∎

<EMAIL_ADDRESS><EMAIL_ADDRESS>
11institutetext: Mukesh Patel School of Technology Management and Engineering,
NMIMS University, Mumbai-400056, India 22institutetext: Institute of Nuclear
Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342 Krakow,
Poland

# Dynamically screened strongly quantized electron transport in binary
neutron-star merger

Sreemoyee Sarkare1,addr1  Souvik Priyam Adhya e2,addr2

(Received: date / Accepted: date)

###### Abstract

We examine electron-transport coefficients in magnetized hot and dense
electron-ion plasma relevant in binary neutron star merger simulation. We
calculate electrical and thermal conductivities in low density, high
temperature, highly magnetized plasma of binary neutron star mergers where
quantum oscillatory behavior of electrons emerge. For pronounced thermodynamic
effects, we consider zeroth Landau level population of electrons for the
calculation of conductivity. We solve Boltzmann equation in presence of
magnetic field to obtain the dissipative components of electrical and thermal
conductivities. The dissipative coefficients are formulated considering
frequency dependent dynamical screening in the quantized electron-ion
scattering rate. Numerical estimations show that the effect of dynamical
screening of photon propagator on electrical and thermal conductivities is
pronounced. We observe that dynamical screening reduces the maxima of both the
electrical and thermal conductivities by factors of thirty one and twenty
respectively leading to a reduction in the corresponding time scales of these
coefficients. The common scaling factor between electrical and thermal
conductivity is also observed to follow cubic relationship with temperature
violating Wiedemann-Franz law.

###### Keywords:

Hard dense loop electrical and thermal conductivities neutron star
quantization Ohmic decay and thermal equilibration time scales

††journal: Eur. Phys. J. C

##  1 Introduction

Binary neutron star mergers and collider experiments are the sources of most
extreme states of matter in the universe with densities several times nuclear
saturation density and temperatures upto hundreds of MeV Dexheimer:2020zzs .
The recent detection of gravitational wave signal GW170817 originating from
binary neutron star (BNS) merger by the LIGO and Virgo detectors have opened
up a new era in multi-messenger astronomy LIGOScientific:2017vwq ;
LIGOScientific:2017zic ; LIGOScientific:2017ync . Additionally, short gamma-
ray bursts (SGRBs) were also detected by the Fermi satellite GRB170817A
indicating the presence of huge magnetic field in the merging event
Paschalidis:2016agf ; Kawamura:2016nmk ; Ruiz:2016rai . These mergers are
unique astrophysical objects of significant sources of gravitational
radiation, electromagnetic as well as neutrino emission Palenzuela:2013hu .
They offer a novel avenue to study highly non-linear gravitational effects
blended with complex micro-physical processes; serving as Einstein’s richest
natural laboratory Baiotti:2016qnr .

In the event post merging, a remnant neutron star is created and if the
remnant possess a mass beyond Tolman–Oppenheimer–Volkoff (TOV) limiting mass,
the merged object survives for 10-100 milliseconds and collapses thereafter.
The description of neutron star mergers requires the knowledge of General
Relativistic Magneto Hydro-dynamics (GRMHD)Anderson:2008zp ; Palenzuela:2008sf
; Liu:2008xy ; Dionysopoulou:2012zv ; Palenzuela:2013kra ; Kiuchi:2015qua ;
Dionysopoulou:2015tda ; Kiuchi:2017zzg ; Ruiz:2017due . Most of these general-
relativistic simulations account for ideal Magneto Hydro-dynamics (MHD) limit.
In a recent work Ref.Harutyunyan:2018mpe , the authors have pointed out that
Hall effect plays an important role in magnetic field decay of the merged
object, hence, the Hall effect needs to be incorporated in the ideal MHD
merger simulation. In this Ref.Harutyunyan:2018mpe  the authors have
considered the electrical conductivity ($\sigma$) of warm neutron star crust
in the non-quantizing scenario calculated in Ref.Harutyunyan:2016rxm . Apart
from these studies, in Ref.Alford:2017rxf  authors have analysed the relevance
of thermal conductivity ($\kappa$), viscous coefficients in BNS mergers in the
non-magnetic scenario. In view of these recent studies Harutyunyan:2016rxm ;
Harutyunyan:2018mpe ; Alford:2017rxf , we analyse the importance of
dissipative mechanism in the merger simulation by evaluating quantized
electrical and thermal conductivity coefficients with magnetically modified
many-body effects in the present paper. We calculate quantized dissipative
coefficients with frequency dependent screening in both hot and dense plasma
relevant in binary neutron star merger. The results can be implemented in
analysing the magnetothermal evolution Aguilera:2007xk  of the merged compact
object as well.

We consider fully ionized plasma of electrons and ions. Heat and charge in
this medium are transported by electrons. The dominant electron transport
mechanism is scattering on ions in the liquid phase. In presence of extreme
magnetic field (B) (B$\sim 10^{16}$G) and density ($\rho$) $(\rho\sim 10^{12}$
g cm${}^{-3})$, the classical description of electrons breaks down. Therefore,
one should incorporate Landau quantization of energy levels in the formalism.
This quantization occurs for a particular set of temperature, density and
magnetic field in case of neutron star. Thus, the inclusion of Landau
quantization eventually modifies the non-magnetic electrical conductivity to
great extentYakovlev1980 ; Hernquist1984 ; Potekhin:1996zh ; Potekhin:1996hu .
In the present paper, we focus mainly on the strongly quantizing case, since
in this domain, the transport coefficients receive major modification due to
the magnetic field.

The calculation of electrical and thermal conductivities by solving Boltzmannn
equation in ultra-compressed plasma have been studied by several authors over
the last few decades 1964Abrikosov ; 1966ApJ…146..858H ; Lampe:1968zz ;
1976ApJ…206..218F , see for a review Schmitt:2017efp . This requires the
information of scattering rate of plasma constituents. The calculations of
scattering rates considering screened Coulomb potential have already been
observed in different Refs.Yakovlev1980 ; 1984MNRAS.209..511N ; Hernquist1984
; Potekhin:1996zh ; Potekhin:1996hu . In all these calculations it has been
assumed that ions are static scatterers. This formulation can not be easily
transported to the relativistic domain where dynamical effects are important
for reliable description of transport coefficients. Medium modified Hard-
Thermal-Loop (HTL) propagators for hot and Hard-Dense-Loop (HDL) for dense
plasma include dynamical effects of relativistic medium through frequency
dependent screening Braaten:1989mz ; Braaten:1990az ; Altherr:1992mf ;
LeBellac:1996kr ; Manuel:2000mk ; Kalikotay:2020snc . While Debye screening in
plasma is related to the static/longitudinal photon exchange, the exchange of
magnetic/ transverse photons contribute to dynamical frequency dependent
screening of the plasma particles. It is observed in different studies
LeBellac:1996kr ; Manuel:2000mk ; Heiselberg:1993cr ; Sarkar:2010bv ;
Sarkar:2012ww ; Adhya:2012sq ; Adhya:2013ima  that for ultra-degenerate case,
both in Quantum Chromodynamics (QCD) and Quantum Electrodynamics (QED)
plasmas, the transverse interactions not only become important but they
dominate over their longitudinal interaction. In a recent calculation
Harutyunyan:2016rxm , the authors have included many-body effects through the
HTL modified propagator in the calculation of non-quantized electrical
conductivity in magnetized, warm neutron star crust. Motivated by all these
calculations of dynamical screening in different coefficients, we include HDL
modified photon propagator in quantized $\sigma$ and $\kappa$ in the context
of BNS merger in the present paper. We incorporate plasma screening through
magnetic Debye mass. The current calculation is important in two ways. First,
in this paper we consider the quantized transport coefficients in estimating
dissipation coefficients in BNS merger. Second, this formulation includes
Landau damping in quantized electron-ion interaction rate in both hot and
dense relativistic, magnetized plasma. Here, we perform the calculations of
$\sigma$ and $\kappa$ in an extreme scenario of temperature $\sim 12$ MeV,
density $\sim 10^{12}$ g cm-3 and magnetic field $B\sim 10^{16}$ G. Finally,
with the strongly quantized $\sigma$ and $\kappa$, we estimate the relevant
dissipative time scales and compare it with the survival time period of the
post-merger object. Validity of Wiedemann-Franz law has also been discussed.

