Figure 1. The first two steps in the construction: the $s_{i}$ points are
represented in blue, while the $t_{i}$ points in red. Observe that in the
second step all red points (and corresponding blue points not shown in the
picture) between $s_{i_{j}-1}$ and $s_{i_{j}}$ are removed, so that in future
steps new intervals are picked.

###  5.3. Proof of Theorem 1.1.

By Theorem 4.2, we already know that
$\limsup_{\alpha\to\infty}\alpha^{4}\sigma^{2}(\alpha)\leq K$. Now, by
definition of $C_{5}$ in (5.21), we have
${\tilde{C}}_{1}C_{5}(C)=C_{4}(C)-\frac{{\tilde{C}}_{1}{\tilde{C}}_{3}}{C}$,
and by (5.14), $\lim_{C\uparrow\infty}C_{4}(C)=[{\rm e}^{-1}-{\rm
e}^{-2}]\int_{\mathbb{R}^{3}}|\psi_{0}(x)|^{2}{\rm d}x$, while
$\frac{{\tilde{C}}_{1}{\tilde{C}}_{3}}{C}\downarrow 0$ as $C\uparrow\infty$.
Hence, there is $C_{*}\in(0,\infty)$ such that for all $\alpha>0$,

$\alpha^{4}\sigma^{2}(\alpha)\leq\frac{1}{C_{*}\int_{\mathbb{R}^{3}}|\psi_{0}(x)|^{2}{\rm
d}x},\qquad\mbox{or}\qquad\frac{m(\alpha)}{\alpha^{4}}\geq
C_{*}\int_{\mathbb{R}^{3}}|\nabla\psi_{0}(x)|^{2}{\rm d}x,$

proving Theorem 1.1. ∎

Acknowledgement: The first two authors are supported by the Deutsche
Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC 2044 -
390685587, Mathematics Münster: Dynamics - Geometry - Structure.

## References

  * [1] F. Baccelli and P. Brémaud  Elements of queueing theory  Springer-Verlag, Berlin (2003).
  * [2] V. Betz and S. Polzer. A functional central limit theorem for polaron path measures.  Comm. Pure Appl. Math. 75, 2345-2392 (2022)
  * [3] V. Betz and S. Polzer. Effective mass of the Polaron: a lower bound.  Comm. Math. Phys. 399, 173-188 (2023)
  * [4] E. Bolthausen, W. König and C. Mukherjee.  Mean field interaction of Brownian occupation measures, II.: Rigorous construction of the Pekar process.  Comm. Pure Appl. Math., 70 1598-1629, (2017)
  * [5] M. Brooks and R. Seiringer. The Frölich Polaron at Strong Coupling-Part II: Energy-Momentum Relation and Effective Mass. arXiv: 2211.03353 (2022)
  * [6] D.J. Daley and D. Vere-Jones  An introduction to the theory of point processes. Vol. I. Elementary Theory and Methods.  Springer (2003).
  * [7] D.J. Daley and D. Vere-Jones  An introduction to the theory of point processes. Vol. II. General theory and structure.  Springer (2008).
  * [8] M. D. Donsker and S. R S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time, IV Comm. Pure Appl. Math. 36 (1983), 183-212.
  * [9] M. Donsker and S. R. S. Varadhan.  Asymptotics for the Polaron.  Comm. Pure Appl. Math. 505-528 (1983).
  * [10] W. Dybalski and H. Spohn. Effective Mass of the Polaron-Revisited. Annales Henri Poincaré 21, 1573- 1594 (2020).
  * [11] R. Feynman. Statistical Mechanics, Benjamin, Reading (1972).
  * [12] W. König and C. Mukherjee.  Mean-field interaction of Brownian occupation measures. I: Uniform tube property of the Coulomb functional.  Ann. Inst. H. Poincaré Probab. Statist, 53, 2214-2228, (2017), arXiv: 1509.06672
  * [13] L. D. Landau and S. I. Pekar. Effective mass of a polaron. Zh.Eksp.Teor.Fiz. 18, 419- 423 (1948)
  * [14] E. H. Lieb.  Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation.  Studies in Appl. Math. 57, 93-105 (1976).
  * [15] E. H. Lieb and R. Seiringer Equivalence of Two Definitions of the Effective Mass of a Polaron, J. Stat. Phys. 154, Issue 1-2, 51-57 (2014).
  * [16] E. H. Lieb and R. Seiringer Divergence of the Effective Mass of a Polaron in the Strong Coupling Limit, J. Stat. Phys. 180, 23-33 (2020)
  * [17] E. H. Lieb and L. Thomas. Exact ground state energy of the strong-coupling Polaron. Comm. Math. Phys., 183, (1997), 511-519.
  * [18] C. Mukherjee and S. R. S. Varadhan.  Brownian occupations measures, compactness and large deviations.  Ann. Probab., 44 3934-3964, (2016)
  * [19] C. Mukherjee and S. R. S. Varadhan.  Identification of the Polaron measure I: fixed coupling regime and the central limit theorem for large times.  Commun. Pure Appl. Math. 73, no. 3, 350-383 (2020). arXiv:1802.05696
  * [20] C. Mukherjee and S. R. S. Varadhan.  Identification of the Polaron measure in strong coupling and the Pekar variational formula.  Ann. Probab. 48, 5, 2119-2144 (2020). arXiv:1812.06927
  * [21] C. Mukherjee and S. R. S. Varadhan.  Corrigendum and Addendum: Identification of the Polaron measure I: fixed coupling regime and the central limit theorem for large times.  Commun. Pure Appl. Math. 75, no. 7, 1642-1653 (2022). (Proof of Theorem 4.5 in arXiv:1802.05696)
  * [22] S. I. Pekar. Theory of polarons, Zh. Eksperim. i Teor. Fiz. 19, (1949).
  * [23] M. Sellke. Almost Quartic Lower Bound for the Fröhlich Polaron’s Effective Mass via Gaussian Domination Preprint, arXiv: 2212.14023, December 2022
  * [24] H. Spohn. Effective mass of the polaron: A functional integral approach. Ann. Phys. 175, (1987), 278-318.


