# Deeply virtual Compton scattering in the tensor-pomeron approach

Piotr Lebiedowicz<EMAIL_ADDRESS>Institute of Nuclear Physics
Polish Academy of Sciences, Radzikowskiego 152, PL-31342 Kraków, Poland  Otto
Nachtmann<EMAIL_ADDRESS>Institut für Theoretische
Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany
Antoni Szczurek 111Also at College of Natural Sciences, Institute of Physics,
University of Rzeszów, ul. Pigonia 1, PL-35310 Rzeszów, Poland.
<EMAIL_ADDRESS>Institute of Nuclear Physics Polish Academy of
Sciences, Radzikowskiego 152, PL-31342 Kraków, Poland

###### Abstract

The two-tensor-pomeron model proposed previously to describe low $x$ DIS data
is applied to real and virtual Compton scattering on a proton. The model
includes two tensor pomerons, a soft and a hard one, and tensor reggeons. We
include contributions of both transverse and longitudinal virtual photons. We
show that this model gives a very good description of experimental data at
small Bjorken $x$ on deeply virtual Compton scattering (DVCS) from HERA. The
reggeon exchange term is particularly relevant for describing the real-photon-
proton scattering measured at lower $\gamma p$ energies at FNAL. We present
two fits which differ somewhat in the strength of the hard pomeron
contribution. In both fits we find that the interference between soft- and
hard-pomeron exchange plays an important role. We find that in DVCS the soft-
pomeron contribution is considerable up to $Q^{2}\sim 20$ GeV2. Our model
allows to study the transition from the small-$Q^{2}$ regime, including the
photoproduction ($Q^{2}=0$) limit, to the large-$Q^{2}$ regime, the DIS limit.
We also discuss the ratio of cross sections for longitudinally and
transversely polarized virtual photons in $\gamma^{*}p\to\gamma p$ as a
function of $|t|$ and $Q^{2}$. The ratio
$\tilde{R}(Q^{2},W^{2},t)=(d\sigma_{\rm L}/dt)/(d\sigma_{\rm T}/dt)$ strongly
increases with $t$. Our findings may be checked in future lepton-nucleon
scattering experiments in the low-$x$ regime, for instance, at a future
Electron-Ion Collider at the BNL (EIC), and, if LHeC is realized, at the LHC.

##  I Introduction

In this Letter we apply the two-tensor-pomeron model Britzger:2019lvc  to
deeply virtual Compton scattering (DVCS) on the proton ($\gamma^{*}p\to\gamma
p$). This Regge type model can be used for large $\gamma^{*}p$ center-of-mass
energy $W\gg m_{p},\sqrt{|t|}$, $|t|\lesssim 1$ GeV2, and small Bjorken-$x$,
say $x=Q^{2}/(W^{2}+Q^{2}-m_{p}^{2})<0.02$. Here $m_{p}$ is the proton mass,
$Q^{2}$ is the photon virtuality, and $t$ is the squared momentum transfer at
the proton vertex.

The DVCS has been a subject of extensive experimental and theoretical
research; see e.g. Ji:1996nm ; Diehl:1997bu ; Bartels:1981jh ; Diehl:2003ny ;
Diehl:2005pc ; Machado:2008tp ; Kopeliovich:2008ct ; Braun:2014sta ;
Kriesten:2019jep ; Dutrieux:2021nlz  for theory papers. For a review and many
references see Diehl:2003ny . Recently, the possibility of probing the
nonlinear (saturation) effects on the DVCS process was considered in
Bendova:2022xhw ; Xie:2022sjm ; Goncalves:2022wzq . A comprehensive
experimental overview of DVCS is presented in dHose:2016mda ; see also Fig. 5
of Aschenauer:2013hhw  which shows the kinematic coverage in the $x$-$Q^{2}$
plane for existing DVCS measurements, as well as planned ones. The H1
H1:2005gdw ; H1:2009wnw  and ZEUS ZEUS:2003pwh ; ZEUS:2008hcd  Collaborations
measured the DVCS cross section over a broad range of $W$ and $Q^{2}$ at low
Bjorken-$x$. Also the differential cross section $d\sigma/dt$ was extracted.
In H1:2009wnw  the beam charge asymmetry due to the interference between the
DVCS and the purely electromagnetic Bethe-Heitler process was measured. The
HERMES Collaboration HERMES:2011bou ; HERMES:2012gbh ; HERMES:2012idp
performed measurements of single- and double-spin DVCS asymmetries. The
Jefferson Lab CLAS Collaboration measured DVCS beam-spin asymmetries and cross
sections CLAS:2007clm ; CLAS:2015uuo ; CLAS:2018bgk ; CLAS:2021gwi  and
longitudinally polarized target-spin asymmetries CLAS:2006krx ; CLAS:2014qtk ;
CLAS:2015bqi . The COMPASS Collaboration COMPASS:2018pup  measured the DVCS
cross section by studying exclusive single-photon production in muon-proton
scattering, $\mu p\to\mu p\gamma$. The Jefferson Lab Hall A Collaboration
measured the photon electroproduction cross section in the valence quark
region JeffersonLabHallA:2006prd ; JeffersonLabHallA:2015dwe , as well as the
DVCS off the neutron JeffersonLabHallA:2007jdm , and recently
JeffersonLabHallA:2022pnx  the DVCS off the proton at high values of
Bjorken-$x$. Experimental programs at Jefferson Lab, at the Electron Ion
Collider (EIC) under construction at BNL Accardi:2012qut ; Aschenauer:2013hhw
; Aschenauer:2017jsk , and, if realized, at a Large Hadron Electron Collider
(LHeC) at the LHC LHeCStudyGroup:2012zhm ; LHeC:2020van , are expected to
improve our knowledge of DVCS in a wide kinematic range.

DVCS is a prime playground for the application of the generalized parton-
distribution (GPD) concept based on perturbative QCD (pQCD), cf. Diehl:2003ny
for a review. However, for small values of Bjorken $x$ the application of pQCD
concepts faces the problem that the effective expansion parameter is there
$\alpha_{s}\ln(1/x)$ which is large for small $x$. More phenomenological
models are then frequently used, based e.g. on the color-dipole approach or
the Regge approach. In our present work we use the latter approach where the
scattering is described using exchange objects. For high energies $W$ the
pomeron will be the most important exchange object. For general reviews of
pomeron physics see, e.g., Donnachie:2002en ; Caneschi . Applications of Regge
theory to DVCS have been given in Capua:2006ij ; Fazio:2013hza  and we shall
comment on these papers below. In our work we shall use the model of
Britzger:2019lvc  comprising a soft and a hard pomeron which couple to
particles like symmetric spin 2 tensors. It was shown in Britzger:2019lvc
that a vector nature of these couplings is excluded. In Ewerz:2016onn  it was
shown that scalar couplings of the pomeron can be excluded through the
experimental results of Adamczyk:2012kn . For some remarks on the history of
tensor-pomeron concepts and corresponding references see Ewerz:2016onn .

In the following we shall use the tensor-pomeron model as introduced for soft
high-energy reactions in Ewerz:2013kda  and extended to hard reactions in
Britzger:2019lvc . The soft and hard pomeron and the charge-conjugation $C=+1$
reggeons ($\mathbb{R}_{+}=f_{2\mathbb{R}},a_{2\mathbb{R}}$) are described as
effective rank-2 symmetric tensor exchanges, the odderon and the $C=-1$
reggeons ($\mathbb{R}_{-}=\rho_{\mathbb{R}},\omega_{\mathbb{R}}$) are
described as effective vector exchanges. Applications of this tensor-pomeron
concept were given for photoproduction of pion pairs in Bolz:2014mya , for a
number of exclusive central-production reactions, see for instance
Lebiedowicz:2013ika ; Lebiedowicz:2014bea ; Lebiedowicz:2016ioh ;
Lebiedowicz:2019boz ; Lebiedowicz:2019jru ; Lebiedowicz:2020yre , and for
soft-photon radiation (bremsstrahlung) in hadronic collisions
Lebiedowicz:2021byo ; Lebiedowicz:2022nnn .

In Britzger:2019lvc  the two-tensor-pomeron approach was applied to describe
simultaneously the experimental data on the inclusive cross sections for DIS
at low-$x$ H1:2015ubc  and for photoproduction. The global fit performed there
was based on the hard and soft pomeron, and the reggeon
$f_{2\mathbb{R}}+a_{2\mathbb{R}}$. The model describes the transition from the
low-$Q^{2}$ regime, including the $Q^{2}=0$ photoproduction limit, where the
real or virtual photon behaves hadronlike and the soft pomeron dominates, to
the large-$Q^{2}$ regime which is the domain of the hard pomeron. For the
photoproduction cross section $\sigma_{\gamma p}(W)$ no significant
contribution from the hard pomeron was found. In this model $\sigma_{\gamma
p}(W)$ is, in the range 6 GeV $<W<209$ GeV, dominated by soft-pomeron exchange
with a significant reggeon contribution for $W<30$ GeV. The transition region
where both pomerons contribute significantly was found to be roughly
$0<Q^{2}<20$ GeV2. But as $Q^{2}$ increases the hard pomeron component becomes
more and more important.

Our paper is organized as follows. In the next section we introduce the
tensor-pomeron approach for real and deeply virtual Compton scattering.
Section III presents the comparison of our model results with experimental
data. There we also discuss theoretical uncertainties and limitations of the
model. Section IV contains our conclusions.

##  II Theoretical formalism

We investigate the real and deeply virtual Compton scattering on a proton

$\gamma^{(*)}(q,\epsilon)+p(p,\lambda)\to\gamma(q^{\prime},\epsilon^{\prime})+p(p^{\prime},\lambda^{\prime})\,.$
(1)

The momenta are indicated in brackets,
$\lambda,\lambda^{\prime}\in\\{1/2,-1/2\\}$ are the proton helicities, and
$\epsilon,\epsilon^{\prime}$ are the photon polarization vectors.

For an initial virtual photon $\gamma^{*}$ the reaction (1) is extracted from
$ep\to ep\gamma$ scattering (see Fig. 1)

$e(k)+p(p,\lambda)\to
e(k^{\prime})+\gamma(q^{\prime},\epsilon^{\prime})+p(p^{\prime},\lambda^{\prime})\,.$
(2)

Figure 1:  DVCS contribution to $ep\to ep\gamma$ (2).