The paper is organised as follows. In section II, we derive the longitudinal
electrical and thermal conductivities in a dynamically screened QED plasma.
Next, in section III we present the constraints on temperature, magnetic field
and density of hot and dense plasma to become relativistic and strongly
quantized in the BNS merger scenario. We present numerical variations for
$\sigma$ and $\kappa$ with temperature, magnetic field and density for the
dense, relativistic plasma along with estimation of corresponding dissipative
time scales in section III. Finally, we summarize and discuss the impact of
dynamical screening on both the coefficients and decay time scales in section
IV. Throughout the manuscript we will use following notation for four vectors
$p=(\epsilon_{p},p_{z},\vec{p}_{\perp})$ and $k=(\epsilon_{k},\vec{k})$. 111We
have used $c=k_{B}=\hslash=1$.

##  2 Electrical conduction from Transport Theory

In the current paper, we consider fully ionized plasma of two components:
electrons ($e$) and positive ions of charge $Ze$ ($Z$ atomic number of the
nucleus). In compact objects, the huge magnetic field quantizes the motion of
electrons in the QED plasma. In this section we derive the coefficients for
electrical conduction, electrical conductivity and coefficients for thermal
conduction, thermal conductivity, in magnetized electron-ion plasma from
transport theory. In presence of magnetic field both the coefficients are
anisotropic and the conductivity tensor is given below,

$\sigma/\kappa=\Bigg{(}\begin{matrix}\sigma_{\perp}/\kappa_{\perp}&-\sigma_{H}/\kappa_{H}&0\\\
\sigma_{H}/\kappa_{H}&\sigma_{\perp}/\kappa_{\perp}&0\\\
0&0&\sigma_{\parallel}/\kappa_{\parallel}\end{matrix}\Bigg{)}.$ (1)

Figure 1: Feynman diagram describing the electron-ion scattering amplitude in
a static medium (left panel). Feynman diagram contributing to the electron-ion
scattering amplitude via exchange of resummed HDL photon propagator (right
panel).

In the above matrix equation, $\sigma_{\parallel}/\kappa_{\parallel}$ and
$\sigma_{\perp}$/$\kappa_{\perp}$ are the parallel and perpendicular
components of $\sigma$ and $\kappa$ respectively in presence of external
magnetic field along $z$ direction. $\sigma_{H}$/ $\kappa_{H}$, are the Hall
components of the conductivities. In the current paper we present the
calculation of quantized $\sigma_{\parallel}/\kappa_{\parallel}$ in electron
ion plasma. For the rest of the paper, we re-define $\sigma_{\parallel}$,
$\kappa_{\parallel}$ as $\sigma$ and $\kappa$ respectively.

The $\sigma$ is related to the electric current density ($j$) and satisfies
the constitutive relation $j=\sigma E$ where $E$ is the electric field. For
thermal conductivity $Q=-\kappa\nabla T$, where, Q is the amount of charge
transported through the plasma under the temperature gradient $\nabla T$. $j$
and $Q$ are related to displacement of the electronic distributions from their
equilibrium configuration due to the presence of electric field and
temperature gradient in the plasma respectively and can be calculated from
kinetic theory,

$\displaystyle j$ $\displaystyle=$ $\displaystyle
2e\int\frac{d^{3}p}{(2\pi)^{3}}v_{p}\Phi$ (2) $\displaystyle Q$
$\displaystyle=$ $\displaystyle
2\int\frac{d^{3}p}{(2\pi)^{3}}(\epsilon_{p}-\mu)v_{p}\Phi.$ (3)

In the above equation $e$ is the charge of an electron, $v_{p}$ is the
velocity of the electrons, $\epsilon_{p}$ is the energy of electrons, $\mu$ is
the chemical potential of electrons. $\Phi$ is the off-equilibrium
distribution function which arises due to the presence of electromagnetic
field in plasma. $\Phi$ is obtained by solving Boltzmann equation in presence
of magnetic field. In presence of perturbation along $z$ direction, the
distribution function ($f_{n,p_{z},s}$) evolves according to the magnetically
modified Boltzmann equation as given below Hernquist1984 ,

$\displaystyle\frac{\partial f_{n,p_{z},s}}{\partial t}+v_{z}\frac{\partial
f_{n,p_{z},s}}{\partial z}-eE\frac{\partial f_{n,p_{z},s}}{\partial
p_{z}}=\mathcal{C}[f_{n,p_{z},s}].$ (4)

In the above equation, $f_{n,p_{z},s}$ describes the population of electrons
defined by the quantum state $n,s,p_{z}$, $n$ is the number of the Landau
level, $s$ is the spin and $p_{z}$ is the $z$ component of electron’s
momentum. $v_{z}$ is the $z$ component of the velocity of the particle. The
third term in the LHS of eq.(4) arises from the Lorentz force term
$\vec{F}=e(\vec{E}+\vec{v_{p}}\times\vec{B})$. In absence of external magnetic
field, the Lorentz force term vanishes.

The RHS of Eq.(4) contains the information of scattering rate of electrons
with the ions present in the medium,

$\displaystyle\mathcal{C}[f_{n,p_{z},s}]=\frac{\partial
f_{n,p_{z},s}}{\partial
t}\Bigg{|}_{coll}=\sum_{f}I_{fi}\big{(}f_{n,p_{z},s\rightarrow
n^{\prime},p^{\prime}_{z},s^{\prime}}\big{)},$ (5)

where, sum is over final state quantum numbers
$n^{\prime},p_{z}^{\prime},s^{\prime}$. $I_{fi}$ is the electron-ion
scattering rate from initial state ($i$) to the final state ($f$) in presence
of $B$. $f_{n,p_{z},s}$ is the initial state distribution function and
$f_{n^{\prime},p_{z}^{\prime},s^{\prime}}$ is the scattered state distribution
function. The distribution function of electrons has two parts in presence of
electromagnetic field,

$\displaystyle f_{n,p_{z},s}$ $\displaystyle=$ $\displaystyle
f_{0}(\epsilon_{p})+\Phi_{n,p_{z},s},$ (6)

where, $f_{0}(\epsilon_{p})$ is the equilibrium Fermi-distribution functions
and $\Phi_{n,p_{z},s}$ is the perturbation due to the electromagnetic field.

We now proceed to calculate the collision integral considering strongly
quantizing magnetic field. To calculate the interaction rate, we consider an
electron with momentum $p=(\epsilon_{p},p_{z},\vec{p_{\perp}})$ and mass $m$
exchanges a virtual photon of momentum $q=$ ($q_{0}$, $\vec{q})$ with an in-
medium ion of momentum $k=(\epsilon_{k},\vec{k})$ and mass $M$. The electron
emerges with momentum
$p^{\prime}=(\epsilon_{p^{\prime}},p^{\prime}_{z},\vec{p^{\prime}_{\perp}})$
and ion with momentum $k^{\prime}=(\epsilon_{k^{\prime}},\vec{k^{\prime}})$
(Fig.(1)). In order to obtain finite interaction rate, we use the HDL re-
summed photon propagator with transverse and longitudinal components.

We start by re-visiting the expression of the interaction rate ($I_{fi}$)
without electromagnetic modification Hernquist1984 ,

$\displaystyle
I_{fi}=\frac{1}{2\epsilon_{p}}\int\frac{d^{3}p^{\prime}}{(2\pi)^{3}2\epsilon_{p^{\prime}}}\int\frac{d^{3}k}{(2\pi)^{3}2\epsilon_{k}}\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}2\epsilon_{k^{\prime}}}$
$\displaystyle[f_{0}(\epsilon_{p})g_{0}(\epsilon_{k})\left(1-f_{0}(\epsilon_{p}^{\prime})\right)-f_{0}(\epsilon_{p}^{\prime})g_{0}(\epsilon_{k}^{\prime})\left(1-f_{0}(\epsilon_{p})\right)]$
$\displaystyle\left(2\pi\right)^{4}\delta\left(\epsilon_{p}+\epsilon_{k}-\epsilon_{p^{\prime}}-\epsilon_{k^{\prime}}\right)\delta^{3}\left(\vec{p}+\vec{k}-\vec{p^{\prime}}-\vec{k^{\prime}}\right)|{\cal
M}_{fi}|^{2}$

where, ${\cal M}_{fi}$ is the electron-ion scattering matrix. ${\cal M}_{fi}$
can be written asHarutyunyan:2016rxm ,

$\displaystyle{\cal
M}_{fi}=-\Delta_{L}J_{0}J^{\prime}_{0}+\Delta_{T}\bm{J}_{t}\bm{J}^{\prime}_{t}=-{\cal
M}_{L}+{\cal M}_{T},$ (8)

where,

$\displaystyle J^{\mu}$ $\displaystyle=$
$\displaystyle-e^{*}\bar{u}(p^{\prime})\gamma^{\mu}u(p),$ $\displaystyle
J^{\prime\mu}$ $\displaystyle=$ $\displaystyle
Ze^{*}v_{k}^{\mu}=Ze^{*}(1,\vec{k}/M),$ (9)

are the components of currents. $e^{\star}=\sqrt{4\pi}e$, $v_{k}$ is the
velocity of ion with momentum $k$. The $\Delta_{T}$ and $\Delta_{L}$ are
transverse and longitudinal HDL effective photon propagators respectively. The
form of the electronic spinors ($u(p),u(p^{\prime})$) are given in the
Appendix. A.