* Mizuno, (1980) Mizuno, H. (1980).  Formation of the Giant Planets.  Progress of Theoretical Physics, 64(2):544–557.
  * Nakagawa et al., (1986) Nakagawa, Y., Sekiya, M., and Hayashi, C. (1986).  Settling and growth of dust particles in a laminar phase of a low-mass solar nebula.  Icarus, 67(3):375–390.
  * Nelson et al., (2013) Nelson, R. P., Gressel, O., and Umurhan, O. M. (2013).  Linear and non-linear evolution of the vertical shear instability in accretion discs.  MNRAS, 435(3):2610–2632.
  * Onishi and Sekiya, (2017) Onishi, I. K. and Sekiya, M. (2017).  Planetesimal formation by an axisymmetric radial bump of the column density of the gas in a protoplanetary disk.  Earth, Planets, and Space, 69(1):50.
  * Ono et al., (2016) Ono, T., Muto, T., Takeuchi, T., and Nomura, H. (2016).  Parametric Study of the Rossby Wave Instability in a Two-dimensional Barotropic Disk.  ApJ, 823(2):84.
  * Paardekooper et al., (2010) Paardekooper, S.-J., Lesur, G., and Papaloizou, J. C. B. (2010).  Vortex Migration in Protoplanetary Disks.  ApJ, 725(1):146–158.
  * Paardekooper et al., (2020) Paardekooper, S.-J., McNally, C. P., and Lovascio, F. (2020).  Polydisperse streaming instability - I. Tightly coupled particles and the terminal velocity approximation.  MNRAS, 499(3):4223–4238.
  * (81) Petersen, M. R., Julien, K., and Stewart, G. R. (2007a).  Baroclinic Vorticity Production in Protoplanetary Disks. I. Vortex Formation.  ApJ, 658:1236–1251.
  * (82) Petersen, M. R., Stewart, G. R., and Julien, K. (2007b).  Baroclinic Vorticity Production in Protoplanetary Disks. II. Vortex Growth and Longevity.  ApJ, 658:1252–1263.
  * Pfeil and Klahr, (2021) Pfeil, T. and Klahr, H. (2021).  The Sandwich Mode for Vertical Shear Instability in Protoplanetary Disks.  ApJ, 915(2):130.
  * Picogna et al., (2018) Picogna, G., Stoll, M. H. R., and Kley, W. (2018).  Particle accretion onto planets in discs with hydrodynamic turbulence.  A&A, 616:A116.
  * Raettig et al., (2015) Raettig, N., Klahr, H., and Lyra, W. (2015).  Particle Trapping and Streaming Instability in Vortices in Protoplanetary Disks.  ApJ, 804(1):35.
  * Raettig et al., (2021) Raettig, N., Lyra, W., and Klahr, H. (2021).  Pebble Trapping in Vortices: Three-dimensional Simulations.  ApJ, 913(2):92.
  * Ragusa et al., (2017) Ragusa, E., Dipierro, G., Lodato, G., Laibe, G., and Price, D. J. (2017).  On the origin of horseshoes in transitional discs.  MNRAS, 464(2):1449–1455.
  * Railton and Papaloizou, (2014) Railton, A. D. and Papaloizou, J. C. B. (2014).  On the local stability of vortices in differentially rotating discs.  MNRAS, 445(4):4409–4426.
  * Richard et al., (2016) Richard, S., Nelson, R. P., and Umurhan, O. M. (2016).  Vortex formation in protoplanetary discs induced by the vertical shear instability.  MNRAS, 456(4):3571–3584.
  * Safronov, (1972) Safronov, V. S. (1972).  Evolution of the protoplanetary cloud and formation of the earth and planets.
  * Schäfer et al., (2020) Schäfer, U., Johansen, A., and Banerjee, R. (2020).  The coexistence of the streaming instability and the vertical shear instability in protoplanetary disks.  A&A, 635:A190.
  * Schaffer et al., (2021) Schaffer, N., Johansen, A., and Lambrechts, M. (2021).  Streaming instability of multiple particle species. II. Numerical convergence with increasing particle number.  A&A, 653:A14.
  * Shakura and Sunyaev, (1973) Shakura, N. I. and Sunyaev, R. A. (1973).  Reprint of 1973A&A….24..337S. Black holes in binary systems. Observational appearance.  A&A, 500:33–51.
  * Shi and Chiang, (2013) Shi, J.-M. and Chiang, E. (2013).  From Dust to Planetesimals: Criteria for Gravitational Instability of Small Particles in Gas.  ApJ, 764(1):20.
  * Simon et al., (2016) Simon, J. B., Armitage, P. J., Li, R., and Youdin, A. N. (2016).  The Mass and Size Distribution of Planetesimals Formed by the Streaming Instability. I. The Role of Self-gravity.  ApJ, 822(1):55.
  * Squire and Hopkins, (2018) Squire, J. and Hopkins, P. F. (2018).  Resonant drag instabilities in protoplanetary discs: the streaming instability and new, faster growing instabilities.  MNRAS, 477(4):5011–5040.
  * Stoll and Kley, (2014) Stoll, M. H. R. and Kley, W. (2014).  Vertical shear instability in accretion disc models with radiation transport.  A&A, 572:A77.
  * Stoll and Kley, (2016) Stoll, M. H. R. and Kley, W. (2016).  Particle dynamics in discs with turbulence generated by the vertical shear instability.  A&A, 594:A57.
  * Stoll et al., (2017) Stoll, M. H. R., Kley, W., and Picogna, G. (2017).  Anisotropic hydrodynamic turbulence in accretion disks.  A&A, 599:L6.
  * Surville and Mayer, (2019) Surville, C. and Mayer, L. (2019).  Dust-vortex Instability in the Regime of Well-coupled Grains.  ApJ, 883(2):176.
  * Takeuchi and Lin, (2002) Takeuchi, T. and Lin, D. N. C. (2002).  Radial Flow of Dust Particles in Accretion Disks.  ApJ, 581(2):1344–1355.
  * Taki et al., (2016) Taki, T., Fujimoto, M., and Ida, S. (2016).  Dust and gas density evolution at a radial pressure bump in protoplanetary disks.  A&A, 591:A86.
  * Tassoul, (1978) Tassoul, J. (1978).  Theory of rotating stars.
  * Turner and Drake, (2009) Turner, N. J. and Drake, J. F. (2009).  Energetic Protons, Radionuclides, and Magnetic Activity in Protostellar Disks.  ApJ, 703(2):2152–2159.
  * Turner et al., (2014) Turner, N. J., Fromang, S., Gammie, C., Klahr, H., Lesur, G., Wardle, M., and Bai, X. N. (2014).  Transport and Accretion in Planet-Forming Disks.  In Beuther, H., Klessen, R. S., Dullemond, C. P., and Henning, T., editors, Protostars and Planets VI, page 411.
  * Urpin, (2003) Urpin, V. (2003).  A comparison study of the vertical and magnetic shear instabilities in accretion discs.  A&A, 404:397–403.
  * Urpin and Brandenburg, (1998) Urpin, V. and Brandenburg, A. (1998).  Magnetic and vertical shear instabilities in accretion discs.  MNRAS, 294(3):399–406.
  * van der Marel et al., (2021) van der Marel, N., Birnstiel, T., Garufi, A., Ragusa, E., Christiaens, V., Price, D. J., Sallum, S., Muley, D., Francis, L., and Dong, R. (2021).  On the Diversity of Asymmetries in Gapped Protoplanetary Disks.  AJ, 161(1):33.
  * van der Marel et al., (2013) van der Marel, N., van Dishoeck, E. F., Bruderer, S., Birnstiel, T., Pinilla, P., Dullemond, C. P., van Kempen, T. A., Schmalzl, M., Brown, J. M., Herczeg, G. J., Mathews, G. S., and Geers, V. (2013).  A Major Asymmetric Dust Trap in a Transition Disk.  Science, 340(6137):1199–1202.
  * Weidenschilling, (1977) Weidenschilling, S. J. (1977).  Aerodynamics of solid bodies in the solar nebula.  MNRAS, 180:57–70.
  * Weidenschilling and Cuzzi, (1993) Weidenschilling, S. J. and Cuzzi, J. N. (1993).  Formation of Planetesimals in the Solar Nebula.  In Levy, E. H. and Lunine, J. I., editors, Protostars and Planets III, page 1031.
  * Wetherill, (1990) Wetherill, G. W. (1990).  Formation of the Earth and Terrestrial Planets.  In Bulletin of the American Astronomical Society, volume 22, page 1335.
  * Yang et al., (2017) Yang, C. C., Johansen, A., and Carrera, D. (2017).  Concentrating small particles in protoplanetary disks through the streaming instability.  A&A, 606:A80.
  * Youdin and Johansen, (2007) Youdin, A. and Johansen, A. (2007).  Protoplanetary Disk Turbulence Driven by the Streaming Instability: Linear Evolution and Numerical Methods.  ApJ, 662(1):613–626.
  * Youdin and Goodman, (2005) Youdin, A. N. and Goodman, J. (2005).  Streaming Instabilities in Protoplanetary Disks.  ApJ, 620(1):459–469.
  * Zhu and Stone, (2014) Zhu, Z. and Stone, J. M. (2014).  Dust Trapping by Vortices in Transitional Disks: Evidence for Non-ideal Magnetohydrodynamic Effects in Protoplanetary Disks.  ApJ, 795(1):53.
  * Zhu and Yang, (2021) Zhu, Z. and Yang, C.-C. (2021).  Streaming instability with multiple dust species - I. Favourable conditions for the linear growth.  MNRAS, 501(1):467–482.

##  Appendix A Influence of Temperature Slope $q$ on Dusty Gas Instability

In Section 3.3 we described how the VSI leads to the emergence of a dusty gas
instability by inducing a corrugation motion of the dusty mid-plane layer. To
further substantiate this hypothesis we conducted additional 2D axisymmetric
simulations including an initial pressure bump with $A=0.4$ and with reduced
values of the temperature slope $q$ as this should result in a weaker VSI and
hence a weaker dusty gas instability. The results of these simulations are
presented in Figures 24 and 25 for the cases $\tau_{0}=6\cdot 10^{-3}$ with
$z=0.01$ and $z=0.03$, respectively. From left to right the simulations adopt
a temperature slope $q=1$ (the fiducial value), $q=0.75$, $q=0.5$ and $q=0.1$.
According to observational studies (Andrews et al.,, 2009) the two
intermediate values $q=0.75$ and $q=0.5$ should be the most realistic ones to
model the outer parts of PPDs. With decreasing value of $q$ we find that the
inward drifting motion as well as the splitting of dust rings is increasingly
mitigated. However, the inward drift remains significant for all but the
smallest value of $q$ considered here. The simulations with the smallest value
$q=0.1$ are almost entirely laminar with only very weak VSI activity.
Additional unstratified 2D simulations as well as 1D simulations with various
values of $q\leq 1$ that we carried out are very similar to the cases with
$q=0.1$ shown here and will therefore not be presented.

Figure 24: Same as Figure 4 but now the temperature slope $q$ is varied from
left to right as indicated. The runs adopted a metallicity $Z=0.01$, Stokes
number $\tau_{0}=6\cdot 10^{-3}$ and an initial pressure bump with $A=0.4$.
Figure 25: Same as Figure 24 except that $Z=0.03$.

##  Appendix B Influence of Density Slope $p$ on Vortex Evolution

As described in Section 5.1, we hypothesize that the size of large vortices
depends on their capability to absorb small vortices within their vicinity.
Based on the study of Paardekooper et al., (2010) we expect this capability to
be increased for flatter surface density profiles where small vortices should
be more efficiently attracted by the mild pressure maximum corresponding to a
large vortex (cf. Figure 23). To test this hypothesis we conducted 3D dust-
free simulations with varying gas surface mass density slope $s$ (and hence
varying mid-plane gas density slope $p$). These simulations are presented in
Figure 26. The results in the left panels correspond to $\sim 1000$ orbits and
illustrate that for $|p|\leq 1$ vortices are substantially stronger and more
numerous than in the simulation with $p=3.5$. In this regard our fiducial run
with $p=2.5$ (lower left panel of Figure 18) is similar to the case $p=3.5$.
Inspection of the time evolution of the mid-plane gas density in the right
panels of Figure 26 shows the development of bumps and dips, which again are
believed to be a consequence of the VSI turbulent mass transport and which are
more prominent in the simulations with smaller $|p|$. We find that large
vortices in simulations with $|p|\leq 1$ migrate slowly, in some cases without
a detectable preference direction. On the other hand, the vortices in the
simulation with $p=3.5$ undergo inward migration similar to the vortices in
Figure 18 where $p=2.5$. This should be a consequence of the overall steeper
density profiles in the latter simulations. Typical migration rates in the
simulations with $p=2.5$ (Figure 18) are $\sim 0.25-0.3$ $H_{g0}$/(100
orbits), whereas in the simulation with $p=3.5$ we find values $\sim 0.4$
$H_{g0}$/(100 orbits). Moreover, due to the similar results of the simulations
with $|p|\leq 1$ this suggests that it is the mid-plane gas volume density
which determines the migration speed of vortices (and therefore their possible
growth by absorption of smaller vortices), rather than the surface mass
density. That is, for $p={-1,0,1,2.5,3.5}$ the gas surface mass density
$\Sigma_{g}$ follows a power-law with indices $s={-2,-1,0,1.5,2.5}$,
respectively. Thus, if $\Sigma_{g}$ were the relevant quantity to vortex
migration we would expect the strongest vortices to occur for $p=0,1,2.5$ and
only little difference between the simulations with $p=0,2.5$ ($p=2.5$ being
the fiducial case), which is not what we observe. These results are also
compatible with the finding of Manger et al., (2020) that vortices in their
simulations with $p=0.66$ and $p=1.5$ appeared very similar, although the
vortices in the top panels ($p=0.66$) of their Figure 8 do appear slightly
larger than those in the bottom panels ($p=1.5$).