Here the Bethe-Heitler process and DVCS contribute with the latter
corresponding to electroproduction of the $\gamma p$ state. A detailed
kinematic analysis of electroproduction reactions can e.g. be found in
Arens:1996xw  where, in particular, many relations of variables in the rest
system of the original proton and in the system used in HERA experiments are
given. The standard kinematic variables for (1) and (2), setting the electron
mass to zero, are (see Table 1 of Arens:1996xw )

$\displaystyle q=k-k^{\prime}\,,\quad q^{2}=-Q^{2}\,,$ $\displaystyle
s=(p+k)^{2}\,,$ $\displaystyle t=(p-p^{\prime})^{2}\,,\quad
t\leqslant-|t|_{\rm min}\,,$ $\displaystyle W^{2}=(p+q)^{2}\,,$ $\displaystyle
x=\frac{Q^{2}}{2p\cdot q}=\frac{Q^{2}}{W^{2}+Q^{2}-m_{p}^{2}}\,,$
$\displaystyle y=\frac{p\cdot q}{p\cdot
k}=\frac{W^{2}+Q^{2}-m_{p}^{2}}{s-m_{p}^{2}}\,,$
$\displaystyle\varepsilon=\frac{2(1-y)-2xym_{p}^{2}(s-m_{p}^{2})^{-1}}{1+(1-y)^{2}+2xym_{p}^{2}(s-m_{p}^{2})^{-1}}\,.$
(3)

Here $\varepsilon$ is the ratio of longitudinal and transverse polarization
strengths of the virtual photon $\gamma^{*}$ and $t=-|t|_{\rm min}$
corresponds to forward scattering. For $Q^{2}=0$ we have $|t|_{\rm min}=0$ and
for $W^{2}\gg Q^{2},m_{p}^{2}$ we find (see also Appendix A of Arens:1996xw )

$|t|_{\rm min}=Q^{2}\left[\frac{Q^{2}m_{p}^{2}}{W^{4}}+{\cal
O}\left(\frac{Q^{6}}{W^{6}},\frac{Q^{4}m_{p}^{2}}{W^{6}},\frac{Q^{2}m_{p}^{4}}{W^{6}},\frac{m_{p}^{6}}{W^{6}}\right)\right]\,.$
(4)

We assume for (2) unpolarized initial particles and no observation of the
polarization of final state particles.

Adapting (3.20) and (3.21) of Arens:1996xw  to the reaction (2) and
integrating over the azimuthal angle $\varphi$ defined in (2.1) of
Arens:1996xw  we get for the DVCS part of (2)

$\displaystyle\frac{d\sigma(ep\to
ep\gamma)}{dydQ^{2}dt}=\Gamma_{ep\gamma}\left(\frac{d\sigma_{\rm
T}}{dt}(Q^{2},W^{2},t)+\varepsilon\frac{d\sigma_{\rm
L}}{dt}(Q^{2},W^{2},t)\right)\,.$ (5)

Here, with Hand’s convention Hand:1963bb , the $\gamma^{*}$ flux factor reads

$\displaystyle\Gamma_{ep\gamma}=\frac{\alpha}{\pi
yQ^{2}}\left(1-y+\frac{1}{2}y^{2}+xy\frac{m_{p}^{2}}{s-m_{p}^{2}}\right)\frac{W^{2}-m_{p}^{2}}{W^{2}-m_{p}^{2}+Q^{2}+4xm_{p}^{2}}\,.$
(6)

Furthermore, $d\sigma_{\rm T}/dt$ and $d\sigma_{\rm L}/dt$ are the
differential cross sections for $\gamma^{*}p\to\gamma p$ for transverse and
longitudinal polarization of the $\gamma^{*}$,

$\displaystyle\frac{d\sigma_{\rm
T}}{dt}(Q^{2},W^{2},t)=\frac{1}{2}\left(\frac{d\sigma_{++}}{dt}(Q^{2},W^{2},t)+\frac{d\sigma_{--}}{dt}(Q^{2},W^{2},t)\right)\,,$
$\displaystyle\frac{d\sigma_{\rm
L}}{dt}(Q^{2},W^{2},t)=\frac{d\sigma_{00}}{dt}(Q^{2},W^{2},t)\,,$ (7)

where

$\displaystyle\frac{d\sigma_{mm}}{dt}(Q^{2},W^{2},t)$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi(W^{2}-m_{p}^{2})\sqrt{(W^{2}-m_{p}^{2}+Q^{2})^{2}+4m_{p}^{2}Q^{2})}}$
(8)
$\displaystyle\times\frac{1}{2}\sum_{\lambda,\lambda^{\prime},a}\big{|}\Braket{\gamma(q^{\prime},\epsilon^{\prime}_{a}),p(p^{\prime},\lambda^{\prime}),{\rm
out}}{eJ_{\mu}(0)\epsilon_{m}^{\mu}}{p(p,\lambda)}\big{|}^{2}\,.$

Here $eJ_{\mu}$ is the electromagnetic current, $\epsilon_{m}(m=\pm,0)$ are
the standard $\gamma^{*}$ polarization vectors for right and left circular and
longitudinal polarization [see (3.11)–(3.14) of Arens:1996xw ] and
$\epsilon^{\prime}_{a}(a=1,2)$ are the polarization vectors of the real photon
in the final state.

So far, everything is completely general. Now we consider the small $x$ region
of DVCS:

$\displaystyle W\geqslant m_{p},\sqrt{|t|}\,,\quad|t|\lesssim 1\;{\rm
GeV}^{2}\,,\quad x\leqslant 0.02\,.$ (9)

There we describe the amplitude for (1) in terms of the exchanges of soft
($\mathbb{P}_{1}$) and hard ($\mathbb{P}_{0}$) pomeron and the reggeons
$f_{2\mathbb{R}}$ and $a_{2\mathbb{R}}$. Only $C=+1$ exchanges can contribute
in (1). The properties of $\mathbb{P}_{0},\mathbb{P}_{1},f_{2\mathbb{R}}$, and
$a_{2\mathbb{R}}$, and their couplings to protons and photons will be taken as
in Britzger:2019lvc . In this way we get

$\displaystyle\Braket{\gamma(q^{\prime},\epsilon^{\prime}),p(p^{\prime},\lambda^{\prime}),{\rm
out}}{eJ_{\nu}(0)\epsilon^{\nu}}{p(p,\lambda)}\equiv(\epsilon^{\prime\mu})^{*}{\cal
M}_{\mu\nu,\lambda^{\prime}\lambda}\,\epsilon^{\nu}$
$\displaystyle=-(\epsilon^{\prime\mu})^{*}\sum_{j=0,1}\Gamma_{\mu\nu\kappa\rho}^{(\mathbb{P}_{j}\gamma^{*}\gamma^{*})}(q^{\prime},q)\,\epsilon^{\nu}\,\Delta^{(\mathbb{P}_{j})\,\kappa\rho,\alpha\beta}(W^{2},t)\,\bar{u}_{\lambda^{\prime}}(p^{\prime})\Gamma_{\alpha\beta}^{(\mathbb{P}_{j}pp)}(p^{\prime},p)u_{\lambda}(p)$
$\displaystyle\quad+(\mathbb{P}_{j}\to f_{2\mathbb{R}},a_{2\mathbb{R}})\,.$
(10)

Here $\Delta^{(\mathbb{P}_{j})}$ and $\Gamma^{(\mathbb{P}_{j}pp)}$ denote the
effective propagator and proton vertex function, respectively, for the tensor
pomeron $\mathbb{P}_{j}$. The corresponding expressions, as given in Appendix
A of Britzger:2019lvc , are as follows

$\displaystyle
i\Delta^{(\mathbb{P}_{j})}_{\mu\nu,\kappa\lambda}(W^{2},t)=\frac{1}{4W^{2}}\left(g_{\mu\kappa}g_{\nu\lambda}+g_{\mu\lambda}g_{\nu\kappa}-\frac{1}{2}g_{\mu\nu}g_{\kappa\lambda}\right)(-iW^{2}\tilde{\alpha}^{\prime}_{j})^{\alpha_{j}(t)-1}\,,$
(11) $\displaystyle
i\Gamma_{\mu\nu}^{(\mathbb{P}_{j}pp)}(p^{\prime},p)=-i3\beta_{jpp}F_{1}^{(j)}(t)\left\\{\frac{1}{2}\left[\gamma_{\mu}(p^{\prime}+p)_{\nu}+\gamma_{\nu}(p^{\prime}+p)_{\mu}\right]-\frac{1}{4}g_{\mu\nu}(\not\\!p^{\prime}+\not\\!p)\right\\}.$
(12)

The coupling constants $\beta_{jpp}$ of the pomerons to protons are
$\beta_{0pp}=\beta_{1pp}=1.87$ GeV-1. The Ansätze for effective propagators
and vertices for the $f_{2\mathbb{R}}$ and $a_{2\mathbb{R}}$ reggeons are
analogous to (11) and (12), respectively. We shall assume identical
trajectories for $f_{2\mathbb{R}}$ and $a_{2\mathbb{R}}$ and combine their
contribution in (10) to a reggeon $\mathbb{R}_{+}$ term labeled $j=2$ as
explained in Appendix A of Britzger:2019lvc ; see (A20)–(A31) of
Britzger:2019lvc .

The pomeron and reggeon trajectory functions are assumed to be of linear form

$\displaystyle\alpha_{j}(t)=\alpha_{j}(0)+\alpha^{\prime}_{j}\,t\,,\quad\alpha_{j}(0)=1+\epsilon_{j}\,,\quad(j=0,1,2)\,.$
(13)

In (11) the parameters multiplying the squared energy $W^{2}$ in the effective
propagators are taken as $\tilde{\alpha}^{\prime}_{j}=\alpha^{\prime}_{j}$.
The slope parameters $\alpha^{\prime}_{j}$ are taken as the default values
from Britzger:2019lvc : $\alpha^{\prime}_{1}=\alpha^{\prime}_{0}=0.25\;{\rm
GeV}^{-2}$, $\alpha^{\prime}_{2}=0.9\;{\rm GeV}^{-2}$. The intercept
parameters of the trajectories (13) were determined from detailed comparison
of the two-tensor-pomeron model with the DIS HERA data and photoproduction
data in Ref. Britzger:2019lvc :

$\displaystyle{\rm soft\;pomeron}\;\mathbb{P}_{1}\mathrel{\mathop{\mathchar
58\relax}}\quad$ $\displaystyle\epsilon_{1}=0.0935(^{+76}_{-64})\,,$ (14)
$\displaystyle{\rm hard\;pomeron}\;\mathbb{P}_{0}\mathrel{\mathop{\mathchar
58\relax}}\quad$ $\displaystyle\epsilon_{0}=0.3008(^{+73}_{-84})\,,$ (15)
$\displaystyle{\rm reggeon}\;\mathbb{R}_{+}\mathrel{\mathop{\mathchar
58\relax}}\quad$ $\displaystyle\alpha_{2}(0)=0.485(^{+88}_{-90})\,.$ (16)

The Ansätze for the $\mathbb{P}\gamma^{*}\gamma^{*}$ and
$\mathbb{R}_{+}\gamma^{*}\gamma^{*}$ coupling functions for both real and
virtual photons are discussed in Britzger:2019lvc . The
$\mathbb{P}_{j}\gamma^{*}\gamma^{*}$ vertex, coupling the soft ($j=0$) and
hard ($j=1$) pomeron to two (virtual or real) photons, reads [see (A18) of
Britzger:2019lvc ]

$\displaystyle
i\Gamma^{(\mathbb{P}_{j}\gamma^{*}\gamma^{*})}_{\mu\nu\kappa\rho}(q^{\prime},q)=i\left[2a_{j\gamma^{*}\gamma^{*}}(q^{2},q^{\prime
2},t)\,\Gamma^{(0)}_{\mu\nu\kappa\rho}(q^{\prime},-q)\,-b_{j\gamma^{*}\gamma^{*}}(q^{2},q^{\prime
2},t)\,\Gamma^{(2)}_{\mu\nu\kappa\rho}(q^{\prime},-q)\right],$
$\displaystyle{\rm where}\;t=(q-q^{\prime})^{2},\quad j=0,1\,.$ (17)

The coupling functions $a_{j\gamma^{*}\gamma^{*}}$ and
$b_{j\gamma^{*}\gamma^{*}}$ have dimensions GeV-3 and GeV-1, respectively. The
rank-4 tensor functions $\Gamma^{(i)}_{\mu\nu\kappa\rho}$ ($i=0,2$) are
defined in (A13)–(A17) of Britzger:2019lvc . The
$\mathbb{R}_{+}\gamma^{*}\gamma^{*}$ vertex for real and virtual photons has
the same structure as (17) and is labeled with $j=2$; see (A27)–(A31) of
Britzger:2019lvc .

