-2)x+f_{c*}(x)$ are
permutation polynomials.

Proof: Using arguments similar to the ones employed in the proof of Theorem 1
we can easily arrive at

$\displaystyle kx+f_{c*}(x)$
$\displaystyle=\left(k+1\right)f_{c^{\prime}*}(x)$

for $c^{\prime}=c/(k+1)$. Again we will be able to conclude that
$kx+f_{c*}\left(x\right)$ is also permutation polynomial for $0<k<(p-1)$.
$\square$

Next we move on to finding maximally complete permutation polynomials. First
we need a temporary definition of a middle subfield of a field: by that we
mean a proper subfield which is not the prime subfield. Middle subfields exist
in finite fields of order $p^{n}$ iff $n$ is a composite number.

###### Theorem 3

Let $\mathbf{F}_{q}$ be a finite field admitting at least one middle subfield.
Choose $b,c\in\mathbf{F}_{q}$ such that $b$ is in some middle subfield and $c$
is in any proper subfield of $\mathbf{F}_{q}$ Then the polynomials
$bf_{c+}(x)$ and $bf_{c*}(x)$ are maximally complete permutation polynomials.

Proof: As $c$ is in a proper subfield of $\mathbf{F}_{q}$, Lemma 1 ensures
$f_{c+}(x)$ is a permutation polynomial and hence the scalar multiple
$bf_{c+}(x)$ is too. We will now show for any $k=1,2,\ldots,p-1$ that
$bf_{c+}(x)+kx$ is a permutation polynomial.

We simply rewrite this as

$bf_{c+}(x)+kx=(b+k)\bigg{(}x+\frac{bc}{b+k}\sum_{j}x^{j(q-1)}\bigg{)}$

And the latter polynomial is the same as $(b+k)f_{c^{\prime}+}(x)$ where
$c^{\prime}=bc/(b+k)$, and hence a permutation polynomial. We need $b+k$ to be
nonzero for this to be true. That follows from the hypothesis that $b$ is not
in the prime field while $k$ is always in the prime field. This completes the
proof that $bf_{c+}(x)$ is maximally complete.

Same arguments work for proving maximal completeness property for the
polynomials $bf_{c*}$ and so omitted.

Next we discuss inter- and intra-relationship among the members of two
families of polynomials $f_{c+}(x)$ and $f_{c*}(x)$.

###### Theorem 4

The two collections of functions $f_{c+}(x)$, and $f_{c*}(x)$ behave well
under composition. In fact,

  * (i)

$\\{\,f_{c+}(x)\mid c\in\mathbf{F}_{q}\,\\}$ is an abelian group under
composition, and is isomorphic to the additive group of $\mathbf{F}_{q}$.

  * (ii)

$\big{\\{}\,f_{c*}(x)\mid c\in\mathbf{F}_{q}-\\{1\\}\,\big{\\}}$ is also an
abelian group under composition, and is isomorphic to the multiplicative group
$\mathbf{F}_{q}^{*}$.

  * (iii)

$f_{c+}(x)$ and $f_{c*}(x)$ have the following relationship:

$x\big{(}f_{c+}(x)-x+1\big{)}=f_{c*}(x)$

Proof:

Verifying (i) is straightforward.

For part (iii) the given relationship between $f_{c+}(x)$ and $f_{c*}(x)$
follows directly from their definitions.

For (ii), to see that it is a group one can easily check that

$\displaystyle f_{c*}\circ f_{d*}(x)$
$\displaystyle=f_{(c+d-cd)*}=\left\\{\begin{array}[]{lll}(\,1-c-d+cd\,)x&\mbox{for
}&x\in\mathbf{F}_{q^{n}}\setminus\mathbf{F}_{q}\\\ x&\mbox{for
}&x\in\mathbf{F}_{q}\end{array}\right.$

and that the compositional inverse of $f_{c*}(x)$ is

$\displaystyle f_{c*}^{-1}(x)$
$\displaystyle=\left\\{\begin{array}[]{lll}cx/(c-1)&\mbox{for
}&x\in\mathbf{F}_{q^{n}}\setminus\mathbf{F}_{q}\\\ x&\mbox{for
}&x\in\mathbf{F}_{q}\end{array}\right.$

Now the statement that is it isomophic to $\mathbf{F}_{q}^{*}$ can be deduced
from the following lemma whose proof is a simple exercise:

###### Lemma 3

For any field $K$, the set $K\setminus\\{1\\}$ is a group under the binary
operation $a*b=a+b-ab$ and this is isomorphic to $K^{*}$ under the map
$a\mapsto 1-a$.

Examples: First two polynomials given below are 3-complete permutation
polynomials over $\mathbf{F}_{25}$, and the other are 5-complete over
$\mathbf{F}_{49}$.

  * •

$f_{2+}(x)=2x^{20}+2x^{16}+2x^{12}+2x^{8}+2x^{4}+x\in\mathbf{F}_{5^{2}}[x]$

  * •

$f_{4*}(x)=4x^{21}+4x^{17}+4x^{13}+4x^{9}+4x^{5}+x\in\mathbf{F}_{5^{2}}[x]$

  * •

$f_{4+}(x)=4x^{42}+4x^{36}+4x^{30}+4x^{24}+4x^{18}+4x^{12}+4x^{6}+x\in\mathbf{F}_{7^{2}}[x]$

  * •

$f_{6*}(x)=6x^{43}+6x^{37}+6x^{31}+6x^{25}+6x^{19}+6x^{13}+6x^{7}+x\in\mathbf{F}_{7^{2}}[x]$

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