Figure 26: _Left_ : Same as Figure 18 but now $A=0$ and the mid-plane gas
density powerlaw index $p$ is varied through corresponding variation of the
powerlaw index $s$ of the surface mass density. The snapshots correspond to
$\sim 1000$ orbits. _Right_ : Same as Figure 19 but now the gas density slope
$p$ is varied through corresponding variation of the slope $s$ of the surface
mass density. Moreover, the dashed lines indicate the locations of three large
vortices at $\sim 1000$ orbits.


# Entanglement duality in spin-spin interactions

Vahid Azimi Mousolou111Electronic address<EMAIL_ADDRESS>Department of
Applied Mathematics and Computer Science, Faculty of Mathematics and
Statistics, University of Isfahan, Isfahan 81746-73441, Iran  Department of
Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden
Anders Bergman  Department of Physics and Astronomy, Uppsala University, Box
516, SE-751 20 Uppsala, Sweden  Anna Delin  Department of Physics and
Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden  Department
of Applied Physics, School of Engineering Sciences, KTH Royal Institute of
Technology, AlbaNova University Center, SE-10691 Stockholm, Sweden  Swedish
e-Science Research Center (SeRC), KTH Royal Institute of Technology, SE-10044
Stockholm, Sweden  Olle Eriksson  Department of Physics and Astronomy, Uppsala
University, Box 516, SE-751 20 Uppsala, Sweden  School of Science and
Technology, Örebro University, SE-701 82, Örebro, Sweden  Manuel Pereiro
Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20
Uppsala, Sweden  Danny Thonig  School of Science and Technology, Örebro
University, SE-701 82, Örebro, Sweden  Erik Sjöqvist222Electronic address:
<EMAIL_ADDRESS>Department of Physics and Astronomy, Uppsala
University, Box 516, SE-751 20 Uppsala, Sweden

###### Abstract

We examine entanglement of thermal states for spin-$\frac{1}{2}$ dimers in
external magnetic fields. Entanglement transition in the temperature-magnetic
field plane demonstrates a duality in spin-spin interactions. This identifies
dual categories of symmetric and anti-symmetric dimers with each category
classified into toric entanglement classes. The entanglement transition line
is preserved from each toric entanglement class to its dual toric class. The
toric classification is an indication of topological nature of the
entanglement.

##  I Introduction

The classification concept has been incorporated into different fields of
studies ranging from biological systems to abstract topology, in order to
categorize relevant objects based on shared characteristics. Scientific
classification schemes have not only led to new discoveries of materials and
resources, such as in the topological classification of matter chiu16  and of
entangled quantum states johansson14 , but have also significantly helped to
find the most efficient and robust approaches to technological advances.

One of the fundamental characteristics of quantum mechanics is the quantum
correlation or the entanglement, which lies at the heart of the difference
between the classical and quantum world einstein1935 ; schrodinger1935 ;
bell1964 . It is widely believed to be much stronger than classical
correlations and over the years has become a critically important resource for
many applications in quantum technology including quantum computing, quantum
cryptography, quantum communication, and hyper-sensitive measurements
nielsen2010 . As a primary attribute of quantum mechanics, quantum
entanglement is also much more involved with the foundations, predictions and
interpretations of quantum phenomena. It is strongly linked to the concepts of
quantum phase transition and quantum geometric phases osterloh2002 ;
osborne2002 ; wei2005 ; orus2008 ; son2011 ; azimi-mousolou2013 . Nonetheless,
the mysterious characteristics of the quantum entanglement still needs to be
explored in-depth.

Here we use entanglement in terms of concurrence hill1997 ; wootters98  from a
new perspective, i.e. classification, in order to analyze spin-spin
interactions in dimers and classify them into entanglement classes. For this,
we focus on the entanglement transition line for thermal state of the general
traceless spin-pair model in the external parameter space specified by
temperature and applied magnetic field. With this we categorize dimers into
dual categories of symmetric and anti-symmetric dimers. Each category is
classified into toric entanglement classes, where each class together with its
dual are distinguished by the same entanglement transition line in
temperature-magnetic field plane. As a nontrivial example, we introduce dual
symmetric and anti-symmetric Heisenberg spin-pair interactions and specify
their toric entanglement classes. We note that in Ref. arnesen01 , the
entanglement between two spins in a one dimensional Heisenberg chain has also
been studied as a function of temperature and external magnetic field, but not
from the classification approach, which is the main concern here.

##  II General Model Hamiltonian

We start with the general traceless spin-pair model described by the
Hamiltonian ($\hbar=1$ throughout the paper)

$\displaystyle
H(\omega,\mathbb{J})=\frac{\omega_{+}}{2}\sigma_{z}^{(1)}+\frac{\omega_{-}}{2}\sigma_{z}^{(2)}+\boldsymbol{\sigma}^{(1)}\cdot\mathbb{J}\boldsymbol{\sigma}^{(2)},$
(1)

with the real-valued matrix

$\displaystyle\mathbb{J}=\left(\begin{array}[]{ccc}J_{xx}&J_{xy}&0\\\
J_{yx}&J_{yy}&0\\\ 0&0&J_{zz}\end{array}\right),$ (5)

where

$\displaystyle J_{xx}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(J+r),\
J_{yy}=\frac{1}{2}(J-r),$ $\displaystyle J_{xy}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(K-D\right),\
J_{yx}=\frac{1}{2}\left(K+D\right),$ $\displaystyle\omega_{\pm}$
$\displaystyle=$ $\displaystyle\omega\pm\Delta.$ (6)

This model accounts for a wide range of important spin-spin interaction
systems, including Heisenberg ($J=J_{zz}$), Dzyaloshinskii–Moriya (D), and
symmetric exchange ($K$), as well as XY anisotropy ($r$). It further allows
for the spins to interact differently with the external magetic field
($\Delta$), due to field inhomogenties or differences in magnetic moments.

The two-qubit Hamiltonian is chosen so as to ensures that the thermal
equilibrium state at temperature $T$ is of a $X$-type:

$\displaystyle\varrho_{T}=\frac{1}{\mathcal{Z}}e^{-H/T}=\left(\begin{array}[]{cccc}\rho_{11}&0&0&\rho_{14}\\\
0&\rho_{22}&\rho_{23}&0\\\ 0&\rho_{32}&\rho_{33}&0\\\
\rho_{41}&0&0&\rho_{44}\end{array}\right).$ (11)

in the ordered product qubit-qubit basis
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$ and we put Boltzmann’s
constant $k_{B}=1$ from now. Explicitly, we have

$\displaystyle\rho_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{Z}}e^{-\frac{J_{zz}}{T}}[\cosh\frac{\epsilon_{1}}{T}-\sinh\frac{\epsilon_{1}}{T}\cos\vartheta],$
$\displaystyle\rho_{44}$ $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{Z}}e^{-\frac{J_{zz}}{T}}[\cosh\frac{\epsilon_{1}}{T}+\sinh\frac{\epsilon_{1}}{T}\cos\vartheta],$
$\displaystyle\rho_{14}$ $\displaystyle=$
$\displaystyle\rho_{41}^{*}=-\frac{1}{\mathcal{Z}}e^{-\frac{J_{zz}}{T}}e^{-i\varphi}\sinh\frac{\epsilon_{1}}{T}\sin\vartheta,$
$\displaystyle\rho_{22}$ $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{Z}}e^{\frac{J_{zz}}{T}}[\cosh\frac{\epsilon_{2}}{T}-\sinh\frac{\epsilon_{2}}{T}\cos\theta],$
$\displaystyle\rho_{33}$ $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{Z}}e^{\frac{J_{zz}}{T}}[\cosh\frac{\epsilon_{2}}{T}+\sinh\frac{\epsilon_{2}}{T}\cos\theta],$
$\displaystyle\rho_{23}$ $\displaystyle=$
$\displaystyle\rho_{32}^{*}=-\frac{1}{\mathcal{Z}}e^{\frac{J_{zz}}{T}}e^{-i\phi}\sinh\frac{\epsilon_{2}}{T}\sin\theta,$
(12)

where

$\displaystyle\tan\vartheta$ $\displaystyle=$
$\displaystyle\frac{\sqrt{r^{2}+K^{2}}}{\omega},\ \ \ \ \ \ \ \
\tan\varphi=\frac{K}{r},$ (13) $\displaystyle\tan\theta$ $\displaystyle=$
$\displaystyle\frac{\sqrt{J^{2}+D^{2}}}{\Delta},\ \ \ \ \ \ \ \
\tan\phi=\frac{D}{J},$ $\displaystyle\epsilon_{1}$ $\displaystyle=$
$\displaystyle\sqrt{\omega^{2}+r^{2}+K^{2}},\ \ \ \ \ \
\epsilon_{2}=\sqrt{\Delta^{2}+J^{2}+D^{2}},\ \ $

and
$\mathcal{Z}=2\left(e^{-\frac{J_{zz}}{T}}\cosh\frac{\epsilon_{1}}{T}+e^{\frac{J_{zz}}{T}}\cosh\frac{\epsilon_{2}}{T}\right)$
the partition function.