Inserting (11), (12) and (17) in (10) we get

$\displaystyle{\cal M}_{\mu\nu,\lambda^{\prime}\lambda}$ $\displaystyle=$
$\displaystyle-i\sum_{j=0,1,2}\left[2a_{j\gamma^{*}\gamma^{*}}(q^{2},0,t)\,\Gamma^{(0)}_{\mu\nu\kappa\rho}(q^{\prime},-q)\,-b_{j\gamma^{*}\gamma^{*}}(q^{2},0,t)\,\Gamma^{(2)}_{\mu\nu\kappa\rho}(q^{\prime},-q)\right]$
(18)
$\displaystyle\times\frac{1}{2W^{2}}(-iW^{2}\alpha^{\prime}_{j})^{\alpha_{j}(t)-1}\;3\beta_{jpp}F_{1}^{(j)}(t)\,\bar{u}_{\lambda^{\prime}}(p^{\prime})\gamma^{\kappa}(p^{\prime}+p)^{\rho}u_{\lambda}(p)\,.$

From (18) we can read off the parameters of our model. Taking the pomeron and
reggeon parameters as specified above for granted ${\cal
M}_{\mu\nu,\lambda^{\prime}\lambda}$ is parametrized by the functions

$\displaystyle
a_{j\gamma^{*}\gamma^{*}}(q^{2},0,t)\,\beta_{jpp}F_{1}^{(j)}(t)\,,$
$\displaystyle
b_{j\gamma^{*}\gamma^{*}}(q^{2},0,t)\,\beta_{jpp}F_{1}^{(j)}(t)\,,$
$\displaystyle j=0,1,2\,.$ (19)

Here we consider the $\beta_{jpp}$ as normalization constants with values
$\beta_{0pp}=\beta_{1pp}=1.87$ GeV-1, $\beta_{2pp}=3.68$ GeV-1, as given in
Appendix A of Britzger:2019lvc .

For forward virtual Compton scattering, $q^{2}=q^{\prime 2}=-Q^{2}$ and $t=0$,
we have from (A19), (A30), and (A31) of Britzger:2019lvc

$\displaystyle
a_{j\gamma^{*}\gamma^{*}}(-Q^{2},-Q^{2},0)=e^{2}\hat{a}_{j}(Q^{2})\,,$
$\displaystyle
b_{j\gamma^{*}\gamma^{*}}(-Q^{2},-Q^{2},0)=e^{2}\hat{b}_{j}(Q^{2})\,,$
$\displaystyle j=0,1,2\,.$ (20)

The coupling functions $\hat{a}_{j}(Q^{2})$ and $\hat{b}_{j}(Q^{2})$ were
determined from the global fit to HERA inclusive DIS data for $Q^{2}<50$ GeV2
and $x<0.01$ and the ($Q^{2}=0$) photoproduction data; see Appendix C and
Table IV of Britzger:2019lvc .

The $Q^{2}$ and $t$ dependencies of the coupling functions in the
$\mathbb{P}_{j}\gamma^{*}\gamma^{*}$ and $\mathbb{R}_{+}\gamma^{*}\gamma^{*}$
vertices in (18) must be determined from a comparison to experimental data. As
we shall show in Sec. III we obtain quite satisfactory descriptions of the
DVCS HERA and elastic $\gamma p$-scattering FNAL data with the following
assumptions:

$\displaystyle
a_{j\gamma^{*}\gamma^{*}}(q^{2},0,t)=e^{2}\hat{a}_{j}(Q^{2})F^{(j)}(t)\,,\quad(j=0,1,2)\,,$
$\displaystyle
b_{j\gamma^{*}\gamma^{*}}(q^{2},0,t)=e^{2}\hat{b}_{j}(Q^{2})F^{(j)}(t)\,,\quad(j=2)\,,$
(21)

and for alternative fits 1 and 2

$\displaystyle{\rm FIT~{}1}\mathrel{\mathop{\mathchar 58\relax}}\;$
$\displaystyle
b_{1\gamma^{*}\gamma^{*}}(q^{2},0,t)=e^{2}\hat{b}_{1}(0)(1+Q^{2}/\Lambda_{1}^{2})^{-1.2}\,F^{(1)}(t)\,,\quad\Lambda_{1}=1.4\;{\rm
GeV}\,,$ (22) $\displaystyle
b_{0\gamma^{*}\gamma^{*}}(q^{2},0,t)=e^{2}\hat{b}_{0}(Q^{2})F^{(0)}(t)\,,$
$\displaystyle{\rm FIT~{}2}\mathrel{\mathop{\mathchar 58\relax}}\;$
$\displaystyle
b_{1\gamma^{*}\gamma^{*}}(q^{2},0,t)=e^{2}\hat{b}_{1}(0)(1+Q^{2}/\Lambda_{2}^{2})^{-2.0}\,F^{(1)}(t)\,,\quad\Lambda_{2}=2.0\;{\rm
GeV}\,,$ (26) $\displaystyle
b_{0\gamma^{*}\gamma^{*}}(q^{2},0,t)=\left\\{\begin{array}[]{lr}e^{2}\hat{b}_{0}(Q^{2})\,F^{(0)}(t)&{\rm
for}\ Q^{2}<1.5~{}{\rm GeV}^{2}\\\
e^{2}\Lambda_{0}(1+Q^{2}/\Lambda_{3}^{2})^{-0.6}\,F^{(0)}(t)&{\rm for}\
Q^{2}\geq 1.5~{}{\rm GeV}^{2}\end{array}\right.\,,$
$\displaystyle\Lambda_{0}=9.46\times 10^{-3}\;{\rm
GeV}^{-1}\,,\quad\Lambda_{3}=2.3\;{\rm GeV}\,.$

The coupling functions as a function of $Q^{2}$ used in the calculations are
shown in Fig. 2. Shown are only the middle values of the pomeron- and
reggeon-$\gamma^{*}\gamma^{*}$ coupling functions up to $Q^{2}=50$ GeV2
determined from a global fit in Britzger:2019lvc . According to
Britzger:2019lvc , for the $\mathbb{R}_{+}$-reggeon term, the function
$\hat{a}_{2}(Q^{2})$ is assumed to be zero while the function
$\hat{b}_{2}(Q^{2})$ (see the green dotted line) vanishes rapidly with
increasing $Q^{2}$. We see that the soft pomeron function $\hat{b}_{1}$ is
larger than the corresponding hard one $\hat{b}_{0}$ up to $Q^{2}\simeq 20$
GeV2. The function $\hat{b}_{0}$ (see the red long-dash-dotted line) is very
small for $Q^{2}\to 0$ and reaches a maximum at $Q^{2}=1.27$ GeV2 with
amplitude $\hat{b}_{0}=0.0082$ GeV-1. This function is important in the
large-$Q^{2}$ region. Thus, it can be expected that the soft pomeron
contribution will be essential for understanding the HERA DVCS data. We
consider two different parametrizations: (1) FIT 1 (22) with the modification
of only one pomeron-$\gamma^{*}\gamma^{*}$ coupling function
$\hat{b}_{1}(Q^{2})$, corresponding to a “soft term”; (2) FIT 2 (26) with the
modification of both $\hat{b}_{1}(Q^{2})$ and $\hat{b}_{0}(Q^{2})$. Here we
increase the hard component. All other coupling functions, in both our fits,
are fixed in accord with those from the fit to DIS of Britzger:2019lvc .

We assume, furthermore, the following combined function to describe the $t$
dependence of the $\gamma^{(*)}p\to\gamma p$ cross sections

$\displaystyle F_{\rm eff}^{(j)}(t)=F^{(j)}(t)\times
F_{1}^{(j)}(t)=\exp(-b_{j}|t|/2)\,.$ (27)

For our study here we assume $b_{1}=b_{2}$. We take, for both FIT 1 and FIT 2
the same parameters, $b_{1}=b_{2}=5.0\;{\rm GeV}^{-2}$ and $b_{0}=1.0\;{\rm
GeV}^{-2}$ found by comparison of the theoretical curves to the experimental
data for $d\sigma/dt$.

With this we have specified our model and given the parameter values for our
two fits which we shall compare to the data in Sec. III.

Figure 2:  The pomeron- and reggeon-$\gamma^{*}\gamma^{*}$ coupling functions
$\hat{a}_{j}(Q^{2})$ and $\hat{b}_{j}(Q^{2})$ from (20), for $\mathbb{R}_{+}$
reggeon ($j=2$, green dotted line) soft pomeron ($j=1$, black solid and dashed
lines), and hard pomeron ($j=0$, red long-dash-dotted and short-dashed lines),
as determined in Britzger:2019lvc . The experimental uncertainties of the fit
are not shown here. For this we refer to Figs. 12–15 of Britzger:2019lvc . The
blue dash-dotted line corresponds to the $b_{1}(-Q^{2},0,0)/e^{2}$ coupling
function used in the FIT 1 (22). The azure dash-dot-dotted line and the
magenta long-dashed line are for $b_{1}(-Q^{2},0,0)/e^{2}$ and
$b_{0}(-Q^{2},0,0)/e^{2}$, respectively, in the FIT 2 (26). The $a$ and $b$
parameters are plotted in units GeV-3 and GeV-1, respectively.

##  III Results

In this section we compare the tensor-pomeron model to the FNAL data
Breakstone:1981wm  on real-photon-proton scattering ($\gamma p\to\gamma p$),
and to HERA data ZEUS:2003pwh ; H1:2005gdw ; ZEUS:2008hcd ; H1:2009wnw  on
DVCS ($\gamma^{*}(Q^{2})p\to\gamma p$) for different averaged $W$ and $Q^{2}$.
We shall restrict our discussion to experimental results that satisfy the
conditions $x\approx Q^{2}/W^{2}<0.02$ and $|t|\lesssim 1$ GeV2 where our
model should be reliable. We also have to make a comment on what quantity is
measured in H1:2005gdw . For this we compare Eqs. (6)–(8) of H1:2005gdw  with
Eqs. (5)–(7). We cannot see that longitudinally polarized $\gamma^{*}$’s do
not contribute after integration over azimuthal angles, as is claimed in
footnote 7 of H1:2005gdw . Thus, to the best of our understanding, the cross
section measured in H1:2005gdw  must be

$\frac{d\sigma}{dt}=\frac{d\sigma_{\rm T}}{dt}+\varepsilon\frac{d\sigma_{\rm
L}}{dt}\,,$ (28)

where $\varepsilon\approx 1$ for the HERA kinematic region. In the following
we suppose that this is the case. We shall use the notation

$\sigma(Q^{2},W^{2})=\int dt\,\frac{d\sigma}{dt}(Q^{2},W^{2},t)$ (29)

and similarly for the $\rm T$ and $\rm L$ components.