To proceed further, here, we describe the screening mechanism of electron-ion
plasma. In earlier calculations 1964Abrikosov ; 1966ApJ…146..858H ;
Lampe:1968zz ; 1976ApJ…206..218F , the authors have implemented longitudinal
component of photon propagator to screen the Coulomb potential,

$D_{\vec{q}}=\frac{1}{\vec{q}^{2}+m_{D}^{2}}.$ (10)

In the above equation $m_{D}^{2}=e^{2}dn_{e}/d\mu$, where, $n_{e}$ is the
number density of electrons. Following linear response theory, there exists
additional weak dynamical screening, related to the energy transfer to the
constituents of plasma known as Landau damping. This arises because of non-
zero frequency of the plasma. We implement the effects of non-zero frequency
in both the electric and magnetic components of the photon propagator computed
within the HDL formalism,

$\displaystyle\Delta^{\mu\nu}(q)=P^{\mu\nu}(q)\Delta_{L}+Q^{\mu\nu}(q)\Delta_{T},$
(11)

where, $P_{\mu\nu}$ and $Q_{\mu\nu}$ are the transverse and longitudinal
projectors, respectively,

$\displaystyle P^{ij}(q)$ $\displaystyle=$
$\displaystyle-\delta^{ij}+\frac{q^{i}q^{j}}{q^{2}},$ (12) $\displaystyle
Q^{00}(q)$ $\displaystyle=$
$\displaystyle-\frac{q^{2}}{|\vec{q}|^{2}}=1-\frac{q_{0}^{2}}{|\vec{q}|^{2}}=1-y^{2},$
(13)

and the effective longitudinal ($\Delta_{L}$) and transverse ($\Delta_{T}$)
propagators are given by the following expressions,

$\displaystyle\Delta_{L}=\frac{1}{q^{2}-\Pi_{L}},$
$\displaystyle\Delta_{T}=\frac{1}{q^{2}-\Pi_{T}}.$ (14)

In the above equations $\Pi_{T}$ and $\Pi_{L}$ are the transverse and
longitudinal HDL photon self-energies and are given by,

$\displaystyle\Pi_{T}(q)$ $\displaystyle=$ $\displaystyle
3m_{D}^{2}\left[\frac{y^{2}}{2}+\frac{y(1{-}y^{2})}{2}\ln\left(\frac{y{+}1}{y{-}1}\right)\right],$
$\displaystyle\Pi_{L}(q)$ $\displaystyle=$ $\displaystyle
3m_{D}^{2}\left[1-y^{2}-\frac{y(1-y^{2})}{2}\ln\left(\frac{y{+}1}{y{-}1}\right)\right].$
(15)

In presence of strong magnetic field, electron density present in the plasma
changes, leading to a modification in the screening. At low temperature and
strong magnetic field, the presence of sharp Fermi surface modifies the nature
of screening. In the relativistic domain $m\ll\mu$ , the Debye mass is given
by Sharma:2010bx ,

$m_{D}^{2}=\left(\frac{e}{\pi}\right)^{2}\left(\frac{eB}{2}\right).$ (16)

One obtains eqn.(16) using $m_{D}^{2}=e^{2}dn_{e}/d\mu$, where, $n_{e}$ in the
relativistic domain $(m\ll\mu)$ is given by $eB\mu/2\pi^{2}$. In the screening
mass, we have ignored the finite temperature correction. This assumption is
valid as we consider only the degenerate domain ($T<\mu$) of the plasma. In
addition, the conductivities are weakly dependent on the screening mass as
discussed in Hernquist1984 . Hence, inclusion of finite temperature Debye mass
would mark negligible correction in the final results of the conductivities.

In order to proceed further, we compute the phase space factor in the
interaction rate given in eq.(2). We do not consider the change of momentum of
ions in the phase space factor. Hence, the linearized phase space factor can
be written as,

$\displaystyle\left[f_{n,p_{z},s}g_{k}(1-f_{n^{\prime},p^{\prime}_{z},s^{\prime}})-f_{n^{\prime},p^{\prime}_{z},s^{\prime}}g_{k^{\prime}}(1-f_{n,p_{z},s})\right]$
$\displaystyle\simeq
g_{k}\left(f_{n,p_{z},s}-f_{n^{\prime},p^{\prime}_{z},s^{\prime}}\right).$
(17)

The final expression for the interaction rate thus becomes (the details of the
derivation are given in Appendix A),

$\displaystyle I_{fi}$ $\displaystyle=$
$\displaystyle\frac{n_{i}}{2}\sum_{n^{\prime},p_{z}^{\prime},s^{\prime}}\int
du\left(\Phi_{n,p_{z},s}-\Phi_{n^{\prime},p^{\prime}_{z},s^{\prime}}\right)$
(18)
$\displaystyle\left[\frac{1}{3(u+\frac{\zeta}{3})(u+\zeta)}-\frac{v_{k}^{2}}{6u(u+\frac{\zeta}{3})}\right]{\cal
F},$

$n_{i}$ is the number density of ions and $u$, $\zeta$ and ${\cal F}$ are
given in the Appendix A. In order to find the transport coefficients, it is
useful to define a dimensionless scattering rate $a$ and a dimensionless
perturbation to the distribution function ($\Psi$) defined as,

$\displaystyle\frac{I_{fi}}{n_{i}v_{z}\sigma_{0}}=a,$
$\displaystyle\frac{eE}{\sigma_{0}n_{i}}\frac{\partial
f_{0}}{\partial\epsilon_{p}}\Psi=\Phi,$ (19)

where, $\sigma_{0}=\pi Z^{2}e^{4}/\omega_{B}^{2}$ and $\omega_{B}=eB/m$. The
electric charge $e$ is related to the fine structure constant by
$\alpha=e^{2}/(4\pi)=1/137$. Using the above two equations, we obtain the
dimensionless form of the linearized Boltzmann equation as given below
Yakovlev84 ,

$\sum_{n^{\prime}~{}s^{\prime}\gamma}a(ns\rightarrow
n^{\prime}s^{\prime})\big{(}\Psi_{n^{\prime}s^{\prime}}-\gamma\Psi_{ns}\big{)}=1.$
(20)

$\gamma=\pm$ denotes the scattering channel for forward ($+$) and backward
reactions ($-$). In the current paper, we present the results for the strongly
quantizing scenario (i.e. zeroth Landau level) which provides the maximum
effect with finite magnetic field in contrast to the non-magnetic scenario.
For zeroth Landau level, $n=n^{\prime}=0$, spin degeneracy is absent and
backward scattering ($\gamma=-1$) is the only allowed channel for scattering.
Hence, after solving the dimensionless Boltzmann equation the off equilibrium
distribution function is obtained as,

$\displaystyle\Psi$ $\displaystyle=$ $\displaystyle\frac{E^{2}-1}{2Q_{2}},$
(21)

where, $E=\epsilon_{p}/m$, $\Psi\equiv\Psi_{0,-1}$ and

$\displaystyle Q_{2}=\int_{0}^{\infty}e^{-u}$
$\displaystyle\times\left[\frac{2}{3\left(u+\frac{\zeta}{3}\right)\left(u+\zeta\right)}-\frac{v_{k}^{2}}{6u\left(u+\frac{\zeta}{3}\right)}\right]du.$
(22)