The above $X$-state form is suitable in our analysis form two points of views:
(i) as pointed out before, it accounts for thermal states of several important
spin-spin interaction models, and (ii) the entanglement measure concurrence
$C(\varrho_{T})$ hill1997 ; wootters98  can be calculated analytically.
Indeed, one finds mazzola10

$\displaystyle C(\varrho_{T})=2\max\big{\\{}C_{1},C_{2},0\big{\\}},$ (14)

where

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle|\rho_{14}|-\sqrt{\rho_{22}\rho_{33}},$ $\displaystyle C_{2}$
$\displaystyle=$ $\displaystyle|\rho_{23}|-\sqrt{\rho_{11}\rho_{44}}.$ (15)

We pursue by exploring the entanglement transition, for which one may solve

$\displaystyle\max\big{\\{}C_{1},C_{2}\big{\\}}=0$ (16)

to extract the critical line in the temperature-magnetic field parameter
space. Eq. (16) gives rise to the following duality

$\displaystyle(\text{I})\ \ \ \ \ C_{2}\leq C_{1}$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle(\text{II})\ \ \ \ \ C_{1}\leq C_{2}$
$\displaystyle=$ $\displaystyle 0,$ (17)

which becomes

$\displaystyle(\text{I})\ \ \ \ \
e^{-\frac{2J_{zz}}{T}}p^{2}-e^{\frac{2J_{zz}}{T}}q^{2}$ $\displaystyle=$
$\displaystyle e^{\frac{2J_{zz}}{T}}$ $\displaystyle(\text{II})\ \ \ \ \
e^{\frac{2J_{zz}}{T}}q^{2}-e^{-\frac{2J_{zz}}{T}}p^{2}$ $\displaystyle=$
$\displaystyle e^{-\frac{2J_{zz}}{T}},$ (18)

for our model system with $p=f(\epsilon_{1},\nu)$ and
$q=f(\epsilon_{2},\theta)$ given by the function $f(x,y)=\sinh\frac{x}{T}\sin
y$.

This duality equation allows us to categorize the spin-spin interactions in
dimers into a dual categories of interactions. Here we focus on symmetric and
antisymmetric dimers in the sense that the two spins of same size forming the
dimer possess parallel or antiparallel orientations of the respective magnetic
moments. For each case the corresponding Hamiltonian would have the following
form

  * •

Symmetric: Here with parallel magnetic moments we have
$\omega_{+}=\omega_{-}=B$ and therefore

$\displaystyle H_{\text{s}}$ $\displaystyle=$
$\displaystyle\frac{B}{2}[\sigma_{z}^{(1)}+\sigma_{z}^{(2)}]+J_{zz}\sigma^{(1)}_{z}\sigma^{(2)}_{z}$
$\displaystyle+\frac{J}{2}[\sigma^{(1)}_{x}\sigma^{(2)}_{x}+\sigma^{(1)}_{y}\sigma^{(2)}_{y}]-\frac{D}{2}[\sigma^{(1)}_{x}\sigma^{(2)}_{y}-\sigma^{(1)}_{y}\sigma^{(2)}_{x}]$
$\displaystyle+\frac{r}{2}[\sigma^{(1)}_{x}\sigma^{(2)}_{x}-\sigma^{(1)}_{y}\sigma^{(2)}_{y}]+\frac{K}{2}[\sigma^{(1)}_{x}\sigma^{(2)}_{y}+\sigma^{(1)}_{y}\sigma^{(2)}_{x}].$

Note that we have $\Delta=0$ and $\theta=\pi/2$.

  * •

Antisymmetric: In this case we have antiparallel magnetic moments, i.e.,
$\omega_{+}=-\omega_{-}=B$ and therefore

$\displaystyle H_{\text{as}}$ $\displaystyle=$
$\displaystyle\frac{B}{2}[\sigma_{z}^{(1)}-\sigma_{z}^{(2)}]-J_{zz}\sigma^{(1)}_{z}\sigma^{(2)}_{z}$
$\displaystyle+\frac{J}{2}[\sigma^{(1)}_{x}\sigma^{(2)}_{x}+\sigma^{(1)}_{y}\sigma^{(2)}_{y}]-\frac{D}{2}[\sigma^{(1)}_{x}\sigma^{(2)}_{y}-\sigma^{(1)}_{y}\sigma^{(2)}_{x}]$
$\displaystyle+\frac{r}{2}[\sigma^{(1)}_{x}\sigma^{(2)}_{x}-\sigma^{(1)}_{y}\sigma^{(2)}_{y}]+\frac{K}{2}[\sigma^{(1)}_{x}\sigma^{(2)}_{y}+\sigma^{(1)}_{y}\sigma^{(2)}_{x}],$
. (20)

where in contrast to symmetric dimers we have $\omega=0$ and thus $\nu=\pi/2$.

We note that (I) and (II) in Eq. (18) classify the above two categories of
symmetric and antisymmetric dimers, each within its own category, into
equivalence classes of dimers in a way that all the dimers in the same class
have the same entanglement phase diagram in the temperature-magnetic field
parameter space. Explicitly, if we solve Eq. (18) for the critical line in the
$(B,T)$ plane, across which the thermal state changes its entangling feature
from entangled to separable, then we find out that two dimers specified with
interaction parameters $(J,D,r,K)$ and $(J^{\prime},D^{\prime},r^{\prime},K)$
give rise to the same critical line in the $(B,T)$ plane or they are in the
same class if and only if

$\displaystyle\left\\{\begin{array}[]{ll}J^{2}+D^{2}=J^{\prime 2}+D^{\prime
2}\\\ r^{2}+K^{2}=r^{\prime 2}+K^{\prime 2}.\end{array}\right.$ (23)

In other words, the above symmetric and antisymmetric categories consist of
equivalence classes of dimers, where each class can be represented
topologically as a two dimensional torus in the $(J,D,r,K)$ parameter space as
shown in Fig. 1. Here, we may refer to an equivalence class of symmetric
dimers as an S-class and an equivalence class of antisymmetric dimers as an
AS-class.

Figure 1: (Color online) Schematic picture of a S-class or AS-class as a two
dimensional torus in the $(J,D,r,K)$ parameter space.

Having each category of dimers classified into equivalent toric classes of
entangled spins, we further notice that each class in one category has its
one-to-one correspondence dual class in the opposite category. To see this we
realize that the two equations in the duality equation Eq. (18) are indeed two
sides of the same coin and actually are related by flipping the sign of the
$J_{zz}$ coupling parameter and exchanging the $p$ and $q$ functions.
Therefore, if a given toric S-class characterized by the coupling parameters
$(J,D,r,K)$ is obtained by one of the equations in Eq. (18), then its
corresponding dual AS-class characterized by the coupling parameters
$(\tilde{J},\tilde{D},\tilde{r},\tilde{K})$ can be obtained from the other
equation in Eq. (18) and vice versa. The two dual toric S-class and AS-class
are equivalent in a sense that they give rise to the same entanglement phase
diagram in the $(B,T)$ plane if their characterisation parameters satisfy the
following relations

$\displaystyle\left\\{\begin{array}[]{ll}J^{2}+D^{2}=\tilde{r}^{2}+\tilde{K}^{2}\\\
r^{2}+K^{2}=\tilde{J}^{2}+\tilde{D}^{2}.\end{array}\right.$ (26)

These relations compared to the ones given in Eq. (23) indicate that the dual
symmetric and antisymmetric entanglement classes have the same toric
characteristic but with their radial or meridian circles in swapped order as
depicted in Fig 2.

Figure 2: (Color online) Toric visualisation of dual symmetric and
antisymmetric entanglement classes. Each class has the same toric
characteristic as its dual but with radial or meridian circles in swapped
order.

Note that although the above analysis focuses on concurrence, we have found
exactly the same classification and entanglement duality using non-locality
horodecki1995 ; horst13  and negativity Vidal2002 . Other than the
entanglement duality in spin-spin interactions demonstrated above, our
analysis reveals the geometric and topological nature of quantum entanglement
in a sense that entanglement phase diagram in dimers provide a clear geometric
foliation of coupling parameter manifold into two dimensional compact torus
leaves.

To further clarify our point and show that the entanglement analysis above
provide a non-trivial classification and duality, we consider two examples in
the following section.

##  III Examples

Consider two identical spin-$\frac{1}{2}$ particles with Heisenberg spin-spin
exchange interaction of strength $J$ in an external magnetic field, as given
by the Hamiltonian

$\displaystyle H^{\rm{Heisenberg}}$ $\displaystyle=$ $\displaystyle
B\left(s_{z}^{(1)}+s_{z}^{(2)}\right)+J{\bf s}^{(1)}\cdot{\bf s}^{(2)}.$ (27)

Here, ${\bf s}=\frac{1}{2}\boldsymbol{\sigma}$ and $B$ is the homogeneous
field strength. We assume antiferromagnetic coupling $J>0$ as entanglement
cannot occur for ferromagnetic coupling arnesen01 .

According to the above classification the Heisenberg model describes a class
of symmetric dimers, which includes all the spin-spin interaction models
following the general form of the Hamiltonian

$\displaystyle H^{\rm{Heisenberg}}_{\text{s}}$ $\displaystyle=$ $\displaystyle
B\left(s_{z}^{(1)}+s_{z}^{(2)}\right)+Js^{(1)}_{z}s^{(2)}_{z}$
$\displaystyle+J^{\prime}[s^{(1)}_{x}s^{(2)}_{x}+s^{(1)}_{y}s^{(2)}_{y}]$
$\displaystyle-D^{\prime}[s^{(1)}_{x}s^{(2)}_{y}-s^{(1)}_{y}s^{(2)}_{x}]$

such that $J^{\prime 2}+D^{\prime 2}=J^{2}$. Note that
$H^{\rm{Heisenberg}}_{\text{s}}=H_{\rm{Heisenberg}}$ for $D^{\prime}=0$ and
$J^{\prime}=J$. Below we show all dimers in this class have the same
entanglement phase diagram in $(B,T)$ plane.