In Fig. 3 we show the $\gamma^{(*)}p\to\gamma p$ total cross sections as a
function of the c.m. energy $W$. In the top panels we compare our fits, FIT 1
(left panel) and FIT 2 (right panel), to the FNAL result from
Breakstone:1981wm 222Note that in Breakstone:1981wm  the value of
$\sigma=88\pm 4$ nb for the elastic $\gamma p$ cross section was given. This
value was obtained by summing the beam-energy-averaged differential cross
sections measured at c.m. energies $W=9.73-15.645$ GeV for $|t|\geqslant 0.07$
GeV2 with those from the preferred fit II of Breakstone:1981wm  for $|t|<0.07$
GeV2. where $Q^{2}=0$ and to the HERA data from ZEUS:2003pwh ; H1:2005gdw ;
ZEUS:2008hcd ; H1:2009wnw  for different values of $Q^{2}$ (the average
values). The complete cross section is a coherent sum of soft and hard
components in the amplitude. These components have different energy
dependence. For real Compton scattering ($Q^{2}=0$) the cross section is
dominated by soft-pomeron exchange with an additional contribution from
reggeon exchange at lower energies. The hard-pomeron contribution is
negligibly small there. The cross sections rise with energy as
$W^{2\epsilon_{1}}$ at $Q^{2}=0$ and change to $W^{2\epsilon_{0}}$ for very
high $Q^{2}$. Here $\epsilon_{1}\approx 0.09$ and $\epsilon_{0}\approx 0.30$
are the intercept parameters of the soft and hard pomerons, respectively, see
(14), (15). In our model the dominant contribution comes from the $b$-type
coupling functions $b_{j\gamma^{*}\gamma^{*}}$ ($j=0,1$). Their size, as shown
in Fig. 2, differs in the two fits. We see from the bottom panels of Fig. 3
that for higher $Q^{2}$ the soft component slowly decreases relative to the
hard one. A significant constructive interference effect between the soft and
hard components is clearly visible. Here and in the following, the
interference term is calculated as the difference of coherent and incoherent
cross sections of the $\mathbb{P}_{1}$, $\mathbb{P}_{0}$, and $\mathbb{R}$
contributions. The Fits 1 and 2 hardly differ for the $W$ region where there
are data. But for higher $W$ values FIT 2, where the contribution from the
hard pomeron is enhanced, gives a steeper rise of the cross section with $W$
and especially so for larger $Q^{2}$. Only future experiments will tell us
what happens in nature.

Figure 3:  Top panels: Total cross sections as a function of the c.m. energy
$W$ for FIT 1 (left) and FIT 2 (right). Comparison of theoretical results to
the FNAL data from Breakstone:1981wm  for real Compton scattering ($Q^{2}=0$)
and to the DVCS HERA data is shown. The upper black solid line is for
$Q^{2}=0$, the orange long-dashed-dotted line is for $Q^{2}=1$ GeV2. The
remaining lines correspond to the values $Q^{2}=2.4,3.2,4,6.2,8,10,15.5,25$
GeV2 (from top to bottom) and should be compared with the HERA data from
ZEUS:2003pwh ; H1:2005gdw ; ZEUS:2008hcd ; H1:2009wnw . Bottom panels: Our fit
results for $Q^{2}=8$ GeV2 together with the H1 data H1:2009wnw . We show the
contributions for soft and hard pomeron separately, see the blue long-dashed
line and the red dashed line, and their coherent sum (total), see the black
solid line. The interference term is shown separately by the green dotted
line.

In Fig. 4 we show how the cross section depends on $Q^{2}$ for two c.m.
energies $W=82$ and 104 GeV. Complete results for our two fits and the soft
and hard components separately are shown together with the HERA data. From the
top panels (for FIT 1) we see that the soft pomeron survives to relatively
large $Q^{2}$ and at $Q^{2}\simeq 50$ GeV2 the interference term plays an
important role in the description of the data. From the bottom panels (for FIT
2) one can observe that at $Q^{2}\simeq 50$ GeV2 the hard pomeron alone is
able to describe the data. Here the contribution from the interference term is
considerable for intermediate values of $Q^{2}\sim 10$ GeV2. It is interesting
to note that the interference effect is also large when the hard component
reaches a maximum. For our fits this is at $Q^{2}\cong 1.3$ GeV2.

Figure 4:  Comparison of the cross sections for the
$\gamma^{(*)}(Q^{2})p\to\gamma p$ reaction as a function of $Q^{2}$ to the
experimental data for $\langle W\rangle=82$ GeV from H1:2005gdw ; H1:2009wnw
(left panels) and for $\langle W\rangle=104$ GeV from ZEUS:2008hcd  (right
panels). Results on top panels correspond to FIT 1 and those on the bottom
panels to FIT 2. The meaning of the lines is the same as in the bottom panels
of Fig. 3.

In Fig. 5 we show the differential cross sections $d\sigma/dt$ from (28) with
$\varepsilon=1$ for different $\langle W\rangle$ and $\langle Q^{2}\rangle$.
In the top panels, the upper line corresponds to $W=12.7$ GeV and $Q^{2}=0$
and should be compared to the averaged FNAL data (top data points) for the
$\gamma p\to\gamma p$ reaction Breakstone:1981wm . In this kinematic range,
for $Q^{2}=0$ and at intermediate $W$, the reggeon plus soft-$\mathbb{P}$
contributions dominate and the hard-$\mathbb{P}$ exchange gives negligible
contribution. As expected, there is a significant interference between the
reggeon and soft-pomeron components. The slope parameters $b_{2}$ and $b_{1}$
in (27) are adjusted to the FNAL $d\sigma/dt$ data on the real-photon-proton
scattering. At higher $W$ and $Q^{2}$ measured at HERA the hard pomeron plays
an increasingly important role. The slope parameter $b_{0}$ for the hard-
pomeron exchange is adjusted to the DVCS HERA data. As we noted above,
$d\sigma(\gamma^{*}p\to\gamma p)/dt$ is the sum of two contributions
$d\sigma_{\rm T}/dt$ and $d\sigma_{\rm L}/dt$ with the latter term becoming
very small for $|t|\to 0$. This is understandable since for
$\gamma^{*}p\to\gamma p$ forward scattering only double-helicity-flip
amplitudes can contribute to $d\sigma_{\rm L}/dt$. Furthermore, $d\sigma_{\rm
T}/dt$ is dominated by the $b$-type couplings and $d\sigma_{\rm L}/dt$ is
dominated by the $a$-type couplings; see Eq. (17).

Figure 5:  Top panels: The differential cross sections $d\sigma/dt$ compared
to experimental data for different $\gamma^{*}p$ c.m. energy and photon
virtuality. The upper line corresponds to $W=12.7$ GeV and $Q^{2}=0$ and
averaged FNAL data Breakstone:1981wm . The theoretical results for
$\gamma^{*}p\to\gamma p$ at higher $W$ and $Q^{2}$ for transverse (long-dashed
lines) and longitudinal (short-dashed lines) polarization of the $\gamma^{*}$
and their sum (solid lines) are shown together with experimental data. Data
for $\langle W\rangle=104$ GeV are from ZEUS:2008hcd  and for $\langle
W\rangle=82$ GeV from H1:2009wnw . Bottom panels: Comparison of the fit
results to the H1 data H1:2009wnw  for $\langle Q^{2}\rangle=10$ GeV2 and
$\langle W\rangle=40$, 70, and 100 GeV (from bottom to top). Here we show the
results scaled by a factor $\rm k$ (specified in the figure legend) for
displaying purposes.

In Fig. 6 we show the complete theoretical result and individual components
contributing to the cross section $d\sigma/dt$, see (28), for $W=82$ GeV and
$Q^{2}=8$ GeV2 together with the H1 data H1:2009wnw . The constructive
interference of the soft and hard pomeron terms is again a salient feature.

We note that our complete result for $d\sigma/dt$ does not have a simple
exponential $t$ dependence as is frequently assumed in other models. This is
caused by the interference of soft- and hard-pomeron terms (each with
different $t$ dependence) and by the longitudinal contribution which is
important for large $|t|$.

Figure 6:  Comparison of the FIT 1 (left) and the FIT 2 (right) to DVCS H1
data H1:2009wnw  for $W=82$ GeV and $Q^{2}=8$ GeV2. The results for
$\gamma^{*}p\to\gamma p$ for transverse (${\rm T}$) and longitudinal (${\rm
L}$) polarization of the $\gamma^{*}$ individually and their sum ${\rm T+L}$
(see the upper solid lines) are shown. The contributions of soft $\mathbb{P}$
(the blue short-dashed lines), hard $\mathbb{P}$ (the red long-dashed lines),
the interference term (the green dotted lines), and their sum total (the thin
full lines) for ${\rm T}$ and ${\rm L}$ components individually are also
shown.

Figure 7 shows the ratios of the $\gamma^{*}p\to\gamma p$ cross sections for
longitudinally and transversely polarized virtual photons,

$\displaystyle R(Q^{2},W^{2})=\frac{\sigma_{\rm L}(Q^{2},W^{2})}{\sigma_{\rm
T}(Q^{2},W^{2})}\,,$ (30)
$\displaystyle\tilde{R}(Q^{2},W^{2},t)=\frac{\frac{d\sigma_{\rm
L}}{dt}(Q^{2},W^{2},t)}{\frac{d\sigma_{\rm T}}{dt}(Q^{2},W^{2},t)}\,,$ (31)

as functions of $Q^{2}$ and $|t|$, respectively. The cross section
$\sigma_{\rm L}$ vanishes proportionally to $Q^{2}$ for $Q^{2}\to 0$. The
ratio $\tilde{R}(Q^{2},W^{2},t)$ strongly grows with $|t|$. We must emphasize
that this behaviour of $\tilde{R}(Q^{2},W^{2},t)$ depends crucially on our
(reasonable) assumption that $a$ and $b$ couplings in the
$\mathbb{P}\gamma^{*}\gamma^{*}$ vertex functions have the same $t$ dependence
for a given $j$ $(j=0,1,2)$; see (18), (19), and (27). But it is clear from
the right panel of Fig. 7 that a very small ratio $\tilde{R}(Q^{2},W^{2},t)$
for all $|t|\lesssim 1.0$ GeV2 could, in our model, only be achieved if the
$a$ and $b$ couplings would have drastically different $t$ dependences.

Figure 7:  The ratios of cross sections $R(Q^{2},W^{2})$ (30) and
$\tilde{R}(Q^{2},W^{2},t)$ (31) for the $\gamma^{*}p\to\gamma p$ reaction for
longitudinally and transversely polarized $\gamma^{*}$. Note that the meaning
of the lines on these two panels is different.

##  IV Conclusions and discussion

In the present paper we have applied the tensor-pomeron approach to deeply
virtual Compton scattering (DVCS) for high c.m. energies $W$ and small
Bjorken-$x$, $x\lesssim 0.02$.

We have made a comparison of the two-tensor-pomeron model to the DVCS data
measured at HERA. As a starting point we have used the fit parameters of the
intercepts of the two pomerons and of the reggeon $\mathbb{R}_{+}$, and their
coupling functions to real and virtual photons determined in Britzger:2019lvc
from deep inelastic scattering (DIS) and real photoabsorption cross sections.
To our surprise, with a ’minimal’, but reasonable, modification of the $Q^{2}$
dependence of only one $\gamma^{*}(Q^{2})\gamma\mathbb{P}$ coupling function
[FIT 1 (22)] we have already got a good fit to the experimental data. We could
describe the $W$, $Q^{2}$ and $t$ dependences of $d\sigma(\gamma^{*}p\to\gamma
p)/dt$ measured at HERA and of the elastic photon-proton cross section
measured at FNAL. The good description of the DVCS data is achieved due to a
sizeable interference of soft and hard pomeron contributions. We have
considered also FIT 2 (26) in which the size of the hard component was
increased. We kept here, on purpose, the same parameters of the form factors
(27) as in FIT 1. The FIT 2 better describes the data at larger $|t|$ for
$Q^{2}\gtrsim 8$ GeV2 (see Fig. 5). Note that in our two-tensor-pomeron model
the soft component and also the interference of soft and hard terms are very
important up to at least $Q^{2}\simeq 20$ GeV2.