We obtain the expressions for $\sigma$ and $\kappa$ with only zeroth Landau
level population by inserting the value of $\Phi$ in the equations below,

$\displaystyle j$ $\displaystyle=$
$\displaystyle\frac{em\omega_{B}}{4\pi^{2}}\int_{m}^{\infty}\Phi
d\epsilon_{p},$ $\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{m\omega_{B}}{4\pi^{2}}\int_{m}^{\infty}(\epsilon_{p}-\mu)\Phi
d\epsilon_{p}.$ (23)

We then insert $j$ and $q$ in the constitutive relations $j=\sigma E$ and
$Q=-\kappa\nabla T$ to obtain both $\sigma$ and $\kappa$. The final form of
electrical conductivity is given below,

$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{\delta_{0}}{\theta}\int_{1}^{\infty}\Psi
f_{0}(1-f_{0})dE,$ (24)

here, $\delta_{0}$ is a constant and is given by
$\delta_{0}=m^{4}b^{2}/8\pi^{3}Z^{2}e^{2}n_{i}$, $\theta=T/m$ and $b=B/B_{c}$,
$B_{c}$ is the critical field given by $4.413\times 10^{13}$ G.
$f_{0}(1-f_{0})/T\equiv-\partial f_{0}/\partial\epsilon_{p}$.

The thermal conductivity coefficient can be obtained by integrating following
expression,

$\displaystyle\kappa$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}T}{3e^{2}}\Big{[}\frac{\delta_{0}}{\theta}\int_{1}^{\infty}\frac{(\epsilon_{p}-\mu)}{T^{2}}^{2}\Psi
f_{0}(1-f_{0})dE$ (25) $\displaystyle-$
$\displaystyle\frac{\delta_{0}}{T\theta}\frac{\int_{1}^{\infty}(\epsilon_{p}-\mu)\Psi
f_{0}(1-f_{0})dE}{\int_{1}^{\infty}\Psi f_{0}(1-f_{0})dE}\Big{]}.$

From the above two equations it is evident that at low temperature satisfying
$(\epsilon_{p}-\mu)\sim T$, $\partial
f_{0}/\partial\epsilon_{p}=\delta(\epsilon_{p}-\mu)$. After integration thus
$\sigma$ becomes temperature independent and $\kappa$ varies linearly with
temperature $T$. Hence, $\kappa/\sigma$ varies linearly with temperature till
$(\epsilon_{p}-\mu)\sim T$ is satisfied. This linear relationship of
$\sigma/\kappa$ with temperature is known as Wiedemann-Franz law.

In the next section, we study the variation of $\sigma$ and $\kappa$ with
different parameters and the effects of the frequency dependent screening
derived in the current section.

##  3 Results

In this section we describe the behaviour of $\sigma$ and $\kappa$ with
density, temperature, magnetic field in the hot and dense QED plasma. First we
present the physical conditions of the plasma for the calculation of the
transport coefficients.

###  3.1 Physical conditions

Physical properties of the BNS merger, which forms an unstable configuration
are different from isolated neutron stars. We consider simplest possible
constituents of post-merger object of electron-ion plasma with fully ionized
ions and free mobile electrons in the low density (up to $10^{12}\mbox{g
cm}^{-3}$), high magnetic field (up to $10^{17}\mbox{G}$) and high temperature
($T\sim 15$ MeV) regime of BNS merger. Electron density $n_{e}$ is related to
ion density $n_{i}$ via $n_{e}=Zn_{i}$. We consider the magnetic field present
along the $z$ direction. Scattering of electrons with ions only contribute in
electrical conductivity. In the absence of magnetic field the electron density
can be written as,

$\displaystyle n_{e}$ $\displaystyle=$
$\displaystyle\frac{2}{(2\pi)^{3}}\int_{0}^{\infty}f_{0}\left(\epsilon_{p}\right)d^{3}p,$
(26)

where, $f_{0}(\epsilon_{p})=1/(exp(\frac{\epsilon_{p}-\mu}{T})+1)$, $T$ is the
temperature. In absence of magnetic field energy of the electrons are given by
$\epsilon_{p}=\sqrt{p_{f}^{2}+m^{2}}$, where, $p_{f}$ is the Fermi momentum.

The magnetically modified electronic energy states are obtained as solutions
of Dirac equations in presence of finite magnetic field Akhiezer ; Lifshitz .
The positive energy states are denoted by quantum numbers
$\epsilon,p_{z},n,s$. $p_{z}$ is the electron momentum along the field which
we consider along $z$ direction, $s=\pm 1$ is the helicity, and $n=0,1,2$
enumerates the Landau levels. For non-zero $B$,
$\mu\equiv\mu(B)=(2\pi^{2}n_{e})/m\omega_{B}$. The energy of the relativistic
electrons in presence of magnetic field is
$\epsilon_{p}=\sqrt{p_{z}^{2}+m^{2}+2n\omega_{B}m}.$

The ground Landau level is non-degenerate with respect to spin while the
higher levels are doubly degenerate. The number density of electrons in
presence of magnetic field is written as,

$\displaystyle
n_{e}=\frac{m\omega_{B}}{(2\pi)^{2}}\int_{-\infty}^{\infty}dp_{z}\sum_{n,s}f(\epsilon_{p}),$
(27)

where, the sums are over $n,s$. The magnetic field strongly quantizes the
motion of electrons and different transport coefficients receive significant
contribution when the electrons are confined to the zeroth Landau level. We do
not consider the situation when ions receive quantum modifications due to the
magnetic field. Parameters which determine zeroth level population are as
follows Chamel:2008ca ,

$\displaystyle T_{\mathrm{c}e}\approx 1.343\times
10^{8}\;B_{12}~{}\mathrm{K},$ $\displaystyle\rho_{B}=7.045\times 10^{3}{A\over
Z}\;({B_{12}})^{3/2}\mathrm{\ g\ cm}^{-3}.$ (28)

In the above equation, $B_{12}=B/10^{12}$, $\omega_{ce}$ is the cyclotron
frequency for electrons. $B$ is strongly quantizing if $\rho<\rho_{B}$ and
$T\ll T_{ce}$.

It is convenient to introduce the relativistic parameters $x_{r}=p_{F}/m\sim
1.008(\frac{\rho_{6}Z}{A})^{1/3}$ (where $\rho_{6}=\rho/10^{6}$), $T_{r}\sim
5.930\times 10^{9}$ K. The electron-ion plasma is relativistic for $x_{r}\gg
1$ and $T\gg T_{r}$. Thus the electrons become relativistic when $T>5\times
10^{9}$ K and density $\rho\sim 10^{6}$ $g$ $cm^{-3}$.

The momentum of an electron is related to the energy via the relation
$p_{nz}/m=\sqrt{(\epsilon_{p}/m)^{2}-2bn-1}$. From this expression one can
obtain the maximum Landau level that the electrons can populate and is given
by the integer part of $\nu=(E^{2}-1)/2b$. The energy of the electrons is
constrained by the relation $(E^{2}-2b)<1$ to meet the condition of lowest
Landau level. This is an important condition for the plots of $\sigma$ and
$\kappa$ as we describe later in this section. The parameters for density,
temperature and magnetic field are appropriately chosen for relativistic
quantized electrons to simultaneously meet the physical conditions applicable
for the merging scenario also. We consider Fe and Mo for the numerical
analysis of both the coefficients. The reason behind choosing these elements
is given in the Appendix.B.