For the Hamiltonian in Eq. (LABEL:eq:GHeisenberg), the thermal state is
characterized by the following nonzero elements:

$\displaystyle\rho^{\prime}_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{Z}}e^{-B/T},\ \
\rho^{\prime}_{44}=\frac{1}{\mathcal{Z}^{\prime}}e^{B/T},$
$\displaystyle\rho^{\prime}_{22}$ $\displaystyle=$
$\displaystyle\rho^{\prime}_{33}=\frac{e^{J/2T}}{\mathcal{Z}^{\prime}}\cosh\left(\Gamma^{\prime}/2T\right),$
$\displaystyle\rho^{\prime}_{23}$ $\displaystyle=$
$\displaystyle(\rho^{\prime}_{32})^{\ast}=-\frac{e^{J/2T}}{\mathcal{Z}^{\prime}}e^{-i\phi}\sinh\left(\Gamma^{\prime}/2T\right)$
(29)

with partition function

$\displaystyle\mathcal{Z}^{\prime}$ $\displaystyle=$ $\displaystyle
2e^{J/2T}\cosh\left(\Gamma^{\prime}/2T\right)+2\cosh\left(B/T\right),$ (30)

where $\Gamma^{\prime}=\sqrt{J^{\prime 2}+D^{\prime 2}}=J$. The concurrence
functions are then given by

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle-\frac{e^{J/2T}}{\mathcal{Z}^{\prime}}\cosh\left(\Gamma^{\prime}/T\right)=-\frac{\left(1+e^{J/T}\right)}{2\mathcal{Z}^{\prime}}<0,$
$\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{Z}}\left[e^{J/2T}\sinh\left(\Gamma^{\prime}/T\right)-1\right]=\frac{e^{J/T}-3}{2\mathcal{Z}^{\prime}}$
(31)

which result in

$\displaystyle
C(\varrho^{\prime}_{T})=\max\left\\{\frac{e^{J/T}-3}{\mathcal{Z}^{\prime}},0\right\\}.$
(32)

Thus, $\varrho^{\prime}_{T}$ is entangled if $e^{J/T}-3>0$. This defines a
critical temperature

$\displaystyle T^{\prime}_{c}=\sqrt{J^{\prime 2}+D^{\prime 2}}/\ln 3=J/\ln
3\approx 0.91J,$ (33)

which is independent of the external magnetic field $B$, for all dimers in the
Heisenberg class denoted here by $[H^{\rm{Heisenberg}}]$. As shown in Fig. 3,
above the critical temperature the thermal state ceases to be entangled. The
same critical temperature as in Eq. (33) has been obtained in Ref. arnesen01
for antiferromagnetic Heisenberg exchange without the Dzyaloshinskii–Moriya
interaction, i.e., $D^{\prime}=0$. Note that the toric characteristic equation

$\displaystyle J^{\prime 2}+D^{\prime 2}=J^{2}$ (34)

identifies each symmetric Heisenberg entanglement class as a one dimensional
torus (one-sphere) in the $(J^{\prime},D)$ plane with the radius given by the
strength of the Heisenberg coupling constant $J$. In other words, symmetric
Heisenberg entanglement classes foliates the $(J^{\prime},D^{\prime})$
parameter space into circles (see Fig. 3).

On the opposite category of antisymmetric dimers, the dual Heisenberg model is
described by the antisymmetric Hamiltonian

$\displaystyle H^{\rm{Heisenberg}}_{\text{as}}$ $\displaystyle=$
$\displaystyle B\left(s_{z}^{(1)}-s_{z}^{(2)}\right)-Js^{(1)}_{z}s^{(2)}_{z}$
$\displaystyle+\tilde{r}[s^{(1)}_{x}s^{(2)}_{x}-s^{(1)}_{y}s^{(2)}_{y}]$
$\displaystyle+\tilde{K}[s^{(1)}_{x}s^{(2)}_{y}+s^{(1)}_{y}s^{(2)}_{x}],$ (35)

where

$\displaystyle\tilde{r}^{2}+\tilde{K}^{2}=J^{2}$ (36)

with $J$ being the Heisenberg coupling constant in Eq. (27). In this case, the
dual thermal state is given by the nonzero elements

$\displaystyle\tilde{\rho}_{11}$ $\displaystyle=$
$\displaystyle\tilde{\rho}_{44}=\frac{e^{J/2T}}{\tilde{\mathcal{Z}}}\cosh\left(\tilde{\Gamma}/2T\right),$
$\displaystyle\tilde{\rho}_{22}$ $\displaystyle=$
$\displaystyle\frac{1}{\tilde{\mathcal{Z}}}e^{-B/T},\ \
\tilde{\rho}_{33}=\frac{1}{\tilde{\mathcal{Z}}}e^{B/T},$
$\displaystyle\tilde{\rho}_{14}$ $\displaystyle=$
$\displaystyle\tilde{\rho}_{41}^{\ast}=-\frac{e^{J/2T}}{\tilde{\mathcal{Z}}}e^{-i\varphi}\sinh\left(\tilde{\Gamma}/2T\right)$
(37)

with partition function

$\displaystyle\tilde{\mathcal{Z}}$ $\displaystyle=$ $\displaystyle
2e^{J/2T}\cosh\left(\tilde{\Gamma}/T\right)+2\cosh\left(B/T\right),$ (38)

where $\tilde{\Gamma}=\sqrt{\tilde{r}^{2}+\tilde{K}^{2}}=J$. The concurrence
functions are given by

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\tilde{\mathcal{Z}}}\left[e^{J/2T}\sinh\left(\tilde{\Gamma}/T\right)-1\right]=\frac{e^{J/T}-3}{2\tilde{\mathcal{Z}}},$
$\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle-\frac{e^{J/2T}}{\tilde{\mathcal{Z}}}\cosh\left(\tilde{\Gamma}/T\right)=-\frac{\left(1+e^{J/T}\right)}{2\tilde{\mathcal{Z}}}<0,$
(39)

which compared to its symmetric counterparts in Eq. (31) the $C_{1}$ and
$C_{1}$ functions are swapped. This results in

$\displaystyle
C(\tilde{\varrho}_{T})=\max\left\\{\frac{e^{J/T}-3}{\tilde{\mathcal{Z}}},0\right\\}.$
(40)

Figure 3: (Color online) Left panel shows the entanglement phase diagram for
symmetric and dual antisymmetric Heisenberg dimers. For each of the Heisenberg
type of dimers, the entanglement undergoes a sudden change at the relative
critical temperature $T_{c}=T^{\prime}_{c}=\tilde{T_{c}}=J/\ln 3\approx 0.91J$
independent of the applied magnetic field strength $B$, which is an indication
of quantum phase transition. Right panels illustrate toric visualisation of
symmetric (S) and dual antisymmetric (AS) Heisenberg entanglement classes.
While each circle of radius $J$ in the $(J^{\prime},D^{\prime})$ plane
represent a Heisenberg S-class corresponding to a given isotropic Heisenberg
exchange coupling $J$, the equivalent dual AS-class is represented by a circle
in the $(\tilde{r},\tilde{K})$ plane with the same radius of $J$.

Similar to the symmetric Heisenberg dimers, the dual thermal state
$\tilde{\varrho}_{T}$ is also entangled if $e^{J/T}-3>0$. This indicates the
entanglement phase diagram with the critical temperature

$\displaystyle\tilde{T}_{c}=\sqrt{\tilde{r}^{2}+\tilde{K}^{2}}/\ln 3=J/\ln
3\approx 0.91J,$ (41)

for antisymmetric Heisenberg dimers to be the same as one obtained for
symmetric Heisenberg dimers above. However, in the dual antisymmetric case the
toric characteristic equation given by Eq. (36), as shown in Fig. 3 foliates
instead the $(\tilde{r},\tilde{K})$ parameter space into circles of radii
specified by the Heisenberg exchange parameter $J$.

Figure 4: (Color online) Upper panel shows the entanglement transition curve
for symmetric and dual antisymmetric XY dimers for different values of
anisotropy parameter $\gamma$. While the isotropic limit $\gamma=0$ represents
the narrowest area of entanglement with magnetic field independent critical
temperature, the critical temperature becomes magnetic field dependent for
$\gamma\neq 0$ and the entanglement area increases with $\gamma$ so that the
Ising limit $\gamma=1$ represents the broadest entanglement area in the
temperature-magnetic field plane. Each value of $\gamma$ distinguishes
exclusive dual toric entanglement classes depicted in the lower panel.

In addition, Fig. 4 illustrates the classification of the XY interaction,
given by the Hamiltonian

$\displaystyle H^{\rm XY}_{\text{s/as}}$ $\displaystyle=$
$\displaystyle(1+\gamma)s_{x}^{(1)}s_{x}^{(2)}+(1-\gamma)s_{y}^{(1)}s_{y}^{(2)}$
(42) $\displaystyle+B(s_{z}^{(1)}\pm s_{z}^{(2)}),$

into dual toric entanglement classes. Here $\gamma\in[0,1]$ is the
dimensionless anisotropy parameter controlling the cylindrical asymmetry of
the spin-spin interaction and $+$ ($-$) is for symmetric (antisymmetric) XY
model. The XY model defines interactions from the isotropic limit $\gamma=0$
with additional symmetry $[H^{\rm XY}_{\text{s/as}},s^{z}]=0$ to the opposite
limit $\gamma=1$, which corresponds to the Ising model with totally ordered
Neel ground state. By comparing the Hamiltonian in Eq. (42) with the dual
symmetric and antisymmetric Hamiltonians in Eqs. (LABEL:SH) and (20) for
$J_{zz}=0$ and then applying the above classification procedure to the XY
model, we fine dual characteristic equations as

$\displaystyle\left\\{\begin{array}[]{ll}J^{2}+D^{2}=1/4\\\
r^{2}+K^{2}=\gamma^{2}/4\end{array}\right.\equiv\left\\{\begin{array}[]{ll}J^{2}+D^{2}=\gamma^{2}/4\\\
r^{2}+K^{2}=1/4\end{array}\right.$ (47)

which specify dual toric XY entanglement classes as depicted in the lower
panel of Fig. 4. In fact, it is the anisotropy parameter $\gamma$ that
controls the entanglement classes and the spin-spin duality. Note that the
critical temperature depends on applied magnetic field for all $\gamma\neq 0$.

##  IV Conclusions

In conclusion, we have demonstrated an entanglement duality of a wide class of
physically important spin-spin interaction dimer models by analyzing
entanglement transition of thermal states in the temperature-magnetic field
plane. This duality allows us to foliate coupling parameter space into dual
symmetric and antisymmetric toric entanglement classes. This classification is
an indication of topological nature of the quantum correlations and hope it
can contribute to our deeper understanding of the various aspects of the
mysterious concept of the quantum correlations.

##  V Acknowledgment

The authors acknowledge financial support from the Knut and Alice Wallenberg
Foundation through Grant No. 2018.0060. E.S. acknowledges financial support
from the Swedish Research Council (VR) through Grant No. 2017-03832.