Now we comment on Refs. Capua:2006ij ; Fazio:2013hza ; Jenkovszky:2018itd
where also Regge theory is applied to DVCS. These authors consider, as far as
we can see, only leading helicity amplitudes. For DVCS this means that only
transversely polarized initial $\gamma^{*}$’s are considered. In contrast, in
our tensor-pomeron approach we present a complete model, including all
helicity amplitudes. Therefore, we could make, e.g., predictions for
$\sigma_{\rm L}/\sigma_{\rm T}$ which can and should be checked by
experiments. Also, our model for DVCS gives different relations between soft
and hard terms compared to the analysis of Fazio:2013hza ; see Sec. IV therein
and the review Jenkovszky:2018itd  where more results are presented. The
contribution from the interference term found in Fazio:2013hza  is
considerable for intermediate values of $Q^{2}$, but smaller than our findings
(FIT 1 and FIT 2).

In our calculations we have included the contributions of both transverse and
longitudinal virtual photons. The longitudinal cross section $d\sigma_{\rm
L}(Q^{2},W^{2},t)/dt$ is predicted to be very small for $|t|\to 0$ but to be
sizeable for $0.5\;{\rm GeV}^{2}\lesssim|t|\lesssim 1.0\;{\rm GeV}^{2}$ (see
Figs. 5 and 6). The corresponding ratio of ${\rm L/T}$ grows strongly with
$|t|$ (see the right panel of Fig. 7). We have also shown the $Q^{2}$
dependence of this ratio for different c.m. energies of the $\gamma^{*}p$
system.

To summarize, we have presented predictions for low-$x$ DVCS of the two-
tensor-pomeron model which previously was successfully applied to low $x$ DIS
in Britzger:2019lvc . The model provides amplitudes for all helicity
configurations and, thus, can be checked by experimentalists in many ways. We
are looking forward to further tests of the non-perturbative QCD dynamics
embodied in our tensor-pomeron exchanges in future electron-hadron collisions
in the low-$x$ regime at the EIC and LHeC colliders.

###### Acknowledgements.

The authors are grateful to Markus Diehl for correspondence and comments and
to Carlo Ewerz for useful discussions. This work was partially supported by
the Polish National Science Centre Grant No. 2018/31/B/ST2/03537. P. L. was
supported by the Bekker Program of the Polish National Agency for Academic
Exchange, Project No. BPN/BEK/2021/2/00009/U/00001.

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# On the conclusive detection of Majorana zero modes: conductance
spectroscopy, disconnected entanglement entropy and the fermion parity noise

Arnav Arora  Department of Physics, Indian Institute of Technology Roorkee,
Roorkee, Uttarakhand-247667, India  Abhishek Kejriwal  Department of Physics,
Indian Institute of Technology Bombay, Powai, Mumbai-400076, India  Bhaskaran
Muralidharan  Department of Electrical Engineering, Indian Institute of
Technology Bombay, Powai, Mumbai-400076, India  Centre of Excellence in
Quantum Information, Computation, Science and Technology, Indian Institute of
Technology Bombay, Powai, Mumbai-400076, India<EMAIL_ADDRESS>
###### Abstract

Semiconducting nanowires with strong Rashba spin-orbit coupling in the
proximity with a superconductor and under a strong Zeeman field can
potentially manifest Majorana zero modes at their edges and are a topical
candidate for topological superconductivity. However, protocols for their
detection based on the local and the non-local conductance spectroscopy have
been subject to intense scrutiny. In this work, by taking current experimental
setups into account, we detail mathematical ideas related to the entanglement
entropy and the fermion parity fluctuations to faithfully distinguish between
true Majorana zero modes and trivial quasi-Majorana zero modes. We demonstrate
that the disconnected entanglement entropy, derived from the von Neumann
entanglement entropy, provides a distinct and robust signature of the
topological phase transition which is immune to system parameters, size and
disorders. In order to understand the entanglement entropy of the Rashba
nanowire system, we establish its connection to a model of interacting
spinfull Kitaev chains. Moreover, we relate the entanglement entropy to the
fermionic parity fluctuation, and show that it behaves concordantly with
entanglement entropy, hence making it a suitable metric for the detection of
Majorana zero modes. In connection with the topological gap protocol that is
based on the conductance spectra, the aforesaid metrics can reliably point
toward the topological transitions even in realistic setups.

##  I  Introduction

Majorana zero modes (MZMs) [1, 2, 3, 4] in condensed matter systems have been
a crucial talking point, tending to their exotic attributes, such as non-
Abelian quantum statistics, which maybe leveraged for futuristic quantum
technologies like fault-tolerant topological quantum computation [5].
Semiconductor-superconductor heterostructures composed of Rashba nanowires
proximitised with superconductors are a prime solid-state candidate for
realizing MZMs [1, 2, 3, 4, 6]. Howbeit, conclusive and unambiguous detection
of MZMs in these systems has been a matter of intense scrutiny and open
deliberation due to controversies involving non-topological origins of MZM-
like signatures. Major experimental efforts for the detection MZMs have been
based on observing zero-bias conductance peaks (ZBCPs) in the local
differential conductance of normal-topological superconductor (N-TS) setups
[7, 8, 9, 10, 11, 12, 11]. However, disorder-induced near zero-energy Andreev
bound states (ABS), often known as quasi-MZMs, imitate local conductance
signatures of MZMs to a great extent [13, 14, 15, 16, 17, 18, 19, 20, 21],
rendering ZBCP based detection of MZMs ambiguous.

Figure 1: Device schematics. (a) The N-TS-N system considered in this work.
The semiconducting nanowire SM (in blue) is proximitised with a superconductor
SC (in purple) which acts as a topological superconductor under a strong spin-
orbit coupling and magnetic field. Two normal contacts (in pink) $L$ and $R$
are used to probe the TS system. (b) The pristine setup, with no disorder
potential and a homogeneous on-site potential constant throughout the
nanowire. (c) The disordered system where the on-site potential varies along
the nanowire due to local inhomogeneities.

Recent protocols have thus focused on exploiting the non-locality of MZMs
through non-local transport spectroscopy using the normal metal-topological
superconductor-normal metal (N-TS-N) hybrid setups [22, 23, 24, 25, 26, 27,
28, 6]. The non-local differential conductance probes the bulk-gap closing and
reopening and is presumed to be uninfluenced by the trivial modes [23]. Ideas
proposing measurements of the full conductance matrix, which includes both
local and non-local measurements, to identify topological regimes have also
been presented [29, 30]. Nonetheless, numerical [31, 32] and experimental [25]
studies have portrayed several issues with the non-local conductance
signatures as well, with the signals becoming faint and sputtered [32]
accompanied by a premature gap closing [31, 25] in the presence of disorders.
This motivates exploring other possibilities that exploit non-local
correlations to classify topological orders. One such attribute which can be
leveraged to ascertain non-locality in topological phases [33, 34, 35] is the
degree of entanglement in the system, which may be quantified using metrics
derived from the von Neumann entropy, such as the disconnected entanglement
entropy [31].
In this work, we provide an in-depth analysis of various entanglement entropy
signatures of MZMs in SM-SC heterostructures. We illustrate that the
disconnected entanglement entropy (DEE) [36] distills out the topological
contributions of $ln2$ units to the entanglement entropy [37, 36, 38] and is
hence apt for identifying topological phase transitions, unlike the bipartite
entanglement entropy (BEE), which is padded with volume and area contributions
[31, 39]. In the process, we develop a new pedagogical model, ‘the spinfull
Kitaev chain’, to elucidate the entropy signatures of $ln2$ units in Rashba
nanowires from a more first-principles perspective. We demonstrate that the
entanglement entropy remains robust and quantized over a wide range of
controllable parameters and withstands the test of disorder and quasi-MZMs.
Further, we inquire into fermion parity noise, a physical observable closely
related to entanglement entropy [40] that maybe be potentially used for its
measurement. We conclude by demonstrating the superiority of DEE over the
local and non-local conductance spectroscopy [31] for the conclusive detection
of MZMs.

##  II  Methods

The three terminal N-TS-N setup [29, 30, 25] that we consider is shown in Fig.
1(a). The setup comprises a semiconducting nanowire (in blue) with strong
Rashba type spin-orbit coupling with an epitaxial layer of s-wave
superconductor (in purple), placed in contact with metallic leads ‘L’ and ‘R’
(in pink). The Hamiltonian of the nanowire system in the Bogoliubov-de Gennes
(BdG) representation is

$\displaystyle\mathcal{H}=\frac{1}{2}\int
dx\Psi_{BdG}^{\dagger}(x)H_{BdG}\Psi_{BdG}(x)$ (1)

where
$\Psi_{BdG}(x)=(\psi_{\uparrow}(x),\>\psi_{\downarrow}(x),\psi_{\uparrow}^{\dagger}(x),\>\psi_{\downarrow}^{\dagger}(x))^{T}$
represents the Nambu spinor, and $H_{BdG}$ represents the BdG Hamiltonian [18]
described by:

$\displaystyle
H_{BdG}=\left(-\frac{\hbar^{2}}{2m}\partial_{x}^{2}-\mu+V(x)\right)\tau_{z}\sigma_{0}+i\alpha_{R}\tau_{z}\sigma_{y}\partial_{x}$
(2)
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+V_{z}\tau_{z}\sigma_{x}+\Delta\tau_{y}\sigma_{y},$

where, $\sigma_{i}$ are the set of Pauli-matrices in the spin basis, with
$\sigma_{0}$ being the identity matrix and $\tau_{i}$ are the set of Nambu-
matrices in the particle-hole basis. $\mu$ is the electrochemical potential,
$\alpha_{R}$ is the Rashba spin-orbit coupling strength,
$V_{z}=\frac{g\mu_{B}B}{2}$ is the Zeeman field, $B$ is the magnetic field and
$\Delta$ is the proximity-induced superconducting gap. In this work, we
elucidate two cases. First, the pristine nanowire case with $V(x)=0$,
portrayed in Fig. 1(b). Here, both the superconducting gap and the on-site
potential are homogeneous and constant throughout the nanowire. As stressed
upon earlier, accounting for disorder is crucial for studying conclusive MZM-
identification techniques. In the second case, we study the one with disorder
where $V(x)$ is non-zero and inhomogeneous. Figure 1(c) shows the disordered
setup where the gap-parameter remains homogeneous but the on-site potential
varies along the nanowire due to disorder-induced local inhomogeneities.
We analyze smooth inhomogeneous potentials at the ends of the nanowire [41,
42, 32]. In experiments, such disorders occur as Schottky barrier formations
due to Fermi energy mismatches with the contacts [43] or due to charge
inhomogeneities in the environment or the leads [44, 45]. For our numerical
simulations, we model this smooth disorder as a half-Gaussian at each end of
the nanowire with a peak value of $V_{max}$ and variance $\sigma$. For our
numerical calculations, we discretize [46, 47] the Hamiltonian (LABEL:eq1)
which yields a tight-binding Hamiltonian:

$\displaystyle\mathcal{H}$
$\displaystyle=\sum_{i}\mathcal{C}_{i}^{\dagger}\bigg{[}\left(2t-\mu+V_{i}\right)\tau_{z}\sigma_{0}+V_{z}\tau_{z}\sigma_{x}+\Delta\tau_{y}\sigma_{y}\bigg{]}\mathcal{C}_{i}$
(3) $\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\sum_{\langle
ij\rangle}\mathcal{C}_{i}^{\dagger}\bigg{[}-t\tau_{z}\sigma_{0}+it_{SO}\tau_{z}\sigma_{y}\bigg{]}\mathcal{C}_{j},$

where,
$\mathcal{C}_{i}=(c_{i\uparrow},c_{i\downarrow},c_{i\uparrow}^{\dagger},c_{i\downarrow}^{\dagger})$
is the Nambu spinor at site ‘i’, $t_{SO}=\alpha_{R}/2a$, $t=\hbar^{2}/2ma^{2}$
and $a$ is the lattice spacing. A direct diagonalization of the tight-binding
Hamiltonian (3) is used to obtain the eigenspectra.