###  3.2 Variation with density

Fig.2 shows the variation of $\sigma$ and $\kappa$ given in Eqs.(24) and (25)
with $\rho$ for different temperatures and different magnetic fields for
different elements Fe and Mo. In order to consider electrons to be
relativistic, the density and temperatures are chosen as $\rho\gg 10^{6}$ $g$
$cm^{-3}$ and $T\gg 5\times 10^{9}$ $K$ respectively. For fixed $B$, $\mu$
increases with $\rho$ and electrons start to populate higher Landau levels.
Since, we are interested in population of the zeroth Landau level, the density
and temperature should also satisfy $\rho<\rho_{B}$ and $T\ll T_{ce}$ as given
in Eq.(28). With these two conditions, both the coefficients have been
obtained by numerically integrating the expressions in Eqns. 24 and 25. In
fig.2, the upper and lower panel plots are for the variation of $\sigma$ and
$\kappa$ with $\rho$ respectively. The left panel plots have been drawn
considering magnetic field B=$10^{17}$ G and $7.5\times 10^{16}$ G in the
right panel. Each plot has three curves corresponding to three different
temperatures $1.4\times 10^{11}K$, $2\times 10^{11}K$ and $5\times 10^{11}K$
for each element. This is observed from the figure that at temperatures
$5\times 10^{10}$ K and $2\times 10^{10}$ K, prominent humps are present in
both the coefficients. The origin of the hump is due to the fulfillment of the
weak degeneracy condition ($|\epsilon_{p}-\mu|\sim T$) of electron
distribution function. The nature of the curve resembles differentiated Fermi
function at $T\ll\mu$. As the temperature increases, the hump gets flattened
since electrons start becoming non-degenerate. In each plot of the figure
curves are drawn for two different materials Fe (solid lines) and Mo (dotted
lines).

###  3.3 Variation with temperature and magnetic field

Fig.(3) shows variation of $\sigma$ and $\kappa$ with $T$ for different
densities. The $\sigma$ in fig.3(a) can be fitted as $\sigma=(a+b\times
T^{c})^{-1}$, with $a=1.45\times 10^{-25}$, $b=1.05\times 10^{-46}$ and
$c=1.919$. At low temperature the effect of $T^{c}$ is very small, hence,
$\sigma$ is constant. On the other hand at high temperature $\sigma\propto
T^{-c}$ and decreases with temperature. Thus, at higher temperatures, the
electrons become classical obeying the inverse dependence of temperature.

The temperature dependence of $\kappa$ can be attributed to low temperature
behaviour of Fermi function whose derivative shows a hump when
$|\epsilon_{p}-\mu|\sim T$. In each plot of fig.(3), we have three curves of
each Fe (solid lines) and Mo (dotted lines) for three different densities
$3\times 10^{11}$ gcm-3, $3.5\times 10^{11}$ gcm-3 and $4\times 10^{11}$
gcm-3.

Fig.(4) shows the variation of $\sigma$ and $\kappa$ with $B$ for different
densities. The fitting parameters for the $\sigma$ with $B$ is given by,
$\sigma=5.15\times 10^{27}-2.71\times 10^{11}B+5.30\times 10^{-6}B^{2}$.
Increasing the magnetic field, $\sigma$ increase with $B$ and saturates.
Variation of $\kappa$ with $B$ is given by the polynomial equation
$\kappa=2.43\times 10^{23}-81.2\times 10^{5}B+8.85\times
10^{-11}B^{2}-3.13\times 10^{-28}B^{3}$. $\kappa$ shows similar trend like
$\sigma$ with $B$.

(a) Variation of $\sigma$ with $\rho$

(b) Variation of $\kappa$ with $\rho$

Figure 2: The comparison of $\sigma$ (upper plots) and $\kappa$ (lower plots)
with $\rho$ for Fe (solid lines) and Mo (dotted lines). The left panel plots
are for magnetic field $10^{17}$ G and the right ones are for $B=7.5\times
10^{16}$ G. In each plot we have three curves for each element corresponding
to three different temperatures $1.4\times 10^{11}$ K, $2\times 10^{11}$ K,
$5\times 10^{11}$ K.

(a) Variation of $\sigma$ with $T$

(b) Variation of $\kappa$ with $T$

Figure 3: The comparison of $\sigma$ (upper panel plots) and $\kappa$ (lower
panel plots) with T for Fe (solid lines) and Mo (dotted lines). The left panel
plots are for magnetic field $10^{17}$ G and the right ones are for
$B=7.5\times 10^{16}$ G. In each plot we have three curves of each element
corresponding to three different densities $3\times 10^{11}$ gcm-3, $3.5\times
10^{11}$ gcm-3, $4\times 10^{11}$ gcm-3.

(a) Variation of $\sigma$ with $B$

(b) Variation of $\kappa$ with $B$

Figure 4: The comparison of $\sigma$ (upper panel) and $\kappa$ (lower panel)
with B for different densities. The temperatures chosen are $2\times 10^{10}$
K (left panel) and $5\times 10^{10}$K (right panel). The choice of elements
are Fe (solid lines) and Mo (dotted lines).

(a) Effect of HDL propagator on $\sigma$

(b) Effect of HDL propagator on $\kappa$

Figure 5: Plots of $\sigma$ (upper panel) and $\kappa$ (lower panel) with
$\rho$ comparing static, Coulomb screening in non-relativistic plasma (blue)
and dynamical screening in relativistic plasma (black) at fixed temperature of
$2\times 10^{10}$ K. The left panel curves are for B$=10^{17}$ G and the right
curves are for B$=5\times 10^{16}$ G. The choice of elements are Mo (dotted
lines) and Fe (solid lines).

Figure 6: The variation of $\kappa/\sigma$ with $T$ for different values of
densities $3\times 10^{11}$ gcm-3, $3.5\times 10^{11}$ gcm-3, $4\times
10^{11}$ gcm-3. The chosen magnetic field is $10^{17}$ G.

###  3.4 Effect of HDL modified propagator

In the Fig.5 we explicitly estimate the modification induced in $\sigma$ and
$\kappa$ when electron-ion interact through HDL photon propagator and also
compare the case when these plasma constituents interact through static,
longitudinal photon propagator. The upper panel is for variation of $\sigma$
and the lower panel is for $\kappa$. The left panel considers magnetic field
of B=$10^{17}$ G and the right panel B=$5\times 10^{16}$ G. In all the four
plots we consider $T=2\times 10^{10}$K. Blue curves represent the coefficients
when the interaction rate is screened through static, longitudinal photon
propagator relevant in non-relativistic plasma and black curves when the
constituents interact through frequency dependent HDL photon propagator
required in the relativistic plasma. We find that the inclusion of HDL
propagator reduces the values of both the coefficients substantially. For
$\sigma$ the maximum value of the curve decreases by a factor of $31$ in
comparison to the case when the interaction rate is Debye screened, whereas,
for $\kappa$ the factor of reduction is $20$.

It is known that modifications to different equilibrium and non-equilibrium
properties of plasma due to inclusion of HDL propagators emerge from frequency
dependent photon propagator. The reduction in both the transport coefficients
shown above arises due to the increase in the interaction rate caused by non-
zero frequency in both the longitudinal and transverse components of the
photon propagator. The electromagnetic perturbation to the distribution
function ($\Psi$) is inversely proportional to $I_{fi}$ and $\sigma,\kappa$
are directly proportional to $\Psi$. Hence, increment in $\Psi$ decreases both
the coefficients. The increment in $I_{fi}$ with the inclusion of HDL photon
propagator can be seen from Eq.(2). The second term in Eq.(2) is dependent on
$v_{k}<1$. $v_{k}$ reduces the numerator of second term in Eq.(2). In the
denominator of the same equation $q_{\perp}^{2}<2m\omega_{B}$ and
$m_{D}^{2}<2m\omega_{B}$ lead to $u<1$, $\zeta<1$ respectively. Hence, the
reduction due to small value of $v_{k}$ gets compensated and interaction rate
increases due to frequency dependent dynamical screening.

###  3.5 Estimation of dissipation time scales

In this paper we give quantitative estimations of both the time scales related
to electrical and thermal transport coefficients. The magnetic field decay (or
diffusion) timescale due to Ohmic dissipation is given by the well-known
expression Harutyunyan:2018mpe ; Rezzolla_book:2013 ,
$\tau_{\sigma}=4\pi\sigma\lambda^{2}_{B}/c^{2}$. The time scale is dependent
on the two factors magnetic field scale height ($\lambda_{B}$) and $\sigma$.
For a typical value of $\lambda_{B}\sim 10^{-4}$ c.m (
$\lambda_{B}\geq\lambda_{mfp}\mbox{(electron mean free path)}\sim 10^{-4}$cm,
$\lambda_{B}<\lambda_{lim}\mbox{(limiting magnetic field scale height)}\sim
10^{-3}$cm) we obtain $\tau_{\sigma}\sim 15$ $ms$ which is well within the
range of survival time period of neutron star merger. Choice of $\lambda_{B}$
satisfies both the conditions specified by Ref.Harutyunyan:2018mpe ,
$\lambda_{mfp}<\lambda_{B}>\lambda_{lim}$. With the same choice of
$\lambda_{B}$, $\tau_{\sigma}\sim 600$ $ms$ in the non-relativistic scenario.
We emphasize on the reduction of time scale due to inclusion of dynamical
screening in the calculation in contrast to the scenario when electrons-ions
interact through longitudinal plasmon.