## References

  * (1) C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016).
  * (2) M. Johansson, M. Ericsson, E. Sjöqvist, and A. Osterloh, Classification scheme of pure multipartite states based on topological phases, Phys. Rev. A 89, 012320 (2014).
  * (3) A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?, Phys. Rev. 47, 777 (1935).
  * (4) E. Schrödinger, The Present Status of Quantum Mechanics, Naturwissenschaften 23, 807 (1935).
  * (5) J. S. Bell, On the Einstein Podolsky Rosen paradox, Physics 1, 195 (1964).
  * (6) M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th Ed. (Cambridge University Press, Cambridge, England, 2010).
  * (7) A. Osterloh, L. Amico, G. Falci, and R. Fazio, Scaling of entanglement close to a quantum phase transition, Nature (London) 416, 608 (2002).
  * (8) T. J. Osborne and M. A. Nielsen, Entanglement in a simple quantum phase transition, Phys. Rev. A 66, 032110 (2002).
  * (9) T.-C. Wei, D. Das, S. Mukhopadyay, S. Vishveshwara, and P. M. Goldbart, Global entanglement and quantum criticality in spin chains, Phys. Rev. A 71, 060305(R) (2005).
  * (10) R. Orús, Universal Geometric Entanglement Close to Quantum Phase Transitions, Phys. Rev. Lett. 100, 130502 (2008).
  * (11) W. Son, L. Amico, R. Fazio, A. Hamma, S. Pascazio, and V. Vedral, Quantum phase transition between cluster and antiferromagnetic states, Europhys. Lett. 95, 50001 (2011).
  * (12) V. Azimi-Mousolou, C. M. Canali, and E. Sjöqvist, Unifying geometric entanglement and geometric phase in a quantum phase transition, Phys. Rev. A 88, 012310 (2013).
  * (13) S. Hill and W. K. Wootters, Entanglement of a Pair of Quantum Bits, Phys. Rev. Lett. 78, 5022 (1997).
  * (14) W. K. Wootters, Entanglement of Formation of an Arbitrary State of Two Qubits, Phys. Rev. Lett. 80, 2245 (1998).
  * (15) M. C. Arnesen, S. Bose, and V. Vedral, Natural Thermal and Magnetic Entanglement in the 1D Heisenberg Model, Phys. Rev. Lett. 87, 017901 (2001).
  * (16) L. Mazzola, B. Bellomo, R. Lo Franco, and G. Compagno, Connection among entanglement, mixedness, and nonlocality in a dynamical context, Phys. Rev. A 81, 052116 (2010).
  * (17) R. Horodecki, P. Horodecki, and M. Horodecki, Violating Bell inequality by mixed spin-$\frac{1}{2}$ states: necessary and sufficient condition, Phys. Lett. A 200, 340 (1995).
  * (18) B. Horst, K. Bartkiewicz, and A. Miranowicz, Two-qubit mixed states more entangled than pure states: Comparison of the relative entropy of entanglement for a given nonlocality, Phys. Rev. A 87, 042108 (2013).
  * (19) G. Vidal and R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002).


15cm(7cm,0.7cm) MS-TP-21-21, KA-TP-18-2021, P3H-21-052, IFJPAN-IV-2021-11

Impact of W and Z Production Data and Compatibility of Neutrino DIS Data in
Nuclear Parton Distribution Functions

K.F. Muzakka1$\star$, P. Duwentäster1, T.J. Hobbs2,3,4, T. Ježo5, M. Klasen1,
K. Kovařík 1, A. Kusina6, J.G. Morfín7, F. I. Olness2, R. Ruiz6, I.
Schienbein8, J.Y. Yu8

1 Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster,
Wilhelm-Klemm-Straße 9, D-48149 Münster, Germany

2 Southern Methodist University, Dallas, TX 75275, USA

3 Jefferson Lab, Newport News, VA 23606, USA

4 Department of Physics, Illinois Institute of Technology, Chicago, Illinois
60616, USA

5 Institute for Theoretical Physics, KIT, D-76131 Karlsruhe, Germany

6 Institute of Nuclear Physics Polish Academy of Sciences, 31342 Kraków,
Poland

7 FNAL, Batavia, IL 60510, USA

8 Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-
Alpes, CNRS/IN2P3, 53 avenue des Martyrs, 38026 Grenoble, France

<EMAIL_ADDRESS>
Proceedings for the XXVIII International Workshop
on Deep-Inelastic Scattering and Related Subjects,
Stony Brook University, New York, USA, 12-16 April 2021
10.21468/SciPostPhysProc.?

## Abstract

Vector boson production and neutrino deep-inelastic scattering (DIS) data are
crucial for constraining the strange quark parton distribution function (PDF)
and more generally for flavor decomposition in PDF extractions. We extend the
nCTEQ15 nuclear PDFs (nPDFs) by adding the recent $W$ and $Z$ production data
from the LHC in a global nPDF fit. The new nPDF set, referred to as nCTEQ15WZ,
is used as a starting point for a follow-up study in which we assess the
compatibility of neutrino DIS data with charged lepton DIS data. Specifically,
we re-analyze neutrino DIS data from NuTeV, Chorus, and CDHSW, as well as
dimuon data from CCFR and NuTeV. To scrutinize the level of compatibility,
different kinematic regions of the neutrino data are investigated. Fits to the
neutrino data alone and a preliminary global fit are performed and compared to
nCTEQ15WZ.

##  1 Introduction

Nuclear parton distribution functions (nPDFs) are key inputs for any
factorization-based theory prediction that involve a nucleus in the initial
state[1, 2, 3]. Due to their non-perturbative nature, nPDFs are determined
from a QCD global analysis of various nuclear data, such as charged lepton or
neutrino deep-inelastic scattering (DIS), or Drell-Yan (DY) lepton pair
production data. Most of these data are sensitive particularly to (anti-)up
and (anti-)down quark nPDFs, while the strange quark and gluon nPDFs are
traditionally the least constrained and thus have larger uncertainties.

It has been shown in a reweighting study [4] that the recent $W$ and $Z$ boson
production data from the LHC[5, 6, 7, 8, 9, 10, 11] provide better constraints
on the strange and gluon nPDFs. However, other $W$ boson-mediated processes,
such as neutrino DIS and the more exclusive observable of charm dimuon
production, are also very sensitive to the strange quark PDF[12, 13].

Including neutrino DIS data in a global fit is not straightforward as these
data have been shown to display tensions with the charged lepton DIS data[14,
15, 16, 17]. To make the situation even more complicated, while tension was
observed in Ref. [14] (which used correlated systematic uncertainties), a
separate analysis [18] of $F_{2}$ and $F_{3}$ neutrino data from NuTeV,
Chorus, and CDHSW (without correlated systematic uncertainties) found
compatibility with the charged lepton DIS and DY data. Additionally, another
study [19] normalized the integrated differential cross section for each
incident neutrino energy $E$ bin to suppress the $E$-dependence fluctuation
and, without correlated systematic uncertainties, obtained compatibility.
Clearly, there remains important questions to address regarding these data
sets.

In this analysis, we review the impact of including $W$ and $Z$ boson
production data from LHC on the fitted nPDFs (referred to as nCTEQ15WZ);
additional details can be found in [20]. We include neutrino data from
NuTeV[21], CDHSW[22], and Chorus[23], as well as charm dimuon data from CCFR
and NuTeV[24]. We first analyze the neutrino data separately and then perform
a global fit combining all the data sets used in the nCTEQ15WZ analysis with
the neutrino data.

##  2 Theoretical Framework and Data Sets

In all our fits, the same nCTEQ15 fitting framework is employed. In
particular, the PDFs of a proton in the nuclear medium are parametrized in a
polynomial form [1], and the mass number ($A$) dependence is parametrized at
the level of the PDF parameters. All theory calculations are performed at
next-to-leading order (NLO) in perturbative QCD. The ACOT scheme[25, 26] is
used for the treatment of heavy quarks. To minimize higher twist effects and
avoid the resonance region, we apply the standard kinematic cuts of $Q>2$ GeV
and $W>3.5$ GeV for all the DIS data, including the neutrino sets. After these
kinematic cuts, the remaining number of data points in the nCTEQ15WZ set is
853, plus 2366 from NuTeV, 929 from CDHSW, 824 from Chorus, and 174 from the
charm dimuon data. In the nCTEQ15WZ fit as well as the new fits with neutrino
data, 19 PDF parameters are actively varied during the minimization process
including: four $u_{v}$ parameters
($a^{u_{v}}_{1},\,a^{u_{v}}_{2},\,a^{u_{v}}_{4},\,a^{u_{v}}_{5}$), three
$d_{v}$ parameters ($a^{d_{v}}_{1},\,a^{d_{v}}_{2},\,a^{d_{v}}_{5}$), seven
gluon parameters
($a^{g}_{1},\,a^{g}_{4},\,a^{g}_{5},\,b^{g}_{0},\,b^{g}_{1},\,b^{g}_{2},\,b^{g}_{5}$),
two $\bar{u}+\bar{d}$ parameters
($a^{\bar{u}+\bar{d}}_{1},\,a^{\bar{u}+\bar{d}}_{5},\,$) and three $s+\bar{s}$
PDF parameters
($a^{s+\bar{s}}_{0},\,a^{s+\bar{s}}_{1},\,a^{s+\bar{s}}_{2},\,$). The rest of
the PDF parameters are fixed to the same values as those in nCTEQ15 PDFs. We
have checked that opening more parameters in the minimization yields little to
no improvement in the obtained $\chi^{2}$ values.

##  3 The nCTEQ15WZ fit

To study the impact of the LHC data on the extracted nPDFs, we performed a
global analysis with the nCTEQ15WZ data set. In addition to varying 19 PDF
parameters, we also fit three normalization parameters for the LHC $W/Z$ data
sets, specifically for i) CMS Run I data, ii) CMS Run 2 data, and iii) ATLAS
data.