Figure 2:  Partitions used for entanglement entropy calculations. (a) A
partition in the middle of the wire used for computing the bipartite
entanglement entropy. (b) Two distinct partitions, $A$ (in green), and $B$ (in
orange), used for computing the disconnected entanglement entropy. The region
$A$ is a continuous region formed by partitioning the wire in the middle and
$B$ is a disconnected region formed by partitioning the chain into two non-
intersecting regions.

Entanglement Entropy: For calculating the entanglement entropy, we leverage
its intricate connection with the correlation matrix [48, 33], which is
defined as :

$C_{nm}=\left(\begin{array}[]{cc}\left\langle
c^{\dagger}c\right\rangle&\left\langle c^{\dagger}c^{\dagger}\right\rangle\\\
\langle cc\rangle&\left\langle
cc^{\dagger}\right\rangle\end{array}\right)_{nm},$ (4)

The matrix elements are calculated using the expectation values of fermionic
operators in the BCS ground-state. The truncated correlation matrix of any
sub-region of the system, say A, can be defined as:

$\tilde{C}_{nm}=C_{(nm)\in A}$ (5)

With a comprehensive calculation [48], we can prove that the von Neumann
entropy of sub-region A can be expressed in terms of the eigenvalues of the
truncated correlation matrix as:

$S=-\sum_{k=1}^{M}\left[\xi_{k}\log\xi_{k}+\left(1-\xi_{k}\right)\log\left(1-\xi_{k}\right)\right]$
(6)

When the system is partitioned into two subsystems, the von Neumann entropy of
one of the sub-systems is called the bi-partite entanglement entropy (BEE). In
this work, we calculate the BEE by partitioning the system into two equal
halves, as shown in Fig. 2(a).
The disconnected entanglement entropy (DEE) [36, 31] is computed using linear
combinations of von Neumann entropy of disconnected regions of the system, an
idea similar to the topological entanglement entropy in 2D-systems [49]. The
disconnected regions used in this work can be seen in Fig. 2(b), we call the
green region $A$ and the orange region $B$. The DEE $S_{D}$ is then calculated
as:

$S_{D}=S_{A}+S_{B}-S_{A\cap B}-S_{A\cup B}\>\>,$ (7)

where, $S_{A}$, $S_{B}$, $S_{A\cap B}$, $S_{A\cup B}$ are the von Neumann
entropies of the corresponding regions. The DEE dissects out area and volume
contributions [31, 36, 34] to the entanglement entropy hence distilling out
only the topological contributions to it. As we will see further, BEE
signatures are poised with the issue of volume and area terms but DEE
signatures are clean and quantized [31, 36].
Fermion Parity Noise: We also relate the entanglement entropy to a well known
conserved physical observable of the system – Fermion parity [40]. The parity
operator is given by $\mathcal{P}=(-1)^{\sum_{i}c_{i}^{\dagger}c_{i}}$. The
fermion parity noise is the statistical measurement uncertainty in
$\mathcal{P}$ of the BCS ground-state, defined by:

$\langle
P\rangle_{2}=\langle\mathcal{P}^{2}\rangle-\langle\mathcal{P}\rangle^{2}$ (8)

where, the expectation value is calculated in the BCS ground-state. It can be
shown that the fermion parity noise of any sub-region of a superconducting
system can be related to the eigenvalues of the truncated correlation matrix
[48] of that region by:

$\langle P\rangle_{2}=\sum_{k=1}^{M}\xi_{k}(1-\xi_{k})$ (9)

We then define disconnected fermion parity noise $\delta P$ on the lines of
the DEE, as a linear combination of fermion parity noise of distinct
disconnected regions of the nanowire, given by:

$\delta P=\langle P_{A}\rangle_{2}+\langle P_{B}\rangle_{2}-\langle P_{A\cap
B}\rangle_{2}-\langle P_{A\cup B}\rangle_{2}.$ (10)

Conductance: The local and non-local conductance signatures are calculated
using the Keldysh non-equilibrium Green’s function (NEGF) formalism [31]. This
requires evaluating the retarded (advanced) Green’s function $G^{r(a)}$ of the
system by incorporating the contacts as self energies $\Sigma_{L/R}$ along
with the device Hamiltonian $H_{BdG}$. In this work, we use the wide-band
approximation to define the broadening matrices $\Gamma_{L/R}$ and
subsequently the self energies $\Sigma_{L/R}$ with coupling energy
$\gamma=0.01\Delta$. The retarded (advanced) Green’s functions along with the
broadening matrices are then used to calculate the energy resolved Andreev
transmission ($T_{A}(E)$), the crossed-Andreev transmission ($T_{CA}(E)$), and
the direct transmission ($T_{D}(E)$) [48], which are consequently used to
calculate the local and non-local conductance in the low temperature limit
using the Landauer-Büttiker formalism as:

$\begin{array}[]{r}\left.G_{LL}(V)\right|_{T\rightarrow
0}\equiv\frac{e^{2}}{h}\left[T_{A}(E=eV)+T_{A}(E=-eV)+\right.\\\
\left.T_{CAR}(E=eV)+T_{D}(E=eV)\right]\end{array}$ (11)
$\left.G_{LR}(V)\right|_{T\rightarrow
0}\equiv\frac{e^{2}}{h}\left[T_{D}(E=eV)-T_{CAR}(E=-eV)\right],$ (12)

where $V$ is the bias voltage applied at either contact in the three terminal
configuration [26].

Table 1: Parameters used in the analysis, unless otherwise stated. Parameter |  | Value
---|---|---
Effective mass | $m^{*}$ | $0.015m_{\mathrm{e}}$
Induced order parameter | $\Delta$ | $0.25\mathrm{meV}$
Tight-binding hopping parameter | $t_{0}$ | $10\mathrm{meV}$
Rashba spin-orbit coupling | $\alpha_{\mathrm{R}}$ | $20\mathrm{meV}\mathrm{nm}$
Chemical potential | $\mu$ | $0.5\mathrm{meV}$
Magnetic g-factor | $g$ | 40
Critical field | $B_{c}$ | $0.58\mathrm{~{}T}$
Figure 3: Topological phase space and the entanglement entropy. (a) The BEE
(in units of $\ln 2$) of a $4\mu m$ pristine nanowire. (b) The BEE (in units
of $\ln 2$) of an $8\mu m$ pristine nanowire. (c) The DEE (in units of $\ln
2$) of a $4\mu m$ pristine nanowire. (d) The DEE (in units of $\ln 2$) of an
$8\mu m$ nanowire. TIn all the cases the entropies are plotted against the
magnetic field $B$. The dashed red line is at the theoretically predicted
critical field $B_{c}$ for topological phase transition. The BEE shows no
characteristic quantization with a minimal change in its value over the phase
transition. The DEE, however, stays close to zero in the trivial regime:
$B<B_{c}$ and is quantized at a value of $\ln 2$ in the topological regime:
$B>B_{c}$. (e) DEE phase diagram (in units of ln2) of a $4\mu m$ nanowire. (f)
DEE phase diagram (in units of ln2) of an $8\mu m$ nanowire. In (e) and (f)
the dashed black curve represents the theoretical phase boundary. The DEE
stays zero in the trivial regime and attains $ln2$ value in the topological
regime for a good range of $mu$ and $B$.  Figure 4:  Decomposition of the
spinfull Kitaev chain into four interacting Majorana chains. The blue and pink
spheres represent $A$ type and $B$ type Majoranas respectively of the $\pm$
chains. (a) the green bonds represent couplings due to $\mu_{+}$ and
$\mu_{-}$, (b) the orange bonds represent couplings due to
$(t_{+}-\Delta_{+})$ and $(t_{-}-\Delta_{-})$, (c) the violet bonds represent
couplings due to $(t_{+}+\Delta_{+})$ and $(t_{-}+\Delta_{-})$ and (d) the
black bonds represent couplings due to $\Delta_{s}$

##  III  Results

We present the results obtained from numerical simulations of the system
described in Fig. 1 based on the experimentally relevant system parameters
presented in Tab. 1. We start by calculating the entanglement entropy
signatures for the pristine nanowire. Figure 3 focuses on the entanglement
entropy as a function of the Zeeman field for short ($4\mu m$) and long ($8\mu
m$) nanowires.
We start by evaluating the BEE signatures for the $4\mu m$ and $8\mu m$
nanowires. The BEE shows no signs of quantization and exhibits a minimal
change in its value over the phase transition for both $4\mu m$ and $8\mu m$
nanowires, as shown in Figs. 3(a), and 3(b), respectively, where the red
dashed lines mark the critical magnetic field for the topological phase
transition. The critical magnetic field $B_{c}$ is evaluated using the well-
known relation: $B_{c}=1/g\sqrt{(\mu^{2}+\Delta^{2})}$. The non-zero values of
the BEE in the trivial phase and lack of quantization can be attributed to the
area and volume contributions that arise from the bulk of the nanowire over
the topological contributions from the ends. These contributions render it
ineffective for the identification of the topological phase transition.
Figures 3(c) and 3(d) present the DEE signatures for the $4\mu m$ and $8\mu m$
nanowires respectively. The red dashed line, as earlier, represents
$B_{c}/\Delta$. Unlike the BEE, the DEE remains zero in the trivial regime
$B<B_{c}$. At the critical point, the DEE shows a sudden jump and attains a
non-zero quantized value of $ln2$ in the topological regime $B>B_{c}$. The DEE
signature is robust to the system size and shows similar behavior for both
short and long nanowires.
The quantization towards the value of $ln2$ becomes more prominent for larger
system size, as is clear from Fig. 3(d). We also observe other prominent
features of the SM-SC hybrid system in the DEE signatures. The oscillations in
the DEE, as seen in Fig. 3(c), correspond to the well-known parity crossings
of the MZMs near zero energy. The amplitude of the parity crossings reduces as
the system size increases. This aligns with the reduced oscillatory behavior
of the longer nanowire in Fig. 3(d) than the shorter nanowire in Fig. 3(c).
The near-zero value of the DEE in the trivial regime and a quantized non-zero
value in the topological regime, irrespective of system size, indicates its
capability to faithfully detect the topological phase transition in the
nanowire and distinguish the topological phase from the trivial phase.
In Fig. 3(e) and Fig. 3(f), we highlight the robustness of DEE as a metric to
distinguish between the topological and trivial phases and detect a
topological phase transition by presenting the topological phase diagram for
the system. Figure 3(e) depicts the value of DEE versus an experimentally
relevant range of the electrochemical potential and the Zeeman field for a
pristine wire of length 4 $\mu m$. The dashed black curve denotes the
theoretically evaluated topological phase boundary for a fixed value of
$\Delta$ and is defined by the relation
$B_{c}=\frac{1}{g}\sqrt{(\mu^{2}+\Delta^{2})}$. The region to the right of the
curve represents the theoretically expected topological phase. On calculating
DEE for each value of $\mu$ and $B$ in this range, we observe that DEE attains
a non-zero quantized value of $\ln 2$ in the region to the right of the dashed
curve, whereas it stays close to zero to the left of the dashed curve. This
shows that the transition in the value of DEE exactly matches the
theoretically expected topological phase boundary, and the value of DEE shows
a quantization in the topological regime for a wide range of experimentally
relevant parameters. A precisely similar feature is observed for a longer
nanowire of length $8\mu m$, as shown in Fig. 3(f), highlighting the
robustness of DEE as a metric to the system size.