Thermal conduction time scale is defined as $\tau_{\kappa}=c_{v}\Delta
z^{2}/6\kappa$, where, $c_{v}$ is the specific heat and $\Delta z$ is the
region which is hotter than surrounding by a temperature difference $\Delta
T$. The contribution of degenerate relativistic electrons in specific heat is,
$c_{v}=5.4\times 10^{19}\left(\frac{n_{e}}{n_{0}}\right)^{2/3}T_{9}$, $n_{0}$
is the nuclear saturation density and $T_{9}=T/10^{9}$. For $n_{e}=0.57\times
10^{35}\mbox{cm}^{-3}$, temperature $T=2\times 10^{10}$K and $\Delta z\sim 25$
c.m, thermal equilibration time scale becomes $\sim 50$ ms which is of the
order of time scale of the merged object. For $\Delta z\sim 1$ km, the time
scale is much greater than the survival time period of the merged compact
star. However, the estimation of time scales presented here demands a more
detailed hydrodynamic calculation to assess the inclusion of dissipation and
resistivity in magneto-hydrodynamic simulation of BNS mergers.

###  3.6 Validity of Wiedemann-Franz law

In a typical degenerate plasma, electrons’ conduction dominates system’s
dissipation coefficients through electrical and thermal coefficients. This
suggests linear increment in $\sigma/\kappa$ with $T$ known as Wiedemann-Franz
law. From the plot of $\sigma$ in fig.(3), we see that when $T<5\times
10^{11}$ K (for $\rho=4\times 10^{11}$gm cm-3) $\sigma$ is temperature
independent. For $T>5\times 10^{11}$ K, $\sigma$ decreases with temperature.
However, $\kappa$ increases with temperature and after the mentioned
temperature, it decreases. Hence, $\kappa/\sigma$ first increases with
temperature and after $5\times 10^{11}$ K it decreases, violating Wiedemann-
Franz law. From eq.(24) and eq.(25) it is evident that if
$(\epsilon_{p}-\mu)=T$ linear scaling of temperature exists between $\kappa$
and $\sigma$. At temperatures less than $T<5\times 10^{10}$ K,
$(\epsilon_{p}-\mu)<T$, hence, $\sigma/\kappa\propto T^{3}$. In both the
temperature domains of $T<5\times 10^{10}$K and $T>5\times 10^{10}$K,
degeneracy condition $|\epsilon_{p}-\mu|\ll\mu$ gets satisfied.

##  4 Summary and conclusions

In this work, we have calculated the quantized longitudinal electronic
transport coefficients in dynamically screened hot and dense magnetized QED
plasma involved in binary neutron star merger simulations. The calculation
considers scattering of electrons with ions through screened electromagnetic
force in electron-ion plasma. We have presented the plots for the variation of
transport coefficients with density ( $\sim 10^{12}$ gm cm-3), temperature
($\sim 10^{10}$ K) and magnetic field ($\sim 10^{17}$ G) for two elements Mo
and Fe. The scales for the generation of the plots are chosen so that they
obey the conditions of relativistic electrons at density $\rho>10^{6}$ g cm-3,
temperature $T>5.93\times 10^{9}$ K and zeroth Landau level population by
obeying $\sqrt{E^{2}-1}/2b\ll 1$. These two constraints indicate, high
magnetic field and low density regime of BNS mergers as the valid physical
domain of our calculation.

For the calculation of both electrical and thermal conductivities, we have
assumed particles are slightly out of equilibrium which allows us to solve the
Boltzmann equation numerically. We calculate electron-ion scattering amplitude
with screened electromagnetic interaction of magnetically modified spinors.
The off-equilibrium distribution function has been obtained by solving the
Boltzmann kinetic equation in relaxation time approximation. However, we have
not considered the finite size of the nuclei and ion structure function for
the calculation of the relaxation rate.

The electromagnetic interaction between electrons and ions have been
incorporated through HDL propagator in the calculation. The calculation should
account for magnetically modified anisotropic photon propagator; however in
the present paper, we have considered only isotropic HDL propagator. For the
screening mass zero temperature magnetically modified Debye mass has been
incorporated but in the hot and dense plasma, inclusion of finite temperature
screening mass would be more relevant.

We have found that the inclusion of the HDL propagator in the relativistic
plasma reduces the values of both $\sigma$ and $\kappa$ in contrast to static
screening in the non-relativistic plasma. The frequency dependent screening in
the HDL propagator enhances the interaction rate. The off equilibrium
distribution function is inversely proportional to the interaction rate and
the transport coefficients are directly proportional to $\Phi$. Hence,
enhancement in the interaction rate decrease both the transport coefficients
at high density in the degenerate regime. We also observe cubic temperature
scaling between $\kappa$ and $\sigma$ violating the Wiedemann-Franz law.

A rough estimation of diffusion and thermal equilibration time scales from the
expressions of quantized $\sigma$, $\kappa$ including frequency dependent
screening have also been attempted. The obtained Ohmic time scale has found
out to be of the same order as the survival time of the merged compact object
due to reduction in the value of $\sigma$ (for $\lambda_{B}\ll$ cm). The
thermal equilibration time scale can also be found to match with the time
scale of the merged object when $\Delta z\sim$ cm. One can infer from the
analysis that many-body effects play an important role in determining the
dissipative time scales relevant in the neutron star merger in the low
density, high temperature and high magnetic field regime if the length-scales
are of the order mentioned above. This estimation of dissipation time scales
of the transport coefficients requires rigorous hydrodynamic formulation for
more realistic values of $\lambda_{B}$ and $\Delta z$. The current calculation
of $\sigma$ and $\kappa$ can be implemented in modeling the magneto-thermal
evolution of the merged compact object as well.

In the current paper, we have considered only longitudinal component of the
transport coefficients (and ignored all the other tensorial components) in the
equation for magnetic-field evolution in plasma. The realistic estimate of
$\tau_{\sigma}$ and $\tau_{\kappa}$ can only be obtained if all the components
of conductivity tensor are known in the background of relevant equation of
state for neutron star merger. However, our calculations presented in this
paper provide a significant step towards conjoining the complex microphysical
effects in plasma with the GRMHD simulations. The present formalism of
transport theory can be extended in relativistic, magnetised QCD matter with
certain modifications like inclusion of QCD coupling constant, diagrams of
strong interaction and proper vertex corrections.

### Acknowledgments

Authors would also like to thank and acknowledge R. Nandi for fruitful
discussions regarding various aspects of this work. Authors acknowledge
fruitful discussion with T. Mazumder regarding numerical analysis of the
current project. SPA acknowledges the Polish Academy of Sciences through grant
agreement
PAN.BFD.S.BDN.612.022.2021-PASIFIC 1, QGPAnatomy. This work received funding
from the European Union’s Horizon 2020 research and innovation program under
the Maria Sklodowska-Curie grant agreement No. 847639 and from the Polish
Ministry of Education and Science.