The result of the fit, in terms of the $\chi^{2}$ values per number of data
points, is reported in Fig.1. The figure shows that an excellent overall fit
was obtained, and good descriptions of the data was observed for all
experiments [20]111Note that Fig.1 does not include the $\chi^{2}$/pt from the
three $W/Z$ normalization parameters, as they are all within ${\sim}1\sigma$,
so that they contribute to the total $\chi^{2}$ less than three units.. The
LHC data is well described and no significant tension was observed with the
charged lepton DIS and DY data. Considering the fact that the LHC $W/Z$ data
involves inclusive cross sections of both charged and neutral current
processes, this suggests the existence of universal nPDFs that can explain
both charged and neutral current processes, at least for this data subset. In
light of the compatibility of the above data sets, it will be especially
insightful to perform a combined fit of the neutrino data and the data used in
the nCTEQ15WZ fit. Recall that the analysis of Ref. [14] found significant
tension between these data. We revisit these issues in Sec. 4.

Figure 1: $\chi^{2}/$pt for the nCTEQ15WZ fit. The data IDs are defined in
[20], and the number of data points is indicated on top of each bar. For each
process, the corresponding $\chi^{2}/$pt value is shown in the legend.  Figure
2: The strange quark and gluon distributions in lead for a selection of nPDFs
at $Q=2$ GeV.

In Fig. 2, we show the strange quark and gluon PDFs from the nCTEQ15WZ fit and
selected global fits in the literature: nNNPDF2.0[3], EPPS16[2], and nCTEQ15.
For the strange quark PDF, nCTEQ15WZ yields a higher distribution at low $x$,
suggesting an elevated strange sea ratio,
$R_{s}\equiv(s+\bar{s})/(\bar{u}+\bar{d})$. Interestingly, a similar effect
was also observed for the proton PDF using LHC $W/Z$ proton data[27]. Still,
the nPDF sets from the different groups agree well within their respective
uncertainties. Note that the nCTEQ15WZ $s(x)$ uncertainty band is larger
compared to nCTEQ15 because the latter fit constrained the strange quark to be
a fixed fraction of $\bar{u}+\bar{d}$ at the input scale. The nCTEQ15WZ fit
made no such assumption, and thus it is a more accurate representation of the
true $s(x)$ uncertainty. Examining the gluon PDF, we can see a smaller error
band compared to nCTEQ15. The improved sensitivity to the gluon PDF can be
explained by the high scale of the LHC $W/Z$ data, which probe small $x$
values and are sensitive to higher-order, gluon-initiated corrections.

##  4 Neutrino Fits

In this section, we fit neutrino and dimuon data sets alone (referred to as
DimuNeu) and then include these data sets in global analyses together with the
nCTEQ15WZ set to study their compatibility and potential tension.

###  4.1 DimuNeu vs. nCTEQ15WZ

Figure 3: The nuclear ratio $R=F_{2}^{A}/F_{2}^{N}$ computed with nCTEQ15WZ
and DimuNeu for charged lepton (left) and neutrino (right) processes at
$Q^{2}=5$ GeV2. $F_{2}^{CC}$ on the right panel is defined as
$F_{2}^{CC}=(F_{2}^{\nu A}+F_{2}^{\bar{\nu}A})/2$. For the charged lepton case
we compare with BCDMS[28, 29] and SLAC[30] data, and for the neutrino case we
compare with NuTeV data[21]. The SLAC/NMC parameterization is taken from [31].
These figures are preliminary and do not fully account for the fitted
normalizations.

To gauge the compatibility between the charged lepton and neutrino data sets,
in Fig. 3 we display the predicted nuclear ratio ($R=F_{2}^{A}/F_{2}^{N}$)
using the nCTEQ15WZ and DimuNeu nPDF sets and overlaid with experimental data.
The left panel displays the predictions for charged leptons and compared with
BCDMS[28, 29] and SLAC [30] data and the right panel displays the predictions
for neutrinos as compared with NuTeV data[21].

Fig. 3 shows that at a representative case of $Q^{2}=5$ GeV2 both the DimuNeu
and nCTEQ15WZ predictions generally agree (within uncertainties) for
intermediate $x$, but diverge from each other for $x\lesssim 0.1$, and to a
lesser extent in the high $x$ region, i.e., for $x\gtrsim 0.6$. Specifically,
the influence of the neutrino data on the DimuNeu fit with $Q^{2}>4$ GeV2 is
to extend the anti-shadowing region to smaller $x$ values, and the usual
shadowing behavior is not evident.

###  4.2 Compatibility Test

We now assess the compatibility of the nCTEQ15WZ data set ($S$) with the
individual neutrino data sets ($\bar{S}$) using the hypothesis-testing method.
Data sets $S$ and $\bar{S}$ are compatible if the following two conditions are
satisfied: i) When the new data set ($\bar{S}$) is included, the $\chi^{2}$
increase of the original nCTEQ15WZ data set ($\Delta\chi^{2}_{S}$) is less
than some tolerance $T$. ii) The combined fit can describe all the data (both
$S$ and $\bar{S}$) within the 90% confidence level. For the first condition,
we choose a tolerance of $T{=}45$. The second condition can be checked by
examining the percentile $p_{\bar{S}}$.

$\bar{S}$ | $\Delta\chi^{2}_{S}$ | N | $p_{\bar{S}}(\%)$
---|---|---|---
Chorus | 17 | 824 | 83.8
CDHSW | 44 | 929 | 0.0
NuTeV | 92 | 2136 | 93.5
DimuNeu | 106 | 4063 | 99.2
Table 1: $\Delta\chi^{2}_{S}$ and the percentiles $p_{\bar{S}}(\%)$ after
including the $\bar{S}$ neutrino data sets.

$\bar{S}$ | $\Delta\chi^{2}_{S}$ | N | $p_{\bar{S}}(\%)$
---|---|---|---
Chorus[x>0.1] | 9 | 738 | 81.7
CDHSW[x>0.1] | 26 | 845 | 0.0
NuTeV[x>0.1] | 17 | 1737 | 74.6
DimuNeu[x>0.1] | 28 | 3494 | 66.0
Table 2: Same as Table 2, but with the $x{\geq}0.1$ cut imposed.

Table 2 shows the result of individually including each new data set
($\bar{S}$) into the fit with the nCTEQ15WZ data ($S$). We observe that all
the neutrino data sets lead to a noticeable $\chi^{2}$ increase for the
nCTEQ15WZ data ($\Delta\chi^{2}_{S}$), and only the Chorus and CDHSW data
marginally satisfy our compatibility criteria.

Examining these results in more detail, we find that the bulk of the tensions
arise from the small $x$ region, where we observed the divergence of the
nCTEQ15WZ and DimuNeu results in Fig. 3. To examine the impact of the small
$x$ region, we repeat our analysis with a cut of $x\geq 0.1$ on the neutrino
data. The results of this study are shown in Table 2. The difference is
striking and we see that now all the neutrino data sets ($\bar{S}$) satisfy
our compatibility criteria.

While we have succeeded in identifying the kinematic region which generates
the tension in our fits, additional work is required to find the underlying
cause. There are a number of potential sources including: a difference in
shadowing of charged lepton and neutrino nucleus scattering, non-perturbative
contributions from nuclear effects, higher-twist terms (the small $x$ data
typically lies at lower $Q$ values), and the PDF fitting methodology
(including assumptions about the PDF parameterization, isospin symmetry, the
strange and gluon distributions). Additionally, there could be a combination
of multiple factors. Further investigations are ongoing, but we perform a
preliminary combined fit in the following section.

###  4.3 The Combined Fit

Figure 4: We show the strange quark and gluon PDFs from the combined fits at
$Q=2$ GeV.

Having identified the region that generates the tension between data sets, we
perform a global nPDF fit including all the data sets (nCTEQ15WZ, dimuon, and
neutrino, referred to as BaseDimuNeu). To see the impact of low $x$ neutrino
data, we also perform another fit with all the above data, but impose a $x\geq
0.1$ cut on the neutrino data (referred to as BaseDimuNeuX).

The results of these fits are displayed in Fig. 4 for the strange quark and
gluon PDFs, as these are most impacted by the new data. For the strange quark
PDF, we see that both the nCTEQ15WZ and BaseDimuNeu fits yield comparably
large distributions in the small $x$ region. In contrast, both the BaseDimu
and BaseDimuNeuX fits prefer a smaller strange PDF at low $x$. The comparison
of BaseDimuNeu and BaseDimuNeuX clearly highlights the impact of the low $x$
($x\leq 0.1$) neutrino data on the strange quark.

Turning to the gluon PDF (Fig. 4, right) we observe that all the fits
approximately coincide except for BaseDimuNeu which, like the strange quark,
yields an increased PDF in the small $x$ region. Again, this points to the
strong influence of the low $x$ ($x\leq 0.1$) neutrino data. Fig. 4 also
displays the uncertainty bands that show both the increase of the uncertainty
towards lower $x$ values and the general reduction of the uncertainty as we
add new data to the fits.

##  5 Conclusion

We have presented the results from our recent PDF analysis (nCTEQ15WZ), which
includes $W$ and $Z$ boson production data from the LHC, and have extended
this fit to include new data from neutrino DIS experiments. Specifically, our
extended analysis includes inclusive cross section data from NuTeV, CDHSW, and
Chorus, as well as charm dimuon data from CCFR and NuTeV.

Both LHC $\mathit{W/Z}$ data and the neutrino data impose strong constraints
on the strange quark and gluon PDFs. We are able to obtain good fits to the
data sets as measured by their $\chi^{2}$ values. However, there are some
tensions that are most evident by comparing fits (BaseDimuNeu and
BaseDimuNeuX) with and without the low $x$ ($x\leq 0.1$) neutrino data. This
difference is mostly reflected in the resulting strange PDF at low $x$ values.