Figure 5:  Three different cases considered for the spinfull Kitaev chain. (a)
The initial case with $\mu_{+}=0$. In this case the unpaired Majoranas at the
end of both the chains (yellow spheres) hybridise through the inter-chain
bonding (black bonds) due to the s-wave like pairing between the chains. (b)
The intermediate case with $0<\mu_{+}<\infty$. In this case, intra-chain
pairing (green bonds) between Majoranas of the $+$ chain (top chain) has been
seeded. (c) The final case with $\mu_{+}=\infty$. In this case bonds due to
$\mu_{+}$ dominate the $+$ chain making it trivial and isolating it from any
other intra-chain or inter-chain bonds. The bottom chain, however, remains
topological with unpaired Majoranas at the ends. Overall the system has two
MZMs at the ends.

In order to explain the entanglement entropy signatures for the Rashba
nanowire from a much more elementary and first-principles perspective, we we
highlight the parallels between the Rashba nanowire and a toy model consisting
two distinct p-wave superconductors interacting through an s-wave pairing
potential. Starting with the normal part of the Rashba nanowire Hamiltonian,
without proximity-induced superconductivity, in the continuous momentum space:

$\mathcal{H}_{0}=\int
dk\Psi^{\dagger}(k)\bigg{[}\frac{-\hbar^{2}k^{2}}{2m}-\mu+i\alpha_{R}k\sigma_{y}+V_{z}\sigma_{x}\bigg{]}\Psi(k),$
(13)

where, $\alpha_{R}$ is the Rashba spin-orbit coupling strength,
$V_{z}=\frac{g\mu_{B}B}{2}$ is the Zeeman field, $k$ is the momentum and
$\Psi(k)=(\psi_{\uparrow}(k),\>\psi_{\downarrow}(k))^{T}$ is the fermionic
spinor at $k$. The eigenvalues $\varepsilon_{\pm}$ of this Hamiltonian are
given by
$\varepsilon_{k,\pm}=\frac{\hbar^{2}k^{2}}{2m}-\mu\pm\sqrt{V_{z}^{2}+\alpha_{R}^{2}k^{2}}$.
The eigen-spinors in terms of the original spinor are given by:

$\left(\begin{array}[]{c}\psi_{+}(k)\\\
\psi_{-}(k)\end{array}\right)=\left(\begin{array}[]{cc}\frac{\left(-i\alpha_{R}k+V_{z}\right)}{\sqrt{V_{z}^{2}+\alpha_{R}^{2}k^{2}}}&1\\\
\frac{\left(i\alpha_{R}k-V_{z}\right)}{\sqrt{V_{z}^{2}+\alpha_{R}^{2}k^{2}}}&1\end{array}\right)\left(\begin{array}[]{c}\psi_{\uparrow}(k)\\\
\psi_{\downarrow}(k)\end{array}\right).$ (14)

The spinor basis, defined by
$\Phi^{{}^{\prime}}=(\psi_{+}(k),\>\psi_{-}(k))^{T}$ is often called the
helical basis. Now we express the full Hamiltonian of the nanowire, including
the proximity-induced superconducting pairing terms, in the helical basis. The
Hamiltonian in the helical basis [50] is given by :

$\displaystyle\mathcal{H}$ $\displaystyle=\frac{1}{2}\int
dk[\varepsilon_{+}(k)\psi^{\dagger}_{+}(k)\psi_{+}(k)+\varepsilon_{-}(k)\psi^{\dagger}_{-}(k)\psi_{-}(k)]$
(15)
$\displaystyle+[\frac{\Delta_{++}(k)}{2}\psi^{\dagger}_{+}(k)\psi^{\dagger}_{+}(-k)+\frac{\Delta_{--}(k)}{2}\psi^{\dagger}_{-}(k)\psi^{\dagger}_{-}(-k)$
$\displaystyle+\Delta_{+-}\psi^{\dagger}_{+}(k)\psi^{\dagger}_{-}(-k)+h.c.]$

where,

$\displaystyle\varepsilon_{\pm}(k)=\frac{\hbar^{2}k^{2}}{2m}-\mu\pm\sqrt{V_{z}^{2}+\alpha_{R}^{2}k^{2}}$
(16) $\displaystyle\Delta_{--,++}(k)=\frac{\pm
i\alpha_{R}k\Delta}{\sqrt{V_{z}^{2}+\alpha_{R}^{2}k^{2}}}$
$\displaystyle\Delta_{+-}(k)=\frac{V_{z}\Delta}{\sqrt{V_{z}^{2}+\alpha_{R}^{2}k^{2}}}$

Figure 6:  Distillation of the size effects in the spinfull Kitaev model. (a)
The BEE (in units of $\ln 2$) and (b) the DEE (in units of $\ln 2$) of a $100$
site spinfull Kitaev chain plotted against the chemical potential of the $+$
chain $\mu_{+}$. The spinfull Kitaev chain, a pedagogical model of the
nanowire system, develops similar entropy signatures to the nanowire system
explaining the characteristics of the BEE and DEE plots of the nanowire from
elementary principles.  Figure 7: Energy spectra, the DEE and the topological
phase space. (a) The energy dispersion (in units of $\Delta$) of a $4\mu m$
disordered nanowire, (b) the energy dispersion (in units of $\Delta$) of an
$8\mu m$ disordered nanowire, (c) DEE of the $4\mu m$ disordered nanowire and
(d) DEE of the $8\mu m$ disordered nanowire plotted against the magnetic
field. (e) The DEE phase diagram (in units of $ln2$) for the $4\mu m$
disordered nanowire and (f) The DEE phase diagram (in units of $ln2$) for the
$8\mu m$ disordered nanowire. Near zero-energy states before the topological
phase transition (red dashed line) are visible in (a) and (b). Neverthless,
DEE remains faithfull, rising to $ln2$ units only in the topological regime
for both short (c) and long (d) nanowires. DEE also remains robust to varying
system parameters, even in the presence of disorder, as seen through the phase
diagram (e) and (f)

The Hamiltonian in (3) can be interpreted as follows: there are two bands $+$
and $-$ with their normal parts being described by $\varepsilon_{+}(k)$ and
$\varepsilon_{+}(k)$ respectively. Within each band, there exists an intra-
band superconducting pairing given by $\Delta_{--,++}(k)$. These pairings are
an odd function of $k$ and hence manifest as intra-band p-wave pairings. There
also exists an inter-band superconducting pairing between the two bands given
by $\Delta_{+-}(k)$. This pairing is an even function of $k$ and manifests as
an inter-band s-wave pairing. Thus, the helical basis allows us to decompose
the nanowire into two p-wave superconductors $+$ and $-$ interacting via an
s-wave coupling. We notice two things here. First, the effective onsite
potential $\mu_{eff}=\mu\mp\sqrt{V_{z}^{2}+\alpha_{R}^{2}k^{2}}$ depends on a
controllable parameter $V_{z}$ and therefore the magnetic field $B$. It
decreases with $B$ for the $+$ chain while increases with $B$ for the $-$
chain. Second, the p-wave pairings $\Delta_{--,++}$ are of opposite parity for
both bands.
This motivates introducing a new pedagogical model, which we call the
"spinfull Kitaev chain" based on a topical p-wave superconductor – The Kitaev
chain. We consider two distinct Kitaev chains, labelled with $+$ and $-$,
interacting via an s-wave pairing potential $\Delta_{s}$. The chemical
potentials, hopping terms, and p-wave pairings are represented by
$\mu_{\sigma}$, $t_{\sigma}$, $\Delta_{\sigma}$, where $\sigma=+\text{ or }-$.
The Hamiltonian, describing this pedagogical model, in the position space
(with open boundary conditions) is given by:

Figure 8:  The disconnected fermion parity noise of the nanowire plotted
against the magnetic field $B$ for (a) a 4$\mu m$ pristine nanowire, (b) an
8$\mu m$ pristine nanowire, (c) a 4$\mu m$ disordered nanowire and (d) an
8$\mu m$ disordered nanowire. The fermion parity noise of disconnected regions
behaves similar to the DEE, with zero value in the trivial regime and $0.5$
quantization in the topological regime both in pristine as well as disordered
setups.

$\displaystyle\mathcal{H}$
$\displaystyle=\sum_{i=1}^{N}\sum_{\sigma=\pm}\mu_{\sigma}c_{i,\sigma}^{\dagger}\>c_{i,\sigma}$
(17)
$\displaystyle+\sum_{i=1}^{N-1}\sum_{\sigma=\pm}t_{\sigma}c_{i+1,\sigma}^{\dagger}\>c_{i,\sigma}+h.c.$
$\displaystyle+\sum_{i=1}^{N-1}\sum_{\sigma=\pm}\Delta_{\sigma}c_{i+1,\sigma}^{\dagger}\>c_{i,\sigma}^{\dagger}+h.c.$
$\displaystyle+\sum_{i=1}^{N}\Delta_{s}c_{i,+}^{\dagger}\>c_{i,-}^{\dagger}+h.c.,$

where $c_{i,\sigma}^{\dagger}$ creates a fermion at site ‘i’ in the
$\sigma=\pm$ chain. For further analysis of the model, we express our
Hamiltonian in the Majorana basis defined by:

$\displaystyle c_{i,\sigma}=\gamma_{i,\sigma,A}+i\gamma_{i,\sigma,B}$ (18)
$\displaystyle
c_{i,\sigma}^{\dagger}=\gamma_{i,\sigma,A}-i\gamma_{i,\sigma,B}$
$\displaystyle\gamma_{i,\sigma,\tau}^{\dagger}=\gamma_{i,\sigma,\tau}$
$\displaystyle\\{\gamma_{i,\sigma_{i},\tau_{i}},\gamma_{j,\sigma_{j},\tau_{j}}\\}=2\>\delta_{i,j}\>\delta_{\sigma_{i},\sigma_{j}}\>\delta_{\tau_{i},\tau_{j}}$

where $i$ represents the site index, $\sigma=+\text{ or }-$ and $\tau=A\text{
or }B$, where $A$ and $B$ represent two distinct types of Majoranas that
constitute a single fermion. In the Majorana basis, our model comprises four
interacting Majorana chains. The Hamiltonian in this basis is given by:

Figure 9: Phase diagram of fermion parity noise of disconnected regions of the
nanowire plotted against the magnetic field $B$ for (a) a 4$\mu m$ pristine
nanowire, (b) an 8$\mu m$ pristine nanowire, (c) a 4$\mu m$ disordered
nanowire and (d) an 8$\mu m$ disordered nanowire. The fermion parity noise of
disconnected regions faithfully detects topological phase transitions for a
good range of $B$ and $\mu$