## Appendix A

In this appendix, we present the important steps for the evaluation of
electron-ion scattering rate in presence of magnetic field. An electron of
momentum $p\equiv(\epsilon_{p},\vec{p})$ scatters with an ion of momentum
$k\equiv(\epsilon_{k},\vec{k})$ leading to the final momentum states
$p^{\prime}\equiv(\epsilon_{p^{\prime}},\vec{p^{\prime}})$ and
$k^{\prime}\equiv(\epsilon_{k^{\prime}},\vec{k^{\prime}})$. The scattering
rate from initial state to final state in absence of magnetic field is given
by,

$\displaystyle
I_{fi}=\frac{1}{2\epsilon_{p}}\int\frac{d^{3}p^{\prime}}{(2\pi)^{3}2\epsilon_{p^{\prime}}}\int\frac{d^{3}k}{(2\pi)^{3}2\epsilon_{k}}\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}2\epsilon_{k^{\prime}}}$
$\displaystyle[f_{0}(\epsilon_{p})g_{0}(\epsilon_{k})\left(1-f_{0}(\epsilon_{p}^{\prime})\right)-f_{0}(\epsilon_{p}^{\prime})g_{0}(\epsilon_{k}^{\prime})\left(1-f_{0}(\epsilon_{p})\right)]$
$\displaystyle\left(2\pi\right)^{4}\delta\left(\epsilon_{p}+\epsilon_{k}-\epsilon_{p^{\prime}}-\epsilon_{k^{\prime}}\right)\delta^{3}\left(\vec{p}+\vec{k}-\vec{p^{\prime}}-\vec{k^{\prime}}\right)|{\cal
M}_{fi}|^{2}$

The above equation is modified in presence of magnetic field,

$\displaystyle I_{fi}$ $\displaystyle=$
$\displaystyle\frac{eB}{16(2\pi)^{5}}\sum_{n^{\prime},p^{\prime}_{z},s^{\prime}}\int
dp^{\prime}_{z}dy_{B}\int d^{3}kdq_{y}\int\frac{d\Omega_{k}}{4\pi}$ (30)
$\displaystyle
g_{f}(k)\left(\Phi_{n,p_{z},s}-\Phi_{n^{\prime},p^{\prime}_{z},s^{\prime}}\right)\delta(\epsilon_{p}+\epsilon_{k}-\epsilon_{p}^{\prime}-\epsilon_{k}^{\prime})$
$\displaystyle\sum_{q_{x},q_{z}}\delta_{k_{x}^{\prime}-k_{x},q_{x}}\delta_{k_{z}^{\prime}-k_{z},q_{z}}|{\cal
M}_{fi}|^{2},$

where, $y_{B}=p_{x}/m\omega_{B}$ and $\Phi_{n,p_{z},s}$ has already been
defined in the Introduction. We have inserted $\int d\Omega_{k}/4\pi=1$ in the
above equation. The argument of delta function can be written as follows,

$\displaystyle\delta(\epsilon_{p}+\epsilon_{k}-\epsilon_{p^{\prime}}-\epsilon_{k^{\prime}})$
$\displaystyle=$
$\displaystyle\delta(\epsilon_{k}-\epsilon_{k^{\prime}}-\hat{p}.\vec{q})$ (31)

where, we have used
$\epsilon_{|p^{\prime}|}=\epsilon_{|p-q|}=\epsilon_{p}-\hat{p}.\vec{q}$. The
angular integrals of $\vec{k}$ can be expressed as,

$\displaystyle\int\frac{d\Omega_{k}}{4\pi}\delta(\hat{p}.\vec{q}-\vec{v_{k}}.\vec{q})=\frac{1}{2|\vec{q}|}$
$\displaystyle\int\frac{d\Omega_{k}}{4\pi}\delta(\hat{p}.\vec{q}-\vec{v_{k}}.\vec{q})(\hat{p}.\hat{k}-\hat{p}.\hat{q}\hat{q}.\hat{k})^{2}$
$\displaystyle=\frac{1}{4|\vec{q}|}\left(1-\frac{(\hat{p}.\vec{q})^{2}}{\vec{|q|}^{2}}\right)\left(v_{k}^{2}-\frac{(\hat{p}.\vec{q})^{2}}{\vec{|q|}^{2}}\right).$
(32)

Eq.(30) can further be written as,

$\displaystyle I_{fi}$ $\displaystyle=$
$\displaystyle\frac{n_{i}}{32(2\pi)^{2}}\sum_{n^{\prime},p_{z}^{\prime},s^{\prime}}\int
dq_{z}dq_{x}\frac{dq_{y}}{\vec{|q|}}$ (33)
$\displaystyle\left[\Phi_{n,p_{z},s}-\Phi_{n^{\prime},p^{\prime}_{z},s^{\prime}}\right]|{\cal
M}_{fi}|^{2},$

where, we have changed the variable $dy_{B}^{\prime}=dq_{x}/m\omega_{B}$
following momentum conservation $p^{\prime}_{x}-p_{x}=q_{x}$. In the above
equation, $n_{i}$ is the number density of ions which can be expressed in
terms of electron number density as $n_{i}=n_{e}/Z$. Further, $n_{e}$ can be
expressed in terms of the Debye mass as $n_{e}=\mu m_{D}^{2}/3e^{2}$.

Next we introduce a dimensionless variable $y=q_{z}/|\vec{q}|$,

$\displaystyle I_{fi}$ $\displaystyle=$ $\displaystyle\frac{\mu
m_{D}^{2}eB}{3\times
32Ze^{2}(2\pi)^{2}}\sum_{n^{\prime},p_{z}^{\prime},s^{\prime}}\int
dydq_{x}dq_{y}$ (34)
$\displaystyle[\Phi_{n,p_{z},s}-\Phi_{n^{\prime},p^{\prime}_{z},s^{\prime}}]|{\cal
M}_{fi}|^{2}.$

In order to calculate $|{\cal M}|^{2}$, we use following electronic spinor in
presence of magnetic field,

$\Psi(r)=\frac{exp[i(p_{x}x+p_{z}z)]}{\sqrt{L_{x}L_{z}}}\Biggl{(}\begin{matrix}\tilde{\alpha}\tilde{A}H_{n-1}(\xi)\\\
-s\tilde{\alpha}\tilde{\beta}\tilde{H}_{n}(\xi)\\\
s\tilde{\beta}\tilde{A}\tilde{H}_{n-1}(\xi)\\\
\tilde{\beta}\tilde{B}\tilde{H}_{n}(\xi)\end{matrix}\Biggr{)}.$ (35)

Using above spinors and the expression for photon propagator (Eq.(2)) in
Eq.(8) in Section.2 one obtains,

$\displaystyle\sum_{spin}|{\cal M}|^{2}=\left(4\pi
Ze^{2}\right)^{2}\bigg{[}\frac{2\pi
ym_{D}^{2}}{\left(q_{\perp}^{2}+Re\Pi_{L}\right)^{2}+Im\Pi_{L}^{2}}$
$\displaystyle-\frac{2\pi
ym_{D}^{2}v_{k}^{2}}{2(q_{\perp}^{2}+Re\Pi_{T})^{2}+Im\Pi_{T}^{2})}\bigg{]}\sum_{n^{\prime},p_{z}^{\prime},s^{\prime}}\left[ss^{\prime}\tilde{\alpha}^{2}+\tilde{\beta}^{2}\right]$
$\displaystyle\left[ss^{\prime}\tilde{A}\tilde{A}^{\prime}I_{n^{\prime}-1}I_{n-1}+\tilde{A}\tilde{A}^{\prime}I_{n^{\prime}-1}I_{n-1}\right]^{2},$
(36)

where, $s$ and $s^{\prime}$ are $\pm$,

$\Bigg{(}\begin{matrix}\tilde{\alpha}\\\ \tilde{\beta}\\\
\end{matrix}\Bigg{)}=\Bigg{(}\begin{matrix}\sqrt{\frac{1}{2}(1+\frac{m}{\epsilon_{p}})}\\\
\sqrt{\frac{1}{2}(1-\frac{m}{\epsilon_{p}})}\\\ \end{matrix}\Bigg{)},$ (37)
$\Bigg{(}\begin{matrix}\tilde{A}\\\ \tilde{B}\\\
\end{matrix}\Bigg{)}=\Bigg{(}\begin{matrix}[\frac{1}{2}(1+\frac{sp_{z}}{\sqrt{\epsilon_{p}^{2}-m^{2}}})]^{1/2}\\\
[\frac{1}{2}(1-\frac{-sp_{z}}{\sqrt{\epsilon_{p}^{2}-m^{2}}})]^{1/2}\\\
\end{matrix}\Bigg{)}$ (38)

and

$\displaystyle
I_{n^{\prime},n}=\int_{-\infty}^{\infty}exp(iq_{y}y)\tilde{H}_{n^{\prime}}(\xi^{\prime})\tilde{H}_{n}(\xi)dy.$
(39)

$H_{n}(\xi)$ is the Hermite polynomial,

$\displaystyle
H_{n}(\xi)=\frac{m\omega_{B}}{\pi}^{\frac{1}{4}}(2^{n}n!)^{\frac{-1}{2}}exp^{\frac{-\xi^{2}}{2}}H_{n}(\xi),$
(40)