We have conducted this study examining the $\chi^{2}$ for individual data and
individual processes (Fig. 1), together with the nuclear ratios (Fig. 3) and
the compatibility criteria (Sec. 4.2). This powerful combination of tools
allows us to incisively analyze these data sets and identify the source and
impact of the tensions. This analysis represents an important step forward as
improved nPDFs yield both precision predictions for hadronic process and an
enhanced understanding of the underlying nuclear interactions.

## Acknowledgements

We would like to thank Alberto Accardi, Nasim Derakhshanian, Cynthia Keppel,
Chloé Leger, Florian Lyonnet, and Peter Risse for useful discussions. The work
of M.K., K.K. and K.F.M. was funded by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) through the Research Training Group GRK
2149. P.D., M.K. and K.K. also acknowledge the support of the DFG through
project-id 273811115 – SFB 1225. A.K. and R.R. acknowledge the support of
Narodowe Centrum Nauki under Grant No 2019/34/E/ST2/00186. The work of T.J.
was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under grant 396021762 - TRR 257. F.O. acknowledges support through
US DOE grant No. DE-SC0010129 and the National Science Foundation under Grant
No. NSF PHY-1748958. The work of I.S. was supported by the French CNRS via the
IN2P3 project GLUE@NLO.

## References

  * [1] K. Kovařík _et al._ ,  _nCTEQ15 - Global analysis of nuclear parton distributions with uncertainties in the CTEQ framework_ ,  Phys. Rev. D93(8), 085037 (2016),  10.1103/PhysRevD.93.085037,  1509.00792.
  * [2] K. J. Eskola, P. Paakkinen, H. Paukkunen and C. A. Salgado,  _EPPS16: Nuclear parton distributions with LHC data_ ,  Eur. Phys. J. C77(3), 163 (2017),  10.1140/epjc/s10052-017-4725-9,  1612.05741.
  * [3] R. Abdul Khalek, J. J. Ethier, J. Rojo and G. van Weelden,  _nNNPDF2.0: Quark Flavor Separation in Nuclei from LHC Data_ (2020),  2006.14629.
  * [4] A. Kusina, F. Lyonnet, D. B. Clark, E. Godat, T. Jezo, K. Kovařík, F. I. Olness, I. Schienbein and J. Y. Yu,  _Vector boson production in pPb and PbPb collisions at the LHC and its impact on nCTEQ15 PDFs_ ,  Eur. Phys. J. C77(7), 488 (2017),  10.1140/epjc/s10052-017-5036-x,  1610.02925.
  * [5] G. Aad _et al._ ,  _$Z$ boson production in $p+$Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV measured with the ATLAS detector_,  Phys. Rev. C92(4), 044915 (2015),  10.1103/PhysRevC.92.044915,  1507.06232.
  * [6] V. Khachatryan _et al._ ,  _Study of Z boson production in pPb collisions at $\sqrt{s_{NN}}=5.02$ TeV_,  Phys. Lett. B759, 36 (2016),  10.1016/j.physletb.2016.05.044,  1512.06461.
  * [7] V. Khachatryan _et al._ ,  _Study of W boson production in pPb collisions at $\sqrt{s_{\mathrm{NN}}}=$ 5.02 TeV_,  Phys. Lett. B750, 565 (2015),  10.1016/j.physletb.2015.09.057,  1503.05825.
  * [8] A. M. Sirunyan _et al._ ,  _Observation of nuclear modifications in W ± boson production in pPb collisions at $\sqrt{s_{\mathrm{NN}}}=$ 8.16 TeV_,  Phys. Lett. B800, 135048 (2020),  10.1016/j.physletb.2019.135048,  1905.01486.
  * [9] R. Aaij _et al._ ,  _Observation of $Z$ production in proton-lead collisions at LHCb_,  JHEP 09, 030 (2014),  10.1007/JHEP09(2014)030,  1406.2885.
  * [10] K. Senosi,  _Measurement of W-boson production in p-Pb collisions at the LHC with ALICE_ ,  PoS Bormio2015, 042 (2015),  10.22323/1.238.0042,  1511.06398.
  * [11] J. Adam _et al._ ,  _W and Z boson production in p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV_,  JHEP 02, 077 (2017),  10.1007/JHEP02(2017)077,  1611.03002.
  * [12] A. Kusina, T. Stavreva, S. Berge, F. I. Olness, I. Schienbein, K. Kovařík, T. Jezo, J. Y. Yu and K. Park,  _Strange Quark PDFs and Implications for Drell-Yan Boson Production at the LHC_ ,  Phys. Rev. D85, 094028 (2012),  10.1103/PhysRevD.85.094028,  1203.1290.
  * [13] F. Faura, S. Iranipour, E. R. Nocera, J. Rojo and M. Ubiali,  _The Strangest Proton?_ ,  Eur. Phys. J. C 80(12), 1168 (2020),  10.1140/epjc/s10052-020-08749-3,  2009.00014.
  * [14] K. Kovařík, I. Schienbein, F. I. Olness, J. Y. Yu, C. Keppel, J. G. Morfin, J. F. Owens and T. Stavreva,  _Nuclear Corrections in Neutrino-Nucleus DIS and Their Compatibility with Global NPDF Analyses_ ,  Phys. Rev. Lett. 106, 122301 (2011),  10.1103/PhysRevLett.106.122301,  1012.0286.
  * [15] I. Schienbein, J. Y. Yu, K. Kovařík, C. Keppel, J. G. Morfin, F. Olness and J. F. Owens,  _PDF Nuclear Corrections for Charged and Neutral Current Processes_ ,  Phys. Rev. D80, 094004 (2009),  10.1103/PhysRevD.80.094004,  0907.2357.
  * [16] I. Schienbein, J. Yu, C. Keppel, J. Morfin, F. Olness and J. Owens,  _Nuclear parton distribution functions from neutrino deep inelastic scattering_ ,  Phys. Rev. D 77, 054013 (2008),  10.1103/PhysRevD.77.054013,  0710.4897.
  * [17] J. F. Owens, J. Huston, C. E. Keppel, S. Kuhlmann, J. G. Morfin, F. Olness, J. Pumplin and D. Stump,  _The Impact of new neutrino DIS and Drell-Yan data on large-x parton distributions_ ,  Phys. Rev. D75, 054030 (2007),  10.1103/PhysRevD.75.054030,  hep-ph/0702159.
  * [18] D. de Florian, R. Sassot, P. Zurita and M. Stratmann,  _Global Analysis of Nuclear Parton Distributions_ ,  Phys. Rev. D85, 074028 (2012),  10.1103/PhysRevD.85.074028,  1112.6324.
  * [19] H. Paukkunen and C. A. Salgado,  _Agreement of Neutrino Deep Inelastic Scattering Data with Global Fits of Parton Distributions_ ,  Phys. Rev. Lett. 110(21), 212301 (2013),  10.1103/PhysRevLett.110.212301,  1302.2001.
  * [20] A. Kusina _et al._ ,  _Impact of LHC vector boson production in heavy ion collisions on strange PDFs_ ,  Eur. Phys. J. C 80(10), 968 (2020),  10.1140/epjc/s10052-020-08532-4,  2007.09100.
  * [21] M. Tzanov _et al._ ,  _Precise measurement of neutrino and anti-neutrino differential cross sections_ ,  Phys. Rev. D74, 012008 (2006),  10.1103/PhysRevD.74.012008,  hep-ex/0509010.
  * [22] J. Berge _et al._ ,  _A Measurement of Differential Cross-Sections and Nucleon Structure Functions in Charged Current Neutrino Interactions on Iron_ ,  Z. Phys. C 49, 187 (1991),  10.1007/BF01555493.
  * [23] G. Onengut _et al._ ,  _Measurement of nucleon structure functions in neutrino scattering_ ,  Phys. Lett. B 632, 65 (2006),  10.1016/j.physletb.2005.10.062.
  * [24] M. Goncharov _et al._ ,  _Precise Measurement of Dimuon Production Cross-Sections in $\nu_{\mu}$ Fe and $\bar{\nu}_{\mu}$ Fe Deep Inelastic Scattering at the Tevatron._,  Phys. Rev. D64, 112006 (2001),  10.1103/PhysRevD.64.112006,  hep-ex/0102049.
  * [25] M. A. G. Aivazis, F. I. Olness and W.-K. Tung,  _Leptoproduction of heavy quarks. 1. General formalism and kinematics of charged current and neutral current production processes_ ,  Phys. Rev. D 50, 3085 (1994),  10.1103/PhysRevD.50.3085,  hep-ph/9312318.
  * [26] M. A. G. Aivazis, J. C. Collins, F. I. Olness and W.-K. Tung,  _Leptoproduction of heavy quarks. 2. A Unified QCD formulation of charged and neutral current processes from fixed target to collider energies_ ,  Phys. Rev. D 50, 3102 (1994),  10.1103/PhysRevD.50.3102,  hep-ph/9312319.
  * [27] _QCD analysis of ATLAS $W^{\pm}$ boson production data in association with jets_ (2019).
  * [28] G. Bari _et al._ ,  _A Measurement of Nuclear Effects in Deep Inelastic Muon Scattering on Deuterium, Nitrogen and Iron Targets_ ,  Phys. Lett. B 163, 282 (1985),  10.1016/0370-2693(85)90238-2.
  * [29] A. C. Benvenuti _et al._ ,  _Nuclear Effects In Deep Inelastic Muon Scattering On Deuterium And Iron Targets_ (1987).
  * [30] J. Gomez, R. G. Arnold, P. E. Bosted, C. C. Chang, A. T. Katramatou, G. G. Petratos, A. A. Rahbar, S. E. Rock, A. F. Sill, Z. M. Szalata, A. Bodek, N. Giokaris _et al._ ,  _Measurement of the $a$ dependence of deep-inelastic electron scattering_,  Phys. Rev. D 49, 4348 (1994),  10.1103/PhysRevD.49.4348.
  * [31] H. Abramowicz, E. M. Levin, A. Levy and U. Maor,  _A Parametrization of $\sigma_{T}(\gamma^{*}p$) above the resonance region $Q^{2}\geq 0$_,  Phys. Lett. B 269, 465 (1991),  10.1016/0370-2693(91)90202-2.