$\displaystyle H$
$\displaystyle=\sum_{i=1}^{N}\sum_{\sigma=\pm}\mu_{\sigma}+\mu_{\sigma}\>\gamma_{i,\sigma,A}\>\gamma_{i,\sigma,B}$
(19)
$\displaystyle+\sum_{i=1}^{N-1}\sum_{\sigma=\pm}\left(t_{\sigma}+\Delta_{\sigma}\right)\gamma_{i+1,\sigma,A}\>\gamma_{i,\sigma,B}$
$\displaystyle+\sum_{i=1}^{N-1}\sum_{\sigma=\pm}\left(t_{\sigma}-\Delta_{\sigma}\right)\gamma_{i,\sigma,A}\>\gamma_{i+1,\sigma,B}$
$\displaystyle+\sum_{i=1}^{N}\Delta_{s}\>\gamma_{i,-,A}\>\gamma_{i,+,B}+\Delta_{s}\>\gamma_{i,-,B}\>\gamma_{i,+,A}.$

In Fig. 4, we show the decomposition of the spinfull Kitaev chain into four
Majorana chains and show the various pairings of Majoranas caused by different
terms of the Hamiltonian as solid bonds. In all the figures, the blue spheres
with $+$ represent $\gamma_{i,+,A}$, the pink spheres with $+$ represent
$\gamma_{i,+,B}$, the blue spheres with $-$ represent $\gamma_{i,-,A}$, the
pink spheres with $-$ represent $\gamma_{i,-,B}$. The green bonds in Fig. 4(a)
represent the terms $\mu_{+}$ and $\mu_{-}$, the orange bonds in Fig. 4(b)
represent the terms $(t_{+}-\Delta_{+})$ and $(t_{-}-\Delta_{-})$, the violet
bonds in Fig. 4(c) represent the terms $(t_{+}+\Delta_{+})$ and
$(t_{-}+\Delta_{-})$, the black bonds in Fig. 4(d) represent the term
$\Delta_{s}$.

Figure 10:  Conductance spectroscopy and the DEE. A comparison of (a) the
local conductance, (b) the non-local conductance and (c) DEE of an $6\mu m$
intermediate length nanowire. The chemical potential is kept at $3.2\Delta$
and maximum value of disorder is kept at $1\Delta$. Even under a relatively
minimal value of disorder, the local conductance closes before the critical
field (black dashed line in (a)). The non-local conductance becomes faint and
signature becomes obscure. Nonethless, the DEE remains faithful – signaling
the transition with pinpoint accuracy.

Next, we study the spinfull Kitaev chain by setting $\mu_{-}=0$,
$\Delta_{+}=-\Delta_{-}=1$, $t_{+}=t_{-}=1$, $\Delta_{s}=0.2$ and we keep
$\mu_{+}$ as the varying parameter. The motivation of varying $\mu_{+}$ comes
from the nanowire model where the $+$ band’s effective potential is varied
through $B$. Similarly, we set $\Delta_{+}=-\Delta_{-}$ to maintain the
opposite parity of the p-wave couplings, just as in the nanowire case. The
reason for keeping $\mu_{-}=0$ shall be made clear soon. At $\mu_{+}=0$ both
chains are in the topological regime and have two unpaired Majoranas at their
ends. However, due to the presence of an s-wave inter-chain coupling
$\Delta_{s}$, the Majoranas at each end pair up to form a normal fermion,
rendering the spinfull Kitaev chain into a trivial phase. This can be seen in
Fig. 5(a), where the yellow unpaired Majoranas hybridize through the solid
black bonds caused by $\Delta_{s}$.
Figure 5(b) represents an intermediate stage where $\mu_{+}$ is finite and can
be seen as the green bonds in the top two chains. Fig. 5(c) depicts the
$\mu_{+}=\infty$ case. Here, $\mu_{+}$ acts as the only dominating bond in the
$+$ chain and decouples the chain $+$ chain from the $-$ chain. The $+$ chain
goes into a trivial regime while the $-$ chain stays in the topological
regime, allowing there to be effectively two Majoranas at the ends of the
wire. This is what precisely happens in the nanowire system, where the
effective onsite potential being controlled by the magnetic field $B$ of one
band tunes it towards the topological regimes while tunes the other band
towards the trivial regime. Before the transition, there are two p-wave
superconductors hosting Majoranas. However, their interaction effectively
leads to none at the edges. At the topological transition, one band becomes
trivial, leading to two unpaired Majoranas at the ends.
Figure 6 shows the entanglement entropy plots for the spinfull Kitaev chain
under the settings described above. Figure 5 (a) shows the BEE plot for a
100-site chain while Fig. 6 (b) shows the DEE plot for the same chain.
Comparing Fig. 3 (a) and Fig. 6 (a) we see very similar features for both the
spinfull Kitaev chain and the nanowire with no quantization and negligible
change in the entanglement entropy value over the phase transition. Similarly,
comparing Fig. 3 (b) and Fig. 6 (b) we see that the DEE is zero in the trivial
regime and quantized to $ln2$ in the topological regime in both the spinfull
Kitaev chain and the nanowire. A DEE value of $ln2$ in the topological regime
is logically expected in the spinfull Kitaev chain since chains are now
independent of any inter-chain interaction, the $+$ Kitaev chain, being in its
trivial regime, contributes $0$ units to the DEE while the $-$ Kitaev chain,
being topological, contributes $ln2$ units to the DEE.

Next, we focus on the effects of disorder in the nanowire system and its
entanglement entropy. As mentioned earlier, the disorder that we study in this
work, i.e., smooth inhomogeneous potentials at the ends of the nanowire,
causes the appearance of near zero-energy sub-gap states called Andreev bound
state (ABS) or quasi-MZMs. These states are trivial in nature and the near
zero-energy behavior of these trivial states over wide ranges of $B$ and $\mu$
hinders the conclusive detection of MZMs through present protocols.

First, in Fig. 7, we show the energy eigenspectrum of the nanowire in the
presence of smooth potentials $V(x)$ at the ends. Figure 7(a) and Fig. 7(b)
show the energy eigenspectrum for a 4$\mu m$ and an 8$\mu m$ wire,
respectively. As shown in in Fig. 7(a), the system hosts zero-energy modes
before the topological phase transition (highlighted by the dashed red line),
depicting the presence of disorder-induced trivial zero-energy modes. These
states become more prominent and appear earlier as the system length
increases, as is visible in the longer nanowire case (Fig. 7(b)).
In Fig. 7(c) and Fig. 7(d), we show the DEE of the 4$\mu m$ and an 8$\mu m$
disordered nanowires corresponding to the energy spectra in Figs. 7(a) and
7(b). For both the nanowire lengths, the DEE is zero in the trivial phase and
rises to a quantized value of $ln2$ in the topological phase only, similar to
the pristine nanowire case. The presence of near-zero energy trivial states
has only a very subtle effect on DEE. It causes it to slightly overshoot at
the phase transition, which may be related to a very generic peaking behavior
of entanglement entropy at quantum critical points in one-dimensional systems.
Nevertheless, the DEE remains faithful in signalling MZMs and the topological
phase transition with clarity in cases where near-zero energy states are
present. We also plot the phase diagram of DEE in the disordered case. Figures
7(e) and 7(f) depict the phase diagram of DEE for a 4$\mu m$ and an 8$\mu m$
wire, respectively, and highlight that the transition in DEE exactly matches
the theoretically estimated topological phase boundary. Thus, the DEE
conclusively detects topological phase transitions, even in the presence of
disorder, for a wide range of parameters $B$ and $\mu$.
The disconnected fermion parity noise $\delta P$, a physical observable
closely related to the DEE, is a possible candidate to effectively gauge DEE
in experiments. Figure 8 depicts the variation of $\delta P$ as a function of
the Zeeman field for the nanowire. In Fig. 8(a) and 8(b), we plot the
disconnected fermion parity noise of a pristine $4\mu m$ nanowire and a
pristine $8\mu m$ nanowire against the magnetic field. For both nanowire
lengths, we observe that the signatures closely match the DEE signature, thus
pointing out the connection between DEE and the fermion parity noise.
To study the robustness of this physically observable metric to disorder, we
show the corresponding plots for the disordered nanowire case in Figs. 8(c)
and 8(d). As expected, we observe that the signatures stay robust to the
presence of trivial zero-energy modes and closely match the DEE signatures. We
also plot the phase diagrams of the disconnected fermion parity noise for Fig.
9(a) a $4\mu m$ pristine nanowire, Fig. 9(b) an $8\mu m$ pristine nanowire,
Fig. 9(c) a $4\mu m$ disordered nanowire, Fig. 9(d) an $8\mu m$ disordered
nanowire and show that the fermion parity noise signature remains clean and
robust over a good range of controllable parameters $\mu$ and $B$ hence acting
as a faithful detector of topological phase transitions.
Lastly, we find it worthwhile to demonstrate the superiority of entanglement
entropy (and hence observables such as the fermion parity) over techniques
with current experimental attention, such as local and non-local differential
conductance. In Fig. 10, we depict the local conductance ($G_{LL}$), non-local
conductance ($G_{LR}$), and the DEE ($S_{D}$) of a three-terminal N-TS-N setup
involving an intermediate length $6\mu m$ nanowire with minimal to modest
disorder. Even under low disorder, the local conductance fails to detect the
phase transition with a premature closure, as shown in Fig. 10(a). The non-
local conductance signatures, on the other hand, become obscure and faint
under experimentally relevant measurement precision, unable to pinpoint the
critical field, as depicted in Fig. 10(b). The DEE, however, remains faithful
and accurately signals the phase transition, as shown in Fig. 10(c).

##  IV Conclusion

We proposed and analyzed in depth the use of entanglement entropy and its
close relation with topological order for a fool-proof signalling of MZMs in
Rashba nanowires. We showed that although the BEE is incapable of determining
topological phase transitions, owing to volume and area contributions, the DEE
reliably pinpoints topological phase transitions following the distillation of
the size contributions. The DEE signatures stay robust to system size and
controllable parameters such as chemical potential and magnetic field, even in
the presence of disorders such as smooth inhomogeneous potentials at the ends
of the nanowire relevant to experimental setups. We explained the quantization
of the entanglement entropy in a bottom-up fashion by deconstructing Rashba
nanowires as interacting Kitaev Chains. Further, we connected the entanglement
entropy signatures to a conserved observable of the system, the fermion parity
noise, and illustrated that it displays signatures similar to the entanglement
entropy. Finally, we concluded by elucidating a simple case with minimal
disorder and moderate nanowire lengths where the local conductance signature
closes prematurely, and the non-local conductance signature is obscure, but
the DEE signature is conclusive and faithfully signals the topological
transition. The theoretical methods developed here are of significant
relevance toward making experimental progress for the conclusive detection of
MZMs, and looking beyond conductance spectroscopy measurements.
Acknowedgements: The authors acknowledge the Visvesvaraya PhD Scheme of the
Ministry of Electronics and Information Technology (MEITY), Government of
India, as implemented by Digital India Corporation (formerly Media Lab Asia).
This work is also supported by the Science and Engineering Research Board
(SERB), Government of India, Grant No. Grant No. STR/2019/000030, the Ministry
of Human Resource Development (MHRD), Government of India, Grant No.
STARS/APR2019/NS/226/FS under the STARS scheme.

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