$\xi=\sqrt{m\omega_{B}}$. Inserting $I_{n^{\prime},n}$ in Eq.(36) and
performing the $y$ integration we obtain,

$\displaystyle\sum_{spin}|{\cal M}|^{2}=\left(4\pi
Ze^{2}\right)^{2}\Big{[}\frac{2\pi
ym_{D}^{2}}{(q_{\perp}^{2}+Re\Pi_{L})^{2}+Im\Pi_{L}^{2}}$ (41)
$\displaystyle-$ $\displaystyle\frac{2\pi
ym_{D}^{2}v_{k}^{2}}{2(q_{\perp}^{2}+Re\Pi_{T})^{2}+Im\Pi_{T}^{2})}\Big{]}\sum_{n^{\prime},p_{z}^{\prime},s^{\prime}}\Big{[}1+ss^{\prime}$
$\displaystyle+\frac{m^{2}}{\epsilon_{p}^{2}}(1-ss^{\prime})\Big{[}1+ss^{\prime}-\frac{1}{2}\frac{ss^{\prime}}{\epsilon_{p}^{2}-m^{2}}(\eta^{\prime}p_{n^{\prime}}-p_{z})^{2}\Big{]}$
$\displaystyle\Big{[}F^{2}_{n^{\prime},n}(u)+F^{2}_{n^{\prime}-1,n-1}(u)]\Big{]}-\frac{ss^{\prime}u\omega_{B}m}{\epsilon_{p}^{2}-m^{2}}$
$\displaystyle\Big{[}F^{2}_{n^{\prime}-1,n}(u)+F^{2}_{n^{\prime},n-1}(u)\Big{]}$
$\displaystyle-\frac{sp_{z}+s^{\prime}\eta^{\prime}p_{n}^{\prime}}{\sqrt{\epsilon_{p}^{2}-m^{2}}}\Big{[}F^{2}_{n^{\prime},n}(u)-F^{2}_{n^{\prime}-1,n-1}(u)\Big{]}$

where, $\eta^{\prime}=\pm$,
$F_{n^{\prime},n}(u)=exp^{-u/2}u^{\frac{n-n^{\prime}}{2}}\sqrt{\frac{n^{\prime}!}{n!}}L_{n^{\prime}}^{n-n^{\prime}}$
and $u=\frac{1}{2m\omega_{B}}(q_{x}^{2}+q_{y}^{2})$. The functions
$L_{n^{\prime}}^{n-n^{\prime}}(u)$ are Laguerre polynomials and
$F_{n^{\prime},n}(u)$ are normalized to
$\int_{0}^{\infty}F_{n^{\prime},n}^{2}du=1$.

Now, to perform the integration in $y$ in Eq.(34) we use the following sum
rule,

$\displaystyle\int_{-1}^{1}\frac{dy}{y}\,\frac{1}{2\pi}\,\frac{2\,{\rm
Im}\,\Pi_{L}(y)}{(q_{\perp}^{2}{+}{\rm Re}\,\Pi_{L}(y))^{2}+({\rm
Im}\,\Pi_{L}(y))^{2}}$ $\displaystyle-\frac{2v_{k}^{2}\,{\rm
Im}\,\Pi_{T}(y)}{(q_{\perp}^{2}{+}{\rm Re}\,\Pi_{T}(y))^{2}+({\rm
Im}\,\Pi_{T}(y))^{2}}$ $\displaystyle=\left(\frac{1}{q_{\perp}^{2}+{\rm
Re}\,\Pi_{T,L}(y{=}\infty)}-\frac{v_{k}^{2}}{q_{\perp}^{2}+{\rm
Re}\,\Pi_{T,L}(y{=}0)}\right).$

In the limiting case, ${\rm Re}\,\Pi_{T,L}(y{=}\infty)=m_{D}^{2}/3$,
${\rm Re}\,\Pi_{T}(y{=}0)=0,{\rm Re}\,\Pi_{L}(y{=}0)=m_{D}^{2}$.

Using the above relations the interaction rate becomes,

$\displaystyle I_{fi}=\frac{n_{i}}{32(2\pi)^{2}}\int
dq_{x}dq_{y}\left(\Phi_{n,p_{z},s}-\Phi_{n^{\prime},p^{\prime}_{z},s^{\prime}}\right)$
(43)
$\displaystyle\left[\frac{2}{3(q_{\perp}^{2}+\frac{m_{D}^{2}}{3})(q_{\perp}^{2}+m_{D}^{2})}-\frac{v_{k}^{2}}{6q_{\perp}^{2}(q_{\perp}^{2}+\frac{m_{D}^{2}}{3})}\right]{\cal
F},$

where,

$\displaystyle{\cal
F}=\frac{4\pi\sigma_{0}}{m^{2}}\sum_{n^{\prime},p_{z}^{\prime},s^{\prime}}[1+ss^{\prime}+\frac{m^{2}}{\epsilon_{p}^{2}}(1-ss^{\prime})$
$\displaystyle[1+ss^{\prime}-\frac{1}{2}\frac{ss^{\prime}}{\epsilon_{p}^{2}-m^{2}}(\eta^{\prime}p_{n^{\prime}}-p_{z})^{2}]$
$\displaystyle[F^{2}_{n^{\prime},n}(u)+F^{2}_{n^{\prime}-1,n-1}(u)]]$
$\displaystyle-\frac{ss^{\prime}u\omega_{B}m}{\epsilon_{p}^{2}-m^{2}}[F^{2}_{n^{\prime}-1,n}(u)+F^{2}_{n^{\prime},n-1}(u)]$
$\displaystyle-\frac{sp_{z}+s^{\prime}\eta^{\prime}p_{n}^{\prime}}{\sqrt{\epsilon_{p}^{2}-m^{2}}}[F^{2}_{n^{\prime},n}(u)-F^{2}_{n^{\prime}-1,n-1}(u)].$
(44)

We change the variable $q_{y}$ to $u$ and perform the integration as follows,

$\displaystyle\int dq_{x}dq_{y}$ $\displaystyle=$ $\displaystyle
m\omega_{B}\int\frac{dq_{x}du}{\sqrt{2m\omega_{B}u-q_{x}^{2}}}$ (45)
$\displaystyle=$ $\displaystyle m\omega_{B}\pi\int du$

Finally, the particle scattering rate becomes,

$\displaystyle I_{fi}$ $\displaystyle=$
$\displaystyle\frac{n_{i}}{2}\sum_{n^{\prime},p_{z}^{\prime},s^{\prime}}\int
du\left(\Phi_{n,p_{z},s}-\Phi_{n^{\prime},p^{\prime}_{z},s^{\prime}}\right)$
(46)
$\displaystyle\left[\frac{1}{3(u+\frac{\zeta}{3})(u+\zeta)}-\frac{v_{k}^{2}}{6u(u+\frac{\zeta}{3})}\right]{\cal
F},$

where, $\zeta=m_{D}^{2}/2m\omega_{B}$.

## Appendix B

In this appendix, we show the compariosn of the conductivities considering two
equation of states (EOSs), BPS (Baym, Pethick and Surtherland) Baym:1971pw
and magnetic BPS model Nandi:2010fp  and comment on considering Fe and Mo for
numerical analysis in the paper. In ref.Baym:1971pw  the equation of state of
zero-temperature matter in complete nuclear equilibrium is given for mass
densities below $5\times 10^{14}$ g cm-3. In Ref.Nandi:2010fp , the BPS
Baym:1971pw  has been extended to include the physical parameters for a low
density plasma in presence of high magnetic field relevant for neutron star
crust. In the fig.7, we plot $\sigma$ and $\kappa$ with $\rho$ for these two
different EOSs. From the plot fig.7 it is evident that the BPS curve matches
well with the Mo curve for both $\sigma$ and $\kappa$. We also include Fe for
reference.

Figure 7: The variation of $\sigma$ (left) and $\kappa$ (right) with $\rho$
for different elements. The magnetic field is chosen as $10^{17}$G and
temperature as $2\times 10^{10}$ T. We show the behaviour of the transport
coefficients for two different EOS s given in Refs. Baym:1971pw ; Nandi:2010fp

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